Eshbach's Handbook of Engineering Fundamentals

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Eshbach’s Handbook of Engineering Fundamentals, Fifth Edition Edited by...

Author: Myer Kutz

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ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS

Eshbach’s Handbook of Engineering Fundamentals, Fifth Edition Edited by Myer Kutz Copyright © 2009 by John Wiley & Sons, Inc.

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS, FIFTH EDITION

Edited by

Myer Kutz

John Wiley & Sons, Inc.

This book is printed on acid-free paper. Copyright  2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Eshbach, Ovid W. (Ovid Wallace), 1893–1958. Eshbach’s handbook of engineering fundamentals / edited by Myer Kutz.—5th ed. p. cm. Includes bibliograhical references. ISBN 978-0-470-08578-3 (cloth: alk. paper) 1. Engineering—Handbooks, manuals, etc. I. Kutz, Myer. II. Title. III. Title: Handbook of engineering fundamentals. TA151.E8 2009 620—dc22 2008041561 Printed in the United States of America. 10 9

8 7

6 5 4

3 2

1

To Ovid W. Eshbach (1893–1958), educator and editor

CONTENTS

Preface Contributors 1. Mathematical and Physical Units, Standards, and Tables Jack H. Westbrook 1. Symbols and Abbreviations 2. Mathematical Tables 3. Statistical Tables 4. Units and Standards 5. Tables of Conversion Factors 6. Standard Sizes 7. Standard Screws 2. Mathematics J. N. Reddy 1. Arithmetic 2. Algebra 3. Set Algebra 4. Statistics and Probability 5. Geometry 6. Trigonometry 7. Plane Analytic Geometry 8. Solid Analytic Geometry 9. Differential Calculus 10. Integral Calculus 11. Differential Equations 12. Finite-Element Method 13. Laplace Transformation 14. Complex Analysis 15. Vector Analysis Bibliography 3. Mechanics of Rigid Bodies Wallace Fowler 1. Definitions 2. Statics 3. Kinematics 4. Kinetics 5. Friction Bibliography 4. Selection of Metals for Structural Design Matthew J. Donachie 1. Introduction 2. Common Alloy Systems

xiii xvii 1 3 23 42 51 69 97 142 159 160 163 181 182 191 213 221 230 238 248 258 269 286 288 303 306 308 308 309 322 335 352 357 358 359 359 vii

viii

CONTENTS

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 5.

6.

7.

8.

What Are Alloys and What Affects Their Use? What Are the Properties of Alloys and How Are Alloys Strengthened? Manufacture of Alloy Articles Alloy Information Metals at Lower Temperatures Metals at High Temperatures Melting and Casting Practices Forging, Forming, Powder Metallurgy, and Joining of Alloys Surface Protection of Materials PostService Refurbishment and Repair Alloy Selection: A Look at Possibilities Level of Property Data Thoughts on Alloy Systems Selected Alloy Information Sources Bibliography

Plastics: Information and Properties of Polymeric Materials Edward N. Peters 1. Introduction 2. Polyolefinic Thermoplastics 3. Side-Chain-Substituted Vinyl Thermoplastics 4. Polyurethane and Cellulosic Resins 5. Engineering Thermoplastics: Condensation Polymers 6. High-Performance Materials 7. Fluorinated Thermoplastics 8. Thermosets 9. General-Purpose Elastomers 10. Specialty Elastomers References Overview of Ceramic Materials, Design, and Application R. Nathan Katz 1. Introduction 2. Processing of Advanced Ceramics 3. Brittleness and Brittle Materials Design 4. Applications 5. Information Sources 6. Future Trends References Mechanics of Deformable Bodies Neal F. Enke and Bela I. Sandor 1. Introduction to Stress and Strain 2. Beams and Bending 3. Torsion and Shafts 4. Plates, Shells, and Contact Stresses 5. Nonlinear Response of Materials 6. Energy Methods 7. Composite Materials 8. Theories of Strength and Failure References Nondestructive Inspection Robert L. Crane and Jeremy S. Knopp 1. Introduction 2. Liquid Penetrants 3. Radiography

359 360 363 363 373 373 376 379 381 383 384 385 385 390 391 392 393 395 396 401 402 409 414 416 420 420 420 422 422 423 424 425 431 432 432 434 434 447 460 464 474 485 488 492 506 509 509 511 513

CONTENTS

4. 5. 6. 7.

Ultrasonic Methods Magnetic Particle Method Thermal Methods Eddy Current Methods Appendix: Ultrasonic Properties of Common Materials References

9. Mechanics of Incompressible Fluids Egemen Ol Ogretim and Wade W. Huebsch 1. Introduction 2. Fluid Properties 3. Fluid Statics 4. Ideal (Inviscid) Fluid Dynamics 5. Viscous Fluid Dynamics 6. Similitude and Dimensional Analysis 7. Flow in Closed Conduits 8. Flow in Open Channels 9. Flow About Immersed Objects 10. Fluid Measurements References Bibliography 10. Aerodynamics of Wings Warren F. Phillips 1. Introduction and Notation 2. Boundary Layer Concept 3. Inviscid Aerodynamics 4. Incompressible Flow over Airfoils 5. Trailing-Edge Flaps and Section Flap Effectiveness 6. Incompressible Flow over Finite Wings 7. Flow over Multiple Lifting Surfaces 8. Wing Stall and Maximum Lift Coefficient 9. Inviscid Compressible Aerodynamics 10. Compressible Subsonic Flow 11. Supersonic Flow References 11. Steady One-Dimensional Gas Dynamics D. H. Daley with contributions by J. B. Wissler 1. Generalized One-Dimensional Gas Dynamics 2. Simple Flows 3. Nozzle Operating Characteristics 4. Normal Shock Waves 5. Plane Oblique Shock Waves 6. Conical Shock Waves 7. Prandtl–Meyer Expansion References 12. Mathematical Models of Dynamic Physical Systems K. Preston White, Jr. 1. Rationale 2. Ideal Elements 3. System Structure and Interconnection Laws 4. Standard Forms for Linear Models 5. Approaches to Linear Systems Analysis 6. State-Variable Methods 7. Simulation

ix

519 526 527 528 533 550 552 553 553 561 566 574 579 581 600 604 605 618 618 619 619 622 623 625 632 636 656 664 673 675 677 680 682 682 683 688 689 690 696 696 697 698 698 699 707 709 714 735 737

x

CONTENTS

8.

Model Classifications References Bibliography

13. Basic Control Systems Design William J. Palm III 1. Introduction 2. Control System Structure 3. Transducers and Error Detectors 4. Actuators 5. Control Laws 6. Controller Hardware 7. Further Criteria for Gain Selection 8. Compensation and Alternative Control Structures 9. Graphical Design Methods 10. Principles of Digital Control 11. Uniquely Digital Algorithms 12. Hardware and Software for Digital Control 13. Software Support for Control System Design 14. Future Trends in Control Systems References 14. Thermodynamics Fundamentals Adrian Bejan 1. Introduction 2. First Law of Thermodynamics for Closed Systems 3. Second Law of Thermodynamics for Closed Systems 4. Energy-Minimum Principle 5. Laws of Thermodynamics for Open Systems 6. Relations among Thermodynamic Properties 7. Analysis of Engineering System Components References 15. Heat Transfer Fundamentals G. P. Peterson 1. Conduction Heat Transfer 2. Convection Heat Transfer 3. Radiation Heat Transfer 4. Boiling and Condensation Heat Transfer References Bibliography 16. Electric Circuits Albert J. Rosa 1. Introduction 2. Direct-Current (DC) Circuits 3. Linear Active Circuits 4. AC Circuits 5. Transient Response of Circuits 6. Frequency Response References 17. Electronics 1. Bipolar Transistors John D. Cressler

741 758 758 760 761 761 765 767 771 776 778 782 785 789 791 795 798 799 801 802 802 803 805 807 807 808 815 817 818 819 834 844 858 868 869 870 870 879 891 905 928 935 948 949 950

CONTENTS

2. Data Acquisition and Conversion Kavita Nair, Chris Zillmer, Dennis Polla, and Ramesh Harjani 3. Data Analysis Arbee L. P. Chen and Yi-Hung Wu 4. Diodes Konstantinos Misiakos 5. Electronic Components Clarence W. de Silva 6. Input Devices George Grinstein and Marjan Trutschl 7. Instruments Halit Eren 8. Integrated Circuits N. Ranganathan and Raju D. Venkataramana 9. Microprocessors Robert P. Colwell 10. Oscilloscopes Andrew Rusek 11. Power Devices Alex Q. Huang and Bo Zhang References Bibliography 18. Light and Radiation M. Parker Givens 1. Introduction 2. Geometric Optics 3. Physical Optics 4. Light Sources 5. Lasers 6. The Eye and Vision 7. Detectors or Optical Transducers References Bibliography 19. Acoustics Jonathan Blotter, Scott Sommerfeldt, and Kent L. Gee 1. Introduction 2. Sound Power, Sound Intensity, and Sound Pressure 3. Decibel and Other Scales 4. Weighting Filters 5. Impedance 6. Theory of Sound 7. Reflection, Transmission, and Absorption 8. Hearing Loss 9. Passive Noise Control 10. Active Noise Control 11. Architectural Acoustics 12. Community and Environmental Noise 13. Sound Quality Analysis 14. Nonlinear Acoustics 15. Human Ear and Hearing

xi

964 979 990 1003 1022 1026 1042 1060 1066 1077 1103 1109 1111 1111 1113 1119 1128 1132 1134 1147 1149 1149 1151 1152 1152 1154 1155 1157 1158 1162 1165 1165 1173 1178 1179 1184 1189 1192

xii

CONTENTS

16.

Microphones and Loudspeakers References Suggested Further Readings

1195 1198 1199

20. Chemistry D. A. Kohl 1. Atomic Structure and Periodic Table 2. Molecular Structure and Chemical Bonding 3. Chemical Reactions and Stoichiometry 4. Chemical Thermodynamics 5. Thermochemistry 6. Chemical Equilibrium 7. Phase Equilibria 8. Chemical Reaction Rates 9. Electrochemistry 10. Organic Chemistry References Bibliography

1200

21. Engineering Economy Kate D. Abel 1. Introduction 2. Cash Flows and Time Value of Money 3. Equivalence 4. Single Sum and Uniform, Gradient, and Geometric Series 5. Comparing Alternatives: Defining Options 6. Comparing Alternatives through Figures of Merit 7. Additional Analyses in Selection Process 8. Capital Recovery, Capital Cost, and Replacement Studies 9. Conclusion References

1246

1200 1203 1205 1209 1213 1219 1221 1228 1231 1240 1245 1245

1246 1246 1247 1249 1251 1252 1256 1257 1257 1258

22. Sources of Materials Data J. G. Kaufman 1. Introduction and Scope 2. Intended Uses for Data 3. Types of Data 4. Subjects of Data Sources 5. Data Quality and Reliability 6. Platforms: Types of Data Sources 7. Specific Data Sources References

1259 1259 1259 1262 1263 1264 1265 1265 1268

Index

1271

PREFACE

In the years 1934–1936, when Ovid Wallace Eshbach (1893–1958) was preparing the first edition of the handbook that still bears his name, he was employed as special assistant in the Personnel Relations Department of AT&T. An electrical engineering graduate with honors from Lehigh University in 1915, he was well known in engineering education circles, particularly at schools which offered a cooperative option to their undergraduates. He coordinated the Bell System–MIT Cooperative Plan, an option in the Electrical Engineering Department at MIT, which permitted selected students to alternate study terms at MIT with terms of work, either with the Bell System or with the General Electric Company. In a memoir (available on the Northwestern University web site), to which I am indebted for this information, Eshbach’s son wrote that his father, in addition to interviewing, hiring, and placing students within the Bell System, monitored their progress, counseled them, and followed their careers. He was also an adjunct MIT professor and taught electrical engineering courses for students co-oping at Bell. Eshbach served on committees of the Society for the Promotion of Engineering Education and the American Institute of Electrical Engineers. He was a member of the Regional Accrediting Committee of the Engineers’ Council for Professional Development as well as the Special Advisory Committee to the President’s Committee on Civil Service Improvement. In 1932 he had directed a survey of adult technical education for the Chamber of Commerce of the State of New York. Several years after he published his handbook, Eshbach was approached to become dean of the Northwestern engineering school. Northwestern had established a new engineering school in the early 1900s, initially as a department within the College of Liberal Arts. In the mid-1920s the College of Engineering became the autonomous School of Engineering, with faculty members devoted exclusively to engineering. There was a crisis in engineering education at Northwestern in 1937 when, after years of declining enrollments, the school was denied accreditation during a national survey of engineering schools carried out by the Engineers’ Council for Professional Development. A major criticism was that the curriculum was too heavily weighted with nonprofessional courses. But in 1939, Walter P. Murphy, a wealthy inventor of railroad equipment, donated $6.7 million for the construction of

Northwestern’s Technological Institute building. When the construction of Tech, as the engineering school was then known, was completed in 1942, Northwestern received an additional bequest of $28 million from Murphy’s estate to provide for an engineering school “second to none.” Although Murphy insisted that the school not be named for him—he would not appear in public or on programs of ceremonies, such as at the cornerstone laying or the dedication of the new building—the cooperative engineering education program bears his name to commemorate his interest in “practical education.” Over the next 45 years cooperative engineering education remained a constant requirement at Tech, now known as the Robert R. McCormick School of Engineering and Applied Science. Eshbach remained Tech’s dean for the rest of his life as far as I can tell. His son reports in his memoir that Eshbach always had himself assigned to teach an undergraduate quiz section, usually in physics. And his name lives on at Northwestern. There is the Ovid W. Eshbach Society, in which alumni and other donors provide funds to strengthen undergraduate engineering education through support for such needs as laboratory equipment, undergraduate research, design competitions, and instructional software. There is also the Ovid W. Eshbach Award, established in 1948 by Tech’s first graduating class, which is awarded for overall excellence in scholarship and leadership. Each spring, nominations are accepted from the graduating class on who they feel most closely typifies the ideal engineering student. The team that Ovid W. Eshbach put together for the first edition of his handbook, which was called Handbook of Engineering Fundamentals, included 40 representatives from academia, industry, and government, most of them based in the northeast and some in the midwest. The handbook was the first volume in the Wiley Engineering Handbook Series, which also included the eleventh edition of the two-volume Kent’s Mechanical Engineers’ Handbook (one volume covered power, the second design and shop practice); the third edition of the two-volume Handbook for Electrical Engineers (one volume covered electric power, the second communications and electronics); and the third edition of the one-volume Mining Engineers’ Handbook. Tables of contents for all handbooks in the series xiii

xiv

were included on pages following the index of the Eshbach volume. The Handbook of Engineering Fundamentals, published in 1936 jointly by Wiley in New York and Chapman & Hall in London, contains 13 sections (chapters) and 1081 pages. Eshbach wrote in the Editor’s Preface: “This handbook has been prepared for the purpose of embodying in a single volume those fundamental laws and theories of science which are basic to engineering practice. It is essentially a summary of the principles of mathematics, physics, and chemistry, the properties and uses of engineering materials, the mechanics of solids and fluids, and the commonly used mathematics and physical tables, to which has been added a discussion of contract relations. Thus, with the exception of the technics of surveying and drawing, there is included the fundamental technology common to all engineering curricula.” The second edition of Handbook of Engineering Fundamentals was published in 1952. It was still part of the Wiley Engineering Handbook Series, to which had been added Handbook of Mineral Dressing. Again, it was jointly published by Wiley and Chapman & Hall. The copy that I have is from the fourth printing, May 1954. On the front cover, COLLEGE EDITION is stamped underneath the name ESHBACH. Eshbach made numerous changes for the second edition. He went west to find contributors—one from Texas and four from California were among the 38 contributors to this edition. With a new section on aerodynamics, he increased the number of sections to 14. He expanded the contracts section and renamed it Engineering Law. In addition, he enlarged the engineering tables to include standard structural sizes for aluminum and data on tangents and offsets for civil engineers; revised the mathematics section to eliminate “simple and commonly known items previously introduced for completeness” and put greater stress on “statistics, determinants, and vector analysis”; thoroughly revised the sections on solid and fluid mechanics; completely revised the section on electricity and magnetism; and in the sections on metallic and nonmetallic materials, “much material, more detailed, and of interest to special groups, has been eliminated to keep the volume within practical size.” By 1975, when the third edition was published, Eshbach had been dead for 17 years. Dr. Mott Souders, a chemical engineer from Piedmont, California, had taken over the editorship, although Eshbach’s name was the only one stamped on the spine and front cover of the book. Souders, too, had died, in 1974, before the book was published, this time solely by Wiley, which now had offices in London, Sydney, and Toronto as well as New York. The handbook was still part of the Wiley Engineering Handbook Series. The center of gravity of contributor locations had shifted further west. In addition to seven contributors from the West Coast and one from Texas, the roster of 40 contributors included 18 on the staff of the U.S. Air

PREFACE

Force Academy, who contributed a section of over 180 pages on aeronautics and astronautics. The third edition has 16 sections and 1562 pages. In his Preface, written in February 1974, Souders noted that the handbook contained new sections on astronautics, heat transfer, electronics, automatic control, and engineering economy. The sections on aeronautics and chemistry had been completely rewritten. New material had been added to the sections on mathematical and physical tables; mathematics, including an article on elements of Fortran; physical units and standards; as well as radiation, light, and acoustics. In the single section on properties of materials, all text was eliminated to provide space for more charts and tables. Souders also eliminated the section on engineering law. But the third edition did feature, on two pages following the Preface, canons of ethics of engineers approved by the Engineer’s Council for Professional Development on September 30, 1963. By the latter 1980s, the handbook’s editorship had passed to Byron Tapley, a professor in the Department of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin. The fourth edition’s size and scope increased dramatically. Whereas the trim size of the previous three editions had been 5 12 by 8 38 inches, the new edition was 7 by 10. The number of sections remained the same, at 16, but the number of pages increased dramatically to close to 2100. The number of contributors nearly doubled, to 77 and included, for the first time, one from overseas, in Athens, Greece. The rest were located throughout the United States—the East Coast orientation of the first edition was a thing of the distant past. As a result of the increased scope and complexity of the undertaking, a recently retired Wiley employee, Thurman Poston, was brought on board to assist Tapley in preparing the new edition. The fourth edition, published in 1990, also had a new name. It was now called Eshbach’s Handbook of Engineering Fundamentals. Also, major topic areas were placed into “chapters” and the term “sections” was now being used for subtopics. The most important changes to the handbook were undertaken in “recognition,” Tapley wrote in his Preface, in November 1989, “given to the dramatic change that computers and computer technology have made in the way we generate, receive, and display information.” Tapley continued: “The handbook has been modified to account for this impact in three substantial ways: (1) the chapter on mathematical and trigonometric tables has been reduced substantially in recognition of the fact that both small handheld computers and desktop personal computers allow a rapid generation of much of the information contained in this chapter, (2) a specific chapter dealing with computers and computer science has been added, and (3) specific applications where computers are useful have been included in many of the chapters.” Tapley added sections on differential equations and the finite-element method; expanded the control theory chapter; split the aeronautics and

PREFACE

astronautics chapter into two distinct chapters (due, I have been told, to usage of the handbook by students at the U.S. Air Force Academy for some years); and extensively revised the chapters on electromagnetics and circuits, electronics, radiation, light, acoustics, and engineering economics. In addition, international standard units were adopted throughout the handbook. My approach to the fifth edition, which is being published nearly two decades after the appearance of the previous edition, has been to revise or update the chapters where there has been substantial change over the intervening years, but the scope of those chapters does not require substantial expansion or alteration; add new chapters in areas where the scope was insufficient and engineers need more basic information; and eliminate chapters superseded by the ubiquity of the digital environment. So the overall goal has been to add more knowledge essential to engineers while reducing the size of the handbook. As a result, there are fewer pages but more chapters. The chapters that have been substantially updated and revised, but where the scope has remained unaltered for the most part, include those on mechanics of incompressible fluids, electromagnetics and circuits, acoustic, and engineering economy. All except the electromagnetics and circuits chapter have new contributors. There are numerous chapters that either cover topics new to the handbook or replace chapters, or sections of chapters, where more basic information is essential for practicing engineers and students at

xv

any level. These chapters include Selection of Metals for Design; Plastics: Thermoplastics, Thermosets, and Elastomers; Ceramics; Nondestructive Testing; Aerodynamics of Wings; Mathematical Models of Dynamic Physical Systems; Basic Control Systems Design; Thermodynamics Fundamentals; Heat Transfer Fundamentals; and Electronics (with sections on bipolar transistors, data acquisition and conversion, data analysis, diodes, electronic components, input devices, instruments, integrated circuits, microprocessors, oscilloscopes, and power devices). I have eliminated the chapter on computers and computer science, inasmuch as contributors now routinely absorb the digital world into their work whenever appropriate, as well as the over 250 pages of materials properties data, which have been replaced by a chapter, Sources of Materials Data, which is a current description of where and how to find reliable materials properties data on the Internet, the standard practice in this digital age. In addition, I have left alone those chapters which contain basic and theoretical information that does not change. Eshbach has gone through a great many iterations in its long life, yet the handbook remains true to its creator’s original vision. My thanks to him as well as to the legion of contributors whose efforts have graced the pages of the five editions of this great reference work. Myer Kutz Delmar, New York

CONTRIBUTORS

Kate D. Abel School of Systems and Enterprises, Stevens Institute of Technology, Hoboken, New Jersey

M. Parker Givens Institute of Optics, University of Rochester, Rochester, New York

Adrian Bejan Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina

Georges Grinstein University of Massachusetts Lowell, Lowell, Massachusetts

Jonathan Blotter Department of Mechanical Engineering, Brigham Young University, Provo, Utah Arbee L. P. Chen National Tsing Hua University, Hsinchu, Taiwan, Republic of China Robert P. Colwell Intel Corporation, Hillsboro, Oregon Robert L. Crane Air Force Research Laboratory, Materials Directorate, Wright Patterson Air Force Base, Dayton, Ohio John D. Cressler Georgia Institute of Technology, Atlanta, Georgia Clarence W. de Silva University of British Columbia, Vancouver, British Columbia, Canada D. H. Daley Department of Aeronautics, United States Air Force Academy, Colorado Springs, Colorado Matthew J. Donachie Rensselaer at Hartford, Hartford, Connecticut Neil F. Enke Department of Engineering Mechanics, University of Wisconsin, Madison, Wisconsin Halit Eren Curtin University of Technology, Bentley, Western Australia, Australia Wallace Fowler Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas Kent L. Gee Department of Mechanical Engineering, Brigham Young University, Provo, Utah

Ramesh Harjani University of Minnesota, Minneapolis, Minnesota Alex Q. Huang Virginia Polytechnic Institute and State University, Blacksburg, Virginia Wade W. Huebsch Department of Mechanical and Aerospace Engineering, College of Engineering and Mineral Resources, West Virginia University, Morgantown, West Virginia R. Nathan Katz Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts J. G. Kaufman Kaufman Associates, Inc., Columbus, Ohio Jeremy S. Knopp Air Force Research Laboratory, Materials Directorate, Wright Patterson Air Force Base, Dayton, Ohio D. A. Kohl The University of Texas at Austin, Austin, Texas J. G. Kaufman Kaufman Associates, Inc., Columbus, Ohio Konstantinos Misiakos NCSR “Demokritos,” Athens, Greece Kavita Nair University of Minnesota, Minneapolis, Minnesota Egemen Ol Ogretim Department of Civil and Environmental Engineering, College of Engineering and Mineral Resources, West Virginia University, Morgantown, West Virginia xvii

xviii

William J. Palm III Department of Mechanical Engineering, University of Rhode Island, Kingston, Rhode Island Edward N. Peters General Electric Company, Selkirk, New York G. P. Peterson Rensselaer Polytechnic Institute, Troy, New York Warren F. Phillips Department of Mechanical and Aerospace Engineering, Utah State University, Logan, Utah Dennis Polla University of Minnesota, Minneapolis, Minnesota N. Ranganathan University of South Florida, Tampa, Florida

CONTRIBUTORS

Bela I. Sandor Department of Engineering Mechanics, University of Wisconsin, Madison, Wisconsin Scott Sommerfeldt Department of Mechanical Engineering, Brigham Young University, Provo, Utah Marjan Trutschl University of Massachusetts Lowell, Lowell, Massachusetts Raju D. Venkataramana University of South Florida, Tampa, Florida Jack H. Westbrook Ballston Spa, New York K. Preston White, Jr. Department of Systems and Information Engineering, University of Virginia, Charlottesville, Virginia J. B. Wissler Department of Aeronautics, United States Air Force Academy, Colorado Springs, Colorado

J. N. Reddy Department of Mechanical Engineering, Texas A&M University, College Station, Texas

Yi-Hung Wu National Tsing Hua University, Hsinchu, Taiwan, Republic of China

Albert J. Rosa Professor Emeritus, University of Denver, Denver, Colorado

Bo Zhang Virginia Polytechnic Institute and State University, Blacksburg, Virginia

Andrew Rusek Oakland University, Rochester, Michigan

Chris Zillmer University of Minnesota, Minneapolis, Minnesota

CHAPTER 1 MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES∗ Jack H. Westbrook Ballston Spa, New York 1 SYMBOLS AND ABBREVIATIONS

3

Table 20

Complete Elliptic Integrals

39

Table 1

Greek Alphabet

3

Table 21

Gamma Functions

41

Table 2

Symbols for Mathematical Operations

3

Table 22

Bessel Functions

41

Table 3

Abbreviations for Scientific and Engineering Terms

Table 4

Symbols for Physical Quantities

Table 5

Graphic Symbols (after Dreyfus, 1972)

Table 6 Table 7 Table 8

Personal Computer Numeric Codes for Characters and Symbols Conversions for Number Systems of Different Bases Computer Graphics Codes and Standards

2 MATHEMATICAL TABLES

5

3

8 13

∗

STATISTICAL TABLES

Binomial Coefficients

42

Table 24

Probability Functions

43

Table 25

Factors for Computing Probable Errors

48

Table 26

Statistics and Probability Formulas

50

19 20

4

42

Table 23

UNITS AND STANDARDS

22

4.1

23

51

Physical Quantities and Their Relations

51

4.2

Dimensions and Dimension Systems

52

Table 9

Certain Constants Containing e and π

23

4.3

Dimension and Unit Systems

53

Table 10

Factorials

23

4.4

The International System of Units

53

Table 11

Common and Natural Logarithms of Numbers

23

4.5

Length, Mass, and Time

57

4.6

Force, Energy, and Power

59

Table 12

Circular Arcs, Chords, and Segments

25

Table 13

Values of Degrees, Minutes, and Seconds in Radians

4.7

Thermal Units and Standards

61

27

4.8

Chemical Units and Standards

63

Table 14

Values of Radians in Degrees

28

4.9

Table 15

Decimals of a Degree in Minutes and Seconds

Theoretical, or Absolute, Electrical Units

64

28

Table 16

Minutes in Decimals of a Degree

29

Internationally Adopted Electrical Units and Standards

67

Table 17

Seconds in Decimals of a Degree

29

Table 18

Table of Integrals

29

Table 27

Temperature Conversion

69

Table 19

Haversines

38

Table 28

Length [L]

70

4.10 5

TABLES OF CONVERSION FACTORS

Table 29 ∗ This

chapter is a revision and extension of Sections 1 and 3 of the third edition, which were written by Mott Souders and Ernst Weber, respectively. Section 4.4 is derived principally from ASTM’s Standard for Metric Practice, ASTM E38082, Philadelphia, 1982 (with permission). Section 6.1 is derived from MIS Newsletter, General Electric Co., 1980 (with permission).

Area

[L2 ] [L3 ]

69

72

Table 30

Volume

Table 31

Plane Angle (No Dimensions)

Table 32

Solid Angle (No Dimensions)

76

Table 33

Time [T ]

77


74 76

1

2

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Table 34

Linear Velocity [LT −1 ]

78

Table 62

Thermal Conductivity [LMT −3 t −1 ]

94

Table 35

Angular Velocity [T −1 ]

79

Table 63

Photometric Units

95

Table 36

Linear Accelerationa [LT −2 ]

79

Table 64

Specific Gravity Conversions

95

−2

Table 37

Angular Acceleration [T

Table 38

Mass [M] and Weighta

80

Table 39

Density or Mass per Unit Volume [ML−3 ]

81

Table 40

Force [MLT −2 ] or [F ]

Table 41 Table 42

]

79

STANDARD SIZES

97

6.1

Preferred Numbers

97

Table 65

Basic Series of Preferred Numbers: R5, R10, R20, and R40 Series

98

82

Table 66

Pressure or Force per Unit Area [ML−1 T −2 ] or [F L−2 ]

Basic Series of Preferred Numbers: R80 Series

99

83

Table 67

Expansion of R5 Series

99

Torque or Moment of Force [ML2 T −2 ] or [F L]a

84

Table 68

Rounding of Preferred Numbers

6.2

Gages

100

Table 69

U.S. Standard Gage for Sheet and Plate Iron and Steel and Its Extension

100

Table 70

American Wire Gage: Weights of Copper, Aluminum, and Brass Sheets and Plates

101

Table 71

Comparison of Wire Gage Diameters in Mils

102

6.3

Paper Sizes

104

Table 72

Standard Engineering Drawing Sizes

104

Table 73

Eleven International Paper Sizes

104

Table 43

Moment of Inertia [ML2 ]

84

Table 44

Energy, Work and Heata [ML2 T −2 ] or [FL]

85

Power or Rate of Doing Work [ML2 T −3 ] or [F LT −1 ]

86

Quantity of Electricity and Dielectric Flux [Q]

87

Charge per Unit Area and Electric Flux Density [QL−2 ]

87

Table 48

Electric Current [QT −1 ]

87

Table 49

Current Density [QT −1 L−2 ]

88

Table 50

Electric Potential and Electromotive Force [MQ−1 L2 T −2 ] or [F Q−1 L]

88

Table 45 Table 46 Table 47

Table 51 Table 52 Table 53 Table 54 Table 55 Table 56 Table 57 Table 58 Table 59 Table 60

Table 61

Electric Field Intensity and Potential Gradient [MQ−1 LT −2 ] or [F Q−1 ] Electric Resistance or [F Q−2 LT ]

89

6

6.4

Sieve Sizes

105

Table 74

Tyler Standard Screen Scale Sieves

105

Table 75

Nominal Dimensions, Permissible Variations, and Limits for Woven Wire Cloth of Standard Sieves, U.S. Series, ASTM Standard

106

6.5

Standard Structural Sizes—Steel

106

[MQ−2 L2 T −1 ] 90 −2

3

Electric Resistivity [MQ L T or [F Q−2 L2 T ]

−1

] 90

99

Electric Conductivity [M −1 Q2 L−3 T ] or [F −1 Q2 L−2 T −1 ]

Table 76

Properties of Wide-Flange Sections

107

90

Table 77

Capacitance [M −1 Q2 L−2 T 2 ] or [F −1 Q2 L−1 ]

Properties of American Standard Beams

110

91

Table 78

Inductance [MQ−2 L2 ] or [F Q−2 LT 2 ]

91

Properties of American Standard Channels

111

Table 79

Properties of Angles with Equal Legs

112

Properties of Angles with Unequal Legs

113

Table 81

Properties and Dimensions of Tees

116

Table 82

Properties and Dimensions of Zees

119

Table 83

Properties and Dimensions of H Bearing Piles

120

Table 84

Square and Round Bars

120

Table 85

Dimensions of Ferrous Pipe

122

Magnetic Flux [MQ [FQ −1 LT ]

−1 2

L T

−1

] or 91

Magnetic Flux Density [MQ −1 T −1 ] or [FQ −1 L−1 T ]

92

Magnetic Potential and Magnetomotive Force [QT −1 ]

92

Table 80

Magnetic Field Intensity, Potential Gradient, and Magnetizing Force [QL−1 T −1 ]

92

Specific Heat [L2 T −2 t −1 ] (t = temperature)

93

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES Table 86

Properties and Dimensions of Steel Pipe Standard Structural Shapes— Aluminum

Table 87

Aluminum Association Standard Channels—Dimensions, Areas, Weights, and Section Properties

Table 88

Table 89

Aluminum Pipe—Diameters, Wall Thicknesses, and Weights

138

Table 99

Aluminum Electrical Conduit—Designed Dimensions and Weights

140

Table 100

Equivalent Resistivity Values

140

Table 101

Property Limits—Wire (Up to 0.374 in. Diameter)

141

129

129

Aluminum Association Standard I Beams—Dimensions, Areas, Weights, and Section Properties

130

Standard Structural Shapes—Equal Angles

7

131

Table 90

Standard Structural Shapes—Unequal Angles

132

Table 91

Channels, American Standard

134

Channels, Shipbuilding, and Carbuilding

135

Table 93

H Beams

135

Table 94

I Beams

136

Table 95

Wide-Flange Beams

136

Table 96

Tees

137

Table 97

Zees

137

Table 92

Table 98

128

6.6

3

STANDARD SCREWS∗

142

Table 102

Standard Screw Threads

143

Table 103

ASA Standard Bolts and Nuts

145

Table 104

Holding Power of Flat or Cup Point Set Screws

155

Table 105

Lag Screws

156

Table 106

Recommended Diameters of Pilot Hole for Types of Wood

156

Nominal and Minimum Dressed Sizes of American Standard Lumber

157

Table 107

American Standard Wood Screws

157

Table 108

Nominal and Minimum Dressed Sizes of American Standard Lumber

158

7.1

1 SYMBOLS AND ABBREVIATIONS

aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaaaaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa

Table 1 Greek Alphabet A B E Z

α β γ δ ε ζ

Alpha Beta Gamma Delta Epsilon Zeta

H I K M

η ϑ ι κ λ µ

θ

Eta Theta Iota Kappa Lambda Mu

N

O P

ν ξ o π ρ σ

ς

Nu Xi Omicron Pi Rho Sigma

T ϒ X "

τ υ φ χ ψ ω

Tau Upsilon Phi Chi Psi Omega

Table 2 Symbols for Mathematical Operationsa Addition and Subtraction a + b, a plus b a − b, a minus b a ± b, a plus or minus b a ∓ b, a minus or plus b Multiplication and Division a × b, or a · b, or ab, a times b a a ÷ b, or , or a/b, a divided by b b Symbols of Aggregation ( ) parentheses [ ] brackets { } braces

a = b, a is not equal to b a > b, a is greater than b a < b, a is less than b a b, a much larger than b a b, a much smaller than b a b, a equals or is greater than b a b, a is less than or equals b a ≡ b, a is identical to b a → b, or a = b, a approaches b as a limit Proportion a/b = c/d, or a : b :: c : d, a is to b as c is to d a ∝ b, a ∼ b, a varies directly as b %, percent Powers and Roots

–vinculum Equalities and Inequalities a = b, a equals b a ≈ b, a approximately equals b

a2 , a squared an , a raised to the nth power √ a, square root of a √ 3 a, cube root of a √ n a, or a1/n , nth root of a (Continues)

4


Table 2 −n

(Continued ) n

a , 1/a 3.14 × 104 = 31,400 3.14 × 10−4 = 0.000314

e, eccentricity in conics p, semi latus rectum in conics l = cos α, m = cos β, n = cos γ , direction cosines

Miscellaneous a, mean value of a a!, = 1 · 2 · 3 . . . a, factorial a |a| = absolute value of a P(n, r) = n(n − 1)(n − 2) · · · (n − r + 1) n P(n, r) = = binomial coefficients C(n, r) = r √ r! i (or j) = −1, imaginary unit π = 3.1416, ratio of the circumference to the diameter of a circle ∞, infinity Plane Geometry 9.2

pH + 0.009 (4.0 − pH)

for pH < 4.0

4.9 Theoretical, or Absolute, Electrical Units

With the general adoption of SI as the form of metric system that is preferred for all applications, further use of cgs units of electricity and magnetism is deprecated. Nonetheless, for historical reasons as well as for comprehensiveness, a brief review is included in this section and section 4.10. The definitions of the theoretical, or “absolute,” units are based on a particular choice of the numerical value of either ke , the constant in Coulomb’s, electrostatic force law, or km , the constant in Ampere’s electrodynamic force law. The designation absolute units is generally used because of historical tradition; an interesting account of the history can be found in Glazebrook’s Handbook for Applied Physics, Vol. II, “Electricity,” pp. 211 ff., 1922. Because of the theoretical background of the unit definitions, they have also been designated as “theoretical” units, which is in good contradistinction to practical units based on physical standards. Theoretical Electrostatic Units The theoretical electrostatic units are based on the cgs system of mechanical units and the choice of the numerical value unity for kev in Coulomb’s law. They are frequently referred to as the cgs electrostatic units, but no specific unit names are available. In order to avoid the cumbersome writing, for example, one “theoretical electrostatic unit of charge,” it had been proposed to use the theoretical “practical” unit names and prefix them with either stat or E.S. as, for example, statcoulomb, or E.S. coulomb. The first alternative will be used here.

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES

The absolute dielectric constant (permittivity) of free space is the reciprocal of the Coulomb constant kev and is chosen as the fourth fundamental quantity in the theoretical electrostatic system of units. Its numerical value is defined as unity, and it is identical with one statfarad per centimeter if use is made of prefixing the corresponding unit of the “practical” series. The theoretical electrostatic unit of charge, or the statcoulomb, is defined as the quantity of electricity that, when concentrated at a point and placed at one centimeter distance from an equal quantity of electricity similarly concentrated, will experience a mechanical force of one dyne in free space. An alternative definition, based on the concept of field lines, gives the theoretical electrostatic unit of charge as a positive charge from which in free space exactly 4π displacement lines emerge. The theoretical electrostatic unit of displacement flux (dielectric flux) is the “line of displacement flux,” or 14 π of the theoretical electrostatic unit of charge. This definition provides the basis for graphical field mapping insofar as it gives a definite rule for the selection of displacement lines to represent the distribution of the field quantitatively. The theoretical electrostatic unit of displacement, or dielectric flux density, is chosen as one displacement line per square centimeter area perpendicular to the direction of the displacement lines. It can be given also as 14 π statcoulomb per square centimeter (according to Gauss’s law). In isotropic media the displacement has the same direction as the potential gradient, and the surfaces perpendicular to the field lines become the equipotential surfaces; the theoretical electrostatic unit of displacement can then be defined as one displacement line per square centimeter of equipotential surface. The theoretical electrostatic unit of electrostatic potential, or the statvolt, is defined as existing at a point in an electrostatic field, if the work done to bring the theoretical electrostatic unit of charge, or the statcoulomb, from infinity to this point equals one erg. This customary definition implies, however, that the potential vanishes at infinite distances and has, therefore, only restricted validity. As it is fundamentally impossible to give absolute values of potential, the use of potential difference and its unit (see below) should be preferred. The theoretical electrostatic unit of electrical potential difference or voltage, is the statvolt and is defined as existing between two points in space if the work done to bring the theoretical electrostatic unit of charge, or the statcoulomb, from one of these points to the other equals one erg. Potential difference is counted positive in the direction in which a negative quantity of electricity would be moved by the electrostatic field. The theoretical electrostatic unit of capacitance, or the statfarad, is defined as the capacitance that maintains an electrical potential difference of one statvolt between two conductors charged with equal and

65

opposite electrical charges of one statcoulomb. In the older literature, the cgs electrostatic unit of capacitance is identified with the “centimeter”; this was replaced by statfarad to avoid confusion. The theoretical electrostatic unit of electric potential gradient, or field strength (field intensity), is defined to exist at a point in an electric field if the mechanical force exerted upon the theoretical electrostatic unit of charge concentrated at this point is equal to one dyne. It is expressed as one statvolt per centimeter. The theoretical electrostatic unit of current, or the statampere, is defined as the time rate of transfer of the theoretical electrostatic unit of charge and is identical with the statcoulomb per second. The theoretical electrostatic unit of electrical resistance, or the statohm, is defined as the resistance of a conductor in which a current of one statampere is produced if a potential difference of one statvolt is applied at its ends. The theoretical electrostatic unit of electromotive force (emf) is defined as equivalent to the theoretical electrostatic unit of potential difference if it produces a current of one statampere in a conductor of one statohm resistance. It is identical with the statvolt but, according to its concept, requires an independent definition. The theoretical electrostatic unit of magnetic intensity is defined as the magnetic intensity at the center of a circle of 4π centimeters diameter in which a current of one statampere is flowing. This unit is equal to 4π statamperes per centimeter but has no name as the factor 4π excludes the possibility of using the prefixed “practical” unit name. The theoretical electrostatic unit of magnetic flux, or the statweben, is defined as the magnetic flux whose time rate of change through a linear conductor loop (linear conductor is used to designate a conductor of infinitely small cross section) produces in this loop an emf of one statvolt. The theoretical electrostatic unit of magnetic flux density, or induction, is defined as the electrostatic unit of magnetic flux per square centimeter area, or the statweber per square centimeter. The absolute magnetic permeability of free space is defined as the ratio of magnetic induction to the magnetic intensity. Its unit is the stathenry per centimeter as a derived unit. The theoretical electrostatic unit of inductance, or the stathenry, is defined as connected with a conductor loop carrying a steady current of one statampere that produces a magnetic flux of one statweber. A more general definition, applicable to varying fields with nonlinear relation between magnetic flux and current, gives the stathenry as connected with a conductor loop in which a time rate of change in the current of one statcoulomb produces a time rate of change in the magnetic flux of one statweber per second.

66


Theoretical Electromagnetic Units The theoretical electromagnetic units are based on the cgs system of mechanical units and Coulomb’s law of mechanical force action between two isolated magnetic quantities m1 and m2 (approximately true for very long bar magnets) that must be written as

Fm =

km m1 m2 2 r2

(7)

where km is the proportionality constant of Ampere’s law for force action between parallel currents that is more basic, and amenable to much more accurate measurement, than (7). The factor 12 appears here because of the three-dimensional character of the field distribution around point magnets as compared with the two-dimensional field of two parallel currents. The theoretical electromagnetic units are obtained by defining the numerical value of kmv /2 (for vacuum) as unity; they are frequently referred to as the cgs electromagnetic units. Only a few specific unit names are available. In order to avoid cumbersome writing, for example, one “theoretical electromagnetic unit of charge,” it had been proposed to use the theoretical “practical” unit names and prefix them with either abor E.M. as, for example, abcoulomb, or E.M. coulomb. The first alternative will be used here. The absolute magnetic permeability of free space is the value kmv /2 in (7) and is chosen as the fourth fundamental quantity in the theoretical electromagnetic system of units. Its numerical value is assumed as unity, and it is identical with one abhenry per centimeter if use is made of prefixing the corresponding unit of the “practical” series. The theoretical electromagnetic unit of magnetic quantity is defined as the magnetic quantity that, when concentrated at a point and placed at one centimeter distance from an equal magnetic quantity similarly concentrated, will experience a mechanical force of one dyne in free space. An alternative definition, based on the concept of magnetic intensity lines, gives the theoretical electromagnetic unit of magnetic quantity as a positive magnetic quantity from which, in free space, exactly 4π magnetic intensity lines emerge. The theoretical electromagnetic unit of magnetic moment is defined as the magnetic moment possessed by a magnet formed by two theoretical electromagnetic units of magnetic quantity of opposite sign, concentrated at two points one centimeter apart. As a vector, its positive direction is defined from the negative to the positive magnetic quantity along the center line. The theoretical electromagnetic unit of magnetic induction (magnetic flux density), or the gauss, is defined to exist at a point in a magnetic field, if the mechanical torque exerted upon a magnet with theoretical electromagnetic unit of magnetic moment and directed perpendicular to the magnetic field is equal to one dyne-centimeter. The lines to which the vector of

magnetic induction is tangent at every point are called induction lines or magnetic flux lines; on the basis of this flux concept, magnetic induction is identical with magnetic flux density. The theoretical electromagnetic unit of magnetic flux, or the maxwell, is the “field line” or line of magnetic induction. In free space, the theoretical electromagnetic unit of magnetic quantity issues 4π induction lines; the unit of magnetic flux, or the maxwell, is then 1/4π of the theoretical electromagnetic unit of magnetic quantity times the absolute permeability of free space. The theoretical electromagnetic unit of magnetic intensity (magnetizing force), or the oersted, is defined to exist at a point in a magnetic field in free space where one measures a magnetic induction of one gauss. The theoretical electromagnetic unit of current, or the abampere, is defined as the current that flows in a circle of one centimeter diameter and produces at the center of this circle a magnetic intensity of one oersted. The theoretical electromagnetic unit of inductance, or the abhenry, is defined as connected with a conductor loop in which a time rate of change of one maxwell per second in the magnetic flux produces a time rate of change in the current of one abampere per second. In the older literature, the cgs electromagnetic unit of inductance is identified with the “centimeter”; this should be replaced by a henry to avoid confusion. The theoretical electromagnetic unit of magnetomotive force (mmf) is defined as the magnetic driving force produced by a conductor loop carrying a steady current of 14 π abamperes; it has the name one gilbert. The concept of magnetomotive force as the driving force in a “magnetic circuit” permits an alternative definition of the gilbert as the magnetomotive force that produces a uniform magnetic intensity of one oersted over a length of one centimeter in the magnetic circuit. Obviously, one gilbert equals one oersted-centimeter. The theoretical electromagnetic unit of magnetostatic potential is defined as the potential existing at a point in a magnetic field if the work done to bring the theoretical electromagnetic unit of magnetic quantity from infinity to this point equals one erg. This customary definition implies, however, that the potential vanishes at infinite distances, and the definition has therefore only restricted validity. The unit, thus defined, is identical with one gilbert. The difference in magnetostatic potential between any two points is usually called magnetomotive force (mmf). The theoretical electromagnetic unit of reluctance is defined as the reluctance of a magnetic circuit in which a magnetomotive force of one gilbert produces a magnetic flux of one maxwell. The theoretical electromagnetic unit of electric charge, or the abcoulomb, is defined as the quantity of electricity that passes through any section of an electric circuit in one second if the current is one abampere. The theoretical electromagnetic unit of displacement flux (dielectric flux) is the “line of displacement


flux,” or 14 π of the theoretical electromagnetic unit of electric charge. This definition provides the basis for graphical field mapping insofar as it gives a definite rule for the selection of displacement lines to represent the character of the field. The theoretical electromagnetic unit of displacement, or dielectric flux density, is chosen as one displacement line per square centimeter area perpendicular to the direction of the displacement lines. It can also be given as 14 π abcoulombs per square centimeter (according to Gauss’s law). In isotropic media the theoretical electromagnetic unit of displacement can be defined as one displacement line per square centimeter of equipotential surface. (See discussion on theoretical electrostatic unit of displacement.) The theoretical electromagnetic unit of electrical potential difference, or voltage, is the abvolt and is defined as the potential difference existing between two points in space if the work done in bringing the theoretical electromagnetic unit of charge, or the abcoulomb, from one of these points to the other equals one erg. Potential difference is counted positive in the direction in which a negative quantity of electricity would be moved by the electrostatic field. The theoretical electromagnetic unit of capacitance, or the abfarad, is defined as the capacitance that maintains an electrical potential difference of one abvolt between two conductors charged with equal and opposite electrical quantities of one abcoulomb. The theoretical electromagnetic unit of potential gradient, or field strength (field intensity), is defined to exist at a point in an electric field if the mechanical force exerted upon the theoretical electromagnetic unit of charge concentrated at this point is equal to one dyne. It is expressed as one abvolt per centimeter. The theoretical electromagnetic unit of resistance, or the abohm, is defined as the resistance of a conductor in which a current of one abampere is produced if a potential difference of one abvolt is applied at its ends. The theoretical electromagnetic unit of electromotive force (emf) is defined as the electromotive force acting in an electric circuit in which a current of one abampere is flowing and electrical energy is converted into other kinds of energy at the rate of one erg per second. This unit is identical with the abvolt. The absolute dielectric constant of free space is defined as the ratio of displacement to the electric field intensity. Its unit is the abfarad per centimeter, a derived unit. Theoretical Electrodynamic Units The theoretical electrodynamic units are based on the cgs system of mechanical units and are therefore frequently referred to as the cgs electrodynamic units. In contradistinction to the theoretical electromagnetic units, these units are derived from a significant experimental law, Ampere’s

67

experiment on the mechanical force between two parallel currents. The units as proposed by Ampere and used by W. Weber differ from the electromagnetic units by factors of 2 and multiples thereof. They can be made to coincide with the theoretical electromagnetic units by proper definition of the fundamental unit of current. Some of the important definitions will be given for this latter case only. For the absolute magnetic permeability of free space, see discussion on theoretical electromagnetic units. The theoretical electrodynamic unit of current, or the abampere, is defined as the current flowing in a circuit consisting of two infinitely long parallel wires one centimeter apart when the electrodynamic force of repulsion between the two wires is two dynes per centimeter length in free space. If the more natural choice of one dyne per centimeter length is made, the original proposal of Ampere is obtained and the unit √ of current becomes 1/ 2 abampere. The theoretical electrodynamic unit of magnetic induction is defined as the magnetic induction inducing an electromotive force of one abvolt in a conductor of one centimeter length and moving with a velocity of one centimeter per second if the conductor, its velocity, and the magnetic induction are mutually perpendicular. The unit thus defined is called one gauss. The theoretical electrodynamic unit of magnetic flux, or the maxwell, is defined as the magnetic flux represented by a uniform magnetic induction of one gauss over an area of one square centimeter perpendicular to the direction of the magnetic induction. The theoretical electrodynamic unit of magnetic intensity, or the oersted, is defined as the magnetic intensity at the center of a circle of 4π centimeters diameter in which a current of one abampere is flowing. All the other unit definitions, which do not pertain to magnetic quantities, are identical with the definitions for the theoretical electromagnetic units. 4.10 Internationally Adopted Electrical Units and Standards

In October 1946, at Paris, the International Committee on Weights and Measures decided to abandon the socalled international practical units based on physical standards (see below) and to adopt effective January 1, 1948, the so-called absolute practical units for international use. Adopted Absolute Practical Units By a series of international actions, the “absolute” practical electrical units are defined as exact powers of 10 of corresponding theoretical electrodynamic and electromagnetic units because they are based on the choice of the proportionality constant in Ampère’s law for free space as kmv = 2 × 10−7 H/m

68


The absolute practical unit of current, or the absolute is defined as the current flowing in a circuit consisting of two very long parallel thin wires spaced 1 m apart in free space if the electrodynamic force action between the wires is 2 × 10−7 N = 0.02 dyne per meter length. It is 10−1 of the theoretical or absolute electrodynamic or electromagnetic unit of current and was adopted internationally in 1881. The absolute practical unit of electric charge, or the absolute coulomb, is defined as the quantity of electricity that passes through a cross-sectional surface in one second if the current is one absolute ampere. It is 10−1 of the theoretical or absolute electromagnetic unit of electric charge and was adopted internationally in 1881. The absolute practical unit of electric potential difference, or the absolute volt, is defined as the potential difference existing between two points in space if the work done in bringing an electric charge of one absolute coulomb from one of these points to another is equal to one absolute joule = 107 ergs. It is 108 of the theoretical or absolute electromagnetic unit of potential difference and was adopted internationally in 1881. The absolute practical unit of resistance, or the absolute ohm, is defined as the resistance of a conductor in which a current of one absolute ampere is produced if a potential difference of one absolute volt is applied at its ends. It is 109 of the theoretical or absolute electromagnetic unit of resistance and was adopted internationally in 1881. The absolute practical unit of magnetic flux, or the absolute weber, is defined to be linked with a closed loop of thin wire of total resistance one absolute ohm if upon removing the wire loop from the magnetic field a total charge of one absolute coulomb is passed through any cross section of the wire. It is 108 of the theoretical or absolute electromagnetic unit of magnetic flux, the maxwell, and was adopted internationally in 1933. The absolute practical unit of inductance, or the absolute henry, is defined as connected with a closed loop of thin wire in which a time rate of change of one absolute weber per second in the magnetic flux produces a time rate of change in the current of one absolute ampere. It is 109 of the theoretical or absolute electromagnetic unit of inductance and was adopted internationally in 1893. The absolute practical unit of capacitance, or the absolute farad, is defined as the capacitance that maintains an electric potential difference of one absolute volt between two conductors charged with equal and opposite electrical quantities of one coulomb. It is 10−9 of the theoretical or absolute electromagnetic unit of capacitance and was adopted internationally in 1881.

Abandoned International Practical Units The International System of electrical and magnetic units is a system for electrical and magnetic quantities that takes as the four fundamental quantities resistance, current, length, and time. The units of resistance and current are defined by physical standards that were originally aimed to be exact replicas of the “absolute” practical units, namely the absolute ampere and the absolute ohm. On account of long-range variations in the physical standards, it proved impossible to rely upon them for international use and they recently have been replaced by the absolute practical units. The international practical standards are defined as follows: The international ohm is the resistance at 0 ◦ C of a column of mercury of uniform cross section having a length of 106.300 cm and a mass of 14.4521 g. The international ampere is defined as the current that will deposit silver at the rate of 0.00111800 g/sec. From these fundamental units, all other electrical and magnetic units can be defined in a manner similar to the absolute practical units. Because of the inconvenience of the silver voltameter as a standard, the various national laboratories actually used a volt, defining its value in terms of the other two standards. At its conference in October 1946 in Paris, the International Committee on Weights and Measures accepted as the best relations between the international and the absolute practical units the following:

1 mean international ohm = 1.00049 absolute ohms 1 mean international volt = 1.00034 absolute volts These mean values are the averages of values measured in six different national laboratories. On the basis of these mean values, the specific unit relation for converting international units appearing on certificates of the National Bureau of Standards, Washington, DC, into absolute practical units are as follows: 1 international ampere = 0.999835 absolute ampere 1 international coulomb = 0.999835 absolute coulomb 1 international henry = 1.000495 absolute henries 1 international farad = 0.999505 absolute farad 1 international watt = 1.000165 absolute watts 1 international joule = 1.000165 absolute joules


BIBLIOGRAPHY FOR UNITS AND MEASUREMENTS Cohen, E. R., and Taylor, B. N., “The 1986 Adjustment of the Fundamental Physical Constants,” Report of the CODATA Task Group on Fundamental Constants, November 1986, CODATA Bulletin No. 63, International Council of Scientific Unions, Committee on Data for Science and Technology, Pergamon, 1986. Hvistendahl, H. S., Engineering Units and Physical Quantities, Macmillan, London, 1964. Jerrard, H. G., and McNeill, D. B., A Dictionary of Scientific Units, 2nd ed., Chapman & Hall, London, 1964. Letter Symbols for Units of Measurement, ANSI/IEEE Std. 260-1978, Institute of Electrical and Electronic Engineers, New York, 1978. Quantities, Units, Symbols, Conversion Factors, and Conversion Tables, ISO Reference 31, 15 sections, International Organization for Standardization Geneva, 1973–1979. Standard for Metric Practice, ASTM E 380-82, American Society for Testing and Materials, Philadelphia, 1982.

69

Young, L., System of Units in Electricity and Magnetism, Oliver and Boyd, Edinburgh, 1969. Young, L., Research Concerning Metrology and Fundamental Constants, National Academy Press, Washington, DC, 1983.

5 TABLES OF CONVERSION FACTORS∗ J. G. Brainerd (revised and extended by J. H. Westbrook) Table 27 ◦ ◦ ◦ ◦

∗

Temperature Conversion

F = (◦ C × 95 ) + 32 = (◦ C + 40) × ◦

C = ( F − 32) × R =◦F + 459.69 K = C + 273.16

5 9

◦

= ( F + 40) ×

9 5 5 9

− 40 − 40

Boldface units in Tables 28–63 are SI.

to

Centimeters Feet Inches Kilometers Nautical Miles Meters Mils Miles Millimeters Yards

↓

Obtain

by →

of →

Number

Multiply

Length [L]

Centimeters 0.01 393.7 6.214 × 10−6 10 1.094 × 10−2

1 3.281 × 10−2 0.3937 10−5

Feet 30.48 1 12 3.048 × 10−4 1.645 × 10−4 0.3048 1.2 × 104 1.894 × 10−4 304.8 0.3333

Inches 2.540 8.333 × 10−2 1 2.540 × 10−5 — 2.540 × 10−2 1000 1.578 × 10−5 25.40 2.778 × 10−2

1.853 × 105 6080.27 7.296 × 104 1.853 1 1853 — 1.1516 — 2027

Kilometers

Table 28

Nautical Miles

70 105 3281 3.937 × 104 1 0.5396 1000 3.937 × 107 0.6214 106 1094

Meters 100 3.281 39.37 0.001 5.396 × 10−4 1 3.937 × 104 6.214 × 10−4 1000 1.094

Mils 1 — 2.540 × 10−2 2.778 × 10−5

2.540 × 10−3 8.333 × 10−5 0.001 2.540 × 10−8

Miles 1.609 × 105 5280 6.336 × 104 1.609 0.8684 1609 — 1 — 1760

Millimeters 0.1 3.281 × 10−3 3.937 × 10−2 10−6 — 0.001 39.37 6.214 × 10−7 1 1.094 × 10−3

91.44 3 36 9.144 × 10−4 4.934 × 10−4 0.9144 3.6 × 104 5.682 × 10−4 914.4 1

Yards

71

120 fathoms = 1 cable length

3 nautical miles = 1 league (U.S.) 3 statute miles = 1 league (Gr. Britain)

6080.27 feet = 1 nautical mile = 1.15156 statute miles

2 yards = 1 fathom

8 furlongs = 1 mile = 5280 feet = 1760 yards = 8000 links = 320 rods = 80 chains

10 chains = 1 furlong = 660 feet = 220 yards = 1000 links = 40 rods

4 rods = 1 chain (Gunther’s) = 66 feet = 22 yards = 100 links

25 links = 1 rod = 16.5 feet = 5.5 yards (1 rod = 1 pole = 1 perch)

7.92 inches = 1 link

Miscellaneous

9 inches = 1 span 2 12 feet = 1 military pace

3 inches = 1 palm 4 inches = 1 hand

(Note: A nautical mile is the length of a minute of longitude of the earth at the equator at sea level. The British Admiralty uses the round figure of 6080 feet. The word “knot” is used to denote “nautical miles per hour.”)

Nautical Measure

Ropes and Cables

Land Measure

Length

72 by →

of →

Number

Multiply

Area [L2 ]

Acres Circular Mils Square Centimeters Square Feet Square Inches Square Kilometers Square Meters Square Miles Square Millimeters Square Yards

↓

Obtain

to

Table 29

Acres 1 — — 4.356 × 104 6,272,640 4.047 × 10−3 4047 1.562 × 10−3 — 4840

Circular Mils — 1 5.067 × 10−6 — 7.854 × 10−7 — — — 5.067 × 10−4 —

Centimeters

Square — 1.973 × 105 1 1.076 × 10−3 0.1550 10−10 0.0001 3.861 × 10−11 100 1.196 × 10−4

Feet

Square 2.296 × 10−5 1.833 × 108 929.0 1 144 9.290 × 10−8 9.290 × 10−2 3.587 × 10−8 9.290 × 104 0.1111

Inches

Square — 1.273 × 106 6.452 6.944 × 10−3 1 6.452 × 10−10 6.452 × 10−4 — 645.2 7.716 × 10−4

Kilometers

Square 247.1 — 1010 1.076 × 107 1.550 × 109 1 106 0.3861 1012 1.196 × 106

Meters

Square 2.471 × 10−4 1.973 × 109 104 10.76 1550 10−6 1 3.861 × 10−7 106 1.196

Miles

Square 640 — 2.590 × 1010 2.788 × 107 4.015 × 109 2.590 2.590 × 106 1 — 3.098 × 106

Millimeters Square

— 1973 0.01 1.076 × 10−5 1.550 × 10−3 10−12 10−6 3.861 × 10−13 1 1.196 × 10−6

Square

2.066 × 10−4 — 8361 9 1296 8.361 × 10−7 0.8361 3.228 × 10−7 8.361 × 105 1

Yards

73

640 acres = 1 square mile = 2560 roods = 102,400 square rods

4 roods = 1 acre = 10 square chains = 160 square rods

square chains = 1 rood = 40 square rods = 1210 square yards

1 square inch = 1.2732 × 106 circular mils = 106 square mils

1 circular inch = 106 circular mils = 0.7854 × 106 square mils

1 square mil = 1.2732 circular mils

A circular mil is the area of a circle 1 mil (or 0.001 inch) in diameter = 0.7854 square mil

1 square inch = 1.2732 circular inches

A circular inch is the area of a circle 1 inch in diameter = 0.7854 square inch

100 square feet = 1 square

1 section of land = 1 square mile; 1 quarter section = 160 acres

2 12

16 square rods = 1 square chain = 484 square yards = 4356 square feet

30 14 square yards = 1 square rod = 272 14 square feet

Circular Inch and Circular Mil

Architect’s Measure

Land Measure

Area

to by

Bushels (Dry) Cubic Centimeters Cubic Feet Cubic Inches Cubic Meters Cubic Yards Gallons (Liquid) Liters Pints (Liquid) Quarts (Liquid)

↓

Obtain

→

of →

Number

Multiply

Volume [L3 ]

Bushels (Dry) 1 3.524 × 104 1.2445 2150.4 3.524 × 10−2 — — 35.24 — —

Cubic Centimeters — 1 3.531 × 10−5 6.102 × 10−2 10−6 1.308 × 10−6 2.642 × 10−4 0.001 2.113 × 10−3 1.057 × 10−3

Cubic Feet 0.8036 2.832 × 104 1 1728 2.832 × 10−2 3.704 × 10−2 7.481 28.32 59.84 29.92

28.38 106 35.31 6.102 × 104 1 1.308 264.2 1000 2113 1057

Cubic Inches

Table 30

Cubic Meters

74 4.651 × 10−4 16.39 5.787 × 10−4 1 1.639 × 10−5 2.143 × 10−5 4.329 × 10−3 1.639 × 10−2 3.463 × 10−2 1.732 × 10−2

Cubic Yards — 7.646 × 105 27 46,656 0.7646 1 202.0 764.6 1616 807.9

Gallons (Liquid) — 3785 0.1337 231 3.785 × 10−3 4.951 × 10−3 1 3.785 8 4

Liters 2.838 × 10−2 1000 3.531 × 10−2 61.02 0.001 1.308 × 10−3 0.2642 1 2.113 1.057

Pints (Liquid) 473.2 1.671 × 10−2 28.87 4.732 × 10−4 6.189 × 10−4 0.125 0.4732 1 0.5

946.4 3.342 × 10−2 57.75 9.464 × 10−4 1.238 × 10−3 0.25 0.9464 2 1

Quarts (Liquid)


Volume

75

1 British Imperial gallon =

1 8

Imperial bushel

and contains

Cubic Measure

277.42 cubic inches. 1 Winchester bushel = 0.9694 Imperial bushel

1 cord of wood = pile cut 4 feet long piled 4 feet high and 8 feet on the ground

1 Imperial bushel = 1.032 Winchester bushels

= 128 cubic feet 1 perch of stone = quantity1 21 feet thick, 1 foot high, and16 21 feet long = 24 34 cubic feet

Same relations as before maintain for gallons (dry measure). [Note: 1 U.S. gallon (dry) = 1.164 U. S. gallons (liquid)).] U.S. UNITS∗

(Note: A perch of stone is, however, often computed differently in different localities; thus, in most if not all of the states west of the Mississippi, stonemasons figure rubble by the perch of 16 12 cubic feet. In Philadelphia, 22 cubic feet is called a perch. In Chicago, stone is measured by the cord of 100 cubic feet. Check should be made against local practice.)

2 pints = 1 quart = 67.2 cubic inches 4 quarts = 1 gallon = 8 pints = 268.8 cubic inches 2 gallons = 1 peck = 16 pints = 8 quarts = 537.6 cubic inches 4 pecks = 1 bushel = 64 pints = 32 quarts = 8 gallons = 2150.42 cubic inches 1 cubic foot contains 6.428 gallons (dry measure)

Board Measure. In assumed to be one inch board measure of a stick feet × breadth in feet ×

Liquid Measure. One U.S. gallon (liquid measure) contains 231 cubic inches. It holds 8.336 pounds distilled water at 62◦ F. One British Imperial gallon contains 277.42 cubic inches. It holds 10 pounds distilled water at 62◦ F.

board measure, boards are in thickness. Therefore, feet of square timber = length in thickness in inches.

Shipping Measure. For register tonnage or measurement of the entire internal capacity of a vessel, it is arbitrarily assumed, to facilitate computation, that

100 cubic feet = 1 register ton For the measurement of cargo:

1 U.S. gallon (liquid) = 0.8327 Imperial gallon 1 Imperial gallon = 1.201 U.S. gallons (liquid) [Note: 1 U.S. gallon (liquid) = 0.8594 U.S. gallon (dry).] U.S. UNITS

40 cubic feet = 1 U.S. shipping ton = 32.143 U.S. bushels 42 cubic feet = 1 British shipping ton = 32.703 Imperial bushels Dry Measure. One U.S. Winchester bushel contains 1.2445 cubic feet or 2150.42 cubic inches. It holds 77.601 pounds distilled water at 62◦ F. (Note: This is a struck bushel. A heaped bushel in general equals 1 41 struck bushels, although for apples and pears it contains 1.2731 struck bushels = 2737.72 cubic inches.) One U. S. gallon (dry measure) = 18 bushel and contains 268.8 cubic inches. (Note: This is not a legal U.S. dry measure and therefore is given for comparison only.) One British Imperial bushel contains 1.2843 cubic feet or 2219.36 cubic inches. It holds 80 pounds distilled water at 62◦ F.

4 2 4 1

gills = 1 pint = 16 fluid ounces pints = 1 quart = 8 gills = 32 fluid ounces quarts = 1 gallon = 32 gills = 8 pints = 128 fluid ounces cubic foot contains 7.4805 gallons (liquid measure)

Apothecaries’ Fluid Measure

60 minims = 1 fluid drachm 8 drachms = 1 fluid ounce In the United States a fluid ounce is the 128th part of a U.S. gallon, or 1.805 cubic inches or 29.58 cubic centimeters. It contains 455.8 grains of water at 62◦ F. In Great Britain the fluid ounce is 1.732 cubic inches and contains 1 ounce avoirdupois (or 437.5 grains) of water at 62◦ F.

∗

The gallon is not a U.S. legal dry measure.

76 by →

Multiply Number of →

by →

sphere is the total solid angle about a point. steradians = 1 sphere by definition.

b 4π

aA

Hemispheres Spheresa Spherical Right Angles Steradiansb

to Obtain ↓


Solid Angle (No Dimensions)

Hemispheres 1 0.5 4 6.283

4.630 × 10−5 60

2.778 × 10−3 3600

rad = 1 circumference = 360◦ by definition.

Table 32

a 2π

90 5400 1 1.571

1.667 × 10−2 1 1.852 × 10−4 2.909 × 10−4

1 60 1.111 × 10−2 1.745 × 10−2

Spheresa 2 1 8 12.57

0.25 3.24 × 105

Quadrants

Minutes

Degrees

Plane Angle (No Dimensions)

Degrees Minutes Quadrants Radiansa Revolutionsa (Circumferences) Seconds

to Obtain ↓

Table 31

0.1591 2.063 × 105

57.30 3438 0.6366 1

Radiansa

Spherical Right Angles 0.25 0.125 1 1.571

1 1.296 × 106

360 2.16 × 104 4 6.283

Revolutionsa (Circumferences) Seconds

Steradiansb 0.1592 7.958 × 10−2 0.6366 1

7.716 × 10−7 1

2.778 × 10−4 1.667 × 10−2 3.087 × 10−6 4.848 × 10−6

77

by →

a One

1 24 1440 3.288 × 10−2 8.64 × 104 0.1429

Days 30.42 730.0 4.380 × 10−4 1 2.628 × 106 4.344

Months (Average)a

of a common year.

6.944 × 10−4 1.667 × 10−2 1 2.283 × 10−5 60 9.921 × 10−5

4.167 × 10−2 1 60 1.370 × 10−3 3600 5.952 × 10−3 1 12

Minutes

Hours

common year = 365 days; one leap year = 366 days; one average month =

Days Hours Minutes Months (Average)a Seconds Weeks

to Obtain ↓


Table 33 Time [T]

Seconds 1.157 × 10−5 2.778 × 10−4 1.667 × 10−2 3.806 × 10−7 1 1.654 × 10−6

7 168 1.008 × 104 0.2302 6.048 × 105 1

Weeks

78 by →

Number of →

Multiply

a Nautical miles per hour.

1 1.969 3.281 × 10−2 0.036 0.0006 1.943 × 10−2 0.6 0.01 2.237 × 10−2 3.728 × 10−4

Centimeters per Second

Linear Velocity [LT −1 ]

Centimeters per Second Feet per Minute Feet per Second Kilometers per Hour Kilometers per Minute Knotsa Meters per Minute Meters per Second Miles per Hour Miles per Minute

to Obtain ↓

Table 34

0.5080 1 1.667 × 10−2 1.829 × 10−2 3.048 × 10−4 9.868 × 10−3 0.3048 5.080 × 10−3 1.136 × 10−2 1.892 × 10−4

Feet per Minute 30.48 60 1 1.097 1.829 × 10−2 0.5921 18.29 0.3048 0.6818 1.136 × 10−2

Feet per Second 27.78 54.68 0.9113 1 1.667 × 10−2 0.5396 16.67 0.2778 0.6214 1.036 × 10−2

Kilometers per Hour 1667 3281 54.68 60 1 32.38 1000 16.67 37.28 0.6214

Kilometers per Minute 51.48 101.3 1.689 1.853 3.088 × 10−2 1 30.88 0.5148 1.152 1.919 × 10−2

Knotsa 1.667 3.281 5.468 × 10−2 0.06 0.001 3.238 × 10−2 1 1.667 × 10−2 3.728 × 10−2 6.214 × 10−4

Meters per Minute 100 196.8 3.281 3.6 0.06 1.943 60 1 2.237 3.728 × 10−2

Meters per Second

44.70 88 1.467 1.609 2.682 × 10−2 0.8684 26.82 0.4470 1 1.667 × 10−2

Miles per Hour

2682 5280 88 96.54 1.609 52.10 1609 26.82 60 1

Miles per Minute


79

and the inch varies from 1.36 to 1.73 ft3/min, but the most common measurement is through an aperture 2 in. high and whatever length is required and through a plank 1 14 in. thick. The lower edge of the aperture should be 2 in. above the bottom of the measuring box and the plank 5 in. high above the aperture, thus making a 6-in. head above the center of the stream. Each square inch of this opening represents a miner’s inch, which is equal to a flow of 1.5 ft3 / min.

Linear Velocity The Miner’s Inch. The miner’s inch is used in measuring flow of water. An act of the California legislature, May 23, 1901, makes the standard miner’s inch 1.5 ft3 / min, measured through any aperture or orifice. The term miner’s inch is more or less indefinite, for the reason that California water companies do not all use the same head above the center of the aperture, Table 35 Angular Velocity [T −1 ] Multiply Number of → to Obtain ↓

by

→

Degrees per Second Radians per Second Revolutions per Minute Revolutions per Second

Degrees per Second

Radians per Second

Revolutions per Minute

1 1.745 × 10−2 0.1667 2.778 × 10−3

57.30 1 9.549 0.1592

6 0.1047 1 1.667 × 10−2

Revolutions per Second 360 6.283 60 1

Table 36 Linear Accelerationa [LT −2 ] Multiply Number of → to Obtain ↓

by

→

Centimeters per Second per Second Feet per Second per Second Kilometers per Hour per Second Meters per Second per Second Miles per Hour per Second a The

(standard) 21.94 mph/sec.

acceleration

due

Centimeters per Second per Second

Feet per Second per Second

Kilometers per Hour per Second

Meters per Second per Second

1

30.48

27.78

100

44.70

3.281 × 10−2

0.036

1 1.097

0.9113 1

3.281 3.6

1.467 1.609

0.01

0.3048

0.2778

1

0.4470

2.237 × 10−2

0.6818

0.6214

2.237

1

to

gravity

Miles per Hour per Second

(g0 ) = 980.7 cm/sec sec, = 32.17 ft/sec sec = 35.30 km/hr sec = 9.807 m/sec sec =

Table 37 Angular Acceleration [T −2 ] Multiply Number of → to Obtain ↓

by

→

Radians per Second per Second Revolutions per Minute per Minute Revolutions per Minute per Second Revolutions per Second per Second

Radians per Second per Second

Revolutions per Minute per Minute

Revolutions per Minute per Second

1 573.0 9.549 0.1592

1.745 × 10−3 1 1.667 × 10−2 2.778 × 10−4

0.1047 60 1 1.667 × 10−2

Revolutions per Second per Second 6.283 3600 60 1

80 →

1 6.481 × 10−2 6.481 × 10−5 64.81 2.286 × 10−3 1.429 × 10−4 — — —

Grains 15.43 1 0.001 1000 3.527 × 10−2 2.205 × 10−3 9.842 × 10−7 10−6 1.102 × 10−6

Grams

Ouncesb 437.5 28.35 2.835 × 10−2 2.835 × 104 1 6.250 × 10−2 2.790 × 10−5 2.835 × 10−5 3.125 × 10−5

Milligrams 1.543 × 10−2 0.001 10−6 1 3.527 × 10−5 2.205 × 10−6 9.842 × 10−10 10−9 1.102 × 10−9

Kilograms 1.543 × 104 1000 1 106 35.27 2.205 9.842 × 10−4 0.001 1.102 × 10−3

7000 453.6 0.4536 4.536 × 105 16 1 4.464 × 10−4 4.536 × 10−4 0.0005

Poundsb

1.016 × 106 1016 1.016 × 109 3.584 × 104 2240 1 1.016 1.120

Tons (Long)

×106 1000 109 3.527 × 104 2205 0.9842 1 1.102

Tons (Metric)

9.072 × 105 907.2 9.072 × 108 3.2 × 104 2000 0.8929 0.9072 1

Tons (Short)

a These same conversion factors apply to the gravitational units of force having the corresponding names. The dimensions of these units when used as gravitational units of force are MLT −2 ; see Table 40. b Avoirdupois pounds and ounces.

by

Number of →

Multiply

Mass [M] and Weighta

Grains Grams Kilograms Milligrams Ouncesb Poundsb Tons (Long) Tons (Metric) Tons (Short)

to Obtain ↓

Table 38


81

Table 39 Density or Mass per Unit Volume [ML−3 ] Multiply Number of →

to Obtain ↓

by

→

Grams per Cubic Centimeter

Kilograms per Cubic Meter

Pounds per Cubic Foot

1 1000 62.43 3.613 × 10−2 3.405 × 10−7

0.001 1 6.243 × 10−2 3.613 × 10−5 3.405 × 10−10

1.602 × 10−2 16.02 1 5.787 × 10−4 5.456 × 10−9

Grams per Cubic Centimeter Kilograms per Cubic Meter Pounds per Cubic Foot Pounds per Cubic Inch Pounds per Mil Foota a Unit

Pounds per Cubic Inch 27.68 2.768 × 104 1728 1 9.425 × 10−6

of volume is a volume one foot long and one circular mil in cross-sectional area.

Avoirdupois Weight.

Used Commercially.

27.343 grains = 1 drachm 16 drachms = 1 ounce (oz) = 437.5 grains 16 ounces = 1 pound (lb) = 7000 grains 28 pounds = 1 quarter (qr) 4 quarters = 1 hundredweight (cwt) = 112 pounds 20 hundredweight = 1 gross or long ton∗ 200 pounds = 1 net or short ton 14 pounds = 1 stone100 pounds = 1 quintal Troy Weight.

Used in weighing gold or silver.

24 grains = 1 pennyweight (dwt) 20 pennyweights = 1 ounce (oz) = 480 grains 12 ounces = 1 pound (lb) = 5760 grains

∗ The long ton is used by the U.S. custom houses in collecting duties upon foreign goods. It is also used in freighting coal and selling it wholesale.

The grain is the same in avoirdupois, troy, and apothecaries’ weights. A carat, for weighing diamonds, = 3.086 grains = 0.200 gram (International Standard, 1913.) 1 pound troy = 0.8229 pound avoirdupois 1 pound avoirdupois = 1.2153 pounds troy Apothecaries’ medicines.

Weight.

Used in compounding

20 grains = 1 scruple() 3 scruples = 1 drachm() = 60 grains 8 drachms = 1 ounce() = 480 grains 12 ounces = 1 pound(lb) = 5760 grains The grain is the same in avoirdupois, troy, and apothecaries’ weights. 1 pound apothecaries = 0.82286 pound avoirdupois 1 pound avoirdupois = 1.2153 pounds apothecaries

82 by →

a

Dynes 1 1.020 × 10−3 10−7 10−5 1.020 × 10−6 2.248 × 10−6 7.233 × 10−5

Grams 980.7 1 9.807 × 10−5 9.807 × 10−3 0.001 2.205 × 10−3 7.093 × 10−2

Joules per Centimeter 107 1.020 × 104 1 100 10.20 22.48 723.3

Newtons, or Joules per Meter 105 102.0 0.01 1 0.1020 0.2248 7.233 Kilograms 9.807 × 105 1000 9.807 × 10−2 9.807 1 2.205 70.93

Pounds 4.448 × 105 453.6 4.448 × 10−2 4.448 0.4536 1 32.17

Conversion factors between absolute and gravitational units apply only under standard acceleration due to gravity conditions. (See Section 4.)

Dynes Grams Joules per Centimeter Newtons, or Joules per Meter Kilograms Pounds Poundals

to Obtain ↓


Table 40 Forcea [MLT −2 ] or [F]

Poundals 1.383 × 104 14.10 1.383 × 10−3 0.1383 1.410 × 10−2 3.108 × 10−2 1

83

by →

1.316 × 10−2 1.333 × 104

1 0.3937 5.354 136.0 27.85 0.1934 1.392 × 10−2 1.333 × 103

9.869 × 10−7 1 7.501 × 10−5 2.953 × 10−5 4.015 × 10−4 1.020 × 10−2 2.089 × 10−3 1.450 × 10−5 1.044 × 10−6 10−1

1 1.013 × 106 76.00 29.92 406.8 1.033 × 104 2117 14.70 1.058 1.013 × 105

Atmospheresa

Centimeters of Mercury at 0◦ Cb

Baryes or Dynes per Square Centimeter

3.386 × 103

70.73 0.4912 3.536 × 10−2

13.60 345.3

1

2.540

3.342 × 10−2 3.386 × 104

Inches of Mercury at 0◦ Cb

2.491 × 10−4

5.204 3.613 × 10−2 2.601 × 10−3

1 25.40

7.355 × 10−2

0.1868

2.458 × 10−3 2.491 × 10−3

Inches of Water at 4◦ C

9.807

0.2048 1.422 × 10−3 1.024 × 10−4

3.937 × 10−2 1

2.896 × 10−3

7.356 × 10−3

9.678 × 10−5 98.07

Kilograms per Square Meterc

47.88

1 6.944 × 10−3 0.0005

0.1922 4.882

1.414 × 10−2

3.591 × 10−2

4.725 × 10−4 478.8

Pounds per Square Foot

6.895 × 103

144 1 0.072

27.68 703.1

2.036

5.171

6.804 × 10−2 6.895 × 104

Pounds per Square Inch

9.576 × 104

2000 13.89 1

384.5 9765

28.28

71.83

0.9450 9.576 × 105

Tons (Short) per Square Foot

Pascal

1

2.089 × 10−2 1.450 × 10−4 1.044 × 10−5

4.015 × 10−8 0.1020

2.953 × 10−4

7.501 × 10−4

9.869 × 10−6 10

c 1 g/cm2 = 10 kg/m2 .

a Definition: One atmosphere (standard) = 76 cm of mercury at 0◦ C. b To convert height h of a column of mercury at t degrees Centigrade to the equivalent height h at 0◦ C use h = h{1 − (m − l)t/(1 + mt)}, where m = 0.0001818 and l = 18.4 × 10−6 if 0 0 the scale is engraved on brass; l = 8.5 × 10−6 if on glass. This assumes the scale is correct at 0◦ C; for other cases (any liquid) see International Critical Tables, Vol. 1, p. 68.

Baryes or Dynes per Square Centimeter Centimeters of Mercury at 0◦ Cb Inches of Mercury at 0 ◦ Cb Inches of Water at 4◦ C Kilograms per Square Meterc Pounds per Square Foot Pounds per Square Inch Tons (Short) per Square Foot Pascal

Atmospheresa

to Obtain ↓

Number of →

Multiply

Table 41 Pressure or Force per Unit Area [ML−1 T −2 ] or [FL−2 ]

84 Table 42

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Torque or Moment of Force [ML2 T −2 ] or [FL]a Multiply Number of →

by

to Obtain ↓

→

Dyne-Centimeters Gram-Centimeters Kilogram-Meters Pound-Feet Newton-Meter

DyneCentimeters

GramCentimeters

KilogramMeters

Pound-Feet

NewtonMeter

1 1.020 × 10−3 1.020 × 10−8 7.376 × 10−8 10−7

980.7 1 10−5 7.233 × 10−5 9.807 × 10−4

9.807 × 107 105 1 7.233 9.807

1.356 × 107 1.383 × 104 0.1383 1 1.356

107 1.020 × 104 0.1020 0.7376 1

a Same dimensions as energy; more properly torque should be expressed as newton-meters per radian to avoid this confusion.

Table 43

Moment of Inertia [ML2 ] Multiply Number of →

to Obtain ↓

by

→

Gram-Centimeters Squared Kilogram-Meters Squared Pound-Inches Squared Pound-Feet Squared Slug-Feet Squared

GramCentimeters Squared

KilogramMeters Squared

PoundInches Squared

PoundFeet Squared

1

107

2.9266 × 103

4.21434 × 105

1.3559 × 107

10−7

1

2.9266 × 10−4

4.21434 × 10−2

1.3559

1

144

4.63304 × 103

6.944 × 10−3 2.15841 × 10−4

1 3.10811 × 10−2

32.1739 1

3.4169 ×

10−4

2.37285 × 10−6 7.37507 × 10−8

3.4169 ×

103

23.7285 0.737507

SlugFeet Squared

85

by →

7.367 × 10−8 3.722 × 10−14 10−7 2.389 × 10−11

7.233 × 10−5 3.654 × 10−11 9.807 × 10−5 2.343 × 10−8 2.724 × 10−11 10−5 2.724 × 10−8

778.0 3.929 × 10−4 1054.8 0.2520 2.930 × 10−4 107.6 0.2930

2.778 × 10−14 1.020 × 10−8 2.778 × 10−11

1.383 × 104 1.356 × 107

1.020 × 10−3 1

1 980.7

1.076 × 107 1.055 × 1010

3.766 × 10−7 0.1383 3.766 × 10−4

3.239 × 10−4

1 5.050 × 10−7 1.356

1.285 × 10−3

9.480 × 10−11

9.297 × 10−8

1

FootPounds

Ergs or CentimeterDynes

CentimeterGrams

British Thermal Unitsb

0.7457 2.737 × 105 745.7

641.3

1.98 × 106 1 2.684 × 106

2.737 × 1010 2.684 × 1012

2545

HorsepowerHours

2.778 × 10−7 0.1020 2.778 × 10−4

2.389 × 10−4

0.7376 3.722 × 10−7 1

1.020 × 104 107

9.480 × 10−4

Joules, c or WattSeconds

1.163 × 10−3 426.9 1.163

1

3087 1.559 × 10−3 4186

4.269 × 107 4.186 × 1010

3.969

KilogramCaloriesb

1 3.671 × 105 1000

860.0

2.655 × 106 1.341 3.6 × 106

3.671 × 1010 3.6 × 1013

3413

KilowattHours

2.724 × 10−6 1 2.724 × 10−3

2.343 × 10−3

7.233 3.653 × 10−6 9.807

105 9.807 × 107

9.297 × 10−3

MeterKilograms

0.001 367.1 1

0.8600

2655 1.341 × 10−3 3600

3.671 × 107 3.6 × 1010

3.413

WattHours

The IT cal, 1000 international steam table calories, has been defined as the 1/860th part of the international kilowatthour (see Mechanical Engineering, Nov. 1935, p. 710). Its value is very nearly equal to the mean kilogram-calorie, 1 IT cal-1.00037 kilogram-calories (mean). 1 Btu = 251.996 IT cal. c Absolute joule, defined as 107 ergs. The international joule, based on the international ohm and ampere, equals 1.0003 absolute joules.

a See note at the bottom of Table 45. b Mean calorie and Btu used throughout. One gram-calorie = 0.001 kilogram-calorie; one Ostwald calorie = 0.1 kilogram-calorie.

British Thermal Unitsb Centimeter-Grams Ergs or CentimeterDynes Foot-Pounds Horsepower-Hours Joules,c or Watt-Seconds KilogramCaloriesb Kilowatt-Hours Meter-Kilograms Watt-Hours

to Obtain ↓

Number of →

Multiply

Table 44 Energy, Work and Heata [ML2 T −2 ] or [FL]

86 by →

Number of →

Multiply

1 4.426 × 10−6 7.376 × 10−8 1.341 × 10−10 1.433 × 10−9 10−10 1.360 × 10−10 10−7

1 1.758 × 108 778.0 12.97 2.357 × 10−2 0.2520 1.758 × 10−2 2.390 × 10−2 17.58

1.356 × 10−3 1.843 × 10−3 1.356

0.7457 1.014 745.7

7.457 × 109 3.3 × 104 550 1 10.69

42.41

Horsepowera

1 Poncelet = 100 kilogram-meters per second

1 Cheval-vapeur = 75 kilogram-meters per second

2.260 × 10−5 3.072 × 10−5 2.260 × 10−2

1.356 × 107 60 1 1.818 × 10−3 1.943 × 10−2

7.712 × 10−2

Foot-Pounds per Second

6.977 × 10−2 9.485 × 10−2 69.77

6.977 × 108 3087 51.44 9.355 × 10−2 1

3.969

KilogramCalories per Minute

1 1.360 1000

1010 4.426 × 104 737.6 1.341 14.33

56.89

Kilowatts

0.7355 1 735.5

7.355 × 109 3.255 × 104 542.5 0.9863 10.54

41.83

Metric Horsepower

10−3 1.360 × 10−3 1

107 44.26 0.7376 1.341 × 10−3 1.433 × 10−2

5.689 × 10−2

Watts

horsepower equals 736 watts (continental Europe). Neither of these latter definitions is equivalent to the first; the ‘‘horsepowers’’ defined in these latter definitions are widely used in the rating of electrical machinery.

a The ‘‘horsepower’’ used in these tables is equal to 550 foot-pounds per second by definition. Other definitions are one horsepower equals 746 watts (U.S. and Great Britain) and one

Note:

1.285 × 10−3

5.689 × 10−9 2.259 × 105 1 1.667 × 10−2 3.030 × 10−5 3.239 × 10−4

Foot-Pounds per Minute

Ergs per Second

British Thermal Units per Minute

Power or Rate of Doing Worka [ML2 T −3 ] or [FLT −1 ]

British Thermal Units per Minute Ergs per Second Foot-Pounds per Minute Foot-Pounds per Second Horsepowera Kilogram-Calories per Minute Kilowatts Metric Horsepower Watts

to Obtain ↓

Table 45


87

Table 46 Quantity of Electricity and Dielectric Flux [Q] Multiply Number of → to Obtain ↓

by

→

Abcoulombs Ampere-Hours Coulombs Faradays Statcoulombs

Abcoulombs

AmpereHours

Coulombs

Faradays

Stat coulombs

1 2.778 × 10−3 10 1.036 × 10−4 2.998 × 1010

360 1 3600 3.731 × 10−2 1.080 × 1013

0.1 2.778 × 10−4 1 1.036 × 10−5 2.998 × 109

9649 26.80 9.649 × 104 1 2.893 × 1014

3.335 × 10−11 9.259 × 10−14 3.335 × 10−10 3.457 × 10−15 1

Table 47 Charge per Unit Area and Electric Flux Density [QL−2 ]

to Obtain ↓

Multiply Number of → by

→

Abcoulombs per Square Centimeter Coulombs per Square Centimeter Coulombs per Square Inch Statcoulombs per Square Centimeter Coulombs per Square Meter

Abcoulombs per Square Centimeter

Coulombs per Square Centimeter

Coulombs per Square Inch

Statcoulombs per Square Centimeter

Coulombs per Square Meter

1 10 64.52 2.998 × 1010 105

0.1 1 6.452 2.998 × 109 104

1.550 × 10−2 0.1550 1 4.647 × 108 1550

3.335 × 10−11 3.335 × 10−10 2.151 × 10−9 1 3.335 × 10−6

10−5 10−4 6.452 × 10−4 2.998 × 105 1

Table 48 Electric Current [QT −1 ]

to Obtain ↓

Abamperes Amperes Statamperes


→

Abamperes

Amperes

1 10 2.998 × 1010

0.1 1 2.998 × 109

Statamperes 3.335 × 10−11 3.335 × 10−10 1

88


Table 49

Current Density [QT −1 L−2 ] Multiply Number of →

to Obtain ↓

by

→

Abamperes per Square Centimeter Amperes per Square Centimeter Amperes per Square Inch Statamperes per Square Centimeter Amperes per Square Meter

Table 50

Abamperes per Square Centimeter

Amperes per Square Centimeter

Amperes per Square Inch

Statamperes per Square Centimeter

Amperes per Square Meter

1 10 64.52 2.998×1010 105

0.1 1 6.452 2.998×109 104

1.550 × 10−2 0.1550 1 4.647×108 1550

3.335 × 10−11 3.335 × 10−10 2.151 × 10−9 1 3.335 × 10−6

10−5 10−4 6.452 × 10−4 2.998×105 1

Electric Potential and Electromotive Force [MQ−1 L2 T −2 ] or [FQ−1 L] Multiply Number of →

to Obtain ↓

by

→ Abvolts

Abvolts Microvolts Millivolts Statvolts Volts

1 0.01 10−5 3.335 × 10−11 10−8

Microvolts

Millivolts

Statvolts

100 1 0.001 3.335 × 10−9 10−6

105

2.998 × 2.998 × 108 2.998×105 1 299.8

1000 1 3.335 × 10−6 0.001

1010

Volts 108 106 1000 3.335 × 10−3 1

89

by →

Abvolts per Centimeter Microvolts per Meter Millivolts per Meter Statvolts per Centimeter Volts per Centimeter Kilovolts per Centimeter Volts per Inch Volts per Mil Volts per Meter

to Obtain ↓

Number of →

Multiply

1 1 0.001 3.335 × 10−11 10−8 10−11 2.540 × 10−8 2.540 × 10−11 10−6

Abvolts per Centimeter 1 1 0.001 3.335 × 10−11 10−8 10−11 2.540 × 10−8 2.540 × 10−11 10−6

Microvolts per Meter 1000 1000 1 3.335 × 10−8 10−5 10−8 2.540 × 10−5 2.540 × 10−8 10−3

Millivolts per Meter

Volts per Centimeter 108 108 105 3.335 × 10−3 1 0.001 2.540 2.540 × 10−3 100

Statvolts per Centimeter 2.998 × 1010 2.998 × 1010 2.998 × 107 1 299.8 0.2998 761.6 0.7616 2.998 × 104

Table 51 Electric Field Intensity and Potential Gradient [MQ−1 LT −2 ] or [FQ−1 ]

1011 1011 108 3.335 1000 1 2540 2.540 105

Kilovolts per Centimeter 3.937 × 107 3.937×107 3.937×104 1.313 × 10−3 0.3937 3.937 × 10−4 1 0.001 39.37

Volts per Inch

3.937×1010 3.937×1010 3.937×107 1.313 393.7 0.3937 1000 1 3.937 × 104

Volts per Mil

106 106 1000 3.335 × 10−5 10−2 10−5 2.540 × 10−2 2.540 × 10−5 1

Volts per Meter

90


Table 52

Electric Resistance [MQ−2 L2 T −1 ] or [FQ−2 LT] Multiply Number of →

to Obtain ↓

by

→

Abohms Megohms Microhms Ohms Statohms

Abohms

Megohms

Microhms

Ohms

1 10−15 0.001 10−9 1.112 × 10−21

1015 1 1012 106 1.112 × 10−6

1000 10−12 1 10−6 1.112 × 10−18

109 10−6 106 1 1.112 × 10−12

Statohms 8.988 × 1020 8.988 × 105 8.988 × 1017 8.988 × 1011 1

Note: Electric Conductance [F −1 Q2 L−1 T −1 ]. 1 Siemens = 1 mho = 1 ohm−1 = 10−6 megmho = 106 micromho.

Table 53

Electric Resistivitya [MQ−2 L3 T −1 ] or [FQ−2 L2 T] Multiply Number of →

to Obtain ↓

by

→

Abohm-Centimeters Microhm-Centimeters Microhm-Inches Ohms (Mil, Foot) Ohms (Meter, Gram)b Ohm-Meters

AbohmCentimeters

MicrohmCentimeters

MicrohmInches

Ohms (Mil, Foot)

Ohms (Meter, Gram)b

1 0.001 3.937 × 10−4 6.015 × 10−3 10−5 δ 10−11

1000 1 0.3937 6.015 0.01δ 10−8

2540 2.540 1 15.28 2.540 × 10−2 δ 2.540 × 10−8

166.2 0.1662 6.545 × 10−2 1 1.662 × 10−3 δ 1.662 × 10−9

105 /δ 100/δ 39.37/δ 601.5/δ 1 10−6 /δ

OhmMeters 1011 108 3.937 × 107 6.015 × 108 10−6 δ 1

a In this table δ is density in grams per cubic-centimeters. The following names, corresponding respectively to those at the tops of columns, are sometimes used: abohms per centimeter cube; microhms per centimeter cube; microhms per inch cube; ohms per milfoot; ohms per meter-gram. The first four columns are headed by units of volume resistivity, the last by a unit of mass resistivity. The dimensions of the latter are Q−2 L6 T −1 , not those given in the heading of the table. b One ohm (meter, gram) = 5710 ohms (mile, pound).

Table 54

Electric Conductivitya [M−1 Q2 L−3 T] or [F −1 Q2 L−2 T −1 ] Multiply Number of →

to Obtain ↓

by

→

Abmhos per Centimeter Mhos (Mil, Foot) Mhos (Meter, Gram) Micromhos per Centimeter Micromhos per Inch Siemens per Meter

Abmhos per Centimeter

Mhos (Mil, Foot)

Mhos (Meter, Gram)

Micromhos per Centimeter

Micromhos per Inch

1 166.2 105 /δ 1000 2540 1011

6.015 × 10−3 1 601.5/δ 6.015 15.28 6.015 × 108

10−5 δ 1.662 × 10−3 δ 1 0.01δ 2.540 × 10−2 δ 106 δ

0.001 0.1662 100/δ 1 2.540 108

3.937 × 10−4 6.524 × 10−2 39.37/δ 0.3937 1 3.937 × 107

Siemens per Meter 10−11 1.662 × 10−9 10−6 /δ 10−8 2.54 × 10−8 1

a See footnote of Table 53. Names sometimes used are abmho per centimeter cube, mho per mil-foot, etc. Dimensions of mass conductivity

are Q2 L−6 T .


91

Table 55 Capacitance [M−1 Q2 L−2 T 2 ] or [F −1 Q2 L−1 ] Multiply Number of →

to Obtain ↓

by

→

Abfarads Farads Microfarads Statfarads

Abfarads

Farads

Microfarads

Statfarads

1 109 1015 8.988 × 1020

10−9 1 106 8.988 × 1011

10−15 10−6 1 8.988 × 105

1.112 × 10−21 1.112 × 10−12 1.112 × 10−6 1

Table 56 Inductance [MQ−2 L2 ] or [FQ−2 LT 2 ] Multiply Number of → to Obtain ↓

by

→

Abhenriesa Henries Microhenries Millihenries Stathenries a

Abhenriesa

Henries

1 10−9 0.001 10−6 1.112 × 10−21

109 1 106 1000 1.112 × 10−12

Microhenries

Millihenries

1000 10−6 1 0.001 1.112 × 10−18

106 0.001 1000 1 1.112 × 10−15

Stathenries 8.988 × 1020 8.988 × 1011 8.988 × 1017 8.988 × 1014 1 1

An abhenry is sometimes called a ‘‘centimeter.’’

Table 57 Magnetic Flux [MQ−1 L2 T −1 ] or [FQ−1 LT]

to Obtain ↓ Kilolines Maxwells (or Lines) Webers


→

Kilolines

Maxwells (or Lines)

1 1000 10−5

0.001 1 10−8

Webers 105 108 1

92


Table 58

Magnetic Flux Density [MQ−1 T −1 ] or [FQ−1 L−1 T] Multiply Number of →

to Obtain ↓

by

→

Gausses (or Lines per Square Centimeter) Lines per Square Inch Webers per Square Centimeter Webers per Square Inch Tesla (Webers per Square Meter)

Table 59

Gausses (or Lines per Square Centimeter) 1 6.452 10−8 6.452 × 10−8 10−4

Tesla (Webers per Square Meter)

0.1550 1 1.550 × 10−9 10−8 1.550 × 10−5

108 6.452 × 108 1 6.452 104

1.550 × 107 108 0.1550 1 1550

104 6.452 × 104 10−4 6.452 × 10−4 1


→

Abampere-Turns

Ampere-Turns

1 10 12.57

0.1 1 1.257

Abampere-Turns Ampere-Turns Gilberts

to Obtain ↓

Webers per Square Inch

Magnetic Potential and Magnetomotive Force [QT −1 ]

to Obtain ↓

Table 60

Lines per Square Inch

Webers per Square Centimeter

Gilberts 7.958 × 10−2 0.7958 1

Magnetic Field Intensity, Potential Gradient, and Magnetizing Force [QL−1 T −1 ] Multiply Number of → by

→

Abampere-Turns per Centimeter Ampere-Turns per Centimeter Ampere-Turns per Inch Oersteds (Gilberts per Centimeter) Ampere-Turns per Meter

AbampereTurns per Centimeter

AmpereTurns per Centimeter

AmpereTurns per Inch

Oersteds (Gilberts per Centimeter)

1 10 25.40 12.57 103

0.1 1 2.540 1.257 102

3.937 × 10−2 0.3937 1 0.4950 39.37

7.958 × 10−2 0.7958 2.021 1 79.58

AmpereTurns per Meter 10−3 10−2 2.54 × 10−2 1.257 × 10−2 1


93

Table 61 Specific Heat [L2 T −2 t−1 ] (t = temperature) To change specific heat in gram-calories per gram per degree Centigrade to the units given in any line of the following table, multiply by the factor in the last column. Unit of Heat or Energy

Unit of Mass

Temperature Scalea

Gram-calories Kilogram-calories British thermal units British thermal units Joules Joules Joules Kilowatt-hours Kilowatt-hours

Gram Kilogram Pound Pound Gram Pound Kilogram Kilogram Pound

Centigrade Centigrade Centigrade Fahrenheit Centigrade Fahrenheit Kelvin Centigrade Fahrenheit

a Temperature

conversion formulas: tc = temperature in Centigrade degrees tf = temperature in Fahrenheit degrees tK = temperature in Kelvin degrees 1F=

5◦ 9 C ◦

1K=1 C tc = 59 (tf − 32) tf = 95 tc + 32 tK = tc + 273

Factor 1 1 1.800 1.000 4.186 1055 4.187 × 103 1.163 × 10−3 2.930 × 10−4

94

a

→

1 12 3.333 × 10−3 1.731 1.483 1.731 × 105 4.134 × 10−4 4.134 × 10−3 5.276 × 10−1 1.731

Btu · ft/

h · ft2 ·◦ F

8.333 × 10−2 1 2.778 × 10−4 1.442 × 10−1 1.240 × 10−1 1.442 × 104 3.445 × 10−5 3.445 × 10−4 4.395 × 10−2 1.442 × 10−1

Btu · in./

h · ft2 ·◦ F

3.0 × 102 3.6 × 103 1 5.192 × 102 4.465 × 102 5.192 × 107 1.240 × 10−1 1.240 1.582 × 102 5.192 × 102

Btu · in./

/sec · ft2 ·◦ F

J/

5.778 × 10−1 6.933 1.926 × 10−3 1 8.599 × 10−1 1.0 × 105 2.388 × 10−4 2.388 × 10−3 3.048 × 10−1 1.0

m · s ·◦ C m · h ·◦ C

kcal/

6.720 × 10−1 8.064 2.240 × 10−3 1.163 1 1.163 × 105 2.778 × 10−4 2.778 × 10−3 3.545 × 10−1 1.163

International Table Btu = 1.055056 × 103 joules and International Table cal = 4.1868 J are used throughout.

by

of →

Number

Multiply

Thermal Conductivitya [LMT −3 t−1 ]

Btu · ft/h · ft2 ·◦ F Btu · in./h · ft2 ·◦ F Btu · in./s · ft2 ·◦ F J/m · s ·◦ C kcal/m · h ·◦ C erg/cm · s ·◦ C kcal/m · s ·◦ C cal/cm · s ·◦ C W/ft ·◦ C W/m · K

↓

Obtain

to

Table 62

5.778 × 10−6 6.933 × 10−5 1.926 × 10−8 1.000 × 10−5 8.599 × 10−6 1 2.388 × 10−9 2.388 × 10−8 3.048 × 10−6 1.00 × 10−5

cm · s ·◦ C

erg/

2.419 × 103 2.903 × 104 8.064 4.187 × 103 3.6 × 103 4.187 × 108 1 10 1.276 × 103 4.187 × 103

m · s ·◦ C

kcal/

2.419 × 102 2.903 × 103 8.064 × 10−1 4.187 × 102 3.6 × 102 4.187 × 107 1.0 × 10−1 1 1.276 × 102 4.187 × 102

cm · s ·◦ C

cal/

1.895 2.275 × 101 6.319 × 10−3 3.281 2.821 3.281 × 105 7.835 × 10−4 7.835 × 10−3 1 3.281

ft ·◦ C

W/

5.778 × 10−1 6.933 1.926 × 10−3 1.0 8.599 × 10−1 1.0 × 105 2.388 × 10−4 2.388 × 10−3 3.048 × 10−1 1

m·K

W/


95

Table 63 Photometric Units Common Unit Luminous intensity Luminance

Luminous flux Quantity of light flux Luminous exitancea Illuminanceb

Multiply by

to Get SI Unit

10−1

International candle cd/in.2 cd/cm2 Foot · lambert cd · sr Candle power (spher.)

9.81 × 1.550 × 103 1 × 104 3.4263 1.0000 12.566

lm Foot candles lmft2 lx Phots

3.103 × 103 1.0764 × 10 1.0764 × 10 1.000 1 × 104

cd cd/m2 cd/m2 cd/m2 lm lm lm· lm/m2 cd/m2 lm/m2 lm/m2 lm/m2 lm/m2 lm/W

Luminous efficacy a b

Luminous emittance. Luminous flux density.

Table 64 Specific Gravity Conversions Specific Gravity 60◦ /60◦

◦

Be

◦

API

lb/gal 60◦ F, wt in air

lb/ft3 at 60◦ F, wt in air

0.600 0.605 0.610 0.615 0.620 0.625 0.630 0.635 0.640 0.645 0.650 0.655 0.660 0.665 0.670 0.675 0.680 0.685 0.690 0.695

103.33 101.40 99.51 97.64 95.81 94.00 92.22 90.47 88.75 87.05 85.38 83.74 82.12 80.53 78.96 77.41 75.88 74.38 72.90 71.44

104.33 102.38 100.47 98.58 96.73 94.90 93.10 91.33 89.59 87.88 86.19 84.53 82.89 81.28 79.69 78.13 76.59 75.07 73.57 72.10

4.9929 5.0346 5.0763 5.1180 5.1597 5.2014 5.2431 5.2848 5.3265 5.3682 5.4098 5.4515 5.4932 5.5349 5.5766 5.6183 5.6600 5.7017 5.7434 5.7851

37.350 37.662 37.973 38.285 38.597 39.910 39.222 39.534 39.845 40.157 40.468 40.780 41.092 41.404 41.716 42.028 42.340 42.652 42.963 43.275

0.700 0.705 0.710 0.715 0.720 0.725 0.730 0.735 0.740

70.00 68.58 67.18 65.80 64.44 63.10 61.78 60.48 59.19

70.64 69.21 67.80 66.40 65.03 63.67 62.34 61.02 59.72

5.8268 5.8685 5.9101 5.9518 5.9935 6.0352 6.0769 6.1186 6.1603

43.587 43.899 44.211 44.523 44.834 45.146 45.458 45.770 46.082

Specific Gravity 60◦ /60◦

◦

Be

◦

API



0.745 0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0.790 0.795

57.92 56.67 55.43 54.21 53.01 51.82 50.65 49.49 48.34 47.22 46.10

58.43 57.17 55.92 54.68 53.47 52.27 51.08 49.91 48.75 47.61 46.49

6.2020 6.2437 6.2854 6.3271 6.3688 6.4104 6.4521 6.4938 6.5355 6.5772 6.6189

46.394 46.706 47.018 47.330 47.642 47.953 48.265 48.577 48.889 49.201 49.513

0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 0.845 0.850 0.855 0.860 0.865 0.870 0.875 0.880 0.885

45.00 43.91 42.84 41.78 40.73 39.70 38.67 37.66 36.67 35.68 34.71 33.74 32.79 31.85 30.92 30.00 29.09 28.19

45.38 44.28 43.19 42.12 41.06 40.02 38.98 37.96 36.95 35.96 34.97 34.00 33.03 32.08 31.14 30.21 29.30 28.38

6.6606 6.7023 6.7440 6.7857 6.8274 6.8691 6.9108 6.9525 6.9941 7.0358 7.0775 7.1192 7.1609 7.2026 7.2443 7.2860 7.3277 7.3694

49.825 50.137 50.448 50.760 51.072 51.384 51.696 52.008 52.320 52.632 52.943 53.225 53.567 53.879 54.191 54.503 54.815 55.127 (Continues)

96 Table 64 Specific Gravity 60◦ /60◦

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued )

◦

Be

◦

TW



0.890 0.895

27.30 26.42

27.49 26.60

7.4111 7.4528

55.438 55.750

0.900 0.905 0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995

25.76 24.70 23.85 23.01 22.17 21.35 20.54 19.73 18.94 18.15 17.37 16.60 15.83 15.08 14.33 13.59 12.86 12.13 11.41 10.70

25.72 24.85 23.99 23.14 22.30 21.47 20.65 19.84 19.03 18.24 17.45 16.67 15.90 15.13 14.38 13.63 12.89 12.15 11.43 10.71

7.4944 7.5361 7.5777 7.6194 7.6612 7.7029 7.7446 7.7863 7.8280 7.8697 7.9114 7.9531 7.9947 8.0364 8.0780 8.1197 8.1615 8.2032 8.2449 8.2866

56.062 56.374 56.685 56.997 57.410 57.622 57.934 58.246 58.557 58.869 59.181 59.493 59.805 60.117 60.428 60.740 61.052 61.364 61.676 61.988 lb/ft3 at 60◦ F, wt in air

TW


1.000 1.005 1.010 1.015 1.020 1.025 1.030 1.035 1.040 1.045 1.050 1.055 1.060 1.065 1.070 1.075 1.080 1.085 1.090 1.095

10.00 0.72 1.44 2.14 2.84 3.54 4.22 4.90 5.58 6.24 6.91 7.56 8.21 8.85 9.49 10.12 10.74 11.36 11.97 12.58

10.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

8.3283 8.3700 8.4117 8.4534 8.4950 8.5367 8.5784 8.6201 8.6618 8.7035 8.7452 8.7869 8.8286 8.8703 8.9120 8.9537 8.9954 9.0371 9.0787 9.1204

62.300 62.612 62.924 63.236 63.547 63.859 64.171 64.483 64.795 65.107 65.419 65.731 66.042 66.354 66.666 66.978 67.290 67.602 67.914 68.226

1.100 1.105 1.110 1.115 1.120 1.125 1.130 1.135

13.18 13.78 14.37 14.96 15.54 16.11 16.68 17.25

20 21 22 23 24 25 26 27

9.1621 9.2038 9.2455 9.2872 9.3289 9.3706 9.4123 9.4540

68.537 68.849 69.161 69.473 69.785 70.097 70.409 70.721


◦

Be

◦


◦

Be

◦

TW



1.140 1.145 1.150 1.155 1.160 1.165 1.170 1.175 1.180 1.185 1.190 1.195

17.81 18.36 18.91 19.46 20.00 20.54 21.07 21.60 22.12 22.64 23.15 23.66

28 29 30 31 32 33 34 35 36 37 38 39

9.4957 9.5374 9.5790 9.6207 9.6624 9.7041 9.7458 9.7875 9.8292 9.8709 9.9126 9.9543

71.032 71.344 71.656 71.968 72.280 72.592 72.904 73.216 73.528 73.840 74.151 74.463

1.200 1.205 1.210 1.215 1.220 1.225 1.230 1.235 1.240 1.245 1.250 1.255 1.260 1.265 1.270 1.275 1.280 1.285 1.290 1.295

24.17 24.67 25.17 25.66 26.15 26.63 27.11 27.59 28.06 28.53 29.00 29.46 29.92 30.38 30.83 31.27 31.72 32.16 32.60 33.03

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

9.9960 10.0377 10.0793 10.1210 10.1627 10.2044 10.2461 10.2878 10.3295 10.3712 10.4129 10.4546 10.4963 10.5380 10.5797 10.6214 10.6630 10.7047 10.7464 10.7881

74.775 75.087 75.399 75.711 76.022 76.334 76.646 76.958 77.270 77.582 77.894 78.206 78.518 78.830 79.141 79.453 79.765 80.077 80.389 80.701

1.300 1.305 1.310 1.315 1.320 1.325 1.330 1.335 1.340 1.345 1.350 1.355 1.360 1.365 1.370 1.375 1.380 1.385 1.390 1.395

33.46 33.89 34.31 34.73 35.15 35.57 35.98 36.39 36.79 37.19 37.59 37.99 38.38 38.77 39.16 39.55 39.93 40.31 40.68 41.06

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

10.8298 10.8715 10.9132 10.9549 10.9966 11.0383 11.0800 11.1217 11.1634 11.2051 11.2467 11.2884 11.3301 11.3718 11.4135 11.4552 11.4969 11.5386 11.5803 11.6220

81.013 81.325 81.636 81.948 82.260 82.572 82.884 83.196 83.508 83.820 84.131 84.443 84.755 85.067 85.379 85.691 86.003 86.315 86.626 86.938

1.400 1.405 1.410

41.43 41.80 42.16

80 81 82

11.6637 11.7054 11.7471

87.250 87.562 87.874


97

Table 64 (Continued ) Specific Gravity 60◦ /60◦

a b

◦

Be

◦

TW


lb/ft3 at 60◦ F, Specific wt in Gravity air 60◦ /60◦

1.415 1.420 1.425 1.430 1.435 1.440 1.445 1.450 1.455 1.460 1.465 1.470 1.475 1.480 1.485 1.490

42.53 42.89 43.25 43.60 43.95 44.31 44.65 45.00 45.34 45.68 46.02 46.36 46.69 47.03 47.36 47.68

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

11.7888 11.8304 11.8721 11.9138 11.9555 11.9972 12.0389 12.0806 12.1223 12.1640 12.2057 12.2473 12.2890 12.3307 12.3724 12.4141

88.186 88.498 88.810 89.121 89.433 89.745 90.057 90.369 90.681 90.993 91.305 91.616 91.928 92.240 92.552 92.864

1.495 1.500 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59

48.01 48.33 48.97 49.61 50.23 50.84 51.45 52.05 52.64 53.23 53.81

99 100 102 104 106 108 110 112 114 116 118

12.4558 12.4975 12.581 12.644 12.748 12.831 12.914 12.998 13.081 13.165 13.248

93.176 93.488 94.11 94.79 95.36 95.98 96.61 97.23 97.85 98.48 99.10

1.60 1.61 1.62 1.63 1.64 1.65 1.66

54.38 54.94 55.49 56.04 56.59 57.12 57.65

120 122 124 126 128 130 132

13.331 13.415 13.498 13.582 13.665 13.748 13.832

99.73 100.35 100.97 101.60 102.22 102.84 103.47

◦

Be

◦

TW



1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79

58.17 58.69 59.20 59.71 60.20 60.70 61.18 61.67 62.14 62.61 63.08 63.54 63.99

134 136 138 140 142 144 146 148 150 152 154 156 158

13.915 13.998 14.082 14.165 14.249 14.332 14.415 14.499 14.582 14.665 14.749 14.832 14.916

104.09 104.72 105.34 105.96 106.59 107.21 107.83 108.46 109.08 109.71 110.32 110.95 111.58

1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89

64.44 64.89 65.33 65.77 66.20 66.62 67.04 67.46 67.87 68.28

160 162 164 166 168 170 172 174 176 178

14.999 15.082 15.166 15.249 15.333 15.416 15.499 15.583 15.666 15.750

112.20 112.82 113.45 114.07 114.70 115.31 115.94 116.56 117.19 117.81

1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

68.68 69.08 69.48 69.87 70.26 70.64 71.02 71.40 71.77 72.14 72.50

180 182 184 186 188 190 192 194 196 198 200

15.832 15.916 16.000 16.083 16.166 16.250 16.333 16.417 16.500 16.583 16.667

118.43 119.06 119.68 120.31 120.93 121.56 122.18 122.80 123.43 124.05 124.68

Baume´ scale. Twaddell scale.

6 STANDARD SIZES 6.1 Preferred Numbers

Selection of standard sizes or ratings of many diverse products can be performed advantageously through the use of a geometrically based progression introduced by C. Renard. He originally adopted as a basis a rule that would yield a 10th multiple of the value a after every 5th step of the series: a × q 5 = 10a

or

q=

√ 5 10

√ √ where the √numerical series a, a[ 5 10], a[ 5 10]2 , √ 5 5 a[ 10]3 , a[ 10]4 , 10a, the values of which, to five

significant figures, are a, 1.5849a, 2.5119a, 3.9811a, 6.309a, 10a. Renard’s idea was to substitute, for these values, more rounded but more practical values. He adopted as a a power of 10, positive, nil, or negative, obtaining the series 10, 16, 25, 40, 63, 100, which may be continued in both directions. From this series, designated by the symbol R5, the R10, R20, R40 series were formed, each adopted √ √ratio being the square root of the preceding one: 10 10, 20 10, √ 40 10. Thus each series provided Renard with twice as many steps in a decade as the preceding one. Preferred numbers are immediately applicable to commercial sizes and ratings of products. It is advantageous to minimize the number of initial sizes and

98


also to have adequate provision for logical expansion if and when additional sizes are required. By making the initial sizes correspond to a coarse series such as R5, unnecessary expense can be avoided if subsequent demand for the product is disappointing. If, on the other hand, the product is accepted, intermediate sizes may be selected in a rational manner by using the next finer series R10, and so on. Such a procedure assures a justifiable relationship between successive sizes and is a decided contrast to haphazard selection. The application of preferred numbers to raw material sizes and to the dimensions of parts also has enormously important potentialities. Under present conditions, commercial sizes of material are the result of a great many dissimilar gauge systems. The current trend in internationally acceptable metric sizing is to use preferred numbers. Even here, though, in the midst of the greatest opportunity for worldwide standardization through the acceptance of Renard series, we have fallen prey to our individualistic nature. The preferred number 1.6 is used by most nations as a standard 1.6 mm material thickness. German manufacturers, however, like 1.5 mm of the International Organization for Standardization (ISO) 497 for a more rounded preferred number. Similarly in metric screw sizes, 6.3 mm is consistent with the preferred number Table 65

series; yet, 6.0 mm (more rounded) has been adopted as a standard fastener diameter. The International Electrochemical Commission (IEC) used preferred numbers to establish standard current ratings in amperes as follows: 1, 1.25, 1.6, 2.5, 3.15, 4.5, 6.3. Notice that R10 series is used except for 4.5, which is a third step R20 series. The American Wire Gauge size for copper wire is based on a geometric series. However, instead of using √ 1.1220, the rounded value of 20 10, in a × q 20 = 10a, the q chosen is 1.123. A special series of preferred numbers is used for designating the characteristic values of capacitors, resistors, inductors, and other electronic products. Instead of using the Renard series R5, R10, R20, R40, R80 as derived from the geometric series of numbers 10N/5 , 10N/10 , 10N/20 , 10N/40 , 10N/80 , the geometric series used is 10N/6 , 10N/12 , 10N/24 , 10N/48 , 10N/96 , 10N/192 , which are designated respectively E6, E12, E24, E48, E96, E192. It should be evident that any series of preferred numbers can be generated to serve any specific case. Examples taken from the American National Standards Institute (ANSI) and ISO standards are reproduced in Tables 65–68.

Basic Series of Preferred Numbers: R5, R10, R20, and R40 Series Theoretical Values

R5

R10

R20

R40

Mantissas of Logarithms

1.00

1.00

1.00

1.00 1.06 1.12 1.18 1.25 1.32 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.12 2.24 2.36 2.50 2.65 2.80 3.00 3.15 3.35 3.55 3.75

000 025 050 075 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575

1.12 1.25

1.25 1.40

1.60

1.60

1.60 1.80

2.00

2.00 2.24

2.50

2.50

2.50 2.80

3.15

3.15 3.55

Calculated Values

Differences between Basic Series and Calculated Values (%)

1.0000 1.0593 1.1220 1.1885 1.2589 1.3335 1.4125 1.4962 1.5849 1.6788 1.7783 1.8836 1.9953 2.1135 2.2387 2.3714 2.5119 2.6607 2.8184 2.9854 3.1623 3.3497 3.5481 3.7584

0 +0.07 −0.18 −0.71 −0.71 −1.01 −0.88 +0.25 +0.95 +1.26 +1.22 +0.87 +0.24 +0.31 +0.06 −0.48 −0.47 −0.40 −0.65 +0.49 −0.39 +0.01 +0.05 −0.22


99

Table 65 (Continued ) Theoretical Values R5

R10

R20

R40

Mantissas of Logarithms

4.00

4.00

4.00

4.00 4.25 4.50 4.75 5.00 5.30 5.60 6.00 6.30 6.70 7.10 7.50 8.00 8.50 9.00 9.50 10.00

600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 000

4.50 5.00

5.00 5.60

6.30

6.30

6.30 7.10

8.00

8.00 9.00

10.00

10.00

10.00

Table 66 Basic Series of Preferred Numbers: R80 Series 1.00 1.03 1.06 1.09 1.12 1.15 1.18 1.22 1.25 1.28 1.32 1.36 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

1.80 1.85 1.90 1.95 2.00 2.06 2.12 2.18 2.24 2.30 2.36 2.43 2.50 2.58 2.65 2.72 2.80 2.90 3.00 3.07

3.15 3.25 3.35 3.45 3.55 3.65 3.75 3.87 4.00 4.12 4.25 4.37 4.50 4.62 4.75 4.87 5.00 5.15 5.20 5.45

5.60 5.80 6.00 6.15 6.30 6.50 6.70 6.90 7.10 7.30 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaaaaaaa Applicable Documents Adoption of Renard’s preferred number system by international standardization bodies resulted in a host of national standards being generated for particular applications. The current organization in the United States that is charged with

Table 67 Preferred Number 1.0 1.6 2.5 4.0 6.3

Calculated Values

Differences between Basic Series and Calculated Values (%)

3.9811 4.2170 4.4668 4.7315 5.0119 5.3088 5.6234 5.9566 6.3096 6.6834 7.0795 7.4989 7.9433 8.4140 8.9125 9.4406 10.0000

+0.47 +0.78 +0.74 +0.39 −0.24 −0.17 −0.42 +0.73 −0.15 +0.25 +0.29 +0.01 +0.71 +1.02 +0.98 +0.63 0

Expansion of R5 Series Divided by 10

Multiplied by 10

Multiplied by 100

Multiplied by 1000

0.10 0.16 0.25 0.40 0.63

10 16 25 40 63

100 160 250 400 630

1000 1600 2500 4000 6300

Table 68

Rounding of Preferred Numbersa

Preferred Number

First Rounding

Second Rounding

1.1 1.25 1.6 2.2 3.2 3.6 5.6 6.3 7.1

1.1 1.2 1.5a 2.2 3.0 3.5 5.5 6.0 7.0

1.12 1.25 1.60 2.24 3.15 3.55 5.60 6.30 7.10 a

Rounded only when using the R5 or R10 series.

generating American national standards is the ANSI. Accordingly, the following national and international standards are in use in the United States.

100


ANSI Z17.1-1973 ANSI C83.2-1971 EIA Standard RS-385

ISO 3-1973 ISO 17-1973

Table 69

American National Standard for Preferred Numbers American National Standard Preferred Values for Components for Electronic Equipment Preferred Values for Components for Electronic Equipment (issued by the Electronics Industries Association; Same as ANSI C83.2-1971) Preferred numbers—series of preferred numbers Guide to the use of preferred numbers and of series of preferred numbers

ISO 497-1973

Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers

Table 67 shows the expansibility of preferred numbers in the positive direction. The same expansibility can be made in the negative direction. Table 68 shows a deviation by roundings for cases where adhering to a basic preferred number would be absurd as in 31.5 teeth in a gear when clearly 32 makes sense. 6.2 Gages

U.S. Standard Gagea for Sheet and Plate Iron and Steel and Its Extensionb

Gage Number

Weight per Square Foot oz. lb

Weight per Square Meter kg

0000000 000000 00000 0000 000 00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

320 300 280 260 240 220 200 180 170 160 150 140 130 120 110 100 90 80 70 60 50 45 40 36 32 28 24 22 20 18 16 14 12

97.65 91.55 85.44 79.34 73.24 67.13 61.03 54.93 51.88 48.82 45.77 42.72 39.67 36.62 33.57 30.52 27.46 24.41 21.36 18.31 15.26 13.73 12.21 10.99 9.765 8.544 7.324 6.713 6.103 5.493 4.882 4.272 3.662

20.00 18.75 17.50 16.25 15.00 13.75 12.50 11.25 10.62 10.00 9.375 8.750 8.125 7.500 6.875 6.250 5.625 5.000 4.375 3.750 3.125 2.812 2.500 2.250 2.000 1.750 1.500 1.375 1.250 1.125 1.000 0.8750 0.7500

Approximate Thickness Wrought Iron, 480 lb/ft3 in. mm 0.500 0.469 0.438 0.406 0.375 0.344 0.312 0.2812 0.2656 0.2500 0.2344 0.2188 0.2031 0.1875 0.1719 0.1562 0.1406 0.1250 0.1094 0.0938 0.0781 0.0703 0.0625 0.0562 0.0500 0.0438 0.0375 0.0344 0.0312 0.0281 0.0250 0.0219 0.0188

12.70 11.91 11.11 10.32 9.52 8.73 7.94 7.14 6.75 6.35 5.95 5.56 5.16 4.76 4.37 3.97 3.57 3.18 2.778 2.381 1.984 1.786 1.588 1.429 1.270 1.111 0.952 0.873 0.794 0.714 0.635 0.556 0.476

Steel and openhearth Iron, 489.6 lb/ft3 in. mm 0.490 0.460 0.429 0.398 0.368 0.337 0.306 0.2757 0.2604 0.2451 0.2298 0.2145 0.1991 0.1838 0.1685 0.1532 0.1379 0.1225 0.1072 0.0919 0.0766 0.0689 0.0613 0.0551 0.0490 0.0429 0.0368 0.0337 0.0306 0.0276 0.0245 0.0214 0.0184

12.45 11.67 10.90 10.12 9.34 8.56 7.78 7.00 6.62 6.23 5.84 5.45 5.06 4.67 4.28 3.89 3.50 3.11 2.724 2.335 1.946 1.751 1.557 1.400 1.245 1.090 0.934 0.856 0.778 0.700 0.623 0.545 0.467


101

Table 69 (Continued )

Gage Number

Weight per Square Meter kg

Weight per Square Foot oz. lb

Approximate Thickness Wrought Iron, 480 lb/ft3 in. mm

Steel and openhearth Iron, 489.6 lb/ft3 in. mm

27 28 29 30 31 32 33 34 35 36 37 38 39

11 10 9 8 7 6 12 6 5 12 5 4 12 4 14 4 3 34

0.6875 0.6250 0.5625 0.5000 0.4375 0.4062 0.3750 0.3438 0.3125 0.2812 0.2656 0.2500 0.2344

3.357 3.052 2.746 2.441 2.136 1.983 1.831 1.678 1.526 1.373 1.297 1.221 1.144

0.0172 0.0156 0.0141 0.0125 0.0109 0.0102 0.0094 0.0086 0.0078 0.0070 0.0066 0.0062 0.0059

0.437 0.397 0.357 0.318 0.278 0.258 0.238 0.218 0.198 0.179 0.169 0.159 0.149

0.0169 0.0153 0.0138 0.0123 0.0107 0.0100 0.0092 0.0084 0.0077 0.0069 0.0065 0.0061 0.0057

0.428 0.389 0.350 0.311 0.272 0.253 0.233 0.214 0.195 0.175 0.165 0.156 0.146

40 41 42 43 44

3 12 3 38 3 14 3 18 3

0.2188 0.2109 0.2031 0.1953 0.1875

1.068 1.030 0.9917 0.9536 0.9155

0.0055 0.0053 0.0051 0.0049 0.0047

0.139 0.134 0.129 0.124 0.119

0.0054 0.0052 0.0050 0.0048 0.0046

0.136 0.131 0.126 0.122 0.117

a For the Galvanized Sheet Gage, add 2.5 oz to the weight per square foot as given in the table. Gage numbers below 8 and above 34 are not used in the Galvanized Sheet Gage. b Gage numbers greater than 38 were not in the standard as set up by law but are in general use.

Table 70 American Wire Gage: Weights of Copper, Aluminum, and Brass Sheets and Plates Approximate Weight,a lb/ft2

Thickness Gage Number 0000 000 00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

in.

mm

Copper

Aluminum

0.4600 0.4096 0.3648 0.3249 0.2893 0.2576 0.2294 0.2043 0.1819 0.1620 0.1443 0.1285 0.1144 0.1019 0.0907 0.0808 0.0720 0.0641 0.0571

11.68 10.40 9.266 8.252 7.348 6.544 5.827 5.189 4.621 4.115 3.665 3.264 2.906 2.588 2.305 2.053 1.828 1.628 1.450

21.27 18.94 16.87 15.03 13.38 11.91 10.61 9.45 8.41 7.49 6.67 5.94 5.29 4.713 4.195 3.737 3.330 2.965 2.641

6.49 5.78 5.14 4.58 4.08 3.632 3.234 2.880 2.565 2.284 2.034 1.812 1.613 1.437 1.279 1.139 1.015 0.904 0.805

Commercial (High) Brass 20.27 18.05 16.07 14.32 12.75 11.35 10.11 9.00 8.01 7.14 6.36 5.66 5.04 4.490 3.996 3.560 3.172 2.824 2.516 (Continues)

102 Table 70

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Approximate Weight,a lb/ft2

Thickness Gage Number 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a

in. 0.0508 0.0453 0.0403 0.0359 0.0320 0.0285 0.0253 0.0226 0.0201 0.0179 0.0159 0.0142 0.0126 0.0113 0.0100 0.00893 0.00795 0.00708 0.00630 0.00561 0.00500 0.00445 0.00397 0.00353 0.00314

mm

Copper

Aluminum

1.291 1.150 1.024 0.9116 0.8118 0.7230 0.6438 0.5733 0.5106 0.4547 0.4049 0.3606 0.3211 0.2859 0.2546 0.2268 0.2019 0.1798 0.1601 0.1426 0.1270 0.1131 0.1007 0.0897 0.0799

2.349 2.095 1.864 1.660 1.480 1.318 1.170 1.045 0.930 0.828 0.735 0.657 0.583 0.523 0.4625 0.4130 0.3677 0.3274 0.2914 0.2595 0.2312 0.2058 0.1836 0.1633 0.1452

0.716 0.639 0.568 0.506 0.451 0.402 0.3567 0.3186 0.2834 0.2524 0.2242 0.2002 0.1776 0.1593 0.1410 0.1259 0.1121 0.0998 0.0888 0.0791 0.0705 0.0627 0.0560 0.0498 0.0443

Commercial (High) Brass 2.238 1.996 1.776 1.582 1.410 1.256 1.115 0.996 0.886 0.789 0.701 0.626 0.555 0.498 0.4406 0.3935 0.3503 0.3119 0.2776 0.2472 0.2203 0.1961 0.1749 0.1555 0.1383

Assumed specific gravities or densities in grams per cubic centimeter; copper, 8.89; aluminum, 2.71; brass, 8.47.

Wire Gages The sizes of wires having a diameter less than 12 in. are usually stated in terms of certain arbitrary scales called “gages.” The size or gage number of a solid wire refers to the cross section of the wire perpendicular to its length; the size or gage number of a stranded wire refers to the total cross section Table 71

of the constituent wires, irrespective of the pitch of the spiraling. Larger wires are usually described in terms of their area expressed in circular mils. A circular mil is the area of a circle 1 mil in diameter, and the area of any circle in circular mils is equal to the square of its diameter in mils.

Comparison of Wire Gage Diameters in Milsa

Gage No.

American Wire Gage (Brown & Sharpe)

Steel Wire Gage

Birmingham Wire Gage (Stubs’)

Old English Wire Gage (London)

7–0 6–0 5–0 4–0 3–0 2–0 0 1

— — — 460 410 365 325 289

490.0 461.5 430.5 393.8 362.5 331.0 306.5 283.0

— — — 454 425 380 340 300

— — — 454 425 380 340 300

Stubs’ Steel Wire Gage — — — — — — — 227

(British) Standard Wire Gage 500 464 432 400 372 348 324 300

Metric Gageb — — — — — — — 3.94


103


Gage No.

American Wire Gage (B. & S.)

Steel Wire Gage

Birmingham Wire Gage (Stubs’)

Old English Wire Gage (London)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

258 229 204 182 162 144 128 114 102 91 81 72 64 57 51 45 40 36 32 28.5 25.3 22.6 20.1 17.9 15.9 14.2 12.6 11.3 10.0 8.9 8.0 7.1 6.3 5.6 5.0 4.5 4.0 3.5 3.1 — — — — — — — — — —

262.5 243.7 225.3 207.0 192.0 177.0 162.0 148.3 135.0 120.5 105.5 91.5 80.0 72.0 62.5 54.0 47.5 41.0 34.8 31.7 28.6 25.8 23.0 20.4 18.1 17.3 16.2 15.0 14.0 13.2 12.8 11.8 10.4 9.5 9.0 8.5 8.0 7.5 7.0 6.6 6.2 6.0 5.8 5.5 5.2 5.0 4.8 4.6 4.4

284 259 238 220 203 180 165 148 134 120 109 95 83 72 65 58 49 42 35 32 28 25 22 20 18 16 14 13 12 10 9 8 7 5 4 — — — — — — — — — — — — — —

284 259 238 220 203 180 165 148 134 120 109 95 83 72 65 58 49 42 35 31.5 29.5 27.0 25.0 23.0 20.5 18.75 16.50 15.50 13.75 12.25 11.25 10.25 9.50 9.00 7.50 6.50 5.75 5.00 4.50 — — — — — — — — — —

a

Stubs’ Steel Wire Gage 219 212 207 204 201 199 197 194 191 188 185 182 180 178 175 172 168 164 161 157 155 153 151 148 146 143 139 134 127 120 115 112 110 108 106 103 101 99 97 95 92 88 85 81 79 77 75 72 69

(British) Standard Wire Gage 276 252 232 212 192 176 160 144 128 116 104 92 80 72 64 56 48 40 36 32 28 24 22 20 18 16.4 14.8 13.6 12.4 11.6 10.8 10.0 9.2 8.4 7.6 6.8 6.0 5.2 4.8 4.4 4.0 3.6 3.2 2.8 2.4 2.0 1.6 1.2 1.0

Metric Gageb 7.87 11.8 15.7 19.7 23.6 27.6 31.5 35.4 39.4 — 47.2 — 55.1 — 63.0 — 70.9 — 78.7 — — — — 98.4 — — — — 118 — — — — 138 — — — — 157 — — — — 177 — — — — 197

Bureau of Standards, Circulars No. 31 and No. 67. For diameters corresponding to metric gage numbers, 1.2, 1.4, 1.6, 1.8, 2.5, 3.5, and 4.5, divide those of 12, 14, etc., by 10.

b

104


6.3 Paper Sizes Table 72

Standard Engineering Drawing Sizesa Flat Sizesb Margin

Size Designation

Widthc (Vertical)

Length (Horizontal)

Horizontal

Vertical

A (horizontal) A (vertical) B C D E F

8.5 11.0 11.0 17.0 22.0 34.0 28.0

11.0 8.5 17.0 22.0 34.0 44.0 40.0

0.38 0.25 0.38 0.75 0.50 1.00 0.50

0.25 0.38 0.62 0.50 1.00 0.50 0.50

Roll Sizes

Widthb

Size Designation G H J K a b c

(Vertical) 11.0 28.0 34.0 40.0

Lengthc (Horizontal) Min 22.5 44.0 55.0 55.0

Marginc Max 90.0 143.0 176.0 143.0

Horizontal 0.38 0.50 0.50 0.50

Vertical 0.50 0.50 0.50 0.50

See ANSI Y14.1-1980. All dimensions are in inches. Not including added protective margins.

International Paper Sizes Countries that are committed to the International System of Units (SI) have a standard series of paper sizes for printing, writing, and drafting. These paper sizes are called the “international paper sizes.” The advantages of the international paper sizes are as follows:

Table 73

1. The ratio of width to length remains constant for every size, namely: 1 Width = √ Length 2

or

1 approximately 1.414

Since this is the same ratio as the D aperture in the unitized 35-mm microfilm frame, the advantages are apparent. √ 2. If a sheet is cut in half, that is, if the 2 length is cut in half, the two√halves retain the constant widthto-length ratio of 1/ 2. No other ratio could do this. 3. All international sizes are created from the A-0 size by single cuts without waste. In storing or stacking they fit together like parts of a jigsaw puzzle—without waste.

Eleven International Paper Sizes

International Paper Size

Millimeters

Inches, Approximate

A-0

841 × 1189

33 81 × 46 43

A-1

594 × 841

23 83 × 33 81

A-2

420 × 594

16 21 × 23 83

A-3

297 × 420

11 43 × 16 21

A-4

210 × 297

8 14 × 11 43

A-5

148 × 210

5 78 × 8 14

A-6

105 × 148

4 18 × 5 78

A-7

74 × 105

2 78 × 4 18

A-8

52 × 74

2 × 2 78

A-9

37 × 52

A-10

26 × 37

1 12 × 2 1 × 1 12


105

6.4 Sieve Sizes Table 74 Tyler Standard Screen Scale Sieves This screen scale has as its base an opening of 0.0029 in., which is the opening in 200-mesh 0.0021-in. wire, the standard sieve, as adopted by the Bureau of Standards of the U.S. government, the openings increasing in the ratio of the square root of 2 or 1.414. Where a closer sizing is required, column 5 shows the Tyler Standard Screen Scale with intermediate sieves. In this series the sieve openings increase in the ratio of the fourth root of 2, or 1.189. Tyler Standard Screen Scale √ 2 or 1.414 Openings (in.) (1)

Every Other Sieve from 0.0029 to 0.742 in., Ratio of 2 to 1 (2)

Every Other Sieve from 0.0041 to 1.050 in., Ratio of 2 to 1 (3)

1.050

—

1.050

—

—

—

—

—

0.742

Every Fourth Sieve from 0.0029 to 0.742 in., Ratio of 4 to 1 (4)

For Closer Sizing Sieves from 0.0029 to 1.050 in., Ratio √ 4 2 or 1.189 (5)

openings (mm) (6)

Openings in Fractions of inch (approx.) (7)

1.050

26.67

1

—

0.148

0.883

22.43

7 8 3 4 5 8 1 2 7 16 3 8 5 16 1 4 7 32 3 16 5 32 1 8 7 64 3 32 5 84 1 16

—

0.135

—

0.135

—

0.120

—

0.105

—

0.105

—

0.092

0.742

—

0.742

0.742

18.85

—

—

—

—

0.624

15.85

0.525

—

0.525

—

0.525

13.33

—

—

—

—

0.441

11.20

0.371

—

—

0.371

9.423

—

—

—

—

0.312

7.925

0.263

—

0.263

—

0.263

6.680

—

—

—

—

0.221

5.613

0.185

—

0.185

0.185

4.699

0.371

0.185 —

—

—

—

0.156

3.962

0.131

—

0.131

—

0.131

3.327

—

—

—

—

0.110

2.794

0.093

—

—

0.093

2.362

—

—

—

—

0.078

1.981

0.065 —

— —

0.065 —

— —

0.065 0.055

1.651 1.397

—

0.046 —

0.046 —

— —

0.046 —

0.046 0.0390

1.168 0.991

—

0.0328 — 0.0232 —

— — 0.0232 —

0.0328 — — —

— — — —

0.0328 0.0276 0.0232 0.0195

0.833 0.701 0.589 0.495

— — —

0.0164 — 0.0116 — 0.0082 — 0.0058 — 0.0041 — 0.0029

— — 0.0116 — — — 0.0058 — — — 0.0029

0.0164 — — — 0.0082 — — — 0.0041 — —

— — 0.0116 — — — — — — — 0.0029

0.0164 0.0138 0.0116 0.0097 0.0082 0.0069 0.0058 0.0049 0.0041 0.0035 0.0029

0.417 0.351 0.295 0.246 0.208 0.175 0.147 0.124 0.104 0.088 0.074

— — — — — — — — — —

0.093

3 64 1 32

1 64

Mesh (8)

Diameter of Wire (9)

2 21

0.088

3

0.070

3 21

0.065

4

0.065

5

0.044

6

0.036

7

0.0328

8

0.032

9

0.033

10 12

0.035 0.028

14 16

0.025 0.0235

20 24 28 32

0.0172 0.0141 0.0125 0.0118

35 42 48 60 65 80 100 115 150 170 200

0.0122 0.0100 0.0092 0.0070 0.0072 0.0056 0.0042 0.0038 0.0026 0.0024 0.0021

106


Table 75 Nominal Dimensions, Permissible Variations, and Limits for Woven Wire Cloth of Standard Sieves, U.S. Series, ASTM Standarda Sieve Opening

mm

in. (approx. equivalents)

Permissible Variations in Average Opening (±%)

Permissible Variations in Maximum Opening (±%)

mm

in. (approx. equivalents)

5.66 4.76 4.00 3.36 2.83 2.38 2.00 1.68 1.41 1.19 1.00 0.84 0.71 0.59 0.50 0.42 0.35 0.297 0.250 0.210 0.177 0.149 0.125 0.105 0.088 0.074 0.062 0.053 0.044 0.037

0.233 0.187 0.157 0.132 0.111 0.0937 0.0787 0.0661 0.0555 0.0469 0.0394 0.0331 0.0280 0.0232 0.0197 0.0165 0.0138 0.0117 0.0098 0.0083 0.0070 0.0059 0.0049 0.0041 0.0035 0.0029 0.0024 0.0021 0.0017 0.0015

3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7

10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 25 25 25 25 25 40 40 40 40 40 60 90 90 90 90

1.28–1.90 1.14–1.68 1.00–1.47 0.87–1.32 0.80–1.20 0.74–1.10 0.68–1.00 0.62–0.90 0.56–0.80 0.50–0.70 0.43–0.62 0.38–0.55 0.33–0.48 0.29–0.42 0.26–0.37 0.23–0.33 0.20–0.29 0.170–0.253 0.149–0.220 0.130–0.187 0.114–0.154 0.096–0.125 0.079–0.103 0.063–0.087 0.054–0.073 0.045–0.061 0.039–0.052 0.035–0.046 0.031–0.040 0.023–0.035

0.050–0.075 0.045–0.066 0.039–0.058 0.034–0.052 0.031–0.047 0.0291–0.0433 0.0268–0.0394 0.0244–0.0354 0.0220–0.0315 0.0197–0.0276 0.0169–0.0244 0.0150–0.0217 0.0130–0.0189 0.0114–0.0165 0.0102–0.0146 0.0091–0.0130 0.0079–0.0114 0.0067–0.0100 0.0059–0.0087 0.0051–0.0074 0.0045–0.0061 0.0038–0.0049 0.0031–0.0041 0.0025–0.0034 0.0021–0.0029 0.0018–0.0024 0.0015–0.0020 0.0014–0.0018 0.0012–0.0016 0.0009–0.0014

Size or Sieve Designation µm 5660 4760 4000 3360 2830 2380 2000 1680 1410 1190 1000 840 710 590 500 420 350 297 250 210 177 149 125 105 88 74 62 53 44 37

No. 3 12 4 5 6 7 8 10 12 14 16 18 20 25 30 35 40 45 50 60 70 80 100 120 140 170 200 230 270 325 400

Wire Diameter

a For sieves from the 1000-µm (No. 18) to the 37-µm (No. 400) size, inclusive, not more than 5% of the openings shall exceed the nominal opening by more than one-half of the permissible variation in the maximum opening.

6.5 Standard Structural Sizes—Steel Steel Sections. Tables 76–83 give the dimensions, weights, and properties of rolled steel structural sections, including wide-flange sections, American standard beams, channels, angles, tees, and zees. The values for the various structural forms, taken from the eighth edition, 1980, of Steel Construction, by the kind permission of the publisher, the American Institute of Steel Construction, give the section specifications required in designing steel structures. The theory of design is covered in Section 4—Mechanics of Deformable Bodies.

Most of the sections can be supplied promptly steel mills. Owing to variations in the rolling practice of the different mills, their products are not identical, although their divergence from the values given in the tables is practically negligible. For standardization, only the lesser values are given, and therefore they are on the side of safety. Further information on sections listed in the tables, together with information on other products and on the requirements for placing orders, may be gathered from mill catalogs.


107

Table 76 Properties of Wide-Flange Sections

Nominal Size (in.)

Weight per Foot (lb)

Area (in.2 )

Depth (in.)

Flange Width Thickness (in.) (in.)

36 × 16 21

300 280 260 245 230 194 182 170 160 150 240 220 200 152 141 130 210 190 172 132 124 116 108 177 160 145 114 102 94 160 145 130 120 110 100 94 84 76 142 127 112 96 82

88.17 82.32 76.56 72.03 67.73 57.11 53.54 49.98 47.09 44.16 70.52 64.73 58.79 44.71 41.51 38.26 61.78 55.90 50.65 38.83 36.45 34.13 31.77 52.10 47.04 42.68 33.53 30.01 27.65 47.04 42.62 38.21 35.29 32.36 29.43 27.63 24.71 22.37 41.76 37.34 32.93 28.21 24.10

36.72 36.50 36.24 36.06 35.88 36.48 36.32 36.16 36.00 35.84 33.50 33.25 33.00 33.50 33.31 33.10 30.38 30.12 29.88 30.30 30.16 30.00 29.82 27.31 27.08 26.88 27.28 27.07 26.91 24.72 24.49 24.25 24.31 24.16 24.00 24.29 24.09 23.91 21.46 21.24 21.00 21.14 20.86

16.655 16.595 16.555 16.512 16.475 12.117 12.072 12.027 12.000 11.972 15.865 15.810 15.750 11.565 11.535 11.510 15.105 15.040 14.985 10.551 10.521 10.500 10.484 14.090 14.023 13.965 10.070 10.018 9.990 14.091 14.043 14.000 12.088 12.042 12.000 9.061 9.015 8.985 13.132 13.061 13.000 9.038 8.962

36 × 12

33 × 15 43 33 × 11 21 30 × 15 30 × 10 21

27 × 14 27 × 10 24 × 14 24 × 12 24 × 9 21 × 13 21 × 9

1.680 1.570 1.440 1.350 1.260 1.260 1.180 1.100 1.020 0.940 1.400 1.275 1.150 1.055 0.960 0.855 1.315 1.185 1.065 1.000 0.930 0.850 0.760 1.190 1.075 0.975 0.932 0.827 0.747 1.135 1.020 0.900 0.930 0.855 0.775 0.872 0.772 0.682 1.095 0.985 0.865 0.935 0.795

Axis X–X S (in.3 )

Web Thickness (in.)

I (in.4 )

0.945 0.885 0.845 0.802 0.765 0.770 0.725 0.680 0.653 0.625 0.830 0.775 0.715 0.635 0.605 0.580 0.775 0.710 0.655 0.615 0.585 0.564 0.548 0.725 0.658 0.600 0.570 0.518 0.490 0.656 0.608 0.565 0.556 0.510 0.468 0.516 0.470 0.440 0.659 0.588 0.527 0.575 0.499

20290.2 18819.3 17233.8 16092.2 14988.4 12103.4 11281.5 10470.0 9738.8 9012.1 13585.1 12312.1 11048.2 8147.6 7442.2 6699.0 9872.4 8825.9 7891.5 5753.1 5347.1 4919.1 4461.0 6728.6 6018.6 5414.3 4080.5 3604.1 3266.7 5110.3 4561.0 4009.5 3635.3 3315.0 2987.3 2683.0 2364.3 2096.4 3403.1 3017.2 2620.6 2088.9 1752.4

1105.1 1031.2 951.1 892.5 835.5 663.6 621.2 579.1 541.0 502.9 811.1 740.6 669.6 486.4 446.8 404.8 649.9 586.1 528.2 379.7 354.6 327.9 299.2 492.8 444.5 402.9 299.2 266.3 242.8 413.5 372.5 330.7 299.1 274.4 248.9 220.9 196.3 175.4 317.2 284.1 249.6 197.6 168.0

Axis Y–Y S (in.3 )

r (in.)

I (in.4 )

15.17 15.12 15.00 14.95 14.88 14.56 14.52 14.47 14.38 14.29 13.88 13.79 13.71 13.50 13.39 13.23 12.64 12.57 12.48 12.17 12.11 12.00 11.85 11.36 11.31 11.26 11.03 10.96 10.87 10.42 10.34 10.24 10.15 10.12 10.08 9.85 9.78 9.68 9.03 8.99 8.92 8.60 8.53

1225.2 1127.5 1020.6 944.7 870.9 355.4 327.7 300.6 275.4 250.4 874.3 782.4 691.7 256.1 229.7 201.4 707.9 624.6 550.1 185.0 169.7 153.2 135.1 518.9 458.0 406.9 149.6 129.5 115.1 492.6 434.3 375.2 254.0 229.1 203.5 102.2 88.3 76.5 385.9 338.6 289.7 109.3 89.6

147.1 135.9 123.3 114.4 105.7 58.7 54.3 50.0 45.9 41.8 110.2 99.0 87.8 44.3 39.8 35.0 93.7 83.1 73.4 35.1 32.3 29.2 25.8 73.7 65.3 58.3 29.7 25.9 23.0 69.9 61.8 53.6 42.0 38.0 33.9 22.6 19.6 17.0 58.8 51.8 44.6 24.2 20.0

r (in.) 3.73 3.70 3.65 3.62 3.59 2.49 2.47 2.45 2.42 2.38 3.52 3.48 3.43 2.39 2.35 2.29 3.38 3.34 3.30 2.18 2.16 2.12 2.06 3.16 3.12 3.09 2.11 2.08 2.04 3.23 3.19 3.13 2.68 2.66 2.63 1.92 1.89 1.85 3.04 3.01 2.96 1.97 1.93

(Continues)

108


Table 76

(Continued )

Nominal Size (in.)


Area (in.2 )

Depth (in.)

Flange Web Width Thickness Thickness (in.) (in.) (in.)

I (in.4 )

21 × 8 14

73 68 62 114 105 96 85 77 70 64 60 55 50 96 88 78 71 64 58 50 45 40 36 426 398 370 342 314 287 264 246 237 228 219 211 202 193 184 176 167 158 150 142 320a 136 127 119 111 103 95 87 84 78 74 68 61

21.46 20.02 18.23 33.51 30.86 28.22 24.97 22.63 20.56 18.80 17.64 16.19 14.71 28.22 25.87 22.92 20.86 18.80 17.04 14.70 13.24 11.77 10.59 125.25 116.98 108.78 100.59 92.30 84.37 77.63 72.33 69.69 67.06 64.36 62.07 59.39 56.73 54.07 51.73 49.09 46.47 44.08 41.85 94.12 39.98 37.33 34.99 32.65 30.26 27.94 25.56 24.71 22.94 21.76 20.00 17.94

21.24 21.13 20.99 18.48 18.32 18.16 18.32 18.16 18.00 17.87 18.25 18.12 18.00 16.32 16.16 16.32 16.16 16.00 15.86 16.25 16.12 16.00 15.85 18.69 18.31 17.94 17.56 17.19 16.81 16.50 16.25 16.12 16.00 15.87 15.75 15.63 15.50 15.38 15.25 15.12 15.00 14.88 14.75 16.81 14.75 14.62 14.50 14.37 14.25 14.12 14.00 14.18 14.06 14.19 14.06 13.91

8.295 8.270 8.240 11.833 11.792 11.750 8.838 8.787 8.750 8.715 7.558 7.532 7.500 11.533 11.502 8.586 8.543 8.500 8.464 7.073 7.039 7.000 6.992 16.695 16.590 16.475 16.365 16.235 16.130 16.025 15.945 15.910 15.865 15.825 15.800 15.750 15.710 15.660 15.640 15.600 15.550 15.515 15.500 16.710 14.740 14.690 14.650 14.620 14.575 14.545 14.500 12.023 12.000 10.072 10.040 10.000

100.3 1478.3 1326.8 2033.8 1852.5 1674.7 1429.9 1286.8 1153.9 1045.8 984.0 889.9 800.6 1355.1 1222.6 1042.6 936.9 833.8 746.4 655.4 583.3 515.5 446.3 6610.3 6013.7 5454.2 4911.5 4399.4 3912.1 3526.0 3228.9 3080.9 2942.4 2798.2 2671.4 2538.8 2402.4 2274.8 2149.6 2020.8 1900.6 1786.9 1672.2 4141.7 1593.0 1476.7 1373.1 1266.5 1165.8 1063.5 966.9 928.4 851.2 796.8 724.1 641.5

18 × 11 34 18 × 8 34

18 × 7 12 16 × 11 12 16 × 8 12

16 × 7

14 × 16

14 × 14 12

14 × 12 14 × 10

0.740 0.685 0.615 0.991 0.911 0.831 0.911 0.831 0.751 0.686 0.695 0.630 0.570 0.875 0.795 0.875 0.795 0.715 0.645 0.628 0.563 0.503 0.428 3.033 2.843 2.658 2.468 2.283 2.093 1.938 1.813 1.748 1.688 1.623 1.563 1.503 1.438 1.378 1.313 1.248 1.188 1.128 1.063 2.093 1.063 0.998 0.938 0.873 0.813 0.748 0.688 0.778 0.718 0.783 0.718 0.643

0.455 0.430 0.400 0.595 0.554 0.512 0.526 0.475 0.438 0.403 0.416 0.390 0.358 0.535 0.504 0.529 0.486 0.443 0.407 0.380 0.346 0.307 0.299 1.875 1.770 1.655 1.545 1.415 1.310 1.205 1.125 1.090 1.045 1.005 0.980 0.930 0.890 0.840 0.820 0.780 0.730 0.695 0.680 1.890 0.660 0.610 0.570 0.540 0.495 0.465 0.420 0.451 0.428 0.450 0.418 0.378

Axis X–X S (in.3 ) 150.7 139.9 126.4 220.1 202.2 184.4 156.1 141.7 128.2 117.0 107.8 98.2 89.0 166.1 151.3 127.8 115.9 104.2 94.1 80.7 72.4 64.4 56.3 707.4 656.9 608.1 559.4 511.9 465.5 427.4 397.4 382.2 367.8 352.6 339.2 324.9 310.0 295.8 281.9 267.3 253.4 240.2 226.7 492.8 216.0 202.0 189.4 176.3 163.6 150.6 138.1 130.9 121.1 112.3 103.0 92.2

Axis Y–Y S r (in.3 ) (in.)

r (in.)

I (in.4 )

8.64 8.59 8.53 7.79 7.75 7.70 7.57 7.54 7.49 7.46 7.47 7.41 7.38 6.93 6.87 6.74 6.70 6.66 6.62 6.68 6.64 6.62 6.49 7.26 7.17 7.08 6.99 6.90 6.81 6.74 6.68 6.65 6.62 6.59 6.56 6.54 6.51 6.49 6.45 6.42 6.40 6.37 6.32 6.63 6.31 6.29 6.26 6.23 6.21 6.17 6.15 6.13 6.09 6.05 6.02 5.98

66.2 60.4 53.1 255.6 231.0 206.8 99.4 88.6 78.5 70.3 47.1 42.0 37.2 207.2 185.2 87.5 77.9 68.4 60.5 34.8 30.5 26.5 22.1 2359.5 2169.7 1986.0 1806.9 1631.4 1466.5 1331.2 1226.6 1174.8 1124.8 1073.2 1028.6 979.7 930.1 882.7 837.9 790.2 745.0 702.5 660.1 1635.1 567.7 527.6 491.8 454.9 419.7 383.7 349.7 225.5 206.9 133.5 121.2 107.3

16.0 14.6 12.9 43.2 39.2 35.2 22.5 20.2 17.9 16.1 12.5 11.1 9.9 35.9 32.2 20.4 18.2 16.1 14.3 9.8 8.7 7.6 6.3 282.7 261.6 241.1 220.8 201.0 181.8 166.1 153.9 147.7 141.8 135.6 130.2 124.4 118.4 112.7 107.1 101.3 95.8 90.6 85.2 195.7 77.0 71.8 67.1 62.2 57.6 52.8 48.2 37.5 34.5 26.5 24.1 21.5

1.76 1.74 1.71 2.76 2.73 2.71 2.00 1.98 1.95 1.93 1.63 1.61 1.59 2.71 2.67 1.95 1.93 1.91 1.88 1.54 1.52 1.50 1.45 4.34 4.31 4.27 4.24 4.20 4.17 4.14 4.12 4.11 4.10 4.08 4.07 4.06 4.05 4.04 4.02 4.01 4.00 3.99 3.97 4.17 3.77 3.76 3.75 3.73 3.72 3.71 3.70 3.02 3.00 2.48 2.46 2.45


109

Table 76 (Continued ) Nominal Size (in.) 14 × 8 14 × 6 43 12 × 12

12 × 10 12 × 8 12 × 6 21 10 × 10

10 × 8 10 × 5 43 8×8

8 × 6 21 8 × 5 41 a


Area (in.2 )

Depth (in.)

Flange Width Thickness (in.) (in.)

53 48 43 38 34 30 190 161 133 120 106 99 92 85 79 72 65 58 53 50 45 40 36 31 27 112 100 89 77 72 66 60 54 49 45 39 33 29 25 21 67 58 48 40 35 31 28 24 20 17

15.59 14.11 12.65 11.17 10.00 8.81 55.86 47.38 39.11 35.31 31.19 29.09 27.06 24.98 23.22 21.16 19.11 17.06 15.59 14.71 13.24 11.77 10.59 9.12 7.97 32.92 29.43 26.19 22.67 21.18 19.41 17.66 15.88 14.40 13.24 11.48 9.71 8.53 7.35 6.19 19.70 17.06 14.11 11.76 10.30 9.12 8.23 7.06 5.88 5.00

13.94 13.81 13.68 14.12 14.00 13.86 14.38 13.88 13.38 13.12 12.88 12.75 12.62 12.50 12.38 12.25 12.12 12.19 12.06 12.19 12.06 11.94 12.24 12.09 11.95 11.38 11.12 10.88 10.62 10.50 10.38 10.25 10.12 10.00 10.12 9.94 9.75 10.22 10.08 9.90 9.00 8.75 8.50 8.25 8.12 8.00 8.06 7.93 8.14 8.00

8.062 8.031 8.000 6.776 6.750 6.733 12.670 12.515 12.365 12.320 12.230 12.190 12.155 12.105 12.080 12.040 12.000 10.014 10.000 8.077 8.042 8.000 6.565 6.525 6.500 10.415 10.345 10.275 10.195 10.170 10.117 10.075 10.028 10.000 8.022 7.990 7.964 5.799 5.762 5.750 8.287 8.222 8.117 8.077 8.027 8.000 6.540 6.500 5.268 5.250

Column core section.

0.658 0.593 0.528 0.513 0.453 0.383 1.736 1.486 1.236 1.106 0.986 0.921 0.856 0.796 0.736 0.671 0.606 0.641 0.576 0.641 0.576 0.516 0.540 0.465 0.400 1.248 1.118 0.998 0.868 0.808 0.748 0.683 0.618 0.558 0.618 0.528 0.433 0.500 0.430 0.340 0.933 0.808 0.683 0.558 0.493 0.433 0.463 0.398 0.378 0.308


Web Thickness (in.)

I (in.4 )

0.370 0.339 0.308 0.313 0.287 0.270 1.060 0.905 0.755 0.710 0.620 0.580 0.545 0.495 0.470 0.430 0.390 0.359 0.345 0.371 0.336 0.294 0.305 0.265 0.240 0.755 0.685 0.615 0.535 0.510 0.457 0.415 0.368 0.340 0.350 0.318 0.292 0.289 0.252 0.240 0.575 0.510 0.405 0.365 0.315 0.288 0.285 0.245 0.248 0.230

542.1 484.9 429.0 385.3 339.2 289.6 1892.5 1541.8 1221.2 1071.7 930.7 858.5 788.9 723.3 663.0 597.4 533.4 476.1 426.2 394.5 350.8 310.1 280.8 238.4 204.1 718.7 625.0 542.4 457.2 420.7 382.5 343.7 305.7 272.9 248.6 209.7 170.9 157.3 133.2 106.3 271.8 227.3 183.7 146.3 126.5 109.7 97.8 82.5 69.2 56.4

77.8 70.2 62.7 54.6 48.5 41.8 263.2 222.2 182.5 163.4 144.5 134.7 125.0 115.7 107.1 97.5 88.0 78.1 70.7 64.7 58.2 51.9 45.9 39.4 34.1 126.3 112.4 99.7 86.1 80.1 73.7 67.1 60.4 54.6 49.1 42.2 35.0 30.8 26.4 21.5 60.4 52.0 43.2 35.5 31.1 27.4 24.3 20.8 17.0 14.1

Axis Y–Y S (in.3 )

r (in.)

I (in.4 )

5.90 5.86 5.82 5.87 5.83 5.73 5.82 5.70 5.59 5.51 5.46 5.43 5.40 5.38 5.34 5.31 5.28 5.28 5.23 5.18 5.15 5.13 5.15 5.11 5.06 4.67 4.61 4.55 4.49 4.46 4.44 4.41 4.39 4.35 4.33 4.27 4.20 4.29 4.26 4.14 3.71 3.65 3.61 3.53 3.50 3.47 3.45 3.42 3.43 3.36

57.5 51.3 45.1 24.6 21.3 17.5 589.7 486.2 389.9 345.1 300.9 278.2 256.4 235.5 216.4 195.3 174.6 107.4 96.1 56.4 50.0 44.1 23.7 19.8 16.6 235.4 206.6 180.6 153.4 141.8 129.2 116.5 103.9 93.0 53.2 44.9 36.5 15.2 12.7 9.7 88.6 74.9 60.9 49.0 42.5 37.0 21.6 18.2 8.5 6.7

14.3 12.8 11.3 7.3 6.3 5.2 93.1 77.7 63.1 56.0 49.2 45.7 42.2 38.9 35.8 32.4 29.1 21.4 19.2 14.0 12.4 11.0 7.2 6.1 5.1 45.2 39.9 35.2 30.1 27.9 25.5 23.1 20.7 18.6 13.3 11.2 9.2 5.2 4.4 3.4 21.4 18.2 15.0 12.1 10.6 9.2 6.6 5.6 3.2 2.6

r (in.) 1.92 1.91 1.89 1.49 1.46 1.41 3.25 3.20 3.16 3.13 3.11 3.09 3.08 3.07 3.05 3.04 3.02 2.51 2.48 1.96 1.94 1.94 1.50 1.47 1.44 2.67 2.65 2.63 2.60 2.59 2.58 2.57 2.56 2.54 2.00 1.98 1.94 1.34 1.31 1.25 2.12 2.10 2.08 2.04 2.03 2.01 1.62 1.61 1.20 1.16

110


Table 77

Properties of American Standard Beams

Nominal Size (in.)


Area (in.2 )

Depth (in.)

Width (in.)

24 × 7 78

120.0 105.9 100.0 90.0 79.9 95.0 85.0 75.0 65.4 70.0 54.7 50.0 42.9 50.0 40.8 35.0 31.8 35.0 25.4 23.0 18.4 20.0 15.3 17.25 12.5 14.75 10.0 9.5 7.7 7.5 5.7

35.13 30.98 29.25 26.30 23.33 27.74 24.80 21.90 19.08 20.46 15.94 14.59 12.49 14.57 11.84 10.20 9.26 10.22 7.38 6.71 5.34 5.83 4.43 5.02 3.61 4.29 2.87 2.76 2.21 2.17 1.64

24.00 24.00 24.00 24.00 24.00 20.00 20.00 20.00 20.00 18.00 18.00 15.00 15.00 12.00 12.00 12.00 12.00 10.00 10.00 8.00 8.00 7.00 7.00 6.00 6.00 5.00 5.00 4.00 4.00 3.00 3.00

8.048 7.875 7.247 7.124 7.000 7.200 7.053 6.391 6.250 6.251 6.000 5.640 5.500 5.477 5.250 5.078 5.000 4.944 4.660 4.171 4.000 3.860 3.660 3.565 3.330 3.284 3.000 2.796 2.660 2.509 2.330

24 × 7 20 × 7 20 × 6 14 18 × 6 15 × 5 12 12 × 5 14 12 × 5 10 × 4 58 8×4 7 × 3 58 6 × 3 38 5×3 4 × 2 58 3 × 2 38

Flange Thickness (in.) 1.102 1.102 0.871 0.871 0.871 0.916 0.916 0.789 0.789 0.691 0.691 0.622 0.622 0.659 0.659 0.544 0.544 0.491 0.491 0.425 0.425 0.392 0.392 0.359 0.359 0.326 0.326 0.293 0.293 0.260 0.260


Web Thickness (in.)

I (in.4 )

0.798 0.625 0.747 0.624 0.500 0.800 0.653 0.641 0.500 0.711 0.460 0.550 0.410 0.687 0.460 0.428 0.350 0.594 0.310 0.441 0.270 0.450 0.250 0.465 0.230 0.494 0.210 0.326 0.190 0.349 0.170

3010.8 2811.5 2371.8 2230.1 2087.2 1599.7 1501.7 1263.5 1169.5 917.5 795.5 481.1 441.8 301.6 268.9 227.0 215.8 145.8 122.1 64.2 56.9 41.9 36.2 26.0 21.8 15.0 12.1 6.7 6.0 2.9 2.5

250.9 234.3 197.6 185.8 173.9 160.0 150.2 126.3 116.9 101.9 88.4 64.2 58.9 50.3 44.8 37.8 36.0 29.2 24.4 16.0 14.2 12.0 10.4 8.7 7.3 6.0 4.8 3.3 3.0 1.9 1.7


r (in.)

I (in.4 )

9.26 9.53 9.05 9.21 9.46 7.59 7.78 7.60 7.83 6.70 7.07 5.74 5.95 4.55 4.77 4.72 4.83 3.78 4.07 3.09 3.26 2.68 2.86 2.28 2.46 1.87 2.05 1.56 1.64 1.15 1.23

84.9 78.9 48.4 45.5 42.9 50.5 47.0 30.1 27.9 24.5 21.2 16.0 14.6 16.0 13.8 10.0 9.5 8.5 6.9 4.4 3.8 3.1 2.7 2.3 1.8 1.7 1.2 0.91 0.77 0.59 0.46

21.1 20.0 13.4 12.8 12.2 14.0 13.3 9.4 8.9 7.8 7.1 5.7 5.3 5.8 5.3 3.9 3.8 3.4 3.0 2.1 1.9 1.6 1.5 1.3 1.1 1.0 0.82 0.65 0.58 0.47 0.40

1.56 1.60 1.29 1.32 1.36 1.35 1.38 1.17 1.21 1.09 1.15 1.05 1.08 1.05 1.08 0.99 1.01 0.91 0.97 0.81 0.84 0.74 0.78 0.68 0.72 0.63 0.65 0.58 0.59 0.52 0.53


111

Table 78 Properties of American Standard Channels

Nominal Weight Size per Foot (in.) (lb) 18 × 4a

15 × 3 83 12 × 3 10 × 2 85

9 × 2 21 8 × 2 41 7 × 2 81 6×2 5 × 1 43 4 × 1 85 3 × 1 21

a

58.0 51.9 45.8 42.7 50.0 40.0 33.9 30.0 25.0 20.7 30.0 25.0 20.0 15.3 20.0 15.0 13.4 18.75 13.75 11.5 14.75 12.25 9.8 13.0 10.5 8.2 9.0 6.7 7.25 5.4 6.0 5.0 4.1

Area (in.2 )

Flange Web Depth Width Average Thickness Thickness (in.) (in.) (in.) (in.)

I (in.4 )

S (in.3 )

r (in.)

I S (in.4 ) (in.3 )

r (in.)

x (in.)

16.98 15.18 13.38 12.48 14.64 11.70 9.90 8.79 7.32 6.03 8.80 7.33 5.86 4.47 5.86 4.39 3.89 5.49 4.02 3.36 4.32 3.58 2.85 3.81 3.07 2.39 2.63 1.95 2.12 1.56 1.75 1.46 1.19

18.00 18.00 18.00 18.00 15.00 15.00 15.00 12.00 12.00 12.00 10.00 10.00 10.00 10.00 9.00 9.00 9.00 8.00 8.00 8.00 7.00 7.00 7.00 6.00 6.00 6.00 5.00 5.00 4.00 4.00 3.00 3.00 3.00

670.7 622.1 573.5 549.2 401.4 346.3 312.6 161.2 143.5 128.1 103.0 90.7 78.5 66.9 60.6 50.7 47.3 43.7 35.8 32.3 27.1 24.1 21.1 17.3 15.1 13.0 8.8 7.4 4.5 3.8 2.1 1.8 1.6

74.5 69.1 63.7 61.0 53.6 46.2 41.7 26.9 23.9 21.4 20.6 18.1 15.7 13.4 13.5 11.3 10.5 10.9 9.0 8.1 7.7 6.9 6.0 5.8 5.0 4.3 3.5 3.0 2.3 1.9 1.4 1.2 1.1

6.29 6.40 6.55 6.64 5.24 5.44 5.62 4.28 4.43 4.61 3.42 3.52 3.66 3.87 3.22 3.40 3.49 2.82 2.99 3.10 2.51 2.59 2.72 2.13 2.22 2.34 1.83 1.95 1.47 1.56 1.08 1.12 1.17

18.5 17.1 15.8 15.0 11.2 9.3 8.2 5.2 4.5 3.9 4.0 3.4 2.8 2.3 2.4 1.9 1.8 2.0 1.5 1.3 1.4 1.2 0.98 1.1 0.87 0.70 0.64 0.48 0.44 0.32 0.31 0.25 0.20

1.04 1.06 1.09 1.10 0.87 0.89 0.91 0.77 0.79 0.81 0.67 0.68 0.70 0.72 0.65 0.67 0.67 0.60 0.62 0.63 0.57 0.58 0.59 0.53 0.53 0.54 0.49 0.50 0.46 0.45 0.42 0.41 0.41

0.88 0.87 0.89 0.90 0.80 0.78 0.79 0.68 0.68 0.70 0.65 0.62 0.61 0.64 0.59 0.59 0.61 0.57 0.56 0.58 0.53 0.53 0.55 0.52 0.50 0.52 0.48 0.49 0.46 0.46 0.46 0.44 0.44

4.200 4.100 4.000 3.950 3.716 3.520 3.400 3.170 3.047 2.940 3.033 2.886 2.739 2.600 2.648 2.485 2.430 2.527 2.343 2.260 2.299 2.194 2.090 2.157 2.034 1.920 1.885 1.750 1.720 1.580 1.596 1.498 1.410

0.625 0.625 0.625 0.625 0.650 0.650 0.650 0.501 0.501 0.501 0.436 0.436 0.436 0.436 0.413 0.413 0.413 0.390 0.390 0.390 0.366 0.366 0.366 0.343 0.343 0.343 0.320 0.320 0.296 0.296 0.273 0.273 0.273

Car and Shipbuilding Channel; not an American standard.

0.700 0.600 0.500 0.450 0.716 0.520 0.400 0.510 0.387 0.280 0.673 0.526 0.379 0.240 0.448 0.285 0.230 0.487 0.303 0.220 0.419 0.314 0.210 0.437 0.314 0.200 0.325 0.190 0.320 0.180 0.356 0.258 0.170

Axis X–X

Axis Y–Y

5.6 5.3 5.1 4.9 3.8 3.4 3.2 2.1 1.9 1.7 1.7 1.5 1.3 1.2 1.2 1.0 0.97 1.0 0.86 0.79 0.79 0.71 0.63 0.65 0.57 0.50 0.45 0.38 0.35 0.29 0.27 0.24 0.21

112 Table 79

Size (in.) 8×8

6×6

5×5

4×4

3 12 × 3 21

3×3

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Properties of Angles with Equal Legs

Thickness (in.) 1 18 1 7 8 3 4 5 8 9 16 1 2

1 7 8 3 4 5 8 9 16 1 2 7 16 3 8 5 16 7 8 3 4 5 8 1 2 7 16 3 8 5 16 3 4 5 8 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16

Axis X–X and Axis Y–Y S r (in.3 ) (in.)

x or y (in.)

Axis Z–Z r (in.)

2.42 2.44 2.45

2.41 2.37 2.32

1.56 1.56 1.57

12.2

2.47

2.28

1.57

10.3

2.49

2.23

1.58

54.1

9.3

2.50

2.21

1.58

7.75 11.00 9.73

48.6 35.5 31.9

8.4 8.6 7.6

2.50 1.80 1.81

2.19 1.86 1.82

1.59 1.17 1.17

28.7

8.44

28.2

6.7

1.83

1.78

1.17

24.2

7.11

24.2

5.7

1.84

1.73

1.18

21.9

6.43

22.1

5.1

1.85

1.71

1.18

19.6

5.75

19.9

4.6

1.86

1.68

1.18

17.2

5.06

17.7

4.1

1.87

1.66

1.19

14.9

4.36

15.4

3.5

1.88

1.64

1.19

12.5

3.66

13.0

3.0

1.89

1.61

1.19

27.2

7.98

17.8

5.2

1.49

1.57

0.97

23.6

6.94

15.7

4.5

1.51

1.52

0.97

20.0

5.86

13.6

3.9

1.52

1.48

0.98

16.2

4.75

11.3

3.2

1.54

1.43

0.98

14.3

4.18

10.0

2.8

1.55

1.41

0.98

12.3

3.61

8.7

2.4

1.56

1.39

0.99

10.3

3.03

7.4

2.0

1.57

1.37

0.99

18.5

5.44

7.7

2.8

1.19

1.27

0.78

15.7

4.61

6.7

2.4

1.20

1.23

0.78

12.8

3.75

5.6

2.0

1.22

1.18

0.78

11.3

3.31

5.0

1.8

1.23

1.16

0.78

9.8

2.86

4.4

1.5

1.23

1.14

0.79

8.2

2.40

3.7

1.3

1.24

1.12

0.79

6.6

1.94

3.0

1.1

1.25

1.09

0.80

11.1

3.25

3.6

1.5

1.06

1.06

0.68

9.8

2.87

3.3

1.3

1.07

1.04

0.68

8.5

2.48

2.9

1.2

1.07

1.01

0.69

7.2

2.09

2.5

0.98

1.08

0.99

0.69

5.8

1.69

2.0

0.79

1.09

0.97

0.69

9.4

2.75

2.2

1.1

0.90

0.93

0.58

8.3

2.43

2.0

0.95

0.91

0.91

0.58

7.2

2.11

1.8

0.83

0.91

0.89

0.58

6.1

1.78

1.5

0.71

0.92

0.87

0.59


Area (in.2 )

56.9 51.0 45.0

16.73 15.00 13.23

98.0 89.0 79.6

17.5 15.8 14.0

38.9

11.44

69.7

32.7

9.61

59.4

29.6

8.68

26.4 37.4 33.1

I (in.4 )


113

Table 79 (Continued ) Size (in.)

Thickness (in.) 1 4 3 16 1 2 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8 1 4 3 16 1 8

2 21 × 2 12

2×2

1 43 × 1 34

1 21 × 1 12

1 41 × 1 14

1×1


Axis X–X and Axis Y–Y S r (in.3 ) (in.)

x or y (in.)

Axis Z–Z r (in.)

Area (in.2 )

I (in.4 )

4.9

1.44

1.2

0.58

0.93

0.84

0.59

3.71

1.09

0.96

0.44

0.94

0.82

0.59

7.7

2.25

1.2

0.72

0.74

0.81

0.49

5.9

1.73

0.98

0.57

0.75

0.76

0.49

5.0

1.47

0.85

0.48

0.76

0.74

0.49

4.1

1.19

0.70

0.39

0.77

0.72

0.49

3.07

0.90

0.55

0.30

0.78

0.69

0.49

4.7

1.36

0.48

0.35

0.59

0.64

0.39

3.92

1.15

0.42

0.30

0.60

0.61

0.39

3.19

0.94

0.35

0.25

0.61

0.59

0.39

2.44

0.71

0.27

0.19

0.62

0.57

0.39

1.65

0.48

0.19

0.13

0.63

0.55

0.40

2.77

0.81

0.23

0.19

0.53

0.53

0.34

2.12

0.62

0.18

0.14

0.54

0.51

0.34

1.44

0.42

0.13

0.10

0.55

0.48

0.35

2.34

0.69

0.14

0.13

0.45

0.47

0.29

1.80

0.53

0.11

0.10

0.46

0.44

0.29

1.23

0.36

0.08

0.07

0.47

0.42

0.30

1.92

0.56

0.08

0.09

0.37

0.40

0.24

1.48

0.43

0.06

0.07

0.38

0.38

0.24

1.01

0.30

0.04

0.05

0.38

0.36

0.25

1.49

0.44

0.04

0.06

0.29

0.34

0.20

1.16

0.34

0.03

0.04

0.30

0.32

0.19

0.80

0.23

0.02

0.03

0.30

0.30

0.20

Table 80 Properties of Angles with Unequal Legs

Size (in.) 9×4

Thickness (in.) 1 7 8 3 4 5 8 9 16 1 2


Area (in.2 )

I (in.4 )

Axis X–X S r (in.3 ) (in.)

y (in.)

I (in.4 )


x (in.)

Axis Z–Z r (in.) tan α

40.8

12.00

97.0

17.6

2.84

3.50

12.0

4.0

1.00

1.00

0.83

0.203

36.1

10.61

86.8

15.7

2.86

3.45

10.8

3.6

1.01

0.95

0.84

0.208

31.3

9.19

76.1

13.6

2.88

3.41

9.6

3.1

1.02

0.91

0.84

0.212

26.3

7.73

64.9

11.5

2.90

3.36

8.3

2.6

1.04

0.86

0.85

0.216

23.8

7.00

59.1

10.4

2.91

3.33

7.6

2.4

1.04

0.83

0.85

0.218

21.3

6.25

53.2

9.3

2.92

3.31

6.9

2.2

1.05

0.81

0.85

0.220

(Continues)

114 Table 80 Size (in.) 8×6

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Thickness (in.) 1 7 8 3 4 5 8 9 16 1 2 7 16

8×4

7×4

6×4

6 × 3 12

5 × 3 12

5×3

4 × 3 12

1 7 8 3 4 5 8 9 16 1 2 7 16 7 8 3 4 5 8 9 16 1 2 7 16 3 8 7 8 3 4 5 8 9 16 1 2 7 16 3 8 5 16 1 2 3 8 5 16 1 4 3 4 5 8 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 5 8 1 2


Area (in.2 )

I (in.4 )


y (in.)

I (in.4 )


x (in.)

Axis Z–Z r (in.) tan α

44.2

13.00

80.8

15.1

2.49

2.65

38.8

8.9

1.73

1.65

1.28

0.543

39.1

11.48

72.3

13.4

2.51

2.61

34.9

7.9

1.74

1.61

1.28

0.547

33.8

9.94

63.4

11.7

2.53

2.56

30.7

6.9

1.76

1.56

1.29

0.551

28.5

8.36

54.1

9.9

2.54

2.52

26.3

5.9

1.77

1.52

1.29

0.554

25.7

7.56

49.3

9.0

2.55

2.50

24.0

5.3

1.78

1.50

1.30

0.556

23.0

6.75

44.3

8.0

2.56

2.47

21.7

4.8

1.79

1.47

1.30

0.558

20.2

5.93

39.2

7.1

2.57

2.45

19.3

4.2

1.80

1.45

1.31

0.560

37.4 33.1

11.00 9.73

69.6 62.5

14.1 12.5

2.52 2.53

3.05 3.00

11.6 10.5

3.9 3.5

1.03 1.04

1.05 1.00

0.85 0.85

0.247 0.253

28.7

8.44

54.9

10.9

2.55

2.95

9.4

3.1

1.05

0.95

0.85

0.258

24.2

7.11

46.9

9.2

2.57

2.91

8.1

2.6

1.07

0.91

0.86

0.262

21.9

6.43

42.8

8.4

2.58

2.88

7.4

2.4

1.07

0.88

0.86

0.265

19.6

5.75

38.5

7.5

2.59

2.86

6.7

2.2

1.08

0.86

0.86

0.267

17.2

5.06

34.1

6.6

2.60

2.83

6.0

1.9

1.09

0.83

0.87

0.269

30.2

8.86

42.9

9.7

2.20

2.55

10.2

3.5

1.07

1.05

0.86

0.318

26.2

7.69

37.8

8.4

2.22

2.51

9.1

3.0

1.09

1.01

0.86

0.324

22.1

6.48

32.4

7.1

2.24

2.46

7.8

2.6

1.10

0.96

0.86

0.329

20.0

5.87

29.6

6.5

2.24

2.44

7.2

2.4

1.11

0.94

0.87

0.332

17.9

5.25

26.7

5.8

2.25

2.42

6.5

2.1

1.11

0.92

0.87

0.335

15.8

4.62

23.7

5.1

2.26

2.39

5.8

1.9

1.12

0.89

0.88

0.337

13.6

3.98

20.6

4.4

2.27

2.37

5.1

1.6

1.13

0.87

0.88

0.339

27.2

7.98

27.7

7.2

1.86

2.12

9.8

3.4

1.11

1.12

0.86

0.421

23.6

6.94

24.5

6.3

1.88

2.08

8.7

3.0

1.12

1.08

0.86

0.428

20.0

5.86

21.1

5.3

1.90

2.03

7.5

2.5

1.13

1.03

0.86

0.435

18.1

5.31

19.3

4.8

1.90

2.01

6.9

2.3

1.14

1.01

0.87

0.438

16.2

4.75

17.4

4.3

1.91

1.99

6.3

2.1

1.15

0.99

0.87

0.440

14.3

4.18

15.5

3.8

1.92

1.96

5.6

1.9

1.16

0.96

0.87

0.443

12.3

3.61

13.5

3.3

1.93

1.94

4.9

1.6

1.17

0.94

0.88

0.446

10.3

3.03

11.4

2.8

1.94

1.92

4.2

1.4

1.17

0.92

0.88

0.449

15.3

4.50

16.6

4.2

1.92

2.08

4.3

1.6

0.97

0.83

0.76

0.344

11.7

3.42

12.9

3.2

1.94

2.04

3.3

1.2

0.99

0.79

0.77

0.350

9.8

2.87

10.9

2.7

1.95

2.01

2.9

1.0

1.00

0.76

0.77

0.352

7.9

2.31

8.9

2.2

1.96

1.99

2.3

0.85

1.01

0.74

0.78

0.355

19.8

5.81

13.9

4.3

1.55

1.75

5.6

2.2

0.98

1.00

0.75

0.464

16.8

4.92

12.0

3.7

1.56

1.70

4.8

1.9

0.99

0.95

0.75

0.472

13.6

4.00

10.0

3.0

1.58

1.66

4.1

1.6

1.01

0.91

0.75

0.479

12.0

3.53

8.9

2.6

1.59

1.63

3.6

1.4

1.01

0.88

0.76

0.482

10.4

3.05

7.8

2.3

1.60

1.61

3.2

1.2

1.02

0.86

0.76

0.486

8.7

2.56

6.6

1.9

1.61

1.59

2.7

1.0

1.03

0.84

0.76

0.489

7.0

2.06

5.4

1.6

1.61

1.56

2.2

0.83

1.04

0.81

0.76

0.492

12.8

3.75

9.5

2.9

1.59

1.75

2.6

1.1

0.83

0.75

0.65

0.357

11.3

3.31

8.4

2.6

1.60

1.73

2.3

1.0

0.84

0.73

0.65

0.361

9.8

2.86

7.4

2.2

1.61

1.70

2.0

0.89

0.84

0.70

0.65

0.364

8.2

2.40

6.3

1.9

1.61

1.68

1.8

0.75

0.85

0.68

0.66

0.368

6.6

1.94

5.1

1.5

1.62

1.66

1.4

0.61

0.86

0.66

0.66

0.371

14.7

4.30

6.4

2.4

1.22

1.29

4.5

1.8

1.03

1.04

0.72

0.745

11.9

3.50

5.3

1.9

1.23

1.25

3.8

1.5

1.04

1.00

0.72

0.750


115

Table 80 (Continued ) Size (in.)

4×3

3 12 × 3

3 12 × 2 12

3 × 2 12

3×2

2 12 × 2

2 12 × 1 12

2 × 1 12

1 34 × 1 14

Thickness (in.) 7 16 3 8 5 16 1 4 5 8 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 1 2 7 16 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 3 8 5 16 1 4 3 16 1 4 3 16 1 8 1 4 3 16 1 8


Area (in.2 )

I (in.4 )


y (in.)

I (in.4 )


x (in.)

Axis Z–Z r tan α (in.)

10.6

3.09

4.8

1.7

1.24

1.23

3.4

1.4

1.05

0.98

0.72

0.753

9.1

2.67

4.2

1.5

1.25

1.21

3.0

1.2

1.06

0.96

0.73

0.755

7.7

2.25

3.6

1.3

1.26

1.18

2.6

1.0

1.07

0.93

0.73

0.757

6.2

1.81

2.9

1.0

1.27

1.16

2.1

0.81

1.07

0.91

0.73

0.759

13.6

3.98

6.0

2.3

1.23

1.37

2.9

1.4

0.85

0.87

0.64

0.534

11.1

3.25

5.1

1.9

1.25

1.33

2.4

1.1

0.86

0.83

0.64

0.543

9.8

2.87

4.5

1.7

1.25

1.30

2.2

1.0

0.87

0.80

0.64

0.547

8.5

2.48

4.0

1.5

1.26

1.28

1.9

0.87

0.88

0.78

0.64

0.551

7.2

2.09

3.4

1.2

1.27

1.26

1.7

0.73

0.89

0.76

0.65

0.554

5.8

1.69

2.8

1.0

1.28

1.24

1.4

0.60

0.90

0.74

0.65

0.558

10.2

3.00

3.5

1.5

1.07

1.13

2.3

1.1

0.88

0.88

0.62

0.714

9.1

2.65

3.1

1.3

1.08

1.10

2.1

0.98

0.89

0.85

0.62

0.718

7.9

2.30

2.7

1.1

1.09

1.08

1.9

0.85

0.90

0.83

0.62

0.721

6.6

1.93

2.3

0.95

1.10

1.06

1.6

0.72

0.90

0.81

0.63

0.724

5.4

1.56

1.9

0.78

1.11

1.04

1.3

0.59

0.91

0.79

0.63

0.727

9.4

2.75

3.2

1.4

1.09

1.20

1.4

0.76

0.70

0.70

0.53

0.486

8.3

2.43

2.9

1.3

1.09

1.18

1.2

0.68

0.71

0.68

0.54

0.491

7.2

2.11

2.6

1.1

1.10

1.16

1.1

0.59

0.72

0.66

0.54

0.496

6.1

1.78

2.2

0.93

1.11

1.14

0.94

0.50

0.73

0.64

0.54

0.501

4.9

1.44

1.8

0.75

1.12

1.11

0.78

0.41

0.74

0.61

0.54

0.506

8.5

2.50

2.1

1.0

0.91

1.00

1.3

0.74

0.72

0.75

0.52

0.667

7.6

2.21

1.9

0.93

0.92

0.98

1.2

0.66

0.73

0.73

0.52

0.672

6.6

1.92

1.7

0.81

0.93

0.96

1.0

0.58

0.74

0.71

0.52

0.676

5.6

1.62

1.4

0.69

0.94

0.93

0.90

0.49

0.74

0.68

0.53

0.680

4.5

1.31

1.2

0.56

0.95

0.91

0.74

0.40

0.75

0.66

0.53

0.684

7.7

2.25

1.9

1.0

0.92

1.08

0.67

0.47

0.55

0.58

0.43

0.414

6.8

2.00

1.7

0.89

0.93

1.06

0.61

0.42

0.55

0.56

0.43

0.421

5.9

1.73

1.5

0.78

0.94

1.04

0.54

0.37

0.56

0.54

0.43

0.428

5.0

1.47

1.3

0.66

0.95

1.02

0.47

0.32

0.57

0.52

0.43

0.435

4.1

1.19

1.1

0.54

0.95

0.99

0.39

0.26

0.57

0.49

0.43

0.440

3.07

0.90

0.84

0.41

0.97

0.97

0.31

0.20

0.58

0.47

0.44

0.446

5.3

1.55

0.91

0.55

0.77

0.83

0.51

0.36

0.58

0.58

0.42

0.614

4.5

1.31

0.79

0.47

0.78

0.81

0.45

0.31

0.58

0.56

0.42

0.620

3.62

1.06

0.65

0.38

0.78

0.79

0.37

0.25

0.59

0.54

0.42

0.626 0.631

2.75

0.81

0.51

0.29

0.79

0.76

0.29

0.20

0.60

0.51

0.43

4.7

1.36

0.82

0.52

0.78

0.92

0.22

0.20

0.40

0.42

0.32

0.340

3.92

1.15

0.71

0.44

0.79

0.90

0.19

0.17

0.41

0.40

0.32

0.349

3.19

0.94

0.59

0.36

0.79

0.88

0.16

0.14

0.41

0.38

0.32

0.357

2.44

0.72

0.46

0.28

0.80

0.85

0.13

0.11

0.42

0.35

0.33

0.364

2.77

0.81

0.32

0.24

0.62

0.66

0.15

0.14

0.43

0.41

0.32

0.543

2.12

0.62

0.25

0.18

0.63

0.64

0.12

0.11

0.44

0.39

0.32

0.551

1.44

0.42

0.17

0.13

0.64

0.62

0.09

0.08

0.45

0.37

0.33

0.558

2.34

0.69

0.20

0.18

0.54

0.60

0.09

0.10

0.35

0.35

0.27

0.486

1.80

0.53

0.16

0.14

0.55

0.58

0.07

0.08

0.36

0.33

0.27

0.496

1.23

0.36

0.11

0.09

0.56

0.56

0.05

0.05

0.37

0.31

0.27

0.506

116 Table 81

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Properties and Dimensions of Tees

Tees are seldom used as structural framing members. When so used they are generally employed on short spans in flexure. This table lists a few selected sizes, the range of whose section moduli will cover all ordinary conditions. For sizes not listed, the catalogs of the respective rolling mills should be consulted. Flange Section Number ST 18 WFa

ST 18 WF

ST 16 WF ST 16 WF ST 15 WF ST 15 WF

ST 13 WF ST 13 WF ST 12 WF ST 12 WF ST 12 WF ST 10 WF ST 10 WFa

Weight per Foot (lb) 150 140 130 122.5 115 97 91 85 80 75 120 110 100 76 70.5 65 105 95 86 66 62 58.0 54.0 88.5 80 72.5 57 51 47 80 72.5 65 60 55 50 47 42 38 71 63.5 56 48 41

Area (in.2 )

Depth of Tee (in.)

Width (in.)

44.09 41.16 38.28 36.01 33.86 28.56 26.77 24.99 23.54 22.08 35.26 32.36 29.40 22.35 20.76 19.13 30.89 27.95 25.32 19.41 18.22 17.07 15.88 26.05 23.72 21.34 16.77 15.01 13.83 23.54 21.31 19.11 17.64 16.18 14.71 13.81 12.35 11.18 20.88 18.67 16.47 14.11 12.05

18.36 18.25 18.12 18.03 17.94 18.24 18.16 18.08 18.00 17.92 16.75 16.63 16.50 16.75 16.66 16.55 15.19 15.06 14.94 15.15 15.08 15.00 14.91 13.66 13.54 13.44 13.64 13.53 13.45 12.36 12.24 12.13 12.16 12.08 12.00 12.15 12.04 11.95 10.73 10.62 10.50 10.57 10.43

16.655 16.595 16.555 16.512 16.475 12.117 12.072 12.027 12.000 11.972 15.865 15.810 15.750 11.565 11.535 11.510 15.105 15.040 14.985 10.551 10.521 10.500 10.484 14.090 14.023 13.965 10.070 10.018 9.990 14.091 14.043 14.000 12.088 12.042 12.000 9.061 9.015 8.985 13.132 13.061 13.000 9.038 8.962

Average Stem Thickness Thickness (in.) (in.) 1.680 1.570 1.440 1.350 1.260 1.260 1.180 1.100 1.020 0.940 1.400 1.275 1.150 1.055 0.960 0.855 1.315 1.185 1.065 1.000 0.930 0.850 0.760 1.190 1.075 0.975 0.932 0.827 0.747 1.135 1.020 0.900 0.930 0.855 0.775 0.872 0.772 0.682 1.095 0.985 0.865 0.935 0.795

0.945 0.885 0.845 0.802 0.765 0.770 0.725 0.680 0.653 0.625 0.830 0.775 0.715 0.635 0.603 0.580 0.775 0.710 0.655 0.615 0.585 0.564 0.548 0.725 0.658 0.600 0.570 0.518 0.490 0.656 0.608 0.565 0.556 0.510 0.468 0.516 0.470 0.440 0.659 0.588 0.527 0.575 0.499

Axis X –X

Axis Y –Y

I (in.4 )

S (in.3 )

r (in.)

y (in.)

I (in.4 )

S (in.3 )

r (in.)

1222.7 1133.3 1059.2 994.3 935.8 904.0 844.0 784.7 741.0 696.7 822.5 754.1 683.6 591.9 551.8 513.0 578.0 520.4 471.0 420.7 394.8 371.8 349.5 391.8 351.4 316.3 288.9 257.7 238.5 271.6 246.2 222.6 213.6 195.2 176.7 185.9 165.9 151.1 177.3 155.8 136.4 137.1 115.4

85.9 79.9 75.4 71.1 67.2 67.3 63.0 58.8 56.0 53.0 63.2 58.4 53.3 47.4 44.7 42.1 48.7 44.1 40.2 37.4 35.3 33.6 32.1 36.7 33.1 29.9 28.3 25.4 23.7 27.6 25.2 23.1 22.4 20.5 18.7 20.3 18.3 16.9 20.8 18.3 16.2 17.1 14.5

5.27 5.25 5.26 5.25 5.26 5.63 5.61 5.60 5.61 5.62 4.83 4.83 4.82 5.15 5.16 5.18 4.33 4.31 4.31 4.66 4.65 4.67 4.69 3.88 3.87 3.85 4.15 4.14 4.15 3.40 3.40 3.41 3.48 3.47 3.46 3.67 3.66 3.68 2.91 2.89 2.88 3.11 3.09

4.13 4.07 4.07 4.04 4.02 4.81 4.77 4.74 4.76 4.79 3.73 3.71 3.67 4.26 4.30 4.37 3.31 3.26 3.23 3.90 3.90 3.94 4.03 2.97 2.91 2.85 3.42 3.39 3.41 2.51 2.48 2.47 2.62 2.57 2.54 2.99 2.97 3.00 2.18 2.11 2.06 2.55 2.48

612.6 563.7 510.3 472.3 435.5 177.7 163.9 150.3 137.7 125.2 437.2 391.2 345.8 128.1 114.9 100.7 354.0 312.3 275.1 92.5 84.8 76.6 67.6 259.4 229.0 203.5 74.8 64.8 57.5 246.3 217.1 187.6 127.0 114.5 101.8 51.1 44.2 38.3 193.0 169.3 144.8 54.7 44.8

73.6 67.9 61.6 57.2 52.9 29.3 27.1 25.0 22.9 20.9 55.1 49.5 43.9 22.1 19.9 17.5 46.9 41.5 36.7 17.5 16.1 14.6 12.9 36.8 32.7 29.1 14.9 12.9 11.5 35.0 30.9 26.8 21.0 19.0 17.0 11.3 9.8 8.5 29.4 25.9 22.3 12.1 10.0

3.73 3.70 3.65 3.62 3.59 2.49 2.47 2.45 2.42 2.38 3.52 3.48 3.43 2.39 2.35 2.29 3.38 3.34 3.30 2.18 2.16 2.12 2.06 3.16 3.12 3.09 2.11 2.08 2.04 3.23 3.19 3.13 2.68 2.66 2.63 1.92 1.89 1.85 3.04 3.01 2.96 1.97 1.93


117

Table 81 (Continued ) Flange Section Number ST 10 WF ST 9 WF ST 9 WF

ST 9 WF ST 8 WF ST 8 WF

ST 8 WF

ST 7 WF

ST 7 WF

ST 7 WF ST 7 WF ST 7 WF ST 7 WFa ST 6 WF

Weight per Foot (lb) 36.5 34 31 57 52.5 48 42.5 38.5 35 32 30 27.5 25 48 44 39 35.5 32 29 25 22.5 20 18 105.5 101 96.5 92 88 83.5 79 75 71 68 63.5 59.5 55.5 51.5 47.5 43.5 42 39 37 34 30.5 26.5 24 21.5 19 17 15 80.5 66.5 60 53 49.5

Area (in.2 )

Depth of Tee (in.)

Width (in.)

10.73 10.01 9.12 16.77 15.43 14.11 12.49 11.32 10.28 9.40 8.82 8.09 7.35 14.11 12.94 11.46 10.43 9.40 8.52 7.35 6.62 5.88 5.30 31.04 29.70 28.36 27.04 25.87 24.55 23.24 22.04 20.92 19.99 18.67 17.49 16.33 15.13 13.97 12.78 12.36 11.47 10.88 10.00 8.97 7.79 7.06 6.32 5.59 5.00 4.41 23.69 19.56 17.65 15.59 14.54

10.62 10.57 10.49 9.24 9.16 9.08 9.16 9.08 9.00 8.94 9.12 9.06 9.00 8.16 8.08 8.16 8.08 8.00 7.93 8.13 8.06 8.00 7.93 7.88 7.82 7.75 7.69 7.63 7.56 7.50 7.44 7.38 7.38 7.31 7.25 7.19 7.13 7.06 7.00 7.09 7.03 7.10 7.03 6.96 6.97 6.91 6.84 7.06 7.00 6.93 6.94 6.69 6.56 6.44 6.38

8.295 8.270 8.240 11.833 11.792 11.750 8.838 8.787 8.750 8.715 7.558 7.532 7.500 11.533 11.502 8.586 8.543 8.500 8.464 7.073 7.039 7.000 6.992 15.800 15.750 15.710 15.660 15.640 15.600 15.550 15.515 15.500 14.740 14.690 14.650 14.620 14.575 14.545 14.5 12.023 12.000 10.072 10.040 10.000 8.062 8.031 8.000 6.776 6.750 6.733 12.515 12.365 12.320 12.230 12.190

Average Stem Thickness Thickness (in.) (in.) 0.740 0.685 0.615 0.991 0.911 0.831 0.911 0.831 0.751 0.686 0.695 0.630 0.570 0.875 0.795 0.875 0.795 0.715 0.645 0.628 0.563 0.503 0.428 1.563 1.503 1.438 1.378 1.313 1.248 1.188 1.128 1.063 1.063 0.998 0.938 0.873 0.813 0.748 0.688 0.778 0.718 0.783 0.718 0.643 0.658 0.593 0.528 0.513 0.453 0.383 1.486 1.236 1.106 0.986 0.921

0.455 0.430 0.400 0.595 0.554 0.512 0.526 0.475 0.438 0.403 0.416 0.390 0.358 0.535 0.504 0.529 0.486 0.443 0.407 0.380 0.346 0.307 0.299 0.980 0.930 0.890 0.840 0.820 0.780 0.730 0.695 0.680 0.660 0.610 0.570 0.540 0.495 0.465 0.420 0.451 0.428 0.450 0.418 0.378 0.370 0.339 0.308 0.313 0.287 0.270 0.905 0.755 0.710 0.620 0.580

Axis X –X

Axis Y –Y

I (in.4 )

S (in.3 )

r (in.)

y (in.)

I (in.4 )

S (in.3 )

r (in.)

110.2 102.8 93.7 102.6 93.9 85.3 84.4 75.3 68.1 61.8 64.8 59.6 53.9 64.7 59.5 60.0 54.0 48.3 43.6 42.2 37.8 33.2 30.7 102.2 95.7 90.1 83.9 80.2 75.0 69.3 64.9 62.1 60.0 54.7 50.4 46.7 42.4 39.1 34.9 37.4 34.8 36.1 33.0 29.2 27.7 24.9 22.2 23.5 21.1 19.0 62.6 48.4 43.4 36.7 33.7

13.7 12.9 11.9 13.9 12.8 11.7 11.9 10.6 9.67 8.82 9.32 8.63 7.85 9.82 9.11 9.45 8.57 7.71 7.00 6.77 6.10 5.37 5.10 16.2 15.2 14.4 13.4 12.9 12.1 11.3 10.6 10.2 9.89 9.04 8.36 7.80 7.10 6.58 5.88 6.36 5.96 6.26 5.74 5.13 4.95 4.49 4.02 4.27 3.86 3.55 11.5 9.03 8.22 7.01 6.46

3.21 3.20 3.21 2.47 2.47 2.46 2.60 2.58 2.57 2.56 2.71 2.71 2.71 2.14 2.14 2.28 2.28 2.27 2.26 2.40 2.39 2.37 2.41 1.81 1.80 1.78 1.76 1.76 1.75 1.73 1.72 1.72 1.73 1.71 1.70 1.69 1.67 1.67 1.65 1.74 1.74 1.82 1.81 1.80 1.88 1.88 1.87 2.05 2.05 2.08 1.63 1.57 1.57 1.53 1.52

2.60 2.59 2.59 1.85 1.82 1.78 2.05 1.99 1.96 1.93 2.17 2.16 2.14 1.57 1.55 1.81 1.77 1.73 1.70 1.89 1.87 1.82 1.90 1.57 1.53 1.49 1.45 1.42 1.39 1.34 1.31 1.29 1.31 1.26 1.22 1.19 1.15 1.12 1.08 1.21 1.19 1.32 1.29 1.25 1.38 1.35 1.33 1.56 1.55 1.59 1.47 1.33 1.28 1.20 1.16

33.1 30.2 26.6 127.8 115.5 103.4 49.7 44.3 39.2 35.2 23.5 21.0 18.6 103.6 92.6 43.8 38.9 34.2 30.2 17.4 15.2 13.3 11.1 514.3 489.8 465.1 441.4 418.9 395.1 372.5 351.3 330.1 283.9 263.8 245.9 227.4 209.9 191.9 174.8 112.7 103.5 66.7 60.6 53.6 28.8 25.6 22.6 12.3 10.6 8.77 243.1 195.0 172.5 150.4 139.1

7.98 7.30 6.45 21.6 19.6 17.6 11.3 10.1 8.97 8.07 6.23 5.57 4.96 18.0 16.1 10.2 9.11 8.05 7.14 4.92 4.33 3.79 3.17 65.1 62.2 59.2 56.4 53.6 50.7 47.9 45.3 42.6 38.5 35.9 33.6 31.1 28.8 26.4 24.1 18.8 17.2 13.3 12.1 10.7 7.14 6.38 5.64 3.64 3.15 2.61 38.9 31.5 28.0 24.6 22.8

1.76 1.74 1.71 2.76 2.73 2.71 2.00 1.98 1.95 1.93 1.63 1.61 1.59 2.71 2.67 1.95 1.93 1.91 1.88 1.54 1.52 1.50 1.45 4.07 4.06 4.05 4.04 4.02 4.01 4.00 3.99 3.97 3.77 3.76 3.75 3.73 3.72 3.71 3.70 3.02 3.00 2.48 2.46 2.45 1.92 1.91 1.89 1.49 1.46 1.41 3.20 3.16 3.13 3.11 3.09

118 Table 81

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Flange

Section Number

ST 6 WF ST 6 WF ST 6 WF ST 6 WF ST 6 Ib ST 6 I ST 5 I ST 4 I ST 3.5 I ST 3 I ST 5 WF

ST 5 WF ST 5 WFa ST 4 WF

ST 4 WF ST 4 WF

Weight per Foot (lb) 46 42.5 39.5 36 32.5 29 26.5 25 22.5 20 18 15.5 13.5 7 25 20.4 17.5 15.9 17.5 12.7 11.5 9.2 10 7.65 8.625 6.25 56 50 44.5 38.5 36 33 30 27 24.5 22.5 19.5 16.5 14.5 12.5 10.5 33.5 29 24 20 17.5 15.5 14 12 10 8.5

Area (in.2 )

Depth of Tee (in.)

13.53 12.49 11.61 10.58 9.55 8.53 7.80 7.36 6.62 5.89 5.29 4.56 3.98 2.07 7.29 5.92 5.10 4.63 5.11 3.69 3.36 2.67 2.92 2.22 2.51 1.81 16.46 14.72 13.09 11.33 10.59 9.70 8.83 7.94 7.20 6.62 5.74 4.85 4.27 3.67 3.10 9.85 8.53 7.06 5.88 5.15 4.56 4.11 3.53 2.94 2.50

6.31 6.25 6.19 6.13 6.06 6.10 6.03 6.10 6.03 5.97 6.12 6.04 5.98 5.96 6.00 6.00 6.00 6.00 5.00 5.00 4.00 4.00 3.50 3.50 3.00 3.00 5.69 5.56 5.44 5.31 5.25 5.19 5.13 5.06 5.00 5.06 4.97 4.88 5.11 5.04 4.95 4.50 4.38 4.25 4.13 4.06 4.00 4.03 3.97 4.07 4.00

Axis X –X

Axis Y –Y

Width (in.)

Average Thickness (in.)

Stem Thickness (in.)

I (in.4 )

S (in.3 )

r (in.)

y (in.)

I (in.4 )

S (in.3 )

r (in.)

12.155 12.105 12.080 12.040 12.000 10.014 10.000 8.077 8.042 8.000 6.565 6.525 6.500 3.970 5.477 5.250 5.078 5.000 4.944 4.660 4.171 4.000 3.860 3.660 3.565 3.330 10.415 10.345 10.275 10.195 10.170 10.117 10.075 10.028 10.000 8.022 7.990 7.964 5.799 5.762 5.750 8.287 8.222 8.117 8.077 8.027 8.000 6.540 6.500 5.268 5.250

0.856 0.796 0.736 0.671 0.606 0.641 0.576 0.641 0.576 0.516 0.540 0.465 0.400 0.224 0.660 0.660 0.544 0.544 0.491 0.491 0.425 0.425 0.392 0.392 0.359 0.359 1.248 1.118 0.998 0.868 0.808 0.748 0.683 0.618 0.558 0.618 0.528 0.433 0.500 0.430 0.340 0.933 0.808 0.683 0.558 0.493 0.433 0.463 0.398 0.378 0.308

0.545 0.495 0.470 0.430 0.390 0.359 0.345 0.371 0.336 0.294 0.305 0.265 0.240 0.200 0.687 0.460 0.428 0.350 0.594 0.310 0.441 0.270 0.450 0.250 0.465 0.230 0.755 0.685 0.615 0.535 0.510 0.457 0.415 0.368 0.340 0.350 0.318 0.292 0.289 0.252 0.240 0.575 0.510 0.405 0.365 0.315 0.288 0.285 0.245 0.248 0.230

31.0 27.8 25.8 23.1 20.6 19.0 17.7 18.7 16.6 14.4 15.3 13.0 11.4 7.66 25.2 18.8 17.2 14.9 12.5 7.81 5.03 3.50 3.36 2.18 2.13 1.27 28.8 24.8 21.3 17.7 16.4 14.5 12.8 11.2 10.1 10.3 8.96 7.80 8.38 7.12 6.31 10.94 9.11 6.92 5.80 4.88 4.31 4.22 3.53 3.66 3.21

5.98 5.38 5.02 4.53 4.06 3.75 3.54 3.80 3.40 2.94 3.14 2.69 2.39 1.83 6.05 4.26 3.95 3.31 3.63 2.05 1.77 1.14 1.36 0.81 1.02 0.55 6.42 5.62 4.88 4.10 3.83 3.39 3.02 2.64 2.40 2.48 2.19 1.95 2.07 1.77 1.62 3.07 2.60 2.00 1.71 1.45 1.30 1.28 1.08 1.13 1.01

1.51 1.49 1.48 1.48 1.47 1.49 1.51 1.60 1.59 1.56 1.70 1.69 1.69 1.92 1.85 1.77 1.83 1.78 1.56 1.45 1.22 1.14 1.07 0.99 0.92 0.83 1.32 1.30 1.28 1.25 1.24 1.22 1.21 1.18 1.18 1.25 1.25 1.27 1.40 1.39 1.43 1.05 1.03 0.99 0.99 0.97 0.97 1.01 1.00 1.12 1.13

1.13 1.08 1.06 1.02 0.98 1.03 1.02 1.17 1.13 1.08 1.26 1.22 1.21 1.76 1.84 1.57 1.65 1.51 1.56 1.20 1.15 0.94 1.04 0.81 0.91 0.69 1.21 1.14 1.07 1.00 0.97 0.92 0.88 0.84 0.81 0.91 0.88 0.88 1.05 1.02 1.06 0.94 0.87 0.78 0.74 0.69 0.67 0.73 0.70 0.83 0.84

128.2 117.7 108.2 97.6 87.3 53.7 48.0 28.2 25.0 22.0 11.9 9.9 8.3 1.13 7.85 6.77 4.93 4.68 4.18 3.39 2.15 1.86 1.58 1.32 1.15 0.93 117.7 103.3 90.3 76.7 70.9 64.6 58.2 51.95 46.5 26.6 22.5 18.2 7.61 6.34 4.87 44.3 37.5 30.45 24.5 21.25 18.5 10.8 9.10 4.25 3.36

21.1 19.5 17.9 16.2 14.6 10.7 9.60 6.98 6.20 5.50 3.62 3.04 2.55 0.57 2.87 2.58 1.94 1.87 1.69 1.46 1.03 0.93 0.82 0.72 0.65 0.56 22.6 20.0 17.6 15.1 13.9 12.8 11.6 10.4 9.30 6.63 5.62 4.58 2.62 2.20 1.69 10.7 9.10 7.50 6.05 5.30 4.60 3.30 2.80 1.61 1.28

3.08 3.07 3.05 3.04 3.02 2.51 2.48 1.96 1.94 1.94 1.50 1.47 1.44 0.74 1.03 1.06 0.98 1.00 0.90 0.95 0.80 0.83 0.73 0.77 0.67 0.71 2.67 2.65 2.63 2.60 2.59 2.58 2.57 2.56 2.54 2.00 1.98 1.94 1.34 1.31 1.25 2.12 2.10 2.08 2.04 2.03 2.01 1.62 1.61 1.20 1.16


119

Table 81 (Continued ) Dimensions Nominal Size (in.)


Depth (in.)

Width Flange (in.)

Minimum Flange (in.)

Thickness Stem (in.)

Area (in.2 )

5 × 3 81 5×3

13.6 11.5

4.00 3.37

3 18 3

5 5

11.2 13.5 9.2 8.5

3.29 3.97 2.68 2.48

4 12 4 3 2 12

4 4 4 4

3×3 3×3 3 × 2 21

7.8 6.7 6.1

2.29 1.97 1.77

3 3 2 12

3 3 3

2 21 × 2 12 2 21 × 2 12 2 41 × 2 14

6.4 4.6 4.1

1.87 1.33 1.19

2 12 2 12 2 14

2 12 2 12 2 14

4.3 3.56

1.26 1.05

2 2

2 2

1 2 3 8 3 8 1 2 3 8 3 8 3 8 5 16 5 16 3 8 1 4 1 4 5 16 1 4

13 32 13 32 3 8 1 2 3 8 3 8 3 8 5 16 5 16 3 8 1 4 1 4 5 16 1 4

4 × 4 21 4×4 4×3 4 × 2 21

2×2 2×2 a b

Axis X –X

Axis Y –Y

I (in.4 )

S (in.3 )

r (in.)

y (in.)

I (in.4 )

S (in.3 )

r (in.)

2.7 2.4

1.1 1.1

0.82 0.84

0.76 0.76

5.2 3.9

2.1 1.6

1.14 1.10

6.3 5.7 2.0 1.2

2.0 2.0 0.90 0.62

1.39 1.20 0.86 0.69

1.31 1.18 0.78 0.62

2.1 2.8 2.1 2.1

1.1 1.4 1.1 1.0

0.80 0.84 0.89 0.92

1.84 1.61 0.94

0.86 0.74 0.51

0.89 0.90 0.73

0.88 0.85 0.68

0.89 0.75 0.75

0.60 0.50 0.50

0.63 0.62 0.65

1.0 0.74 0.52

0.59 0.42 0.32

0.74 0.75 0.66

0.76 0.71 0.65

0.52 0.34 0.25

0.42 0.27 0.22

0.53 0.51 0.46

0.44 0.37

0.31 0.26

0.59 0.59

0.61 0.59

0.23 0.18

0.23 0.18

0.43 0.42

WF indicates structural tee cut from wide-flange section. I indicates structural tee cut from standard beam section.

Table 82 Properties and Dimensions of Zees

Zees are seldom used as structural framing members. When so used they are generally employed on short spans in flexure. This table lists a few selected sizes, the range of whose section moduli will cover all ordinary conditions. For sizes not listed, the catalogs of the respective rolling mills should be consulted. Dimensions Nominal Size (in.)


Area (in.2 )

Depth (in.)

Width of Flange (in.)

6 × 3 12

21.1 15.7

6.19 4.59

6 18 6

3 58 3 12

5 × 3 14

17.9 16.4 14.0 11.6

5.25 4.81 4.10 3.40

5 5 18 1 5 16 5

3 14 3 38 5 3 16 3 14

1 4 × 3 16

15.9 12.5 10.3

4.66 3.66 3.03

1 4 16 4 18 1 4 16

3 18 3 3 16 3 18

8.2 12.6 9.8

2.41 3.69 2.86

4 3 3

1 3 16 2 11 16 2 11 16

6.7

1.97

3

2 11 16

3 × 2 11 16

Axis X –X

Axis Y –Y

Axis Z –Z

Thickness (in.)

I (in.4 )

S (in.3 )

r (in.)

I (in.4 )

S (in.3 )

r (in.)

r (in.)

1 2 3 8 1 2 7 16 3 8 5 16 1 2 3 8 5 16 1 4 1 2 3 8 1 4

34.4 25.3

11.2 8.4

2.36 2.35

12.9 9.1

3.8 2.8

1.44 1.41

0.84 0.83

19.2 19.1 16.2 13.4

7.7 7.4 6.4 5.3

1.91 1.99 1.99 1.98

9.1 9.2 7.7 6.2

3.0 2.9 2.5 2.0

1.31 1.38 1.37 1.35

0.74 0.77 0.76 0.75

11.2 9.6 7.9

5.5 4.7 3.9

1.55 1.62 1.62

8.0 6.8 5.5

2.8 2.3 1.8

1.31 1.36 1.34

0.67 0.69 0.68

6.3 4.6 3.9

3.1 3.1 2.6

1.62 1.12 1.16

4.2 4.9 3.9

1.4 2.0 1.6

1.33 1.15 1.17

0.67 0.53 0.54

2.9

1.9

1.21

2.8

1.1

1.19

0.55

120


Table 83

Properties and Dimensions of H Bearing Piles

Flange

Section Number and Nominal Size

Width b (in.)

Thickness t (in.)

Thickness W (in.)

I (in.4 )

S (in.3 )

r (in.)

(in.4 )

S (in.3 )

r (in.)

117 102 89 73 74 53 57 42

34.44 30.01 26.19 21.46 21.76 15.58 16.76 12.35

14.234 14.032 13.856 13.636 12.122 11.780 10.012 9.720

14.885 14.784 14.696 14.586 12.217 12.046 10.224 10.078

0.805 0.704 0.616 0.506 0.607 0.436 0.564 0.418

0.805 0.704 0.616 0.506 0.607 0.436 0.564 0.418

1228.5 1055.1 909.1 733.1 566.5 394.8 294.7 210.8

172.6 150.4 131.2 107.5 93.5 67.0 58.9 43.4

5.97 5.93 5.89 5.85 5.10 5.03 4.19 4.13

443.1 379.6 326.2 261.9 184.7 127.3 100.6 71.4

59.5 51.3 44.4 35.9 30.2 21.2 19.7 14.2

3.59 3.56 3.53 3.49 2.91 2.86 2.45 2.40

36

10.60

8.026

8.158

0.446

0.446

119.8

29.9

3.36

40.4

9.9

1.95

Weight/ft (lb)

Square

Round Area (in.2 )

Weight/ft (lb)

Area (in.2 )

0

1 1 16 1 8 3 16 1 4

I

Square and Round Barsa Square

1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

Axis Y–Y

Depth d (in.)

BP 12, 12 × 12 BP 10, 10 × 10 BP 8, 8×8

Size (in.)

Axis X–X

Area A (in.2 )

BP 14, 14 × 14 12

Table 84

Web


0.013

0.0039

0.010

0.0031

0.053

0.0156

0.042

0.0123

0.120

0.0352

0.094

0.0276

0.213

0.0625

0.167

0.0491

0.332

0.0977

0.261

0.0767

0.478

0.1406

0.376

0.1105

0.651

0.1914

0.511

0.1503

0.850

0.2500

0.668

0.1963

1.076

0.3164

0.845

0.2485

1.328

0.3906

1.043

0.3068

1.607

0.4727

1.262

0.3712

1.913

0.5625

1.502

0.4418

2.245

0.6602

1.763

0.5185

2.603

0.7656

2.044

0.6013

2.988 3.400 3.838

0.8789 1.0000 1.1289

2.347 2.670 3.015

0.6903 0.7854 0.8866

4.303

1.2656

3.380

0.9940

4.795

1.4102

3.766

1.1075

5.313

1.5625

4.172

1.2272

Size (in.) 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

2 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16

Round

Weight/ft (lb)

Area (in.2 )

Weight/ft (lb)

Area (in.2 )

5.857

1.7227

4.600

1.3530

6.428

1.8906

5.049

1.4849

7.026

2.0664

5.518

1.6230

7.650

2.2500

6.008

1.7671

8.301

2.4414

6.519

1.9175

8.978

2.6406

7.051

2.0739

9.682

2.8477

7.604

2.2365

10.413

3.0625

8.178

2.4053

11.170

3.2852

8.773

2.5802

11.953

3.5156

9.388

2.7612

12.763 13.600 14.463

3.7539 4.0000 4.2539

10.024 10.681 11.359

2.9483 3.1416 3.3410

15.353

4.5156

12.058

3.5466

16.270

4.7852

12.778

3.7583

17.213

5.0625

13.519

3.9761

18.182

5.3477

14.280

4.2000

19.178

5.6406

15.062

4.4301

20.201

5.9414

15.866

4.6664

21.250

6.2500

16.690

4.9087

22.326

6.5664

17.534

5.1572


121

Table 84 (Continued ) Square Size (in.) 5 8 11 16 3 4 13 16 7 8 15 16

3 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

4 1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

5 1 16 1 8 3 16 1 4 5 16 a

Weight/ft (lb)

Square

Round Area (in.2 )

Weight/ft (lb)

Area (in.2 )

Size (in.) 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

23.428

6.8906

18.400

5.4119

24.557

7.2227

19.287

5.6727

25.713

7.5625

20.195

5.9396

26.895

7.9102

21.123

6.2126

28.103

8.2656

22.072

6.4918

29.338 30.60 31.89

8.6289 9.000 9.379

23.042 24.03 25.05

6.7771 7.069 7.366

33.20

9.766

26.08

7.670

34.54

10.160

27.13

7.980

35.91

10.563

28.21

8.296

6

37.31

10.973

29.30

8.618

38.73

11.391

30.42

8.946

40.18

11.816

31.55

9.281

41.65

12.250

32.71

9.621

43.15

12.691

33.89

9.968

44.68

13.141

35.09

10.321

46.23

13.598

36.31

10.680

47.81

14.063

37.55

11.045

49.42

14.535

38.81

11.416

51.05

15.016

40.10

11.793

52.71 54.40 56.11

15.504 16.000 16.504

41.40 42.73 44.07

12.177 12.566 12.962

57.85

17.016

45.44

13.364

59.62

17.535

46.83

13.772

61.41

18.063

48.23

14.186

1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

7

63.23

18.598

49.66

14.607

65.08

19.141

51.11

15.033

66.95

19.691

52.58

15.466

68.85

20.250

54.07

15.904

70.78

20.816

55.59

16.349

72.73

21.391

57.12

16.800

74.71

21.973

58.67

17.257

76.71

22.563

60.25

17.721

78.74

23.160

61.85

18.190

80.80

23.766

63.46

18.665

82.89 85.00 87.14

24.379 25.000 25.629

65.10 66.76 68.44

19.147 19.635 20.129

89.30

26.266

70.14

20.629

91.49

26.910

71.86

21.135

93.71

27.563

73.60

21.648

1 16 1 8 3 16 1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 13 16 7 8 15 16

95.96

28.223

75.36

22.166

One cubic inch of rolled steel is assumed to weigh 0.2833 lb.

8

Weight/ft (lb)

Round Area (in.2 )

Weight/ft (lb)

Area (in.2 )

98.23

28.891

77.15

22.691

100.53

29.566

78.95

23.221

102.85

30.250

80.78

23.758

105.20

30.941

82.62

24.301

107.58

31.641

84.49

24.850

109.98

32.348

86.38

25.406

112.41

33.063

88.29

25.967

114.87

33.785

90.22

26.535

117.35

34.516

92.17

27.109

119.86 122.40 124.96

35.254 36.000 36.754

94.14 96.13 98.15

27.688 28.274 28.866

127.55

37.516

100.18

29.465

130.17

38.285

102.23

30.069

132.81

39.063

104.31

30.680

135.48

39.848

106.41

31.296

138.18

40.641

108.53

31.919

140.90

41.441

110.66

32.548

143.65

42.250

112.82

33.183

146.43

43.066

115.00

33.824

149.23

43.891

117.20

34.472

152.06

44.723

119.43

35.125

154.91

45.563

121.67

35.785

157.79

46.410

123.93

36.450

160.70

47.266

126.22

37.122

163.64 166.60 169.59

48.129 49.000 49.879

128.52 130.85 133.19

37.800 38.485 39.175

172.60

50.766

135.56

39.871

175.64

51.660

137.95

40.574

178.71

52.563

140.36

41.282

181.81

53.473

142.79

41.997

184.93

54.391

145.24

42.718

188.07

55.316

147.71

43.445

191.25

56.250

150.21

44.179

194.45

57.191

152.72

44.918

197.68

58.141

155.26

45.664

200.93

59.098

157.81

46.415

204.21

60.063

160.39

47.173

207.52

61.035

162.99

47.937

210.85

62.016

165.60

48.707

214.21 217.60

63.004 64.000

168.24 170.90

49.483 50.265

122 0.405

0.540

0.675

0.840

1.050

1.315

1.660

1 8

1 4

3 8

1 2

3 4

1

1 14 5S 10S 40ST, 40S 80XS, 80S 160 XX

5S 10S 40ST, 40S 80XS, 80S 160 XX

5S 10S 40ST, 40S 80XS, 80S 160 XX

5S 10S 40ST, 40S 80XS, 80S 160 XX

10S 40ST, 40S 80XS, 80S

10S 40ST, 40S 80XS, 80S

10S 40ST, 40S 80XS, 80S

Schedule No.

Outside Diameter (in.)

Nominal Pipe Size (in.)

0.065 0.109 0.140 0.191 0.250 0.382

0.065 0.109 0.133 0.179 0.250 0.358

0.065 0.083 0.113 0.154 0.219 0.308

0.065 0.083 0.109 0.147 0.188 0.294

0.065 0.091 0.126

0.065 0.088 0.119

0.049 0.068 0.095

Wall Thickness (in.)

Dimensions of Ferrous Pipe

Table 85

1.530 1.442 1.380 1.278 1.160 0.896

1.185 1.097 1.049 0.957 0.815 0.599

0.920 0.884 0.824 0.742 0.612 0.434

0.710 0.674 0.622 0.546 0.464 0.252

0.545 0.493 0.423

0.410 0.364 0.302

0.307 0.269 0.215

Inside Diameter (in.)

0.326 0.531 0.668 0.881 1.107 1.534

0.255 0.413 0.494 0.639 0.836 1.076

0.201 0.252 0.333 0.433 0.572 0.718

0.158 0.197 0.250 0.320 0.385 0.504

0.125 0.167 0.217

0.097 0.125 0.157

0.055 0.072 0.093

Metal (in.2 )

0.01277 0.01134 0.01040 0.00891 0.00734 0.00438

0.00768 0.00656 0.00600 0.00499 0.00362 0.00196

0.00461 0.00426 0.00371 0.00300 0.00204 0.00103

0.00275 0.00248 0.00211 0.00163 0.00117 0.00035

0.00162 0.00133 0.00098

0.00092 0.00072 0.00050

0.00051 0.00040 0.00025

Flow (ft2 )

Cross-Sectional Area

0.435 0.435 0.435 0.435 0.435 0.435

0.344 0.344 0.344 0.344 0.344 0.344

0.275 0.275 0.275 0.275 0.275 0.275

0.220 0.220 0.220 0.220 0.220 0.220

0.177 0.177 0.177

0.141 0.141 0.141

0.106 0.106 0.106

Outside

0.401 0.378 0.361 0.335 0.304 0.235

0.310 0.287 0.275 0.250 0.213 0.157

0.241 0.231 0.216 0.194 0.160 0.114

0.186 0.176 0.163 0.143 0.122 0.066

0.143 0.129 0.111

0.107 0.095 0.079

0.0804 0.0705 0.0563

Inside

Circumference, ft, or surface, ft2 / ft of Length

5.73 5.09 4.57 3.99 3.29 1.97

3.449 2.946 2.690 2.240 1.625 0.878

2.072 1.903 1.665 1.345 0.917 0.461

1.234 1.112 0.945 0.730 0.527 0.155

0.727 0.596 0.440

0.412 0.323 0.224

0.231 0.179 0.113

U.S. gal/ min

2865 2545 2285 1995 1645 985

1725 1473 1345 1120 812.5 439.0

1036.0 951.5 832.5 672.5 458.5 230.5

617.0 556.0 472.0 365.0 263.5 77.5

363.5 298.0 220.0

206.5 161.5 112.0

115.5 89.5 56.5

lb/hr water

Capacity at 1 ft/sec Velocity

1.11 1.81 2.27 3.00 3.76 5.21

0.87 1.40 1.68 2.17 2.84 3.66

0.69 0.86 1.13 1.47 1.94 2.44

0.54 0.67 0.85 1.09 1.31 1.71

0.42 0.57 0.74

0.33 0.42 0.54

0.19 0.24 0.31

Weight of Plain-End Pipe (lb/ft)

123

1.900

2.375

2.875

3.500

4.0

4.5

1 21

2

2 21

3

3 21

4

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

5S 10S 40ST, 40S 80XS, 80S

5S 10S 40ST, 40S 80XS, 80S 160 XX

5S 10S 40ST, 40S 80XS, 80S 160 XX

5S 10S 40ST, 40S 80ST, 80S 160 XX

5S 10S 40ST, 40S 80SX, 80S 160 XX

0.083 0.120 0.237 0.337 0.438 0.531 0.674

0.083 0.120 0.226 0.318

0.083 0.120 0.216 0.300 0.438 0.600

0.083 0.120 0.203 0.276 0.375 0.552

0.065 0.109 0.154 0.218 0.344 0.436

0.065 0.109 0.145 0.200 0.281 0.400

4.334 4.260 4.026 3.826 3.624 3.438 3.152

3.834 3.760 3.548 3.364

3.334 3.260 3.068 2.900 2.624 2.300

2.709 2.635 2.469 2.323 2.125 1.771

2.245 2.157 2.067 1.939 1.687 1.503

1.770 1.682 1.610 1.500 1.338 1.100

1.152 1.651 3.17 4.41 5.58 6.62 8.10

1.021 1.463 2.680 3.678

0.891 1.274 2.228 3.016 4.213 5.466

0.728 1.039 1.704 2.254 2.945 4.028

0.472 0.776 1.075 1.477 2.195 2.656

0.375 0.614 0.800 1.069 1.429 1.885

0.10245 0.09898 0.08840 0.07986 0.07170 0.06647 0.05419

0.08017 0.07711 0.06870 0.06170

0.06063 0.05796 0.05130 0.04587 0.03755 0.02885

0.04003 0.03787 0.03322 0.02942 0.02463 0.01711

0.02749 0.02538 0.02330 0.02050 0.01552 0.01232

0.01709 0.01543 0.01414 0.01225 0.00976 0.00660

1.178 1.178 1.178 1.178 1.178 1.178 1.178

1.047 1.047 1.047 1.047

0.916 0.916 0.916 0.916 0.916 0.916

0.753 0.753 0.753 0.753 0.753 0.753

0.622 0.622 0.622 0.622 0.622 0.622

0.497 0.497 0.497 0.497 0.497 0.497

1.135 1.115 1.054 1.002 0.949 0.900 0.825

1.004 0.984 0.929 0.881

0.873 0.853 0.803 0.759 0.687 0.602

0.709 0.690 0.647 0.608 0.556 0.464

0.588 0.565 0.541 0.508 0.436 0.393

0.463 0.440 0.421 0.393 0.350 0.288

46.0 44.4 39.6 35.8 32.2 28.9 24.3

35.98 34.61 30.80 27.70

27.21 26.02 23.00 20.55 16.86 12.95

17.97 17.00 14.92 13.20 11.07 7.68

12.34 11.39 10.45 9.20 6.97 5.53

7.67 6.94 6.34 5.49 4.38 2.96

23,000 22,200 19,800 17,900 16,100 14,450 12,150

17.990 17,305 15,400 13,850

13,605 13,010 11,500 10,275 8430 6475

8985 8500 7460 6600 5535 3840

6170 5695 5225 4600 3485 2765

3835 3465 3170 2745 2190 1480

(Continues)

3.92 5.61 10.79 14.98 19.01 22.52 27.54

3.48 4.97 9.11 12.51

3.03 4.33 7.58 10.25 14.31 18.58

2.48 3.53 5.79 7.66 10.01 13.70

1.61 2.64 3.65 5.02 7.46 9.03

1.28 2.09 2.72 3.63 4.86 6.41

124 (Continued ) Outside Diameter (in.) 5.563

6.625

8.625

10.75

Table 85


5

6

8

10

5S 10S 20 30 40ST, 40S 80S, 60XS 80 100 120 140, XX 160

5S 10S 20 30 40ST, 40S 60 80XS, 80S 100 120 140 XX 160

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

Schedule No.

0.134 0.165 0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

0.109 0.148 0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

0.109 0.134 0.280 0.432 0.562 0.719 0.864

0.109 0.134 0.258 0.375 0.500 0.625 0.750


10.842 10.420 10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

6.407 6.357 6.065 5.761 5.501 5.187 4.897

5.345 5.295 5.047 4.813 4.563 4.313 4.063


4.47 5.49 8.25 10.07 11.91 16.10 18.95 22.66 26.27 30.63 34.02

2.915 3.941 6.578 7.260 8.396 10.48 12.76 14.99 17.86 19.93 21.30 21.97

2.23 2.73 5.58 8.40 10.70 13.34 15.64

1.87 2.29 4.30 6.11 7.95 9.70 11.34

Metal (in.2 )

0.5993 0.5922 0.5731 0.5603 0.5475 0.5185 0.4987 0.4729 0.4479 0.4176 0.3941

0.3855 0.3784 0.3601 0.3553 0.3474 0.3329 0.3171 0.3017 0.2817 0.2673 0.2578 0.2532

0.2239 0.2204 0.2006 0.1810 0.1650 0.1467 0.1308

0.1558 0.1529 0.1390 0.1263 0.1136 0.1015 0.0900

Flow (ft2 )


2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814

2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258

1.734 1.734 1.734 1.734 1.734 1.734 1.734

1.456 1.456 1.456 1.456 1.456 1.456 1.456

Outside

2.744 2.728 2.685 2.655 2.620 2.550 2.503 2.438 2.372 2.291 2.225

2.201 2.180 2.127 2.113 2.089 2.045 1.996 1.947 1.882 1.833 1.800 1.784

1.677 1.664 1.588 1.508 1.440 1.358 1.282

1.399 1.386 1.321 1.260 1.195 1.129 1.064

Inside


269.0 265.8 257.0 252.0 246.0 233.0 223.4 212.3 201.0 188.0 177.0

173.0 169.8 161.5 159.4 155.7 149.4 142.3 135.4 126.4 120.0 115.7 113.5

100.5 98.9 90.0 81.1 73.9 65.9 58.7

69.9 68.6 62.3 57.7 51.0 45.5 40.4

U.S. gal/ min

134,500 132,900 128,500 126,000 123,000 116,500 111,700 106,150 100,500 94,000 88,500

86,500 84,900 80,750 79,700 77,850 74,700 71,150 67,700 63,200 60,000 57,850 56,750

50,250 49,450 45,000 40,550 36,950 32,950 29,350

34,950 34,300 31,150 28,850 25,500 22,750 20,200

lb/hr water


15.23 18.70 28.04 34.24 40.48 54.74 64.40 77.00 89.27 104.13 115.65

9.93 13.40 22.36 24.70 28.55 35.66 43.39 50.93 60.69 67.79 72.42 74.71

7.60 9.29 18.97 28.57 36.42 45.34 53.16

6.36 7.77 14.62 20.78 27.04 32.96 38.55


125

12.75

14

16

12

14

16

5S 10S 10 20 30, ST 40, XS 60 80 100 120 140 160

5S 10S 10 20 30, ST 40 XS 60 80 100 120 140 160

5S 10S 20 30 ST, 40S 40 XS, 80S 60 80 100 120, XX 140 160

0.165 0.188 0.250 0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

0.156 0.188 0.250 0.312 0.375 0.438 0.500 0.594 0.750 0.938 1.094 1.250 1.406

0.156 0.180 0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.000 1.125 1.312

15.670 15.624 15.500 15.376 15.250 15.000 14.688 14.312 13.938 13.562 13.124 12.812

13.688 13.624 13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.812 11.500 11.188

12.438 12.390 12.250 12.090 12.000 11.938 11.750 11.626 11.374 11.062 10.750 10.500 10.126

8.18 9.34 12.37 15.38 18.41 24.35 31.62 40.19 48.48 56.61 65.79 72.14

6.78 8.16 10.80 13.42 16.05 18.66 21.21 25.02 31.22 38.49 44.36 50.07 55.63

6.17 7.11 9.82 12.88 14.58 15.74 19.24 21.52 26.07 31.57 36.91 41.09 47.14

1.3393 1.3314 1.3104 1.2985 1.2680 1.2272 1.1766 1.1171 1.0596 1.0032 0.9394 0.8953

1.0219 1.0125 0.9940 0.9750 0.9575 0.9397 0.9218 0.8957 0.8522 0.8017 0.7610 0.7213 0.6827

0.8438 0.8373 0.8185 0.7972 0.7854 0.7773 0.7530 0.7372 0.7056 0.6674 0.6303 0.6013 0.5592

4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189

3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665

3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338

4.10 4.09 4.06 4.03 3.99 3.93 3.85 3.75 3.65 3.55 3.44 3.35

3.58 3.57 3.53 3.50 3.47 3.44 3.40 3.35 3.27 3.17 3.09 3.01 2.93

3.26 3.24 3.21 3.17 3.14 3.13 3.08 3.04 2.98 2.90 2.81 2.75 2.65

601 598 587 578 568 550 528 501 474 450 422 402

459 454 446 438 430 422 414 402 382 360 342 324 306

378.7 375.8 367.0 358.0 352.5 349.0 338.0 331.0 316.7 299.6 283.0 270.0 251.0

300,500 299,000 293,500 289,000 284,000 275,000 264,000 250,500 237,000 225,000 211,000 201,000

229,500 227,000 223,000 219,000 215,000 211,000 207,000 201,000 191,000 180,000 171,000 162,000 153,000

189,350 187,900 183,500 179,000 176,250 174,500 169,000 165,500 158,350 149,800 141,500 135,000 125,500

(Continues)

27.87 31.62 42.05 52.36 62.58 82.77 107.54 136.58 164.86 192.40 223.57 245.22

22.76 27.70 36.71 45.68 54.57 63.37 72.09 85.01 106.13 130.79 150.76 170.22 189.12

22.22 24.20 33.38 43.77 49.56 53.56 65.42 73.22 88.57 107.29 125.49 139.68 160.33

126 (Continued ) Outside Diameter (in.) 18

20

Table 85


18

20

5S 10S 10 20, ST 30, XS 40 60 80 100 120 140 160

5S 10S 10 20 ST 30 XS 40 60 80 100 120 140 160

Schedule No.

0.188 0.218 0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969

0.165 0.188 0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781


19.624 19.564 19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.062

17.670 17.624 17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.250 14.876 14.438


11.70 13.55 15.51 23.12 30.63 36.21 48.95 61.44 75.33 87.18 100.3 111.5

9.25 10.52 13.94 17.34 20.76 24.16 27.49 30.79 40.64 50.28 61.17 71.82 80.66 90.75

Metal (in.2 )

2.1004 2.0878 2.0740 2.0211 1.9689 1.9302 1.8417 1.7550 1.6585 1.5763 1.4849 1.4071

1.7029 1.6941 1.6703 1.6468 1.6230 1.5993 1.5763 1.5533 1.4849 1.4180 1.3423 1.2684 1.2070 1.1370

Flow (ft2 )


5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236

4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712

Outside

5.14 5.12 5.11 5.04 4.97 4.92 4.81 4.70 4.57 4.45 4.32 4.21

4.63 4.61 4.58 4.55 4.52 4.48 4.45 4.42 4.32 4.22 4.11 3.99 3.89 3.78

Inside


943 937 930 902 883 866 826 787 744 707 665 632

74 760 750 739 728 718 707 697 666 636 602 569 540 510

U.S. gal/ min

471,500 467,500 465,000 451,000 441,500 433,000 413,000 393,500 372,000 353,500 332,500 316,000

382,000 379,400 375,000 369,500 364,000 359,000 353,500 348,500 333,000 318,000 301,000 284,500 270,000 255,000

lb/hr water


39.76 45.98 52.73 78.60 104.13 123.06 166.50 208.92 256.15 296.37 341.10 379.14

31.32 35.48 47.39 59.03 70.59 82.06 93.45 104.76 138.17 170.75 208.00 244.14 274.30 308.55


127

30

30

5S 10, 10S ST 20, XS 30

5S 10, 10S 20, ST XS 30 40 60 80 100 120 140 160 0.250 0.312 0.375 0.500 0.625

0.218 0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.531 1.812 2.062 2.344 29.500 29.376 29.250 29.000 28.750

23.564 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.876 19.312 23.37 29.10 34.90 46.34 57.68

16.29 18.65 27.83 36.90 41.39 50.39 70.11 87.24 108.1 126.3 142.1 159.5 4.746 4.707 4.666 4.587 4.508

3.0285 3.012 2.948 2.885 2.854 2.792 2.655 2.536 2.391 2.264 2.155 2.034 7.854 7.854 7.854 7.854 7.854

6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 7.72 7.69 7.66 7.59 7.53

6.17 6.15 6.09 6.02 5.99 5.92 5.78 5.64 5.48 5.33 5.20 5.06 2130 2110 2094 2055 2020

1359 1350 1325 1295 1281 1253 1192 1138 1073 1016 965 913 1,065,000 1,055,000 1,048,000 1,027,500 1,010,000

679,500 675,000 662,500 642,500 640,500 626,500 596,000 569,000 536,500 508,000 482,500 456,500 79.43 99.08 118.65 157.53 196.08

55.08 63.41 94.62 125.49 140.80 171.17 238.29 296.53 367.45 429.50 483.24 542.09

Schedule Nos. 5S, 10S, and 40S American National Standards Institute (ANSI)/American Society of Mechanical Engineers (ASME) B.36.19-1985, ‘‘Stainless Steel Pipe.’’ ST = standard wall, XS = extra strong wall, XX = double extra strong wall are all taken from ANSI/ASME, B.36.10M-1985, ‘‘Welded and Seamless Wrought-steel Pipe.’’ Wrought-iron pipe has slightly thicker walls, approximately 3%, but the same weight per foot, because of lower density. Decimal thicknesses for respective pipe sizes represent their nominal or average wall dimensions. Mill tolerances as high as 12 21 % are permitted. Plain-end pipe is produced by a square cut. Pipe is also shipped from the mills threaded, with a threaded coupling on one end, or with the ends beveled for welding, or grooved or sized for patented couplings. Weights per foot for threaded and coupled pipe are slightly greater because of the weight of the coupling, but it is not available larger than 12 in., or lighter than Schedule 30 sizes 8 through 12 in., or Schedule 40 6 in. and smaller. Source: From Chemical Engineer’s Handbook, 4th ed., New York, McGraw-Hill, 1963. Used by permission.

24

24

128 Table 86

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Properties and Dimensions of Steel Pipea Dimensions

Nominal Diameter (in.)



Thickness (in.)

Couplings Weight per Foot (lb) Thread Plain and Ends Coupling

Threads per Inch


Length (in.)

Properties

Weight (lb)

I (in.4 )

A (in.2 )

k (in.)

Schedule 40ST 1 8 1 4 3 8 1 2 3 4

7 8

0.405

0.269

0.068

0.24

0.25

27

0.562

0.03

0.001

0.540

0.364

0.088

0.42

0.43

18

0.685

1

0.04

0.003

0.072 0.12 0.125 0.16

0.675

0.493

0.091

0.57

0.57

18

0.848

1 18

0.07

0.007

0.167 0.21

0.840

0.622

0.109

0.85

0.85

14

1.024

1 38

0.12

0.017

0.250 0.26

1.050

0.824

0.113

1.13

1.13

14

1.281

1 58

0.21

0.037

0.333 0.33

1

1.315

1.049

0.133

1.68

1.68

11 12

1.576

1 78

0.35

0.087

0.494 0.42

1 14

1.660

1.380

0.140

2.27

2.28

11 12

1.950

2 18

0.55

0.195

0.669 0.54

1 12

1.900

1.610

0.145

2.72

2.73

11 12

2.218

2 38

0.76

0.310

0.799 0.62

2

2.375

2.067

0.154

3.65

3.68

11 12

2.760

2 58

1.23

0.666

1.075 0.79

2 12

2.875

2.469

0.203

5.79

5.82

8

3.276

2 78

1.76

1.530

1.704 0.95

3

3.500

3.068

0.216

7.58

7.62

8

3.948

3 18

2.55

3.017

2.228 1.16

3 12

4.000

3.548

0.226

9.11

9.20

8

4.591

3 58

4.33

4.788

2.680 1.34

4

4.500

4.026

0.237

10.79

10.89

8

5.091

3 58

5.41

7.233

3.174 1.51

5

5.563

5.047

0.258

14.62

14.81

8

6.296

4 18

9.16

15.16

4.300 1.88

6

6.625

6.065

0.280

18.97

19.19

8

7.358

4 18

10.82

28.14

5.581 2.25

8

8.625

8.071

0.277

24.70

25.00

8

9.420

4 58

15.84

63.35

7.265 2.95

8

8.625

7.981

0.322

28.55

28.81

8

9.420

4 58

15.84

72.49

8.399 2.94

10

10.750

10.192

0.279

31.20

32.00

8

11.721

6 18

33.92

125.4

10

10.750

10.136

0.307

34.24

35.00

8

11.721

6 18

33.92

137.4

10.07

3.69

10

10.750

10.020

0.365

40.48

41.13

8

11.721

6 18

33.92

160.7

11.91

3.67

12

12.750

12.090

0.330

43.77

45.00

8

13.958

6 18

48.27

248.5

12.88

4.39

12

12.750

12.000

0.375

49.56

50.71

8

13.958

6 18

48.27

279.3

14.38

4.38

9.178 3.70

Schedule 80XS 1 8 1 4 3 8 1 2 3 4

0.405

0.215

0.095

0.31

0.32

27

0.582

1 18

0.05

0.001

0.093 0.12

0.540

0.302

0.119

0.54

0.54

18

0.724

1 38

0.07

0.004

0.157 0.16

0.675

0.423

0.126

0.74

0.75

18

0.898

1 58

0.13

0.009

0.217 0.20

0.840

0.546

0.147

1.09

1.10

14

1.085

1 78

0.22

0.020

0.320 0.25

1.050

0.742

0.154

1.47

1.49

14

1.316

2 18

0.33

0.045

0.433 0.32

1

1.315

0.957

0.179

2.17

2.20

11 12

1.575

2 38

0.47

0.106

0.639 0.41

1 14

1.660

1.278

0.191

3.00

3.05

11 12

2.054

2 78

1.04

0.242

0.881 0.52

1 12

1.900

1.500

0.200

3.63

3.69

11 12

2.294

2 78

1.17

0.391

1.068 0.61

2

2.375

1.939

0.218

5.02

5.13

11 12

2.870

3 58

2.17

0.868

1.477 0.77

2 12

2.875

2.323

0.276

7.66

7.83

8

3.389

4 18

3.43

1.924

2.254 0.92

3

3.500

2.900

0.300

10.25

10.46

8

4.014

4 18

4.13

3.894

3.016 1.14

3 12

4.000

3.364

0.318

12.51

12.82

8

4.628

4 58

6.29

6.280

3.678 1.31

4

4.500

3.826

0.337

14.98

15.39

8

5.233

4 58

8.16

9.610

4.407 1.48

5

5.563

4.813

0.375

20.78

21.42

8

6.420

5 18

12.87

20.67

6

6.625

5.761

0.432

28.57

29.33

8

7.482

5 18

15.18

40.49

8

8.625

7.625

0.500

43.39

44.72

8

9.596

6 18

26.63

105.7

12.76

2.88

10

10.750

9.750

0.500

54.74

56.94

8

11.958

6 58

44.16

211.9

16.10

3.63

12

12.750

11.750

0.500

65.42

68.02

8

13.958

6 58

51.99

361.5

19.24

4.34

6.112 1.84 8.405 2.20


129

Table 86 (Continued ) Dimensions

Nominal Diameter (in.)



Thickness (in.)

Couplings Weight per Foot (lb) Thread Plain and Ends Coupling

Threads per Inch


Length (in.)

Properties

Weight (lb)

I (in.4 )

A (in.2 )

k (in.)

Schedule XX 1 2 3 4

0.840

0.252

0.294

1.71

1.73

14

1.085

1 78

0.22

0.024

0.504 0.22

1.050

0.434

0.308

2.44

2.46

14

1.316

2 18

0.33

0.058

0.718 0.28

1

1.315

0.599

0.358

3.66

3.68

11 12

1.575

2 38

0.47

0.140

1.076 0.36

1 14

1.660

0.896

0.382

5.21

5.27

11 12

2.054

2 78

1.04

0.341

1.534 0.47

1 12

1.900

1.100

0.400

6.41

6.47

11 12

2.294

2 78

1.17

0.568

1.885 0.55

2

2.375

1.503

0.436

9.03

9.14

11 12

2.870

3 58

2.17

1.311

2.656 0.70

2 12

2.875

1.771

0.552

13.70

13.87

8

3.389

4 18

3.43

2.871

4.028 0.84

3

3.500

2.300

0.600

18.58

18.79

8

4.014

4 18

4.13

5.992

5.466 1.05

3 12

4.000

2.728

0.636

22.85

23.16

8

4.628

4 58

6.29

9.848

6.721 1.21

4

4.500

3.152

0.674

27.54

27.95

8

5.233

4 58

8.16

15.28

5

5.563

4.063

0.750

38.55

39.20

8

6.420

5 18

12.87

33.64

11.34

1.72

6

6.625

4.897

0.864

53.16

53.92

8

7.482

5 18

15.18

66.33

15.64

2.06

8

8.625

6.875

0.875

72.42

73.76

8

9.596

6 18

26.63

21.30

2.76

162.0

8.101 1.37

Large Outside Diameter Pipe Pipe 14 in. and larger is sold by actual outside step diameter and thickness. 1 in. from 1 to 1 in., inclusive. Sizes 14, 15, and 16 in. are available regularly in thicknesses varying by 16 4 All pipe is furnished random length unless otherwise ordered, viz: 12–22 ft with privilege of furnishing 5 % in 6–12-ft lengths. Pipe railing is most economically detailed with slip joints and random lengths between couplings. a Steel Construction, 1980, A.I.S.C.

6.6 Standard Structural Shapes— Aluminum∗ Table 87 Aluminum Association Standard Channels—Dimensions, Areas, Weights, and Section Propertiesa

Size Depth A (in.)

Width B (in.)

2.00 2.00 3.00 3.00 4.00 4.00 5.00 5.00

1.00 1.25 1.50 1.75 2.00 2.25 2.25 2.75

Areab (in.2 )

Weightc (lb/ft)

0.491 0.911 0.965 1.358 1.478 1.982 1.881 2.627

0.557 1.071 1.135 1.597 1.738 2.331 2.212 3.089

Section Propertiesd

Flange Thickness t1 (in.)

Web Thickness t (in.)

Fillet Radius R (in.)

I (in.4 )

Axis X –X S r (in.3 ) (in.)

I (in.4 )

S (in.3 )

Axis Y –Y r (in.)

x (in.)

0.13 0.26 0.20 0.26 0.23 0.29 0.26 0.32

0.13 0.17 0.13 0.17 0.15 0.19 0.15 0.19

0.10 0.15 0.25 0.25 0.25 0.25 0.30 0.30

0.288 0.546 1.41 1.97 3.91 5.21 7.88 11.14

0.288 0.546 0.94 1.31 1.95 2.60 3.15 4.45

0.045 0.139 0.22 0.42 0.60 1.02 0.98 2.05

0.064 0.178 0.22 0.37 0.45 0.69 0.64 1.14

0.303 0.391 0.47 0.55 0.64 0.72 0.72 0.88

0.298 0.471 0.49 0.62 0.65 0.78 0.73 0.95

0.766 0.774 1.21 1.20 1.63 1.62 2.05 2.06

(Continues) Tables 87–101 are from Aluminum Standards and Data. Copyright  1984 The Aluminum Association.

130


Table 87

(Continued ) Section Propertiesd

Size Depth A (in.)

Width B (in.)

Areab (in.2 )

Weightc (lb/ft)


6.00 6.00 7.00 7.00 8.00 8.00 9.00 9.00 10.00 10.00 12.00 12.00

2.50 3.25 2.75 3.50 3.00 3.75 3.25 4.00 3.50 4.25 4.00 5.00

2.410 3.427 2.725 4.009 3.526 4.923 4.237 5.927 5.218 7.109 7.036 10.053

2.834 4.030 3.205 4.715 4.147 5.789 4.983 6.970 6.136 8.360 8.274 11.822

0.29 0.35 0.29 0.38 0.35 0.41 0.35 0.44 0.41 0.50 0.47 0.62



I (in.4 )

0.17 0.21 0.17 0.21 0.19 0.25 0.23 0.29 0.25 0.31 0.29 0.35

0.30 0.30 0.30 0.30 0.30 0.35 0.35 0.35 0.35 0.40 0.40 0.45

14.35 21.04 22.09 33.79 37.40 52.69 54.41 78.31 83.22 116.15 159.76 239.69

Axis X –X S r (in.3 ) (in.)

I (in.4 )

Axis Y –Y S r x (in.3 ) (in.) (in.)

4.78 7.01 6.31 9.65 9.35 13.17 12.09 17.40 16.64 23.23 26.63 39.95

1.53 3.76 2.10 5.13 3.25 7.13 4.40 9.61 6.33 13.02 11.03 25.74

0.90 1.76 1.10 2.23 1.57 2.82 1.89 3.49 2.56 4.47 3.86 7.60

2.44 2.48 2.85 2.90 3.26 3.27 3.58 3.63 3.99 4.04 4.77 4.88

0.80 1.05 0.88 1.13 0.96 1.20 1.02 1.27 1.10 1.35 1.25 1.60

0.79 1.12 0.84 1.20 0.93 1.22 0.93 1.25 1.02 1.34 1.14 1.61

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061. d I = moment of inertia; S = section modulus; r = radius of gyration. b

Aluminum Association Standard I Beams—Dimensions, Areas, Weights, and Section Propertiesa

Table 88

Section Propertiesd

Size Depth A (in.)

Width B (in.)

Areab (in.2 )

Weightc (lb/ft)


3.00 3.00 4.00 4.00 5.00 6.00 6.00 7.00 8.00 8.00 9.00 10.00 10.00 12.00 12.00

2.50 2.50 3.00 3.00 3.50 4.00 4.00 4.50 5.00 5.00 5.50 6.00 6.00 7.00 7.00

1.392 1.726 1.965 2.375 3.146 3.427 3.990 4.932 5.256 5.972 7.110 7.352 8.747 9.925 12.153

1.637 2.030 2.311 2.793 3.700 4.030 4.692 5.800 6.181 7.023 8.361 8.646 10.286 11.672 14.292

0.20 0.26 0.23 0.29 0.32 0.29 0.35 0.38 0.35 0.41 0.44 0.41 0.50 0.47 0.62

a



I (in.4 )

0.13 0.15 0.15 0.17 0.19 0.19 0.21 0.23 0.23 0.25 0.27 0.25 0.29 0.29 0.31

0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40

2.24 2.71 5.62 6.71 13.94 21.99 25.50 42.89 59.69 67.78 102.02 132.09 155.79 255.57 317.33

Axis X –X S (in.3 ) 1.49 1.81 2.81 3.36 5.58 7.33 8.50 12.25 14.92 16.94 22.67 26.42 31.16 42.60 52.89

r (in.)

I (in.4 )

1.27 1.25 1.69 1.68 2.11 2.53 2.53 2.95 3.37 3.37 3.79 4.24 4.22 5.07 5.11

0.52 0.68 1.04 1.31 2.29 3.10 3.74 5.78 7.30 8.55 12.22 14.78 18.03 26.90 35.48

Axis Y –Y S r (in.3 ) (in.) 0.42 0.54 0.69 0.87 1.31 1.55 1.87 2.57 2.92 3.42 4.44 4.93 6.01 7.69 10.14

0.61 0.63 0.73 0.74 0.85 0.95 0.97 1.08 1.18 1.20 1.31 1.42 1.44 1.65 1.71

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061. d I = moment of inertia; S = section modulus; r = radius of gyration. b


131

Table 89 Standard Structural Shapes—Equal Angles a

A 3 4 3 4

1 1 1 1 1 41 1 41 1 41 1 21 1 21 1 21 1 43 1 43 1 43 1 43 2 2 2 2 2 2 21 2 21 2 21 2 21 2 21 3 3 3 3 3 3 3 21 3 21 3 21 3 21

t

R

R1

1 8 3 16 3 32 1 8 3 16 1 4 1 8 3 16 1 4 1 8 3 16 1 4 1 8 3 16 1 4 5 16 1 8 3 16 1 4 5 16 3 8 1 8 3 16 1 4 5 16 3 8 3 16 1 4 5 16 3 8 7 16 1 2 1 4 5 16 3 8 1 2

1 8 1 8 1 8 1 8 1 8 1 8 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 3 16 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8

3 32 3 32 3 32 3 32 3 32 3 32 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4

Areab (in.2 )

Weight per Footc (lb)

0.171

0.201

0.246

0.289

0.179

0.211

0.234

0.275

0.340

0.400

0.437

0.514

0.292

0.343

0.434

0.510

0.558

0.656

0.360

0.423

0.529

0.619

0.688

0.809

0.423

0.497

0.622

0.731

0.813

0.956

0.996

1.171

0.491

0.577

0.723

0.850

0.944

1.110

1.160

1.364

1.366

1.606

0.616

0.724

0.910

1.070

1.194

1.404

1.470

1.729

1.714

2.047

1.084

1.275

1.432

1.684

1.770

2.082

2.104

2.474

2.428

2.855

2.744

3.227

1.691

1.989

2.093

2.461

2.488

2.926

3.253

3.826 (Continues)

132 Table 89


A

t

R

R1

4

8

1 4 5 16 3 8 7 16 1 2 9 16 5 8 11 16 3 4 3 8 7 16 1 2 5 8 3 8 7 16 1 2 5 8 1 2 3 4

8

1

3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 5 8 5 8 5 8

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8

4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 8

a b c

Areab (in.2 )


1.941

2.283

2.406

2.829

2.862

3.366

3.310

3.893

3.753

4.414

4.187

4.924

4.613

5.425

5.032

5.918

5.441

6.399

3.603

4.237

4.177

4.912

4.743

5.578

5.853

6.883

4.353

5.119

5.052

5.941

5.743

6.754

7.102

8.352

7.773

9.141

11.461

13.478

15.023

17.667

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061.

Table 90

Standard Structural Shapes—Unequal Anglesa

A

B

1 14

3 4

1 14 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 34

1 3 4 3 4

1 1 1 14 1 14 1 14 1 14

t

R

R1

3 32 1 8 1 8 3 16 5 32 1 4 1 8 3 16 1 4 1 8

3 32 1 8 1 8 1 8 5 32 3 16 3 16 3 16 3 16 3 16

3 64 1 16 1 16 3 32 5 64 1 8 1 8 1 8 1 8 1 8

Areab (in.2 )


0.180

0.212

0.267

0.314

0.267

0.314

0.386

0.454

0.368

0.433

0.563

0.662

0.329

0.387

0.481

0.566

0.624

0.734

0.358

0.421


133

Table 90 (Continued ) A

B

t

R

R1

1 43

1 41

1 43

2 21

1 41 1 21 1 21 1 21 1 21 1 21 1 21 1 21

2 21

2

2 21 2 21 2 21 2 21

2

3

2

3

2

3

2

3

2

3

2

3

2 21

3

2 21

3

2 21

3 21 3 21 3 21 3 21 3 21 3 21 3 21 3 21

2 21 2 21 2 21 2 21

4

3

4

3

4

3

4

3

4

3

4

3

4

3 21

4

3 21

5

3

5

3

5

3 21

5

3 21

5

3 21

3 16 1 4 1 8 3 16 1 4 3 8 3 16 1 4 5 16 1 8 3 16 1 4 5 16 3 8 3 16 1 4 5 16 3 8 7 16 1 4 5 16 3 8 1 4 5 16 3 8 1 2 1 4 5 16 3 8 1 2 1 4 5 16 3 8 7 16 1 2 5 8 3 8 1 2 3 8 1 2 5 16 3 8 7 16

3 16 3 16 3 16 3 16 3 16 3 16 1 4 1 4 3 16 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 3 8 7 16 7 16 7 16

1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 3 16 3 16 3 16 3 16 3 16 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 5 16 5 16 5 16 5 16 5 16 5 16 5 16

2 2 2 2 2 21 2 21

2 2 2

3 3 3 3

Areab (in.2 )


0.528

0.621

0.688

0.809

0.422

0.496

0.622

0.731

0.813

0.956

1.172

1.378

0.723

0.850

0.944

1.110

1.152

1.355

0.554

0.652

0.817

0.961

1.069

1.257

1.314

1.545

1.554

1.828

0.911

1.071

1.193

1.403

1.471

1.730

1.740

2.046

2.001

2.353

1.307

1.537

1.614

1.898

1.916

2.253

1.432

1.684

1.770

2.082

2.104

2.474

2.744

3.227

1.566

1.842

1.937

2.278

2.300

2.705

3.003

3.532

1.691

1.988

2.091

2.459

2.488

2.926

2.874

3.380

3.253

3.826

3.988

4.690

2.660

3.128

3.488

4.102

2.848

3.349

3.738

4.396

2.558

3.008

3.046

3.582

3.527

4.148 (Continues)

134 Table 90


A

B

t

R

R1

5

3 12

5

6

3 12 3 12 3 12 3 12

6

4

6

4

6

4

6

4

6

4

6

4

8

6

8

6

8

6

1 2 5 8 5 16 3 8 1 2 3 8 7 16 1 2 9 16 5 8 3 4 5 8 11 16 3 4

7 16 7 16 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

5 16 5 16 5 16 5 16 5 16 3 8 3 8 3 8 3 8 3 8 3 8 5 16 3 8 3 8

6 6

a b c

Areab (in.2 )


4.000

4.704

4.921

5.787

2.878

3.385

3.433

4.037

4.512

5.306

3.603

4.237

4.179

4.915

4.743

5.578

5.298

6.230

5.853

6.883

6.931

8.151

8.371

9.844

9.152

10.763

9.931

11.679

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. Weights per foot are based on nominal dimensions and a density of 0.098lb/in.3 , which is the density of alloy 6061.

Table 91

Channels, American Standarda

A

B

C

t

t1

R

R1

Areab (in.2 )


3

1.410

1 34

0.170

0.170

0.270

0.100

1.205

1.417

3

1.498

1 34

0.258

0.170

0.270

0.100

1.470

1.729

3

1.596

1 34

0.356

0.170

0.270

0.100

1.764

2.074

4

1.580

2 34

0.180

0.180

0.280

0.110

1.570

1.846

4

1.647

2 34

0.247

0.180

0.280

0.110

1.838

2.161

4

1.720

2 34

0.320

0.180

0.280

0.110

2.129

2.504

5

1.750

3 34

0.190

0.190

0.290

0.110

1.969

2.316

5

1.885

3 34

0.325

0.190

0.290

0.110

2.643

3.108

5

2.032

3 34

0.472

0.190

0.290

0.110

3.380

3.975

6

1.920

4 12

0.200

0.200

0.300

0.120

2.403

2.826

6

1.945

4 12

0.225

0.200

0.300

0.120

2.553

3.002

6

2.034

4 12

0.314

0.200

0.300

0.120

3.088

3.631

6

2.157

4 12

0.437

0.200

0.300

0.120

3.825

4.498

7

2.110

5 12

0.230

0.210

0.310

0.130

3.011

3.541

7

2.194

5 12

0.314

0.210

0.310

0.130

3.599

4.232

7

2.299

5 12

0.419

0.210

0.310

0.130

4.334

5.097

8

2.290

6 14

0.250

0.220

0.320

0.130

3.616

4.252

8

2.343

6 14

0.303

0.220

0.320

0.130

4.040

4.751

8

2.435

6 14

0.395

0.220

0.320

0.130

4.776

5.617

8

2.527

6 14

0.487

0.220

0.320

0.130

5.514

6.484


135

Table 91 (Continued ) A

B

C

t

t1

R

R1

Areab (in.2 )


9

2.430

7 14

0.230

0.230

0.330

0.140

3.915

4.604

9

2.648

7 14

0.448

0.230

0.330

0.140

5.877

6.911

10

2.600

8 14

0.240

0.240

0.340

0.140

4.488

5.278

10

2.886

8 14

0.526

0.240

0.340

0.140

7.348

8.641

12

2.960

10

0.300

0.280

0.380

0.170

6.302

7.411

12

3.047

10

0.387

0.280

0.380

0.170

7.346

8.639

12

3.170

10

0.510

0.280

0.380

0.170

8.822

10.374

15

3.400

12 38

0.400

0.400

0.500

0.240

9.956

11.708

15

3.716

12 38

0.716

0.400

0.500

0.240

14.696

17.282

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061.

Table 92 Channels, Shipbuilding, and Carbuildinga

A

B

Areab (in.2 )


C

t

t1

R

R1

Slope

3

2

1 34

0.250

0.250

0.250

0

12:12.1

1.900

2.234

3

2

1 78

0.375

0.375

0.188

0.375

0

2.298

2.702

4

2 12

2 38

0.318

0.313

0.375

0.125

1:34.9

2.825

3.322

5

2 78

3

0.438

0.438

0.250

0.094

1:9.8

4.950

5.821

6

3

4 12

0.500

0.375

0.375

0.250

0

4.909

5.773

6

3 12

4

0.375

0.412

0.480

0.420

1:49.6

5.044

5.932

8

3

5 34

0.380

0.380

0.550

0.220

1:14.43

5.600

6.586

8

3 12

5 34

0.425

0.471

0.525

0.375

1:28.5

6.682

7.858

10

3 12

7 12

0.375

0.375

0.625

0.188

1:9

7.298

8.581

10

9 3 16

7 12

0.438

0.375

0.625

0.188

1:9

7.928

9.323

10

3 58

7 12

0.500

0.375

0.625

0.188

1:9

8.548

10.052


Table 93 H Beamsa

A

B

C

t

t1

R

R1

Slope

Areab (in.2 )

4

4

2 38

0.313

0.290

0.313

0.145

1:11.3

4.046

4.758

5

5

0.313

0.330

0.313

0.165

1:13.6

5.522

6.494

6

5.938

0.250

0.360

0.313

0.180

1:15.6

6.678

7.853

8

7.938

0.313

0.358

0.313

0.179

1:18.9

9.554

11.263

8

8.125

0.500

0.358

0.313

0.179

1:18.9

11.050

12.995

a

3 38 4 38 6 14 6 14

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. b Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061.

136 Table 94

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS I Beamsa

A


B

C

t

t1

R

R1

Areab (in.2 )

0.170

0.170

0.270

0.100

1.669

1.963

0.349

0.170

0.270

0.100

2.203

2.591

0.190

0.190

0.290

0.110

2.249

2.644

0.326

0.190

0.290

0.110

2.792

3.283

0.210

0.210

0.310

0.130

2.917

3.430

0.494

0.210

0.310

0.130

4.337

5.100

0.230

0.230

0.330

0.140

3.658

4.302

0.343

0.230

0.330

0.140

4.336

5.099

0.345

0.250

0.350

0.150

5.147

6.053

0.270

0.270

0.370

0.160

5.398

6.348

0.532

0.270

0.370

0.160

7.494

8.813

3

2.330

3

2.509

4

2.660

4

2.796

5

3

5

3.284

6

3.330

6

3.443

7

3.755

8

4

8

4.262

1 34 1 34 2 34 2 34 3 12 3 12 4 12 4 12 5 14 6 14 6 14

10

4.660

8

0.310

0.310

0.410

0.190

7.452

8.764

12

5

9 34

0.350

0.350

0.450

0.210

9.349

10.994

a b c


Table 95

A 6.000 6.000 8.000 8.000 8.000 9.750 9.900 11.940 12.060 a b c

Wide-Flange Beamsa

B

t

t1

R

R1

Areab (in.2 )

4.000 6.000 5.250 6.500 8.000 7.964 5.750 8.000 10.000

0.230 0.240 0.230 0.245 0.288 0.292 0.240 0.294 0.345

0.279 0.269 0.308 0.398 0.433 0.433 0.340 0.516 0.576

0.250 0.250 0.320 0.400 0.400 0.500 0.312 0.600 0.600

— — — — — — 0.031 — —

3.538 4.593 5.020 7.076 9.120 9.706 6.205 11.772 15.593

Weight per Footc (lb) 4.161 5.401 5.904 8.321 10.725 11.414 7.297 13.844 18.337



137

Table 96 Teesa

C

D

t

R

Areab (in.2 )


2

0.312

0.312

0.250

0.250

1.071

1.259

2 41

0.312

0.312

0.250

0.250

1.208

1.421

2 21

2 21

0.375

0.375

0.312

0.250

1.626

1.912

3 4

3 4

0.438 0.438

0.438 0.438

0.375 0.375

0.312 0.500

2.310 3.183

2.717 3.743

A

B

2 2 41

a

Users are encouraged to ascertain current availability of particular structural shapes through inquiries to their suppliers. Areas listed are based on nominal dimensions. c Weights per foot are based on nominal dimensions and a density of 0.098 lb/in.3 , which is the density of alloy 6061. b

Table 97 Zeesa


A

B

t

R

R1

Areab (in.2 )

3

2 11 16 2 11 16 1 3 16 3 18 3 3 16 3 14 5 3 16

0.250

0.312

0.250

1.984

2.333

0.375

0.312

0.250

2.875

3.381

0.250

0.312

0.250

2.422

2.848

0.312

0.312

0.250

3.040

3.575

0.375

0.312

0.250

3.672

4.318

0.500

0.312

0.250

5.265

6.192

0.375

0.312

0.250

4.093

4.813

3 4 1 4 16

4 41 5 1 5 16 a b c


138 Table 98 Nominal Pipe Sizea (in.) 1 8 1 4 3 8

1 2

3 4

1

1 14

1 12

2

2 12

3 3 12

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Aluminum Pipe—Diameters, Wall Thicknesses, and Weights Outside Diameter (in.)



Schedule Numbera

Noma

Minb,c

Maxb,c

Nom

Noma

Minb

Maxb

40 80 40 80 40 80 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80 160 5 10 40 80

0.405 0.405 0.540 0.540 0.675 0.675 0.840 0.840 0.840 0.840 0.840 1.050 1.050 1.050 1.050 1.050 1.315 1.315 1.315 1.315 1.315 1.660 1.660 1.660 1.660 1.660 1.900 1.900 1.900 1.900 1.900 2.375 2.375 2.375 2.375 2.375 2.875 2.875 2.875 2.875 2.875 3.500 3.500 3.500 3.500 3.500 4.000 4.000 4.000 4.000

0.374 0.374 0.509 0.509 0.644 0.644 0.809 0.809 0.809 0.809 0.809 1.019 1.019 1.019 1.019 1.019 1.284 1.284 1.284 1.284 1.284 1.629 1.629 1.629 1.629 1.629 1.869 1.869 1.869 1.869 1.869 2.344 2.344 2.351 2.351 2.351 2.844 2.844 2.846 2.846 2.846 3.469 3.469 3.465 3.465 3.465 3.969 3.969 3.960 3.960

0.420 0.420 0.555 0.555 0.690 0.690 0.855 0.855 0.855 0.855 0.855 1.065 1.065 1.065 1.065 1.065 1.330 1.330 1.330 1.330 1.330 1.675 1.675 1.675 1.675 1.675 1.915 1.915 1.915 1.915 1.915 2.406 2.406 2.399 2.399 2.399 2.906 2.906 2.904 2.904 2.904 3.531 3.531 3.535 3.535 3.535 4.031 4.031 4.040 4.040

0.269 0.215 0.364 0.302 0.493 0.493 0.710 0.674 0.622 0.546 0.464 0.920 0.884 0.824 0.742 0.612 1.185 1.097 1.049 0.957 0.815 1.530 1.442 1.380 1.278 1.160 1.770 1.682 1.610 1.500 1.338 2.245 2.157 2.067 1.939 1.687 2.709 2.635 2.469 2.323 2.125 3.334 3.260 3.068 2.900 2.624 3.834 3.760 3.548 3.364

0.068 0.095 0.088 0.119 0.091 0.091 0.065 0.083 0.109 0.147 0.188 0.065 0.083 0.113 0.154 0.219 0.065 0.109 0.133 0.179 0.250 0.065 0.109 0.140 0.191 0.250 0.065 0.109 0.145 0.200 0.281 0.065 0.109 0.154 0.218 0.344 0.083 0.120 0.203 0.276 0.375 0.083 0.120 0.216 0.300 0.438 0.083 0.120 0.226 0.318

0.060 0.083 0.077 0.104 0.080 0.080 0.053 0.071 0.095 0.129 0.164 0.053 0.071 0.099 0.135 0.192 0.053 0.095 0.116 0.157 0.219 0.053 0.095 0.122 0.167 0.219 0.053 0.095 0.127 0.175 0.246 0.053 0.095 0.135 0.191 0.301 0.071 0.105 0.178 0.242 0.328 0.071 0.105 0.189 0.262 0.383 0.071 0.105 0.198 0.278

— — — — — — 0.077 0.095 — — — 0.077 0.095 — — — 0.077 0.123 — — — 0.077 0.123 — — — 0.077 0.123 — — — 0.077 0.123 — — — 0.095 0.135 — — — 0.095 0.135 — — — 0.095 0.135 — —

Weight per Foot (lb) Nomd 0.085 0.109 0.147 0.185 0.196 0.196 0.186 0.232 0.294 0.376 0.453 0.237 0.297 0.391 0.510 0.672 0.300 0.486 0.581 0.751 0.984 0.383 0.625 0.786 1.037 1.302 0.441 0.721 0.940 1.256 1.681 0.555 0.913 1.264 1.737 2.581 0.856 1.221 2.004 2.650 3.464 1.048 1.498 2.621 3.547 4.955 1.201 1.720 3.151 4.326

Minb,d 0.091 0.118 0.159 0.200 0.212 0.212 — — 0.318 0.406 0.489 — — 0.422 0.551 0.726 — — 0.627 0.811 1.062 — — 0.849 1.120 1.407 — — 1.015 1.357 1.815 — — 1.365 1.876 2.788 — — 2.164 2.862 3.741 — — 2.830 3.830 5.351 — — 3.403 4.672


139

Table 98 (Continued ) Nominal Pipe Sizea (in.) 4

5

6

8

10

12

a





Schedule Numbera

Noma

Minb,c

Maxb,c

Nom

Noma

Minb

Maxb

Nomd

Minb,d

5 10 40 80 120 160 5.563 10 40 80 120 160 5 10 40 80 120 160 5 10 20 30 40 60 80 100 120 140 160 5 10 20 30 40 60 80 100 5 10 20 30 40 60 80

4.500 4.500 4.500 4.500 4.500 4.500 5.532 5.563 5.563 5.563 5.563 5.563 6.625 6.625 6.625 6.625 6.625 6.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 8.625 10.750 10.750 10.750 10.750 10.750 10.750 10.750 10.750 12.750 12.750 12.750 12.750 12.750 12.750 12.750

4.469 4.469 4.455 4.455 4.455 4.455 5.625 5.532 5.507 5.507 5.507 5.507 6.594 6.594 6.559 6.559 6.559 6.559 8.594 8.594 8.539 8.539 8.539 8.539 8.539 8.539 8.539 8.539 8.539 10.719 10.719 10.642 10.642 10.642 10.642 10.642 10.642 12.719 12.719 12.622 12.622 12.622 12.622 12.622

4.531 4.531 4.545 4.545 4.545 4.545 5.345 5.625 5.619 5.619 5.619 5.619 6.687 6.687 6.691 6.691 6.691 6.691 8.718 8.718 8.711 8.711 8.711 8.711 8.711 8.711 8.711 8.711 8.711 10.843 10.843 10.858 10.858 10.858 10.858 10.858 10.858 12.843 12.843 12.878 12.878 12.878 12.878 12.878

4.334 4.160 4.026 3.826 3.624 3.438 0.109 5.295 5.047 4.813 4.563 4.313 6.407 6.357 6.065 5.761 5.501 5.187 8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.813 10.482 10.420 10.250 10.136 10.020 9.750 9.562 9.312 12.438 12.390 12.250 12.090 11.938 11.626 11.374

0.083 0.120 0.237 0.337 0.438 0.531 0.095 0.134 0.258 0.375 0.500 0.625 0.109 0.134 0.280 0.432 0.562 0.719 0.109 0.148 0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.906 0.134 0.165 0.250 0.307 0.365 0.500 0.594 0.719 0.156 0.180 0.250 0.330 0.406 0.562 0.688

0.071 0.105 0.207 0.295 0.383 0.465 0.123 0.117 0.226 0.328 0.438 0.547 0.095 0.117 0.245 0.378 0.492 0.629 0.095 0.130 0.219 0.242 0.282 0.355 0.438 0.520 0.629 0.710 0.793 0.117 0.144 0.219 0.269 0.319 0.438 0.520 0.629 0.136 0.158 0.219 0.289 0.355 0.492 0.602

0.095 0.135 — — — — 2.196 0.151 — — — — 0.123 0.151 — — — — 0.123 0.166 — — — — — — — — — 0.151 0.186 — — — — — — 0.176 0.202 — — — — —

1.354 1.942 3.733 5.183 6.573 7.786 — 2.688 7.057 7.188 9.353 11.40 2.624 3.213 6.564 9.884 12.59 15.69 3.429 4.635 7.735 8.543 9.878 12.33 15.01 17.62 21.00 23.44 25.84 5.256 6.453 9.698 11.84 14.00 18.93 22.29 26.65 7.258 8.359 11.55 15.14 18.52 25.31 30.66

— — 4.031 5.598 7.099 8.409 — — 5.461 7.763 10.10 12.31 — — 7.089 10.67 13.60 16.94 — — 8.354 9.227 10.67 13.31 16.21 19.03 22.68 25.31 27.90 — — 10.47 12.69 15.12 24.07 28.78 28.78 — — 12.47 16.35 20.00 27.33 33.11

In accordance with ANSI Standards B36.10 and B36.19. on standard tolerances for pipe. c For schedules 5 and 10 these values apply to mean outside diameters. d Based on nominal dimensions, plain ends, and a density of 0.098 lb/in.3 , the density of 6061 alloy. For alloy 6063 multiply by 0.99, and for alloy 3003 multiply by 1.01. b Based

140


Table 99

Aluminum Electrical Conduit—Designed Dimensions and Weights

Nominal or Trade Size of Conduit (in.) 1 4 3 8 1 2 3 4

Nominal Inside Diameter (in.)


Nominal Wall Thickness (in.)

Length without Coupling (ft and in.)

Minimum Weight of 10 Unit Lengths with Couplings Attached (lb)

0.364

0.540

0.088

9–11 12

13.3

9–11 12 9–11 14 9–11 14

17.8

0.493

0.675

0.091

0.622

0.840

0.109

1

0.824 1.049

1.050 1.315

0.113 0.133

9–11

36.4 53.0

1 14

1.380

1.660

0.140

9–11

69.6

1 12 2

1.610 2.067

1.900 2.375

0.145 0.154

9–11 9–11

86.2 115.7

2 12

2.469

2.875

0.203

9–10 12

182.5

3

3.068

3.500

0.216

9–10 12

238.9

3 12

3.548

4.000

0.226

9–10 14

287.7

4 5 6

4.026 5.047 6.065

4.500 5.563 6.625

0.237 0.258 0.280

9–10 14 9–10 9–10

340.0 465.4 612.5

27.4

Table 100 Equivalent Resistivity Values Equivalent Resistivity at 68◦ F Volume

Volume Conductivity, Percent International Amended Copper Standard at 68◦ F

Ohm—Circular Mil/ft

Microhm—in.

52.5 53.5 53.8 53.9 54.0 54.3 55.0 56.0 56.5 57.0 59.0 59.5 61.0 61.2 61.3 61.4 61.5 61.8 62.0 62.1 62.2 62.3 62.4

19.754 19.385 19.277 19.241 19.206 19.099 18.856 18.520 18.356 18.195 17.578 17.430 17.002 16.946 16.918 16.891 16.863 16.782 16.727 16.700 16.674 16.647 16.620

1.2929 1.2687 1.2617 1.2593 1.2570 1.2501 1.2341 1.2121 1.2014 1.1908 1.1505 1.1408 1.1128 1.1091 1.1073 1.1055 1.1037 1.0983 1.0948 1.0931 1.0913 1.0896 1.0878


141

Table 101 Property Limits—Wire (Up to 0.374 in. Diameter) Ultimate Strength (ksi)

Alloy and Temper

Min

1350-O 1350-H12 and H22 1350-H14 and H24 1350-H16 and H26

8.5 12.0 15.0 17.0

8017-H212b

15.0

Max

Electrical Conductivitya percent IACS at 68◦ F min

14.0 17.0 20.0 22.0

61.8 61.0 61.0 61.0

21.0

61.0

22.0

61.0

20.0

61.0

1350

8017

8030 8030-H221

15.0 8176

8176-H24

15.0 8177

8177-H221

15.0

22.0

Alloy

Specified

and Temper

Diameter (in.)

Ultimate Strength (ksi min) Individuala

Averaged

0.0105–0.0500 0.0501–0.0600 0.0601–0.0700 0.0701–0.0800 0.0801–0.0900 0.0901–0.1000 0.1001–0.1100 0.1101–0.1200 0.1201–0.1400 0.1401–0.1500 0.1501–0.1800 0.1801–0.2100 0.2101–0.2600

23.0 27.0 27.0 26.5 26.0 25.5 24.5 24.0 23.5 23.5 23.0 23.0 22.5

25.0 29.0 28.5 28.0 27.5 27.0 26.0 25.5 25.0 24.5 24.0 24.0 23.5

0.0601–0.0700 0.0701–0.0800 0.0801–0.0900 0.0901–0.1000 0.1001–0.1100 0.1101–0.1200 0.1201–0.1400 0.1401–0.1500 0.1501–0.1600 0.1601–0.2100 0.2101–0.2600

38.0 37.5 37.0 36.5 36.0 35.5 35.0 35.0 34.5 32.5 31.5

40.0 39.5 39.0 38.5 38.0 37.5 37.0 36.5 36.0 34.0 33.0

6201-T81

0.0612–0.1327 0.1328–0.1878

46.0 44.0

48.0 46.0

8176-H24

0.0500–0.2040

15.0

17.0

61.0 Electrical Conductivitya min percent IACS

Elongation Percent min in 10 in.

at 68◦ F

Individuala

Averaged

— 1.2 1.3 1.4 1.5 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.2

— 1.4 1.5 1.6 1.6 1.6 1.6 1.7 1.8 1.9 2.0 2.1 2.3

1.3 1.4 1.5 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.2

— — — — — — — — — — —

3.0 3.0

— —

52.5

10.0

—

61.0

1350 1350-H19

                                      

61.0

5005 5005-H19

5005-H19

6201

                              

53.5

8176

a To convert conductivity to maximum resistivity use Table 100. b Applicable up to 0.250 in. c Any test in a lot. d Average of all tests in a lot.

142

7


STANDARD SCREWS∗

Standard Screw Threads The Unified and American Screw Threads included in Table 102 are taken from the publication of the American Standards Association, ASA B1.1—1949. The coarse-thread series is the former United States Standard Series. It is recommended for general use in engineering work where conditions do not require the use of a fine thread. The fine-thread series is the former “Regular Screw Thread Series” established by the Society of Automotive Engineers (SAE). The fine-thread series is recommended for general use in automotive and aircraft work and where special conditions require a fine thread. The extra-fine-thread series is the same as the former SAE fine series and the present SAE extra-fine series. It is used particularly in aircraft and aeronautical equipment where (a) thin-walled material is to be threaded; (b) thread depth of nuts clearing ferrules, coupling flanges, and so on, must be held to a minimum; and (c) a maximum practicable number of threads is required within a given thread length. The method of designating a screw thread is by the use of the initial letters of the thread series, preceded by the nominal size (diameter in inches or the screw

This section is extracted, with permission, from EMPIS Materials Selector. Copyright  1982 General Electric Co.

number) and number of threads per inch, all in Arabic numerals, and followed by the classification designation, with or without the pitch diameter tolerances or limits of size. An example of an external thread designation and its meaning is as follows: Example 1 1/4′′—20UNC —2A Class of screw thread Thread series Number of threads per inch (n) Nominal size

A left-hand thread must be identified by the letters LH following the class designation. If no such designation is used, the thread is assumed to be right hand. Classes of thread are distinguished from each other by the amounts of tolerance and allowance specified in ASA B1.1—1949.


143

Table 102 Standard Screw Threads

Sizes

Basic Major Diameter D (in.)

Threads per Inch n

Basic Pitch Diametera E (in.)

Minor Diameter External Threads Ks (in.)

Minor Diameter Internal Threads Kn (in.)

Section at Minor Diameter at D − 2hb ) (in.2 )

Stress Areab (in.2 )

Coarse-thread Series—UNC and NC (Basic Dimensions) 1 (0.073) 2 (0.086) 3 (0.099) 4 (0.112)

0.0730 0.0860 0.0990 0.1120

64 56 48 40

0.0629 0.0744 0.0855 0.0958

0.0538 0.0641 0.0734 0.0813

0.0561 0.0667 0.0764 0.0849

0.0022 0.0031 0.0041 0.0050

0.0026 0.0036 0.0048 0.0060

5 (0.125) 6 (0.138) 8 (0.164) 10 (0.190) 12 (0.216)

0.1250 0.1380 0.1640 0.1900 0.2160

40 32 32 24 24

0.1088 0.1177 0.1437 0.1629 0.1889

0.0943 0.0997 0.1257 0.1389 0.1649

0.0979 0.1042 0.1302 0.1449 0.1709

0.0067 0.0075 0.0120 0.0145 0.0206

0.0079 0.0090 0.0139 0.0174 0.0240

1 4 5 16 3 8 7 16

0.2500

20

0.2175

0.1887

0.1959

0.0269

0.0317

0.3125

18

0.2764

0.2443

0.2524

0.0454

0.0522

0.3750

16

0.3344

0.2983

0.3073

0.0678

0.0773

0.4375

14

0.3911

0.3499

0.3602

0.0933

0.1060

1 2 1 2 9 16 5 8 3 4 7 8

0.5000

13

0.4500

0.4056

0.4167

0.1257

0.1416

0.5000

12

0.4459

0.3978

0.4098

0.1205

0.1374

0.5625

12

0.5084

0.4603

0.4723

0.1620

0.1816

0.6250

11

0.5660

0.5135

0.5266

0.2018

0.2256

0.7500

10

0.6850

0.6273

0.6417

0.3020

0.3340

0.8750

9

0.8028

0.7387

0.7547

0.4193

0.4612

1

1.0000

8

0.9188

0.8466

0.8647

0.5510

0.6051

1 18

1.1250

7

1.0322

0.9497

0.9704

0.6931

0.7627

1 14

1.2500

7

1.1572

1.0747

1.0954

0.8898

0.9684

1 38

1.3750

6

1.2667

1.1705

1.1946

1.0541

1.1538

1 12

1.5000

6

1.3917

1.2955

1.3196

1.2938

1.4041

1 34

1.7500

5

1.6201

1.5046

1.5335

1.7441

1.8983

2

2.0000

4 12

1.8557

1.7274

1.7594

2.3001

2.4971

2 14 2 12 2 34

2.2500

4 12

2.1057

1.9774

2.0094

3.0212

3.2464

2.5000

4

2.3376

2.1933

2.2294

3.7161

3.9976

2.7500

4

2.5876

2.4433

2.4794

4.6194

4.9326

3

3.0000

4

2.8376

2.6933

2.7294

5.6209

5.9659

3 14

3.2500

4

3.0876

2.9433

2.9794

6.7205

7.0992

3 12

3.5000

4

3.3376

3.1933

3.2294

7.9183

8.3268

3 34

3.7500 4.0000

4 4

3.5876 3.8376

3.4433 3.6933

3.4794 3.7294

9.2143 10.6084

9.6546 11.0805

4

(Continues)

144 Table 102

Sizes

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Basic Major Diameter D (in.)

Threads per Inch n






Fine-Thread Series—UNF and NF (Basic Dimensions) 0 (0.060) 1 (0.073) 2 (0.086) 3 (0.099) 4 (0.112)

0.0600 0.0730 0.0860 0.0990 0.1120

80 72 64 56 48

0.0519 0.0640 0.0759 0.0874 0.0985

0.0447 0.0560 0.0668 0.0771 0.0864

0.0465 0.0580 0.0691 0.0797 0.0894

0.0015 0.0024 0.0034 0.0045 0.0057

0.0018 0.0027 0.0039 0.0052 0.0065

5 (0.125) 6 (0.138) 8 (0.164) 10 (0.190) 12 (0.216)

0.1250 0.1380 0.1640 0.1900 0.2160

44 40 36 32 28

0.1102 0.1218 0.1460 0.1697 0.1928

0.0971 0.1073 0.1299 0.1517 0.1722

0.1004 0.1109 0.1339 0.1562 0.1773

0.0072 0.0087 0.0128 0.0175 0.0226

0.0082 0.0101 0.0146 0.0199 0.0257

1 4 5 16 3 8 7 16

0.2500

28

0.2268

0.2062

0.2113

0.0326

0.0362

0.3125

24

0.2854

0.2614

0.2674

0.0524

0.0579

0.3750

24

0.3479

0.3239

0.3299

0.0809

0.0876

0.4375

20

0.4050

0.3762

0.3834

0.1090

0.1185

1 2 9 16 5 8 3 4 7 8

0.5000

20

0.4675

0.4387

0.4459

0.1486

0.1597

0.5625

18

0.5264

0.4943

0.5024

0.1888

0.2026

0.6250

18

0.5889

0.5568

0.5649

0.2400

0.2555

0.7500

16

0.7094

0.6733

0.6823

0.3513

0.3724

0.8750

14

0.8286

0.7874

0.7977

0.4805

0.5088

1

1.0000

12

0.9459

0.8978

0.9098

0.6245

0.6624

1 18

1.1250

12

1.0709

1.0228

1.0348

0.8118

0.8549

1 14

1.2500

12

1.1959

1.1478

1.1598

1.0237

1.0721

1 38

1.3750

12

1.3209

1.2728

1.2848

1.2602

1.3137

1 12

1.5000

12

1.4459

1.3978

1.4098

1.5212

1.5799

Extra-Fine-Thread Series—NEF (Basic Dimensions) 12 (0.216)

0.2160

32

0.1957

0.1777

0.1822

0.0242

0.0269

1 4 5 16 3 8 7 16

0.2500

32

0.2297

0.2117

0.2162

0.0344

0.0377

0.3125

32

0.2922

0.2742

0.2787

0.0581

0.0622

0.3750

32

0.3547

0.3367

0.3412

0.0878

0.0929

0.4375

28

0.4143

0.3937

0.3988

0.1201

0.1270

1 2 9 16 5 8 11 16

0.5000

28

0.4768

0.4562

0.4613

0.1616

0.1695

0.5625

24

0.5354

0.5114

0.5174

0.2030

0.2134

0.6250

24

0.5979

0.5739

0.5799

0.2560

0.2676

0.6875

24

0.6604

0.6364

0.6424

0.3151

0.3280

3 4 13 16 7 8 15 16

0.7500

20

0.7175

0.6887

0.6959

0.3685

0.3855

0.8125

20

0.7800

0.7512

0.7584

0.4388

0.4573

0.8750

20

0.8425

0.8137

0.8209

0.5153

0.5352

0.9375

20

0.9050

0.8762

0.8834

0.5979

0.6194


145


Sizes

Basic Major Diameter D (in.)


Threads per Inch n





Fine-Thread Series—UNF and NF (Basic Dimensions) 1

1.0000

20

0.9675

0.9387

0.9459

0.6866

0.7095

1 1 16

1.0625

18

1.0264

0.9943

1.0024

0.7702

0.7973

1 18

1.1250

18

1.0889

1.0568

1.0649

0.8705

0.8993

3 1 16

1.1875

18

1.1514

1.1193

1.1274

0.9770

1.0074

1 14

1.2500

18

1.2139

1.1818

1.1899

1.0895

1.1216

5 1 16

1.3125

18

1.2764

1.2443

1.2524

1.2082

1.2420

1 38

1.3750

18

1.3389

1.3068

1.3149

1.3330

1.3684

7 1 16

1.4375

18

1.4014

1.3693

1.3774

1.4640

1.5010

1 12

1.5000

18

1.4639

1.4318

1.4399

1.6011

1.6397

9 1 16

1.5625

18

1.5264

1.4943

1.5024

1.7444

1.7846

1 58

1.6250

18

1.5889

1.5568

1.5649

1.8937

1.9357

1 11 16

1.6875

18

1.6514

1.6193

1.6274

2.0493

2.0929

1 34

1.7500 2.0000

16 16

1.7094 1.9594

1.6733 1.9233

1.6823 1.9323

2.1873 2.8917

2.2382 2.9501

2

Note: Bold type indicates unified threads—UNC and UNF. a British: effective diameter. b The stress area is the assumed area of an externally threaded part which is used for the purpose of computing the tensile strength.

Table 103 ASAa Standard Bolts and Nuts

Nominal Size

Across Flats (in.)

Across Square Corners (in.)

Across Hex Corners (in.)

Thickness Unfinished (in.)

Semifinished (in.)

11 64 13 64 1 4 19 64 21 64 3 8 27 64 1 2 19 32 21 32 3 4

5 32 3 16 15 64 9 32 19 64 11 32 25 64 15 32 9 16 19 32 11 16

Regular Bolt Heads 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

1 1 81

3 8 1 2 9 16 5 8 3 4 7 8 15 16 1 18 5 1 16 1 12 1 11 16

0.498

0.413

0.665

0.552

0.747

0.620

0.828

0.687

0.995

0.826

1.163

0.966

1.244

1.033

1.494

1.240

1.742

1.447

1.991

1.653

2.239

1.859

(Continues)

146 Table 103

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Across Square Corners (in.)

Across Hex Corners (in.)

Nominal Size

Across Flats (in.)

1 14

1 87

2.489

2.066

1 38 1 12 1 58 1 34 1 78

1 2 16 1 24 7 2 16 2 85 13 2 16

2.738

2.273

2.986

2.480

1

3.235

2.686

3 1 32

3.485

2.893

5 1 32

3.733

3.100

1 14

2

3

3.982

3.306

1 11 32

2 14 2 12 2 34

3 83 3 43 4 81 4 21

4.479

3.719

1 12

4.977

4.133

1 21 32

5.476

4.546

1 53 64

5.973

4.959

2

1.167

0.969

1.249

1.037

1.416

1.175

1.665

1.383

1.914

1.589

2.162

1.796

1 18

7 8 15 16 1 1 16 1 14 7 1 16 5 18 13 1 16

2.411

2.002

1 14

2

2.661

2.209

1

1 38

3 2 16

2.909

2.416

3 1 32

1 12

2 83

3.158

2.622

3 1 16

1 58

9 2 16

3.406

2.828

9 1 32

1 34 1 78

2 43 15 2 16 3 81 3 21 3 87 4 41 4 85

3.655

3.036

1 38

3.905

3.242

1 15 32

4.153

3.449

9 1 16

4.652

3.862

1 34

5.149

4.275

1 15 16

5.646

4.688

2 18

2

6.144

5.102

5 2 16

3 2 16

3

Thickness Unfinished (in.)

Semifinished (in.)

27 32 29 32

25 32 27 32 15 16 1 1 32 3 1 32 3 1 16 7 1 32 1 38 17 1 32 11 1 16 1 78

Heavy Bolt Heads 1 2 9 16 5 8 3 4 7 8

1

2 2 14 2 12 2 34 3

Nominal Size

Width Across Flats (in.)

Width Across Corners Square Hex (in.) (in.)

7 16 15 32 17 32 5 8 23 32 13 16 29 32

Thickness Unfinished, Regular Nuts Jam Nuts (in.) (in.)

13 32 7 16 1 2 19 32 11 16 3 4 27 32 15 16 1 1 32 1 18 7 1 32 5 1 16 13 1 32 7 1 16 1 58 13 1 16

Thickness Semifinished, Regular Nuts Jam Nuts (in.) (in.)

Regular Nuts and Regular Jam Nuts 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4

7 16 9 16 5 8 3 4 13 16 7 8

0.584

0.484

0.751

0.624

0.832

0.691

1.000

0.830

1.082

0.898

1.163

0.966

1

1.330

1.104

1 18

1.494

1.240

7 32 17 64 21 64 3 8 7 16 1 2 35 64 21 32

5 32 3 16 7 32 1 4 5 16 11 32 3 8 7 16

13 64 1 4 5 16 23 64 27 64 31 64 17 32 41 64

9 64 11 64 13 64 15 64 19 64 21 64 23 64 27 64


147


Nominal Size

Width Across Flats (in.)

Width Across Corners Square Hex (in.) (in.)

Thickness Unfinished, Regular Nuts Jam Nuts (in.) (in.) 1 2 9 16 5 8 3 4 13 16 7 8 15 16

7 8

5 1 16

1.742

1.447

1

1 21

1.991

1.653

49 64 7 8

1 81

11 1 16

2.239

1.859

1

1 41 1 83 1 21 1 85 1 43 1 87

1 87 1 2 16 1 24 7 2 16 5 28 13 2 16

2.489

2.066

3 1 32

2.738

2.273

13 1 64

2.986

2.480

5 1 16

3.235

2.686

27 1 64

3.485

2.893

17 1 32

1

3.733

3.100

41 1 64

1 1 16

2

3

3.982

3.306

1 34

1 18

2 41

3 83

4.479

3.719

31 1 32

1 14

2 21

3 43

4.977

4.133

3 2 16

1 12

2 43

4 81 4 21

5.476

4.546

1 58

5.973

4.959

13 2 32 2 58

0.670

0.556

0.794

0.659

0.919

0.763

1.042

0.865

1.167

0.969

1.249

1.037

1.416

1.175

1.665

1.382

1.914

1.589

2.162

1.796

1

1 81

1 2 19 32 11 16 25 32 7 8 15 16 1 1 16 1 14 7 1 16 5 18 13 1 16

2.411

2.002

1 18

1 41

2

2.661

2.209

1 14

1 83

3 2 16

2.909

2.416

1 38

1 21

2 83

3.158

2.622

1 12

1 85 1 43 1 87

3.406

2.828

1 58

3.656

3.035

1 34

1

3.905

3.242

1 78

1 1 16

4.153

3.449

2

1 18

4.652

3.862

1 14

5.149

4.275

5.646

4.688

2 14 2 12 2 34

3

9 2 16 3 24 15 2 16 3 81 3 21 3 87 4 41 4 85

6.144

5.102

3

1 34

3 41

5

6.643

5.515

3 14

1 78

3 21

5 83

7.140

5.928

3 12

2

3 43

5 43 6 81

7.637

6.341

3 34

2 18

8.135

6.755

4

2 14

3

1 34

Thickness Semifinished, Regular Nuts Jam Nuts (in.) (in.) 3 4 55 64 31 32 1 1 16 11 1 64 9 1 32 25 1 64 1 12 1 39 64 1 23 32 1 59 64 9 2 64 23 2 64 2 37 64

31 64 35 64 39 64 23 32 25 32 27 32 29 32 31 32 1 1 32 3 1 32 1 13 64 1 29 64 1 37 64 1 45 64

15 64 19 64 23 64 27 64 31 64 35 64 39 64 47 64 55 64 63 64 7 1 64 7 1 32 1 11 32 1 15 32 1 19 32 1 23 32 1 27 32 1 31 32 2 13 64 2 29 64 2 45 64 2 61 64 3 3 16 7 3 16 11 3 16 3 15 16

11 64 13 64 15 64 17 64 19 64 21 64 23 64 27 64 31 64 35 64 39 64 23 32 25 32 27 32 29 32 31 32 1 1 32 3 1 32 13 1 64 1 29 64 1 37 64 1 45 64 1 13 16 1 15 16 1 2 16 3 2 16

Heavy Nuts and Heavy Jam Nuts 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

1

2 2 41 2 21 2 43

4

1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

3 16 7 32 1 4 9 32 5 16 11 32 3 8 7 16 1 2 9 16 5 8 3 4 13 16 7 8 15 16

1 12 1 58

148 Table 103

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) Regular Slotted Nuts Semifinished Width

Nominal Size

Across Flats (in.)

Across Corners (in.)

7 16 9 16 5 8 3 4 13 16 7 8

0.485

1

1.104

1 18

1.240

5 1 16

1.447

1

1 12

1.653

1 18 1 14 1 38 1 12 1 58 1 34 1 78

1 11 16 1 78 1 2 16 1 24 7 2 16 5 28 2 13 16

1.859

2

3

3.306

2 14

3 38

3.719

2 12 2 34

3 34 4 18 4 12

1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

3 a

0.624 0.691 0.830 0.898 0.966

2.066 2.273 2.480 0.686 2.893 3.100

4.133 4.546 4.959

Heavy Slotted Nuts Semifinished Width

Thickness (in.) 13 64 1 4 5 16 23 64 27 64 31 64 17 32 41 64 3 4 55 64 31 32 1 1 16 11 1 64 9 1 32 25 1 64 1 12 1 39 64 1 23 32 1 59 64 9 2 64 23 2 64 2 37 64

Across Flats (in.)

Across Corners (in.)

1 2 19 32 11 16 25 32 7 8 15 16 1 1 16 1 14 7 1 16 1 58 1 13 16

0.556

2

2.209

3 2 16

2.416

2 38

2.622

9 2 16

2.828

2 34

3.035

2 15 16

3.242

3 18

3.449

3 12

3.862

3 78 4 14 4 58

4.275

0.659 0.763 0.865 0.969 1.037 1.175 1.382 1.589 1.796 2.002

4.688 5.102

Slot

Thickness (in.)

Width (in.)

Depth (in.)

15 64 19 64 23 64 27 64 31 64 35 64 39 64 47 64 55 64 63 64 7 1 64 7 1 32 11 1 32 1 15 32 1 19 32 1 23 32 1 27 32 1 31 32 2 13 64 2 29 64 2 45 64 2 61 64

5 64 3 32 1 8 1 8 5 32 5 32 3 16 3 16 3 16 1 4 1 4 5 16 5 16 3 8 3 8 7 16 7 16 7 16 7 16 9 16 9 16 5 8

3 32 3 32 1 8 5 32 5 32 3 16 7 32 1 4 1 4 9 32 11 32 3 8 3 8 7 16 7 16 1 2 9 16 9 16 9 16 11 16 11 16 3 4

ANSI standards B18.2.1—1981, B18.2.2—1972 (R1983), B18.6.3—1972 (R1983).

Selection of Screws By definition, a screw is a fastener that is intended to be torqued by the head. Screws are the most widely used method of assembly despite recent technical advances of adhesives, welding, and other joining techniques. Use of screws is essential in those applications that require ease of disassembly for normal maintenance and service. There is no real economy if savings made in factory installation create service problems later. There are many types of screws, and each variety will be treated separately. Material selection is generally common to all types of screws. Material. Not all materials are suitable for the processes used in the manufacture of fasteners. Largevolume users or those with critical requirements can be very selective in their choice of materials. Low-volume users or those with noncritical applications would be

wise to permit a variety of materials in a general category in order to improve availability and lower cost. For example, it is usually desirable to specify lowcarbon steel or 18-8-type stainless steel∗ rather than ask for a specific grade. Low-carbon steel is widely used in the manufacture of fasteners where lowest cost is desirable and tensile strength requirements are ∼50, 000 psi. If corrosion is a problem, these fasteners can be plated either electrically or mechanically. Zinc or cadmium plating is used in most applications. Other finishes include nickel, chromium, copper, tin, and silver electroplating; electroless nickel and other immersion coatings; hot dip galvanizing; and phosphate coatings. ∗ Manufacturer may use UNS—S30200, S30300, S30400, S30500 (AISI type 302, 303, 304, or 305) depending upon quantity, diameter, and manufacturing process.


Medium-carbon steel, quenched, and tempered is widely used in applications requiring tensile strengths from 90,000 to 120,000 psi. Alloy steels are used in applications requiring tensile strengths from 115,000 to 180,000 psi, depending on the grade selected. Where better corrosion resistance is required, 300 series stainless steel can be specified. The 400 series stainless steel is used if it is necessary to have a corrosion-resistant material that can be hardened and tempered by heat treatment. For superior corrosion resistance, materials such as brass, bronze, aluminum, or nickel are sometimes used in the manufacture of fasteners. If strength is no problem, plastics such as nylons are used in severe corrosion applications. Drivability. When selecting a screw, thought must be given to the means of driving for assembly and disassembly as well as the head shape. Most screw heads provide a slot, a recess, or a hexagon shape as a means of driving. The slotted screw is the least preferred driving style and serves only when appearance must be combined with ease of disassembly with a common screwdriver. Only a limited amount of torque can be applied with a screwdriver. A slot can become inoperative after repeated disassembly destroys the edge of the wall that the blade of the screwdriver bears against. The hexagon head is preferred for the following reasons:

Least likely to accidentally spin out (thereby marring the surface of the product) Lowest initial cost Adaptable to high-speed power drive Minimum worker fatigue

Fig. 1

149

Ease of assembly in difficult places Permits higher driving torque, especially in large sizes where strength is important Contains no recess to become clogged with dirt and interfere with driving Contains no recess to weaken the head Unless frequent field disassembly is required, use of the unslotted hex head is preferred. Appearance is the major disadvantage of the hex head, and this one factor is judged sufficient to eliminate it from consideration for the front or top of products. The recessed head fastener is widely used and becomes the first choice for appearance applications. It usually costs more than a slot or a hexagon shape. There are many kinds of recesses. The Phillips and Phillips POZIDRIV are most widely used. To a lesser extent the Frearson, clutch-type, hexagonal, and fluted socket heads are used. For special applications, proprietary types of tamper-resistant heads can be selected (Fig. 1). The recessed head has some of the same advantages as the hex head (see preceding list). It also has improved appearance. The Phillips POZIDRIV is slowly replacing the Phillips recess. The POZIDRIV recess can be readily identified by four radial lines centered between each recess slot. These slots are a slight modification of the conventional Phillips recess. This change improves the fit between the driver and the recess, thus minimizing the possibility of marring a surface from accidental spinout of the driver as well as increasing the life of the driver. The POZIDRIV design is recommended in high-production applications requiring high driving torques. The POZIDRIV

Recessed head fasteners.

150


recess usually sells at a high-production applications requiring high driving torques. The POZIDRIV recess usually sells at a slightly higher price than the conventional Phillips recess, but some suppliers will furnish either at the same price. The savings resulting from longer tool life will usually justify the higher initial cost. A conventional Phillips driver could be used to install or disassemble a POZIDRIV screw. However, a POZIDRIV driver should be used with a POZIDRIV screw in order to take advantage of the many features inherent in the new design. To avoid confusion, it should be clearly understood that a POZIDRIV driver cannot be used to install or remove a conventional Phillips head screw. A Frearson recess is a somewhat different design than a Phillips recess and has the big advantage that one driving tool can be used for all sizes whereas a Phillips may require four driving tools in the range from no. 2 (0.086-in.) to 3/8 (0.375-in.) screw size. This must be balanced against the following disadvantages: Limited availability. Greater penetration of the recess means thinner walls between the bottom of the recess and the outer edge of the screw, which tends to weaken the head. The sharp point of the driver can easily scratch or otherwise mar the surface of the product if it accidently touches. Although one driver can be used for all sizes, for optimum results, different size drivers are recommended for installing various screw sizes, thus minimizing the one real advantage of the Frearson recess. The hexagon and fluted socket head cap screws are only available in expensive high-strength alloy steel. Its unique small outside diameter or cylindrical head is useful on flanges, counterbored holes, or other locations where clearances are restricted. Such special applications may justify the cost of a socket head cap screw. Appreciable savings can be made in other applications by substitution of a hexagon head screw. Despite any claims to the contrary, the dimensional accuracy of hexagon socket head cap screws is no better than that of other cold-headed products, and there is no merit in close-thread tolerances, which are advocated by some manufacturers of these products. The high prices, therefore, should be justified solely on the basis of possible space savings in using the cylindrical head. The fluted socket is not as readily available and should only be considered in the very small sizes where a hexagon key tends to round out the socket. The fluted socket offers spline design so that the key will neither slip nor be subject to excessive wear. Many types of special recesses are tamper resistant. In most of these designs, the recess is an unusual shape

requiring a special tool for assembly and disassembly. A readily available driving tool such as a screwdriver or hexagon key would not fit the recess. The purpose of a tamper-resistant fastener is to prevent unauthorized removal of parts and equipment. Their protection is needed on any product located in public places to discourage vandalism and thievery. They may also be necessary on some consumer products as a safety measure to protect the amateur repairman from injury or to prevent him from causing serious damage to equipment. With product liability mania what it is today, the term “tamperproof” has all but disappeared. Now the fasteners are called “tamper resistant.” They are the same as they were under their previous name, but the new term better reflects their true capabilities. Any skilled thief with ample time and proper tools can saw, drill, blast, or otherwise disassemble any tamper-resistant fastener. Therefore, these fasteners are intended only to discourage the casual thief or amateur tinkerer and make it more difficult for a skilled professional. Whatever the choice of fastener design, it is essential that hardened material be specified. No fastener is ever truly tamperproof, but hardened steel helps. Fasteners made of soft material can be disassembled easily by sawing a slot, hammering with a chisel, or drilling a hole and using an extraction bit. Head Shapes The following information is equally applicable to all types of recesses as well as a slotted head. For simplification only slotted screws are shown. The pan head is the most widely used and is intended to replace the round, binding, and truss heads in order to keep varieties to a minimum. It is preferred because it presents the best combination of appearance with adequate head height to minimize weakness due to depth of penetration of the recess (Fig. 2). The round head was widely used in the past (Fig. 3). It has since been delisted as an American National Standard. Give preference to pan heads on all new designs. Figure 4 shows the superiority of the pan head: The high edge of the pan head at its periphery,

Fig. 2

Fig. 3

Pan head.

Round head.


Fig. 4

Drive-slot engagement.

where driving action is most effective, provides superior driver-slot engagement and reduces the tendency to chew away the metal at the edge of the slot. The flat head is used where a flush surface is required. The countersunk section aids in centering the screw (Fig. 5). The oval head is similar to a flat head except that instead of a flush surface it presents a low silhouette that improves the appearance (Fig. 6). The truss head is similar to the round head except that the head is shallower and has a larger diameter. It is used where extra bearing surface is required for extra holding power or where the clearance hole is oversized or the material is soft. It also presents a low silhouette that improves the appearance (Fig. 7). The binding head is similar to the pan head and is commonly used for electrical connections where an undercut is usually specified to bind and prevent the fraying of stranded wire (Fig. 8). The fillister head has the smallest diameter for a given shank size. It also has a deep slot that allows a higher torque to be applied during assembly. It is not as readily available or as widely used as some of the other head styles (Fig. 9).

Fig. 5

151

Fig. 8

Binding head.

Fig. 9

Fillister head.

Flat head. Fig. 10

Fig. 6

Oval head.

Fig. 7

Truss head.

Hex head.

The advantages of a hex head are listed in the discussion on drivability. This type head is available in eight variations (Fig. 10). The indented design is lowest cost as the hex is completely cold upset in a counterbore die and possesses an identifying depression in the top surface of the head. The trimmed design requires an extra operation to produce clean sharp corners with no indentation. Appearance is improved and there is no pocket on top to collect moisture.

152


The washer design has a larger bearing surface to spread the load over a wider area. The washer is an integral part of the head and also serves to protect the finish of the assembly from wrench disfigurement. The slot is used to facilitate field service. It adds to the cost, weakens the head, and limits the amount of tightening torque that can be applied. A slot is unnecessary in high-production factory installation. Any given location should standardize on one or possibly two of the eight variations. Types of Screws Machine Screws. Machine screws are meant to be assembled in tapped holes, either into a product or into a nut. The screw threads of a machine screw are readily available in American National Standard Unified Inch Coarse and Fine Thread series. They are generally considered for applications where the material to be joined is too hard, too weak, too brittle, or too thick to take a tapping screw. It is also used in applications where the assembly requires a fastener made of a material that cannot be hardened enough to make its own thread, such as brass or nylon machine screws. Applications requiring freedom from dust or particles of any kind cannot use thread-cutting screws and, therefore, must be joined by machine screws or a tapping screw which forms or rolls a thread. There are many combinations of head styles, shapes, and materials. Self-Tapping Screws. There are many different types of self-tapping screws commercially available. The following three types are capable of creating an internal thread by being twisted into a smooth hole:

1. Thread-forming screws 2. Thread-cutting screws 3. Thread-rolling screws The following two types create their own opening before generating the thread:

Types A and AB. Type AB screws have a spaced thread. This means that each thread is spaced further away from its adjacent thread than the popular machine screw series. They also have a gimlet point for ease in entering a predrilled hole. This type of screw is primarily intended to be used in sheet metal with a thickness from 0.015 in. (0.38 mm) to 0.05 in. (1.3 mm), resin-impregnated plywood, natural woods, and asbestos compositions. Type AB screws were introduced several years ago to replace the type A screws. The type A screw is the same as the type AB except for a slightly wider spacing of the threads. Both are still available and can be used interchangeably. The big advantage of the type AB screw is that its threads are spaced exactly as the type B screws to be discussed later. In the interest of standardization it is recommended that type AB screws be used in place of either the type A or the type B series (Fig. 11). Type B. Type B screws have the same spacing as type AB screws. Instead of a gimlet point, they have a blunt point with incomplete threads at the point. This point makes the type B more suitable for thicker metals and blind holes. The type B screws can be used in any of the applications listed under type AB. In addition the type B screw can be used in sheet metal up to a thickness of 0.200 in. (5 mm) and in nonferrous castings (Fig. 12). Type C. Type C screws look like type B screws except that threads are spaced to be exactly the same as a machine screw thread and may be used to replace a machine screw in the field. They are recommended for general use in metal 0.030–0.100 in. (0.76–2.54 mm) thick. It should be recognized that in specific applications, involving long thread engagement or hard materials, this type of screw requires extreme driving torques. 2. Thread-Cutting Screws. Thread-cutting screws create an internal thread by actual removal of material from the internal hole. The design of the cavity to provide space for the chips and the design of the cutting edge differ with each type. They are used in place of the thread-forming type for applications

4. Self-drilling and tapping screws 5. Self-extruding and tapping screws 1. Thread-Forming Screws. Thread-forming screws create an internal thread by forming or squeezing material. They rely on the pressure of the screw thread to force a mating thread into the workpiece. They are applicable in materials where large internal stresses are permissible or desirable to increase resistance to loosening. They are generally used to fasten sheet metal parts. They cannot be used to join brittle materials, such as plastics, because the stresses created in the workpiece can cause cracking. The following types of thread-forming screws are commonly used:

Fig. 11 Type AB.

Fig. 12 Type B.


153

in materials where disruptive internal stresses are undesirable or where excessive driving torques are encountered. The following types of thread-cutting screws are commonly used: Type BT (Formerly Known as Type 25). Type BT screws have a spaced thread and a blunt point similar to the type B screw. In addition they have one cutting edge and a wide chip cavity. These screws are primarily intended for use in very friable plastics such as urea compositions, asbestos, and other similar compositions. In these materials, a larger space between threads is required to produce a satisfactory joint because it reduces the buildup of internal stresses that fracture brittle plastic when a closer spaced thread is used. The wide cutting slot creates a large cutting edge and permits rapid deflection of the chips to produce clean mating threads. For best results all holes should be counterbored to prevent fracturing the plastic. Use of this type screw eliminates the need to use tapped metallic inserts in plastic materials (Fig. 13). Type ABT. Type ABT screws are the same as type BT screws except that they have a gimlet point similar to a type AB screw. This design is not recognized as an American National Standard and should only be selected for large-volume applications (over 50,000 pieces of one size and type). It is primarily intended for use in plastic for the same reasons as listed for type BT screws (Fig. 14). Type D (Formerly Known as Type 1). Type D screws have threads of machine screw diameter–pitch combinations approximating unified form with a blunt point and tapered entering threads. In addition a slot is cut off center with one side on the center line. This radial side of the slot creates the sharp serrated cutting edge such as formed on a tap. The slot leaves a thinner section on one side of the screw that collapses and helps concentrate the pressure on the cutting edge. This screw is suitable for use in all thicknesses of metals (Fig. 15). Type F. Type F screws are identical to type D except that instead of one slot there are several slots cut at a slight angle to the axis of the thread. This screw is

Fig. 13

Fig. 14

Type BT.

Type ABT.

Fig. 15

Type D.

Fig. 16 Type F.

suitable for use in all thicknesses of metals and can be used interchangeably with a type D screw in many applications. However, the type F screw is superior to the type D screw for tapping into cast iron and permits the use of a smaller pilot hole (Fig. 16). Type D or Type F. Because in many applications these two types can be used interchangeably with the concomitant advantages of simpler inventory and increased availability, a combined specification is often issued permitting the supplier to furnish either type. Type T (Formerly Known as Type 23). Type T screws are similar to type D and type F except that they have an acute rake angle cutting edge. The cut in the end of the screw is designed to eliminate a pocket that confines the chips. The shape of the slot is such that the chips are forced ahead of the screw as it is driven. This screw is suitable for plastics and other soft materials when a standard machine screw series thread is desired. It is used in place of type D and type F when more chip room is required because of deep penetration (Fig. 17). Type BF. Type BF screws are intended for use in plastics. The wide thread pitch reduces the buildup of internal stresses that fracture brittle plastics when a smaller thread pitch is used. The screw has a blunt point and tapered entering threads with several cutting edges and chip cavity (Fig. 18).

Fig. 17 Type T.

Fig. 18

Type BF.

154


Fig. 19

Thread-rolling screws.

3. Thread-Rolling Screws. Thread-rolling screws (see Fig. 19) form an internal thread by flowing metal and thus do not cut through or disrupt the grain flow lines of materials as do thread-cutting screws. The screw compacts and work hardens the material, thereby forming strong, smoothly burnished internal threads. The screws have the threads of machine screw diameter–pitch combinations. This type screw is ideal for applications where chips can cause electrical shorting of equipment or jamming of delicate mechanism. Freedom from formation of chips eliminates the costly problem of cleaning the product of chips and burrs as would otherwise be required. The ratio of driving torque to stripping torque is approximately 1 : 8 for a thread-rolling screw as contrasted to 1 : 3 for a conventional tapping screw. This higher ratio permits the driver torque release to be set well over the required driving torque and yet safely below the stripping torque. This increased ratio minimizes poor fastening due to stripped threads or inadequate seating of the screws. Plastite is intended for use in filled or unfilled thermoplastics and some of the thermosetting plastics. The other three types are intended for use in metals. At present, there are no data to prove the superiority of one type over another. 4. Self-Drilling and Tapping Screws. The selfdrilling and tapping screw (Fig. 20) drills its own hole and forms a mating thread, thus making a complete fastening in a single operation. Assembly labor is reduced by eliminating the need to predrill holes at assembly and by solving the problem of hole alignment. These screws must complete their metal-drilling function and fully penetrate the material before the screw thread can engage and begin its advancement. In order to meet this requirement, the unthreaded point length must be equal to or greater than the material thickness to be drilled. Therefore, there is a strict limitation on minimum and maximum material thickness that varies with screw size. There are many different styles and types of self-drilling and tapping screws to meet specific needs. 5. Self-Extruding Screws. Self-extruding screws provide their own extrusion as they are driven into an inexpensively produced punched hole. The resulting extrusion height is several times the base material thickness. This type screw is suitable for material in

Fig. 20 Self-drilling and tapping screws.

Fig. 21

Self-extruding screw.

thicknesses up to 0.048 in. (1.2 mm). By increasing the thread engagement, these screws increase the differential between driving and stripping torque and provide greater pull-out strength. Since they do not produce chips, they are excellent for grounding sheet metal for electrical connections (Fig. 21). There is almost no limit to the variety of head styles, thread forms, and screw materials that are available commercially. The listing only shows representative examples. Users should attempt to keep varieties to a minimum by carefully selecting those variations that best meet the needs of their type of product. Set Screws. Set screws are available in various combinations of head and point style as well as material and are used as locking, locating, and adjustment devices. The common head styles are slotted headless, square head, hexagonal socket, and fluted socket. The slotted headless has the lowest cost and can be used in a counterbored hole to provide a flush surface. The square head is applicable for location or adjustment of static parts where the projecting head is not objectionable. Its use should be avoided on all rotating parts. The hexagonal socket head can be used in a counterbored hole to provide a flush surface. It permits greater torque to be applied than with a slotted headless design. Fluted sockets are useful in very small

MATHEMATICAL AND PHYSICAL UNITS, STANDARDS, AND TABLES Table 104 d (in.) P (lb)

155

Holding Power of Flat or Cup Point Set Screws 1 4

5 16

3 8

7 16

1 2

9 16

5 8

3 4

7 8

100

168

256

366

500

658

840

1280

1830

diameters, that is, no. 6 (0.138 in.) and under, where hexagon keys tend to round out the socket in hexagonal socket set screws. Set screws should not be used to transmit large amounts of torque, particularly under shock torsion loads. Increased torsion loads may be carried by two set screws located 120◦ apart. The following points are available with the head styles discussed: The cup point (Table 104) is the standard stock point for all head shapes and is recommended for all general locking purposes. Flats are recommended on round shafts when close fits are used and it is desirable to avoid interference in disassembling parts because of burrs produced by action of the cup point or when the flats are desired to increase torque transmission. When flats are not used, it is recommended that the minimum shaft diameter be not less than four times the cup diameter since otherwise the whole cup may not be in contact with the shaft. The self-locking cup point has limited availability. It has counterclockwise knurls to prevent the screw from working loose even in poorly tapped holes (Fig. 22). When oval points are used, the surface it contacts should be grooved or spotted to the same general contour as the point to assure good seating. It is used where frequent adjustment is necessary without excessive deformation of the part against which it bears (Fig. 23). When flat points are used, it is customary to grind a flat on the shaft for better point contact. This point is preferred where wall thickness is thin and on top of plugs made of any soft material (Fig. 24). When the cone point is used, it is recommended that the angle of countersink be as nearly as possible the angle of screw point for the best efficiency. Cone point

Fig. 22 Cup point.

Fig. 23 Oval point.

1 2500

Fig. 24

Fig. 25

Fig. 26

1 81 3388

1 14 4198

Flat point.

Cone point.

Half-dog point.

set screws have some application as pivot points. It is used where permanent location of parts is required. Because of penetration, it has the highest axial and torsional holding power of any point (Fig. 25). The half-dog point should be considered in lieu of full-dog points when the usable length of thread is less than the nominal diameter. It is also more readily obtained than the full-dog point. It can be used in place of dowel pins and where end of thread must be protected (Fig. 26). Lag Screws. Lag screws (Table 105) are usually used in wood but also can be used in plastics and with expansion shields in masonry. A 60◦ gimlet point is the most readily available type. A 60◦ cone point, not covered in these drawings, is also available. Some suppliers refer to this item as a lag bolt (Fig. 27). A lag screw is normally used in wood when it is inconvenient or objectionable to use a through bolt and nut. To facilitate the insertion of the screw especially in denser types of wood, it is advisable to use a lubricant on the threads. It is important to have a pilot hole of proper size and following are some recommended hole sizes for commonly used types of wood. Hole sizes for other types of wood should be in proportion

156

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Table 105 Lag Screws Diameter of screw (in.) No. of threads per inch Across flats of hexagon and square heads (in.) Thickness of hexagon and square heads (in.)

1 4

5 16

3 8

7 16

1 2

5 8

3 4

7 8

1

10

9

7

7

6

5

4

3 8 3 16

15 32 1 4

9 16 5 16

21 32 3 8

3 4 7 16

15 16 17 32

4 12 1 18

5 1 16

3 21 1 21

5 8

3 4

7 8

Length of Threads for Screws of All Diameters Length of screw (in.)

1 12

2

2 21

3

3 12

4

4 12

To head

2

2 14

2 12

3

3 12

6

7

8

9

10–12

4 21

5

6

6

7

Length of screw (in.)

5

1 12 5 12

Length of thread (in.)

4

4

Length of thread (in.)

Table 106 Recommended Diameters of Pilot Hole for Types of Wooda Screw Diameter (in.) 0.250 0.312 0.375 0.438 0.500 0.625 0.750 a

White Oak

Southern Yellow Pine, Douglas Fir

Redwood, Northern White Pine

0.160 0.210 0.260 0.320 0.375 0.485 0.600

0.150 0.195 0.250 0.290 0.340 0.437 0.540

0.100 0.132 0.180 0.228 0.280 0.375 0.480

Pilot holes should be slightly larger than listed when lag screws of excessive lengths are to be used.

Fig. 27 Lag screws. Fig. 28 Shoulder screw.

to the relative specific gravity of that wood to the ones listed in Table 106. Shoulder Screws. These screws are also referred to as “stripper bolts.” They are used mainly as locators or retainers for spring strippers in punch and die operations and have found some application as fulcrums or pivots in machine designs that involve links, levers, or other oscillating parts. Consideration should be given to the alternative use of a sleeve bearing and a bolt on the basis of both cost and good design (Fig. 28). Thumb Screws. Thumb screws have a flattened head designed for manual turning without a driver or a wrench. They are useful in applications requiring frequent disassembly or screw adjustment (Fig. 29). Weld Screws. Weld screws come in many different head configurations, all designed to provide one or more projections for welding the screw to a part.

Fig. 29

Thumb screws.

Overhead projections are welded directly to the part. Underhead projections go through a pilot hole. The designs in Figs. 30 and 31 are widely used. In projection welding of carbon steel screws, care should be observed in applications, since optimum weldability is obtained when the sum, for either parent metal or screw, of one-fourth the manganese content


Fig. 30

of seasoned wood is given by the formula P = KD 2 , where P is the lateral resistance per screw (lb), D is the diameter (in.), and K is 4000 for oak (red and white), 3960 for Douglas fir (coast region) and southern pine, and 3240 for cypress (southern) and Douglas fir (inland region). The following rules should be observed: (a) The size of the lead hole in soft (hard) woods should be about 70% (90%) of the core or root diameter of the screw; (b) lubricants such as soap may be used without great loss in holding power; (c) long, slender screws are preferable generally, but in hardwood too slender screws may reach the limit of their tensile strength; and (d) in the screws themselves, holding power is favored by thin sharp threads, rough unpolished surface, full diameter under the head, and shallow slots.

Single-projection weld screw.

Fig. 31 Underhead weld screws.

plus the carbon content does not exceed 0.38. For good weldability with the annular ring type, the height of the weld projection should not exceed half the parent metal thickness as a rule of thumb. Copper flash plating is provided for applications where cleanliness of the screw head is necessary in obtaining good welds. Wood Screws. Wood screws are (Table 107) readily available in lengths from 14 to 5 in. for steel and from 14 to 3 21 in. for brass. Consideration should be given to the use of type AB thread-forming screws, which are lower in cost and more efficient than wood screws for use in wood. Wood screws are made with flat, round, or oval heads. The resistance of wood screws to withdrawal from side grain of seasoned wood is given by the formula P = 2850G2 D, where P is the allowable load on the screw (lb/in. penetration of the threaded portion), G is specific gravity of oven-dry wood, and D is the diameter of the screw (in.). Wood screws should not be designed to be loaded in withdrawal from the end grain. The allowable safe lateral resistance of wood screws embedded seven diameters in the side grain

Table 107

SEMS. The machine and tapping screws can be purchased with washers or lock washers as an integral part of the purchased screws. When thus joined together, the part is known as a SEMS unit. The washer is assembled on a headed screw blank before the threads are rolled. The inside diameter of the washer is of a size that will permit free rotation and yet prevent disassembly from the screw after the threads are rolled. If these screws and washers were purchased separately, there would be an initial cost savings over the preassembled units. However, these preassembled units reduce installation time because only one hand is needed to position them, leaving the other hand free to hold the driving tool. The time required to assemble a loose washer is eliminated. In addition, these assemblies act to minimize installation errors and inspection time because the washer is in place, correctly oriented. Also the use of a single unit, rather than two separate parts, simplifies bookkeeping, handling, inventory, and other related operations. 7.1 Nominal and Minimum Dressed Sizes of American Standard Lumber Table 108 applies to boards, dimensional lumber, and timbers. The thicknesses apply to all widths and all widths to all thicknesses.

American Standard Wood Screwsa

Number Threads per inch Diameter (in.) Number Threads per inch Diameter (in.) a Included

157

0 32 0.060 9 14 0.177

1 28 0.073 10 13 0.190

2 26 0.086 11 12 0.203

angle of flathead = 82◦ ; see Fig. 18.

3 24 0.099 12 11 0.216

4 22 0.112 14 10 0.242

5 20 0.125 16 9 0.268

6 18 0.138 18 8 0.294

7 16 0.151 20 8 0.320

8 15 0.164 24 7 0.372

158


Item

Nominal and Minimum Dressed Sizes of American Standard Lumber Thicknesses Minimum Dressed Drya Green (in.) (in.)

Nominal

Boardsb

3 4

1 1 14

1

1 12

1 14

25 32 1 1 32 9 1 32

Dimension

2

1 12

9 1 16

2 12

2 2 12 3

1 2 16 9 2 16 1 3 16

3 3 12

Dimension

Timbers a Maximum

4

3 12

9 3 16

4 12

4

1 4 16

5 and thicker

1 2

off

Nominal                                                                                                                                 

Face Widths Minimum Dressed Drya Green (in.) (in.)

2 3 4

1 21 2 21 3 21

9 1 16 9 2 16 9 3 16

5 6 7 8

4 21 5 21 6 21 7 41

4 58 5 58 6 58 7 12

9 10 11

8 41 9 41 10 41

8 12 9 12 10 21

12 14 16

11 41 13 41 15 41

11 21 13 21 15 21

2

1 21

9 1 16

3 4 5

2 21 3 21 4 21 5 21 7 41 9 41 11 41 13 41 15 41 1 21 2 21 3 21 4 21 5 21 7 41 9 41 11 41

9 2 16 9 3 16 4 58

6 8 10 12 14 16 2 3 4 5 6 8 10 12

14 16 5 and wider

5 58 7 12 9 12 11 21 13 21 15 21 9 1 16 9 2 16 9 3 16 4 58 5 58

7 12 9 12 11 21 13 21 15 21 1 2 off

moisture content of 19 % or less. less than the minimum thickness for 1 in. nominal but 58 in. or greater thickness dry ( 11 16 in. green) may be regarded as American Standard Lumber, but such boards shall be marked to show the size and condition of seasoning at the time of dressing. They shall also be distinguished from 1-in. boards on invoices and certificates. Source: From American Softwood Lumber Standard, NBS 20–70, National Bureau of Standards, Washington, DC, 1970, amended 1986 (available from Superintendent of Documents). b Boards

CHAPTER 2 MATHEMATICS∗ J. N. Reddy Department of Mechanical Engineering Texas A&M University College Station, Texas 1

2

3

4

ARITHMETIC

160

4.4

Statistical Design of Experiments

186

4.5

Precision of Measurements

186

1.1

Roman Numerals

160

1.2

Roots of Numbers

160

GEOMETRY

191

1.3

Approximate Computation

161

5.1

Geometric Concepts

191

1.4

Interpolation

162

5.2

Mensuration

203

ALGEBRA

163

5.3

Constructions

203

2.1

Numbers

163

TRIGONOMETRY

213

2.2

Identities

163

6.1

Circular Functions of Plane Angles

213

2.3

Binomial Theorem

164

6.2

Solution of Triangles

216

2.4

Approximate Formulas

164

6.3

Spherical Trigonometry

218

2.5

Inequalities

165

6.4

Hyperbolic Trigonometry

218

2.6

Ratio and Proportion

165

6.5

2.7

Progressions

165

Functions of Imaginary and Complex Angles

220

2.8

Partial Fractions

165

PLANE ANALYTIC GEOMETRY

221

2.9

Logarithms

166

7.1

Point and Line

221

2.10

Equations

167

7.2

Transformation of Coordinates

223

2.11

Matrices and Determinants

175

7.3

Conic Sections

223

2.12

Systems of Equations

179

7.4

Higher Plane Curves

226

2.13

Permutations and Combinations

180

SOLID ANALYTIC GEOMETRY

230

2.14

Probability

180

8.1

Coordinate Systems

230

5

6

7

8

SET ALGEBRA

181

8.2

Point, Line, and Plane

231

3.1

Sets

181

8.3

Transformation of Coordinates

235

3.2

Groups

182

8.4

Quadric Surfaces

235

3.3

Rings, Integral Domains, and Fields

182

9

DIFFERENTIAL CALCULUS

238

STATISTICS AND PROBABILITY

182

9.1

Functions and Derivatives

238

4.1

Frequency Distributions of One Variable

182

9.2

Differentiation Formulas

240

4.2

Correlation

184

9.3

Partial Derivatives

240

4.3

Statistical Estimation by Small Samples

185

9.4

Infinite Series

242

9.5

Maxima and Minima

247

∗

This chapter is a revision and extension of Section 2 of the third edition, which was written by John L. Barnes. Eshbach’s Handbook of Engineering Fundamentals, Fifth Edition Edited by Myer Kutz Copyright © 2009 by John Wiley & Sons, Inc.

159

160 10

11

12

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS INTEGRAL CALCULUS

248

12.2

One-Dimensional Problems

270

10.1

Integration

248

12.3

Two-Dimensional Problems

277

10.2

Definite Integrals

250

LAPLACE TRANSFORMATION

286

10.3

Line, Surface, and Volume Integrals

253

13.1

Transformation Principles

286

10.4

Applications of Integration

255

13.2

Procedure

287

13.3

Transform Pairs

288

DIFFERENTIAL EQUATIONS

258

11.1

Definitions

258

COMPLEX ANALYSIS

288

11.2

First-Order Equations

258

14.1

Complex Numbers

288

11.3

Second-Order Equations

260

14.2

Complex Variables

302

11.4

Bessel Functions

260

11.5

Linear Equations

261

11.6

Linear Algebraic Equations

263

11.7

Partial Differential Equations

265

FINITE-ELEMENT METHOD

269

12.1

269

Introduction

The names of Greek letters are found in Table 1, standard mathematical symbols in Table 2, and abbreviations for engineering terms in Table 3 in Section 4.5. 1

ARITHMETIC

1.1 Roman Numerals Roman Notation. This uses seven letters and a bar; a letter with a bar placed over it represents a thousand times as much as it does without the bar. The letters and rules for combining them to represent numbers are as follows:

I 1

13

V 5

X 10

L 50

C 100

D 500

M 1000

L 50,000

Rule 1 If no letter precedes a letter of greater value, add the numbers represented by the letters. Example 1

XXX represents 30; VI represents 6.

Rule 2 If a letter precedes a letter of greater value, subtract the smaller from the greater; add the remainder or remainders thus obtained to the numbers represented by the other letters. Example 2 IV represents 4; XL represents 40; CXLV represents 145. Other illustrations:

IX XIII XIV LV XLII XCVI MDCI IV CCXL 9 13 14 55 42 96 1601 4240

14

15

VECTOR ANALYSIS

303

15.1

Vector Algebra

303

15.2

Differentiation and Integration of Vectors

304

15.3

Theorems and Formulas

305

BIBLIOGRAPHY

306

1.2 Roots of Numbers

Roots can be found by use of Table 7, or logarithms, in Section 2.9. To find an nth root by arithmetic, use a method indicated by the binomial theorem expansion of (a + b)n : n(n − 1) n−2 2 a b 2 n(n − 1)(n − 2) n−3 3 a b + · · · + bn + 3·2 = a n + bD

(a + b)n = a n + na n−1 b +

where D = na n−1 + 12 n(n − 1)a n−2 b + · · · + bn−1 . 1. Point off the given number into periods of n figures each, starting at the decimal point and going both ways. 2. Find the largest nth power in the left-hand period and use its root as the first digit of the result. Subtract this nth power from the left-hand period and bring down the next period. 3. Use the quantity D, in which a is 10 times the first digit since the first digit occupies a higher place than the second, as the divisor to obtain the second digit b. As a trial divisor to estimate b, use the first term in D, since it is the largest. Multiply D by b, subtract, and bring down the next period.

MATHEMATICS

161

4. To get the next digit use 10 times the first two digits as a and proceed as before. Example 3 See the tabulation for Example 3 below. 1.3 Approximate Computation Standard Notation. N = a · 10b , N is a given number; 1 ≤ a < 10, the figures in a being the significant figures in N; b is an integer, positive or negative or zero. Example 4 If N = 2,953,000, in which the first five figures are significant, then N = 2.9530 × 106 . A number is rounded to contain fewer significant figures by dropping figures from the right-hand side. If the figures dropped amount to more than 12 in the last figure kept, this last figure is increased by 1. If the figures dropped amount to 12 , the last figure may or may not be increased. Since the last significant figure used in making a measurement, an estimate, and so on, is not exact but is usually the nearer of two consecutive figures, an approximate number may represent any value in a range from 12 less in its last significant figure to 12 more. The absolute error in an approximate number may be as much as 12 in the last significant figure. Example 5 If N = 2.9530 × 106 is an approximate number, then 2.95295 × 106 ≤ N ≤ 2.95305 × 106 .

The absolute error is between −0.00005 × 106 and 0.00005 × 106 . The size of the absolute error depends on the location of the decimal point. The relative error is the ratio of the absolute error to the number. Its size depends on the number of significant figures. Example 6 The relative error in Example 5 is at most 0.00005 × 106 /2.9530 × 106 , or about 1 in 60,000; the percentage error is at most 100 × (0.00005/2.9530), or less than 0.002%. In the result of a computation with approximate numbers, some figures on the right are doubtful and should be rounded off. It is always possible, by using the bounds of the ranges that approximate numbers represent, to compute exactly the bounds of the range in which the result lies and then round off the uncertain figures. Example 7 Divide the approximate number 536 by the approximate number 217.4:

At least

At most

535.5 536.5 536 = 2.47− = 246+ = 2.47− 217.4 217.45 217.35

Tabulation for Example 3

1. Square root of 302.980652: 3 02. 98 06 52 D = 2a + b = 27 344 34,806

17.406 +

1 202 189 1398 1376 220,652 208,836

2. Cube root of 1,58,252.632929: 53 = Trial divisor = 3a 2 = 3 × 502 = 7,500 3ab = 3 × 50 × 4 = 600 b 2 = 42 = 16 D = 3a 2 + 3ab + b2 = 8,116 3 × 54002 = 8,7480,000 3 × 5400 × 9 = 145,800 92 = 81 87,625,881

158 252 .632 929 125 33,252

32,464 788,632,929

788,632,929

54.09

162


In the quotient the third figure may be in error. It is useless to carry the division further. The following rules usually give the largest number of significant figures that it is reasonable to keep. Addition and Subtraction. Keep as the last significant figure in the result the figure in the last full column. The absolute accuracy of the result is determined by the least absolutely accurate number. Example 8

2.953xx 0.8942x 0.06483 3.912xx Multiplication, Division, Powers, and Roots. Keep no more significant figures in the result than the fewest in any number involved. The relative accuracy of the result is determined by that of the least relatively accurate number. Shortcuts as shown in the examples may be used.

Use of Tables. In using a table to find the value of a function corresponding to an approximate value of an argument, it is usually advisable to retain no more significant figures in the function than there are in the argument, although the accuracy of the function varies considerably, depending inversely on the slope of the curve representing the function. However, there is no need for many-place tables if the values of the argument are known only to a few significant figures. 1 Example 10 52 = 0.019; cos 61.3◦ = 0.877; log 3.74 = 0.573. To investigate the behavior of the error for any given function, the differential approximation is useful. If y = f (x), then dy = f (x) dx approximates the absolute error, and dy/y = f (x) dx/f (x) the relative error. For particular approximate values of the arguments, the bounds of the ranges of the functions can be found directly from a table with arguments given to one additional place.

1.4 Interpolation Example 9

1.

Gregory–Newton Interpolation Formula. Let f (x) be a tabulated function of the argument x, x the constant difference between values of x for which the function is tabulated, and p a proper fraction. To find f (x + px) use the formula

2953 × 413 2953 413 118 12 3 0 9 122 xxxx = 1.22 ×

f (x + p x) = f (x) + p f +p C2 2 f + p C3 3 f + · · · 106 in which

2.

(1.22 × 106 )/2953 413 2953 1,220,000 11,812 295 388 295 30 93 90

In intermediate results keep one additional figure. If there is much difference in the relative accuracy, that is, the number of significant figures, of the numbers involved in a computation, round all of them to one more significant figure than the least accurate number has. This procedure may introduce a small error in the last figure kept in the result. A threedigit number beginning with 8 or 9 has about the same relative accuracy as a four-digit number beginning with 1.

p Cr

=

p(p − 1) · · · (p − r + 1) r!

and r f = r th functional difference. Binomial coefficients for interpolation: p

p C2

p C3

p C4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−0.0450 −0.0800 −0.1050 −0.1200 −0.1250 −0.1200 −0.1050 −0.0800 −0.0450

0.0285 0.0480 0.0595 0.0640 0.0625 0.0560 0.0455 0.0320 0.0165

−0.0207 −0.0336 −0.0402 −0.0416 −0.0391 −0.0336 −0.0262 −0.0176 −0.0087

p C5

0.0161 0.0255 0.0297 0.0300 0.0273 0.0228 0.0173 0.0113 0.0054

In ordinary linear interpolation the first two terms of the formula are used.

MATHEMATICS

163

x

Find √ f (x) = x

15

3.8730

16

4.0000

Example 11

√

15.4. f

2 f

0.1270

−0.0039

0.1231 17

−0.0036

4.1231 0.1195

18

3 f

0.0003

4.2426 x = 1

p = 0.4

f (15 + 0.4 × 1) = 3.8730 + 0.4 × 0.1270 + 0.1200 × 0.0039 + 0.0640 × 0.0003 = 3.9243 2 ALGEBRA 2.1 Numbers Classification 1. Real (positive and negative). (a) Rational, expressible as the quotient of two integers. i. Integers, as −1, 2, 53. ii. Fractions, as 34 , − 52 . (b) Irrational, not expressible as the quotient of √ two integers, as 2, π. 2. Imaginary, a product√of a real number and the imaginary unit i(= −1). Electrical engineers use j to √ avoid confusion with i for current. √ Example: −2 = 2i. 3. Complex, a sum of a real number and an imaginary number, as a + bi (a and b real), −3 + 0.5i. A real number may be regarded as a complex number in which b = 0 and an imaginary number as one in which a = 0. The Absolute Value of: 1. A real number is the number itself if the number is positive and the number with its sign changed if it is negative, as, for example, |3| = | − 3| = 3. √ 2. A complex number a + biis a 2 + b2 , as, for

example, | − 3 + 0.5i| =

2.2 Identities Powers

1. 2. 3. 4.

(−a)n = a n if n is even (−a)n = −a n if n is odd a m · a n = a m+n a m /a n = a m−n

9+

1 4

= 3.04.

5. (ab)n = a n bn 6. (a/b)n = a n /bn = (b/a)−n = b−n /a −n = a n b−n 7. a −n = (1/a)n = a1n 8. (a m )n = a mn 9. a 0 = 1; 0n = 0; 00 is meaningless Roots

√ n a = a 1/n √ √ n ( a)n = n a n = a √ √ √ n ab = n a n b √ √ √ n a/b = n a/ n b √ √ √ mn m+n m a n a = a (1/m)+(1/n) = a √ √ m n a = ( m a)n = a n/m √ √ m √ n a = mn a = n m a = (a 1/m )1/n = a 1/mn √ √ √ 8. a + b = a + b + 2 ab

1. 2. 3. 4. 5. 6. 7.

Products

1. (a ± b)2 = a 2 ± 2ab + b2 2. (a + b)(a − b) = a 2 − b2 . 3. (a + b + c)2 = a 2 + b2 + c2 + 2ab + 2ac +2bc 4. (a ± b)3 = a 3 ± 3a 2 b + 3ab2 ± b3 5. a 3 ± b3 = (a ± b)(a 2 ∓ ab + b2 ) Quotients

1. (a n − bn )/(a − b) = a n−1 + a n−2 b + a n−3 b2 + · · · + abn−2 + bn−1 if a = b 2. (a n + bn )/(a + b) = a n−1 − a n−2 b + a n−3 b2 − · · · − abn−2 + bn−1 if n is odd 3. (a n − bn )/(a + b) = a n−1 − a n−2 b + a n−3 b2 − · · · + abn−2 − bn−1 if n is even Fractions

Signs:

−a −a a a = = =− . b −b b −b

Addition and subtraction: b ad ± bc a b a±b a a a ± = , ± = , ± c d cd c c c c d b c a(d ± c) a + 3 − = cd def e g df 2 =

ae2 fg + bdf 2 − ce3 g de3 f 2 g

164


Multiplication:

Division:

√

1 2 1 1·3 3 x− x ± x 2 2·4 2·4·6 1·3·5 4 1·3·5·7 x ± x5 − · · · − 2·4·6·8 2 · 4 · 6 · 8 · 10 1·3 2 1·3·5 3 1 1 = (1 ± x)−1/2 = 1 ∓ x + x ∓ x √ 2 2 ·4 2·4·6 1±x

c ac a ac a × = , = b d bd b bc

a d ad a a/c a/b = × = , = c/d b c bc b b/c

Series

1 ± x = (1 ± x)1/2 = 1 ±

+ ···

1. 1 + 2 + 3 + 4 + · · · + (n − 1) + n =

1 2 n(n

+ 1)

2. p + (p + 1) + (p + 2) + · · · + (q − 1) + q = 1 2 (q + p)(q − p + 1) 3. 2 + 4 + 6 + 8 + · · · + (2n − 2) + 2n = n(n + 1) 4. 1 + 3 + 5 + 7 + · · · + (2n − 3) + (2n − 1) = n2 5. 12 + 22 + 32 + 42 + · · · + (n − 1)2 + n2 = 1 6 n(n + 1)(2n + 1) 6. 13 + 23 + 33 + 43 + · · · + (n − 1)3 + n3 = 1 2 2 4 n (n + 1) 7. 14 + 24 + 34 + 44 + · · · + (n − 1)4 + n4 = 1 2 30 n(n + 1)(2n + 1)(3n + 3n − 1) 2.3 Binomial Theorem

(a ± b)n = a n ± na n−1 b +

n(n − 1) n−2 2 a b 1·2

n(n − 1)(n − 2) n−3 3 a b + ··· 1·2·3 n(n − 1) · · · (n − r + 1) n−r r a b + ··· + (±1)r r! ±

in which the last term shown is the (r + 1)th; r!, called r factorial, equals 1 · 2 · 3 · · · (r − 1) · r; and 0! = 1. If n is a positive integer, the series is finite; it has n + 1 terms, the last being bn ; and it holds for all values of a and b. If n is fractional or negative, the series is infinite; it converges only for |b| < |a| (see Section 9.4). The coefficients n, n(n − 1)/2!, n(n − 1)(n − 2)/ 3!, . . . are called binomial coefficients. For brevity the coefficient n(n −1)· · · (n − r + 1)/r! of the (r + 1)th n terms is written r or n Cr . If n is a positive integer, the coefficients of the rth term from the beginning and the rth from the end are equal. For any value of n and −1 < x < 1, n(n − 1) 2 n(n − 1)(n − 2) 3 x ± x 1·2 1·2·3 n(n − 1)(n − 2)(n − 3) 4 x ± ··· + 1·2·3·4

(1 ± x)n = 1 ± nx +

1 = (1 ± x)−1 = 1 ∓ x + x 2 ∓ x 3 + x 4 ∓ x 5 + · · · 1±x

2.4 Approximate Formulas

(a) If |x| and |y| are small compared with 1: (1 ± x)2 = 1 ± 2x (1 ± x)1/2 = 1 ± 12 x 1/(1 ± x) = 1 ∓ x (1 + x)(1 + y) = 1 + x + y (1 + x)(1 − y) = 1 + x − y ex = 1 + x + 12 x 2 (where e = 2.71828) 7. loge (1 ± x)

1. 2. 3. 4. 5. 6.

= ±x − x 2 /2 ± x 3 /3 1+x 8. loge 1−x

= 2 x + 13 x 3 + 15 x 5

         (Last     term often   may be    omitted.)       

(b) If |x| is small compared with a and a > 0: 9. a x = 1 + x loge a + 12 x 2 (loge a)2 . term often may be omitted.)

(Last

(c) If a and b are nearly equal and both >0: √ 1 10. ab = (a + b) 2 (d) If b is small compared with a and both >0: √ 11. a 2 ± b = a ± b/2a √ 12. a 2 ± b = a ± b/3a 2 √ 13. a 2 + b2 = 0.960a + 0.398b. This is within 4% of the true value if a > b. √ A closer approximation is a 2 + b2 = 0.9938a 2 + 0.0703b + 0.3567(b /a). √ 14. a 2 + b2 + c2 = 0.939a + 0.389b + 0.297c. This is within 6% of the true value if a > b > c. For instance, for the numbers 43, 42, and 41, the error is 1: 21. e1/n = 1 + 1/(n − 0.5) 22. e−1/n = 1 − 1/(n + 0.5) (h) As n → ∞: 1 1 + 2 + 3 + 4 + 5··· + n → 23. 2 n 2 1 + 22 + 32 + 42 + · · · + n 2 1 24. → n3 3 1 + 23 + 33 + 43 + · · · + n 3 1 25. → n4 4 2.5 Inequalities Laws of Inequalities for Positive Quantities (a) If a > b, then a+c a−c ac a c

> b+c > b−c > bc b > c

b c−a −ca c a

< a < c−b < −cb c < b

Corollary: If a − c > b, then a > b + c. (b) If a > b and c > d, then a + c > b + d; ac > bd; but a − c may be greater than, equal to, or less than b − d; a/c may be greater than, equal to, or less than b/d. 2.6 Ratio and Proportion Laws of Ratio and Proportion

(a) If a/b = c/d, then b a = c d mc + nd ma + nb = pa + qb pc + qd

ad = bc a n b

=

c n d

If also e/f = g/ h, then, ae/bf = cg/dh. (b) If a/b = c/d = e/f = · · ·, then c e pa + qc + re + · · · a = = = ··· = b d f pb + qd + rf + · · ·

Variation

If y = kx, y varies directly as x; that is, y is directly proportional to x. If y = k/x, y varies inversely as x; that is, y is inversely proportional to x. If y = kxz, y varies jointly as x and z. If y = k(x/z), y varies directly as x and inversely as z. The constant k is called the proportionality factor. 2.7 Progressions Arithmetic Progression. This is a sequence in which the difference d of any two consecutive terms is a constant. If n = number of terms, a = first term, l = last term, s = sum of n terms, then l = a + (n − 1)d, and s = (n/2)(a + l). The arithmetic mean A of two quantities m, n is the quantity that placed between them makes with them an arithmetic progression; A = (m + n)/2. Example 12 Given the series 3 + 5 + 7 + · · · to 10 terms. Here n = 10, a = 3, d = 2; hence l = 3 + (10 − 1) × 2 = 21 and s = (10/2)(3 + 21) = 120. Geometric Progression. This is a sequence in which the ratio r of any two consecutive terms is a constant. If n = number of terms, a = first term, l = last term, s = sum of n terms, then l = ar n−1 , s = (rl − a)/(r − 1) = a(1 − r n )/(1 − r). The geometric mean G of two quantities m, n is the quantity that placed between them √ makes with them a geometric progression; G = mn. Example 13 Given the series 3 + 6 + 12 + · · · to six terms. Here n = 6, a = 3, r = 2; hence l = 3 × 26−1 = 96 and s = (2 × 96 − 3)/(2 − 1) = 3(1 − 26 )/ (1 − 2) = 189. If |r| < 1 then, as n → ∞, s → a/(1 − r). Example 14 Given the infinite series 12 + 14 + 18 + · · · . Here a = 12 and r = 12 ; hence s → ( 12 )/(1 − 12 ) = 1 as n → ∞. Harmonic Progression. This is a sequence in which the reciprocals of the terms form an arithmetic progression. The harmonic mean H of two quantities m, n is the quantity that placed between them makes with them a harmonic progression; H = 2mn/(m + n). The relation among the arithmetic, geometric, and harmonic means of two quantities is G2 = AH . 2.8 Partial Fractions A proper algebraic fraction is one in which the numerator is of lower degree than the denominator. An improper fraction can be changed to the sum of a polynomial and a proper fraction by dividing the numerator by the denominator.

166


A proper fraction can be resolved into partial fractions, the denominators of which are factors, prime to each other, of the denominator of the given fraction. Case 1: The denominator can be factored into real linear factors P , Q, R, . . . all different. Let A B C Num = + + + ··· P QR · · · P Q R

Case 3: The denominator can be factored into quadratic factors, P , Q, . . . , all different, which cannot be factored into real linear factors. Let Num Ax + B Cx + D = + + ··· PQ··· P Q Example 17

Ax + B C 3x 2 − 2 = 2 + (x 2 + x + 1)(x + 1) x +x+1 x+1

Example 15

A B C 6x 2 − x + 1 = + + x3 − x x x−1 x+1

Clearing fractions, 3x 2 − 2 = (Ax + B)(x + 1) + C(x 2 + x + 1)

Clearing fractions, 6x 2 − x + 1 = A(x − 1)(x + 1) + Bx(x + 1) + Cx(x − 1)

= (A + C)x 2 + (A + B + C)x + (B + C) (1)

(a) Substitution method. Letting x = 0, A = −1; x = 1, B = 3; and x = −1, C = 4 yields 1 3 4 6x 2 − x + 1 =− + + x3 − x x x−1 x+1 (b) Method of undetermined coefficients. Rewriting Eq. (1),

Use the method of undetermined coefficients to find A, B, C. Case 4: The denominator can be factored into quadratic factors, P, Q, . . . , one or more repeated, which cannot be factored into real linear factors. Let Ax + B Cx + D Ex + F Num = + + P 2 Q3 · · · P P2 Q +

6x − x + 1 = (A + B + C)x + (B − C)x − A 2

2

Equating coefficients of like powers of x, A + B + C = 6, B − C = −1, −A = 1. Solving this system of equations, A = −1, B = 3, C = 4. Case 2: The denominator can be factored into real linear factors, P , Q, . . . , one or more repeated. Let B D A C E Num + 2+ + 2 + 3 + ··· = 2 3 p Q P p Q Q Q Example 16

B C A D x+1 + = + + x(x − 1)3 x x − 1 (x − 1)2 (x − 1)3

Gx + H Ix + J + + ··· Q2 Q3

Example 18

Bx + C A 5x 2 − 4x + 16 + = (x − 3)(x 2 − x + 1)2 x − 3 x2 − x + 1 +

Dx + E (x 2 − x + 1)2

Clearing fractions, 5x 2 − 4x + 16 = A(x 2 − x + 1)2 + (Bx + C)(x − 3) × (x 2 − x + 1) + (Dx + E)(x − 3)

Clearing fractions,

Find A by substituting x = 3. Then use the method of undetermined coefficients to find B, C, D, E.

x + 1 = A(x − 1)3 + Bx(x − 1)2 + Cx(x − 1) + Dx

2.9 Logarithms

A and D can be found by substituting x = 0 and x = 1. After inserting these numerical values for A and D, B and C can be found by the method of undetermined coefficients.

If N = bx , then x is the logarithm of the number N to the base b. For computation, common, or Briggs, logarithms to the base 10 (abbreviated log10 or log) are used. For theoretical work involving calculus, natural, or Naperian, logarithms to the irrational base

MATHEMATICS

167

e = 2.71828 · · · (abbreviated ln, loge , or log) are used. The relation between logarithms of the two systems is loge n =

25.0468 − 10 15.3 9.7468 − 10 = log 0.5582

log10 n log10 n = = 2.303 log10 n log10 e 0.4343

The integral part of a common logarithm, called the characteristic, may be positive, negative, or zero. The decimal part, called the mantissa and given in tables, is always positive. To find the common logarithm of a number, first find the mantissa from Table 10 in Section 9.4, disregarding the decimal point of the number. Then from the location of the decimal point find the characteristic as follows. If the number is greater than 1, the characteristic is positive or zero. It is 1 less than the number of figures preceding the decimal point. For a number expressed in standard notation the characteristic is the exponent of 10. Example 19 log 6.54 = 0.8156, log 6540 = log(6.54 × 103 ) = 3.8156. If the number is less than 1, the characteristic is negative and is numerically 1 greater than the number of zeros immediately following the decimal point. To avoid having a negative integral part and a positive decimal part, the characteristic is written as a difference. Example 20 log 0.654 = log(6.54 ×10−1 ) = 1.8156 = 9.8156 − 10, log 0.000654 = log(6.54 × 10−4 ) = 4.8156 = 6.8156 − 10. To find a number whose logarithm is given, each of the preceding steps is reversed. The cologarithm of a number is the logarithm of its reciprocal. Hence, cologN = log 1/N = log 1 − log N = − log N. Use of Logarithms in Computation

To To To To

To subtract 15.3 from 15.0468, add 10 to 15.0468 and subtract 10 from it:

multiply a and b divide a by b raise a to the nth power find the nth root of a

log ab = log a + log b log a/b = log a − log b log a n = n log a log a 1/n = (1/n) log a

Example 21

1. 68.31 × 0.2754 = 18.81: log 68.31 = 1.8345 log 0.2754 = 9.4400 − 10 11.2745 − 10 = 1.2745 = log 18.81 2. 0.68411.53 = 0.5582: log 0.6831 = 9.8345 − 10 1.53 × (9.8345 − 10) = 15.0468 − 15.3

3.

√ 5

0.6831 = 0.9266: log 0.6831 = 9.8345 − 10

1 5 (49.8345

− 50) = 9.9669 − 10 = log 0.9266

To solve a simple exponential equation of the form a x = b, equate the logarithms of the two sides of the equation: x log a = log b from which x=

log b log a

Example 22

x=

log x = log(log b) − log(log a)

and

0.6831x = 27.54.

1.4400 1.4400 log 27.54 = = = −8.701 log 0.6831 9.8345 − 10 −0.1655

2.10 Equations The equation f (x) = a0 x n + a1 x n−1 + a2 x n−2 + · · · + an = 0, ai real, is a polynomial equation of degree n in one variable. For n = 1, the equation f (x) = ax + b = 0 is linear. It has one root, x1 = −b/a. Quadratic Equation For n = 2, the equation f (x) = ax 2 + bx + c = 0 is quadratic. It has two roots, both real or both complex, given by the formulas √ 2c −b ± b2 − 4ac x1 , x2 = = √ 2a −b ∓ b2 − 4ac √ To avoid loss of precision if b2 − 4ac and |b| are nearly equal, use the form that does not involve the difference. If the quantity b2 − 4ac, called the discriminant, is greater than zero, the roots are real and unequal; if it equals zero, the roots are real and equal; if it is less than zero, the roots are complex. Cubic Equation For n = 3, the equation f (x) = a0 x 3 + a1 x 2 + a2 x + a3 = 0 is cubic. It has three roots, all real or one real and two complex. Algebraic Solution. Write the equation in the form ax 3 + 3bx 2 + 3cx + d = 0. Let

q = ac − b2

and

r = 12 (3abc − a 2 d) − b3

168


Case 2: If q is negative and q 3 + r 2 ≥ 0:

Also let q 3 + r 2 )1/3 s2 = (r − q 3 + r 2 )1/3 s1 = (r +

and

Then the roots are x1 = x2 = x3 =

(s1 + s2 ) − b a − 12 (s1 + s2 ) + − 12 (s1 + s2 ) −

1 2

√

a √ 1 2

−3(s1 − s2 ) − b −3(s1 − s2 ) − b

a

If q 3 + r 2 > 0, there are one real and two complex roots. If q 3 + r 2 = 0, there are three real roots of which at least two are equal. If q 3 + r 2 < 0, there are three real roots, but the numerical solution leads to finding the cube roots of complex quantities. In such a case the trigonometric solution is employed.

√ 1 −1 ±r cosh y1 = ±2 −q cosh 3 −q 3

√ 1 −1 ±r cosh y2 = ∓ −q cosh 3 −q 3

1 −1 ±r cosh + i −3q sinh 3 −q 3

√ 1 −1 ±r cosh y3 = ∓ −q cosh 3 −q 3

1 −1 ±r cosh − i −3q sinh 3 −q 3 Case 3: If q is positive:

1 √ −1 ±r sinh y1 = ±2 q sinh 3 q3

±r 1 √ sinh−1 y2 = ∓ q sinh 3 q3

1 −1 ±r sinh + i 3q cosh 3 q3

1 √ −1 ±r sinh y3 = ∓ q sinh 3 q3

1 −1 ±r sinh − i 3q cosh 3 q3

Example 23 Given the equation x 3 + 12x 2 + 45x + 54 = 0. Here a = 1, b = 4, c = 15, d = 54. Let q = 15 − 16 = −1; r = 12 (180 − 54) − 64 = 1; q 3 + r 2 = −1 + 1 = 0, s1 = s2 = (−1)1/2 = −1; s1 + s2 = −2; s1 − s2 = 0. Hence the roots are x1 = (−2−4) = −6; x2 = x3 = [− 12 (−2) − 4] = −3. Trigonometric Solution. Write the equation in the form ax 3 + 3bx 2 + 3cx + d = 0. Let q = ac − b2 and r = 12 (3abc − a 2 d) − b3 (as in algebraic solution). Then the roots are

x1 =

y1 − b a

x2 =

y2 − b a

x3 =

y3 − b a

where y1 , y2 , and y3 have the following values (upper of alternative signs being used when r is positive and the lower when r is negative): Case 1: If q is negative and q 3 + r 2 ≤ 0: √

±r 1 y1 = ±2 −q cos cos−1 3 −q 3

√ 2π 1 −1 ±r cos + y2 = ±2 −q cos 3 3 −q 3

√ 4π ±r 1 cos−1 + y3 = ±2 −q cos 3 3 −q 3

Example 24 Given the equation x 3 + 6x 2 − 9x − 54 = 0. Here a = 1, b = 2, c = −3, d = −54; q = −3 − 4 = −7; r = 12 (−18 + 54) − 8 = 10; q 3 + r 2 = −343 +100 = −243. Note that q is negative; q 3 + r 2 < 0; r is positive. Therefore use Case 1 with upper signs:

√

10 1 cos−1 √ y1 = 2 7 cos 3 343

√ ◦ = 2 7 cos 19.1 = 5

Hence, one root is x1 = 5 − 2 = 3. The other roots can be similarly determined. Quartic Equation For n = 4, the equation f (x) = a0 x 4 + a1 x 3 + a2 x 2 + a3 x + a4 = 0 is quartic. It has four roots, all real, all complex, or two real and two complex.

MATHEMATICS

169

6. If x is replaced by (a) y/m, (b) −y, (c) y + h, the roots of the resulting equation φ(y) = 0 are (a) m times, (b) the negatives of, (c) less by h than the corresponding roots of f (x) = 0. 7. Descartes’ Rule of Signs. A variation of sign occurs in f (x) = 0 if two consecutive terms have unlike signs. The number of positive roots is either equal to the number of variations of sign or is less by a positive even integer. For negative roots apply the rule to f (−x) = 0. 8. If, for two real numbers a and b, f (a) and f (b) have opposite signs, there is an odd number of roots between a and b. 9. If k is the exponent of the first term with a negative coefficient and G the greatest of the absolute values of the negative coefficients, then √ an upper bound of the real roots is 1 + n−k G/a0 . 10. Sturm’s Theorem. Let the equation f (x) = 0 have no multiple roots. With f0 = f (x) and f1 = f (x), form the sequence f0 , f1 , f2 , . . ., fn as follows:

To solve, first divide the equation by a0 to put it in the form x 4 + ax 3 + bx 2 + cx + d = 0. Find any real root y1 of the cubic equation: 8y 3 − 4by 2 + 2(ac − 4d)y − [c2 + d(a 2 − 4b)] = 0 Then the four roots of the quartic equation are given by the roots of the two quadratic equations: x2 +

1 2a

+

1 2a

−

x2 +

1 2 4a

+ 2y1 − b x + (y1 + y12 − d) = 0

1 2 4a

+ 2y1 − b x + (y1 − y12 − d) = 0

nth-Degree Equation Properties of f(x) = a0 xn + a1 xn−1 + · · · + an = 0. Assume an ’s are real.

1. Remainder Theorem. If f (x) is divided by x − r until a remainder independent of x is obtained, this remainder is equal to f (r), the value of f (x) for x = r. 2. Factor Theorem. If and only if x − r is a factor of f (x), then f (r) = 0. 3. The equation f (x) = 0 has n roots, not necessarily distinct. Complex roots occur in conjugate pairs, a + bi and a − bi. If n is odd, there is at least one real root. 4. The sum of the roots is −a1 /a0 , the sum of the products of the roots taken two at a time is a2 /a0 , the sum of the products of the roots taken three at a time is −a3 /a0 , and so on. The product of all the roots is (−1)n an /a0 . 5. If the ai are integers and p/q is a rational root of f (x) = 0 reduced to its lowest terms, then p is a divisor of an and q of a0 . If a0 is 1, the rational roots are integers.

f0 = q1 f1 − f2

f1 = q2 f2 − f3

f2 = q3 f3 − f4 , . . . , fn−2 = qn−1 fn−1 − fn At any step, a function fi may be multiplied by a positive number to avoid fractions. Let a and b be real numbers, a < b such that f (a) = 0, f (b) = 0, and let V (a) be the number of variations of sign in the nonzero members of the sequence f0 (a), f1 (a), . . . , fn . Then the number of real roots between a and b is V (a) − V (b). If f (x) = 0 has multiple roots, the sequence terminates with the function fm , m < n, when fm−1 = qm fm . For this sequence, V (a) − V (b) is the number of distinct real roots between a and b. Example 25 below.

See the tabulation for Example 25


1. Locate the real roots of x 3 − 7x − 7 = 0. x = −2

−1

0

1

2

3

4

3x −

9 2

x

f0 = x 3 − 7x − 7 f1 = 3x 2 − 7 f2 = 2x + 3

− + −

− − +

− − +

− − +

− + +

− + +

+ 2x + 3 3x 2 − 7 x 3 − 7x − 7 + 6x 2 − 17 3x 3 − 21x − 21 2 + 6x + 9x 3x 2 − 7x

f3 = 1

+

+

+

+

+

+

+

3

1

1

1

1

1

1

V (x) =

V (−2) − V (−1) = 2 −2 < r1 r2 < −1

−9x − 14 − 14x − 21 27 2x + 3 = f 2 2 1 − 2 1 = f3

−9x −

V (3) − V (4) = 1 3 < r3 < 4

170


Tabulation for Example 25 (continued)

2. Locate the real roots of 4x 3 − 3x − 1 = 0. x = −1 f0 = 4x − 3x − 1

−

f1 =

+

3

3(4x 2

− 1)

f2 = 2x + 3 V (x) =

0

1

−

0

+

−

− + 2 1

2x − 1x

2

2x + 1 4x 2 − 1 +

4x 2

+

+ 2x

4x 3 − 3x − 1 4x 3 − x

−2x − 1 −2x − 1

0

− 2x − 1 2x − 1 = f2

V (−1) − V (0) = 2 V (3) − V (4) = 1 0 < r2 < 2 −1 < r1 < 0

Then r1 can be found to be a double root.

Synthetic Division. To divide a polynomial f (x) by x − a, proceed as in Example 25. Divide f (x) = 4x 3 − 7x + 1 by x + 2. Arrange the coefficients in order of descending powers of x, supplying zeros for missing powers. Place a(= −2) to the left. Bring down the first coefficient, multiply it by a, and add the product to the next coefficient. Multiply the sum by a, add the product to the next coefficient, and continue thus:

− 2 4 + 0 − 7 + 1 − 8 − 16 − 18 4 − 8 +

9 − 17

The last number is the remainder. It is the value of the polynomial f (x) = 4x 3 − 7x + 1 for x = −2, or f (−2) = −17. The other numbers in the last line are the coefficients of the quotient 4x 2 − 8x + 9, a polynomial of one degree less than the dividend. Rational Roots. Possible integral and fractional roots can be found by property 5 and tested by synthetic division. If a rational root r is found, then the remaining roots are roots of q(x) ≡ f (x)/(x − r) = 0. Irrational Roots. Horner’s Method This consists of diminishing a root repeatedly toward zero and adding together the amounts by which it is diminished. This sum approximates the original root. The method is explained by an example. A root of x 3 + 4x − 7 = 0 is located between the successive integers 1 and 2, graphically or by synthetic division, using property 8. First, the roots are diminished by 1 (property 6c) to give an equation f (y + 1) ≡ φ(y) = 0, which has a root between 0 and 1. The method of obtaining the coefficients of φ(y) by use of successive synthetic divisions is illustrated. The remainders are the required coefficients. The root between 0 and 1 of φ(y) = 0 is then located between successive tenths. Since its value is small, the last two terms set equal to 0 suffice to estimate that it is between 0.2 and 0.3. Next, diminish the roots by 0.2 to obtain

an equation with a root between 0 and 0.1. To check that the root was between 0.2 and 0.3, note that the first remainder, which is the value of φ(0.2), remains negative when φ(y) is divided by y − 0.2, and that the remainder would be found to be positive if φ(y) were divided by y − 0.3. Repeat the process, using the last two terms to estimate that the root of the new equation is between 0.05 and 0.06, and then diminish by 0.05. At the next stage it is frequently possible to estimate two more figures by using the last two terms. 1+0

+ 4

−

7 1

+1 1+1 +1 1+2 +1 1+3 + 0.2 1 + 3.2 + 0.2 1 + 3.4 + 0.2 1 + 3.6 + 0.05 1 + 3.65 + 0.05 1 + 3.70 + 0.05 1 + 3.75

+ + + +

1 5 2 7

+ −

5 2

+ + + + +

7 0.64 7.64 0.68 8.32

− + − −

+ + + + +8.6875

8.32 0.1825 8.5025 0.185

2 1.528 0.472

−0.472 +0.425125 −0.046875

0.2

0.05

8.6875x − 0.046875 = 0 x = 0.0054

The root is 1.2554. To find a negative irrational root −r by Horner’s method, replace x in f (x) = 0 by −y, find the positive root r of φ(y) = f (−y) = 0, and change its sign. Newton’s Method This can be used to find a root of either an algebraic or a transcendental equation. The

MATHEMATICS

171

Fig. 1

root is first located graphically between α and β, f (α) and f (β) having unlike signs (Fig. 1). Assume that there is no maximum, minimum, or inflection point in the interval (α, β), that is, that neither f (x) nor f (x) equals zero for any point in (α, β). Take as a first approximation a the endpoint α or β for which f (x) and f (x) have the same sign, that is, if the curve is concave up, take the endpoint at which f (x) is positive, and, if concave down, the endpoint at which f (x) is negative. The point a1 = a − f (a)/f (a), at which the tangent to the curve at [a, f (a)] intersects the x axis, is between a and the root. Then, by using a1 instead of a, a still better approximation a2 is obtained, and so forth. If the endpoint for which f (x) and f (x) have opposite signs were used, it could happen that the approximation obtained would be better than a1 , but it might be much worse since the tangent would not cross the x axis between the endpoint used and the root (Fig. 1). Example 26

Find the real root of x 3 + 4x − 7 = 0. f (x) = x 3 + 4x − 7 f (x) = 3x 2 + 4 f (x) = 6x

Graphically (Fig. 2), α = 1.2, β = 1.3. Since f (1.2) = −0.472 and f (1.3) = 0.397, and f (x) is positive in the interval, then a = 1.3. a1 = a −

0.397 f (a) = 1.3 − = 1.3 − 0.044 = 1.256 f (a) 9.07

0.005385 = 1.256 − 0.00062 = 1.25538 a2 = 1.256 − 8.7326 If Newton’s method of using the tangent is not applicable, either because of the presence of a maximum, minimum, or inflection point or because of difficulty in finding f (x), the interpolation method using the chord joining [α, f (α)] and [β, f (β)] can be used. The chord crosses the x axis at a = α − f (α)(β − α)/[f (β) − f (α)], a better approximation than either α or β. Note that this formula differs from Newton’s only in having the difference quotient, which is the slope of the chord, in place of the derivative, which is the slope of the tangent. To get a still better approximation, repeat the procedure, using as one endpoint a

Fig. 2

and as the other either α or β, chosen so that f (x) has opposite signs at the endpoints of the new interval. Graphical Method of Solution This can be used to solve any kind of equation if it gives sufficient accuracy. To solve the equation f (x) = 0, graph the function y = f (x). The x coordinates of the points at which the graph intersects the x axis are roots of f (x) = 0. Another method is to set f (x) equal to any convenient difference f1 (x) − f2 (x) and graph the functions y = f1 (x) and y = f2 (x) on the same axes. The x coordinates of the points of intersection of the two graphs are real roots of f (x) = 0. Also, see section 2.12. Graeffe’s Method for Real and Complex Roots Let x1 , x2 , . . . , xn be the roots of the equation a0 x n + a1 x n−1 + · · · + an = 0, arranged in descending order of absolute values. Form a sequence of equations such that the roots of each are the negatives of the squares of the roots of the preceding equation. Using the negatives of the squares gives more uniform formulas. Let Ai be a coefficient of the equation being formed, and ai a coefficient of the preceding equation:

A0 = a0 = 1 A1 = a12 − 2a0 a2 = a12 − 2a2 A2 = a22 − 2a1 a3 + 2a4 A3 = a32 − 2a2 a4 + 2a1 a5 − 2a6 .. . 2 − 2an−2 an An−1 = an−1

An = an2 Each coefficient is the sum of the square of the preceding and twice the product of all pairs of equidistant coefficients in the preceding equation, taken with alternately minus and plus signs. Missing coefficients

172


are zero. The process is ended when further steps do not affect the nonfluctuating coefficients to the accuracy desired in the roots. As the successive equations are formed, various cases arise depending on the behavior of the coefficients. Among them are: Case 1: Each coefficient approaches the square of the preceding. The roots are real and unequal in absolute value. Let Ai be a coefficient of the equation p p p −xn . Then, approxwhose roots are −x√1 , −x2 , . . . , √ p p imately, x1 = ± A1 , x2 = ± A2 /A1 , . . . , xn = √ ± p An /An−1 . The signs of the roots are determined by substitution in the original equation. It is usually sufficient to find successive integers between which a root is located. Example 27 f (x) = x − 2x − 5x + 4 = 0. the tabulation for Example 27 below. Using synthetic substitution, 3

2

3 1 − 2 − 5 + 4 + 3 + 3 − 6 1 + 1 − 2 − 2

See

4 1 − 2 − 5 + 4 + 4 + 8 + 12 1 + 2 − 3 − 16

we have f (3) = −2, f (4) = 16. Therefore there is a root between 3 and 4, and x1 = 3.177. log x2 = =

1 16 (log 2.136 × 1 16 (12.3296 −

1012 − log 1.080 × 108 )

8.0334)

=

× 4.2962 = 0.2685

1 16

x2 = ±1.856 Using synthetic substitution, f (−2) = −2, f (−1) = 6. Therefore x2 = −1.856. log x3 =

1 16 (log 4.295 ×

=

1 16 (9.6330

=

1 16 (157.3034

− 12.3296) − 160) = 9.8315 − 10

x3 = ±0.678 Since x1 + x2 + x3 = 2, x3 = 0.678. Case 2: A coefficient fluctuates in sign. There is a pair of complex roots. If the sign of Ai fluctuates, − iv are complex. then xi = u + iv and xi+1 = u√ p Ai+1 /Ai−1 , 2u = Let r 2 = u2 + v 2 . Then r 2 = √ −a1 − (sum of real roots), v = r 2 − u2 . Example 28 f (x) = x 4 − 2x 3 − 4x 2 + 5x − 7 = 0. See the tabulation for Example 28 on next page. If, for a fourth-degree equation, alternate coefficients, that is, the second and fourth, fluctuate in sign, all four roots are complex. √ Let the√ roots be u1 ± iv1 , u2 ± iv2 . Then r12 = p A2 , r22 = p A4 /A2 , 2(u1 + u2 ) = −a1 , 2(r22 u1 + r12 u2 ) = −a3 .


x3 1 1

1st 2nd

1 1

4th

1 1

8th

1 1

16th

1

log x1 =

x2 −2 4 10 14 196 −82 1.14 × 102 1.300 × 104 −0.247 × 104 1.053 × 104 1.109 × 108 −0.029 × 108 1.080 × 108 1 16

x −5 25 16 41 1.681 × 103 −0.448 × 103 1.233 × 103 1.520 × 106 −0.058 × 106 1.462 × 106 2.137 × 1012 −0.001 × 1012 2.136 × 1012

log 1.080 × 108 =

x1 = ±3.177

109 − log 2.136 × 1012 )

1 16

x0 4 16 16 256 256 6.554 × 104 6.554 × 104 4.295 × 109 4.295 × 109

× 8.0334 = 0.5021

MATHEMATICS

173


x4 1 1

x3 −2 4 8

2nd

1 1

12 144 −44

4th

1 1

8th

1 1

16th

1 1

32nd

1

1st

100 1.0000 × 104 −0.2652 × 104 7.348 × 103 5.399 × 107 −0.400 × 107 4.999 × 107 2.499 × 1015 −0.001 × 1015 2.498 × 1015

x2 −4 16 20 −14 22 484 744 98 1326 1.758 × 106 0.239 × 106 0.005 × 106 2.002 × 106 4.008 × 1012 0.073 × 1012 4.081 × 1012 1.665 × 1025 1.665 × 1025

Since the sign of A3 fluctuates, x3 and x4 are complex. √ 32 x1 = ± 2.498 × 1015 = ±3.028 15.3976 log(2.498 × 1015 ) = = 0.4812 32 32 x2 = ± 25.2214 15.3976 9.8238 32

1.665 × 1025 = ±2.028 2.498 × 1015

= 0.3070

r2 = 27.0434 25.2214 1.8220 32

32

32

1.105 × 1027 = 1.140 1.665 × 1025

= 0.05694

x3 , x4 = 0.5 ± 0.943i

x0 −7 49

x 5 25 −56 −31 961 −2156

49 2401

−1195 1.428 × 106 −6.367 × 106

2401 5.765 × 106

−4.939 × 106 2.439 × 1013 −2.308 × 1013 1.31 × 1012 0.017 × 1026 −2.713 × 1026 −2.696 × 1026

5.765 × 106 3.324 × 1013

f (3) = −

3.324 × 1013 1.105 × 1027 1.105 × 1027

f (4) = +

x1 = 3.028 f (−3) = +

f (−2) = −

x2 = −2.028 u=

v=

2 − (3.028 − 2.028) = 0.500 2 √

1.140 − 0.250 =

√

0.890 = 0.943

174


Tabulation for Example 29 x4 1 1

x3 −3 9 2

2nd

1 1

11 121 −106

4th

1 1

15 0.225 × 103 −4.466 × 103

8th

1 1

−4.241 × 103 1.799 × 107 −1.126 × 107

16th

1

0.675 × 107

1st

x2 −1 1 24 28 53 2809 −968 392 2.233 × 103 4.986 × 106 0.565 × 106 0.077 × 106 5.628 × 106 3.1674 × 1013 0.1555 × 1013 0.0003 × 1013 3.323 × 1013

x0 14 196

x 4 16 28 44 1936 −20776

196 38416

−1.884 × 104 3.549 × 108 −1.716 × 108

3.842 × 104 1.476 × 109

1.833 × 108 3.360 × 1016 −1.661 × 1016

1.476 × 109 2.178 × 1018

1.699 × 1016

2.178 × 1018

Since A1 and A3 fluctuate in sign, there are four complex roots. r12 =

√

16

3.323 × 1013 = 7.000

13.5215 log(3.323 × 10 ) = = 0.8451 16 16 18 16 2.178 × 10 = 2.000 r22 = 3.323 × 1013 18.3380 13.5215 4.8165 = 0.3010 16 13

2(u1 + u2 ) = 3 2(2u1 + 7u2 ) = −4 u2 = −1 u1 = 2.5 √ r22 − u22 = 2 − 1 = 1 √ √ = r12 − u21 = 7 − 6.25 = 0.75 = 0.866

v2 = v1

x1 , x2 = 2.5 ± 0.866i x3 , x 4 = −1 ± i

Example 29 f (x) = x 4 − 3x 3 − x 2 + 4x + 14 = 0. See the tabulation for Example 29 above.

value. If Ai approaches one-half the square of the √ preceding coefficient, then |xi | = |xi+1 | = 2p Ai+1 /Ai−1 .

Case 3: A coefficient approaches one-half the square of the preceding. There is a double real root or there are two real roots of equal absolute

Example 30 f (x) = x 3 +2.20x 2 −2.95x +0.80 = 0. See the tabulation for Example 30 below.


1st

x3 1 1

2nd

1 1

4th

1 1

8th

1

x2 2.20 4.84 5.90 10.74 1.1535 × 102 −0.1037 × 102 1.050 × 102 1.1025 × 104 −0.0026 × 104 1.100 × 104

x −2.95 8.703 −3.52 5.183 2.686 × 10 −1.375 × 10 1.311 × 10 1.719 × 102 −0.860 × 102 0.859 × 102

x0 0.80 0.64 0.64 0.4096 0.4096 0.1678 0.1678

MATHEMATICS

175

Tabulation for Example 30 Since A2 approaches one-half the square of the preceding coefficient, |x2 | = |x3 |.

√ 8 x1 = ± 1.100 × 104 = ±3.20 4.0414 log(1.100 × 104 ) = = 0.5052 8 8 0.1678 = 0.50 |x2 | = |x3 | = 16 1.100 × 104 9.2248 − 10 4.0414 155.1834 − 160 = 9.6990 − 10 16 For a more extensive treatment of Graeffe’s method, see mathworld.wolfram.com/GraeffesMethod. html (August 2008) and math.fullerton.edu/mathews/ n2003/GraeffesMethodMod/html (August 2008). 2.11 Matrices and Determinants Definitions 1. A matrix is a system of mn quantities, called elements, arranged in a rectangular array of m rows and n columns:   a11 a12 · · · a1n a11 a12 · · · a1n  a21 a22 · · · a2n  a21 a22 · · · a2n = . A= . . . . . . .  . . . ..  . . .. .. .. . .. a am1 am2 · · · amn m1 am2 · · · amn

= (aij ) = ||aij || i = 1, . . . , m

j = 1, . . . , n

2. If m = n, then A is a square matrix of order n. 3. Two matrices are equal if and only if they have the same number of rows and of columns and corresponding elements are equal. 4. Two matrices are transposes (sometimes called conjugates) of each other if either is obtained from the other by interchanging rows and columns. 5. The complex conjugate of a matrix (aij ) with complex elements is the matrix (a ij ). See Section 13.1. 6. A matrix is symmetric if it is equal to its transpose, that is, if aij = aj i , i, j = 1, . . . , n. 7. A matrix is skew symmetric, or antisymmetric, if aij = −aj i , i, j = 1, . . . , n. The diagonal elements aii = 0. 8. A matrix all of whose elements are zero is a zero matrix. 9. If the nondiagonal elements aij , i = j , of a square matrix A are all zero, then A is a diagonal matrix. If, furthermore, the diagonal elements are all equal, the matrix is a scalar matrix ; if they are all 1, it is an identity or unit matrix, denoted by I.

f (−4) = −

f (−3) = +

f (0.5) = 0

f (−0.5) = 0

x1 = −3.20 x2 = x3 = 0.50

10. The determinant |A| of a square matrix (aij ), i, j = 1, . . . , n, is the sum of the n! products a1r1 a2r2 · · · anrn , in which r1 , r2 , . . . , rn is a permutation of 1, 2, . . . , n, and the sign of each product is plus or minus according as the permutation is obtained from 1, 2, . . . , n by an even or an odd number of interchanges of two numbers. Symbols used are a11 a12 · · · a1n a a · · · a 2n 21 22 |A| = .. i, j = 1, . . . , n . = |aij | .. . . . .. . . a a · · · a n1 n2 nn 11. A square matrix (aij ) is singular if its determinant |aij | is zero. 12. The determinants of the square submatrices of any matrix A, obtained by striking out certain rows or columns or both, are called the determinants or minors of A. A matrix is of rank r if it has at least one r-rowed determinant that is not zero while all its determinants of order higher than r are zero. The nullity d of a square matrix of order n is d = n − r. The zero matrix is of rank 0. 13. The minor Dij of the element aij of a square matrix is the determinant of the submatrix obtained by striking out the row and column in which aij lies. The cofactor Aij of the element aij is (−1)i+j Dij . A principal minor is the minor obtained by striking out the same rows as columns. 14. The inverse of the square matrix A is A

An1   |A| |A|   . . .   . . = . ..   . A Ann  1n ··· |A| |A| 11

A−1

...

AA−1 = A−A = I

176


15. The adjoint of A is A

 11 · · · An1 . . . . . ..  adj A =  .. A1n · · · Ann 16. Elementary transformations of a matrix are: a. The interchange of two rows or of two columns b. The addition to the elements of a row (or column) of any constant multiple of the corresponding elements of another row (or column) c. The multiplication of each element of a row (or column) by any nonzero constant 17. Two m × n matrices A and B are equivalent if it is possible to pass from one to the other by a finite number of elementary transformations. a. The matrices A and B are equivalent if and only if there exist two nonsingular square matrices E and F , having m and n rows, respectively, such that EAF = B. b. The matrices A and B are equivalent if and only if they have the same rank. Matrix Operations Addition and Subtraction. The sum or difference of two matrices (aij ) and (bij ) is the matrix (aij ± bij ), i = 1, . . . , m, j = 1, . . . , n. Scalar Multiplication. The product of the scalar k and the matrix (aij ) is the matrix (kaij ). Matrix Multiplication. The product (pik ), i = 1, . . ., m, k = 1, . . . , q, of two matrices (aij ), i = 1, . . . , m, j = 1, . . . , n, and (bj k ), j = 1, . . . , n, k = 1, . . . , q, is the matrix whose elements are

pik =

n

aij bj k = ai1 b1k + ai2 b2k + · · · + ain bnk

j =1

The element in the ith row and kth column of the product is the sum of the n products of the n elements of the ith row of (aij ) by the corresponding n elements of the kth column of (bj k ). Example 31 a11 a12 b11 b12 b13 a21 a22 b21 b22 b23 a b + a12 b21 a11 b12 + a12 b22 a11 b13 + a12 b23 = 11 11 a21 b11 + a22 b21 a21 b12 + a22 b22 a21 b13 + a22 b23

All the laws of ordinary algebra hold for the addition and subtraction of matrices and for scalar multiplication. Multiplication of matrices is not in general commutative, but it is associative and distributive. If the product of two or more matrices is zero, it does not follow that one of the factors is zero. The factors are divisors of zero. Example 32 a 0 0 0 0 0 b 0 c d = 0 0 Linear Dependence

1. The quantities l1 , l2 , . . . , ln are linearly dependent if there exist constants c1 , c2 , . . . , cn , not all zero, such that c1 l1 + c2 l2 + · · · + cn ln = 0 If no such constants exist, the quantities are linearly independent. 2. The linear functions li = ai1 x1 + ai2 x2 + · · · + ain xn

i = 1, 2, . . . , m

are linearly dependent if and only if the matrix of the coefficients is of rank r < m. Exactly r of the li form a linearly independent set. 3. For m > n, any set of m linear functions are linearly dependent. Consistency of Equations

1. The system of homogeneous linear equations ai1 x1 + ai2 x2 + · · · + ain xn = 0

i = 1, 2, . . . , m

has solutions not all zero if the rank r of the matrix (aij ) is less than n. If m < n, there always exist solutions not all zero. If m = n, there exist solutions not all zero if |aij | = 0. If r of the equations are so selected that their matrix is of rank r, they determine uniquely r of the variables as homogeneous linear functions of the remaining n − r variables. A solution of the system is obtained by assigning arbitrary values to the n − r variables and finding the corresponding values of the r variables. 2. The system of linear equations ai1 x1 + ai2 x2 + · · · + ain xn = ki

i = 1, 2, . . . , m

is consistent if and only if the augmented matrix derived from (aij ) by annexing the column k1 , . . . , km has the same rank r as (aij ). As in the case of a system of homogeneous linear equations, r of the variables can be expressed in terms of the remaining n − r variables.

MATHEMATICS

177

Linear Transformations

+ a21 x2 x1 + a22 x22 + · · · + a2n x2 xn + an1 xn x1 + an2 xn x2 + · · · + ann xn2

1. If a linear transformation xi = ai1 x1 + ai2 x2 + · · · + ain xn

i = 1, 2, . . . , n

with matrix (aij ) transforms the variables xi into the variables xi and a linear transformation xi = bi1 x1 + bi2 x2 + · · · + bin xn

i = 1, 2, . . . , n

with matrix (bij ) transforms the variables xi into the variables xi , then the linear transformation with matrix (bij )(aij ) transforms the variables xi into the variables xi directly. 2. A real orthogonal transformation is a linear transformation of the variables xi into the variables xi such that n n 2 xi2 = xi i=1

i=1

A transformation is orthogonal if and only if the transpose of its matrix is the inverse of its matrix. 3. A unitary transformation is a linear transformation of the variables xi into the variables xi such that n

xi x i =

i=1

n

xi x i

i=1

A transformation is uni‘tary if and only if the transpose of the conjugate of its matrix is the inverse of its matrix. Quadratic Forms ables is n

A quadratic form in n vari-

aij xi xj = a11 x12 + a12 x1 x2 + · · · + a1n x1 xn

i,j =1

in which aj i = aij . The symmetric matrix (aij ) of the coefficients is the matrix of the quadratic form and the rank of (aij ) is the rank of the quadratic form. A real quadratic form of rank r can be reduced by a real nonsingular linear transformation to the normal form 2 x12 + · · · + xp2 − xp+1 − · · · − xr2 in which the index p is uniquely determined. If p = r, a quadratic form is positive, and if p = 0, it is negative. If, furthermore, r = n, both are definite. A quadratic form is positive definite if and only if the determinant and all the principal minors of its matrix are positive. A method of reducing a quadratic form to its normal form is illustrated. Example 33 See the tabulation for Example 33 below. The transformation

x = 3x + 2y − z

8 y = − 16 3 y + 3z

3 2 reduces q to 13 x 2 − 16 y . The transformation

x =

√

3x

4 y = √ y 3

z = z

further reduces q to the normal form x 2 − y 2 of rank 2 and index 1. Expressing x, y, z in terms of x , y , z , the real nonsingular linear transformation that reduces q to the normal form is


q = 3x 2 − 4y 2 − z2 + 4xy − 2xz + 4yz   1 2  3x 2 + 2xy − xz  = 3 (3x + 2y − z) + q1 , in which the quantity in parentheses is obtained by factoring x out q = +2xy − 4y 2 + 2yz of the first row   −xz + 2yz − z2 = 13 (9x 2 + 4y 2 + z2 + 12xy − 6xz − 4yz) + q1 q1 = − 34 y12 − 13 z2 + 43 yz − 4y 2 + 4yz − z2 8 2 − 16 3 y + 3 yz 3 8 2 = − 16 = (− 16 3 y + 3 z) + q2 + 83 yz − 43 z2 q2 = 0

z = z

178


a11 a21 a31

√

1 3 x + √ y x= 3 2 3 √ 3 1 y + 2z y=− 4 z = z Hermitian Forms A Hermitian form in n variables is n aij xi x j aj i = a ij i,j =1

The matrix (aij ) is a Hermitian matrix. Its transpose is equal to its conjugate. The rank of (aij ) is the rank of the Hermitian form. A Hermitian form of rank r can be reduced by a nonsingular linear transformation to the normal form x1 x 1 + · · · + xp x p − xp+1 x p+1 − · · · − xr x r in which the index p is uniquely determined. If p = r, the Hermitian form is positive, and, if p = 0, it is negative. If, furthermore, r = n, both are definite Determinants Second- and third-order determinants are formed from their square symbols by taking diagonal products, down from left to right being positive and up negative:

a11 a21

a12 a22 = a11 a22 − a21 a12

a12 a22 a32

a13 a23 = a11 a22 a33 + a12 a23 a31 + a13 a32 a21 a33 − a31 a22 a13 − a32 a23 a11 − a33 a12 a21

Third- and higher order determinants are formed by selecting any row or column and taking the sum of the products of each element and its cofactor. This process is continued until second- or third-order cofactors are reached: a11 a12 a13 a a a a a21 a22 a23 = a11 a22 a23 − a21 a12 a13 32 33 32 33 a31 a32 a33 a a + a31 a12 a13 22 23 The determinant of a matrix A is: 1. Zero if two rows or two columns of A have proportional elements 2. Unchanged if: a. The rows and columns of A are interchanged b. To each element of a row or column of A is added a constant multiple of the corresponding element of another row or column 3. Changed in sing if two rows or two columns of A are interchanged 4. Multiplied by c if each element of any row or column of A is multiplied by c 5. The sum of the determinants of two matrices B and C if A, B, and C have all the same elements except that in one row or column each element of A is the sum of the corresponding elements of B and C Example 34 See the tabulation for Example 34 below.

Tabulation for Example 34 2 2 4 1

9 −3 8 2

9 12 3 6

4 2 8 2 = −5 4 4 1

5 −7 0 0

9 12 3 6

2 4 8 2 = 3 −5 4 1 4

5 −7 0 0

3 4 1 2

4 8 −5 4

Multiply 1st column Factor 3 out of by −2 and add to 2nd 3rd column 2 2 4 8 3 4 1 −5 + 3 × (−7) 4 1 −5 = = 3 × (−5) 4 1 1 2 4 2 4 Expand according to 2nd column 1 = −21 2

4 −5 − (−21) 4 1

0

1st and 3rd rows are proportional

−5 = −21[(4 + 10) − (16 + 5)] = +147 4

Expand according to 1st row

1 −21 4 1

1 1 2

0 −5 4

Subtract 3rd row from 1st

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179

2.12 Systems of Equations Linear Systems (also see Section 11.6) Homogeneous. ai1 x1 + · · · + ain xn = 0, i = 1, . . . , m. Let r = rank of(aij ). For m = n:

r = n, |aij | = 0; one solution, x1 = · · · = xn = 0. r < n, |aij | = 0; infinite number of solutions. Nonhomogeneous. ai1 x1 + · · · + ain xn = ki , i = 1, . . . , m.

Solve

x − 2y + z = −2 3x + 2y + 2z = 7

For m = n: ra = rk ; consistent. (a) ra = rk = n, |aij | = 0; independent. One solution. (b) ra = rk < n, |aij | = 0; dependent. Infinite number of solutions. ra < rk ; inconsistent. No solution. Methods of Solution Elimination is a practical method of solution for a system of two or three linear equations in as many variables. Example 35 1. By addition and subtraction, solve

x − 2y + z = −2

Example 36

2x + y + 3z = 9

Let a = (aij ), an m × n matrix. a . . . aln k1  11 . ..  , .. k = augmented matrix =  .. . . am1 · · · amn km an m × (n + 1) matrix. ra = rank of a. rk = rank of k.

2x + y + 3z = 9

From (8), y = x − 2, and from (9), z = −2x + 1. Substituting for y and z in (7), x − 2x − 4 + 2x − 1 = 5, from which x = 2. Then y = 2 − 2 = 0, z = −4 + 1 = −3. Determinants can be used to solve a system of n nonhomogeneous linear equations in n variables for which |aij | = 0. To solve for xj , form a fraction the denominator of which is the determinant |aij | and the numerator the determinant obtained from |aij | by replacing its j th column by the constants ki .

(2) (3)

3x + 2y + 2z = 7

(4)

(3) + (4)

gives 4x + 3z = 5

(5)

2 × (2) + (3)

gives 5x + 7z = 16

(6)

5 × (5) − 4 × (6)

gives

−13z = −39 or z=3

Putting z = 3 in (5) or (6) gives x = −1. Then from (2), (3), or (4), y = 2. 2. By substitution, solve x + 2y − z = 5 (7) x− y =2 (8) 2x +z =1 (9)

9 1 3 9 1 3 9 1 3 −2 −2 1 −2 −2 1 16 0 7 7 2 2 5 0 3 5 0 3 = = x= 2 1 3 2 1 3 2 1 3 1 −2 1 1 −2 1 5 0 7 3 2 2 4 0 3 4 0 3 =

−(48 − 35) = −1 −(15 − 28)

Miscellaneous Systems To be solvable a system of equations must have as many independent equations as variables. A system of two polynomial equations of degrees m and n has mn solutions, real or complex. For systems in general no statement can be made regarding the number of solutions. Graphical Method of Solution. This is a general method for systems of two equations in two variables. It consists of graphing both equations on the same axes and reading the pairs of coordinates of the points of intersection of the graphs as solutions of the system. This method gives real solutions only. Example 37

Solve y = sin x

x2 + y2 = 2

Solution from the graph (Fig. 3) gives x = 1.1, y = 0.9. From symmetry, x = −1.1, y = −0.9, is also a solution. Method of Elimination of Variables. This is a general method that can be applied to systems composed of any kinds of equations, algebraic or transcendental. However, except in fairly simple cases, practical difficulties are frequently encountered. Example 38

Solve y = sin x

x2 + y2 = 2

180


A combination of n objects taken r at a time is an unarranged selection of any r of the n objects. The number of combinations of n objects taken r at a time is n Cr

=

n Pr

r!

=

n! =n Cn−r r!(n − r)!

In particular, n C1 = n, n Cn = 1. Combinations taken any number at a time, n C1 +n C2 + · · · +n Cn = 2n − 1. Fig. 3

2.14 Probability

Squaring both sides of the first equation and subtracting it from the second to eliminate y, x 2 = 2 − sin2 x. This equation can be solved by Newton’s method. Extraneous solutions introduced by squaring can be eliminated by reference to the graph. There are numerous devices for eliminating variables in special systems. For example, to solve the system of two general quadratics a1 x 2 + b1 xy + c1 y 2 + d1 x + e1 y + f1 = 0

(10)

a2 x 2 + b2 xy + c2 y 2 + d2 x + e2 y + f2 = 0

(11)

eliminate x 2 by multiplying (10) by a2 and (11) by a1 and subtracting, solve the resulting equation for x, substitute this expression in either of the given equations, and clear fractions. The resulting fourthdegree equation in y can be solved by Horner’s method. In a similar manner y could have been eliminated instead of x. 2.13 Permutations and Combinations Fundamental Principle. If in a sequence of s events the first event can occur in n1 ways, the second in n2 , . . ., the s th in ns , then the number of different ways in which the sequence can occur is n1 n2 . . . ns . A permutation of n objects taken r at a time is an arrangement of any r objects selected from the n objects. The number of permutations of n objects taken r at a time is n Pr

= n(n − 1)(n − 2) · · · (n − 4 + 1) =

n! (n − r)!

In particular, n P1 = n, n Pn = n!. Cyclic permutations are n! c c n Pr = n Pn = (n − 1)! r(n − r)! If the n objects are divided into s sets each containing ni objects that are alike, the distinguishable permutations are n = n1 + n2 + · · · + ns

n Pn

=

n! n1 !n2 ! · · · ns !

If, in a set M of m events that are mutually exclusive and equally likely, one event will occur, and if in the set M there is a subset N of n events (n ≤ m), then the a priori probability p that the event that will occur is one of the subset N is n/m. The probability q that the event that will occur does not belong to N is 1 − n/m. Example 39 If the probability of drawing one of the 4 aces from a deck of 52 cards is to be found, then 4 1 m = 52, n = 4, and p = 52 = 13 . The probability of 1 = 12 drawing a card that is not an ace is q = 1 − 13 13 . If, out of a large number r of observations in which a given event might or might not occur, the event has occurred s times, then a useful approximate value of the experimental, or a posteriori, probability of the occurrence of the event under the same conditions is s/r. Example 40 From the American Experience Mortality Table, out of 100,000 persons living at age 10 years 749 died within a year. Here r = 100,000, s = 749, and the probability that a person of age 10 will die within a year is 749/100,000. If p is the probability of receiving an amount A, then the expectation is pA. Addition Rule (either or). The probability that any one of several mutually exclusive events will occur is the sum of their separate probabilities. Example 41 The probability of drawing an ace from 1 a deck of cards is 13 , and the probability of drawing a king is the same. Then the probability of drawing 1 1 2 either an ace or a king is 13 + 13 = 13 . Multiplication Rule (both and). (a) The probability that two (or more) independent events will both (or all) occur is the product of their separate probabilities. (b) If p1 is the probability that an event will occur, and if, after it has occurred, p2 is the probability that another event will occur, then the probability that both will occur in the given order is p1 p2 . This rule can be extended to more than two events.

MATHEMATICS

181

Example 42 (a) The probability of drawing an ace 1 from a deck of cards is 13 , and the probability of 1 . Then the drawing a king from another deck is 13 probability that an ace will be drawn from the first 1 1 1 · 13 = 169 . deck and a king from the second is 13 (b) After an ace has been drawn from a deck of cards, 4 . If two cards are the probability of drawing a king is 51 drawn in succession without the first being replaced, the probability that the first is an ace and the second a 1 4 4 king is 13 · 51 = 663 . Repeated Trials. If p is the probability that an event will occur in a single trial, then the probability that it will occur exactly s times in r trials is the binomial, or Bernoulli, distribution function r Cs p

s

avoid a paradox in logic, these two ideas must be kept distinct. Two sets S 1 , S 2 may be compared as follows. If every element of set S 1 is also an element of S 2 , then S 1 is contained in S 2 . This is written S 1 ⊂ S 2 and is read “S 1 is contained in S 2 ” or “S 1 is a subset of S 2 .” If, in addition, S 2 ⊂ S 1 , then their relation is written S 1 = S 2 . On the other hand, if S4 has at least one element not contained in S 3 but S 3 ⊂ S 4 , S 3 is a proper subset of S 4 . If S 5 can contain all the elements of S 6 , this can be stressed by writing S 5 ⊆ S 6 . Evidently Ø ⊂ S for every set S . If S, called the space, is the largest set concerned in a particular discussion, all the other sets are subsets of S. Thus set A ⊂ S. The complement of A , A c , with respect to space S is the set of elements in S that are not elements of A .

(1 − p)r−s

The probability that it will occur at least s times is p r + r Cr−1 p r−1 (1 − p) + r Cr−2 p r−2 (1 − p)2 + · · · + r Cs p s (1 − p)r−s Example 43 If five cards are drawn, one from each of five decks, the probability that exactly three will 1 3 12 2 be aces is 5 C3 ( 13 ) ( 13 ) . The probability that at 1 5 1 4 12 ) +5 C4 ( 13 ) ( 13 ) + least three will be aces is ( 13 1 3 12 2 C ( ) ( ) . 5 3 13 13 3 SET ALGEBRA 3.1 Sets A set is a collection of objects called elements that are distinguished by a particular characteristic. Examples are a set of engineers, a set of integers, a set of points. Element e belongs to set S is written e ∈ S . If not, e ∈ S . A set can be denoted by including the listed elements, or merely by a typical element, in curly brackets: {2, 4, 6}; {e1 , e2 }, {e}. A set with no elements is called the null set and is denoted by Ø. A set with one element e1 is denoted by {e1 }; and to

Binary Operations for Sets. The union, Sa ∪ Sb , of sets Sa and Sb is the set of elements in Sa or Sb or in both. Note that union differs from the idea of sum since in the union the common elements are counted only once. The intersection, Sa ∩ Sb , of sets Sa and Sb is the set of elements in both S 1 and S 2 . See the tabulation below. Let Sa , Sb , Sc have their elements in space S. Boolean algebra has as one representation the following:

UNICITY. Unique union Sa ∪ Sb ⊂ S. Unique intersection Sa ∩ Sb ⊂ S. COMMUTATIVITY. Sa ∪ Sb = Sb ∪ Sa , Sa ∩ Sb = Sb ∩ Sa . ASSOCIATIVITY. Sa ∪ (Sb ∪ Sc ) = (Sa ∪ Sb ) ∪ Sc , Sa ∩ (Sb ∩ Sc ) = (Sa ∩ Sb ) ∩ Sc . Sa ∪ (Sb ∩ Sc ) = (Sa ∪ Sb )∩ DISTRIBUTIVITY. (Sa ∪ Sc ), Sa ∩ (Sb ∪ Sc ) = (Sa ∩ Sb ) ∪ (Sa ∩ Sc ). IDEMPOTENCY. Sa ∪ Sa = Sa , Sa ∩ Sa = Sa . SPACE. Sa ∪ S = S, Sa ∩ S = Sa . NULL SET. Sa ∪ Ø = Sa , Sa ∩ Ø = Ø. SUBSET. Ø ⊂ Sa ⊂ S, Sa ⊂ (Sa ∪ Sb ), (Sa ∩ Sb ) ⊂ Sa , Sa ⊂ Sb ⇒ Sa ∪ Sb = Sb , and Sa ∩ Sb = Sa .

Tabulation for Binary Operations for Sets S e2

e1 a

∪

e3

S e4

e5 (

b

a

∪

e1

e2

c b)

a

∩

e3

e4

e5

b b

b a

a

(e1, e3, e4, e5) = ( Union

a

Intersection

∩

b)

c

182


COMPLEMENT. To Sa ⊂ S there corresponds unique Sa c ⊂ S; Sa ∪ Sa c = S, Sa ∩ Sa c = Ø. DE MORGAN’S RELATIONS. (Sa ∪ S b )c = Sa c ∩ S bc , (Sa ∩ S b )c = Sa c ∪ S bc . INVARIANT under the duality transformation, ∪ ↔ ∩, ⊂↔⊃, S ↔ Ø, are all the preceding relations. 3.2 Groups

A group is a system composed of a set of elements {a} and a rule of combination of any two of them to form a product, such that: 1. The product of any ordered pair of elements and the square of each element are elements of the set. 2. The associative law holds. 3. The set contains an identity element I such that I a = aI = a for any element a of the set. 4. For any element a of the set there is in the set an inverse a −1 such that aa −1 = a −1 a = I . 5. If, in addition, the commutative law holds, the group is commutative, or Abelian. The order of a group is the number n of elements in the group. 3.3 Rings, Integral Domains, and Fields Rings. Space S consists of a set of elements e1 , e2 , e3 , . . . . These elements are compared for equality and order and combined by the operations of addition and multiplication. These terms are partially defined by the following sets of assumptions. Equality is a term from logic and means that if two expressions have this relation, then one may be substituted for the other. Assumptions of equality E1 . Unicity: either e1 = e2 or e1 = e2 .E2 . Reflexivity: E1 = e1 , E3 . Symmetry: e1 = e2 ⇒ e2 = e1 . E4 . Transitivity: e1 = e2 , e2 = e3 ⇒ e1 = e3 . Assumptions of addition A1 . Closure: e1 + e2 ⊂ S.A2 ; e1 = e2 ⇒ e1 + e3 = e2 + e3 and e3 + e1 = e3 +e2 . (Invariance under addition.) A3 . Associativity: e1 + (e2 + e3 ) = (e1 + e2 ) + e3 . A4 . Identity element: There exists an element z ⊂ S such that e1 + z = e1 , z + e1 = e1 .A5 . Commutativity: e1 + e2 = e2 + e1 . Theorem 1: z is unique. Negative. To each e ⊂ S, there corresponds an e ⊂ S such that e + e = z; e is called the negative of e and written −e. Theorem 2: e or −e is unique. Theorem 3: −(−e) = e. Theorem 4: −z = z. Theorem 5: Equation x + e1 = e2 has the solution x = e2 − e1 . Theorem 6: e1 + e3 = e1 ⇒ e3 = z. Assumptions of multiplication M1 . Closure: e1 · e2 ⊂ S.M2 .e1 = e2 ⇒ e1 · e3 = e2 · e3 and e3 · e1 = e3 · e2 . (Invariance under multiplication.) M3 . Associativity: e1 (e2 · e3 ) = (e1 · e2 )e3 .M4 . Identity element:

There exists an element u ⊂ S such that e1 · u = e1 , u · e1 = e1 .M5 . Commutativity: e1 · e2 = e2 · e1 . Theorem 7: u is unique. Reciprocal. To each element e ⊂ S except z there corresponds an e ⊂ S such that e · e = u; e is called the reciprocal of e and written e−1 . Theorem 8: e or e−1 is unique. M7 . Distributivity: e1 (e2 + e3 ) = e1 · e2 + e1 · e3 . Theorem 9: e · z = z. Theorem 10: e1 (−e2 ) = −(e1 · e2 ) = (−e1 )e2 . Theorem 11: (−e1 )(−e2 ) = e1 · e2 . Theorem 12: If S contains an element besides z, then it is u = z. Theorem 13: e1 · e2 = z ⇒ either e1 = z or e2 = z. A ring is a space S having at least two elements for which assumptions E1 to E4 , A1 to A6 , M1 to M5 , and M7 hold. An example is a residue system modulo 4. Integral Domain. An integral domain is a ring for which, as an assumption, Theorem 13 holds. An example is the set of all integers. Field. A field is an integral domain for which M6 holds. An example of a field is the set of algebraic numbers. Assumptions of (linear) order O1 . (Contains E1 .) If e1 , e2 ⊂ S, then either e1 < e2 , e1 = e2 , or e2 < e1 .O2 .e1 < e2 ⇒ e1 + e3 < e2 + e3 . (Invariance under addition.) O3 . Transitivity: e1 < e2 , e2 < e3 ⇒ e1 < e3 . Negative. If e1 < z, then e1 is called negative. Positive. If z < e2 , then e2 is called positive. O4 .z < e2 z < e3 ⇒ z < e2 · e3 . An ordered integral domain is an integral domain for which O1 to O4 hold. An example is the set of all integers. An ordered field is an ordered integral domain for which M6 holds. An example is the set of all rational numbers. If an additional order assumption, O5 , known as the Dedekind assumption—see a book on real analysis—is included, then the space S for which assumptions E1 to E4 , A1 to A6 , M1 to M7 , and O1 to O5 hold is called the real number space. An example is the set of real numbers. Here z is denoted 0, and u is denoted 1. Another example is the set of points on the real line. 4

STATISTICS AND PROBABILITY

4.1 Frequency Distributions of One Variable Definitions A frequency distribution of statistical data consisting of N values of a variable x is a tabulation by intervals, called classes, showing the number fi , called the frequency or weight, in each class; N = fi . The midvalue xi of a class is the class mark. For equal classes, the class interval is c = xi+1 − xi . The cumulative frequency, cum f , at any class is the sum of the frequencies of all classes up to and including the given class.

MATHEMATICS

183

If x0 = 0, ν1 = x, which is the arithmetic mean. In u units

Graphs Frequency Polygon. Plot the points (xi , fi ) and draw a broken line through them. Histogram. Draw a set of rectangles using as bases intervals representing the classes marked off on a straight line and using altitudes proportional to the frequencies. Frequency Curve. Draw a continuous curve approximating a frequency polygon or such that the region under the curve approximates a histogram. As the class interval c is taken smaller and the total frequency N larger, the approximation becomes better. Ogive.

k 1 fi uri N

r = 0, 1, . . .

u=

i=1

x − x0 c

c = class interval 2. About the mean. In x units µr =

k 1 fi (xi − x)r N

r = 0, 1, . . .

i=1

x = ν1 in x units

This is a graph of cumulative frequencies. In u units

Averages Arithmetic Mean k 1 AM = x = fi xi N i=1

in which N =

k i=1

f

f

f

k 1 log GM = fi log xi N i=1

r = 0, 1, . . .

i=1

u = ν1 in u units

HM = k Root-Mean-Square

k

rms =

µ2 = ν2 − ν12 µ3 = ν3 − 3ν1 ν2 + 2ν13 µr (in x units) = cr µr (in u units)

N

i=1 (fi /xi )

µ1 = 0

µ4 = ν4 − 4ν1 ν3 + 6ν12 ν2 − 3ν14

Harmonic Mean

fi xi2 N

i=1

Median. (a) For continuously varying data, the value of x for which cum f = N/2; (b) for discrete data, the value of x such that there is an equal number of values larger and smaller; for N odd, N = 2k − 1, the median is xk ; for N even, N = 2k, the median may be taken as 1 2 (xk + xk+1 ).

The value of x that occurs most frequently.

Moments 1. About x0 . In x units k 1 fi (xi − x0 )r N i=1

k 1 fi (ui − u)r N

µ0 = 1

GM = (x1 1 · x2 2 · · · xk k )1/N

νr =

µr =

In either x or u units, the µ’s as functions of the ν’s are

fi .

Geometric Mean

Mode.

νr =

r = 0, 1, . . . .

√

In x units, µ2 is the variance; µ2 is the standard deviation σ . Both are used as measures of dispersion. To compute σ , k 2 i=1 fi ui − u2 σ =c N Probable error = 0.6745σ . 3. In standard (deviation) units, α1 = 0 µ3 α3 = 3 σ µ4 α4 = 4 σ

α2 = 1 (a measure of skewness) (a measure of kurtosis)

The moment-generating function, or arbitrary-range inverse real Laplace transform, is r = 0, 1, . . .

b

M(θ ) =

eθx f (x) dx a

184


The rth moment is d r M µr = dθ r θ=0

r = 0, 1, 2, . . .

M Tiles The rth quartile Qr is the value of x for which cum f/N = r/4. The rth percentile Pr is the value of x for which cum f/N = r/100. For r = 10s, Pr = Ds , the sth decile. Other Measures of Shape Dispersion

1. Range of x, the difference between the largest and the smallest values of x. 2. Mean deviation, (1/N) ki=1 fi |xi − x|. 3. Semi-interquartile range, or quartile deviation, Q = 12 |Q3 − Q1 | Skewness. Quartile coefficient of skewness, (Q3 − 2Q2 + Q1 )/Q. Statistical Hypotheses A hypothesis concerning one or more statistical distribution parameters is a statistical hypothesis. A test of such a hypothesis is a procedure leading to a decision to accept or reject the hypothesis. The significance level is the probability value below which a hypothesis is rejected. A type 1 error is made if the hypothesis is correct but the test rejects the hypothesis. A type 2 error is made if the hypothesis is false but the test accepts the hypothesis. If the variable x has a distribution function f (x; θ ), with parameter θ , then the likelihood function, that is, distribution function of a random sample of size n, is P (θ ) = f (x1 ; θ )f (x2 ; θ ) · · · f (xn ; θ ). The use of Pmax (θ ) in the estimation of population parameters is the method of maximum likelihood. It often consists of solving dP /dθ = 0 for θ . Random Sampling A set x1 , x2 , . . . , xn of values of x with distribution function f(x) is a sample of size n drawn from the population described by f(x). If repeated samples of size n drawn from the population have the xr ’s independently distributed in the probability sense and each xr has the same distribution as the population, then the sampling is random. Normal and Nonnormal Distributions The normal distribution function in analytic and tabular form is found in Section 4.5. A linear combination of independent normal variables is normally distributed. The Poisson distribution, P (x) = e−m mx /x!, is the limit approached by the binomial distribution (Section) if the probability p that an event will occur in a single trial approaches zero and the number of trials r

becomes infinite in such a way that rp = m remains constant. If m is the mean of a nonnormal distribution of x, σ the standard deviation, and if the moment-generating function exists, then the variable (x − m)n1/2 /σ , in which x is the mean of a sample of size n, has a distribution that approaches the normal distribution as n → ∞. Nonparametric methods are those that do not involve the estimation of parameters of a distribution function. Tchebycheff’s inequality (Section) provides nonparametric tests for the validity of hypotheses. It leads to the law of large numbers. Let p be the probability of an event occurring in one trial and pn the ratio of the number of occurrences in n trials to the number n. The probability that |pn − p| > ε is ≤ pq/nε; this can be made arbitrarily small, however small ε is, by taking n large enough. The ratio pn converges stochastically to the probability p. Two numbers L1 , L2 between which a large fraction of a population is expected to lie are tolerance limits. If z is the fraction of the population of a variable with a continuous distribution that lies between the extreme values of a random sample of size n from this population, then the distribution of z is f (z) = n(n − 1)zn−2 (1 − z). Statistical Control of Production Processes A chart on which percentage defective in a sample is graphed as a function of output time can be used for control of an industrial process. Horizontal lines are drawn through the mean m and the controls m ± 3σ/n1/2 . The behavior of the graph with respect to these control lines is used as an error signal in a feedback system that controls the process. If the graph goes out of the band bounded by the control lines, the process is stopped until the trouble is located and removed. 4.2 Correlation To discover whether there is a simple relation between two variables, corresponding pairs of values are used as coordinates to plot the points of a scatter diagram. The simplest relation exists if the scatter diagram can be approximated more or less closely by a straight line. Least-Square Straight Line. This line, which minimizes the sum of the squares of the y deviations of the points, is yˆ − y = M(x − x)

in which M=

(x − x)y

(x − x)2

(x, y) is a plotted point, and (x, y) ˆ is a point on this line of regression of y on x. The correlation coefficient "1/2 !

(y − y) ˆ 2 r =± 1−

(y − y)2

MATHEMATICS

185

is a measure of the usefulness of the regression line. If r = 0, the line is useless; if r = ±1, the line gives a perfect estimate. The percentage of the variance of y that has been accounted for by y’s relation to x is equal to r 2 . Polynomial of Degree n−1. This can be passed through n points (xi , yi ). The method of doing this by divided differences is as follows: Example 44 Find the polynomial through (1, 5), (3, 11), (4, 31), (6, 3). Using the first three values of x, assume the polynomial to be of the form y = a1 + a2 (x − 1) + a3 (x − 1)(x − 3) + a4 (x − 1)(x − 3)(x − 4). The ai are the last four numbers in the top diagonal of the following:

4.3

Statistical Estimation by Small Samples A statistic is an unbiased estimate of a population parameter if its expected value is equal to the population parameter. In the problem of estimating a population parameter, such as the mean or variance, the interval within which c percent of the sample parameter values lies is the c percent confidence interval for the parameter. The χ 2 distribution function for ν degrees of freedom is 1 (χ 2 )(ν−2)/2 e−χ2/2 f (χ 2 ) = ν/2 2 (ν/2)

and its moment-generating function is M(φ) = (1 − 2θ )−ν/2 The sum of the squares of n random sample values of x has a χ 2 distribution with n degrees of freedom if x has a normal distribution with zero mean and unit variance. The binomial index of dispersion is χ2 =

k (xr − x)2 x(1 − x/n) r=1

To form the graphic, put the given (xi , yi ) in the first two columns. To find a number in any other column, divide the difference of the two numbers just above and below it immediately to the left by the difference of the x’s in the two diagonals through it. The polynomial is y = 5 + 3(x − 1) + 17 3 (x − 1)(x − (x − 1)(x − 3)(x − 4). 3) − 51 15 Power Formula. y = ax n fits well if the points (xi , yi ) lie approximately on a straight line when plotted on logarithmic (log scales on both horizontal and vertical axes) graph paper. To find a and n use two of the points (x1 , y1 ) and (x2 , y2 ), preferably far apart: log y2 − log y1 n= log x2 − log x1

log a = log y1 − n log x1 Exponential Formula. y = aenx fits well if the points (xi , yi ) lie approximately on a straight line when plotted on semilogarithmic (log scale on vertical axis) graph paper. To find a and n use two of the points (x1 , y1 ) and (x2 , y2 ), preferably far apart:

n=

log y1 − log y2 ln y1 − ln y2 = 2.3026 x1 − x2 x1 − x2

ln a = ln y1 − nx1

or

log a = log y1 − 0.4343nx1

For p small and n large, this reduces to the Poisson index of dispersion kr=1 (xr − x)2 /x. These indices are used to test the hypothesis that k sample frequencies xr came from the same binomial or Poisson population, respectively. Student’s t distribution for the variable t = uν 1/2 / v is −(ν+1)/2 t2 f (t) = c 1 + ν ν degrees of freedom, c constant, if u has a normal distribution with zero mean and unit variance and v 2 has a χ 2 distribution with ν degrees of freedom. The F distribution for the variable F = (u/ν1 )/ (v/ν2 ) is cF (ν1 −2)/2 f (F ) = (ν2 + ν1 F )(ν1 +ν2 )/2 ν1 and ν2 degrees of freedom, c constant, if u and v have independent χ 2 distributions with ν1 and ν2 degrees of freedom, respectively. Analysis of Variance Experimental error is the variation in the basic variable remaining after the effects of controlled variables have been removed (Section 4.5). The analysis of variance means the resolution of the basic sum of squares into the component that measures the part of the variation being tested and the component that measures the experimental error.

186


4.4 Statistical Design of Experiments To get valid conclusions from an experiment, there is need for proper control of the other variables besides those being investigated and also for sufficiently large and random samples. Sampling Inspection. To make an inspection efficient, the cost and usually the amount of sampling should be minimized. It is a common practice in industry for a consumer to accept or reject a lot on the basis of a sample drawn from the lot. There is a maximum fraction of defectives that the consumer will tolerate. This is the lot tolerance fraction defective pt . A random sample of n pieces is selected from a lot of N pieces. The maximum allowable number of defective pieces in an acceptable sample is c. Single sampling means: (a) Inspect a sample of n pieces. (b) Accept the lot if the number of defective pieces is c or less; otherwise inspect the remainder of the lot. (c) Replace all defective pieces found by nondefective ones. The consumer’s risk, that is, the probability that a consumer will accept a lot of quality lower than pt , is

For a more extensive treatment of the elementary theory of statistics, see Applied Statistics and Probability for Engineers by D. C. Montgomery and G. C. Runger (Wiley, Hoboken, New Jersey, 2007). 4.5 Precision of Measurements Observations and Errors The error of an observation is ei = mi − m, i = 1, 2, . . . , n, where the mi are the observed values, the ei the errors, and m the mean value, that is, the arithmetic mean of a very large number (theoretically infinite) of observations. In a large number of measurements random errors are as often negative as positive and have little effect on the arithmetic mean. All other errors are classed as systematic. If due to the same cause, they affect the mean in the same sense and give it a definite bias.

(12)

Best Estimate and Measured Value. If all systematic errors have been eliminated, it is possible to consider the sample of individual repeated measurements of a quantity with a view to securing the “best” estimate of the mean value m and assessing the degree of reproducibility that has been obtained. The final result will then be expressed in the form E ± L, where E is the best estimate of m and L the characteristic limit of variation associated with a certain risk. Not merely E but the entire result E ± L is the value measured.

If a producer has standardized quality at a fractional value p, the process average fraction defective, then the producer’s risk, that is, the probability that a lot will be erroneously rejected, is

Arithmetic Mean. If a large number of measurements have been made to determine directly the mean m of a certain quantity, all measurements having been made with equal skill and care, the best estimate of m from a sample of n is the arithmetic mean m of the measurements in the sample,

Pc =

c

Npt x

x=0

Pp = 1 −

c x=0

N − Npt n−x N n

Np x

N − Np n−x N n

1 mi n n

m=

i=1

(13)

These two risks correspond to errors of type 2 and type 1, respectively. The average number of pieces inspected per lot for single sampling is I = n + (N − n)Pp . The amount of inspection and ordinarily the cost are minimized by finding the pair of values of n and c that satisfy (a) above for an assigned value of Pc and minimize I . Sequential Analysis. An improvement on the fixedsize sampling methods already described results in greater efficiency if the inspection can be conducted on an accumulation-of-information basis. Such sequential methods operate on successive terms of a sequence of observations as they are received. They involve two steps: (a) to accept or reject the hypothesis under test and (b) to continue taking additional observations if the hypothesis is rejected.

Standard deviation is the root-mean-square of the deviations ei of a set of observations from the mean,

σ =

1 2 ei n n

1/2

i=1

Since neither the mean m nor the errors of observation ei are ordinarily known, the deviations from the arithmetic mean, or the residuals, xi = mi − m, i = 1, 2, . . . , n, will be referred to as errors. Likewise, for σ the unbiased value # n $1/2 −1/2 2 (mi − m) σ = (n − 1)

= (n − 1)

1/2

i=1 n i=1

ei2

1/2

MATHEMATICS

187

will be used, in which n is replaced by n − 1 since one degree of freedom is lost by using m instead of m, m being related to the mi . Normal Distribution Relative Frequency of Errors. The Gauss– Laplace, or normal, distribution of frequency of errors is (Fig. 4) 1 2 y = √ e−x /2σ 2 σ 2π

or

Fig. 4

Probability. The fraction of the total number of errors whose values lie between x = −a and x = a is

1 2 2 y = √ he−h x π

h P = √ π

√ where 2h σ = 1, or h = 1/( 2σ ), and y represents the proportionate number of errors of value x. The area under the curve is unity. The dotted curve is also an error distribution curve with a greater value of the precision index h, which measures the concentration of observations about their mean. 2 2

2 Table 1 Values of P = √ π haa 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 a

0

1

ha

e−h

2 x2

+a

e−h

2x2

−a

2 dx = √ π

ha

e−h

2x2

d (hx)

0

(14) that is, P is the probability of an error x having a value between −a and a (see Table 1). Similarly, the shaded area represents the probability of errors between b and c.

d(hx)

0

2

0.01128 0.02256 0.11246 0.12362 0.13476 0.22270 0.23352 0.24430 0.32863 0.33891 0.34913 0.42839 0.43797 0.44747 0.52050 0.52924 0.53790 0.60386 0.61168 0.61941 0.67780 0.68467 0.69143 0.74210 0.74800 0.75381 0.79691 0.80188 0.80677 0.84270 0.84681 0.85084 0.88021 0.88353 0.88679 0.91031 0.91296 0.91553 0.93401 0.93606 0.93807 0.95229 0.95385 0.95538 0.96611 0.96728 0.96841 0.97635 0.97721 0.97804 0.98379 0.98441 0.98500 0.98909 0.98952 0.98994 0.99279 0.99309 0.99338 0.99532 0.99552 0.99572 0.99702 0.99715 0.99728 0.99814 0.99822 0.99831 0.99886 0.99891 0.99897 0.99931 0.99935 0.99938 0.99959 0.99961 0.99963 0.99976 0.99978 0.99979 0.99987 0.99987 0.99988 0.99992 0.99993 0.99993 0.99996 0.99996 0.99996 0.99998 1.0000 1.0000 √ ha = 0.47694(a/r) = (1/ 2)(a/σ ).

3

4

5

6

7

8

9

0.03384 0.14587 0.25502 0.35928 0.45689 0.54646 0.62705 0.69810 0.75952 0.81156 0.85478 0.88997 0.91805 0.94002 0.95686 0.96952 0.97884 0.98558 0.99035 0.99366 0.99591 0.99741 0.99839 0.99902 0.99941 0.99965 0.99980 0.99989 0.99994 0.99997 1.0000

0.04511 0.15695 0.26570 0.36936 0.46623 0.55494 0.63459 0.70468 0.76514 0.81627 0.85865 0.89308 0.92051 0.94191 0.95830 0.97059 0.97962 0.98613 0.99074 0.99392 0.99609 0.99753 0.99846 0.99906 0.99944 0.99967 0.99981 0.99989 0.99994 0.99997

0.05637 0.16800 0.27633 0.37938 0.47548 0.56332 0.64203 0.71116 0.77067 0.82089 0.86244 0.89612 0.92290 0.94376 0.95970 0.97162 0.98038 0.98667 0.99111 0.99418 0.99626 0.99764 0.99854 0.99911 0.99947 0.99969 0.99982 0.99990 0.99994 0.99997

0.06762 0.17901 0.28690 0.38933 0.48466 0.57162 0.64938 0.71754 0.77610 0.82542 0.86614 0.89910 0.92524 0.94556 0.96105 0.97263 0.98110 0.98719 0.99147 0.99443 0.99642 0.99775 0.99861 0.99915 0.99950 0.99971 0.99983 0.99991 0.99995 0.99997

0.07886 0.18999 0.29742 0.39921 0.49375 0.57982 0.65663 0.72382 0.78144 0.82987 0.86977 0.90200 0.92751 0.94731 0.96237 0.97360 0.98181 0.98769 0.99182 0.99466 0.99658 0.99785 0.99867 0.99920 0.99952 0.99972 0.99984 0.99991 0.99995 0.99997

0.09008 0.20094 0.30788 0.40901 0.50275 0.58792 0.66378 0.73001 0.78669 0.83423 0.87333 0.90484 0.92973 0.94902 0.96365 0.97455 0.98249 0.98817 0.99216 0.99489 0.99673 0.99795 0.99874 0.99924 0.99955 0.99974 0.99985 0.99992 0.99995 0.99997

0.10128 0.21184 0.31828 0.41874 0.51167 0.59594 0.67084 0.73610 0.79184 0.83851 0.87680 0.90761 0.93190 0.95067 0.96490 0.97546 0.98315 0.98864 0.99248 0.99511 0.99688 0.99805 0.99880 0.99928 0.99957 0.99975 0.99986 0.99992 0.99996 0.99998

188


Probable Error. Results of measurements are sometimes expressed in the form E ± r, where r is the probable error of a single observation and is defined as the number that the actual error may with equal probability be greater or less than. From (14)

2 √ π and or

hr

e

−h2 x 2

xi2 : 0.0004, 0.0004, 0.0169, 0.0784, 0.1764, 0.0049, 0.0529, 0.0009, 0.0289, 0.0009. Hence 10

d(hx) = 0.50

0

xi2 = 0.3610

10

and

i=1

hr = 0.47694 r = 0.4769 ×

√

2σ = 0.6745σ

Similarly, 5% of the errors x are greater than 2σ and less than 1% are greater than 3σ . For rapid comparisons the following approximate formula due to Peters is useful: r ≈ 0.8453[n(n − 1)]−1/2

n

|xi |

i=1

The standard deviation of the arithmetic mean, σm , as calculated from data, is related to the standard deviation σ by the formula σm = n−1/2 σ = [n(n − 1)]−1/2

n

1/2 xi2

i=1

From this formula and Tables 1 and 2 the limits corresponding to given risks can be determined as indicated previously. It is evident that the stability of the mean increases with n, that is, the effect of the erratic behavior of single cases decreases with increase of n. The probable error of the arithmetic mean as calculated from data, rm , is then given by −1/2

rm = 0.6745[n(n − 1)]

n

1/2

So by the standard formulas, r = 0.6745(9)−1/2 (0.3610)1/2 = 0.13, rm = (10)−1/2 r = 0.042. By the approximate formulas, r ≈ 0.8453(90)−1/2 (1.40) = 0.12, rm ≈ 0.039. For the best estimate of the baseline, the result is 455.330 with probable error ±0.042 (using result given by the standard formula), usually written 455.330 ± 0.042. In any considerable number of observations it should be the case, as it is here, that half of the residuals are less than the probable error. Rounded Numbers. It can be shown that the standard deviation σ of a rounded number (Section 1.3) due to rounding is σ = 0.2887 w, where w is a unit in the last place retained. Consequently, the probable error of a rounded number due to rounding is

r = 0.6745 × 0.2887 w = 0.1947 w Weighted Observations. Sometimes, notwithstanding the care with which observations are taken, there are reasons for believing that certain observations are better than others. In such cases the observations are given different weights, that is, are counted a different numbers of times, the weights or numbers expressing their relative practical worth. If there are n weighted observations mi with weights pi , these being made directly on the same quantity, then the best estimate of the mean value m of the quantity is the weighted arithmetic mean m of the sample,

n i=1 pi mi m≡ n i=1 pi

xi2

i=1

and Peters’s formula for the approximate value is rm ≈ 0.8453[n2 (n − 1)]−1/2

n

|xi |

i=1

Example 45. The following are 10 measurements, mi , of the length of a baseline. The values of the residuals, xi , and their squares are given: m = 455.35, 455.35, 455.20, 455.05, 455.75, 455.40, 455.10, 455.30, 455.50, 455.30.

Arithmetic mean m = 455.330. xi : 0.02, 0.02, −0.13, −0.28, 0.42, 0.07, −0.23, −0.03, −0.17, −0.03.

|xi | = 1.40

i=1

For the set of weighted observations we have

−1/2

r = 0.6745(n − 1)

n

1/2 pi xi2

i=1

as the probable error of an observation of unit weight and # rm = 0.6745 (n − 1)

n i=1

$−1/2 pi

n i=1

1/2 pi xi2

MATHEMATICS

189

Table 2 Values of Functions of n and n − 1 Factors for Computing Actual and Approximate Values of r and rm n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.6745 √ n−1

0.6745 n(n − 1)

0.8453 n(n − 1)

0.6745 0.4769 0.3894 0.3372 0.3016 0.2754 0.2549 0.2385 0.2248 0.2133 0.2034 0.1947 0.1871 0.1803 0.1742 0.1686 0.1636 0.1590 0.1547 0.1508 0.1472 0.1438 0.1406 0.1377 0.1349 0.1323 0.1298 0.1275 0.1252 0.1231 0.1211 0.1192 0.1174 0.1157 0.1140 0.1124 0.1109 0.1094 0.1080 0.1066 0.1053 0.1041 0.1029 0.1017 0.1005 0.0994 0.0984 0.0974 0.0964

0.4769 0.2754 0.1947 0.1508 0.1231 0.1041 0.0901 0.0795 0.0711 0.0643 0.0587 0.0540 0.0500 0.0465 0.0435 0.0409 0.0386 0.0365 0.0346 0.0329 0.0314 0.0300 0.0287 0.0275 0.0265 0.0255 0.0245 0.0237 0.0229 0.0221 0.0214 0.0208 0.0201 0.0196 0.0190 0.0185 0.0180 0.0175 0.0171 0.0167 0.0163 0.0159 0.0155 0.0152 0.0148 0.0145 0.0142 0.0139 0.0136

0.5978 0.3451 0.2440 0.1890 0.1543 0.1304 0.1130 0.0996 0.0891 0.0806 0.0736 0.0677 0.0627 0.0583 0.0546 0.0513 0.0483 0.0457 0.0434 0.0412 0.0393 0.0376 0.0360 0.0345 0.0332 0.0319 0.0307 0.0297 0.0287 0.0277 0.0268 0.0260 0.0252 0.0245 0.0238 0.0232 0.0225 0.0220 0.0214 0.0209 0.0204 0.0199 0.0194 0.0190 0.0186 0.0182 0.0178 0.0174 0.0171

0.8453 √ n n−1 0.4227 0.1993 0.1220 0.0845 0.0630 0.0493 0.0399 0.0332 0.0282 0.0243 0.0212 0.0188 0.0167 0.0151 0.0136 0.0124 0.0114 0.0105 0.0097 0.0090 0.0084 0.0078 0.0073 0.0069 0.0065 0.0061 0.0058 0.0055 0.0052 0.0050 0.0047 0.0045 0.0043 0.0041 0.0040 0.0038 0.0037 0.0035 0.0034 0.0033 0.0031 0.0030 0.0029 0.0028 0.0027 0.0027 0.0026 0.0025 0.0024

n

0.6745 √ n−1

0.6745 n(n − 1)

0.8453 n(n − 1)

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0954 0.0944 0.0935 0.0926 0.0918 0.0909 0.0901 0.0893 0.0886 0.0878 0.0871 0.0864 0.0857 0.0850 0.0843 0.0837 0.0830 0.0824 0.0818 0.0812 0.0806 0.0800 0.0795 0.0789 0.0784 0.0779 0.0774 0.0769 0.0764 0.0759 0.0754 0.0749 0.0745 0.0740 0.0736 0.0732 0.0727 0.0723 0.0719 0.0715 0.0711 0.0707 0.0703 0.0699 0.0696 0.0692 0.0688 0.0685 0.0681 0.0678

0.0134 0.0131 0.0128 0.0126 0.0124 0.0122 0.0119 0.0117 0.0115 0.0113 0.0111 0.0110 0.0108 0.0106 0.0105 0.0103 0.0101 0.0100 0.0098 0.0097 0.0096 0.0094 0.0093 0.0092 0.0091 0.0089 0.0088 0.0087 0.0086 0.0085 0.0084 0.0083 0.0082 0.0081 0.0080 0.0079 0.0078 0.0077 0.0076 0.0075 0.0075 0.0074 0.0073 0.0072 0.0071 0.0071 0.0070 0.0069 0.0068 0.0068

0.0167 0.0164 0.0161 0.0158 0.0155 0.0152 0.0150 0.0147 0.0145 0.0142 0.0140 0.0137 0.0135 0.0133 0.0131 0.0129 0.0127 0.0125 0.0123 0.0122 0.0120 0.0118 0.0117 0.0115 0.0113 0.0112 0.0111 0.0109 0.0108 0.0106 0.0105 0.0104 0.0102 0.0101 0.0100 0.0099 0.0098 0.0097 0.0096 0.0094 0.0093 0.0092 0.0091 0.0090 0.0089 0.0089 0.0088 0.0087 0.0086 0.0085

0.8453 √ n n−1 0.0023 0.0023 0.0022 0.0022 0.0021 0.0020 0.0020 0.0019 0.0019 0.0018 0.0018 0.0017 0.0017 0.0017 0.0016 0.0016 0.0016 0.0015 0.0015 0.0015 0.0014 0.0014 0.0014 0.0013 0.0013 0.0013 0.0013 0.0012 0.0012 0.0012 0.0012 0.0011 0.0011 0.0011 0.0011 0.0011 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009

190


Tabulation for Example 46 pi mi pi mi xi xi2 pi xi2

5 178.26 891.30 0.10 0.010 0.05

4 176.30 705.20 1.86 3.460 13.84

1 181.06 181.06 2.90 8.410 8.41

as the probable error of the arithmetic mean of weighted items, in which mi − ni=1 pi mi n xi ≡ i=1 pi Example 46. Let six observations on the same quantity be made with weights pi , the sum of these weights being 21 (see the tabulationabove). The 6 sum of the weighted observations, i=1 pi mi , is 3741.36. The best estimate of the value of m for the observed quantity is m = 3741.36/21 = 178.16. Subtracting this from each mi gives the residuals xi . The 6 sum 2of the weighted squares of the residuals, i=1 pi xi , is 62.95. Then the preceding formulas give the probable error of an observation of weight unity as r = 2.39 and the probable error of the weighted mean as rm = 0.52. The final result then is 178.16 ± 0.52. Probable Error in a Result Calculated from Means of Several Observed Quantities. Let Z be a sum of n means of observed independent quantities, each taken with a plus or a minus sign. Then, if rj , j = 1, 2, . . . , n, are the probable errors in 1/2 n 2 these means, the probable error in Z is . j =1 rj Let Z = Az, where z is the mean of an observed quantity with probable error r and A an exact number. Then the probable error in Z is Ar. Let Z be any differentiable function of the means of independently observed quantities zj with probable errors rj . Then the probable error in Z is % &1/2 m 2 2 . For example, if Z = z1 z2 , the j =1 (∂Z/∂zj ) rj

probable error in Z is (z12 r22 + z22 r12 )1/2 . Conditions of Applicability. The theory underlying the foregoing development depends on the following assumptions: (a) The sample consists of a large number of observations. (b) The observations have Table 3

4 177.95 711.80 0.21 0.441 0.18

3 176.20 528.60 1.96 3.842 11.53

4 180.85 723.40 2.69 7.230 28.94

been made with equal care and skill so that (i) there are approximately an equal number of readings above and below the mean (except in the case of weighted items), (ii) the individual deviations from the mean are small in most cases, and (iii) the number of deviations diminishes rapidly as their size increases. The extent to which the observed data satisfy these assumptions is a measure of the extent to which we are justified in using the Gauss error distribution curve, which is consistent with the statement that m is the best estimate of the mean value m and which leads to the factor 0.6745 used in computing probable error. Even if we were not justified in assuming the Gaussian distribution of errors, the arithmetic mean still remains the best estimate we have for m. Therefore, there is little difficulty in this regard, especially since “errors” appear to follow the Gaussian distribution as closely as any other we know. Our difficulties enter in connection with the factor 0.6745 and the accuracy of the σ , as estimated from the data. If the number of observations n in a sample is small, the estimate of the standard deviation of the possible infinity of observations with mean m is itself subject to considerable error. For example, for n = 3 the standard error of the standard deviation is as large as the standard deviation itself, and hence the probable error calculated from r = 0.6745σ would not be very reliable. Table 3 will illustrate this. The second and third columns give the probability that the probable error of a single observation should be out 20 and 50%, respectively. From Table 3 it is clear that with 10 observations the odds are only 3 : 2 that the calculated probable error is within 20% of the correct value and about 30 : 1 that it is within 50% of the correct value. Of course, the probable error of the mean will be correspondingly out. The use of Table 2.3 is quite legitimate for 100 < n, and for 30 < n < 100 the table may be used provided σ is multiplied by (n − 3)−1/2 . For n < 30, a rough estimate can be obtained from the fact that the percentage of cases lying outside the range, m ± kσ , is < 100k −2 for 1 < k. A striking property of this inequality due to Tchebycheff is that it is nonparametric,

Combination of Observations

n

20%

50%

n

20%

50%

5 10 15 20

0.64 0.40 0.29 0.21

0.24 0.034 0.008 0.0002

30 40 50 100

0.12 0.076 0.047 0.0050

0.00014 8 × 10−6 6 × 10−7

Source: D. Brunt, The Combination of Observations, Cambridge University Press, 1917.

MATHEMATICS

191

which means independent of the nature of the distribution assumed. 5 GEOMETRY 5.1 Geometric Concepts 1 Plane Angles A degree (◦ ) is 360 of a revolution (or perigon) and is divided into 60 units called minutes ( ) that in turn are divided into 60 units called seconds ( ). A radian is a central angle that intercepts a circular arc equal to its radius. One radian, therefore, equals 360/2π degrees, or 57.295779513◦ , and 1◦ = 0.017453293 radian. An angle of 90◦ is a right angle, and the lines that form it are perpendicular. An angle less than a right angle is acute. An angle greater than a right angle but less than 180◦ is obtuse. If the sum of two angles equals 90◦ , they are complementary to each other, and if their sum is 180◦ , supplementary to each other.

Polygons A polygon, or plane rectilinear figure, is a closed broken line. A triangle is a polygon of three sides. It is isosceles if two sides (and their opposite angles) are equal; it is equilateral if all three sides (and all three angles) are equal. A quadrilateral is a polygon of four sides. This classification includes the trapezium, having no two sides parallel; the trapezoid, having two opposite sides parallel (isosceles trapezoid if the nonparallel sides are equal); and the parallelogram, having both pairs of opposite sides parallel and equal. The parallelogram includes the rhomboid, having no right angles and, in general, adjacent sides not equal; the rhombus, having no right angles but all sides equal; the rectangle, having only right angles and, in general, adjacent sides not equal; and the square, having only right angles and all sides equal. Similar polygons have their respective angles equal and their corresponding sides proportional. A regular polygon has all sides equal and all angles equal. An equilateral triangle and a square are regular polygons. Other polygons classified according to number of sides are (5) pentagon, (6) hexagon, (7) heptagon, (8) octagon, (9) enneagon, or nonagon, (10) decagon, and (12) dodecagon. Two regular polygons of the same number of sides are similar. Properties of Triangles

and equals the sum of the opposite interior angles (i.e., ∠XAB = ∠B + ∠C). A median of a triangle is a line joining a vertex to the midpoint of the opposite side. The three medians meet at the center of gravity, G, and G trisects each median (e.g., AG = 23 AD). Bisectors of angles of a triangle (Fig. 6) meet in a point M equidistant from all sides. M is the center of the inscribed circle (tangent to all sides), or the incenter of the triangle. An angle bisector divides the opposite side into segments proportional to the adjacent sides of the angle (e.g. AK/KC = AB/BC). An altitude of a triangle is a perpendicular from a vertex to the opposite side. The three altitudes meet in a point called the orthocenter. The perpendicular bisectors of the sides of a triangle (Fig. 7) meet in a point O equidistant from all vertices. O is the center of the circumscribed circle (passing through all vertices), or the circumcenter of the triangle. The longest side of a triangle is opposite the largest angle, and vice versa. The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. If two triangles are mutually equiangular, they are similar, and their corresponding sides are proportional. Orthogonal Projection. In Figs. 8 and 9, AE is the orthogonal projection of AB on AC, BE being perpendicular to AC. The square of the side opposite an acute angle equals the sum of the squares of the other two sides diminished by twice the product of one of those sides by the orthogonal projection of the other side upon it. In Fig. 8, a 2 = b2 + c2 − 2b · AE. The

Fig. 6

Fig. 7

General Triangle. The sum of the angles equals 180◦ .∠XAB (Fig. 5) is an exterior angle of ABC Fig. 8

Fig. 5

Fig. 9

192


square of the side opposite an obtuse angle equals the sum of the squares of the other two sides increased by twice the product of one of those sides by the orthogonal projection of the other side upon it. In Fig. 9, a 2 = b2 + c2 + 2b · AE. Fig. 12

Right Triangle. In Fig. 10, let h be the altitude drawn from the vertex of right angle C to the hypotenuse c. Then ∠A + ∠B = 90◦ ; c2 = a 2 + b2 ; h2 = mn; b2 = cm; a 2 = cn; median from C = c/2. Isosceles Triangle. Two sides are equal and their opposite angles are equal. If a straight line from the vertex at which the equal sides meet bisects the base, it also bisects the angle at the vertex and is perpendicular to the base. Circles A circle is a closed plane curve, all the points of which are equidistant from a center point. A chord is a straight line joining two points on a curve, that is, joining the extremities of an arc. A segment of a circle is the part of its plane included between a concave arc and its chord. An angle intercepts an arc cut off by its sides; the arc subtends the angle. A central angle of a circle is one whose vertex is at the center and whose sides are two radii. A sector of a circle is the part of its plane that is included between an arc and two radii drawn to its extremities. A secant of a circle is a straight line intersecting it in two points. Parallel secants (or tangents) intercept equal arcs. A tangent line meets a circle in only one point and is perpendicular to the radius to that point. If a radius is perpendicular to a chord, it bisects both the chord and the arc intercepted by the chord. If two circles are tangent to each other, the line of centers passes through the point of contact; if the circles intersect, the line of centers bisects the common chord at right angles. In Fig. 11, the product of linear segments AC and AE equals the product of linear segments AB and AF. In Fig. 12, the product of the whole secant AB and its external segment AE equals the product of the whole

Fig. 13

secant AC and its external segment AF. In Fig. 13, the product of the whole secant AD and its external segment AC equals the square of tangent AB (or AE ). Also ∠ABE = ∠AEB. Angle Measurement. Considering the arc of a circle to be expressed in terms of the central angle that it subtends, the arc may be said to contain a certain number of degrees and hence be used to express the measurement of other angles related to the circle. On this basis, an entire circle equals 360◦ . The inscribed angle formed by two chords intersecting on a circle equals half the arc intercepted by it. Thus, in Fig. 14, ∠BAC = 12 arc BC. An angle inscribed in a semicircle is a right angle. The angle formed by a tangent to a circle and a chord having one extremity at the point of contact equals half the arc intercepted by the chord. In Fig. 14, ∠BAT = 12 arc BCA. The angle formed by two chords intersecting within a circle equals half the sum of the intercepted arcs. In Fig. 11, ∠BAC (or ∠EAF ) = 12 (arc BC + arc EF ). The angle formed by two secants, or two tangents, or a secant and a tangent, intersecting outside a circle, equals half the difference of the intercepted arcs. In Fig 12, ∠BAC = 12 (arc BC − arc EF ). In Fig. 13, ∠BAE = 1 1 2 (arc BDE − arc BCE), and ∠BAD = 2 (arc BD − arc BC). Coaxal Systems Types

Fig. 10

Fig. 11

1. A set of nonintersecting circles having collinear centers and orthogonal to a given circle with

Fig. 14

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193

The inverse of a circle not passing through the center of inversion is a circle, the inverse of a circle through the center is a straight line not through the center, and the inverse of a straight line through the center is itself. Two intersecting curves invert into curves intersecting at the same angle.

Fig. 15

2. 3. 4. 5. 6.

center also collinear. The endpoints of the diameter of the given circle on the line of centers are the limiting points of the system (Fig. 15, centers on horizontal line). A set of circles through two given points (Fig. 15, centers on vertical line). A set of circles with a common point of tangency. A set of concentric circles. A set of concurrent lines. A set of parallel lines.

Nonplanar Angles A dihedral angle is the opening between two intersecting planes. In Fig. 17, P–BD–Q is a dihedral angle of which the two planes are the faces and their line of intersection DB is the edge. A plane angle that measures a dihedral angle is an angle formed by two lines, one in each face, drawn perpendicular to the edge at the same point (as ∠ABC). A right dihedral angle is one whose plane angle is a right angle. Through a given line oblique or parallel to a given plane, one and only one plane can be passed perpendicular to the given plane. The line of intersection CD (Fig. 18) is the orthogonal projection of line AB upon plane P . The angle between a line and a plane is the angle that the line (produced if necessary) makes with its orthogonal projection on the plane. This angle is the least angle that the line makes with any line in the plane. A polyhedral angle is the opening of three or more planes that meet in a common point. In Fig. 19, O–ABCDE is a polyhedral angle of which the intersections of the planes, as OA, OB, and so on, are the edges; the portions of the planes lying between the edges are the faces; and the common point O is the vertex. Angles formed by adjacent edges, as angles AOB, BOC, and so on, are face angles. A polyhedral angle

Conjugate Systems. Two coaxal systems whose members are mutually orthogonal are conjugate. A conjugate pair may consist of (a) a system of type 1 and one of type 2, with the limiting points of one the common points of the other (Fig. 15); (b) two systems of type 3; (c) a system of type 4 and one of type 5; (d) two systems of type 6. Inversion If the point O is the center of a circle c of radius r, if P and P are collinear with O, and if OP · OP = r 2 , then P and P are inverse to each other with respect to the circle c (Fig. 16). The point O is the center of inversion.

Fig. 17

Fig. 18

Fig. 16

Fig. 19

194


Fig. 20

is called a trihedral angle if it has three faces; a tetrahedral angle if it has four faces; and so on. A solid angle measures the opening between surfaces, either planar or nonplanar, which meet in a common point. The polyhedral angle is a special case. In Fig. 20 the solid angle at any point P , subtended by any surface S, is equal numerically to the portion. A of the surface of a sphere of unit radius that is cut out by a conical surface with vertex at P and having the boundary of S for base. The unit solid angle is the steradian and equals the central solid angle that intercepts a spherical area (of any shape) equal to the radius squared. The total solid angle about a point equals 4π steradians. A spherical angle is the opening between two arcs of great circles drawn on a sphere from the same point (vertex) and is measured by the plane angle formed by tangents to its sides at its vertex. If the planes of the great circles are perpendicular, the angle is a right spherical angle. Polyhedrons A polyhedron is a convex closed surface consisting of parts of four or more planes, called its faces; its faces intersect in straight lines, called its edges; its edges at points, called its vertices. A prism is a polyhedron of which two faces (the bases) are congruent polygons in parallel planes and the other (lateral ) faces are parallelograms whose planes intersect in the lateral edges. Prisms are triangular, rectangular, quadrangular, and so on, according as their bases are triangles, rectangles, quadrilaterals, and so on. A right prism has its lateral edges perpendicular to its bases. A prism whose bases are parallelograms is a parallepiped ; if in addition the edges are perpendicular to the bases, it is a right parallelepiped. A rectangular parallelepiped is a right parallelepiped whose bases are rectangles. A cube is a parallelepiped whose six faces are squares. A truncated prism is that part of a prism included between a base and a section made by a plane oblique to the base. A right section of a prism is a section made by a plane that cuts all the lateral edges perpendicularly. A prismatoid is a polyhedron of which two faces (the bases) are polygons in parallel planes and the other (lateral) faces are triangles or trapezoids with one side common with one base and the opposite vertex or side common with the other base.

A pyramid is a polyhedron of which one face (the base) is a polygon and the other (lateral) faces are triangles meeting in a common point called the vertex of the pyramid and intersecting one another in its lateral edges. Pyramids are triangular, quadrangular, and so on, according as their bases are triangles, quadrilaterals, and so on. A regular pyramid (or right pyramid ) has for its base a regular polygon whose center coincides with the foot of the perpendicular dropped from the vertex to the base. A frustum of a pyramid is the portion of a pyramid included between its base and a section parallel to the base. If the section is not parallel to the base, a truncated pyramid results. A regular polyhedron has all faces formed of congruent regular polygons and all polyhedral angles equal. The only regular polyhedrons possible are the five types discussed in the mensuration table (Table 4). A tetrahedron is a polyhedron of four faces. It may be described also as a triangular pyramid, and any one of its four triangular faces may be considered as the base. The four perpendiculars erected at circumcenters of the four faces meet in a point equidistant from all vertices, which is the center of the circumscribed sphere. The four medians, joining each vertex with the center of gravity of the opposite face, meet in a point, which is the center of gravity of the tetrahedron. This point is three-fourths of the distance from each vertex along a median. The four altitudes meet in a point, called the orthocenter of the tetrahedron. The six planes bisecting the six dihedral angles meet in a point equidistant from all faces, this being the center of the inscribed sphere. Solids Having Curved Surfaces A cylinder is a solid bounded by two parallel plane surfaces (the bases) and a cylindrical lateral surface. A cylindrical surface is a surface generated by the movement of a straight line (the generatrix ) which constantly is parallel to a fixed straight line and touches a fixed curve (the directrix ) not in the plane of the fixed straight line. The generatrix in any position is an element of the cylindrical surface. A circular cylinder is one having circular bases. A right cylinder is one whose elements are perpendicular to its bases. A truncated cylinder is the part of a cylinder included between a base and a section made by a plane oblique to the base. A right section of a cylinder is a section made by a plane which cuts all the elements perpendicularly. A cone is a solid bounded by a conic lateral surface and a plane (the base) that cuts all the elements of the conic surface. A conic surface is a surface generated by the movement of a straight line (the generatrix) that constantly touches a fixed plane curve (the directrix) and passes through a fixed point (the vertex) not in the plane of the fixed curve. The generatrix in any position is an element of the conic surface. A circular cone is one having a circular base. A right cone is a circular cone whose center of the base coincides with the foot of the perpendicular dropped from the vertex to the base. A frustum of a cone is the portion of a

MATHEMATICS

195

Table 4 Mensuration Formulas Approximate Decimal Equivalents (for reference) π = 3.1416 π/2 = 1.5708 π/4 = 0.7854 π/180 = 0.01745 π/360 = 0.00873

1 = 0.318 π 1/2π = 0.159 1/4π = 0.080 180/π = 57.296 360/π = 114.592

√ 2 = 1.414 √ 3√= 1.732 1/√2 = 0.707 1/ 3 = 0.577

1a. Plane Rectilinear Figures Notation. Lines, a, b, c, . . . ; angles, α, β, γ , . . . ; altitude (perpendicular height), h; side, l; diagonals, d, d1 , . . . ; perimeter, p; radius of inscribed circle, r; radius of circumscribed circle, R; area, A. 1. Right triangle

(One angle 90◦ ) p = a + b + c; c2 = a2 + b2 ; A = 12 ab = 12 a2 tan β = 14 c2 sin 2β = c 14 c2 sin 2α. For additional formulas, see general triangle below and also trigonometry.

2. General triangle (and equilateral triangle)

For general triangle: p = a + b + c. Let s = 12 (a + b + c). s(s − a)(s − b)(s − c) a abc r = ; R= = ; s 2 sin α 4rs 2 ab b sin γ sin α abc ah = sin γ = = rs = . A = 2 2 2 sin β 4R Length of median to side c = 12 2(a2 + b2 ) − c2 . ab[(a + b)2 − c2 ] Length of bisector of angle γ = . a+b For equilateral triangle (a = b = c = l and α = β = γ = 60◦ ): (Equal sides and equal angles) l l p = 3l, r = √ ; R = √ = 2r; 2 3 3 √ √ 2 l 3 l 3 2h h = ; l= √ ; A= . 2 4 3 For additional formulas, see trigonometry.

3. Rectangle (and square)

For rectangle: p = 2(a + b); d =

√ a2 + b2 ; A = ab.

For square (a = b = l): √ d d2 p = 4l; d = l 2; l = √ ; A = l2 = . 2 2 4. General parallelogram (and rhombus)

For general parallelogram (rhomboid): (Opposite sides parallel) p = 2(a + b); d1 = a2 + b2 − 2ab cos γ ; d2 = a2 + b2 + 2ab cos γ ; d12 + d22 = 2(a2 + b2 ); A = ah = ab sin γ .

(Continues)

196 Table 4

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) 1a. Plane Rectilinear Figures (Continued) For rhombus (a = b = l): (Opposite sides parallel and all sides equal) p = 4l; d1 = 2l sin 12 γ ; d2 = 2l cos 12 γ ; d12 + d22 = 4l2 ; d1 d2 = 2l2 sin γ ; A = lh = l2 sin γ = 12 (d1 d2 ).

5. General trapezoid (and isosceles trapezoid)

Let midline bisecting nonparallel sides = m. Then m = 12 (a + b). For general trapezoid: (Only one pair of opposite sides parallel) p = a + b + c + d; A = 12 (a + b)h = mh. For isosceles trapezoid (d = c): (Nonparallel sides equal) A =

1 2 (a

+ b)h = mh = 12 (a + b)c sin y

= (a − c cos γ )c sin γ = (b + c cos γ )c sin γ . 6. General quadrilateral (trapezium)

(No sides parallel) p = a + b + c + d. A =

1 2 d1 d2

sin α = sum of areas of the two triangles formed by either

diagonal and the four sides. 7. Quadrilateral inscribed in circle

(Sum of opposite angles = 180◦ ) ac + bd = d1 d2 . Let s = 12 (a + b + c + d) = 12 p and α = angle between sides a and b. A = (s − a)(s − b)(s − c)(s − d) = 12 (ab + cd) sin α.

8. Regular polygon (and general polygon)

For regular polygon: (Equal sides and equal angles) Let n = number of sides. 2π radians; n n−2 Vertex angle = β = π radians. n p = ns; s = 2r tan α = 2R sin α; Central angle = 2α =

r = 12 s cot α; R = 12 s csc α; A = 12 nsr = nr 2 tan α = 12 nR2 sin 2α = equal triangles such as OAB.

1 4

ns2 cot α = sum of areas of the n

For general polygon: A = sum of areas of constituent triangles into which it can be divided.

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197

Table 4 (Continued ) 1b. Plane Curvilinear Figures Notation. Lines, a, b, . . .; radius, r; diameter, d; perimeter, p; circumference, c; central angle n radians, θ; arc, s; chord of arc s, l; chord of half arc s/2, l ; rise, h; area, A. 9. Circle (and circular arc)

For circle: d = 2r; c = 2π r = π d; A = π r 2 =

π d2 c2 = . 4 4π

For circular arc: Let arc PAQ = s; and chord PA = l . Then, s = rθ = 12 dθ; s = 13 (8l − l). (The latter equation is Huygen’s approximate formula. For θ small; error is very small; for θ = 120◦ , error is about 0.25%; for θ = 180◦ , error is less than 1.25%.) √ l = 2r sin 12 θ; l = 2 2hr − h2 (approximate formula) s l 4h2 + l2 = ; r= (approximate formula) θ 2 sin(θ/2) 8h h = r ∓ r 2 − 14 l2 (− ifθ≤180◦ + ifθ≥180◦ ) = r(1 − cos 12 θ) r =

= r versin 12 θ = 2r sin2 14 θ = 12 l tan 14 θ = r + y − √ Side ordinate y = h − r + r 2 − x2 . 10. Circular sector (and semicircle)

√ r 2 − x2 .

For circular sector: A = 12 θr 2 = 12 sr. For semicircle: A = 12 π r 2 .

11. Circular segment

A =

1 2 2 r (θ

− sin θ)

∓ l(r − h)](− ifh ≤ r; + ifh ≥ r). = 2lh h A= or (8l + 6l). (Approximate formulas. For h small compared with r, 3 15 error is very small; for h = 14 r, first formula errs about 3.5% and second less than 1.0%.) 1 2 [sr

12. Annulus

(Region between two concentric circles) A = π (r12 − r22 ) = π (r1 + r2 )(r1 − r2 ); A of sector ABCD =

1 2 2 θ(r1

− r22 ) = 12 θ(r1 + r2 )(r1 − r2 )

=

1 2 t(s1

+ s2 ).

13. Ellipse

p = π (a + b) 1 +

p = π (a + b)

R2 4

+

R4 64

+

R4 a−b + · · · where R = . 256 a+b

64 − 3R4 (approximate formula). 64 − 16R2

A = π ab; A of quadrant AOB = 14 π ab; A of sector AOP =

x x ab ab cos−1 ; A of sector POB = sin−1 ; 2 a 2 a (Continues)

198 Table 4


14. Parabola

1b. Plane Curvilinear Figures (Continued) x A of section BPP B = xy + ab sin−1 ; a x A of segment PAP P = −xy + ab cos−1 . a For additional formulas, see analytic geometry. √ √ 4h + l2 + 16h2 l2 loge . Arc BOC = s = 12 l2 + 16h2 + 8h l h Let R = . Then: l 8R2 32R4 s=l 1+ − + · · · (approximate formula). 3 5 h−d h 2 dl2 ; d = 2 (l − l12 ); l1 = l ; h= 2 l h l − l12 2hl ; 3

l3 − l13 2 A of section ABCD = d 2 . 3 l − l12 A of segment BOC =

15. Hyperbola

For additional formulas, see analytic geometry. x x y = ab cosh−1 ; A of figure OPAP O = ab loge + a b a x y x + = xy − ab cosh−1 . A of segment PAP = xy − ab loge a b a For additional formulas, see analytic geometry.

16. Cycloid

Arc OP = s = 4r(1 − cos 12 φ); arc OMN = 8r; A under curve OMN = 3π r 2 .

17. Epicycloid

18. Hypocycloid

For additional formulas, see analytic geometry. 4r Rφ ; Arc MP = s = (R + r) 1 − cos R 2r Rφ r Rφ Area MOP = A = (R + r)(R + 2r) − sin . 2R r r For additional formulas, see analytic geometry. 4r Rφ ; Arc MP = s = (R − r) 1 − cos R 2r Rφ r Rφ Area MOP = A = (R − r)(R − 2r) − sin . 2R r r For additional formulas, see analytic geometry.

19. Catenary

If d is small compared with l: # $ 2 2d 2 Arc MPN = s = l 1 + (approximately). 3 l For additional formulas, see analytic geometry:

20. Helix (a skew curve)

Let length of helix = s; radius of coil (= radius of cylinder in figure) = r; distance advanced in one revolution = pitch = h; and number of revolutions = n. Then: s = n (2π r)2 + h2 .

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199


21. Spiral of Archimedes

1b. Plane Curvilinear Figures (Continued) r Let a = . Then: φ Arc OP = s = 12 a[φ 1 + φ 2 + loge (φ + 1 + φ 2 )]. For additional formulas, see analytic geometry.

22. Irregular figure

Divide the figure into an even number n of strips by means of n + 1 ordinates yi spaced equal distances w. The area can then be determined approximately by any of the following approximate formulas, which are presented in the order of usual increasing approach to accuracy. In any of the first three cases, the greater the number of strips used, the more nearly accurate will be the result:

Trapezoidal rule

A = w[ 12 (y0 + yn ) + y1 + y2 + · · · + yn−1 ];

Durand’s rule

A = w[0.4(y0 + yn ) + 1.1(y1 + yn−1 ) + y2 + y3 + · · · + yn−2 ];

Simpson’s rule (n must be even)

A =

1 3

w[(y0 + yn ) + 4(y1 + y3 + · · · + yn−1 )

+ 2(y2 + y4 + · · · + yn−2 )]; Weddle’s rule (for 6 strips only)

3w [5(y1 + y5 ) + 6y3 + y0 + y2 + y4 + y6 ]. 10 Areas of irregular regions can often be determined more quickly by such methods as plotting on squared paper and counting the squares; graphical coordinate representation (see analytic geometry); or use of a planimeter.

A=

1c. Solids Having Plane Surfaces Notation. Lines, a, b, c,. . . ; altitude (perpendicular height), h; slant height, s; perimeter of base, ph or pB ; perimeter of a right section, pr ; area of base, Ab or AB ; area of a right section, Ar ; total area of lateral surfaces, Al ; total area of all surfaces, At ; volume, V. 23. Wedge (and right triangular prism)

For wedge: (Narrow-side rectangular); V = 16 ab(2l1 + l2 ). For right triangular prism (or wedge having parallel triangular bases perpendicular to sides): l2 = l1 = l: V = 12 abl.

24. Rectangular prism (or rectangular parallelepiped) (and cube)

For rectangular prism or rectangular parallelepiped: Al = 2c(a + b); At = 2(de + ac + bc); V = Ar c = abc. For cube (letting b = c = a):

25. General prism

√ At = 6a2 ; V = a3 ; diagonal = a 3. Al = hpb = spr = s(a + b + · · · + n); V = hAb = sAr .

26. General truncated prism (and truncated triangular prism)

For general truncated prism: V = Ar · (length of line BC joining centers of gravity of bases). For truncated triangular prism: V = 13 Ar (a + b + c).

27. Prismatoid

Let area of midsection = Am . V = 16 h(AB + Ab + 4Am ).

(Continues)

200 Table 4

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS (Continued ) 1c. Solids Having Plane Surfaces (Continued)

28. Right regular pyramid (and prustum of right regular pyramid)

For right regular pyramid: Al = 12 spB ; V = 13 hAB . For prustum of right regular pyramid: Al = 12 s(pB + pb ); V = 13 h(AB + Ab +

29. General pyramid (and prustum of pyramid)

√

AB Ab ).

For general pyramid: V = 13 hAB . For prustum of general pyramid: √ V = 13 h(AB + Ab + AB Ab ).

30. Regular polyhedrons

Let edge = a, and radius of inscribed sphere = r. Then: r=

3V and: At

Number of Faces

Total Area At

Form of Faces

4 Equilateral triangle 6 Square 8 Equilateral triangle 12 Regular pentagon 20 Equilateral triangle (Factors shown only to four decimal places.)

1.7321a2 6.0000a2 3.4641a2 20.6457a2 8.6603a2

Volume V 0.1179a3 1.0000a3 0.4714a3 7.6631a3 2.1817a3

1d. Solids Having Curved Surfaces Notation. Lines, a, b, c, . . . ; altitude (perpendicular height), h, h1 , . . . ; slant height, s; radius, r; perimeter of base, pb ; perimeter of a right section, pr ; angle in radians, φ; arc, s; chord of segment, l; rise, h; area of base, Ab or AB ; area of a right section, Ar ; total area of convex surface, Al ; total area of all surfaces, At ; volume, V. 31. Right circular cylinder (and truncated right circular cylinder)

For right circular cylinder: Al = 2π rh; At = 2π r(r + h); V = π r 2 h. For truncated right circular cylinder: Al = π r(h1 + h2 ); At = π r[h1 + h2 + r + V =

32. Ungula (wedge) of right circular cylinder

1 2 2 π r (h1

+ h2 ).

2rh Al = [a + (b − r)φ]; b h V = [a(3r 2 − a2 ) + 3r 2 (b − r)φ] 3b # $ hr 3 sin3 φ = sin φ − − φ cos φ . b 3 For semicircular base (letting a = b = r): 2r 2 h . 3 Al = pb h = pr s; Al = 2rh; V =

33. General cylinder

V = Ab h = Ar s.

r 2 + 12 (h1 − h2 )2 ];

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201

Table 4 (Continued ) 1d. Solids Having Curved Surfaces (Continued) 34. Right circular cone (and frustum of right circular cone)

For right circular cone: Al = π rB s = π rB rB2 + h2 ; At = π rB (rB + s); V =

1 2 3 π rB h.

For frustum of right circular cone: s = h21 + (rB − rb )2 ; Al = π s(rB + rb ); V = 35. General cone (and frustum of general cone)

+ rb2 + rB rb ).

1 2 3 π h1 (rB

For general cone: V = 13 AB h. For frustum of general cone: √ V = 13 h1 (AB + Ab + AB Ab ).

36. Sphere

Let diameter = d. At = 4π r 2 = π d2 ; V =

37. Spherical sector (and hemisphere)

4π r 3 π d3 = . 3 6

For spherical sector: πr 2π r 2 h (4h + l); V = . 2 3

At =

For hemisphere (letting h = 12 l = r): 2π r 3 . 3 For spherical zone bounded by two planes: At = 3π r 2 ; V =

38. Spherical zone (and spherical segment)

Al = 2π rh; At = 14 π (8rh + a2 + b2 ). For spherical zone bounded by one plane (b = 0): Al = 2π rh = 14 π (4h2 + a2 ); At =

1 4 π (8rh

+ a2 ) = 12 π (2h2 + a2 ).

For spherical segment with two bases: V=

1 2 24 π h(3a

+ 3b2 + 4h2 ).

For spherical segment with one base (b = 0): V= 39. Spherical polygon (and spherical triangle)

1 2 24 π h(3a

+ 4h2 ) = π h2 (r − 13 h).

For spherical polygon: Let sum of angles in radians = θ and number of sides = n. A = [θ − (n − 2)π ]r 2 [The quantity θ − (n − 2)π is called ‘‘spherical excess.’’] For spherical triangle (n = 3): A = (θ − π )r 2 For additional formulas, see trigonometry. (Continues)

202 Table 4


40. Torus

1d. Solids Having Curved Surfaces (Continued) Al = 4π 2 Rr; V = 2π 2 Rr 2 .

41. Ellipsoid (and spheroids)

For ellipsoid: V = 43 πabc. For prolate spheroid: √ a2 − b2 Let c = b and = e. a −1 4 sin e At = 2πb2 + 2πab ; V = πab2 . e 3

42. Paraboloid of revolution

For oblate spheroid: √ a2 − b2 Let c = a and = e. a 2 πb 1 + e 4 At = 2πa2 + ln ; V = πa2 b. e 1 −#e 3 3/2 3 $ 2 2πl l l 2 . Al of segment DOC = − + h 3h2 16 4 For paraboloidal segment with two bases: V of ABCD =

πd 2 (l + l12 ). 8

For paraboloidal segment with one base (l1 = 0 and d = h): V of DOC =

πhl2 . 8 πh 2 (l + 4l12 ). 24

43. Hyperboloid of revolution

V of segment AOB =

44. Surface and solid of revolution

Let perpendicular distance from axis to center of gravity (G) of curve (or surface) = r. Curve (or surface) must not cross axis. Then, Area of surface generated by curve revolving about axis: Al = 2πrs. Volume of solid generated by surface revolving about axis: V = 2πrA.

45. Irregular solid

One of the following methods can often be employed to determine the volume of an irregular solid with a reasonable approach to accuracy: (a) Divide the solid into prisms, cylinders, etc., and sum their individual volumes. (b) Divide one surface into triangles after replacing curved lines by straight ones and curved surfaces by plane ones. Then multiply the area of each triangle by the mean depth of the section beneath it (which generally approximates the average of the depths at its corners). Sum the volumes thus obtained. (c) If two surfaces are parallel, replace any curved lateral surfaces by plane surfaces best suited to the contour and then employ the prismatoidal formula.

MATHEMATICS

cone included between its base and a section parallel to the base. A sphere is a solid bounded by a surface all points of which are equidistant from a point within called the center. Every plane section of a sphere is a circle. This circle is a great circle if its plane passes through the center of the sphere; otherwise, it is a small circle. Poles of such a circle are the extremities of the diameter of the sphere that is perpendicular to the plane of the circle. Through two points on a spherical surface, not extremities of a diameter, one great circle can be passed. The shortest line that can be drawn on the surface of a sphere between two such points is an arc of a great circle less than a semicircumference joining those points. If two spherical surfaces intersect, their line of intersection is a circle whose plane is perpendicular to the line of centers and whose center lies on this line. A spherical sector is the portion of a sphere generated by the revolution of a circular sector about a diameter of the circle of which the sector is a part. A hemisphere is half of a sphere. A spherical segment is the portion of a sphere contained between two parallel plane sections (the bases), one of which may be tangent to the sphere (in which case there is only one base). The term “segment” also is applied in an analogous manner to various solids of revolution, the planes in such cases being perpendicular to an axis. A zone is the portion of a spherical surface included between two parallel planes. A spherical polygon is a figure on a spherical surface bounded by three or more arcs of great circles. The sum of the angles of a spherical triangle (polygon of three sides) is greater than two right angles and less than six right angles. Other solids appearing in the mensuration table (Table 4), if not sufficiently defined by their figures, may be found discussed in the section on analytic geometry. 5.2 Mensuration

Perimeters of similar figures are proportional to their respective linear dimensions, areas to the squares of their linear dimensions, and volumes of similar solids to the cubes of their linear dimensions (see Table 4). 5.3 Constructions Lines

1.

To draw a line parallel to a given line.

Case 1: At a given distance from the given line (Fig. 21). With the given distance as radius and with any centers m and n on the given line AB, describe arcs xy and zw, respectively. Draw CD touching these arcs. CD is the required parallel line. Case 2: Through a given point (Fig. 22). Let C be the given point and D be any point on the given line AB. Draw CD. With equal radii draw arcs bf and ce

203

Fig. 21

Fig. 22

Fig. 23

with D and C, respectively, as centers. With radius equal to chord bf and with c as center draw an arc cutting arc ce at E. CE is the required parallel line. 2. To bisect a given line (Fig. 23). Let AB be the given line. With any radius greater than 0.5 AB describe two arcs with A and B as centers. The line CD, through points of intersection of the arcs, is the perpendicular bisector of the given line. 3. To divide a given line into a given number of equal parts (Fig. 24). Let AB be the given line and let the number of equal parts be five. Draw line AC at any convenient angle with AB, and step off with dividers five equal lengths from A to b. Connect b with B, and draw parallels to Bb through the other points in AC. The intersections of these parallels with AB determine the required equal parts on the given line. 4. To divide a given line into segments proportional to a number of given unequal parts. Follow the same procedure as under item 3 except make the

Fig. 24

204


Fig. 25

lengths on AC equal to (or proportional to) the lengths of the given unequal parts. 5. To erect a perpendicular to a given line at a given point in the line. Case 1: Point C is at or near the middle of the line AB (Fig. 25). With C as center, describe arcs of equal radii intersecting AB at a and b. With a and b as centers, and any radius greater than Ca, describe arcs intersecting at D. CD is the required perpendicular. Case 2: Point C is at or near the extremity of the line AB (Fig. 26). With any point O as center and radius OC, describe an arc intersecting AB at a. Extend aO to intersect the arc at D. CD is the required perpendicular. 6. To erect a perpendicular to a given line through a given point outside the line. Case 1: Point C is opposite, or nearly opposite, the middle of the line AB (Fig. 27). With C as center, describe an arc intersecting AB at a and b. With a and b as centers, describe arcs of equal radii intersecting at D. CD is the required perpendicular. Case 2: Point C is opposite, or nearly opposite, the extremity of the line AB (Fig. 28). Through C, draw any line intersecting AB at a. Divide line Ca into two equal parts, ab and bC (method given previously). With b as center and radius bC, describe an arc intersecting AB at D. CD is the required perpendicular.

Fig. 28

Angles

7.

To bisect a given angle.

Case 1: Vertex B is accessible (Fig. 29). Let ABC be the given angle. With B as center and a large radius, describe an arc intersecting AB and BC at a and c, respectively. With a and c as centers, describe arcs of equal radii intersecting at D. DB is the required bisector. Case 2: The vertex is inaccessible (Fig. 30). Let the given angle be that between lines AB and BC. Draw lines ab and bc parallel to the given lines, and at equal distances from them, intersecting at b. Construct Db bisecting angle abc (method given previously). Db is the required bisector. 8. To construct an angle equal to a given angle if one new side and the new vertex are given (Fig. 31). Let ABC be the given angle, DE the new side, and E the new vertex. With center B and a convenient radius, describe arc ac. With the same radius and center E, draw arc df. With radius equal to chord ac and with center d draw an arc cutting the arc df at F . Draw EF. Then DEF is the required angle. 9. To construct angles of 60◦ and 30◦ (Fig. 32). About any point A on a line AB, describe with a

Fig. 29

Fig. 26 Fig. 30

Fig. 27

Fig. 31

MATHEMATICS

205

Fig. 32

Fig. 33

convenient radius the arc bc. From b, using an equal radius, describe an arc cutting the former one at C. Draw AC, and drop a perpendicular CD from C to line AB. Then CAD is a 60◦ angle and ACD is a 30◦ angle. 10. To construct an angle of 45◦ (Fig. 33). Set off any distance AB ; draw BC perpendicular and equal to AB and join CA. Angles CAB and ACB are each 45◦ . 11. To draw a line making a given angle with a given line (Fig. 34). Let AB be the given line. With A as the center and with as large a radius as convenient, describe arc bc. Determine from Table 12 in Chapter 1, the length of chord to radius 1, corresponding to the given angle. Multiply this chord by the length of Ab, and with the product as a new radius and b as a center, describe an arc cutting bc at C. Draw AC. This line makes the required angle with AB.

be the given points. With the given radius and these points as centers, describe arcs cutting each other at C. From C, with the same radius, describe arc AB, which is the required arc. 13. To bisect a given arc of a circle. Draw the perpendicular bisector of the chord of the arc. The point in which this bisector meets the arc is the required midpoint. 14. To locate the center of a given circle or circular arc (Fig. 36). Select three points A, B, C on the circle (or arc) located well apart. Draw chords AB and BC and erect their perpendicular bisectors. The point O, where the bisectors intersect, is the required center. 15. To draw a circle through three given points not in the same straight line. Case 1: Radius small and center accessible (Fig. 36). Let A, B, C be the given points. Draw lines AB and BC and erect their perpendicular bisectors. From point O, where the bisectors intersect, describe a circle of radius OA that is the required circle. Case 2: Radius very long or center inaccessible (Fig. 37). Let A, O, A be the given points (O not necessarily midpoint of AOA ). Draw arcs Aa and A a with centers at A and A, respectively; extend AO to determine a and A O to determine a ; point off from a on aA equal parts ab, bc, and so on; lay off a b , b c , and so on, equal to ab; join A with any point as b and A with the corresponding point b ; the intersection P of these joining lines is a point on the required circle. 16. To lay out a circular arc without locating the center of the circle, given the chord and the rise (Fig. 37). Let AA be the chord and QO the rise. (In this case, O is the midpoint of AOA .) The arc can be

Circles

12. To describe through two given points an arc of a circle having a given radius (Fig. 35). Let A and B

Fig. 36 Fig. 34

Fig. 35

Fig. 37

206


Fig. 41 Fig. 38

constructed through the points A, O, A , as under item 15, Case 2. 17. To construct, upon a given chord, a circle in which a given angle can be inscribed (Fig. 38). Let AB be the given chord and α the given angle. Construct angle ABC equal to angle α. Bisect line AB by the perpendicular at D. Draw a perpendicular to BC from point B. With O, the point of intersection of the perpendiculars, as center and OB as radius, describe a circle. The angle AEB, with vertex E located anywhere on the arc AEB, equals α, and therefore the circle just drawn is the one required. 18. To draw a tangent to a given circle through a given point. Case 1: Point A is on the circle (Fig. 39). Draw radius OA. Through A, perpendicular to OA, draw BAC, the required tangent. Case 2: Point A is outside the circle (Fig. 40). Two tangents can be drawn. Join O and A. Bisect OA at D, and with D as center and DO as radius, describe an arc intersecting the given circle at B and C. BA and CA are the required tangents. 19. To draw a common tangent to two given circles. Let the circles have centers O and O and corresponding radii r and r (r > r ).

Case 1: Common internal tangents (when circles do not intersect) (Fig. 41). Construct a circle having the same center O as the larger circle and a radius equal to the sum of the radii of the given circles (r + r ). Construct a tangent O P from center O of the smaller circle to this circle. Construct O N perpendicular to this tangent. Draw OP. The line MN joining the extremities of the radii OM and O N is a common tangent. The figure shows two such common internal tangents. Case 2: Common external tangents (Fig. 42). Construct a circle having the same center O as the larger circle and radius equal to the difference of the radii (r − r ). Construct a tangent to this circle from the center of the smaller circle. The line joining the extremities M, N of the radii of the given circles perpendicular to this tangent is a required common tangent. There are two such tangents. 20. To draw a circle with a given radius that will be tangent to two given circles. (Fig. 43). Let r be the given radius and A and B the given circles. About center of circle A with radius equal to r plus radius of A, and about center of B with radius equal to r plus radius of B, draw two arcs cutting each other in C, which is the center of the required circle. 21. To describe a circular arc touching two given circles, one of them at a given point. (Fig. 44). Let AB, FG be the given circles and F the given point. Draw the radius EF, and produce it both ways. Set off FH

Fig. 42 Fig. 39

Fig. 40

Fig. 43

MATHEMATICS

207

Fig. 47 Fig. 44

equal to the radius AC of the other circle; join CH, and bisect it by the perpendicular LT, cutting EF at T . About center T , with radius TF, describe arc FA as required. 22. To draw a circular arc that will be tangent to two given lines inclined to one another, one tangential point being given (Fig. 45). Let AB and CD be the given lines and E the given point. Draw the line GH, bisecting the angle formed by AB and CD. From E draw EF at right angles to AB ; then F , its intersection with GH, is the center of the required circular arc. 23. To connect two given parallel lines by a reversed curve composed of two circular arcs of equal radius, the curve being tangent to the lines at given points (Fig. 46). Let AD and BE be the given lines and A and B the given points. Join A and B, and bisect the connecting line at C. Bisect CA and CB by perpendiculars. At A and B erect perpendiculars to the given lines, and the intersections a and b are the centers of the arcs composing the required curve. 24. To describe a circular arc that will be tangent to a given line at a given point and pass through another given point outside the line (Fig. 47). Let AB be the given line, A the given point on the line, and C the given point outside it. Draw from

A a line perpendicular to the given line. Connect A and C by a straight line, and bisect this line by the perpendicular ca. The point a where these two perpendiculars intersect is the center of the required circular arc. 25. To draw a circular arc joining two given relatively inclined lines, tangent to the lines, and passing through a given point on the line bisecting their included angle (Fig. 48). Let AB and DE be the given lines and F the given point on the line FC that bisects their included angle. Through F draw DA at right angles to FC ; bisect the angles A and D by lines intersecting at C, and about C as a center, with radius CF, draw the arc HFG required. 26. To draw a series of circles between two given relatively inclined lines touching the lines and touching each other (Fig. 49). Let AB and CD be the given lines. Bisect their included angle by the line NO. From a point P in this line draw the perpendicular PB to the line AB, and on P describe the circle BD, touching the given lines and cutting the center line at E. From E draw EF perpendicular to the center line, cutting AB at F ; and about F as a center describe an arc EG, cutting AB at G. Draw GH parallel to BP, giving H , the center of the next circle, to be described with the radius HE ; and so on for the next circle IN.

Fig. 45 Fig. 48

Fig. 46

Fig. 49

208


Fig. 50 Fig. 54

31. To circumscribe a circle about a given regular polygon.

Fig. 51

27. To circumscribe a circle about a given triangle (Fig. 50). Construct perpendicular bisectors of two sides. Their point of intersection O is the center (circumcenter) of the required circle. 28. To inscribe a circle in a given triangle (Fig. 51). Draw bisectors of two angles intersecting in O (incenter). From O draw OD perpendicular to BC. Then the circle with center O and radius OD is the required circle. 29. To circumscribe a circle about a given square (Fig. 52). Let ACBD be the given square. Draw diagonals AB and CD of the square intersecting at E. On center E, with radius AE, describe the required circle. The same procedure can be used for circumscribing a circle about a given rectangle. 30. To inscribe a circle in a given square (Fig. 53). Let ACBD be the given square. Draw diagonals AB and CD of the square intersecting at E. Drop a perpendicular EF from E to one side. On center E, with radius EF, describe the required circle.

Case 1: The polygon has an even number of sides (Fig. 54). Draw a diagonal AB joining two opposite vertices. Bisect the diagonal by a perpendicular line DE, which is another diagonal or a line bisecting two opposite sides, depending on whether the number of sides is or is not divisible by 4. With the midpoint C as the center and radius CA, describe the required circle. Case 2: The polygon has an odd number of sides (Fig. 55). Bisect two of the sides at D and E by the perpendicular lines DB and EA which pass through the respective opposite vertices and intersect at a point C. With C as the center and radius CA, describe the required circle. 32. To inscribe a circle in a given regular polygon (Figs. 54 and 55). Locate the center C as in item 31. With C as center and radius CD, describe the required circle. Polygons

33. To construct a triangle on a given base, the lengths of the sides being given (Fig. 56). Let AB be the given base and a, b the given lengths of sides. With

Fig. 52

Fig. 55

Fig. 53

Fig. 56

MATHEMATICS

209

Fig. 57

A and B as centers and b and a as respective radii, describe arcs intersecting at C. Draw AC and BC to complete the required triangle. 34. To construct a rectangle of given base and given height (Fig. 57). Let AB be the base and c the height. Erect the perpendicular AC equal to c. With C and B as centers and AB and c as respective radii, describe arcs intersecting at D. Draw BD and CD to complete the required rectangle. 35. To construct a square with a given diagonal (Fig. 58). Let AC be the given diagonal. Draw a circle on AC as diameter and erect the diameter BD perpendicular to AC. Then ABCD is the required square. 36. To inscribe a square in a given circle (Fig. 58). Draw perpendicular diameters AC and BD. Their extremities are the vertices of an inscribed square. 37. To circumscribe a square about a given circle (Fig. 59). Draw perpendicular diameters AC and BD. With A, B, C, D as centers and the radius of the circle as radius, describe the four semicircular arcs shown. Their outer intersections are the vertices of the required square. 38. To inscribe a regular pentagon in a given circle (Fig. 60). Draw perpendicular diameters AC and BD intersecting at O. Bisect AO at E and, with E as center

Fig. 60

and EB as radius, draw an arc cutting AC at F . With B as center and BF as radius, draw an arc cutting the circle at G and H ; also with the same radius, step around the circle to I and K. Join the points thus found to form the pentagon. 39. To inscribe a regular hexagon in a given circle (Fig. 61). Step around the circle with compasses set to the radius and join consecutive divisions thus marked off. 40. To circumscribe a regular hexagon about a given circle (Fig. 62). Draw a diameter ADB and, with center A and radius AD, describe an arc cutting the circle at C. Draw AC and bisect it with the radius DE. Through E, draw FG parallel to AC, cutting diameter AB extended at F . With center D and radius DF, describe the circumscribing circle FH ; within this circle inscribe a regular hexagon as under item 39. This hexagon circumscribes the given circle, as required. 41. To construct a regular hexagon having a side of given length (Fig. 61). Draw a circle with radius equal to the given length of side and inscribe a regular hexagon (see item 39). 42. To construct a regular octagon having a side of given length (Fig. 63). Let AB be the given side. Produce AB in both directions, and draw perpendiculars

Fig. 58 Fig. 61

Fig. 59

Fig. 62

210


Fig. 63

AE and BF. Bisect the external angles at A and B by the lines AH and BC making them equal to AB. Draw CD and HG parallel to AE and equal to AB ; from the centers G, D, with the radius AB, draw arcs cutting the perpendiculars at E, F, and draw EF to complete the octagon. 43. To inscribe a regular octagon in a given circle (Fig. 64). Draw perpendicular diameters AC and BD. Bisect arcs AB, BC, . . . and join Ae, eB, . . . to form the octagon. 44. To inscribe a regular octagon in a given square (Fig. 65). Draw diagonals of the given square intersecting at O. With A, B, C, D as centers and AO as radius, describe arcs cutting the sides of the square at gn, fk, hm, and ol. Join the points thus found to form the octagon. 45. To circumscribe a regular octagon about a given circle (Fig. 66). Describe a square about the given circle. Draw perpendiculars ij, kl, and so on, to the diagonals of the squares, touching the circle. Then ij, jk, kl, and so on, form the octagon. 46. To describe a regular polygon of any given number of sides when one side is given (Fig. 67). Let

Fig. 66

Fig. 67

AB be the given side and let the number of sides be five. Produce the line AB, and with A as center and AB as radius, describe a semicircle. Divide this into as many equal parts as there are to be sides of the polygon—in this case, five. Draw lines from A through the division points a, b, and c (omitting the last). With B and c as centers and AB as radius, cut Aa at C and Ab at D. Draw cD, DC, and CB to complete the polygon. 47. To inscribe a regular polygon of a given number of sides in a given circle. Determine the central angle subtended by any side by dividing 360◦ by the number of sides. Lay off this angle successively round the center of the circle by means of a protractor. The radii thus drawn intersect the circle at vertices of the required polygon. Ellipse An ellipse is a curve for which the sum of the distances of any point on it from two fixed points (the foci ) is constant.

Fig. 64

Fig. 65

48. To describe an ellipse for which the axes are given (Fig. 68). Let AB be the major and RS the minor axis (AB > RS). With O as center and OB and OR as radii, describe circles. From O draw any radial line intersecting the circles at M and N. Through M draw

Fig. 68

MATHEMATICS

211

a line parallel to OR and through N a line parallel to OB. These lines intersect at H , a point on the ellipse. Repeat the construction to obtain other points. 49. To locate the foci of an ellipse, given the axes (Fig. 68). With R as center and radius equal to AO, describe arcs intersecting AB at F and F , the required foci. 50. To describe an ellipse mechanically, given an axis and the foci (Fig. 68). A cord of length equal to the major axis is pinned or fixed at its ends to the foci F and F . With a pencil inside the loop, keeping the cord taut so as to guide the pencil point, trace the outline of the ellipse (Q represents the pencil point and length F QF the cord). If the minor axis RS is given rather than the major axis AB, the length AB (for the cord) is readily determined as F R + RF . 51. To draw a tangent to a given ellipse through a given point. Case 1: Point P is on the curve (Fig. 68). With O as center and OB as radius, describe a circle. Through P draw a line parallel to OR intersecting the circle at K. Through K draw a tangent to the circle intersecting the major axis at T . PT is the required tangent. Case 2: Point P is not on the curve (Fig. 69). With P as center and radius P F , describe an arc. With F as center and radius AB, describe an arc intersecting the first arc at M and N. Draw FM and FN intersecting the ellipse at E and G. PE and PG are the required tangents. 52. To describe an ellipse approximately by means of circular arcs of three radii (Fig. 70). On the major

Fig. 69

Fig. 70

axis AB draw the rectangle BG of altitude equal to half the minor axis, OC ; to the diagonal AC draw the perpendicular GHD; set off OK equal to OC, and describe a semicircle on AK ; produce OC to L; set off OM equal to CL, and from D describe an arc with radius DM ; from A, with radius OL, draw an arc cutting AB at N; from H , with radius HN, draw an arc cutting arc ab at a. Thus the five centers H, a, D, b, H are found, from which the arcs AR, RP, PQ, QS, SB are described. The part of the ellipse below axis AB can be constructed in like manner. Parabola A parabola is a curve for which the distance of any point on it from a fixed line (the directrix ) is equal to its distance from a fixed point (the focus). For a general discussion of its properties, see the section on analytic geometry.

53. To describe a parabole for which the vertex, the axis, and a point of the curve are given (Fig. 71). Let A be the given vertex, AB the given axis, and M the given point. Construct the rectangle ABMC. Divide MC and CA into the same number of equal parts (say four), numbering the divisions consecutively in the manner shown. Connect A1, A2, and A3. Through 1 , 2 , 3 , draw parallels to the axis AB. The intersections I, II, and III of these lines are points on the required curve. A similar construction below the axis will give the other symmetric branch of the curve. 54. To locate the focus and directrix of a parabola, given the vertex, the axis, and a point of the curve (Fig. 71). Let A be the given vertex, AB the given axis, and M the given point. Drop the perpendicular MB from M to AB. Bisect it at E and draw AE. Draw ED perpendicular to AE at E and intersecting the axis at D. With A as center and BD as radius, describe arcs cutting the axis at F and J . Then F is the focus and the line GH, perpendicular to the axis through J , is the directrix. 55. To describe a parabola mechanically, given the focus and directrix (Fig. 72). Let F be the given focus and EN the given directrix. Place a straight edge to the directrix EN, and apply to it a square, LEG. Fasten to the end G one end of a cord equal in length to the edge EG, and attach the other end to the focus F ; slide the square along the straight edge, holding the cord taut

Fig. 71

212


Fig. 74 Fig. 72

against the edge of the square by a pencil D, by which the parabolic curve is described. 56. To draw a tangent to a given parabola through a given point. Case 1: The point is on the curve (Fig. 71). Let II be the given point. Drop a perpendicular from II to the axis, cutting it at b. Make Aa equal to Ab. Then a line through a and II is the required tangent. The line II c perpendicular to the tangent at II is the normal at that point; bc is the subnormal. All subnormals of a given parabola are equal to the distance from the directrix to the focus and hence equal to each other. Thus the subnormal at I is de equal to bc, where d is the foot of the perpendicular dropped from I. The tangent at I can be drawn as a perpendicular to Ie through I. Case 2: The point is off the curve (on the convex side) (Fig. 73). Let P be the given point and F the focus of the parabola. With P as center and PF as radius, draw arcs intersecting the directrix at B and D. Through B and D draw lines parallel to the axis intersecting the parabola at E and H . PE and PH are the required tangents. Hyperbola A hyperbola is a curve for which the difference of the distances of any point on it from two fixed points (the foci) is constant. It has two distinct branches.

57. To describe a hyperbola for which the foci and the difference of the focal radii are given (Fig. 74). Let

F and F’ be the given foci and AOB the given difference of the focal radii. Lay out AOB (the transverse axis) so that AF = F B and AO = OB. A and B are points on the required curve. With centers F and F and any radius greater than FB or F A, describe arcs aa. With the same centers and radius equal to the difference between the first radius and the transverse axis AOB, describe arcs bb, intersecting arcs aa at P, Q, R, and S, points on the required curve. Repeat the construction for additional points. Make BC = BC = OF = OF , and construct the rectangle DEFG; CC is the conjugate axis. The diagonals DF and EG, produced, are called asymptotes. The hyperbola is tangent to its asymptotes at infinity. 58. To locate the foci of a hyperbola, given the axes (Fig. 74). With O as center and radius equal to BC, describe arcs intersecting AB extended at F and F , the required foci. 59. To describe a hyperbola mechanically, having given the foci and the difference of the focal radii (Fig. 75). Let F and F be the given foci and AB the given difference of focal radii. Using a ruler longer than the distance F F , fasten one of its extremities at the focus F . At the other extremity H attach a cord of such a length that the length of the ruler exceeds the length of the cord by the given distance AB. Attach the other extremity of the cord at the focus F . Press a pencil P against the ruler, and keep the cord constantly taut while the ruler is turned around F as a center. The point of the pencil will describe one branch of the curve, and the other can be obtained in like manner. 60. To draw a tangent to a given hyperbola through a given point. Case 1: Point P is on the curve (Fig. 76). Draw lines connecting P with the foci. Bisect the angle F P F . The bisecting line TP is the required tangent.

Fig. 73

Fig. 75

MATHEMATICS

213

Fig. 76

Fig. 79

Involute of a Circle An involute of a circle is a curve generated by the free end of a taut string as it is unwound from a circle.

Fig. 77

Case 2: Point P is off the curve on the convex side (Fig. 77). With P as center and radius P F , describe an arc. With F as center and radius AB, describe an arc intersecting the first arc at M and N. Produce lines FM and FN to intersect the curve at E and G. PE and PG are the required tangents. Cycloid A cycloid is a curve generated by a point on a circle rolling on a straight line.

62. To describe an involute of a given circle (Fig. 79). Let AB be the given circle. Through B draw Bb perpendicular to AB. Make Bb equal in length to half the circumference of the circle. Divide Bb and the semicircumference into the same number of equal parts, say six. From each point of division 1, 2, 3, . . . of the circumference, draw lines to the center C of the circle. Then draw 1a1 perpendicular to C1, 2a2 perpendicular to C2, and so on. Make 1a1 equal to bb1 ; 2a2 equal to bb2 ; 3a3 equal to bb3 ; and so on. Join the points A, a1 , a2 , a3 , etc., by a curve; this curve is the required involute. 6 6.1

61. To describe a cycloid for which the generating circle is given (Fig. 78). Let A be the generating point. Divide the circumference of the generating circle into an even number of equal arcs, as A1, 1–2, . . . , and set off the rectified arcs on the base. Through the points 1, 2, 3, . . . on the circle, draw horizontal lines, and on them set off distances 1a = A1, 2b = A2, 3c = A3, . . . . The points A, a, b, c, . . . are points of the cycloid. An epicycloid is a curve generated by a point on one circle rolling on the outside of another circle. A hypocycloid is a curve generated by the point if the generating circle rolls on the inside of the second circle.

Fig. 78

TRIGONOMETRY Circular Functions of Plane Angles

Definitions and Values Trigonometric Functions. The angle α in Fig. 80 is measured in degrees or radians, as defined in Section 5.1. The ratio of any two of the quantities x, y, or r determines the extent of the opening between the lines OP and OX. Since these ratios are functions of the angle, they may be used to measure or construct it.

Fig. 80

214


The definitions and terms used to designate the functions are as follows: Sine α = Cosine α = Tangent α = Cotangent α = Secant α = Cosecant α = Versine α = Coversine α = Haversine α =

y = sin α r x = cos α r y = tan α x x = cot α y r = sec α x r = csc α y r −x = vers α = 1 − cos α r r −y = covers α = 1 − sin α r r −x = hav α = 12 vers α 2r

Values of Trigonometric Functions. An angle α (Fig. 80), if measured in a counterclockwise direction, is said to be positive; if measured clockwise, negative. Following the convention that x is positive if measured along OX to the right of the OY axis and negative if measured to the left, and similarly, y is positive if measured along OY above the OX axis and negative if measured below, the signs of the trigonometric functions are different for angles in the quadrants I, II, III, and IV (Table 5). Values of trigonometric functions are periodic, the period of the sin, cos, sec, csc being 2π radians, and that of the tan and cot, π radians (Tables 6–8). For example, in Fig. 81 (n an integer)

90◦ ± α 180◦ ± α 270◦ ± α 360◦ ± α

−α sin − sin α cos + cos α tan − tan α cot − cot α sec + sec α csc − csc α

sin

cos

tan

cot

sec

csc

+ + − −

+ − − +

+ − + −

+ − + −

+ − − +

+ + − −

± sin α + cos α ± tan α ± cot α + sec α ± csc α

sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β ∓ sin α sin β tan(α ± β) = 1 ∓ tan α tan β/ tan α ± tan β cot(α ± β) = cot β ± cot α/ cot β cot α ∓ 1 If x is small, say 3◦ or 4◦ , then the following are close approximations, in which the quantity x is to be expressed in radians (1◦ = 0.01745 rad): sin α ≈ α

cos α ≈ 1

tan α ≈ α

sin(α ± x) ≈ sin α ± x cos α cos(α ± x) ≈ cos α ∓ x sin α Functions of Half-Angles

=

I II III IV

− cos α ± sin α ∓ cot α ∓ tan α ± csc α − sec α

Functions of the Sum and Difference of Two Angles

tan(α + πn) = tan α

Quadrant

∓ sin α − cos α ± tan α ± cot α − sec α ∓ csc α

Functional Relations Identities

sin(α + 2πn) = sin α

Signs of Trigonometric Functions

+ cos α ∓ sin α ∓ cot α ∓ tan α ∓ csc α + sec α

While the direct functions (e.g., sine) are single valued, the indirect are many valued; thus sin 30◦ = 0.5, but sin−1 0.5 = 30◦ , 150◦ , . . . .

sin 12 α =

Inverse, or Antifunctions. The symbol sin−1 x means the angle whose sine is x and is read inverse sine of x, antisine of x, or arc sine x. Similarly for cos−1 x, tan−1 x, cot−1 x, sec−1 x, csc−1 x, vers−1 x, the last meaning an angle α such that 1 − cos α = x. Table 5

Table 6 Functions of Angles in Any Quadrant in Terms of Angles in First Quadrant

cos 12 α = =

1 2 (1

√ 1 2

2

1 + sin α −

1 2 (1

√ 1

− cos α) 1 2

√

1 − sin α

+ cos α)

1 + sin α +

1 2

√

1 − sin α

1 − cos α 1 + cos α/1 − cos α = sin α sin α = 1 + cos α 1 + cos α cot 12 α = 1 + cos α/1 − cos α = sin α sin α = 1 − cos α

tan 12 α =

MATHEMATICS

215

Table 7 Functions of Certain Angles 0◦ sin

30◦ 1

0

cos

1

tan cot sec csc

0 ∞ 1 ∞

45◦ √ 1 2 2 √ 1 2 2

2 √ 3 √ 1/3 √ 3 3 √ 2/3 3 2 1 2

60◦ √ 1 2 3

90◦

1 2

√ 3 √ 1/3 3 2√ 2/3 3

1 √1 √2 2

180◦

270◦

360◦

1

0

−1

0

0

−1

0

1

∞ 0 ∞ 1

0 ∞ −1 ∞

∞ 0 ∞ −1

0 ∞ 1 ∞

Table 8 Functions of an Angle in Terms of Each of the Othersa sin α = a

cos α = a

tan α = a

cot α = a

a

√ 1 − a2

a √ 1 + a2 1 √ 1 + a2

√

sin √

cos

√

tan

√ cot √

sec csc a The

1 − a2

a √ 1 − a2 a

a 1 − a2 1 − a2 a

a √ 1 − a2 1 a

1 1− 1 a

a2

1 √ 1 − a2

√

a 1 a √ 1 + a2 √ 1 + a2 a

sec α = a √ a2 − 1 a

1 1 + a2 a

√

1 a

√ a2 − 1 a

1 a

1 + a2 1 a a

√

csc α = a

√ a2 − 1

1 √ a2 − 1

1 √ 2 a −1

√ a2 − 1

1 + a2 a

a

a √ 2 a −1

1 + a2

a √ 2 a −1

a

sign of the radical is to be determined by the quadrant.

Functions of Multiples of Angles

sin 3α = 3 sin α − 4 sin3 α

sin 2α = 2 sin α cos α

cos 3α = 4 cos3 α − 3 cos α

tan 2α =

sin 4α = 8 cos3 α sin α − 4 cos α sin α

2 tan α 1 − tan2 α

cos 4α = 8 cos4 α − 8 cos2 α + 1

cos 2α = cos α − sin α = 2 cos α − 1 = 1 − 2 sin α 2

2

cot2 α − 1 cot 2α = 2 cot α

2

2

sin nα = 2 sin(n − 1)α cos α − sin(n − 2)α = n sin α cosn−1 α − n C3 sin3 α cosn−3 α + n C5 sin5 α cosn−5 α − · · · cos nα = 2 cos(n − 1)α cos α − cos(n − 2)α = cosn α − n C2 sin2 α cosn−2 α + n C4 sin4 α cosn−4 α − · · · (For n Cr , see p. 164.) Products and Powers of Functions

Fig. 81

sin α sin β =

1 2

cos(α − β) −

1 2

cos(α + β)

cos α cos β =

1 2

cos(α − β) +

1 2

cos(α + β)

216


sin α cos β =

1 2

sin(α − β) +

1 2

√

sin(α + β)

tan α cot α = sin α csc α = cos α sec α = 1 sin2 α = 12 (1 − cos 2α)

cos2 α = 12 (1 + cos 2α)

sin3 α = 14 (3 sin α − sin βα)

cos3 α

= 14 (3 cos α + cos 3α) sin4 α = 18 (3 sin 4 cos 2α + cos 4α)

cos4 α

= 18 (3 + 4 cos 2α + cos 4α) sin5 α =

1 16 (10 sin α

sin α =

1 32 (10

cos5 α =

1 16 (10 cos α

cos6 α =

1 32 (10

6

− 5 sin 3α + sin 5α)

− 15 cos 2α + 6 cos 4α − cos 6α) + 5 cos 3α + cos 5α)

+ 15 cos 2α + 6 cos 4α + cos 6α)

Sums and Differences of Functions

sin α + sin β = 2 sin 12 (α + β) cos 12 (α − β) sin α − sin β = 2 cos 12 (α + β) sin 12 (α − β) cos α + cos β = 2 cos 12 (α + β) cos 12 (α − β) cos α − cos β = −2 sin 12 (α + β) sin 12 (α − β) tan α + tan β =

sin(α + β) cos α cos β

=

sin(α + β) sin α sin β

tan α − tan β =

sin(α − β) cos α cos β

=−

cot α + cot β

cot α − cot β

sin(α − β) sin α sin β

sin2 α − sin2 β = sin(α + β) sin(α − β) cos α − cos β = − sin(α + β) sin(α − β) 2

2

cos2 α − sin2 β = cos(α + β) cos(α − β) Antitrigonometric or Inverse Functional Relations. In the following formulas the periodic constant is omitted:

π − cos−1 x 2 x = cos−1 1 − x 2 = tan−1 √ 1 − x2

sin−1 x = − sin−1 (−x) =

1 − x2 1 = csc−1 x x 1 = sec−1 √ 1 − x2 π cos−1 x = π − cos−1 (−x) = − sin−1 x 2 √ 1 −1 2 = 2 cos (2x − 1) = sin−1 1 − x 2 √ 1 − x2 x −1 = cot−1 √ = tan x 1 − x2 = cot−1

1 1 = csc−1 √ x 1 − x2 π tan−1 x = − tan−1 (−x) = − cot−1 x 2 x 1 = sin−1 √ = cos−1 √ 2 1+x 1 + x2 √ 1 + x2 −1 1 −1 −1 2 = sec = cot 1 + x csc x x 1 1 sec−1 x = cos−1 cot−1 x = tan−1 x x 1 csc−1 x = sin−1 x −1 −1 −1 sin x ± sin y = sin (x 1 − y 2 ± y 1 − x 2 ) cos−1 x ± cos−1 y = cos−1 [xy ∓ (1 − x 2 )(1 − y 2 )] sin−1 x ± cos−1 y = sin−1 [xy ± (1 − x 2 )(1 − y 2 )] = cos−1 (y 1 − x 2 ∓ x 1 − y 2 ) = sec−1

tan−1 x ± tan−1 y = tan−1

x±y 1 ∓ xy

tan−1 x ± cot−1 y = tan−1

xy ± 1 y∓x = cot−1 y∓x xy ± 1

6.2 Solution of Triangles Relations between Angles and Sides of Plane Triangles. Let a, b, c = sides of triangle; α, β, γ = angles opposite a, b, c, respectively; A = area of triangle; s = 12 (a + b + c); r = radius of inscribed circle (Fig. 82).

b c a = = (law of sines) sin α sin β sin γ a 2 = b2 + c2 − 2bc cos α (law of cosines) tan 12 (α − β) a−b = a+b tan 12 (α + β) α + β + γ = 180◦

(law of tangents)

MATHEMATICS

217

Given a, b, α: sin β =

Fig. 82

a = b cos γ + c cos β b = c cos α + a cos γ c = a cos β + b cos α √ A = s(s − a)(s − b)(s − c) 2 2 2 A sin β = A sin γ = A sin α = bc ca ab α (s − b)(s − c) sin = 2 bc β (s − c)(s − a) sin = 2 ca γ (s − a)(s − b) sin = 2 ab α s(s − a) cos = 2 bc γ β s(s − b) s(s − c) cos = cos = 2 ca 2 ab (s − b)(s − c) α tan = 2 s(s − a) β (s − c)(s − a) tan = 2 s(s − b) γ (s − a)(s − b) tan = 2 s(s − c) Solution of Plane Oblique Triangles. Given a, b, c (if logarithms are to be used, use 1): (s − a)(s − b)(s − c) , 1. r = s √ A = s(s − a)(s − b)(s − c) = rs, r β r α = tan = , tan 2 s−a 2 s−b r γ = . tan 2 s−c 2 b + c2 − a 2 2. cos α = , 2bc 2 2 2 a +c −b cos β = , 2ac 2 2 2 a +b −c cos γ = or 2ab ◦ γ = 180 − (α + β).

b sin α a

(if a > b, β < π/2 and has only one value; if b > a, β has two values, β1 , β2 = 180◦ − β1 ); γ = 180◦ − (α + β); c = a sin γ / sin α; A = 12 ab sin γ . Given a, α, β: a sin β sin α a sin γ c= sin α

b=

◦

γ = 180 − (α + β) A = 12 ab sin γ

Given a, b, γ (if logarithms are to be used, use 1): a−b cot 12 γ , a+b a sin γ ◦ 1 1 , c= 2 (α + β) = 90 − 2 γ , sin α 1 A = 2 ab sin γ . a sin γ , 2. c = a 2 + b2 − 2ab cos γ , sin α = c ◦ β = 180 − (α + γ ). a sin γ ◦ , β = 180 − (α + γ ), 3. tan α = b − a cos γ a sin γ . c= sin α

1. tan 12 (α − β) =

Mollweide’s Check Formulas

sin 12 (α − β) a−b = c cos 12 γ

cos 12 (α − β) a+b = c sin 12 γ

Solution of Plane Right Triangles. Let γ = 90◦ and c be the hypotenuse. Given any two sides or one side and an acute angle α:

(c + b)(c − b) = b tan α = c sin α a = c cos α b = c2 − a 2 = (c + a)(c − a) = tan α b a c = a 2 + b2 = = sin α cos α a b a ◦ β = 90 − α α = sin−1 = cos−1 = tan−1 c c b a=

A=

c2 − b2 =

ab a2 b2 tan α c2 sin 2α = = = 2 2 tan α 2 4

218


6.3 Spherical Trigonometry Spherical Trigonometry. Let O be the center of the sphere and a, b, c the sides of a triangle on the surface with opposite angles α, β, γ , respectively, the sides being measured by the angle subtended at the center of the sphere. Let s = 12 (a + b + c), σ = 12 (α + β + γ ), E = α + β + γ − 180◦ , the spherical excess. The following formulas are valid usually only for triangles of which the sides and angles are all between 0◦ and 180◦ . To each such triangle there is a polar triangle whose sides are 180◦ − α, 180◦ − β, 180◦ − γ and whose angles are 180◦ − a, 180◦ − b, 180◦ − c. General Formulas sin b sin c sin a = = sin α sin β sin γ

α+β 2 α−β tan 2 c α+β cos cos 2 2 c α+β cos sin 2 2 c α−β sin cos 2 2 c α−β sin sin 2 2 tan

(law of sines)

cos a = cos b cos c + sin b sin c cos α (law of cosines) cos α = − cos β cos γ + sin β sin γ cos a (law of cosines) cos a sin b = sin a cos b cos γ + sin c cos α cot a sin b = sin γ cot α + cos γ cos b

cos[(a − b)/2] γ cot cos[(a + b)/2] 2 sin[(a − b)/2] γ = cot sin[(a + b)/2] 2 a+b γ = cos sin 2 2 a−b γ = cos cos 2 2 a+b γ = sin sin 2 2 a−b γ = sin cos 2 2 =

The Right Spherical Triangle. be the hypotenuse.

Let γ = 90◦ and c

cos c = cos a cos b = cot α cot β

cos a =

cos α sin β

cos β sin α sin a sin α = sin c tan a tan α = sin b

cos α =

tan b tan c

cos b =

cos α sin β = sin γ cos a − sin α cos β cos c cot α sin β = sin c cot a − cos c cos β a = sin 2 sin

α = 2

cos

a = 2

cos

α = 2

tan

a = 2

α = 2 E tan 4 s = tan tan 2 E = cot 2 a+b = tan 2 a−b = tan 2 tan

− cos σ cos(σ − α) sin β sin γ sin(s − b) sin(s − c) sin b sin c cos(σ − β) cos(σ − γ ) sin β sin γ sin s sin(s − a) sin b sin c − cos σ cos(σ − α) cos(σ − β) cos(σ − γ )

6.4 Hyperbolic Trigonometry Hyperbolic Angles. These are defined in a manner similar to circular angles but with reference to an equilateral hyperbola. The comparative relations are shown in Figs. 83 and 84. A circular angle is a central angle measured in radians by the ratio s/r or the ratio 2A/r 2 , where A is the area of the sector included by the angle α and the arc s (Fig. 83). For the hyperbola the radius ρ is not constant and only the value of the differential hyperbolic angle dθ is defined by the ratio ds/ρ. Thus 2A ds = 2 θ= ρ a

sin(s − b) sin(s − c) sin s sin(s − a)

(s − a) (s − b) (s − c) tan tan 2 2 2 cot(a/2) cot(b/2) + cos γ sin γ cos[(α − β)/2] c tan cos[(α + β)/2] 2 sin[(α − β)/2] c tan sin[(α + β)/2] 2

Fig. 83

MATHEMATICS

219

Fig. 84

where A represents the shaded area in Fig. 84. If both s and ρ are measured in the same units, the angle is expressed in hyperbolic radians. Hyperbolic Functions. These are defined by ratios similar to those defining functions of circular angles and also named similarly. Their names and abbreviations are y Hyperbolic sine θ = = sinh θ a x Hyperbolic cosine θ = = cosh θ a y Hyperbolic tangent θ = = tanh θ x x Hyperbolic cotangent θ = = coth θ y a Hyperbolic secant θ = = sech θ x a Hyperbolic cosecant θ = = csch θ y Values and Exponential Equivalents. The values of hyperbolic functions may be computed from their exponential equivalents. The graphs are shown in Fig. 85. Values for increments of 0.01 rad are given in Table 18. −θ

sinh θ =

e −e 2

tanh θ =

e −e eθ + e−θ

θ

θ

cosh θ =

e +e 2 θ

cosh2 θ − sinh2 θ = 1

cosh θ + sinh θ = eθ

sech θ =

1 cosh θ

cosh θ − sinh θ = e−θ

sinh(−θ ) = − sinh θ

cosh(−θ ) = cosh θ

tanh(−θ ) = − tanh θ

coth(−θ ) = − coth θ

sinh(θ1 ± θ2 ) = sinh θ1 cosh θ2 ± cosh θ1 sinh θ2 cosh(θ1 ± θ2 ) = cosh θ1 cosh θ2 ± sinh θ1 sinh θ2 tanh(θ1 ± θ2 ) =

tanh θ1 ± tanh θ2 1 ± tanh θ1 tanh θ2

coth(θ1 ± θ2 ) =

1 ± coth θ1 coth θ2 coth θ1 ± coth θ2

sinh 2θ = 2 sinh θ cosh θ =

2 tanh θ 1 − tanh2 θ

cosh 2θ = sinh2 θ + cosh2 θ = 1 + 2 sinh2 θ = 2 cosh2 θ − 1 = tanh 2θ =

1 + tanh2 θ 1 − tanh2 θ

2 tanh θ 1 + tanh2 θ

1 + coth2 θ 2 coth θ sinh 12 (0) = 21 (cosh θ − 1) cosh 12 (0) = 21 (cosh θ + 1) sinh θ cosh θ − 1 θ = tanh = 2 cosh θ + 1 cosh θ + 1 cosh θ − 1 = sinh θ coth 2θ =

If θ is extremely small, sinh θ ≈ θ, cosh θ ≈ 1, and tanh θ ≈ θ . For large values of θ, sinh θ ≈ cosh θ and tanh θ ≈ coth θ ≈ 1.

sech2 θ = 1 − tanh2 θ

csch2 θ = coth2 θ − 1

−θ

−θ

Fundamental Identities 1 csch θ = sinh θ 1 coth θ = tanh θ

Fig. 85

220


sinh θ1 ± sinh θ2 = 2 sinh 12 (θ1 ± θ2 ) cosh 12 (θ1 ∓ θ2 ) cosh θ1 + cosh θ2 = 2 cosh

1 2 (θ1

+ θ2 ) cosh

1 2 (θ1

− θ2 )

cosh θ1 − cosh θ2 = 2 sinh 12 (θ1 + θ2 ) sinh 12 (θ1 − θ2 ) sinh(θ1 ± θ ) tanh θ1 ± tanh θ2 = cosh θ1 cosh θ2 (cosh θ ± sinh θ )n = cosh nθ ± sinh nθ Antihyperbolic or Inverse Functions. The inverse hyperbolic sine of u is written sinh−1 u. Values of the inverse functions may be computed from their logarithmic equivalents:

√ sinh−1 u = loge (u + u2 + 1) √ cosh−1 u = loge (u + u2 + 1) 1+u tanh−1 u = 12 loge 1−u u+1 1 −1 coth u = 2 loge u−1

Relations of Hyperbolic to Circular Functions. By comparison of the exponential equivalents of hyperbolic and circular functions, the following iden√ tities are established (i = −1):

= −i sinh iα = cosh iα = −i tanh iα = i coth iα = sech iα = i csch iα

sinh β cosh β tanh β coth β sech β csch β

= = = = = =

−i sin iβ cos iβ −i tan iβ i cot iβ sec iβ i csc iβ

Relations between Inverse Functions sin−1 A cos−1 A tan−1 A cot−1 A sec−1 A csc−1 A

= −i sinh−1 iA = −i cosh−1 A = −i tanh−1 iA = i coth−1 iA = −i sech−1 A = i csch−1 iA

sinh−1 B cosh−1 B tanh−1 B coth−1 B sech−1 B csch−1 B

= = = = = =

−i sinh−1 iB −i cos−1 B −i tan−1 iB i cot−1 iB i sec−1 B i csc−1 iB

Functions of a Complex Angle. In complex, notaiθ tion c√= a + ib = |c|(cos √ θ + i sin θ ) = |c|e , where |c| = a 2 + b2 , i = −1, and θ = tan−1 b/a. Freθ quently |c|ei is written c∠θ . iθ Loge |c|e = log |c| + i(θ + 2kπ) and is infinitely many valued. By its principal part will be understood loge |c| + iθ . Some convenient identities are

loge 1 = 0

√ n

cos θ ± i sin θ = cos

θ + 2πk θ + 2πk ± i sin n n

The use of complex angles occurs frequently in electric circuit problems where it is often necessary to express the functions of them as a complex number: sin(α ± iβ) = sin α cosh β ± i cos α sinh β = cosh2 β − cos2 αe±iθ where θ = tan−1 cot α tanh β; cos(α ± iβ) = cos α cosh β ∓ i sin α sinh β = cosh2 β − sin2 αe±iθ where θ = tan−1 tan α tanh β;

6.5 Functions of Imaginary and Complex Angles

sin α cos α tan α cot α sec α csc α

π 3π loge (−i) = i 2 2 (cos θ ± i sin θ )n = cos nθ ± i sin nθ loge i = i

loge (−1) = iπ

sinh(α ± iβ) = sinh α cos β ± i cosh α sin β = sinh2 α + sin2 βe±iθ = cosh2 α + cos2 βe±iθ where θ = tan−1 coth α tan β; cosh(α ± iβ) = cosh α cos β ± i sinh α sin β = sinh2 α + cos2 βe±iθ = cosh2 α + sin2 βe±iθ where θ = tan−1 tanh α tan β; and tan(α ± iβ) =

sin 2α ± i sinh 2β cos 2α + cosh 2β

tanh(α ± iβ) =

sinh 2α ± i sin 2β cosh 2α + cos 2β

The hyperbolic sine and cosine have the period 2πi; the hyperbolic tangent has the period πi: sinh(α + 2kπi) = sinh α tanh(α + kπi) = tanh α

cosh(α + 2kπi) = cosh α coth(α + kπi) = coth α

MATHEMATICS

221

Inverse Functions of Complex Numbers sin−1 (A ± iB) " ! 1 ( B 2 + (1 + A)2 − B 2 + (1 − A)2 ) = sin−1 2 " ! 1 ( B 2 + (1 + A)2 + B 2 + (1 − A)2 ) ± i cosh−1 2 cos−1 (A ± iB) " ! 1 ( B 2 + (1 + A)2 − B 2 + (1 − A)2 ) = cos−1 2 " ! 1 ( B 2 + (1 + A)2 + B 2 + (1 − A)2 ) ∓ i cosh−1 2

Fig. 86

tan−1 (A ± iB) " ! A A 1 π − tan−1 + tan−1 = 2 ±B − 1 ±B + 1 ±i

A2 + (1 ± B)2 1 loge 2 4 A + (1 ∓ B)2

sinh−1 (A ± iB) " ! 1 ( A2 + (1 + B)2 − A2 + (1 − B)2 ) = cosh−1 2 " ! 1 ( A2 + (1 + B)2 − A2 + (1 − B)2 ) ± i sin−1 2

Fig. 87

cosh−1 (A ± iB) " ! 1 ( B 2 + (1 + A)2 + B 2 + (1 − A)2 ) = cosh−1 2 " ! 1 ( B 2 + (1 + A)2 − B 2 + (1 − A)2 ) ± i cos−1 2

Fig. 88

tanh−1 (A ± iB) =

1 2A ±2B 1 tanh−1 + i tan−1 2 1 + A2 + B 2 2 1 − 2A − B 2

7 PLANE ANALYTIC GEOMETRY 7.1 Point and Line Coordinates. The position of a point P1 in a plane is determined if its distance and direction from each of two lines or axes OX and OY which are perpendicular to each other are known. The distances x and y (Fig. 86) perpendicular to the axes are called the Cartesian or rectangular coordinates of the point. The directions to the right of OY and above OX are called positive and opposite directions negative. The point O of intersection of OY and OX is called the origin. The position of a point P is also given by its radial distance r from the origin and the angle θ between the radius r and the horizontal axis OX (Fig. 87). These coordinates r, θ are called polar coordinates.

The distance s between two points P1 (x1 , y1 ) and P2 (x2 , y2 ) (Fig. 88) on a straight line is s=

(x2 − x1 )2 + (y2 − y1 )2

(15)

In polar coordinates the distance s between P1 (r1 , θ1 ) and P2 (r2 , θ2 ) is s=

r12 + r22 − 2r1 r2 cos(θ2 − θ1 )

(16)

The slope m of the line P1 P2 is defined as the tangent of the angle φ which the line makes with OX : m = tan φ =

y2 − y1 x2 − x1

(17)

222


To divide the segment P1 P2 in the ratio c1 /c2 , internally or externally, x=

c2 x1 ± c1 x2 c2 ± c1

y=

c2 y1 ± c1 y2 c2 ± c1

y=k

The midpoint of P1 P2 is x = 12 (x1 + x2 )

√ Ax + By + C = 0 by ± A2 + B 2 . The sign before the radical is taken opposite to that of C if C=0 and the same as that of B if C = 0. Equations of lines parallel to the x and y axes, respectively, are

y = 12 (y1 + y2 )

(18)

where A, B, and C are constants. Other forms of the equation are y = mx + b

(19)

where m is the slope and b is the y intercept; y − y1 = m(x − x1 )

(21)

where (x1 , y1 ) and (x2 , y2 ) are two points on the line; y x + =1 a b

(22)

where a and b are the x and y intercepts, respectively; x cos α + y sin α − p = 0

p1 =

Ax1 + By1 + C √ ± A2 + B 2

(25)

where the sign before the radical is opposite to that of C if C=0 and the same as B if C = 0. Parallel Lines. The two lines y = m1 x + b1 , y = m2 x + b2 are parallel if m1 = m2 . For the form A1 x + B1 y + C1 = 0, A2 x + B2 y + C2 = 0, the lines are parallel if B1 A1 = (26) A2 B2

(20)

where m is the slope and (x1 , y1 ) is a point on the line; x1 − x2 x − x1 = y − y1 y1 − y2

(24)

The perpendicular distance of a point P1 (x1 , y1 ) (Fig. 89) from the line Ax + By + C = 0 is

Equation of a Straight Line. In Cartesian coordinates the equation of a straight line is of the first degree and is expressed as

Ax + By + C = 0

x=k

(23)

where α is the angle between OX and the perpendicular from the origin to the line and p is the length of the perpendicular (Fig. 89). This is called the perpendicular form and is obtained by dividing the general form

The equation of a line through the point (x1 , y1 ) and parallel to the line Ax + By + C = 0 is A(x − x1 ) + B(y − y1 ) = 0

(27)

Perpendicular Lines. The two lines y = m1 x + b1 and y = m2 x + b2 are perpendicular if

m1 = −

1 m2

(28)

For the form A1 x + B1 y + C1 = 0, A2 x + B2 y + C2 = 0, the lines are perpendicular if A1 A2 + B1 B2 = 0

(29)

The equation of a line through the point (x1 , y1 ) perpendicular to the line Ax + By + C = 0 is B(x − x1 ) − A(y − y1 ) = 0

(30)

Intersecting Lines. Let A1 x + B1 y + C1 = 0 and A2 x + B2 y + C2 = 0 be the equations of two intersecting lines and λ an arbitrary real number. Then

(A1 x + B1 y + C1 ) + λ(A2 x + B2 y + C2 ) = 0 (31) Fig. 89

represents the system of lines through the point of intersection.

MATHEMATICS

223

The three lines A1 x + B1 y + C1 = 0, A2 x + B2 y + C2 = 0, A3 x + B3 y + C3 = 0 meet in a point if A1 A2 A3

C1 C2 = 0 C3

B1 B2 B3

(32) Fig. 91

The angle θ between two lines with equations A1 x + B1 y + C1 = 0 and A2 x + B2 y + C2 = 0 can be found from

the relations between the old and the new coordinates under transformation are

A1 B2 − A2 B1 sin θ = (A21 + B12 )(A22 + B22 ) A1 A2 + B1 B2 cos θ = (A21 + B12 )(A22 + B22 ) tan θ =

x = x + h (33)

A1 B2 − A2 B1 A1 A2 − B1 B2

cos θ = tan θ =

Rotation of Axes about the Origin. Let θ (Fig. 91) be the angle through which the axes are rotated. Then

y = x sin θ + y cos θ (36) If the axes are both translated and rotated, x = x cos θ − y = sin θ + h

(37)

y = x sin θ + y cos θ + k

m2 − m1

Coordinate Transformation. The relations between the rectangular coordinates x, y and the polar coordinates r, θ are

(1 + m21 )(1 + m22 ) 1 + m1 m 2

(35)

x = x cos θ − y sin θ

The signs of tan θ and cos θ determine whether the acute or obtuse angle is meant. If the equations are in the form y = m1 x + b1 , y = m2 x + b2 , then sin θ =

y = y + k

(34)

(1 + m21 )(1 + m22 )

x = r cos θ y θ = tan−1 x

m 2 − m1 1 + m1 m2

7.2 Transformation of Coordinates Change of Origin O to O . Let (x, y) denote the coordinates of a point P with respect to the old axes and (x , y ) the coordinates with respect to the new axes (Fig. 90). Then, if the coordinates of the new origin O with respect to the old axes are x = h, y = k,

7.3

y = r sin θ

r=

x2 + y2 (38)

Conic Sections

Conic Section. This is a curve traced by a point P moving in a plane so that the distance PF of the point from a fixed point (focus) is in constant ratio to the distance PM of the point from a fixed line (directrix ) in the plane of the curve. The ratio e = P F /P M is called the eccentricity. If e < 1, the curve is an ellipse; e = 1, a parabola; e > 1, a hyperbola; and e = 0, a circle, which is a special case of an ellipse. Circle.

The equation is (x − x0 )2 + (y − y0 )2 = r 2

(39)

where (x0 , y0 ) is the center and r the radius. If the center is at the origin, Fig. 90

x2 + y2 = r 2

(40)

224


Another form is x 2 + y 2 + 2gx + 2fy + c = 0

(41)

with center (−g, −f ) and radius g 2 + f 2 − c. The equation of the tangent to (41) at a point P1 (x1 , y1 ) is xx1 + yy1 + g(x + x1 ) + f (y + y1 ) + c = 0 Ellipse

(y − y0 ) (x − x0 ) + =1 a2 b2

y2 x2 =1 + 2 2 a a (1 − e2 )

(48)

ax 2 + by 2 + 2gx + 2fy + c = 0

(49)

2

(43)

where (x0 , y0 ) is the center, a the semimajor axis, and b the semiminor axis. In Fig. 92, (x0 , y0 ) = (0, 0). Coordinates of foci are F1 = (−ae, 0), F2 = (ae, 0); e2 = (F1 P )2 /(MP )2 = 1 − b2 /a 2 < 1; and the directrices are the lines x = −a/e, x = a/e. The chord LL through F is called the latus rectum and has the length 2b2 /a = 2a(1 − e2 ). If P1 is any point on the ellipse, F1 P1 = a + ex1 , F2 P1 = a + ex1 , and F1 P1 + F2 P1 = 2a (a constant). The area of the ellipse with semiaxes a and b is A = πab

(44)

The equation of the tangent to the ellipse (Fig. 92) at the point (x1 , y1 ) is yy1 xx1 + 2 =1 a2 b

(45)

the equation of the tangent with slope m is y = mx ±

Other Forms of the Equation of the Ellipse

(42)

(Fig. 92). The equation is 2

Conjugate Diameters. A line through the center of an ellipse is a diameter ; if the slopes m and m of the two diameters y = mx and y = m x are such that mm = −b2 /a 2 each diameter bisects all chords parallel to the other and the diameters are called conjugate.

a 2 m2 + b 2

(46)

The equation of the normal to the ellipse at the point (x1 , y1 ) is a 2 y1 (x − x1 ) − b2 x1 (y − y1 ) = 0

Fig. 92

If a, b, and g 2 /a + f 2 /b − c have the same sign, (49) is an ellipse whose axes are parallel to the coordinate axes. The parametric form is x = a cos φ Hyperbola

y = b sin φ

(50)

(Fig. 93). The equation is (y − y0 )2 (x − x0 )2 − =1 2 a b2

(51)

where (x0 , y0 ) is the center, AA = 2a is the transverse axis, and BB = 2b is the conjugate axis. In Fig. 93, (x0 , y0 ) = (0, 0); e2 =

(F1 P )2 b2 =1+ 2 >1 2 (P M) a

the coordinates of the foci are F1 = (−ae, 0), F2 = (ae, 0); and the directrices are the lines x = −a/e, x = a/e. The chord LL through F is called the latus rectum and has the length 2b2 /a = 2a(e2 − 1). If P1 is any point on the curve, F1 P1 = ex1 − a, F2 P1 = ex1 + a, and |F2 P1 − F1 P1 | = 2a (a constant).

(47)

Fig. 93

MATHEMATICS

225

The equation of the tangent to the hyperbola (Fig. 93) at the point (x1 , y1 ) is yy1 xx1 − 2 =1 a2 b

(52)

The equation of the tangent whose slope is m is y = mx ±

a 2 m2 − b 2

(53)

The equation of the normal to the hyperbola at the point (x1 , y1 ) is a 2 y1 (x − x1 ) + b2 x1 (y − y1 ) = 0

(54)

Conjugate Hyperbolas and Diameters. The two hyperbolas y2 y2 x2 x2 − 2 =1 − 2 =1 2 2 a b b a

are conjugate. The transverse axis of each is the conjugate axis of the other. If the slopes of the two lines y = mx and y = m1 x through the center O are connected by the relation mm1 = b2 /a 2 , each of these lines bisects all chords of the hyperbola that are parallel to the other line. Two such lines are called conjugate diameters. The equation of the hyperbola referred to its conjugate diameters as oblique axes is y’2 x’2 − =1 (55) a12 b12 where 2a1 and 2b1 are the conjugate axes. Asymptotes. The lines y = (b/a)x and y = −(b/a)x are the asymptotes of the hyperbola x 2 /a 2 − y 2 /b2 = 1. The asymptotes are two tangents whose points of contact with the curve are at an infinite distance from the center. The equation of the hyperbola when referred to its asymptotes as oblique axes is

4x y = a 2 + b2

If a and b have unlike signs, (59) is a hyperbola with axes parallel to the coordinate axes. The parametric form is x = a sec φ Parabola.

y = a tan φ

The equation of the parabola is (y − y0 )2 = 4a(x − x0 )

yy1 = 2a(x + x1 )

y = mx +

2a(y − y1 ) + y1 (x − x1 ) = 0

Other Forms of the Equation of the Hyperbola

(59)

(63)

(64)

A diameter of the curve is a straight line parallel to the axis. It bisects all chords parallel to the tangent at the point where the diameter meets the parabola. If P1 T is tangent to the curve at (x1 , y1 ), then T Q = 2x1 is the subtangent, and QN = 2a (a constant) is the subnormal, where P1 N is perpendicular to P1 T . The equation of the form y 2 + 2gx + 2fy + c = 0, where g = 0, is a parabola whose axis is parallel to OX ; and the equation x 2 + 2gx + 2fy + c = 0, where f = 0, is a parabola whose axis is parallel to OY.

is called the rectangular or equilateral hyperbola.

ax 2 + by 2 + 2gx + 2fy + c = 0

a m

The normal to the parabola at the point (x1 , y1 ) is

(57)

(58)

(62)

The equation of the tangent whose slope is m is

(56)

y2 x2 =1 − a2 a 2 (e2 − 1)

(61)

If (x0 , y0 ) = (0, 0), the vertex is at the origin (Fig. 94); the focus F is on OX, called the axis of the parabola, and has the coordinates (a, 0); and the directrix is x = −a. The chord LL through F is the latus rectum and has the length 4a. The eccentricity e = F P /P M = 1. The tangent to the parabola y 2 = 4ax at the point (x1 , y1 ) is

If a = b, the asymptotes are the perpendicular lines y = x, y = −x; the corresponding hyperbola x2 − y2 = a2

(60)

Fig. 94

226


The parabola referred to the tangents at the extremities of its latus rectum as axes of coordinates is x 1/2 ± y 1/2 = b1/2

(65)

Let A + B = a + b, AB = ab − h2 = d, and A − B have the same sign as h. Let c = D/d; then the equation of the conic referred to its axes is y2 x2 + =1 −c /A −c /B

where b is the distance from the origin to each point of tangency. Polar Equations of the Conics. If e is the eccentricity, the directrix is vertical, the focus is at a distance p to the right or left of it, respectively, and the polar origin is taken at the focus, the polar equation is  ep  for ellipse, hyperbola, or parabola   1 ∓ e cos θ      (66)  2 r = a(1 − e ) for ellipse or circle (67)    1 ∓ e cos θ      a(e2 − 1)  for hyperbola (68) 1 ∓ e cos θ

If the directrix is horizontal and the focus is at a distance p above or below it, respectively, the polar equation is  ep  for ellipse, hyperbola, or parabola   1 ∓ e sin θ      (69)  2 r = a(1 − e ) for ellipse or circle (70)    1 ∓ e sin θ    2   a(e − 1)  for hyperbola (71) 1 ∓ e sin θ

(74)

To find the center (x0 , y0 ) of the conic solve the equations ax0 + hy0 + g = 0

hx0 + by0 + f = 0

(75)

To remove the term in xy from (64), rotate the axes about the origin through an angle θ such that tan 2θ = 2h/(a − b). 7.4 Higher Plane Curves Plane Curves. The point (x, y) describes a plane curve if x and y are continuous functions of a variable t (parameter), as x = x(t), y = y(t). The elimination of t from the two equations gives F (x, y) = 0 or in explicit form y = f (x). The angle τ , which a tangent to the curve makes with OX, can be found from

dy ds

sin τ =

cos τ =

dx ds

tan τ =

dy = y dx

(76)

where ds is the element of arc length: ds =

dx 2 + dy 2 = 1 + y 2 dx

(77)

In polar coordinates, General Equation of a Conic Section. equation has the form

ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0 Let a D = h g

h b f

g f c

This (72)

ds = dr 2 + r 2 dθ 2 =

dr dθ

2 θp + r 2

(78)

From (Fig. 95), it may be seen that a d = h

h b

δ =a+b

sin ψ =

r dθ ds

cos ψ =

dr ds

(73) Then the following is a classification of conic sections: 1. A parabola for d = 0, D = 0 2. Two parallel lines (possibly coincident or imaginary) for d = 0, D = 0 3. An ellipse for d > 0, δD < 0 4. No locus (imaginary ellipse) for d > 0, δD > 0 5. Point ellipse for d > 0, D = 0 6. A hyperbola for d < 0, D = 0 7. Two intersecting lines for d < 0, D = 0

Fig. 95

tan =

r dθ dr

(79)

MATHEMATICS

227

The equation of the tangent to the curve F (x, y) = 0 at the point (x1 , y1 ) is

∂F ∂x

+

∂F ∂y

(y − y1 ) = 0

(80)

x=x1 ,y=y1

−

2 −

2 s ≈l 1+ 3

(x − x1 ) x=x1 ,y=y1

∂F ∂x

(y − y1 ) = 0

(81)

x=x1 ,y=y1

The equation of the tangent to the curve y = f (x) at the point (x1 , y1 ) is y − y1 =

dy dx

(x − x1 )

(82)

x=x1

The equation of the normal to the curve y = f (x) at the point (x1 , y1 ) is y − y1 = −

1 (x − x1 ) (dy/dx)x=x1

(83)

The radius of curvature of the curve at the point (x, y) is '

ρ=

∂2F ∂2F ∂x 2 ∂y 2

(86)

Then for D > 0, the curve has a double point with two real different tangents. For D = 0, the curve has a cusp with two coincident tangents. For D < 0, the curve has an isolated point with no real tangents. See Figs. 96–100 for special curves. For l large compared with d, #

∂ 2F ∂x∂y

x=x1 ,y=y1

∂F ∂y

The equation of the normal to the curve F (x, y) = 0 at the point (x1 , y1 ) is

D=

(x − x1 )

Let

'

( 2 3/2

1 + (dy/dx) ds = dτ d 2 y/dx 2

=

Fig. 96

( 2 3/2

1+y y

(84)

The reciprocal 1/ρ is called the curvature of the curve at (x, y). The coordinates (x0 , y0 ) of the center of curvature for the point (x, y) on the curve [the center of the circle of curvature tangent to the curve at (x, y) and of radius ρ] are dy [1 + y 2 ] = x − y x0 = x − ρ ds y y0 = y + ρ

[1 + y 2 ] dx =y+ ds y

A curve has a singular point if, simultaneously, F (x, y) = 0

∂F =0 ∂x

∂F =0 ∂y

(85)

Fig. 97

2d l

2 $

228


Fig. 98

Fig. 100

Fig. 101

Fig. 99

Trochoid. This is a curve traced by a point at a distance b from the center of a circle of radius a as the circle rolls on a straight line:

x = aφ − b sin φ

y = a − b cos φ

See Figs. 101–103 for cycloids. For one arch, arc length = 8a, area = 3πa 2 . Hypotrochoid. This is a curve traced by a point at a distance b from the center of a circle of radius a as the circle rolls on the inside of a fixed circle of radius R:

R−a φ a R−a y = (R − a) sin φ − b sin φ a x = (R − a) cos φ + b cos

Hypocycloid.

b = a (Fig. 104).

Fig. 102

Epitrochoid. This is a curve traced by a point at a distance b from the center of a circle of radius a as the circle rolls on the outside of a fixed circle of radius R. See Figs. 105 and 106. Other forms of the right-hand side of the equation, b + 2a sin θ, b − 2a cos θ, b − 2a sin θ , give curves rotated through 1, 2, 3 right angles, respectively. See Figs. 107–110. In Fig. 111, as θ → ∞, r → 0. The curve winds an indefinite number of times around the origin. As

MATHEMATICS

229

Fig. 103 Fig. 106

Fig. 107

Fig. 104

Fig. 108

Fig. 105

θ → 0, r → ∞. The curve has an asymptote parallel to the polar axis at a distance a. In Fig. 112, the tangent to the curve at any point makes a constant angle α(= cot−1 m) with the radius vector. As θ → −∞, r → 0. The curve winds an indefinite number of times around the origin. Figure 113 illustrates the locus of a point P , the product of whose distances from two fixed points F1

and F2 is equal to the square of half the distance between them, r1 · r2 = c2 . The roses r = a sin nθ and r = a cos nθ have, for n even, 2n leaves; for n odd, n leaves. In Fig. 118, the locus of point P is such that OP = AB. In Fig. 119, if the line AB rotates about A, intersecting the y axis at B, and if P B = BP = OB, the locus of P and P is the strophoid. Figure 123 illustrates the locus of one end P of tangent line of length a as the other end Q is moved along the x axis. In Fig. 126, y = cos π/2t 2 , (t) = πt.

230


Fig. 111

Fig. 109

8

SOLID ANALYTIC GEOMETRY

8.1 Coordinate Systems Right-Hand Rectangular (Fig. 127). The position of a point P(x, y, z) is fixed by its distances x, y, z from the mutually perpendicular planes yz, xz, and xy, respectively. Spherical, or Polar (Fig. 128). The position of a point P (r, θ, φ) is fixed by its distance from a given point O, the origin, and its direction from O, determined by the angles θ and φ. Cylindrical (Fig. 128). The position of a point P (ρ, φ, z) is fixed by its distance z from a given plane and the polar coordinates (ρ, φ) of the projection Q of P on the given plane.

Fig. 110

Fig. 112

Relations among coordinates of the three systems are

x = r sin θ cos φ = ρ cos φ

(88)

y = r sin θ sin φ = ρ sin φ

(89)

z = r cos θ ρ = x 2 + y 2 = r sin θ

(90) (91)

MATHEMATICS

231

Fig. 116

Fig. 113

Fig. 117

Fig. 114

Fig. 115 Fig. 118

φ = tan−1

y x

r = x 2 + y 2 + z2 = ρ 2 + z2 x2 + y2 ρ = tan−1 θ = tan−1 z z

(92)

8.2

(93)

Euclidean Distance between Two Points. This distance between P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) is

(94)

Point, Line, and Plane

s=

(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2

(95)

232


Fig. 121

Fig. 119

Fig. 122

Fig. 120

To divide the segment P1 P2 in the ratio c1 /c2 , internally or externally, c2 x1 ± c1 x2 c2 ± c1 c2 z1 ± c1 zz z= c2 ± c1

x=

y=

c2 y1 ± c1 y2 c2 ± c1

Fig. 123

(96)

The midpoint of P1 P2 is x=

1 2 (x1

+ x2 )

z=

1 2 (z1

+ z2 )

y=

1 2 (y1

Angles. The angles α, β, γ that the line P1 P2 makes with the coordinate directions x, y, z, respectively, are the direction angles of P1 P2 . The consines

x2 − x1 s z2 − z1 cos γ = s cos α =

+ y2 ) (97)

cos β =

y2 − y1 s (98)

MATHEMATICS

233

Fig. 124 Fig. 126

Fig. 127

Fig. 125

are the direction cosines of P1 P2 , and cos2 x + cos2 β + cos2 γ = 1

(99)

If : m : n = cos α : cos β : cos γ , then cos α = √ cos γ = √

l l 2 + m2 + n 2 n l 2 + m2 + n 2

cos β = √

Fig. 128

m l 2 + m2 + n 2

Planes.

A plane is represented by

(100)

The angle θ between two lines in terms of their direction angles α1 , β1 , γ1 and α2 , β2 , γ2 is obtained from cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 (101) If cos θ = 0, the lines are perpendicular to each other.

Ax + By + Cz + D = 0

(102)

If one of the variables is missing, the plane is parallel to the axis of the missing variable. For example, Ax + By + D = 0 represents a plane parallel to the z axis. If two of the variables are missing, the plane is parallel to the plane of the missing variables. For example, z = k represents a plane parallel to the xy plane and k units from it.

234


A plane through three points P1 (x1 , y1 , z1 ), P2 (x2 , y2 , z2 ), and P3 (x3 , y3 , z3 ) has the equation x y z 1 y1 z1 1 x 1 (103) x y2 z2 1 = 0 2 x 3 y3 z3 1 The equation of a plane whose x, y, z intercepts are, respectively, a, b, c (Fig. 129) is y z x + + =1 a b c

(104)

The perpendicular form of the equation of a plane, where OP = p is the perpendicular distance of the plane from the origin O and has the direction angles α, β, γ , is

Parallel Planes. Two planes A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0 are parallel if A1 : B1 : C1 = A2 : B2 : C2 ;

A(x − x1 ) + B(y − y1 ) + C(z − z1 ) = 0

(108)

is a plane through the point P1 (x1 , y1 , z1 ) and parallel to the plane Ax + By + Cz + D = 0. Angle θ between Two Planes. The angle between Ax + By + Cz + D = 0 and A1 x + B1 y + C1 z + D1 = 0 is the angle between two intersecting lines, each perpendicular to one of the planes:

cos θ =

AA1 + BB1 + CC1 (109) 2 ± (A + B 2 + C 2 )(A21 + B12 + C12 )

(105)

The two planes are perpendicular if AA1 + BB1 + CC1 = 0.

To bring the general form Ax + By + Cz + D = 0 into √ the perpendicular form, divide it by ± A2 + B 2 + C 2 , where the sign before the radical is opposite to that of D. The coefficients A, B, C are proportional to the direction cosines λ, µ, ν of a line perpendicular to the plane. Therefore,

Points, Planes, and Lines. Four points, Pk (xk , yk , zk )(k = 1, 2, 3, 4), lie in the same plane if 1 y1 z1 x1 x2 y2 z2 1 (110) 1 x3 y3 z3 = 0 1 x4 y4 z4

x cos α + y cos β + z cos γ − p = 0

A(x − x1 ) + B(y − y1 ) + C(z − z1 ) = 0

(106)

is a plane through P1 (x1 , y1 , z1 ) and perpendicular to a line with direction cosines λ, µ, ν proportional to A, B, C. Perpendicular Distance between Point and Plane. The distance between point P1 from a plane Ax + By + Cz + D = 0 is given by

Ax1 + By1 + Cz1 + D √ P P1 = ± A2 + B 2 + C 2

(107)

where the sign before the radical is opposite to that of D.

Four planes, Ak x + Bk y + Ck z + Dk = 0 (k = 1, 2, 3, 4), pass through the same point if A1 B1 C1 D1 B2 C2 D2 A2 (111) A B3 C3 D3 = 0 3 A4 B4 C4 D4 A straight line is represented as the intersection of two planes by two first-degree equations A1 x + B1 y + C1 z + D1 = 0 A2 x + B2 y + C2 z + D2 = 0

(112)

The three planes through the line perpendicular to the coordinate planes are its projecting planes. The equation of the xy projecting plane is found by eliminating z between the two given equations, and so on. The line can be represented by any two of its projecting planes, for example, y = m1 x + b 1

z = m2 x + b 2

(113)

If the line goes through a point P1 (x1 , y1 , z1 ) and has the direction angles α, β, γ , then

Fig. 129

y − y1 z − z1 x − x1 = = cos α cos β cos γ

(114)

MATHEMATICS

235

The following relations exist:

and cos β m1 = cos α

(1)

cos γ m2 = cos α

The equations of a line through two points (x1 , y1 , z1 ) and (x2 , y2 , z2 ) are y − y1 z − z1 x − x1 = = x2 − x1 y2 − y1 z2 − z1

(2)

(3) (115) (4)

A line through a point P1 perpendicular to a plane Ax + By + Cz + D = 0 has the equations (5) y − y1 z − z1 x − x1 = = A B C

(116) (6)

Line of Intersection of Two Planes. The direction cosines λ, µ, ν of the line of intersection of two planes Ax + By + Cz + D = 0 and A1 x + B1 y + C1 z + D1 = 0 are found from the ratios

B λ : µ : ν = B 1

C C C1 : C 1

A A A1 : A1

B B1 (117)

8.3 Transformation of Coordinates Changing the Origin. Let the coordinates of a point P with respect to the original axes be x, y, z and with respect to the new axes x , y , z . For a parallel displacement of the axes with x0 , y0 , z0 the coordinates of the new origin

x = x0 + x

y = y0 + y

z = z0 + z (118)

Rotation of the Axes about the Origin. Let the cosines of the angles of the new axes x , y , z with the x axis be λ1 , µ1 , ν1 , with the y axis be λ2 , µ2 , ν2 , with the z axis be λ3 , µ3 , ν3 . Then

x = λ1 x + µ1 y + ν1 z

x = λ1 x + λ2 y + λ3 z

y = λ2 x + µ2 y + ν2 z

y = µ1 x + µ2 y + µ3 z

z = λ3 x + µ3 y + ν3 z

z = ν1 x + ν2 y + ν3 z (119)

(7)

(8)

λ21 + µ21 + ν12 λ22 + µ22 + ν22 λ23 + µ23 + ν32 λ21 + λ22 + λ23

= = = =

1 1 1 1

µ21 + µ22 + µ23 ν12 + ν22 + ν32 λ1 λ2 + µ1 µ2 + ν1 ν2 λ2 λ3 + µ2 µ3 + ν2 ν3 λ3 λ1 + µ3 µ1 + ν3 ν1 λ1 µ1 + λ2 µ2 + λ3 µ3 µ1 ν1 + µ2 ν2 + µ3 ν3 ν1 λ1 + ν2 λ2 + ν3 λ3 λ1 µ1 ν1 λ2 µ2 ν2 λ3 µ3 ν3 λ1 µ1 ν1 λ2 µ2 ν2 λ3 µ3 ν3

= = = = = = = = = = = = = = = = =

1 1 0 0 0 0 0 0 µ2 ν3 − ν2 µ3 ν2 λ3 − λ2 ν3 λ2 µ3 − µ2 λ3 ν1 µ3 − µ1 ν3 λ1 ν3 − ν1 λ3 µ1 λ3 − λ1 µ3 µ1 ν2 − ν1 µ2 ν1 λ2 − λ1 ν2 λ1 µ2 − µ1 λ2

= 1

For a combination of displacement and rotation, apply the corresponding equations simultaneously. 8.4 Quadric Surfaces The general form of the equation of a surface of the second degree is

F (x, y, z) ≡ a11 x 2 + 2a12 xy + 2a13 xz + a22 y 2 + 2a23 yz + a33 z2 + 2a14 x + 2a24 y + 2a34 z + a44 = 0

(120)

where the aik are constants and aik = aki , that is, a12 = a21 , and so on. Let a11 a12 a13 a14 a11 a12 a13 a21 a22 a23 a24 D = a d = a21 a22 a23 31 a32 a33 a34 a31 a32 a33 a41 a42 a43 a44 Let I ≡ a11 + a22 + a33 and J ≡ a22 a33 + a33 a11 2 2 2 + a11 a22 − a23 − a13 − a12 . Here, D, d, I, and J are invariant under coordinate transformation. The following is a classification of the quadratic surfaces, so far as they are real and do not degenerate into curves in one plane:

236


Ellipsoid for D < 0, I d > 0, J > 0 Hyperboloid of two sheets for D < 0, I d and J not both >0 Hyperboloid of one sheet for D > 0, I d and J not both >0 Cone for D = 0, d = 0, I d and J not both >0 Elliptic paraboloid for D < 0, d = 0, J > 0 Hyperbolic paraboloid for D > 0, d = 0, J < 0 Cylinder for D = 0, d = 0 Ellipsoid and Hyperboloids. Consider the center of the quadric as the origin and the principal axes of the quadric as the orthogonal coordinate axes. Then

y2 z2 x2 + + =1 a2 b2 c2 y2 z2 x2 + 2 − 2 =1 2 a b c 2

2

[ellipsoid (Fig. 130)]

(121)

[hyperboloid of one sheet (Fig. 131)]

where λ1 , λ2 , λ3 are the real roots of the cubic equation a11 − λ a12 a13 a22 − λ a23 = 0 (125) a12 a13 a23 a33 − λ

(122)

Cone.

[hyperboloid of two sheets (Fig. 132)]

ax 2 + by 2 + cz2 + 2hxy + 2gxz + 2fyz = 0 (126) (123)

where a, b, c are the semiaxes. The length of the semiaxis is found from D λ1 d

The equation

2

y z x + 2 + 2 = −1 a2 b c

a2 = −

Fig. 132

b2 = −

D λ2 d

c2 = −

D (124) λ3 d

represents a cone with vertex at the origin. If the cross section of the cone is an ellipse with axes 2a and 2b whose plane is parallel to the xy plane and at a distance c from the origin, then the equation of the cone with vertex at the origin is y2 z2 x2 + − =0 a2 b2 c2

(127)

If a = b, the cross section is circular and the cone is a cone of revolution. Sphere.

An equation of the form

x 2 + y 2 + z2 + ax + by + cz + d = 0

(128)

represents a sphere with radius r=

Fig. 130

1 2

√

a 2 + b2 + c2 − 4d

(129)

and center x0 = − 12 a

y0 = − 12 b

z0 = − 12 c

(130)

If (x0 , y0 , z0 ) are the coordinates of the center and r is the radius, then the equation of the sphere is (x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2

(131)

If x0 = 0, y0 = 0, z0 = 0, then the equation is Fig. 131

x 2 + y 2 + z2 = r 2

(132)

MATHEMATICS

Paraboloids.

237

are elliptic, hyperbolic, and parabolic cylinders, respectively, with elements or generators parallel to OZ.

The equation y2 x2 + 2 = 2cz 2 a b

(133)

F (x, y, z) ≡ a11 x 2 + 2a12 xy + 2a13 xz + a22 y 2

represents an elliptic paraboloid (Fig. 133). If a = b, the equation is of the form x 2 + y 2 = 2cz

Tangent Plane. The equation of the tangent plane to any quadric

+ 2a23 yz + a33 z2 + 2a14 x

(paraboloid of revolution) (134)

The equation x2 y2 − = 2cz [hyperbolic paraboloid (Fig.134)] a2 b2 (135) Cylinder. The equation of a cylinder perpendicular to the yz, xz, or xy plane is the same as the equation of a section of the cylinder in the corresponding plane. Thus

y2 x2 + 2 =1 2 a b

(136)

x2 y2 − 2 =1 2 a b

(137)

y 2 = 4ax

(138)

+ 2a24 y + 2a34 z + a44 = 0 (139) at the point (x1 , y1 , z1 ) is ∂F (x − x1 ) ∂x x=x1 ,y=y1 ,z=z1 ∂F + (y − y1 ) ∂y x=x1 ,y=y1 ,z=z1 ∂F + (z − z1 ) = 0 ∂z x=x1 ,y=y1 ,z=z1

(140)

Example 47. Find the tangent plane to the hyperboloid of one sheet at point (x1 , y1 , z1 ). Given x 2 /a 2 + y 2 /b2 − z2 /c2 = 1. Then ∂F (x − x1 ) ∂x x=x1 ,y=y1 ,z=z1 ∂F (y − y1 ) + ∂y x=x1 ,y=y1 ,z=z1 ∂F + (z − z1 ) ∂z x=x1 ,y=y1 ,z=z1

=

2x1 (x − x1 ) 2y1 (y − y1 ) 2z1 (z − z1 ) + − =0 a2 b2 c2 x12 y12 z12 yy1 zz1 xx1 + − − − + a2 b2 c2 a2 b2 c2 xx1 yy1 zz1 = 2 + 2 − 2 − 1 = 0 (tangent plane) a b c

Fig. 133

The Normal. The line through a point P1 on a surface and perpendicular to the tangent plane at P1 is called the normal to the surface at P1 . The equations of the normal to the surface F (x, y, z) = 0 at the point (x1 , y1 , z1 ) are

∂F ∂x

x − x1

=

x=x1 ,y=y1 ,z=z1

= Fig. 134

∂F ∂y ∂F ∂z

y − y1 x=x1 ,y=y1 ,z=z1

z − z1 x=x1 ,y=y1 ,z=z1

(141)

238


9 DIFFERENTIAL CALCULUS 9.1 Functions and Derivatives Function. If two variables x and y are so related that to each value of x in a given domain there corresponds a value of y, then y is a function of x in that domain. The variable x is the independent variable and y the dependent variable. The symbols F (x), f (x), φ(x), and so on, are used to represent functions of x; the symbol f (a) represents the value of f (x) for x = a. Limit, Derivative, Differential. The function f (x) approaches the limit 1 as x approaches a if the difference |f (x) − 1| can be made arbitrarily small for all values of x except a within a sufficiently small interval with a as midpoint. In symbols, limx→a f (x) = 1. The symbols limx→a f (x) = ∞ or limx→a f (x) = −∞ mean that, for all values of x except a within a sufficiently small interval with a as midpoint, the values of f (x) can be made arbitrarily large positively or negatively, respectively. The symbols limx→∞ f (x) = 1 and limx→−∞ f (x) = 1 mean that the difference |f (x) − 1| can be made arbitrarily small for all values of x sufficiently large positively or negatively, respectively. A change in x is called an increment of x and is denoted by x. The corresponding change in y is denoted by y. If f (x + x) − f (x) lim x→0 x

exists, it is called the derivative of y with respect to x and is denoted by dy/dx, f (x), or Dx y. The geometric interpretation of f (x) is f (x) =

dy = tan θ dx

(142)

or f (x) is equal to the slope of the tangent to the curve y = f (x) at the point P (x, y) (Fig. 135): RQ y RS = lim = lim x→0 x P R P R→0 P R f (x + x) − f (x) = lim x→0 x dy = f (x) = tan θ = dx

The differentials of x and y, respectively, are dx = x

dy = f (x) dx

Continuity. A function is continuous at x = b if it has a definite value at b and approaches that value as a limit whenever x approaches b as a limit. The notion of continuity at a point suggests that the graph of the function can be drawn without lifting pencil from paper at the point. The analytic conditions that f (x) be continuous at b are that f (b) have a definite value and that for an arbitrarily small positive number ε there exist a δ(ε) such that

|f (x) − f (b)| < ε

for all values of x

for which |x − b| < δ(ε) (144) A function that is continuous at each point of an interval is said to be continuous in that interval. An example of a continuous function is f (x) = x 2 . The function φ(x) = 1/(x − a) is continuous for all values of x except x = a, at which point it becomes infinite. Every differentiable function is continuous, although the reverse is not always true. If, in the preceding definition of continuity, the number δ can be chosen the same for all points in the interval, the function is said to be uniformly continuous in that interval. Derivatives of Higher Order. The derivative of the first derivative of y with respect to x is called the second derivative of y with respect to x and is denoted by

d dx

dy dx

=

d 2y = f (x) = Dx2 y dx 2

By successive differentiations the nth derivative d ny = f (n) (x) = Dxn y dx n

(143)

(145)

(146)

is obtained. The nth differential of y is denoted by d n y = f (n) (x) dx n

(147)

Parametric Differentiation. To find the derivatives of y with respect to x if y = y(t) and x = x(t):

Fig. 135

y =

dy/dt dy = dx dx/dt

(148)

y =

d 2y dy /dt = dx 2 dx/dt

(149)

d ny dy (n−1) /dt = n dx dx/dt

(150)

y (n) =

MATHEMATICS

239

Example 48. Find the derivatives of y with respect to x for the ellipse x = a cos t, y = b sin t:

b cos t b dy = = − cot t y = dx −a sin t a

0 0

∞ ∞

0·∞

(3b/a 2 ) csc3 t cot t 3b dy = = − 3 csc4 t cot t dx −a sin t a

Logarithmic Differentiation for Products and Quotients. If u v m (151) y= wn

take the logarithms of both sides before differentiating: ln y = l ln u + m ln v − n ln w

(152)

l du m dv n dw 1 dy = + − y dx u dx v dx w dx n dw dy l du m dv =y + − dx u dx v dx w dx

lim f (x) = lim

x→a

(153)

x→a

lim

1∞

φ (x) ψ (x)

(1 Hospital’s rule)

(158)

Find the value of sin x/x for x = 0:

lim

Find dy/dx if

sin x cos x = lim =1 x→0 1 x

∞/∞. If f (x) = φ(x)/ψ(x), φ(a) = ∞, and ψ(a) = ∞, then φ(x) φ (x) = lim (159) lim x→a ψ(x) x→a ψ (x)

ln(x 2 − 25) − 3 ln(x − 1) − 2 ln(x + 5)

2x 3 2 1 dy = − − y dx 2(x 2 − 25) x − 1 x + 5 y(−4x 2 + 11x + 65) dy = dx (x 2 − 25)(x − 1) Mean Value Theorem. If f(x) is single valued, continuous in the interval a ≤ x ≤ b, and has a derivative for all values of x between a and b, then there is a value x = ξ, a < ξ < b, such that

f (b) − f (a) = (b − a)f (ξ )

(155)

Another form is f (x + h) = f (x) + hf (x + θ h)

0∞

unless φ (a) = 0 and ψ (a) = 0. In this case, the rule is applied again, and so forth. Example 50.

x 2 − 25 (x − 1)3 (x + 5)2 1 2

◦

φ (a) φ(x) φ (ξ ) = lim = ψ(x) ξ →a ψ (ξ ) ψ (a)

(154)

√

ln y =

∞

(157) If, however, φ (a) = 0 and ψ (a) = 0, the rule is applied again, with the result

x→0

y=

◦

0/0. If f (x) = φ(x)/ψ(x), φ(a) = 0, and ψ(a) = 0, then

x→a

Example 49.

∞−∞ 0

then it may happen that lim f(x) has a definite value. For the determination of this limiting value, if it exists, the following rules can be used:

(b/a) csc2 t b dy = = − 2 csc3 t y = dx −a sin t a y =

appears in one of the meaningless forms

0 1, series (176) diverges; if L = 1, the test fails. Example 54

103 10n 102 + + ··· + + ··· 2! 3! n!

10 +

(1) Since

L = lim

n→∞

10n+1 /(n + 1)! 10 = lim =0 n→∞ n + 1 10n /n!

the series converges. (2)

1 + 1 (1 + 1)(2 + 1) + + ··· 1 + 3 (1 + 3)(2 + 3) +

(1 + 1)(2 + 1) · · · (n + 1) + ··· (1 + 3)(2 + 3) · · · (n + 3)

(176)

n=1

is called an infinite series. Let sn = a1 + a2 + · · · + an . If the partial sums sn approach a limit S as n → ∞, then the series is convergent and S is the sum or value of the series. A series that is not convergent is divergent. If the series of absolute values |a1 | + |a2 | + · · · + |an | + · · · is convergent, then series (176) is absolutely convergent. A series that converges, but not absolutely, is conditionally convergent. The sum of an absolutely convergent series is not changed by rearrangement of its terms.

Since L = lim

n→∞

(n + 1) + 1 =1 (n + 1) + 3

the test fails. Raabe’s test can be used. See Eq. (179). Root Test.

Let L = lim |an |1/n n→∞

(178)

If L < 1, series (176) converges; if L > 1, series (176) diverges; if L = 1, the test fails.

MATHEMATICS

243

Example 55

1+

Convergence of an Alternating Series.

1 1 1 + + ··· + + ··· 2 3 (log 2) (log 3) (log n)n

n→∞

1 =0 log n

the series converges. Integral Test. Let f (n) = an . If f (x) is a positive nonincreasing function of x for x > k, then series converges or diverges with the improper integral )(176) ∞ k f (x) dx. Example 56

1+

1 1 1 + + ··· + + ··· 3 3 2(log 2) 3(log 3) n(log n)3

f (x) =

1 x(log x)3

and

2

Series of Functions A power series is a series of the form ∞

an x n = a0 + a1 x + a2 x 2 + · · · + an x n + · · ·

n=0

(181) If limn→∞ |an−1 /an | = r, the power series converges absolutely for all values of x in the interval −r < x < r. For |x| = r, it is necessary to use one of the convergence tests for series of numerical terms. Example 58

Then

∞

(180)

in which the terms are alternately positive and negative is an alternating series. If, from some term on, |an+1 | ≤ |an | and an → 0 as n → ∞, the series converges. The sum of the first n terms differs numerically from the sum of the series by less than |an+1 |.

Since L = lim

a1 − a2 + a3 − + · · · + (−1)n+1 an + · · ·

A series

dx 1 = lim n→∞ 2 x(log x)3 =

for x ≥ 2

1−

1 1 − (log 2)2 (log n)2

Let

L = lim n n→∞

an −1 an+1

(179)

If L > 1, series (176) converges; if L < 1, series (176) diverges; if L = 1, the test fails.

lim

n→∞

n · 2n =2 (n − 1)2n−1

Taylor’s Series. If f (x) has continuous derivatives in the neighborhood of a point x = a, then

f (x) = f (a) + +

f (a) f (a) (x − a) + (x − a)2 + · · · 1! 2!

f (n−1) (a) (x − a)n−1 + · · · (n − 1)!

(182)

with the remainder after n terms

Example 57

1 + 1 (1 + 1)(2 + 1) + + ··· 1 + 3 (1 + 3)(2 + 3) +

Since

the interval of convergence is −2 < x < 2. For x = 2, the series is a convergent alternating series. For x = −2, it is a divergent p series.

1 2(log 2)2

Since the integral is convergent, the series is also. Raabe’s Test.

n x2 x x3 n x + − + · · · + (−1) + ··· 1 · 2 2 · 22 3 · 23 n · 2n

(1 + 1)(2 + 1) · · · (n + 1) + ··· (1 + 3)(2 + 3) · · · (n + 3)

f (n) (ξ ) (x − a)n n! ξ = a + θ (x − a) 0 0, there exists an N dependent on ε but not on x such that |Rn (x)| < ε for n > N. If a power series converges in the interval −r < x < r, then it converges uniformly in any interval within this interval. The sum of a uniformly convergent series of continuous functions is also a continuous function.

a0 = q0 b0 a1 = q0 b1 + q1 b0 a2 = q0 b2 + q1 b1 + q2 b0 an = q0 bn + q1 bn−1 + · · · + qn b0

(197)

See Table 10 for series expansions of various functions.

246


Table 10 Functions Expanded in Series (log = loge ) n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3 a x + a x + ··· (x2 < a2 ) 2! 3! x2 x3 x4 1+x+ + + + ··· (−∞ < x < ∞) 2! 3! 4! 2 (x log a) (x log a)3 1 + x log a + + + ··· (−∞ < x < ∞) 2! 3! 4 6 8 x x x − + − ··· (−∞ < x < ∞) 1 − x2 + 2! 3! 4! 2 4 5 6 x 3x 8x 3x 56x7 1+x+ − − − + + ··· (−∞ < x < ∞) 2! 4! 5! 6! 7! 2 4 6 x 4x 31x e 1− + − + ··· (−∞ < x < ∞) 2! 4! 6! π x2 3x3 9x4 37x5 π 1+x+ + + + + ··· − <x< 2! 3! 4! 5! 2 2 1 x−1 2 1 x−1 3 x−1 1 + + + ··· x> x 2 x 3 x 2 # $ 3 5 x−1 1 x−1 1 x−1 2 + + ··· (x > 0) + x+1 3 x+1 5 x+1

(a + x)n = an + nan−1 x + ex = ax = e−x

2

=

esin x = ecos x = etan x = log x = log x =

x2 x3 x4 log(1 + x) = x − + − + ··· (−1 < x1) = 2 + 3 + 5 + ··· x−1 x 3x 5x x2 x4 x6 log sin x = log x − − − − ··· (−π < x < π ) 6 180 2835 2 4 6 8 x x 17x π π x − − − − ··· − <x< log cos x = − 2 12 45 2520 2 2 π 7x4 62x6 π x2 + + + ··· − <x< log tan x = log x + 3 90 2835 2 2 x3 x5 x7 sin x = x − + − + ··· (−∞ < x < ∞) 3! 5! 7! 2 4 6 x x x + − + ··· (−∞ < x < ∞) cos x = 1 − 2! 4! 6! π 3 5 7 2x 17x 62x9 π x + + + + ··· − <x< tan x = x + 3 15 315 2835 2 2 1 x x3 2x5 x7 cot x = − − − − − ··· (−π < x < π ) x 3 45 945 4725 2 4 6 5x 61x π π x + + + ··· − <x< sec x = 1 + 2! 4! 6! 2 2 x 7x3 31x5 1 + + + + ··· (−π < x < π ) csc x = x 3! 3 · 5! 3 · 7! 3 5 7 x 3x 3 · 5x sin−1 x = x + + + + ··· (−1 ≤ x≤1) 2·3 2·4·5 2·4·6·7 π −1 −1 − sin x cos x = 2 π 1 1 1 tan−1 x = (x2 ≥ 1) − + 3 − 5 + ··· 2 x 3x 5x x5 x7 x3 + − + ··· (−1 ≤ x ≤ 1) = x− 3 5 7 (Continues)

MATHEMATICS Table 10 cot

−1

x

sec−1 x csc−1 x sinh x cosh x tanh x coth x sechx cschx sinh−1 x sinh−1 x cosh−1 x tanh−1 x coth−1 x sech−1 x csch−1 x

247

(Continued ) π = − tan−1 x 2 1 1 π 3 3·5 − − = − − − ··· (x2 > 1) 2 x 2 · 3x3 2 · 4 · 5x5 2 · 4 · 6 · 7x7 π = − sec−1 x 2 x3 x5 x7 = x+ + + + ··· (−∞ < x < ∞) 3! 5! 7! x4 x6 x8 x2 + + + + ··· (−∞ < x < ∞) = 1+ 2! 4! 6! 8! 3 5 7 x 2x 17x π π = x− + − + ··· − <x< 3 15 315 2 2 1 x x3 2x5 x7 = + − + − + ··· (−π < x < π ) x 3 45 945 4725 π 2 4 6 8 x 5x 61x 1385x π = 1− + − + − ··· − <x< 2! 4! 6! 8! 2 2 1 x 7x3 31x5 = − + − + ··· (−π < x < π ) x 6 360 15,120 3 5 7 3x 3 · 5x x + − + ··· (−1 < x < 1) = x− 2·3 2·4·5 2·4·6·7 1 3 3·5 = log 2x + − + + ··· (x2 > 1) 2 4 2 · 2x 2 · 4 · 4x 2 · 4 · 6 · 6 · x6 1 1·3 1·3·5 = ± log 2x − − − − ··· (x > 1) 2 · 2x2 2 · 4 · 4x4 2 · 4 · 6 · 6x6 3 5 7 x x x = x+ + + + ··· (−1 < x < 1) 3 5 7 1 1 1 1 = + 5 + 7 + ··· (x2 > 1) + x 3x3 5x 7x 2 1·3 1 2 1·3·5 6 = ± log − (0 < x < 1) x − x− x − ··· x 2·2 2·4·4 2·4·6·6 1 1 3 3·5 − = + − + ··· (x2 > 1) x 2 · 3x3 2 · 4 · 5x5 2 · 4 · 6 · 7x7

9.5 Maxima and Minima Function of One Variable. A function f (x) has a relative maximum (minimum) at a point x = a if at every point in some neighborhood of x = a the values of f (x) are all less (greater) than f (a). Either a maximum or a minimum is an extreme. If the derivative exists at a relative extreme, it must be zero, that is, the tangent must be parallel to the x axis. To locate possible extreme points solve the equation f (x) = 0. A solution x = a gives a maximum (minimum) value of f (x) if and only if the derivative is positive (negative) for x < a and negative (positive) for x > a. If the derivative does not change sign, x = a gives a point of inflection. A solution x = a can be tested also by using the higher derivatives of f (x). Let f (n) (x) be the first derivative that does not equal zero for x = a, f (n) (a) = 0. If n is even, there is a maximum (Fig. 136) if f (n) (a) < 0 and a minimum (Fig. 137)

Fig. 136

if f (n) (a) > 0. If n is odd, there is a point of inflection (Figs. 138 and 139). In many problems physical considerations make testing unnecessary. Example 62. A piece of wire of length 30 in. is bent into a rectangle. Find the maximum area. Let x = the base, then 12 (30 − 2x) = the altitude. The area

248


then f (a, b) is an extreme value, which is a maximum if ∂2f ∂2f a2 ) u 2 − a2 2a u + a 1 a−u = ln (u2 < a2 ) 2a a + u 1 u = − tanh−1 (u2 < a2 ) a a u 1 = − coth−1 (u2 > a2 ) a a du u √ = sin−1 a a2 − u 2 u = − cos−1 a √ du √ = ln(u + u2 ± a2 ) u 2 ± a2 u du √ = sinh−1 a u 2 + a2 du −1 u √ = cosh a u 2 − a2 du 1 −1 u = √ sec a a u u 2 − a2 1 −1 u = − csc a a du u = vers−1 √ a 2au − u2 u −1 1− = cos a

sec u du = ln(sec u + tan u) π u + = ln tan 4 2

21.

sinh u du = cosh u

22.

cosh u du = sinh u

csc u du = ln(csc u − cot u)

23.

tanh u du = ln cosh u

24.

coth u du = ln sinh u

25.

sechu du = 2 tan−1 eu

26.

cschu du = ln tanh

du u 2 + a2

u = ln tan 2 u 1 = tan−1 a a 1 −1 u = − cot a a

u 2

The constant of integration is omitted in the above integrals.

Elliptic integral of the first kind: φ

F (φ, k) =

0 x

= 0

dθ 1−

k2

dξ

k2 < 1

1 − k2 ξ 2 dξ 1 − ξ2

x = sin φ

sin θ

k2 < 1

(207)

Elliptic integral of the third kind: (206)

φ

(φ, n, k) =

x

= 0

dθ (1 + n sin θ ) 1 − k 2 sin2 θ dξ (1 + nξ 2 ) (1 − ξ 2 )(1 − k 2 ξ 2 ) 2

0

Elliptic integral of the second kind: φ 1 − k 2 sin2 θ dθ E(φ, k) = 0

0 2

(1 − ξ 2 )(1 − k 2 ξ 2 )

x = sin φ

x

=

x = sin φ

k2 < 1

(208)

250


The “complete” integrals are K=F

π

π ,k = 2 2

+

3·5 2·4·6

#

2 1 3 2 4 2 1+ k + k 2 2·4 $

3

k6 + · · ·

(209)

π! 1 3 2 ,k = 1− k k4 − 2 2 22 22 · 42 " 32 · 5 − 2 2 2 k6 − · · · (210) 2 ·4 ·6 π π K=F , 1 − k2 , 1 − k2 E = E 2 2 (211) They are connected by the relation E=E

π

KE + EK − KK =

1 2π

(212)

The inverse function of u = F (φ, k) is φ = am u (am = amplitude),

Integration of Rational Fractions. If the quotient of two polynomials Pn (x) (217) R(x) = Pd (x)

is not a proper fraction, that is, if the degree of the numerator is not less than that of the denominator, R(x) can be changed, by dividing as indicated, to the sum of a polynomial, which is immediately integrable, and a proper fraction. If the proper fraction cannot be integrated by reference to Table 18, use the methods of Section 2.8 to resolve it, if possible, into partial fractions. These can be integrated from the table. Irrational Functions. These can sometimes be put into integrable forms by rationalizing them by a change of variable. Form

Substitution

f [(ax + b)p/q ] dx f [(ax + b)p/q (ax + b)r/s ] dx

Let ax + b = y q Let ax + b = y n , where n is the LCM of q, s √ Let x 2 + ax + b =y−x √ Let −x 2 + ax + b √ = (α − x)(β + x) = (α − x)y or = (β + x)y x Let tan = y 2 Let x = a sin y

f [x, f [x,

3

√ √

u x ≡ sin φ ≡ sn u = u − (1 + k 2 ) 3! + (1 + 14k 2 + k 4 ) cos φ ≡ cn u = 1 −

u2 2!

− (1 + 44k 2 + 16k 4 )

u5 − ··· 5! + (1 + 4k 2 )

(213) u4 4! (214)

u4 u2 + k 2 (4 + k 2 ) 1 − k 2 x 2 ≡ φ ∼ = dn u = 1 − k 2 2! 4! u6 − k (16 + 44k + k ) + · · · 6! 2

2

4

(215)

f [x,

√

a

u dv = uv −

v du

(u and v functions of x)

(216) is useful in integrating a product if factors of the product are a function of x and the derivative of another function of x. ) Example 63. To find x sin x dx, let u) = x, dv = sin x dx. Then du ) = dx, v = − cos x, and x sin x dx = −x cos x + cos x dx = −x cos x + sin x + c.

x 2 + a 2 ] dx

Let x = a tan y or x = a sinh y

The definite integral of f (x) from a to b is b

The formula

Let x = a sec y or x = a cosh y

10.2 Definite Integrals

Methods of Integration Integration by Parts.

−x 2 + ax + b] dx

f [sin x, cos x] dx √ f [x, a 2 − x 2 ] dx √ f [x, x 2 − a 2 ] dx

6

u + ··· 6!

x 3 + ax + b] dx

f (x) dx =

lim

n

n→∞ ν=1 max xν → 0

f (ξν ) xν

(218)

in which the interval a ≤ x ≤ b is divided into n arbitrary parts xν , ν = 1, 2, . . . , n, and ξν is an arbitrary point in xν (Fig. 140). A sufficient condition that this integral exists is that f (x) be continuous. However, it is necessary and sufficient only that f (x) be bounded and that its points of discontinuity form a set of Lebesgue measure 0. A set of points is of Lebesgue measure 0 if the points can be enclosed in a set of intervals Iν , ν = 1, 2, 3, . . . , finite or infinite in number, such that, for any ε > 0, the sum of the lengths of the Iν is m, an arbitrary positive number, for sufficiently large values of x, the interval diverges. )∞ 2 3/2 Example 65. The integral 0 x dx/(x + x ) exists, since, for k = 2 and M = 1,

[f1 (x) + f2 (x) + · · · + fn (x)] dx b

(219)

If one limit is infinite,

f (x) dx = lim

a

a

=

b−a (y0 + 4y1 + 2y2 + 4y3 + 2y4 3n

b

f (x) dx +

f (x) dx c

a≤c≤b

f (x) dx = (b − a)f (ξ )

a

for some ξ such that a ≤ ξ ≤ b (mean value theorem). Simpson’s Rule ) b for Approximate Integration. To evaluate a f (x) dx approximately, divide the interval from a to b into an even number n of equal

f (x) dx = lim

ε→0 a

b−ε

f (x) dx

0 < ε < (b − a)

(221) The integral exists if there is a number k < 1 and a number M independent of x such that (b − x)k |f (x)| < M for a ≤ x < b. If there is a number k≥1 and a number m such that (b − x)k |f (x)| > m for a ≤ x < b, the integral diverges. )1 Example 66. The integral 0 dx/(1 − x) diverges, since, for k = 1 and m = 12 , (1 − x)/(1 − x) = 1 > 12 . If the integrand is infinite at the lower limit, the tests are analogous. If the integrand is infinite at an intermediate point, use the point to divide the interval into two subintervals and apply the preceding tests.

252


Multiple Integrals. Let f (x, y) be defined in the region R of the xy plane. Divide R into subregions R1 , R2 , . . . , Rn of areas A1 , A2 , . . . , An . Let n (ξi , ηi ) be any point in Ri . If the sum i=1 f (ξi , ηi ) Ai has a limit as n → ∞ and the maximum diameter of the subregions Ri approaches zero, then

f (x, y) dA = lim

n

n→∞

R

f (ξi , ηi ) Ai

(222)

i=1

The double integral is evaluated by two successive single integrations, first with respect to y holding x constant between variable limits of integration and then with respect to x between constant limits (Fig. 142). If f (x, y) is continuous, the order of integration can be reversed, b

f (x, y) dA =

f (x, y) dy dx a

R

y2 (x) y1 (x)

d

=

x2 (y)

f (x, y) dx dy c

(223a)

x1 (y)

In polar coordinates, β

F (r, θ ) dA = R

G(ρ, φ, z) dV =

(226)

Integrals Containing a Parameter. If f (x, y) is a continuous function of x and y in the closed rectangle x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 , and if f (x, y) is integrated with respect to x, with y regarded as fixed and called a parameter, then x1

Differentiation under the Integral Sign. If ∂f/∂y is a continuous function of x and y in a closed rectangle, then

dφ = dy

θ2 (r)

(223b)

θ1 (r)

The analogous triple integrals are evaluated by three single integrations. In rectangular coordinates, f (x, y, z) dV =

f (x, y, z) dx dy dz

R

(224)

In spherical coordinates,

x1 x0

∂f (x, y) dx ∂y

d dφ = dy dy dx =

x1 =g1 (y)

f (x, y) x0 =g0 (y)

g1 (y) g0 (y)

dg0 ∂f (x, y) dx − f (g0 , y) ∂y dy dg1 dy

2

F (r, θ, φ)r sin θ dr dθ dφ (225)

Fig. 142

(228)

If ∂f/∂y is continuous and the limits of integration are differentiable functions of y, then

+ f (g1 , y)

R

(227)

is a continuous function of y. Geometrically, the function f (x, y) may be plotted as a surface z = f (x, y). Then the value of φ(yi ) is the area of the section under the surface made by the plane y = yi (Fig. 143). If the limits of integration are continuous functions of y instead of constants, then φ(y) is continuous.

r2 (θ)

F (r, θ )r dθ dr k

f (x, y) dx = φ(y)

x0

r1 (θ) l

=

G(ρ, φ, z)ρ dρ dφ dz

R

F (r, θ )r dr dθ α

F (r, θ, φ) dV =

In cylindrical coordinates,

Fig. 143

(229)

MATHEMATICS

253

If f (x) is integrable in the interval a ≤ x ≤ b and continuous at a point within ) x the interval, then at that point the function F (x) = a f (ξ ) dξ has a derivative F (x) = f (x). Uniform Convergence and Change of Order of Integration. The improper integral

φ(y) =

∞

f (x, y) dx

(230)

x0

converges uniformly in y in the interval y0 ≤ y ≤ y1 if for any ε > 0 there exists an L dependent on ε but not on y such that

∞ l

f (x1 y) dx < ε

for l ≥ L

(231)

)∞ If x0 f (x, y) dx is uniformly convergent for y0 ≤ y ≤ y1 , then y1

∞

f (x, y) dx dy =

x0

y0

∞

y1

f (x, y) dy dx x0

y0

(232)

Fig. 144

A function f (x) defined in the interval (a, b) is measurable if the set of points x for which y0 ≤ f (x) < y1 is measurable for any values of y0 and y1 . Let u and l be the upper and lower bounds of a measurable function f (x) defined in the interval (a, b) (Fig. 144). Divide the interval (u, l ) into n arbitrary subintervals yν by the points y0 = 1, y1 , . . . , yn = u. Let Sν be the set of points for which yν−1 ≤ f (x) < yν and ην any point in the interval yν . Then the Lebesgue integral of f (x) in the interval (a, b) is

Stieltjes Integral. If f (x) and φ(x) are defined in the interval (a, b), the Stieltjes integral of f (x) with respect to φ(x) is

b

f (x) dx =

a

lim

n

n→∞ ν=1 max yν → 0

ην · m(Sν )

(234)

b

f (x) dφ(x) a

=

lim

n

n→∞ ν=1 max xν → 0

f (ξν )[φ(xν ) − φ(xν−1 )] (233)

in which the interval (a, b) is divided into n arbitrary parts xν = xν − xν−1 by the points x0 = a, x1 , . . . , xn = b, and ξν is an arbitrary point in xν . This limit exists if f (x) is continuous and φ(x) is of bounded variation, that is, can be expressed as the difference of two nonincreasing or two nondecreasing bounded functions. However, it is not necessary that f (x) be continuous, but only that the variation of φ(x) over the set of points of discontinuity of f (x) be zero.

If the Riemann integral in the interval (a, b), defined on p. 250, exists, the Lebesgue integral does also, and the two are equal, but not conversely. 10.3

Line, Surface, and Volume Integrals

Line Integrals. Let P (x, y) and Q(x, y) be functions continuous at all points of a continuous curve C joining the points A and B in the xy plane. Divide the curve C into n arbitrary parts sν by the points (xν , yν ), let (ξν , ην ) be an arbitrary point on sν , and let xν and yν be the projections of Sν on the x and the y axes (Fig. 145). The line integral is B

[P (x, y) dx + Q(x, y) dy] =

A

Lebesgue Integral. Let S be a set of points in the interval (a, b), and C(S) the complement of S, that is, the set of all the points of (a, b) that do not belong to S. Enclose the points of S in a set of intervals Iν , ν = 1, 2, 3, . . ., finite or infinite in number, and let the sum of the lengths of the Iν be L. The greatest lower bound of all possible values of L is the exterior measure m(S) of S. The interior measure of S is m(S) = (b − a) − m[C(S)]. If m(S) = m(S), the set S is measurable and its measure is m(S) = m(S).

×

n

lim

n→∞ max xν , yν → 0

[P (ξν , ην ) xν + Q(ξν , ην ) yν ]

(235)

ν=1

If the equation of the curve C is y = f (x), x = φ(y), or the parametric equations x = x(t), y = y(t), the line integral can be evaluated as a definite integral in the one variable x, y, or t, respectively.

254


Fig. 146 Fig. 145

) 1,3

Example 67. Find the value of 0,0 [y dx + (xy − x 2 ) dy] along the paths (a) y = 3x, (b) y 2 = 9x. 2

Area. The area of a region bounded by a closed curve C such that a line parallel to the x or y axis meets C in no more than two points is

A=

(a) Substitute y = 3x, dy = 3 dx and obtain 1 0 1

(x dy − y dx)

(238)

C

The formula can be applied to any region that can be divided by a finite number of lines into regions satisfying the preceding condition.

[9x 2 + (3x 2 − x 2 )3] dx

=

1 2

15x 2 dx = 5

0

(b) Substitute y 2 = 9x, 2y dy = 9 dx, and obtain 3!

" 3 y4 2 3 y y + − dy 9 9 81 0 "3 ! y5 3 1 4 y − =6 = 12 405 0 20

Surface Integrals. Let P (x, y, z ) be a function continuous at all points of a region S (bounded by a simple closed curve) of a surface z = f (x, y) which has a continuously turning tangent plane except possibly at isolated points or lines. Let A be the projection of S on the xy plane. Divide S into arbitrary subregions Sν and let (ξν , ην , ζν ) be an arbitrary point in Sν (Fig. 147). The surface integral is

lim

n

n→∞ ν=1 max diamSν → 0

P (ξν , ην , ζν ) Sν

A line integral in the xyz space B

= [P (x, y, z) dx + Q(x, y, z) dy

A

+ R(x, y, z) dz]

P (x, y, z) dS S

(236)

P (x, y, z) 1 +

= A

∂z ∂x

2

+

∂z ∂y

2 dx dy (239)

is defined similarly. Applications Work. The work done by a constant force F acting on a particle that moves a distance s along a straight line inclined at an angle θ to the force is W = F s cos θ . If the path is a curve C and the force variable, the differential of work is dW = F cos θ ds, where ds is the differential of the path. Then

W =

dW =

F cos θ ds = C

(X dx + Y dy) C

(237) where X and Y are the x and y components of F (Fig. 146).

Fig. 147

MATHEMATICS

255

If α, β, γ are the direction angles of the normal to S, the form of the surface integral analogous to the line integral (235) is

Since P = x Q = y R = z ∂P ∂Q ∂R = 1 = 1 = 1 ∂x ∂y ∂z

(P dy dz + Q dz dx + R dx dy) A

(P cos α + Q cos β + R cos γ ) dS

=

(240)

and

S

Green’s Theorem. Let P (x, y) and Q(x, y) be continuous functions with continuous partial derivatives ∂P /∂y and ∂Q/∂x in a simply connected region R bounded by a simple closed curve C. Then

R

√

a

∂Q ∂P − ∂x ∂y

dx dy =

(P dx + Q dy) (241) C

Stokes’s Theorem. Let P (x, y, z ), Q(x, y, z ), R(x, y, z ) be continuous functions with continuous first partial derivatives, S a region (bounded by a simple closed curve C) of a surface z = f (x, y), continuous with continuous first partial derivatives. Then ! ∂Q ∂R ∂R ∂P − dy dz + − dz dx ∂y ∂z ∂z ∂x S " ∂Q ∂P + − dx dy ∂x ∂y

(P dx + Q dy + R dz)

(242)

C

The signs are such that an observer standing on the surface with head in the direction of the normal will see the integration around C taken in the positive direction. Divergence, or Gauss’s, Theorem. Let P (x, y, z ), Q(x, y, z ), R(x, y, z ) be continuous functions with continuous first partial derivatives. Let V be a region in the xyz space bounded by a closed surface S with a continuously turning tangent plane except possibly at isolated points or lines. Then ∂Q ∂R ∂P + + dx dy dz ∂x ∂y ∂z V

=

(P dy dz + Q dz dx + R dx dy)

(243)

S

)) Example 68. Evaluate (x dy dz+y dz dx +z dx dy) over the cylinder x2 + y2 = a2

z = ±b

−

√

+b −b

a 2 −x 2

3 dz dy dx = 6πa 2 b

Independence of Path and Exact Differential. Under the conditions of Green’s theorem, the following statements are equivalent:

)

C (P dx + Q dy) = 0 for any closed curve C in the region R. ) (ξ,η) 2. The value of (a,b) (P dx + Q dy) is independent of the curve connecting (a, b) and (ξ, η), any points in R. 3. ∂P /∂y = ∂Q/∂x at all points of R. 4. There exists a function F (x, y) such that dF = P dx + Q dy.

1.

A region is simply connected if any closed curve in the region can be shrunk to a point without passing outside the region.

=

−a

a 2 −x 2

Under the conditions of Stokes’s theorem, the corresponding statements for three dimensions are: )

C (P dx + Q dy + R dz) = 0 for any closed curve C in the region S. ) (ξ,η,ζ ) 2. The value of (a,b,c) (P dx + Q dy + R dz) is independent of the curve connecting (a, b, c) and (ξ, η, ζ ), any points in S. 3. ∂P /∂y = ∂Q/∂x, ∂Q/∂z = ∂R/∂y, ∂R/∂x = ∂P /∂z at all points of S. 4. There exists a function F (x, y, z ) such that dF = P dx + Q dy + R dz.

1.

10.4 Applications of Integration Length of Arc of a Curve. The length s of the arc of a plane curve y = f (x) from the point (a, b) to the point (c, d ) is c

s=

1+

a

dy dx

2

d

dx =

1+

b

dx dy

2 dy

(244) If the equation of the curve is in polar coordinates, r = f (θ ), then the length of the arc from the point (r1 , θ1 ) to the point (r2 , θ2 ) is s=

θ2

r2

θ1

+

dr dθ

2 dθ =

r2 r1

1 + r2

dθ dr

2 dr (245)

256


If the curve is in three dimensions, represented by the equations y = f1 (x), z = f2 (x), the length of arc from x1 = a to x2 = b is 2 2 b dy dz s= 1+ + dx (246) dx dx a

If expressed in polar coordinates, the area by double integration is dθ

A=

f (x) dx

(247)

a

where y has the same sign for all values of x between a and b. In polar coordinates, the area bounded by the curve r = f (θ ) and the two radii θ = α, θ = β (Fig. 148) is A=

1 2

β

r 2 dθ

(248)

α

In rectangular coordinates, if the area is bounded by the two curves y2 = f (x), y1 = φ(x) and the lines x2 = b, x1 = a (Fig. 149), then b

A= a

dy

r dr

A=

ψ(y)

dy c

dx

(250)

or (251)

Area of a Surface Revolution. The area of the surface of a solid of revolution generated by revolving the curve y = f (x) between x = a and x = b is given as 2 b dy y 1+ dx about the x axis (252) 2π dx a 2 d dx 2π x 1+ dy about the y axis (253) dy c

where c = f (a) and d = f (b). Volume.

φ(x)

d

dθ θ1 =φ1 (r)

r1

(249)

If the area is bounded by the two curves x2 = ψ(y), x1 = ξ(y) and the lines y2 = d, y1 = c, then

r dr

θ2 =φ2 (r)

r2

f (x)

dx

r1 =f1 (θ)

θ1

Plane Area. The area bounded by the curve y = f (x), the x axis, and the ordinates at x = a, x = b is b

r2 =f2 (θ)

θ2

A=

V =

By triple integration,

           

dx dy dz

(rectangular coordinates) (254)

2

r sin θ dθ dφ dr (spherical coordinates)

          

(255) ρ dρ dφ dz

(cylindrical coordinates) (256)

ξ(y)

(the limits of integration to be supplied). Volume of a Solid of Revolution. The volume of a solid of revolution generated by revolving the region bounded by the x axis and the curve y = f (x) between x = a and x = b is b

π

y 2 dx

about the x axis

(257a)

x 2 dy

about the y axis

(257b)

a d

Fig. 148

π c

where c = f (a) and d = f (b).

Fig. 149

Surfaces. If the equation of a surface is written in the parametric form x = f1 (u, v), y = f2 (u, v), z = f3 (u, v), the length of arc of a curve u = u(t), v = v(t) on the surface is 2 2 du dv dv du +G E + 2F dt s= dt dt dt dt (258)

MATHEMATICS

257

for a plane curve about the x and y axis and about the origin, respectively;

The area S of a region on the surface is

S= where

E=

∂x ∂u

2

EG − F 2 du dv

+

∂y ∂u

2

+

∂z ∂u

(259) Ix =

y 2 dA Iy =

∂y ∂y ∂z ∂z ∂x ∂x + + ∂u ∂v ∂u ∂v ∂u ∂v 2 2 2 ∂y ∂z ∂x + + G= ∂v ∂v ∂v If the equation of the surface is written as x = u, y = v, z = f (u, v) = f (x, y), the arc length is given as s=

(1 + p2 )

dx dt

2

+ 2pq

dx dy dy + (1 + q 2 ) dt dt dt

dt

and the area as

p=

1 + p 2 + q 2 dx dy

∂z ∂x

q=

y 2 dm

Ixy =

z2 dm

Mxy =

y=b

(266)

z dm

b

ρy dA =

y=a

Moment. The moments of a mass m about the yz, xz, and xy planes are respectively

y dm

Ixz =

(261)

(the limits of integration to be supplied).

x dm Mxz =

x 2 dm

for a solid of mass m about the yz, xz, and xy plane and about the x axis, respectively (the limits of integration to be supplied).

F =

Myz =

Iyz =

Fluid Pressure. The total force F against a plane surface perpendicular to the surface of the liquid and between the depths a and b is

where

∂z ∂y

(x 2 + y 2 ) dA

Ix = Ixz + Ixy , etc.

2 (260)

S=

I0 =

(265) for a plane area about the x and y axis and about the origin, respectively; and

2

F =

x 2 dA

ρyx dy

(267)

a

where ρ is the weight of the liquid per unit volume and y is the depth beneath the surface of the liquid of a horizontal element of area dA. Usually, dA = x dy, where x is the width of the vertical surface expressed as a function of y.

(262) (the limits of integration to be supplied). Center of Gravity. The coordinates of the center of gravity of a mass m are

) x dm x= ) dm

) y dm y= ) dm

) z dm z= ) (263) dm

Center of Pressure. The depth y of the center of pressure against a surface perpendicular to the surface of the liquid and between the depths a and b is y=b

ρy 2 dA y=

y=a y=b

(268) ρy dA

y=a

(the limits of integration to be supplied). Moment of Inertia.

Ix =

y 2 ds

Iy =

The moments of inertia I are x 2 ds

I0 =

(x 2 + y 2 ) ds (264)

Work. The work W done in moving a particle from s = a to s = b against a force whose component expressed as a function of s in the direction of motion is F (s) is

W =

s=b

F (s) ds s=a

(269)

258


11

For the special case where P is a function of x only and Q a function of y only,

DIFFERENTIAL EQUATIONS

11.1

Definitions

A differential equation is an equation containing an unknown function of a set of variables and its derivatives. If the equation has derivatives with respect to one variable only, it is an ordinary differential equation, otherwise it is a partial differential equation.

P (x) dx + Q(y) dy = 0 the variables are separated. The solution is P (x) dx +

Example 69

d 2y + k2 y = 0 dx 2 d 2y dy = 1 + y2 + 2 dx dx y

∂ 2z ∂z ∂2z − = xyz + zx 2 ∂x ∂x ∂y ∂y y−x

dx dy +3 =0 dx dy

(270)

Solve

(272)

This can be written as x dx + y dy = 0 and has the solution

(273)

Separation of Variables. A differential equation of the first order,

(274)

x dx +

y dy = 12 x 2 + 12 y 2 = c

If c = r 2 /2, then x 2 + y 2 = r 2 , a set of concentric circles. There are an infinite number of solutions depending on the value of r. Through each point in the plane there passes one circle and only one. Homogeneous Equations. A function f(x, y) is homogeneous of the nth degree in x and y if f (kx, ky) = k n f (x, y). An equation

P (x, y) dx + Q(x, y) dy = 0

(278)

is homogeneous if the functions P (x, y) and Q(x, y) are homogeneous in x and y. By substituting y = vx, the variables can be separated. Example 71. Solve (x 2 + y 2 ) dx − 2xy dy = 0. This is of the form P (x, y) dx + Q(x, y) dy = 0, where P and Q are homogeneous functions of the second degree. Making the substitution y = vx, the equation becomes (1 + v 2 ) dx − 2v(x dv + v dx) = 0. Separating variables,

2v dx − dv = 0 x 1 − v2 Integrating, loge x(1 − v 2 ) = loge c; replacing v = y/ x, log(1 − y 2 /x 2 )x = loge c; and taking exponentials, x 2 − y 2 = cx. Linear Differential Equation. The differential equation

can be brought into the form P (x, y) dx + Q(x, y) dy = 0

(277)

x dy =− dx y

First-Order Equations

dy =0 f x, y, dx

Example 70.

Q(y) dy = c

(271)

Equations (270), (271), and (273) are ordinary differential equations and (272) is a partial differential equation. The order of a differential equation is the order of the highest derivative involved. Thus in Eqs. (270)– (272), the order is 2; in (273), the order is 1. The degree of a differential equation is the exponent of the highest order appearing in the equation after it is rationalized and cleared of fractions with respect to the derivatives. The degree of (270), (272), and (273) is 1; that of (271) is 2. A solution or integral of a differential equation is a relation among the variables that satisfies the equation identically. A general solution of an ordinary differential equation of the nth order is one that contains n independent constants. Thus, y = sin x + c is a general solution of the equation dy/dx = cos x. A particular solution is one that is derivable from a general solution by assigning fixed values to the arbitrary constants. Thus, y1 = sin x, y2 = sin x + 4 are two particular solutions of the preceding equation. 11.2

(276)

(275)

dy + P (x)y = Q(x) dx

(279)

MATHEMATICS

259

in which y and dy/dx appear only in the first degree and P and Q are functions of x is a linear equation of the first order. This has the general solution y=e

) − p(x) dx

!

)

Q(x)e

is an exact differential equation if its left side is an exact differential du = P dx + Q dy

" dx + c

P (x) dx

(280)

that is, if ∂P /∂y = ∂Q/∂x. Then, !

An equation in the theory of electric

Example 72. networks is

L

(284)

P dx +

di + Ri = E dt

Q−

∂

)

P dx ∂y

" dy = c

(285)

is a solution. where i is the current, L the inductance (a constant), R the resistance (a constant), and E the electromotive force, a function of time or constant. If E = E(t), i=e

−(R/L)t

!

E (R/L)t e dt + c L

"

Example 74. Solve (x 2 − 4xy − y 2 ) dx + (y 2 − 2 2xy − 2x ) dy = 0. This is an exact equation because ∂P /∂y = −4x − 2y = ∂Q/∂x,

(x 2 − 4xy − y 2 ) dx = 13 x 3 − 2x 2 y − xy 2

If E is constant and if i = 0 at t = 0, then

[(y 2 − 2xy − 2x 2 ) − (−2x 2 − 2xy)] dy = 13 y 3

E i = (1 − e−(R/L)t ) R

The general solution is Bernoulli Equation.

This is 1 3 3x

dy + P (x)Y = Q(x)y n dx

(281)

in which n = 1. By making the substitution z = y 1−n , a linear equation is obtained and the general solution is y = e−

)

! P (x) dx

(1 − n)

e(1−n)

)

"1/1−n P (x) dx

Solve the equation dy − xy = xy 2 dx

Substitute z = y −1 and obtain dz/dx + xz = −x. The general integral is z = ce−x

2 /2

−1

or

Exact Differential Equation.

Integrating Factor. If the left member of the differential equation P (x, y) dx + Q(x, y) dy = 0 is not an exact differential, look for a factor v(x, y) such that du = v(P dx + Q dy) is an exact differential. Such an integrating factor satisfies the equation

Q(x) dx + c (282)

Example 73.

y=

Q

∂v ∂v −P + ∂x ∂y

∂Q ∂P − ∂x ∂y

v=0

This is

dy + P (x)y 2 + Q(x)y + R(x) = 0 dx

The equation (283)

(286)

Example 75. The equation (xy 2 − y 3 ) dx + (1 − xy 2 ) dy = 0 when multiplied by v = 1/y 2 becomes (x − y) dx + (1/y 2 − x) dy = 0, of which the left side du = (x − y) dx + (1/y 2 − x) dy is an exact differential since ∂P /∂y = ∂Q/∂x. The integration gives u = x 2 /2 − xy − 1/y. The general solution is u = c or x 2 y − 2xy 2 − 2cy − 2 = 0. Riccati’s Equation.

1 2 ce−x /2 −1

P (x, y) dx + Q(x, y) dy = 0

− 2x 2 y − xy 2 + 13 y 3 = c

(287)

If a particular integral y1 is known, place y = y1 + 1/z and obtain a linear equation in z.

260

11.3


Let p = dy/dx. Then

Second-Order Equations

The differential equation F x, y,

dy d 2 y , dx dx 2

a

=0

(288) By separating variables,

is of the second order. If some of these variables are missing, there is a straightforward method of solution. Case 1: With y and dy/dx missing, d 2y = f (x) dx 2

dx

(289)

f (x) dx + cx + c1

d 2y = f (y) dx 2

d 2y =f dx 2

x=

dy dx

x+c + c1 a

d 2y =f dx 2

(292)

dy ,x dx

(295)

y=

p(x) dx + c

(296)

Case 5: With x missing, d 2y =f dx 2

d 2y dp = dx 2 dx

(293)

dy ,y dx

(297)

Place dy/dx = p and obtain the first-order equation p dp/dy = f (p, y). If this can be solved for p, then (294)

dp +c f (p)

Solve for p, replace p by dy/dx, and solve the resulting first-order equation. Example 76. nary is

dy x+c = sinh dx a

Place dy/dx = p and obtain the first-order equation dp/dx = f (p, x). If this can be solved for p, then

as a solution. Case 3: With x and y missing,

Then

p=

or

Case 4: With y missing, (291)

dy + c1 ) c + 2 f (y) dy

dy =p dx

x+c a

y = a cosh

(290)

Multiply both sides by 2 dy/dx and obtain

Place

sinh−1 p =

Integrating this latter,

Case 2: With x and dy/dx missing,

x=

dx dp = 2 a 1+p which has the solution

This has the solution y=

dp = 1 + p2 dx

The differential equation of the cate 2 d 2y dy a 2 = 1+ dx dx

x=

dy +c p(y)

(298)

11.4 Bessel Functions

Wherever the mathematics of problems having circular or cylindrical symmetry appears, it is usually appropriate to consider the solutions of Bessel’s differential equation (299). Such applications include radiation from a cylindrical antenna, eddy current losses in a cylindrical wire, and sinusoidal angle modulations including phase and frequency modulation, x2

dy d 2y + (x 2 − n2 )y = 0 +x dx 2 dx

(299)

MATHEMATICS

261

A solution of this equation is yk = cerk x

(303)

if rk is a root of the algebraic equation, r n + a1 r n−1 + · · · + an−1 r + an = 0

(304)

If all the n roots r1 , r2 , . . . , rn of (304) are different, then y = c1 er1 x + c2 er2 x + · · · + cn ern x

(305)

Fig. 150 Bessel functions of first kind.

where n is real, possibly integral or fractional, or complex, and the solution y(x) is said to be of the first kind and denoted Jn (x) for 0 ≤ n an integer. Tables of J0 (x) and J1 (x) are available in Table 22 in Chapter 1. Graphs of these are shown in Fig. 150. Bessel functions Jn (x) are almost periodic functions that for increasing x have a zero-crossing half “period” approaching π from below. A sequence of these functions can be used to construct an orthogonal series much in the same way that periodic functions, sine, and cosine waves make up a Fourier series. For an extensive set of tables of Bessel functions of many types, Essentials of Mathematical Methods in Science and Engineering, by S. S. Bayin (Wiley, Hoboken, New Jersey, 2008).

is a general solution of (302). If k of the roots are equal, r1 = r2 = · · · = rk while rk+1 , . . . , rn are different, then y = (c1 + c2 x + · · · + ck x k−1 )er1 x + ck+1 erk+1x + · · · + cn ern x

is a general solution. If r1 = p + iq, r2 = p − iq are conjugate complex roots of (304), then c1 er1 x + c2 er2 x = epx (C1 cos qx + C2 sin qx) (307) Example 77

dy d 2y + 40y = 0 + 13 dx 2 dx

11.5 Linear Equations General Theorem.

has the solution y = c1 e−5x + c2 e−8x .

The differential equation

d n−1 y dy d ny + Pn (x)y + P (x) + · · · + Pn−1 (x) 1 n n−1 dx dx dx = F (x) (300) is called the general nth-order linear differential equation. If F (x) = 0, the equation is homogeneous; otherwise it is nonhomogeneous. If φ(x) is a solution of the nonhomogeneous equation and y1 , y2 , . . . , yn are linearly independent solutions of the homogeneous equation, then the general solution of (300) is y = c1 y1 + c2 y2 + · · · + cn yn + φ(x)

(306)

(301)

The part φ(x) is called the particular integral, and the part c1 y1 + · · · + cn yn is the complementary function. Homogeneous Differential Equation with Constant Coefficients

d ny d n−1 y dy + an y = 0 (302) + a1 n−1 + · · · + an−1 n dx dx dx

Example 78

dy d 2y + 34y = 0 +6 dx 2 dx has the solution y = (c1 cos 5x + c2 sin 5x)e−3x . Nonhomogeneous Differential Equation with Constant Coefficients

d ny d n−1 y dy + an y = F (x) + a1 n−1 + · · · + an−1 n dx dx dx (308) The complementary function is found as previously. To find the particular integral, replace dy by D, dx

d 2y by D 2 , . . . , dx 2

d ny by D n dx n (309)

P (D)y = (D n + a1 D n−1 + · · · + an−1 D + an )y = F (x)

(310)

262


Particular integrals yp , in which Bi , A, B are undetermined coefficients, to be determined by substituting yp in (310) and equating coefficients of like terms, are: (a) If F (x) = x n + b1 x n−1 + · · · + bn−1 x + bn , then yp = x n + B1 x n−1 + · · · + Bn−1 x + Bn . If D m is a factor of P (D), then yp = (x n + B1 x n−1 + · · · + Bn−1 x + Bn )x m . (b) If F (x) = b sin ax or b cos ax, then yp = A sin ax + B cos ax. If (D 2 + a 2 )m is a factor of P (D), then yp = (A sin ax + B cos ax)x m . (c) If F (x) = ceaxam then yp = Aeaxam . If (D − a)m is a factor of P (D), then yp = x m Aeax . (d) If F (x) = g(x)eaxam , place yp = eax w in (308), divide out eax , and solve the equation for wp as a function of x. (e) If F (x) is the sum of a number of these functions, then yp is the sum of the particular integrals corresponding to each of the functions. (f) If F (x) is not of the type (e), try the method of Laplace transformation (Section 13). Example 79. d 2 y/dx 2 + 4y = x 2 + cos x can be written as (D 2 + 4)y = (D + 2i)(D − 2i) = x 2 + cos x. By (307), the complementary function is y = c1 cos 2x + c2 sin 2x. For a particular integral take yp = ax 2 + bx + c + f sin x + g cos x [by (a), (b), (e)]. Then d 2 yp = 2a − f sin x − g cos x dx 2

" ! y d d d x −1 − 2 y, . . . = dx 3 dt dt dt 3d

3

(311) is transformed into a linear homogeneous differential equation with constant coefficients. Depression of Order. If a particular integral of a linear homogeneous differential equation is known, the order of the equation can be lowered. If y1 is a particular integral of

d n−1 y dy d ny + Pn (x) = 0 + P (x) + · · · + Pn−1 (x) 1 dx n dx n−1 dx (313) substitute y = y1 z. The coefficient of z will be zero, and then by placing dz/dx = u, the equation is reduced to the (n − 1)st order.

= x 2 + cos x Equating coefficients, a = 14 , b = 0, c = − 81 , f = 0, g = 13 and the general solution is y = c1 cos 2x + c2 sin 2x + x 2 /4 − 18 + 13 cos x. Euler’s Homogeneous Equation n−1 dy d ny y n−1 d + an y = 0 + ax + · · · + an−1 x dx n dx n−1 dx (311) Place x = et , and since

xn

x

dy dy = dx dt

x2

" ! d 2y d d − 1 y = dx 2 dt dt

Given

Example 80.

dy d 2y + q(x)y = 0 + p(x) dx 2 dx and y1 , a particular integral of this equation. Let y = y1 z. Then dy1 dy dz = y1 +z dx dx dx d 2 y1 dy1 dz d 2z d 2y +z 2 = y1 2 + 2 2 dx dx dx dx dx Substituting in the original equation

and substituting in the original equation d 2 yp + 4yp = 2a − f sin x − g cos x + 4ax 2 dx 2 + 4bx + 4c + 4f sin x + 4g cos x

(312)

y1

d 2 y1 dy1 dz d 2z +z 2 +2 2 dx dx dx dx " ! dy1 dz +z + qy1 z = 0 + p y1 dx dx

and since the coefficient of z is zero, this reduces to dy1 d 2z dz + py1 =0 y1 2 + 2 dx dx dx Writing du dy1 dx + 2 + py1 =0 u dx y1

dz =u dx By integrating, loge u +

p dx + loge y12 = loge c

u=

c exp − y12

P dx

or

MATHEMATICS

263

Any system of linear equations in which all fk are zero is called homogeneous. Consider the following equations associated with the matrix operator A:

Another integration gives z. Then c exp − y12

y = y1

P dx

dx + c1

Aα = 0 (homogeneous equation) ∗

Systems of Linear Differential Equations with Constant Coefficients. For a system of n linear equations with constant coefficients in n dependent variables and one independent variable t, the symbolic algebraic method of solution may be used. If n = 2,

(D n + a1 D n−1 + · · · + an )x + (D + b1 D m

(D + c1 D p

p−1

m−1

+ · · · + bm )y = R(t)

+ · · · + cp )x

(314)

where D = d/dt. The equations may be written as P1 (D)x + Q1 (D)y = R

P2 (D)x + Q2 (D)y = S (315) Treating these as algebraic equations, eliminate either x or y and solve the equation thus obtained. Solve the system,

(a) (b)

dy dx + + 2x + y = 0 dt dt dy + 5x + 3y = 0 dt

By using the symbol D these equations can be written (D + 2)x + (D + 1)y = 0

5x + (D + 3)y = 0

Eliminating x, (D 2 + 1)y = 0. From (307) (a) this has the solution y = c1 cos t + c2 sin t. Substituting this in (b), x=−

3c1 + c2 c1 − 3c2 cos t + sin t 5 5

11.6 Linear Algebraic Equations

Consider the set of linear algebraic equations n

aki αi = fk

(k = 1, 2, . . . , m)

(i = 1, 2, . . . , n)

Equation (316) contains m linear algebraic equations in n unknowns, αi .

(319)

where (·, ·) denotes the inner product in Euclidean space and Ai are the column vectors of the matrix [A]. From Eqs. (316) and (319), we deduce the following result, known as the solvability condition: The nonhomogeneous equation Aα = f possesses a solution α if and only if the vector f is orthogonal to all vectors β that are the solutions of the homogeneous adjoint equation, A∗ β = 0. In analytical form this statement can be expressed as (f,β) = 0

(320)

We now consider two cases of linear equations and discuss the existence and uniqueness of solutions of linear equations. 1. If (317) has only the trivial (i.e., zero) solution, it follows that det A = 0 (otherwise, the trivial solution cannot be determined) and hence det A∗ = 0. Therefore, the adjoint homogeneous equation (318) also has only the trivial solution. Moreover, the solvability conditions are automatically satisfied for any f [since the only solution of (318) is β = 0], and the nonhomogeneous equation 316 has one and only one solution, α = A−1 f, where A−1 is the inverse of the matrix A. 2. If (317) has nontrivial solutions, then det A = 0. This in turn implies that the rows (or columns) of A are linearly dependent. If these linear dependencies are also reflected in the column vector f (e.g., if the third row of A is the sum of the first and second rows, we must have f3 = f1 + f2 in order to have any solutions), then there is a hope of having a solution to the system. If there are r(≥ n) number of independent solutions to (316), A is said to have a r-dimensional null space (i.e., nullity of A is r). It can be shown that A∗ also has a r-dimensional null space, which is in general different from that of A. A necessary and sufficient condition for (316) to have solutions is provided by the solvability condition

(316)

i=1

(318)

where A∗ is the adjoint of A. For the linear algebraic equations, A∗ = AT , the transpose of A. The homogeneous adjoint equations can also be written in the form (Ai , β) = 0

+ (D q + d1 D q−1 + · · · dq )y = S(t)

Example 81.

A β = 0 (adjoint homogeneous equation)

(317)

(f,β) ≡

n

fi βi = 0

i=1

where β is the solution of Eq. (318).

264


Example 82.

This example has three cases:

1. Consider the following pair of equations in two unknowns α1 and α2 : 3α1 − 2α2 = 4 or ! 3 2

−2 1

2α1 + α2 = 5

"* + * + α1 4 α2 = 5

(Aα = f)

We note that det A = 3 + 4 = 7 = 0. The solution is then given by * + # 1 α1 7 α2 = − 2 7

2 7 3 7

$* + * + 4 2 5 = 1

The solution of the adjoint equations is trivial, β = 0, and therefore, the solvability condition is identically satisfied. 2. Next consider the pair of equations 6α1 + 4α2 = 4 or

! 6 3

3α1 + 2α2 = 2

"* + * + 4 α1 4 2 α2 = 2

Aα = f

α (1) = (2, −2) Note that there are many dependent solutions to the pair. For example, (2, −2), (4, −5), (−2, 4), and so on, are solutions of Aα = f . The solution to the adjoint homogeneous equation "* + * + 0 3 β1 2 β2 = 0

is given by β2 = −2β1 . Note that (f, β) ≡ f1 β1 + f2 β2 = 4(− 21 β2 ) + 2β2 = 0; hence the solvability condition is satisfied. 3. Finally, consider the pair of equations 6α1 + 4α2 = 3 or

!

6 4 3 2

3α1 + 2α2 = 2

"* + * + α1 3 α2 = 2

Geometrically, we can interpret these three pairs of equations as pairs of straight lines in R 2 with αi = xi , i = 1, 2 (see Fig. 151). In part 1, the lines represented by the two equations intersect at the point (x1 , x2 ) = (2, 1). In part 2, the lines coincide, or intersect, at an infinite number of points, and hence many solutions exist. In part 3, the lines do not intersect at all showing that no solutions exist. From this geometric interpretation, one can see that the lines are nearly parallel (i.e., the angle θ is nearly zero), the determinant of A is nearly zero [because tan θ = (a11 a22 − a12 a21 )/(a11 a21 + a12 a22 )], and therefore it is difficult to obtain an accurate numerical solution. In such cases the system of equations is said to be ill conditioned. While these observations can be generalized to a system of n equations, the geometric interpretation becomes complicated. Example 83.

This example has two cases:

1. Consider the following set of three equations in three unknowns: α1 + α2 + α3 = 2

We have det A = 0, because row 1 (R1 ) is equal to 2 times row 2 (R2 ). However, we also have 2f2 = f1 . Consequently, we have one linearly independent solution, say α (1) , and the other depends on α (1) :

! 6 4

We note that det A = 0, because 2R2 = R1 . However, 2f2 = f1 . Hence the pair of equations is inconsistent, and therefore no solutions exist.

α1 − α2 − 3α3 = 3

or

Aα = f

3α1 + α2 − α3 = 1 The adjoint homogeneous equations become β1 + β2 + 3β3 = 0 β1 − β2 + β3 = 0

or

A∗ β = 0

β1 − 3β2 − β3 = 0 Solving for β, we obtain β1 = 2β2 = −2β3 . Hence, the null space of A∗ is defined by N (A∗ ) = {(2a, a, −a), a is a real number} Clearly N (A∗ ) is one dimensional. The null space of A is given by N (A) = {(a, −2a, a), a is a real number} Note that N (A∗ ) = N (A), but their dimension is the same. Clearly (2, 1, −1) is a solution of A∗ β = 0 while (1, −2, 1) is a solution of Aα = 0. The solvability condition gives (f, β) = 2 × 2 + 3 × 1 + 1 × (−1) = 0 and therefore Aα = f has no solution. 2. Reconsider the preceding linear equations with f = {−1, 3, 1}T . Then the solvability condition

MATHEMATICS

265

Fig. 151 Geometric interpretation of the solution of two simultaneous algebraic equations in a plane: (a) unique solution; (b) many solutions; (c) no solution.

is clearly satisfied. Hence there is one linearly independent solution to Aα = f (note that −2R1 + R3 = R2 and −2f1 + f3 = f2 ). α ≡ (α1 , α2 , α3 ) = (1, −2, 0) Only one of the three α’s is arbitrary (not determined), and the remaining two α’s are given in terms of the arbitrary α. For example, if α 1 is arbitrary, we have

Definition. If x1 , x2 , . . . , xn are n independent variables, z = z(x1 , x2 , . . . , xn ) the dependent variable, and

∂z ∂z = p1 , . . . , = pn ∂x1 ∂xn

or

Aα = f

3α1 + α2 − α3 = 3 We have det A = 0. It can be easily verified that (A) = (A∗ ) = {(0, 0, 0)}. The unique solution to Aα = f is given by α1 = 34 α2 = 12 α3 = − 14 .

(321)

then F (x1 , x2 , . . . , xn , z, p1 , p2 , . . . , pn ) = 0

α1 + α2 + α3 = 1 α1 + α2 − 3α3 = 2

11.7 Partial Differential Equations First Order

(322)

is a partial differential equation of the first order. An equation f (x1 , x2 , . . . , xn , z, c1 , . . . , cn ) = 0

(323)

with n independent constants is a complete integral of (322) if the elimination of the constants by partial differentiation gives the differential equation (322).

266


Example 86. Given mx − ly. From

Example 84

# F = z2

∂z ∂x

2

+

∂z ∂y

$

2

+ 1 − c2 = 0

(ny − mz)p + (lz − nx)q =

dy dz dx = = ny − mz lz − nx mx − ly

Then f = (x − h)2 + (y − k)2 + z2 − c2 = 0 is a solution, since by differentiating it with respect to x and y, (x − h) + z

∂z =0 ∂x

(y − k) + z

∂z =0 ∂y

and substituting the values of x − h, y − k from the last two equations in f , expression F is obtained. If the eliminant obtained by eliminating c1 , . . . , cn from the equations f = 0, ∂f/∂c1 = 0, . . . , ∂f/∂cn = 0 satisfies the differential equation, it is a singular solution. This differs from a particular integral in that it is usually not obtainable from the complete integral by giving particular values to the constants. Suppose that the equation ∂z ∂z , =0 F x, y, z, ∂x ∂y has the complete integral f (x, y, z, a, b) = 0. Let one of the constants b = φ(a); then f [x, y, z, a, φ(a)] = 0. The general integral is the set of solutions found by eliminating a between f [x, y, z, a, φ(a)] = 0 and dφ/da = 0 for all choices of φ. Linear Differential Equations

P (x, y, z)p + Q(x, y, z)q = R(x, y, z) p=

∂z ∂x

q=

∂z ∂y

General Method of Solution. Given F (x, y, z, p, q) = 0, the partial differential equation to be solved. Since z is a function of x and y, it follows that dz = p dx + q dy. If another relation can be found among x, y, z, p, q, such as f (x, y, z, p, q) = 0, then p and q can be eliminated. The solution of the ordinary differential equation thus formed involving x, y, z will satisfy the given equation, F (x, y, z, p, q) = 0. The unknown function f must satisfy the following linear partial differential equation:

∂F ∂f ∂F ∂F ∂f ∂F ∂f + + p +q ∂p ∂x ∂q ∂y ∂p ∂q ∂z ∂F ∂f ∂F ∂f ∂F ∂F +p − +q =0 − ∂x ∂z ∂p ∂y ∂z ∂q (325) which is satisfied by any of the solutions of the system ∂x ∂y dz = = ∂F /∂p ∂F /∂q p ∂F /∂p + q ∂F /∂q −dq −dp = ∂F /∂x + p ∂F /∂z ∂F /∂y + q ∂F /∂z (326)

=

where (324)

is a linear partial differential equation. From the system of ordinary equations dy dz dx = = P Q R the two independent solutions u(x, y, z) = c1 , v(x, y, z) = c2 are obtained. Then (u, v) = 0, where is an arbitrary function, is the general solution of Pp + Qq = R. Example 85. Given xp + yq = z. The system dx/x = dy/y = dz/z has the solution u = y/x = c1 , v = z/x = c2 . Then the general solution is

y z , =0 (u, v) = x x

by using the multipliers l, m, n and adding the fraction (l dx + m dy + n dz)/0 is obtained. Therefore l dx + m dy + n dz = 0. This has the solution lx + my + nz = c1 . Similarly, x dx + y dy + z dz = 0, or x 2 + y 2 + z2 = c2 . Then the general solution is (x 2 + y 2 + z2 , lx + my + nz) = 0.

Example 87. Solve p(q 2 + 1) + (b − z)q = 0. Here Eqs. (326) reduce to

dp dx dp dz = 2 = = 2 2 pq q 3pq + p + (b − z)q q +1 =

dy −z + b + 2pq

The third fraction, by virtue of the given equation, reduces to dz/2pq 2 . From the first two fractions, by integration, q = cp. This and the original equation determine the values of p and q, namely, √ p=

c1 (z − b) − 1 c1

q=

c1 (z − b) − 1

MATHEMATICS

267

Substitution of these values in dz = p dx + q dy gives dx dz = + dy c1 (z − b) − 1 c1

An example is the equation of a vibrating string,

In this equation the variables are separable; this on inte√ gration gives the complete integral 2 c1 (z − b) − 1 = x + c1 y + c2 . There is no singular solution. In this work, had another pair of ratios been chosen, say dq/q 2 = dx/(q 2 + 1), another complete integral would have been obtained, namely, * + ' (2 1 (z − b) 12 (x + k1 ) − (x + k ) + 1 1 2

where z is the transverse displacement of a point on the string, with abscissa x at time t and a 2 is constant. If B 2 − AC < 0, the equation is of the elliptic type that has the normal form

+ y + k2 = 0

Definitions. A linear partial differential equation of the second order with two independent variables is of the form

L = Ar + 2Bs + Ct + Dp + Eq + F z = f (x, y) (327) where ∂ 2z ∂x 2 p = ∂z/∂x

s = ∂ 2 z/(∂x ∂y)

∂z ∂2z ∂z ∂ 2z +b + cz = 0 + 2 +a 2 ∂ξ ∂z ∂ξ ∂η

(332)

An example is Laplace’s equation ∂2z ∂2z + =0 ∂ξ 2 ∂η2

Second Order

r=

2 ∂2z 2∂ z = a ∂t 2 ∂x 2

t = ∂ 2 z/∂y 2

q = ∂z/∂y

The coefficients A, . . . , F are real continuous functions of the real variables x and y. Let ξ = ξ(x, y), η = η(x, y) be two solutions of the following homogeneous partial differential equation of the first order: Ap 2 + 2Bpq + Cq 2 = 0

(328)

If B 2 − AC = 0, the homogeneous form of (327), L = 0, is called the parabolic type and has the normal form ∂z ∂z ∂ 2z +b + cz = 0 +a ∂ξ 2 ∂ξ ∂η

(329)

where a, b, c are functions of ξ and η. An example is the equation of heat flow, ∂u/∂t = a 2 ∂ 2 u/∂t 2 , where u = u(x, t) is the temperature, t is the time, a 2 is constant. If B 2 − AC > 0 in (328), the homogeneous form of (327) is the hyperbolic type that has as its two normal forms ∂z ∂z ∂2z +a +b + cz = 0 ∂ξ ∂η ∂ξ ∂η

(330)

∂z ∂ 2z ∂z ∂2z +b + cz = 0 − 2 +a 2 ∂ξ ∂η ∂ξ ∂η

(331)

usually written ∇ 2 z = 0. The two solutions of (328) are real in the hyperbolic case and conjugate complex in the elliptic case. That is, in the latter case, ξ = 12 (α + iβ), η = 12 (α − iβ), where α and β are real, and 1 ∂2z ∂2z ∂ 2z = + ∂ξ ∂η 4 ∂α 2 ∂β 2 As in ordinary linear equations, the whole solution consists of the complementary function and the particular integral. Also, if z = z1 , z = z2 , . . . , z = zn are solutions of the homogeneous equation (327), L = 0, then z = c1 z1 + c2 z2 + · · · + cn zn is again a solution. Equations Linear in the Second Derivatives. The general type of second-order equation linear in the second derivatives may be written in the form

Ar + Bs + Ct = V

(333)

where A, B, C, V are functions of x, y, z, p, q. From the equations A dy 2 − B dx dy + C dx 2 = 0

(334)

A dp dy + C dq dx − V dx dy = 0

(335)

p dx + q dy = dz

(336)

it may be possible to derive either one or two relations between x, y, z, p, q, called intermediary integrals, and from these to deduce the solution of (333). To obtain an intermediary integral, resolve (334), supposing the left member is not a perfect square, into the two equations dy − n1 dx = 0, dy − n2 dx = 0. From the first of these and from (335) combined, if necessary, with (336), obtain the two integrals u1 (x, y, z, p, q) = a, v1 (x, y, z, p, q) = b; then u1 = f1 (v1 ), where f1 is an arbitrary function, is now an intermediary integral.

268


In the same way, from dy − n2 dx = 0, obtain another pair of integrals u2 = a1 , v2 = b1 ; then u2 = f2 (v2 ) is an intermediary integral. For the final integral, if n1 = n2 , the intermediary integral may be integrated. If n1 = n2 , solve the two intermediary integrals for p and q, substitute in p dx + q dy = dz, and integrate for the solution. Solve

Example 88.

r 2 − a2t = 0

(337)

Hence y + ax = c1 , y − ax = c2 . Combining = c1 with (338), dp + a dq = 0 is obtained, upon p + aq = c3 = f1 (y + ax). Combining = c1 with (338), dp − a dq = 0 is obtained, upon p − aq = c4 = f2 (y − ax). Solving for p and q, p=

(338) y + ax wherey − ax where-

z = f (y + αx + iβx) + g(y + αx − iβx)

+ ax) + f2 (y − ax)]

Example 91.

which is an exact differential. Integration gives z = φ(y + ax) + ψ(y − ax). Homogeneous Equation with Constant Coefficients

The auxiliary equation is X 2 − 2X + 2 = 0 and m = 1 ± i. The general solution is z = f (y + x + ix) + g(y + x − ix), which can be written as z = f1 (y + x + ix) + f1 (y + x − ix) + i[g1 (y + x + ix) − g1 (y + x − ix)], where f1 and g1 are any twicedifferentiable real functions. If, in particular, f1 = cos u and g1 = cu , it can be shown that z = 2 cos(x + y) cosh x − 2ex+y sin x. Method of Separation of Variables. As an example of this method, the solution will be given to Laplace’s equation

(339) ∇ 2u =

This equation is equivalent to ∂ ∂ − m1 ∂x ∂y

∂ ∂ − m2 ∂x ∂y

Assume that z=0

(340)

where m1 and m2 are roots of the auxiliary equation X 2 + A1 X + A2 = 0. The general solution of (340) is z = f1 (y + m1 x) + f2 (y + m2 x)

Solve

∂ 2z ∂ 2z ∂2z +2 2 =0 −2 2 ∂x ∂x ∂y ∂y

1 [f1 (y + a)(dy + a dx) 2a − f2 (y − ax)(dy − a dx)]

∂2z ∂ 2z ∂2z + A2 2 = 0 + A1 2 ∂x ∂x ∂y ∂y

Solve

The auxiliary equation is X 2 + 6X + 9 = (X + 3) (X + 3) = 0. The general solution is z = f1 (y − 3x) + xf2 (y − 3x). If the coefficients in Eq. (339) are real, the complex roots of the auxiliary equation occur in conjugate pairs. Then the general solution will have the form

Substituting these in p dx + q dy = dz,

∂ 2z ∂2z ∂ 2z − 15 2 = 0 +2 2 ∂x ∂x ∂y ∂y

∂ 2z ∂ 2z ∂2z +9 2 =0 +6 2 ∂x ∂x ∂y ∂y

1 [f1 (y + ax) − f2 (y − ax)] q= 2a

dz =

Solve

The auxiliary equation is 8X 2 + 2X − 15 = (2X + 3)(4X − 5) = 0. Hence m1 = − 32 , m2 = 54 . The general solution is z = f1 (2y − 3x) + f2 (4y + 5x). If the auxiliary equation has multiple factors, the general solution is z = f1 (y + m1 x) + xf2 (y + m1 x).

dy + a dx = 0

dp dy − a 2 dx dq = 0

1 2 [f1 (y

8

Example 90.

the equation for a vibrating string. The auxiliary equations are dy − a dx = 0

Example 89.

(341)

∂2u ∂2u + 2 =0 ∂x 2 ∂y

u = X(x) · Y (y)

(342)

(343)

where X is a function of x only and Y a function of y only. By substitution and dividing by X · Y , (342) becomes 1 d 2Y 1 d 2X =− (344) 2 X dx Y dy 2

MATHEMATICS

269

Since the left side does not contain y, the right side does not contain x, and the two sides are equal, they must equal a constant, say −k 2 : 1 d 2X = −k 2 X dx 2

1 d 2Y = k2 Y dy 2

(345)

The solutions of these homogeneous linear differential equations with constant coefficients are Y = c3 eky + c4 e−ky (346)

X = c1 cos kx + c2 sin kx Hence, from (343),

u = (c1 cos kx + c2 sin kx)(c3 eky + c4 e−ky ) = eky (k1 cos kx + k2 sin kx) + e−ky (k3 cos kx + k4 sin kx)

(347)

Since (342) is linear, the sum of any number of solutions is again a solution. An infinite number of solutions may be taken provided the series converges and may be differentiated term by term. Then u=

∞ [eky (An cos kx + Bn sin kx) n=0

+ e−ky (Dn cos kx + En sin kx)]

(348)

is a solution of (342). The coefficients of (348) are determined by using the series as a Fourier series to fit the boundary conditions. Functions that satisfy Laplace’s equation are harmonic. In polar coordinates (342) becomes ∇ 2u =

∂ 2u 1 ∂ 2 u 1 ∂u =0 + 2 2 + 2 ∂r r ∂θ r ∂r

(349)

In three dimensions, Laplace’s equation in rectangular coordinates is ∇ 2u =

∂ 2u ∂ 2u ∂ 2u + 2 + 2 =0 ∂x 2 ∂y ∂z

(350)

In cylindrical coordinates, ∇ 2u =

1 ∂2u ∂ 2u 1 ∂u ∂2u + + + =0 ∂ρ 2 ρ ∂ρ ρ 2 ∂φ 2 ∂z2

∂2u sin θ ∂φ 2 ∂u 1 ∂ sin θ (352) + 2 r sin θ ∂θ ∂θ 1 ∂ r 2 ∂r

r2

∂u ∂r

+

1

r2

Basic Features The finite-element method is characterized by two distinct features: First, the domain of the problem is viewed as a collection of simple subdomains, called finite elements. By the word domain we refer to a physical structure, system, or region over which the governing equations are to be solved. The collection of the elements is called the finite-element mesh. Second, over each element, the solution of the equations being solved is approximated by interpolation polynomials. The first feature, dividing a whole into parts, called discretization of the domain, allows the analyst to represent any complex system as one of numerous smaller connected elements, each element being of a simpler shape that permits approximation of the solution by a linear combination of algebraic polynomials. The second feature, elementwise polynomial approximation, enables the analyst to represent the solution on an element by polynomials so that the numerical evaluation of integrals becomes easy. The polynomials are typically interpolants of the solution at a preselected number of points, called nodes, in the element. The number and location of the nodes in an element depends on the geometry of the element and the degree of the polynomial, which in turn depends on the equation being solved. Since the solution is represented by polynomials on each element, a continuous approximation of the solution of the whole can only be obtained by imposing the continuity of the finite-element solution, and possibly its derivatives, at element interfaces (i.e., at the nodes common to two elements). The procedure of putting the elements together is called the connectivity or assembly.

(351)

In spherical coordinates, ∇ 2u =

12 FINITE-ELEMENT METHOD 12.1 Introduction The finite-element method is a powerful numerical technique that uses variational methods and interpolation theory for solving differential and integral equations of initial and boundary-value problems. The method is so general that it can be applied to a wide variety of engineering problems, including heat transfer, fluid mechanics, solid mechanics, chemical processing, electrical systems, and a host of other fields. The method is also so systematic and modular that it can be implemented on a digital computer and can be utilized to solve a wide range of practical engineering problems by merely changing the data input to the program. The method is naturally suited for the description of complicated geometries and the modeling and simulation of most physical phenomena.

Finite-Element Approximation Beyond the two features already described, the finite-element method is a variational method, like the Ritz, Galerkin, and weighted-residual methods, in which the approximate solution is sought in the form

2

u ≈ UN =

N j =1

cj φj

270


where φj are preselected functions and cj are parameters that are determined using a variational statement of the equation governing u. However, the finiteelement method typically entails the solution of a very large number of equations for the nodal values of the function being sought. The number of equations is equal to the number of unknown nodal values. In most practical problems the number of unknown nodal values are so large that it is practical only if the calculations are carried on an electronic computer. 12.2

One-Dimensional Problems

The finite-element analysis consists of dividing a domain into simple parts (i.e., elements) that are easier to work with. Over each element the method involves representing the solution in terms of its nodal values and the development of a relationship between the nodal values and their counterparts by means of a variational method. Assembly of these relations and solution of the equations after imposing known boundary and initial conditions completes the analysis. Evaluation of an Integral of the integral

Consider the evaluation

b

I=

f (x) dx

(353)

a

where f (x) is a complicated function whose integration by conventional methods (e.g., exact integration) is not possible. A step-by-step procedure of the numerical evaluation of the integral I by the finite-element method is given later. Discretization of Domain. The area can be approximated by representing the interval (domain) = (a, b) as a finite set of subintervals (see Fig. 152). A typical subinterval (element), e = (xe , xe+1 ), is of length he ≡ xe+1 − xe , with x1 = a, and xN +1 = b, where N is the number of elements. Approximation of Solution. Over each element, the function f (x) is approximated using polynomials of a desired degree. The accuracy increases with increasing N and degree of the approximating polynomial. Over each element e , the function f (x) can be approximated by a linear polynomial (see Fig. 153)

f (x) ≈ Fe (x) = c1e + c2e x c1e

(354)

c2e

where and are constants that can be determined in terms of the values of the function f at the endpoints, xe and xe+1 , called the nodes. Let F1e and F2e denote the values of Fe (x) at nodes 1 and 2 of element e : F1e

= Fe (xe )

F2e

= Fe (xe+1 )

(355)

Fig. 152 Piecewise approximation of integral of function by polynomials.

Now Fe (x) can be expressed in terms of its values at the nodes as xe+1 − x e x − xe e e e F1 + F2 = Fj ψj he he 2

Fe (x) =

j =1

(356) where ψje are called the element interpolation functions (see Fig. 153), ψ1e =

xe+1 − x he

ψ2e =

x − xe he

(357)

Let the approximation of the area I over a typical element e be denoted by Ie , Ie =

xe+1

Fe (x) dx xe

(358)

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271

Assembly of Equations. An approximation of the total area I is given by the sum of the areas Ie , e = 1, 2, . . . , N:

I=

N

xe+1

f (x) dx xe

e=1

≈

N

xe+1

Fe (x) dx xe

e=1

=

N

Ie =

e=1

N he e=1

2

(F1e + F2e )

(360)

Incidentally, Eq. (360) is known as the trapezoidal rule. The accuracy of the approximation can be improved by increasing the number of elements N (see Fig. 152c) or by using higher order approximation of f (x) over each element. Note that the accuracy can also be improved by using unequal intervals, with smaller elements in areas where function f (x) varies rapidly. The quadratic interpolation of f (x) over e is given by f (x) ≈ Fe (x) = F1e ψ1e + F2e ψ2e + F3e ψ3e =

3

Fje ψje

(361)

j =1

where ψje are the quadratic interpolation functions

Fig. 153 Finite-element approximation of function f(x) over typical element.

Substituting Eq. (356) into (358) and integrating, one obtains Ie =

2 j =1

Fje

xe+1 xe

he e (F + F2e ) 2 1

(x − ξ2 )(x − ξ3 ) (ξ1 − ξ2 )(ξ1 − ξ3 )

ψ2e =

(x − ξ1 )(x − ξ3 ) (ξ2 − ξ1 )(ξ2 − ξ3 )

ψ3e =

(x − ξ1 )(x − ξ2 ) (ξ3 − ξ1 )(ξ3 − ξ2 )

(362)

and ξ1 , ξ2 , and ξ3 are the coordinates of the three nodes in e . If nodes are equally spaced within each element (see Fig. 154), (ξ1 , ξ2 , ξ3 ) take the values

ψje dx

ξ1 = x2e−1

* ! " he 1 F1e he xe+1 − (xe+1 + xe ) = he 2 ! "+ he (xe+1 + xe ) − he xe +F2e 2 =

ψ1e =

ξ2 = x2e

ξ3 = x2e+1

(e = 1, 2, . . . , N)

(359)

Thus, the area under the function Fe (x) over the element e is given by the area of the trapezoid of sides F1e and F2e and width he (see Fig. 153b).

Then Eqs. (362) become x 2x ψ1e = −1 −1 he he x 2x −1 ψ3e = he he

ψ2e = −

4x he

x −1 he

(363)

where x = x − x2e−1 and he is the length of the element e .

272


The total area is given by I≈

N e=1

Ie =

N he e=1

6

(F1e + 4F2e + F3e )

This equation is known as the one-third Simpson’s rule. Example 92.

Consider the integral of the function f (x) = sin(2 cos x) sin2 x

over the domain = (0, π/2). Table 12 contains the finite-element solutions obtained using linear and quadratic interpolation. It is clear that the accuracy improves as the number of elements or the degree of polynomial is increased. Solution of a Differential Equation Model Equation. Consider the differential equation ! " du d a(x) − f (x) = 0 0 < x < L (365) − dx dx

which arises in connection with heat transfer in a heat exchanger fin, where a(x) = kA, k is the thermal conductivity, A is the cross-sectional area of the fin, f (x) is the heat source, and u = u(x) is temperature to be determined. Equation (365) also arises in many fields of engineering. In addition to Eq. (365), the function u is required to satisfy certain boundary conditions (i.e., conditions at points x = 0 and x = L). Equation (365), in general, has the following types of boundary conditions: Specify either u or (a du/dx ) at a boundary point Fig. 154 tions.

One-dimensional quadratic interpolation func-

In general, the interpolation functions ψje satisfy the properties * 0 if i = j (364) ψje (ξi ) = 1 if i = j Substituting Eq. (361) into (358) and integrating, one obtains Ie =

3

3 j =1

Approximation. Over a typical element e = (xe , xe+1 ), the function u(x) is approximated by Ue (x), which is assumed to be of the form

Ue (x) = Fje

j =1

=

Discretization. The domain = (0, L) is represented as a collection of line elements, each element having at least two end nodes so that it can be connected to adjacent elements. A two-node element with one unknown per node requires, uniquely, a linear polynomial approximation of the variable over the element (see Fig. 155).

Fje

x2e+1 x2e−1 he 0

ψje (x) dx

ψje (x) dx

= 16 he (F1e + 4F2e + F3e )

n

Uje ψje (x)

(366)

j =1

where Uje denotes the value of Ue (x) at the j th node and ψje are the linear [see Eq. (357)], quadratic [see Eq. (363)], or higher order interpolation functions. The values Uje are to be determined such that Eq. (365), with appropriate boundary conditions, is satisfied in integral sense.

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273

Table 12 Finite-Element Solutions Using Linear and Quadratic Interpolation Number of Elements (1)b

2 4 (2) 6 (3) 8 (4) 10 (5) a (1

Linear Interpolation I 0.38790 0.48149 0.49640 0.50150 0.50384

Errora

Quadratic Interpolation (%)

23.6 5.2 2.3 1.3 0.8

I

Error (%)

Exact

0.51719 0.51268 0.50865 0.50817 0.50805

−1.8 −0.9 −0.1 −0.04 −0.02

0.50797 0.50797 0.50797 0.50797 0.50797

− I/I(100). in parentheses indicate number of equivalent quadratic elements.

b Numbers

Variational Formulation. The variational statement of Eq. (333) over an element e = (xe , xe+1 ) (see Fig. 155) is constructed as follows. Multiply Eq. (333) with an arbitrary but continuous function W and integrate over the domain of the element to obtain " ! xB dU d a − f dx (367) W − 0= dx dx xA

The (Ritz) finite-element model uses a weak form that can be obtained from Eq. (367) by trading differentiation between the weight function W and the variable of approximation U equally: ! " dU xB dx − W a dx xA xA (368) which is obtained by integrating the first term in Eq. (367) by parts. The term weak form is appropriate because the solution U of Eq. (368) requires weaker continuity conditions on ψi than U of Eq. (367). Also, the weak formulation allows the incorporation of the boundary conditions of the “flux” type, dU/dx (the coefficient of the weight function W in the boundary term, called the natural boundary condition), into the variational statement (368). Boundary conditions on U in the same form as the weight function in the boundary terms are called the essential boundary conditions. Identifying the coefficients of the weight function in the boundary terms (i.e., fluxes) as the dual variables, 0=

Fig. 155 One-dimensional domain, finite-element discretization, and finite-element approximation over an element.

Need for a Variational Statement. The difference between the numerical evaluation of an integral and the numerical solution of a differential equation is that in the case of a differential equation one is required to determine a function that satisfies a given differential equation and boundary conditions. It is possible to recast the differential equation as an integral statement, called a variational statement. The variational statement of Eq. (333), with the aid of a variational method of approximation, gives the same number of algebraic equations as the number of unknowns (n) in the approximation (334).

a

xB

a

dW dU − Wf dx dx

dU = −P1e dx x=xe

a

dU = P2e dx x=xe+1

Eq. (368) can be written as xe+1 dW dU a − Wf dx 0= dx dx xe − W (xe )P1e − W (xe+1 )P2e

(369)

Equation (369) represents the variational statement of Eq. (365) for the (Ritz) finite-element model.

274


As a general rule, the essential boundary conditions of the variational form of a problem indicate what interelement continuity conditions are to be imposed on the function U and its derivatives. This in turn dictates the type and degree of approximation and hence the element type. For example, Eq. (369) indicates that U must be continuous in the interval (xe , xe+1 ). A complete continuous polynomial in x is a linear polynomial Ue (x) = c1e + c2e x The constants c1e and c2e are expressed in terms of the values of Ue at nodes 1 and 2, Ue =

2

Uje ψje (x)

For the (n − 1)st-degree polynomial approximation, Ue is of the form n Uje ψje (x) Ue (x) = j =1

(Ritz) Finite-Element Model. In the Ritz model Uje Eq. (369) is satisfied for each W = ψie (i = 1, 2, . . . , n). For each choice of W , an algebraic equation can be obtained:     n e xe+1 dψie e dψj   − ψ1e f  dx a 0= Uj dx dx xe j =1

− ψ1e (xe )P1e − ψ1e (xe+1 )P2e     n e xe+1 dψ2e e dψj   − ψ2e f  dx a 0= Uj dx dx xe j =1

− .. .

xe+1 xe

f ψie dx + ψie (xe )P1e + ψie (xe+1 )P2e

(370b) To be more specific, let ψie be the linear interpolation functions of Eq. (357). Because of the interpolation property (364) of ψje , the Fie of Eq. (371) can be written as xe+1 f ψje dx + Pie ≡ fie + Pie Fie = xe

For elementwise constant values of a and f , the element coefficient matrix [K e ] and source vector {f e } become [K e ] =

j =i

ψ2e (xe )P1e

Fie =

! ae 1 he −1

0=

xe

−

j =1

−

ψne (xe+1 )P2e

The ith equation can be written in compact form as 0=

n

Kije Uje − Fie

j =1

* + he fe 1 1 2

U2e = U1e+1

e = 1, 2, . . . , N

In addition to the continuity of Ue , the balance of the dual variables Pi at interelement nodes is also enforced: P2e + P1e+1 = 0

e = 1, 2, . . . , N

Note that this does not guarantee the continuity of a dUe /dx at interelement nodes. The finite-element approximation on the entire e domain = N e=1 is given by

− ψ2e (xe+1 )P2e

   n e e dψ dψ j  a n  − ψne f  dx Uje dx dx

ψne (xe )P1e

{f e } =

Assembly of Elements. The element equations (370) must be put together to obtain the equations of the whole domain. Geometrically, the elements are connected together by noting that the second node of element e is the same as the first node of element e+1 . Since the solution and hence its approximation are single valued throughout the domain, the geometric continuity also implies the continuity of the approximate solution (see Fig. 156):

 xe+1

" −1 1

(370a)

U=

N

Ue =

e=1

N 2

Uje ψje (x)

e=1 j =1

In view of the continuity conditions and the elementwise definition of the interpolation functions ψje , the finite-element approximation can be written as U=

N +1

UJ J (x)

(371)

J =1

where Kije =

xe+1

a xe

e dψie dψj dx dx dx

where UJ denotes the value of U (x) at the J th (global) node of the mesh and J are the global interpolation

MATHEMATICS

Fig. 156

275

Assembly of finite elements using continuity of finite-element approximation between elements.

functions, related to the local (or element) interpolation functions by

Substitution of Eq. (371) for U and W = I (I = 1, 2, . . . , N + 1) into Eq. (372) gives L

0= 0

j =1



$x=L # N dUe  a − I f dx − f dx

(J = 2, 3, . . . , N) N +1 = ψ2N

  N +1 d d J a I  UJ dx dx 

0 = x1 ≤ x ≤ x2 1 = ψ11 ψ2I −1 xJ −1 ≤ x ≤ xJ J = ψ1J xJ ≤ x ≤ xJ +1 xN ≤ x ≤ xN +1 = L

Note that J are continuous and defined only on the two elements connected at the global node J . Analogous to the variational form (369) for an element e , a variational form for the entire domain can be derived as ! " L dW dU dU x=L a − Wf dx − W a 0= dx dx dx 0 x=0 (372)

e=1

x=0

Since each I is defined on two neighboring elements, this equation becomes 0=

XI xI −1

#

dψ I −1 a 2 dx $

− ψ2I −1 f

dψ I −1 dψ I −1 UI −1 1 + UI 2 dx dx

dx − ψ2I −1 (L)P2I −1

276


! dψ I dψ I dψ I a 1 UI 1 + UI +1 2 dx dx dx xI " − ψ1I f dx − ψ1I (a)P1I

+

=

Equation (375) is in the same form as Eq. (365). Therefore, Eqs. (370) and (374) describe the element and global finite-element models of Eq. (375). For the choice of linear interpolation functions, we have

xI +1

I −1 K21 UI −1

+

I −1 (K22

+

I K11 )UI

+

Kije =

I K12 UI +1

− (f2I −1 + f1I ) − ψ2I −1 (L)P2I −1 − ψ1I (0)P1I (373) Thus, the equations of the connected elements (i.e., the finite-element equations of the entire domain) are given by setting I = 1, 2, . . . , N + 1 in Eq. (373) (set KI0J = FI0 = PI0 = 0):

re+1

2πke r re

where ψie =

re+1 − r re+1 − re

ψ2e =

he = re+1 − re

0

1 1 2 2 1 2 (P2 K21 U1 + (K22 + K11 )U2 + K12 U3 = f21 + f12 + + P1 )

e K11 = 2πke

0

2 2 3 3 2 3 (P2 U2 + (K22 + K11 )U3 + K12 U4 = f22 + f13 + + P1 ) K21

N N UN + K22 Un+1 = f2N + P2N K21

2

 0  0   0   ..   .  N K22

(374)

2

One does not repeat the connectivity procedure des cribed in Eqs. (371)–(374) for every problem but uses the pattern implied in the final equations (374) for all problems described by Eq. (365). Example 93. Heat conduction in a long radially symmetric coaxial cylindrical cable can be described by

! " du d a(r) =0 − dr dr

re

1 2 r − dr he

We have

or, in matrix form, 0 0 3 K12 .. . N K21

re+1

πke = (re+1 + re ) he

.. .

1 1 K12 0 K11  1 1 2 2 K22 + K11 K12 K21  3 2 2  0 K22 + K11 K21   . .. ..  . . .  . 0 0 0    1  f1 + P11  U1             1 2      f + f U    2 1        22   3 U3 = f2 + f1 ×    ..     ..         .      .          N N UN +1 f +P

r − re re+1 − re

e For example, K11 is given by

1 1 K11 U1 + K12 U2 = f11 + P11



dψie dψie dr dr dr

(375)

where u denotes the temperature and a = 2πrk, k being the thermal conductivity of the medium.

! πke 1 (re+1 + re ) −1 he

−1 1

" * e + * e+ P1 u1 = ue2 P2e

where Pie denote the internal heats,

dU dr r=re dU e P2 = 2πke r dr r=re+1 P1e = −2πke

r

The assembled equations for an N-element case are shown in the tabulation at the top of page 277. We now impose the boundary conditions of the problem. Suppose that the domain is the cross section of a coaxial cylinder with two materials (i.e., with different thermal conductivities), as shown in Fig. 157. Let the internal and external radii be r1 = 20 mm and rN +1 = 50 mm and let the thickness of the first material be 11.6 mm and that of the second material be 18.4 mm and the associated material constants (k) be 5 and 1. We assume the boundary conditions to be u(20) = 100◦ C and u(50) = 0.0. These conditions translate to U1 = 100.0 P21 + P12 = 0, . . .

UN +1 = 0.0 P2N −1 + P1N = 0

For a nonuniform mesh of four elements (h1 = 5.1, h2 = 6.5, h3 = 8.2, h4 = 10.2; equivalently, r1 = 20, r2 = 25.1, r3 = 31.6, r4 = 39.8, and r5 = 50.0),

MATHEMATICS

277

Tabulation for Example 93  K K1 1 − 0  h1 h1   K1 K2 K2 K1 − + −  h1 h h h2 1 2   K K3 K 2 2  − +  0  h2 h2 h3   ..  · . ·     ·  ·    · ·



..

.

··· −

KN hN 0

       P11    U1        1 2     U      P + P 2     2 1        3 2      U   3   P2 + P1   = . ..  .       .  .    0              N −1 U N N     + P P      1 2     KN +1  U   N +1 N −  P 2 hN +1   KN +1  hN +1

KN hN KN +1 KN + hN hN +1 KN +1 − hN +1 −

where Ki = ki (ri+1 + ri )π. the assembled equations become 

22.108 −22.108 0 −22.108 43.916 −21.808  0 −21.808 26.162 2π   0 0 − 4.354 0 0 0     P2     1 1 2  U1     P2 + P1   U2  × U3 = P22 + P13        P23 + P14   U4      U5 P24

The boundary and continuity conditions are 0 0 −4.354 8.756 −4.402



0 0   0  −4.402 4.402

U1 = 100.0 P22 + P13 = 0

U5 = 0.0

P21 + P12 = 0

P23 + P13 = 0

The solution for U2 , U3 , and U4 is obtained by solving the second, third, and fourth equations of the assembled system: $ U2 0 −4.354 U3 8.756 U4

#

43.916 −21.808 −21.808 26.162 0 −4.354 22.108U1 0 = 0

or ◦

◦

◦

U2 = 91.745 C U2 = 83.377 C U4 = 41.458 C Table 13 contains a comparison of the finite-element solutions obtained by three different nonuniform meshes with the analytical solution. The numerical convergence and accuracy are apparent from the results. 12.3

Two-Dimensional Problems

As a model equation, consider the following secondorder equation in two dimensions: − Fig. 157 Finite-element representation of radially symmetric problem with two different materials.

∂ ∂x

a11

∂u ∂x

−

∂ ∂y

a22

∂u ∂y

+ a0 u = f in

(376) The coefficients a11 , a22 , and a0 and the source term f are known functions of position (x, y) in the domain .

278 Table 13

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Finite-Element Solutions Obtained by Various Nonuniform Meshes

r

Two Elements

Four Elements

Eight Elements

Analytical Solution

20.0 22.6 25.1 28.4 31.6 35.7 39.8 44.9 50.0

100.000 — — — 83.375 — — — 0.000

100.000 — 91.745 — 83.377 — 41.458 — 0.000

100.00 95.559 91.746 87.258 83.377 61.213 41.457 19.551 0.000

100.000 95.559 91.746 87.257 83.377 61.210 41.457 19.549 0.000

Equation (376) arises in the study of a number of engineering problems, including heat transfer, irrotational flow of a fluid, transverse deflection of a membrane, and torsion of a cylindrical member. Also, the Stokes flow and plane elasticity problems are described by a pair of equations of the same form as the model equation. Thus, the finite-element procedure to be described for Eq. (376) is applicable to any problem that can be formulated as one of solving equations of the form of (376). While the basic ideas are the same as described before, the mathematical complexity for twodimensional problems increases because of the partial differential equations on two-dimensional domains with possibly curved boundaries. It is necessary to approximate not only the solution of a partial differential equation but also the domain by a suitable finite-element mesh. This latter property is what made the finite-element method a more attractive practical analysis tool over other competing methods. Discretization of a Domain Two-dimensional domains can be represented by more than one type of geometric shape. For example, a plane curved domain can be represented by triangular elements or rectangular elements. Without reference to a specific geometric shape, we simply denote a typical element by e and proceed to discuss the approximation of Eq. (376). The choice of the finite-element mesh depends both on the element characteristics (convergence, computational simplicity, etc.) and the ability to represent the domain accurately. The concept of so-called isoparametric formulations allows the representation of the element geometry by the same interpolation as that used in the approximation of the dependent variables. Thus, by identifying nodes on the boundary of the domain, one can approximate the domain by suitable collection of elements to a desired accuracy. Element Equations Variational Formulation. Consider a typical finite element e from the finite-element mesh of the domain (see Fig. 158). Let ψie (i = 1, 2, . . . , n) denote the interpolation functions used to approximate u on e . Multiply Eq. (376) with a weight function W , integrate

over the element domain e , and use the Green–Gauss theorem to trade differentiation to W to obtain the weak variational form ! ∂W ∂W e ∂U e ∂U a11 + a22 0= ∂x ∂x ∂y ∂y e " ! e ∂U W nx a11 + a0e W U − Wfe dx dy − ∂x e " ∂U e ds (377) + ny a22 ∂y where nx and ny are the components (i.e., direction cosines) of the unit normal n, ˆ nˆ = nx iˆ + ny jˆ = cos α iˆ + sin α jˆ on the boundary e and ds is the elemental arc length along the boundary of the element. From an inspection of the boundary term in Eq. (377), it follows that the specification of the coefficient of W , e ∂U e ∂U + ny a22 (378) qne ≡ nx a11 ∂x ∂y constitutes the natural boundary condition. The variable qn is of physical interest in most problems. For example, in the case of the heat transfer through an anisotropic medium (where aij denotes the conductivities of the medium), qn denotes the heat flux across the boundary of the element (see Fig. 158). The variable U is called the primary variable and qn (heat flux) is termed the secondary variable. The variational form in Eq. (377) now becomes ! ∂W ∂W e ∂U e ∂U a11 + a22 0= ∂x ∂x ∂y ∂y e " e W U − Wfe dx dy − W qne ds (379) + a00 e

This variational equation forms the basis of the Ritz finite-element model. The boundary term indicates that W should be continuous at interelement boundaries.

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279

Fig. 158 Finite-element representation of two-dimensional domain with various types of boundary conditions.

Finite-Element Formulation. The variational form in (379) indicates that the approximation chosen for u should be at least bilinear in x and y so that ∂u/∂x and ∂u/∂y are nonzero and the interelement continuity of u can be imposed. Suppose that the temperature is approximated by the expression

u ≈ Ue =

n

Uje ψje

(380)

j =1

Uje

where are the values of Ue at the point (xj , yj ) in e and ψje are the interpolation functions with the property ψie (xj , yj ) = δij The specific form of ψie will be derived later for linear triangular and rectangular elements. Substituting Eq. (380) for Ue and ψie for W into the variational form (379), the ith algebraic equation of the model is obtained, n j =1

Kije Uje = Fie

(i = 1, 2, . . . , n)

(381)

where Kije =

! e

∂ψie ∂x

e a11

"

∂ψje

∂x

+

∂ψie ∂y

∂ψje e a22 ∂y

+ a0e ψie ψje dx dy Fie =

e

fe ψie dx dy +

e

qne ψie ds ≡ fie + Pie

(382) Note that Kije = Kjei (i.e., [K e ] is symmetric). Equation (381) is called the finite-element model of Eq. (376). Assembly of Elements The assembly of finiteelement equations is based on the same principle as that employed in one-dimensional problems. We illustrate the procedure by considering a finite-element mesh consisting of two triangular elements (see f Fig. 159). Let Kije and Kij (i, j = 1, 2, 3) denote the coefficient matrices and {F e } and {F f } denote the column vectors of three-node triangular elements e and f . From the finite-element mesh shown in Fig. 159, the following correspondence between the global and

280


The complete assembled equations for the two-element mesh is given by 

e K11

 K21  K e  31 0 Fig. 159 Assembly (or connectivity) of linear triangular elements.

element nodal values of the temperature is noted: f

U1 = U1e

U2 = U2e = U1

U3 = U3e = Uef

f

U4 = U2

The continuity of U at the interelement nodes guarantees its continuity along the entire interelement boundary. To see this, consider two linear triangular elements (see Fig. 159). The finite-element solution for U is linear along the boundaries of the elements. The interelement boundary is along the line connecting global nodes 2 and 3. Since Ue is linear along side 2–3 of element e , it is uniquely determined by the two values U2e and U3e . Similarly, Uf is uniquely determined along side 1–3 of element f by the two f f f f values U1 and U3 . Since U2e = U1 and U3e = U3 , it follows that Ue = Uf along the interface. Similar arguments can be presented for higher order elements. The coefficient Kije is a representation of a physical property of node i with respect to node j of element e . The assembled coefficient matrix also represents the same property among the global nodes. But the global property comes from the element nodes shared by the global nodes. For example, the coefficient K23 of the global coefficient matrix is the sum of the contributions from nodes 2 and 3 of e and nodes 1 and 3 of f (see Fig. 159): f

e K23 = K23 + K13

f

e K32 = K32 + K31

Similarly, f

e K22 = K22 + K11

f

e K33 = K33 + K33 , . . .

If the global nodes I and J do not correspond to nodes in the same element, then KI J = 0. For example, K14 is zero because global nodes 1 and 4 do not belong to the same element. The column vectors can be assembled using the same logic: f

F2 = F2e + F1

f

F3 = F3e + F3 , . . .

=

e K12 e K22 e K32

+ +

f K11 f K31

e K13 e K23 e K33

f K21  Fe  1       F e + F f  2 1 f e  F +F       3 f 3 F2

+ +

f K13 f K33

f

K23

0







U1   f  K12  U2   f U  K32      3 f U4 K32

Imposition of Boundary Conditions The boundary conditions on the primary variables (temperatures) and secondary variables (heats) are imposed on the assembled equations in the same way as in the onedimensional problems. To understand the physical significance of the P ’s [see Eq. (382)], take a closer look at the definition,

Pie ≡

e

qne ψie (s) ds

(383)

where ψie (s) is the value of ψie (x, y) on the boundary e . The heat flux qne [see Eq. (378)] is an unknown when e is an interior element of the mesh (see Fig. 158a). However, when the element equations are assembled, the contribution of the heat flux qne to the nodes (namely, Pie ) of e get canceled by similar contributions from the adjoining elements (see Fig. 158b). If the element r has any of its sides on the boundary of the domain (see Fig. 158c), then on that side the heat flux qnr is either specified or unspecified. If qnr is specified, then the heat Pir at the nodes on that side can be computed using Eq. (383). If qnr is not specified, then the primary variable Ur is known on that portion of the boundary. The remaining steps of the analysis do not differ from those of one-dimensional problems. Interpolation Functions Linear Triangular Element. The simplest finite element in two dimensions is the triangular element. Since a triangle is defined uniquely by three points that form its vertices, the vertex points are chosen as the nodes (see Fig. 160a). These nodes will be connected to the nodes of adjoining elements in a finite-element mesh. A polynomial in x and y that is uniquely defined by three constants is of the form p(x, y) = c0 + c1 x + c2 y. Hence, assume approximation of ue in the form

Ue = c0e + c1e x + c2e y

(384)

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281

Fig. 160

Typical linear triangular element and associated finite-element interpolations function.

Proceeding as in the case of one-dimensional elements, write Uie

≡ Ue (xi , yi ) =

c0e

+

c1e xi

+

c2e yi

i = 1, 2, 3

where (xi , yi ) denote the global coordinates of the element node i in e . In explicit form this equation becomes  e    1 x1 y1 c0e  U1  U e = 1 x2 y2  c1e  e  2e  1 x3 y3 U3 c2 Note that the element nodes are numbered counterclockwise. Upon solving for c’s and substituting back into Eq. (384), one obtains Ue =

3

Uie ψie (x, y)

i=1

ψie =

1 (α e + βie x + γie y) 2Ae i

where Ae represents the area of the triangle, and αie = xj yk − xk yj i = j = k

βie = yj − yk

γie = xk − xj

i, j, k = 1, 2, 3

and the indices on αie , βie , and γie permute in a natural order. For example, α1e is given by setting i = 1, j = 2, and k = 3: α1e = x2 y3 − x3 y2 The sign of the determinant changes if the node numbering is changed to clockwise. The interpolation functions ψie satisfy the interpolation properties listed in Eq. (364). The shape of these functions is shown in Fig. 160b. Note that the derivative of ψie with respect to x or y is a constant. Hence, the derivatives of the solution evaluated in the postcomputation would be elementwise constant. Also, the coefficient matrix e ∂ψ e e ∂ψ e j j e ∂ψi e ∂ψi + a22 dx dy a11 Kije = ∂x ∂x ∂y ∂y e (385)

282


can be easily evaluated for the linear interpolation functions for a triangle. We have βe ∂ψie = i ∂x 2Ae

γe ∂ψie = i ∂y 2Ae

e e and, for elementwise constant values of a11 and a22 , the coefficients of Kije become

Kije =

1 e e e (a e β e β e + a22 γi γj ) 4A2e 11 i j

=

1 e e e (a e β e β e + a22 γi γj ) 4Ae 11 i j

dx dy e

Linear Rectangular Element. A rectangular element is uniquely defined by the four corner points (see Fig. 161). Therefore, the four-term polynomial can be used to derive the interpolation functions. Express ue in the form

ue = c0e + c1e x + c2e y + c3e xy

Fig. 161

(386)

and obtain  e U1     U e 



1 1 2 = U e  1     3 1 U4

x1 x2 x3 x4

y1 y2 y3 y4

  x1 y1 c0e    e  x2 y2   c1 e  x3 y3  c2     x4 y4 c3e

By inverting the equations for c’s and substituting into Eq. (386), one obtains η η ξ ξ 1− ψ2e = 1− ψ1e = 1 − a b a b ξ ξ η η ψ3e = ψ4e = 1 − ab a b where (ξ, η) are the element coordinates, ξ = x − x1

η = y − y1

The functions are geometrically represented in Fig. 161. In calculating element matrices, one finds that the use

Typical linear rectangular element and associated finite-element interpolation functions.

MATHEMATICS

283



of the local coordinate system (ξ, η) is more convenient than using the global coordinates (x, y). For the linear rectangular element, the derivatives of the shape functions are not constant within the element: ∂ψie = linear in y ∂x

∂ψie = linear in x ∂y

The integration of polynomial expressions over a rectangular element is made simple by the fact that e = (0, a) × (0, b):

e

a

f (x, y) dx dy =

2 b −2 e [K ] =  6a −1 1

u(0, y) = u(1, y) = 0

b

−2 2 1 −1

−1 1 2 −2

 1 −1 −2 2

−1 −2 2 1

 −2 −1 1 2

u(x, 0) = 0

∂u (x, 1) = x ∂y

0

The coefficients in Eq. (385) can be easily evaluated over a linear rectangular element for elementwise cone e and a22 : stant values of a11

1 2 −2 −1

Example 94. Consider a computational example of Eq. (376) for the case where a11 = a22 = 1, f = 0, and is a unit square. Let the boundary conditions be as follows (see Fig. 162a):

f (x, y) dx dy b



2 a  1 +  6b −1 −2

The finite-element model is given by Eq. (381), with Kije =

e

e e ∂ψie ∂ψj ∂ψie ∂ψj + ∂x ∂x ∂y ∂y

dx dy

fie = 0

Triangular Elements. The 2 × 2 mesh of triangular elements is shown in Fig. 162b. The element

Fig. 162 Domain, boundary conditions, and finite-element meshes.

284


Note that the individual fluxes Pie in Eqs. (387) are not zero, but their sum is equal to the values indicated. For example, consider P26 :

coefficient matrices are given by # 1 1 −1 [K ] = [K ] = [K ] = [K ] = 2 0 # 1 1 0 [K 2 ] = [K 4 ] = [K 6 ] = [K 8 ] = 2 −1 1

3

5

7

−1 2 −1 0 1 −1

$ 0 −1 1 $ −1 −1 2

P26 =

qψ26 ds

S6

1.0

=

0.5

The assembled equations are given by (refer to Fig. 162b) the tabulation below where × denotes a zero due to disconnectivity (e.g., K13 = 0 because global nodes 1 and 3 do not belong to the same element). The boundary conditions on the primary variables (i.e., U ’s) are

+

+

+

P34

+

P25

+

P17

+

P18 +

qψ26 (x, 1)(−dx) +

=0

0.5

x2x dx +

0

=

1.0

+

qψ26 (0, y)(−dy)

1.0

2x(1 − x) dx

0.5

− 23 [(1)3 − (0.5)3 ]

qψ26 (x, 1) dx

qψ38 (x, 1) dx

1.0

= 23 (0.5)3 + (1)2 − (0.5)2

0.5 0

0.5

P35 + P26 + P38 =

(because no flux is specified at node 5) P35 + P26 + P38 =

0.5

where ψ26 (x, y) = 2x. The first integral is nonzero but gets canceled by a similar but negative contribution from P35 , the second integral is nonzero and can be evaluated since q = x is known, and the third integral is zero because ψ26 (0, y) = 0. Evaluating the integral in Eq. (387) [with ψ26 (x, y) = 2x and ψ38 = 2(y − x)], we obtain

The known secondary variables are (correspond to nodes 5 and 8) P22

0 0.5

U1 = U2 = U3 = U4 = U6 = U7 = U9 = 0

P31

1 qψ26 (y − 0.5, y) √ dy 2

1 12

+

1 6

=

1 4

(387)

Tabulation for Triangular Elements 1 

2

1+1

    1   2   Symmetric 

×

 U1     U2    U3      U4   U5       U6    U7     U8    U9

3

−1 2+1+1

× −1 2

4 −1 × × 2+1+1

  P11 + P21   1     3 1 4     P + P + P 2   1 2 1       P32   3       5 6 2   4 P + P + P   1 1 3    (P1 + P2 + P4 + P5  5 2 3 2 3 =    +P71 + P81 )          3 4 7       P + P + P  6   3 2 2      6    7 P    3          5 6 8 8    P3 + P2 + P3        9 8 7 P +P                  

3

2

5 0 −1−1 × −1 − 1 2+2+1 +1 + 1 + 1

6

7

8

× 0+0 −1 × −1 − 1

× × × −1 ×

× × × 0+0 −1−1

2+1+1

× 2

× −1 2+1+1

9  × ×  ×   ×   0 + 0   −1   ×   −1 1+1

MATHEMATICS

285

The boundary conditions are given by

To solve for the unknowns U5 and U8 , equations (5) and (8) of the assembled equations are used. This choice is dictated by the fact that the remaining equations contain additional unknowns in P ’s. The solution is given by 1 2 U8 = 14 U5 = 28

P31 + P42 + P23 + P14 = 0

P33 + P44 = 0.25

The condensed equations become ! 1 16 6 −2

The internal heat Pie can be determined from either the element equations (381) or by definition (383). In general, the values computed by the two methods are not the same because Pie determined from the element equations is the internal heat in equilibrium with the heat from the neighboring elements, whereas Pie computed from the gradient of the approximate temperature field is not.

−2 8

"*

+ * + 0 U5 U8 = 14

and the solution is given by U5 =

3 124

U8 =

6 31

The exact solution of the problem is given by

Rectangular Elements. For the 2 × 2 mesh of rectangular elements shown in Fig. 162c, the element matrices are given by   4 −1 −2 −1 1 4 −1 −2 −1 [K − ] = [K 2 ] = −2 −1 4 −1 6 −1 −2 −1 4

u(x, y) =

∞ 2 (−1)n+1 sin nπx sinh nπy π2 n2 cosh nπ n=1

A comparison of the finite-element solutions obtained with 2 × 2 and 4 × 4 meshes of linear rectangular and triangular elements with the series solution is presented in Table 14. The finite-element solution improves as the mesh is refined.

The assembled equations are shown in the tabulation below. Tabulation for Rectangular Elements

     1   6    

1

2

4

−1 4×4

3 × −1 4

4

5

−2 −2 × 4+4

−2 −1 − 1 −2 −1 − 1 4+4 +4 + 4

6

7

8

× −2 −1 × −1 − 1

× × × −1 −2

× × × −2 −1−1

4+4

× 4

−2 −1 4+4

Symmetric

9

     P11    U 1    1    1 2      P + P     U2  2 1 2 ×               P22 ×  U     3 3         3 1 ×    U P + P      4 4      1 4  1   ×  P + P2 + P3 + P4  5    U5   2 3 4 1 −2 =                  −1     2 4     6 U P + P 6    × 3 2            3 7 U     −1  7 P4            4  8     U 3 4 8   P3 + P4          9 U9 4 P3

Table 14 Comparison of Finite-Element Solutions Triangles x 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

Rectangles

y

2×2

4×4

2×2

4×4

Series Solution

0.25 0.25 0.25 0.50 0.50 0.50 0.75 0.75 0.75 1.00 1.00 1.00

— — — — 0.0357 — — — — — 0.1429 —

0.0101 0.0151 0.0114 0.0253 0.0387 0.0305 0.0525 0.0840 0.0719 0.1007 0.1729 0.1729

— — — — 0.0242 — — — — — 0.1936 —

0.0095 0.0136 0.0097 0.0254 0.0370 0.0270 0.0552 0.0882 0.0675 0.1059 0.1851 0.2027

0.0103 0.0152 0.0112 0.0264 0.0400 0.0308 0.0555 0.0894 0.0765 0.1057 0.1846 0.1990

286


Table 15

Basic Steps in Finite-Element Analysis of Typical Problem

1. Discretization of a Domain. Represent the given domain as a collection of a finite number of simple subdomains, called finite elements. The number, shape, and type of element depend on the domain and differential equation being solved. The principal parts of this step include: (a) Number the nodes (see step 2) and elements of the collection, called the finite-element mesh. (b) Generate the coordinates of the nodes in the mesh and the relationship between the element nodes to global nodes (called the connectivity matrix, which indicates the relative position of each element in the mesh). 2. Approximation of the Solution (a) Derivation of the Approximating Functions. For each element in the mesh, derive the approximation functions needed in the variational method. These functions are generally algebraic polynomials generated by interpolating the unknown function in terms of its values at preselected points, the nodes, of the element. (b) Variational Approximation of the Equation. Using the functions derived in step 2a and any appropriate variational method, derive the algebraic equations among the nodal values of the primary and secondary variables. 3. Connectivity (or Assembly) of Elements. Combine the algebraic equations of all elements in the mesh by imposing the continuity of the primary nodal variables (i.e., the values of the primary variables at a node shared by two or more elements are the same). This can be viewed as putting the elements (which were isolated in steps 2a and 2b from the mesh to derive the algebraic equations) back into their original places. This gives the algebraic equations governing the whole problem. 4. Imposition of Boundary Conditions. Impose the boundary conditions, both on primary and secondary variables of the assembled equations. 5. Solution of Equations. Solve the equations for the unknown nodal values of the primary variables. 6. Computation of Additional Quantities. Using the nodal values of the primary variables, compute the secondary variables (via constitutive equations).

In summary, the finite-element method is a numerical technique of solving field problems of engineering. It is endowed with two unique features: The domain in which the equations are defined is represented by a collection of simple parts (finite elements), and over each element the problem is approximated using any one of the variational methods with polynomials for the approximation functions. The first feature allows approximate representation of geometrically complicated domains by simple geometric shapes, while the second feature enables the approximation of the field variables, evaluation of the coefficient matrices, and solution of the finite-element equations on a computer. A list of basic steps of the finite-element analysis is presented in Table 15. 13

LAPLACE TRANSFORMATION

13.1

f (t) a real function of t that equals zero for t < 0, F (s) a function of s, and e the base of the natural logarithms. If the Lebesgue integral ∞

e−st f (t) dt = F (s)

(388)

0

then F (s) is the direct Laplace transform of f (t); in simpler notation L [f (t)] = F (s)

(389)

2. Inverse Laplace Transformation. Under certain conditions the direct transformation can be inverted, giving as one explicit representation

Transformation Principles

The Laplace and Fourier transformation methods and the Heaviside operational calculus are in essence different aspects of the same method. This method simplifies the solving of linear constant-coefficient integrodifferential equations and convolution-type integral equations. For brevity the conditions under which the steps of the method may be validly applied will be omitted. Hence the correctness of a final result should be checked in each case by showing that the formal solution satisfies the given equation and conditions. 1. Direct Laplace Transformation. Let t be a real variable, s a complex variable (Section 14.2),

1 2πi

c+i∞

ets F (s) ds(=)f (t)

(390)

c−i∞

in which c is a real constant chosen so that the path of integration lies to the right of all the singularities of F (s), and (=) means equals except possibly for a set of values of t of measure zero. If this relation holds, then f (t) is the inverse Laplace transform of F (s). In simpler notation the transformation is written L −1 [F (s)](=)f (t)

(391)

MATHEMATICS

287

3. Transformation of nth Derivative. If L [f (t)] = F (s), then " n−1 d n f (t) n = s F (s) − f (k) (0+) · s n−1−k dt n k=0 (392) where f (2) (0+) means d 2 f (t)/dt 2 evaluated for t → 0 and f (0) (0+) means f (0+) and n = 1, 2, 3, . . . . 4. Transformation of nth Integral. If L [f (t)] = F (s), then   n 2 34 5   L  · · · f (t) dt    !

L

=s

−n

F (e) +

−n

f

(k)

(0+) · s

−n−1−k

Step A. Find the Laplace transform of the equation to be solved and express it in terms of the transform of the unknown function. Thus, ! " dy(t) L k1 + k2 y(t) + k3 y(t) dt = L [u(t)] dt By (396) this becomes " ! ! dy(t) + k2 L [y(t)] + k3 L k1 L dt = L [u(t)] By (392) and (393) and the given initial conditions of the problem the equation becomes k1 [sY (s) − y(0)] + k2 Y (s) + k3 [s −1 Y (s) + y (−1) (0) · s −1 ] = L [u(t)]

k=−1

(393) (−2) where n) = (0+) ) 1, 2, 3, . . . . For example, f means f (t)dtdt evaluated for t → 0. 5. Inverse Transformation of Product. If L −1 [F1 (s)] = f1 (t)

L −1 [F2 (s)] = f2 (t) (394)

then t

L −1 [F1 (s) · F2 (s)] =

f1 (t − λ) · f2 (λ) dλ

0

(395) 6. Linear Transformations L and L −1 . Let k1 , k2 be real constants. Then L [k1 f1 (t) + k2 f2 (t)] = k1 L [f1 (t)] + k2 L [f2 (t)] (396) and L −1 [k1 F2 (s) + k2 F2 (s)] = k1 L −1 [F1 (s)] + k2 L −1 [F2 (s)]

(397)

13.2 Procedure To illustrate the application of the rules of procedure the following simple initial-value problem will be solved. Given the equation

k1

dy(t) + k2 y(t) + k3 dt

y(t) dt = u(t)

and initial values y(0), y (−1) (0) where u(t) = 0 for t < 0 and u(t) = 1 for 0 < t and k1 , k2 , k3 are real constants. Assume that y(t) has a Laplace transform Y (s), that is, L [y(t)] = Y (s).

" y(t) dt

Step B. Solve the resulting equation for the transform of the unknown function. Thus, Y (s) =

L [u(t)] + k1 y(0) − y (−1) (0) · s −1 k1 s + k2 + k3 s −1

Step C. Evaluate the direct transform of the given function (right member) in the original equation. Since 1 L [u(t)] = s Y (s) =

k1 y(0) · s − y (−1) (0) + 1 k1 s 2 + k2 s + k3

Step D. Obtain the solution of the problem by evaluating the inverse Laplace transform of the function obtained by the preceding steps. One way to carry out step D is to find the inverse transform from the table of Laplace transforms in Section 12.3. To use the table, the denominator of the fraction should be factored: ! " k1 y(0) · s − y (−1) (0) + 1 y(t) = L −1 [Y (s)] = L −1 k1 s 2 + k2 s + k3 " ! k1 y(0) · s − y (−1) (0) + 1 = L −1 k1 (s + K1 )(s + K2 ) in which K1 ≡

k2 1 2 − (k − 4k1 k3 )1/2 2k1 2k1 2

K2 ≡

k2 1 2 + (k − 4k1 k3 )1/2 2k1 2k1 2

288


To find the result it is necessary to distinguish between two cases. Case 1: If K1 = K2 , [k1 y(0)K1 + y (−1) (0) − 1]e−K1 t −[k1 y(0)K2 + y (−1) (0) − 1]e−K2 t y(t) = k1 (K1 − K2 ) for 0 < t and y(t) = 0 for t < 0. Case 2: If K1 = K2 = K, then K = k2 /2k1 , and y(t) = L −1

!

k1 y(0) · s − y (−1) (0) + 1 k1 (s + K)2

"

k1 y(0)e−Kt − [y (−1) (0) − 1 +k1 y(0)K]te−Kt y(t) = k1 for 0 < t and y(t) = 0 for t < 0. The solutions can be shown to satisfy the original equation and initial conditions. The use of step C can be avoided by using steps E, F, and G in place of steps C and D in the following way. Step E. Factor the transform of the unknown function obtained by step B and evaluate the inverse Laplace transform of each factor. Note. The inverse transform of a rational fraction can be found only if it is a proper fraction. Thus sL [u(t)] k1 y(0) · s − y (−1) (0) + k1 s 2 + k2 s + k3 k1 s 2 + k2 s + k3

Let y1 (t) ≡ L −1

!

k1 y(0) · s − y (−1) (0) k1 (s + K1 )(s + K2 )

"

= [k1 (y)(0)K1 + y (−1) (0)]e−K1 t −

y(t) = y1 (t) + [k1 (K1 − K2 )]−1 t

×

[K1 e−K1 (t−τ ) − K2 e−K2 (t−τ ) ]u(τ ) dτ

0

Step G. Evaluate the (convolution) integral arising from step F. Thus,

From Table 16,

Y (s) =

Step F. Use condition 5 to find the inverse transform of the product. Thus, by condition 6 and step F,

[k1 y(0)K2 + y (−1) (0)]e−K2 t k1 (K1 − K2 )

y(t) = y1 (t) + [k1 (K1 − K2 )]−1 (e−k2 t − e−K1 t ) for 0 < t and y(t) = 0 for t < 0. For the particular problem treated it is much simpler to use steps C and D than steps E, F, and G. However, for a more complicated right member of the original equation it could happen that step G would be easier to carry out than step C, in which case the second method (A, B, E, F, G) should be used rather than the first (A, B, C, D). One physical representation of the initial-value problem that we have used for illustration is the problem of finding the current response of a series electric circuit containing constant lumped inductance, resistance, and capacitance to an applied electromotive force u(t), with an initial current in the inductance and an initial charge on the condenser. The complete method (of which only a part has been given) is not restricted in its field of application to linear equations with constant coefficients, but the solution of this type of equation is most simplified. 13.3 Transform Pairs The Laplace transforms in Table 16 are applicable in the solution of ordinary integrodifferential and difference equations. 14 COMPLEX ANALYSIS 14.1 Complex Numbers A complex number A is a combination of two real numbers a1 , a2 in the ordered pair (a1 , a2 ) = A = a1 + ia2 , where i = (−1)1/2 . Real and imaginary numbers are special cases of complex numbers obtained by placing (a1 , 0) = a1 , (0, a2 ) = ia2 (see Fig. 163).

for 0 < t and y1 (t) = 0 for t < 0. Also L −1

!

" K1 e−K1 t − K2 e−K2 t s = k1 (s + K1 )(s + K2 ) k1 (K1 − K2 )

for 0 < t and y1 (t) = 0 for t < 0. Finally, L −1 {L [u(t)]} = u(t).

Fig. 163

MATHEMATICS

289

Table 16 Laplace Transformsa Unilateral Laplace Operation Transform Pairs f(t), 0 ≤ t

Name Linearity

Real differentiation Multiplication by s

af(t), a is constant or variable independent of t, f1 (t) ± f2 (t) df(t) = f (t) dt f (t) if f(0+) = 0

f(t) dt = f (−1) (t)

Real integration

t

Division by s

f(t) dt = f (−1) (t) − f (−1) (0+) t , f a a is positive constant or positive variable independent of t 0

Scale change

1

Complex multiplication Real translation

Complex translation

Second independent variable Differentiation with respect to second independent variable Final value

Initial value Complex differentiation Complex integration

aF(s), a is constant or variable independent of s, F1 (s) ± F2 (s) sF(s) − f(0+) sF(s) f (−1) (0+) F(s) + s s F(s) s aF(as), a is positive constant or positive variable independent of s

f1 (t − τ )f2 (τ ) dτ = f1 (t) ∗ f2 (t)

F1 (s)F2 (s)

f(t − a) if f(t − a) = 0, 0 < t < a, f(t + a) if f(t + a) = 0, −a < t 0, K > 0). The steady-state output deviation due to a unit step disturbance is −1/K. This deviation can be reduced by choosing K large. The transient √ behavior is indicated by the damping ratio, ζ = c/2 I K. For slight damping, the response to a step input will be very oscillatory and the overshoot large. The situation is aggravated if the gain K is made large to reduce the deviation due to the disturbance. We conclude, therefore, that proportional control of this type of second-order plant is not a good choice unless the damping constant c is large. We will see shortly how to improve the design. 5.2 Integral Control The offset error that occurs with proportional control is a result of the system reaching an equilibrium in which the control signal no longer changes. This allows a constant error to exist. If the controller is modified to produce an increasing signal as long as the error is nonzero, the offset might be eliminated. This is the principle of integral control. In this mode the change in the control signal is proportional to the integral of the error. In the terminology of Fig. 7, this gives

F (s) =

KI E(s) s

(16)

where F (s) is the deviation in the control signal and KI is the integral gain. In the time domain, the relation is t f (t) = KI e(t) dt (17) 0

if f (0) = 0. In this form, it can be seen that the integration cannot continue indefinitely because it would theoretically produce an infinite value of f (t) if e(t) does not change sign. This implies that special care must be taken to reinitialize a controller that uses integral action. Integral Control of a First-Order System Integral control of the velocity in the system of Fig. 20 has the block diagram shown in Fig. 22, where G(s) =

774


K/s, K = K1 KI KT /R. The integrating action of the amplifier is physically obtained by the techniques to be presented in Section 6 or by the digital methods presented in Section 10. The control system is stable if I , c, and K are positive. For a unit step command input, ωss = 1; so the offset error is zero. For a unit step disturbance, the steady-state deviation is zero if the system is stable. Thus, the steady-state performance using integral control is excellent for this √plant with step inputs. The damping ratio is ζ = c/2 I K. For slight damping, the response will be oscillatory rather than exponential as with proportional control. Improved steady-state performance has thus been obtained at the expense of degraded transient performance. The conflict between steady-state and transient specifications is a common theme in control system design. As long as the system is underdamped, the time constant is τ = 2I /c and is not affected by the gain K, which only influences the oscillation frequency in this case. It might be physically possible to make K small enough so that ζ 1, and the nonoscillatory feature of proportional control recovered, but the response would tend to be sluggish. Transient specifications for fast response generally require that ζ < 1. The difficulty with using ζ < 1 is that τ is fixed by c and I . If c and I are such that ζ < 1, then τ is large if I c. Integral Control of a Second-Order System Proportional control of the position servomechanism in Fig. 23 gives a nonzero steady-state deviation due to the disturbance. Integral control [G(s) = K/s] applied to this system results in the command transfer function

K (s) = 3 r (s) I s + cs 2 + K

(18)

With the Routh criterion, we immediately see that the system is not stable because of the missing s term. Integral control is useful in improving steady-state performance, but in general it does not improve and may even degrade transient performance. Improperly applied, it can produce an unstable control system. It is best used in conjunction with other control modes. 5.3 Proportional-Plus-Integral Control

Integral control raised the order of the system by 1 in the preceding examples but did not give a characteristic equation with enough flexibility to achieve acceptable transient behavior. The instantaneous response of proportional control action might introduce enough variability into the coefficients of the characteristic equation to allow both steady-state and transient specifications to be satisfied. This is the basis for using proportional-plus-integral control (PI control). The algorithm for this two-mode control is F (s) = KP E(s) +

KI E(s) s

(19)

The integral action provides an automatic, not manual, reset of the controller in the presence of a disturbance. For this reason, it is often called reset action. The algorithm is sometimes expressed as F (s) = KP

1+

1 TI s

E(s)

(20)

where TI is the reset time. The reset time is the time required for the integral action signal to equal that of the proportional term if a constant error exists (a hypothetical situation). The reciprocal of reset time is expressed as repeats per minute and is the frequency with which the integral action repeats the proportional correction signal. The proportional control gain must be reduced when used with integral action. The integral term does not react instantaneously to a zero-error signal but continues to correct, which tends to cause oscillations if the designer does not take this effect into account. PI Control of a First-Order System PI action applied to the speed controller of Fig. 20 gives the diagram shown in Fig. 21 with G(s) = KP + KI /s. The gains KP and KI are related to the component gains, as before. The system is stable for positive values of KP and KI . For r (s) = 1/s, ωss = 1, and the offset error is zero, as with integral action only. Similarly, the deviation due to a unit step disturbance is zero at√steady state. The damping ratio is ζ = (c + KP )/2 I KI . The presence of KP allows the damping ratio to be selected without fixing the value of the dominant time constant. For example, if the system is underdamped (ζ < 1), the time constant is τ = 2I /(c + KP ). The gain KP can be picked to obtain the desired time constant, while KI is used to set the damping ratio. A similar flexibility exists if ζ = 1. Complete description of the transient response requires that the numerator dynamics present in the transfer functions be accounted for.1,2 PI Control of a Second-Order System Integral control for the position servomechanism of Fig. 23 resulted in a third-order system that is unstable. With proportional action, the diagram becomes that of Fig. 22, with G(s) = KP + KI /s. The steady-state performance is acceptable, as before, if the system is assumed to be stable. This is true if the Routh criterion is satisfied, that is, if I , c, KP , and KI are positive and cKP − I KI > 0. The difficulty here occurs when the damping is slight. For small c, the gain KP must be large in order to satisfy the last condition, and this can be difficult to implement physically. Such a condition can also result in an unsatisfactory time constant. The root-locus method of Section 9 provides the tools for analyzing this design further.

BASIC CONTROL SYSTEMS DESIGN

775

5.4 Derivative Control

Integral action tends to produce a control signal even after the error has vanished, which suggests that the controller be made aware that the error is approaching zero. One way to accomplish this is to design the controller to react to the derivative of the error with derivative control action, which is F (s) = KD sE(s)

(21)

where KD is the derivative gain. This algorithm is also called rate action. It is used to damp out oscillations. Since it depends only on the error rate, derivative control should never be used alone. When used with proportional action, the following proportional-plusderivative (PD) control algorithm results: F (s) = (KP + KD s)E(s) = KP (1 + TD s)E(s) (22) where TD is the rate time or derivative time. With integral action included, the proportional-plus-integralplus-derivative (PID) control law is obtained: KI + KD s E(s) F (s) = KP + s

(23)

This is called a three-mode controller. PD Control of a Second-Order System The presence of integral action reduces steady-state error but tends to make the system less stable. There are applications of the position servomechanism in which a nonzero derivation resulting from the disturbance can be tolerated but an improvement in transient response over the proportional control result is desired. Integral action would not be required, but rate action can be added to improve the transient response. Application of PD control to this system gives the block diagram of Fig. 23 with G(s) = KP + KD s. The system is stable for positive values of KD and KP . The presence of rate action does not affect the steady-state response, and the steady-state results are identical to those with proportional control; namely, zero offset error and a deviation of

Fig. 24

−1/KP , due to the√disturbance. The damping ratio is ζ = (c √+ KD )/2 I KP . For proportional control, ζ = c/2 I KP . Introduction of rate action allows the proportional gain KP to be selected large to reduce the steady-state deviation, while KD can be used to achieve an acceptable damping ratio. The rate action also helps to stabilize the system by adding damping (if c = 0, the system with proportional control is not stable). The equivalent of derivative action can be obtained by using a tachometer to measure the angular velocity of the load. The block diagram is shown in Fig. 24. The gain of the amplifier–motor–potentiometer combination is K1 , and K2 is the tachometer gain. The advantage of this system is that it does not require signal differentiation, which is difficult to implement if signal noise is present. The gains K1 and K2 can be chosen to yield the desired damping ratio and steadystate deviation, as was done with KP and KI . 5.5 PID Control The position servomechanism design with PI control is not completely satisfactory because of the difficulties encountered when the damping c is small. This problem can be solved by the use of the full PID control law, as shown in Fig. 23 with G(s) = KP + KD s + KI /s. A stable system results if all gains are positive and if (c + KD )KP − I KI > 0. The presence of KD relaxes somewhat the requirement that KP be large to achieve stability. The steady-state errors are zero, and the transient response can be improved because three of the coefficients of the characteristic equation can be selected. To make further statements requires the root-locus technique presented in Section 9. Proportional, integral, and derivative actions and their various combinations are not the only control laws possible, but they are the most common. PID controllers will remain for some time the standard against which any new designs must compete. The conclusions reached concerning the performance of the various control laws are strictly true only for the plant model forms considered. These are the first-order model without numerator dynamics and the second-order model with a root at s = 0 and no numerator zeros. The analysis of a control law for any other

Tachometer feedback arrangement to replace PD control for position servo.1

776


linear system follows the preceding pattern. The overall system transfer functions are obtained, and all of the linear system analysis techniques can be applied to predict the system’s performance. If the performance is unsatisfactory, a new control law is tried and the process repeated. When this process fails to achieve an acceptable design, more systematic methods of altering the system’s structure are needed; they are discussed in later sections. We have used step functions as the test signals because they are the most common and perhaps represent the severest test of system performance. Impulse, ramp, and sinusoidal test signals are also employed. The type to use should be made clear in the design specifications. 6

The principle states that a power unit G(s) can be used with a feedback element H (s) to create a desired transfer function T (s). The power unit must have a gain high enough that |G(s)H (s)| 1, and the feedback elements must be selected so that H (s) = 1/T (s). This principle was used in Section 1 to explain the design of a feedback amplifier. 6.2 Electronic Controllers

The operational amplifier (op amp) is a high-gain amplifier with a high input impedance. A diagram of an op amp with feedback and input elements with impedances Tf (s) and Ti (s) is shown in Fig. 25. An approximate relation is

CONTROLLER HARDWARE

The control law must be implemented by a physical device before the control engineer’s task is complete. The earliest devices were purely kinematic and were mechanical elements such as gears, levers, and diaphragms that usually obtained their power from the controlled variable. Most controllers now are analog electronic, hydraulic, pneumatic, or digital electronic devices. We now consider the analog type. Digital controllers are covered starting in Section 10. 6.1 Feedback Compensation and Controller Design

Tf (s) Eo (s) =− Ei (s) Ti (s) The various control modes can be obtained by proper selection of the impedances. A proportional controller can be constructed with a multiplier, which uses two resistors, as shown in Fig. 26. An inverter is a multiplier circuit with Rf = Ri . It is sometimes needed because of the sign reversal property of the op amp. The multiplier circuit can be modified to act as an adder (Fig. 27).

Most controllers that implement versions of the PID algorithm are based on the following feedback principle. Consider the single-loop system shown in Fig. 1. If the open-loop transfer function is large enough that |G(s)H (s)| 1, the closed-loop transfer function is approximately given by T (s) =

G(s) 1 G(s) ≈ = (24) 1 + G(s)H (s) G(s)H (s) H (s)

Fig. 26

Fig. 25

Operational amplifier (op amp).1

Op-amp implementation of proportional control.1


Fig. 27

Fig. 28

Op-amp adder circuit.1

Op-amp implementation of PI control.1

PI control can be implemented with the circuit of Fig. 28. Figure 29 shows a complete system using op amps for PI control. The inverter is needed to

777

create an error detector. Many industrial controllers provide the operator with a choice of control modes, and the operator can switch from one mode to another when the process characteristics or control objectives change. When a switch occurs, it is necessary to provide any integrators with the proper initial voltages or else undesirable transients will occur when the integrator is switched into the system. Commercially available controllers usually have built-in circuits for this purpose. In theory, a differentiator can be created by interchanging the resistance and capacitance in the integrating op amp. The difficulty with this design is that no electrical signal is “pure.” Contamination always exists as a result of voltage spikes, ripple, and other transients generally categorized as “noise.” These highfrequency signals have large slopes compared with the more slowly varying primary signal, and thus they will dominate the output of the differentiator. In practice, this problem is solved by filtering out high-frequency signals, either with a low-pass filter inserted in cascade with the differentiator or by using a redesigned differentiator such as the one shown in Fig. 30. For the ideal PD controller, R1 = 0. The attenuation curve for the ideal controller breaks upward at ω = 1/R2 C with a slope of 20 dB/decade. The curve for the practical controller does the same but then becomes flat

Fig. 29 Implementation of PI controller using op amps. (a) Diagram of system. (b) Diagram showing how op amps are connected.2

778


The gain Kf is known only imprecisely and is sensitive to changes induced by temperature and other environmental factors. Also, the linear region over which Eq. (14) applies is very small. However, the device can be made useful by compensating it with feedback elements, as was illustrated with the electropneumatic valve positioner shown in Fig. 19. 6.4 Hydraulic Controllers

Fig. 30

Practical op-amp implementation of PD control.1

for ω > (R1 + R2 )/R1 R2 C. This provides the required limiting effect at high frequencies. PID control can be implemented by joining the PI and PD controllers in parallel, but this is expensive because of the number of op amps and power supplies required. Instead, the usual implementation is that shown in Fig. 31. The circuit limits the effect of frequencies above ω = 1/βR1 C1 . When R1 = 0, ideal PID control results. This is sometimes called the noninteractive algorithm because the effect of each of the three modes is additive, and they do not interfere with one another. The form given for R1 = 0 is the real or interactive algorithm. This name results from the fact that historically it was difficult to implement noninteractive PID control with mechanical or pneumatic devices. 6.3 Pneumatic Controllers

The nozzle–flapper introduced in Section 4 is a highgain device that is difficult to use without modification.

The basic unit for synthesis of hydraulic controllers is the hydraulic servomotor. The nozzle–flapper concept is also used in hydraulic controllers.5 A PI controller is shown in Fig. 32. It can be modified for proportional action. Derivative action has not seen much use in hydraulic controllers. This action supplies damping to the system, but hydraulic systems are usually highly damped intrinsically because of the viscous working fluid. PI control is the algorithm most commonly implemented with hydraulics. 7 FURTHER CRITERIA FOR GAIN SELECTION

Once the form of the control law has been selected, the gains must be computed in light of the performance specifications. In the examples of the PID family of control laws in Section 5, the damping ratio, dominant time constant, and steady-state error were taken to be the primary indicators of system performance in the interest of simplicity. In practice, the criteria are usually more detailed. For example, the rise time and maximum overshoot, as well as the other transient response specifications of the previous chapter, may be encountered. Requirements can also be stated in terms of frequency response characteristics, such as bandwidth, resonant frequency, and peak amplitude. Whatever specific from they take, a complete set of specifications for control system performance generally should include the following considerations for given forms of the command and disturbance inputs: 1. Equilibrium specifications (a) Stability (b) Steady-state error 2. Transient specifications (a) Speed of response (b) Form of response 3. Sensitivity specifications (a) Sensitivity to parameter variations (b) Sensitivity to model inaccuracies (c) Noise rejection (bandwidth, etc.)

Fig. 31 Practical control.1

op-amp

implementation

of

PID

In addition to these performance stipulations, the usual engineering considerations of initial cost, weight, maintainability, and so on must be taken into account. The considerations are highly specific to the chosen hardware, and it is difficult to deal with such issues in a general way.


779

Fig. 32 Hydraulic implementation of PI control.1

Two approaches exist for designing the controller. The proper one depends on the quality of the analytical description of the plant to be controlled. If an accurate model of the plant is easily developed, we can design a specialized controller for the particular application. The range of adjustment of controller gains in this case can usually be made small because the accurate plant model allows the gains to be precomputed with confidence. This technique reduces the cost of the controller and can often be applied to electromechanical systems. The second approach is used when the plant is relatively difficult to model, which is often the case in process control. A standard controller with several control modes and wide ranges of gains is used, and the proper mode and gain settings are obtained by testing the controller on the process in the field. This approach should be considered when the cost of developing an accurate plant model might exceed the cost of controller tuning in the field. Of course, the plant must be available for testing for this approach to be feasible. 7.1 Performance Indices

The performance criteria encountered thus far require a set of conditions to be specified—for example, one for steady-state error, one for damping ratio, and one for the dominant time constant. If there are many such conditions, and if the system is of high order with several gains to be selected, the design process can get quite complicated because transient and steady-state criteria tend to drive the design in different directions. An alternative approach is to specify the system’s desired performance by means of one analytical expression called a performance index. Powerful analytical and numerical methods are available that allow the gains to be systematically computed by minimizing (or maximizing) this index. To be useful, a performance index must be selective. The index must have a sharply defined extremum in the vicinity of the gain values that give the desired performance. If the numerical value of the index does not change very much for large changes in the

gains from their optimal values, the index will not be selective. Any practical choice of a performance index must be easily computed, either analytically, numerically, or experimentally. Four common choices for an index are the following:  ∞    |e(t)| dt (IAE index) (25)    0    ∞     t|e(t)| dt (ITAE index) (26)   0 J = ∞     [e(t)]2 dt (ISE index) (27)    0   ∞      t[e(t)]2 dt (ITSE index) (28)  0

where e(t) is the system error. This error usually is the difference between the desired and the actual values of the output. However, if e(t) does not approach zero as t → ∞, the preceding indices will not have finite values. In this case, e(t) can be defined as e(t) = c(∞) − c(t), where c(t) is the output variable. If the index is to be computed numerically or experimentally, the infinite upper limit can be replaced by a time tf large enough that e(t) is negligible for t > tf . The integral absolute-error (IAE) criterion (25) expresses mathematically that the designer is not concerned with the sign of the error, only its magnitude. In some applications, the IAE criterion describes the fuel consumption of the system. The index says nothing about the relative importance of an error occurring late in the response versus an error occurring early. Because of this, the index is not as selective as the integral-of-time-multiplied absolute-error (ITAE) criterion (26). Since the multiplier t is small in the early stages of the response, this index weights early errors less heavily than later errors. This makes sense physically. No system can respond instantaneously,

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and the index is lenient accordingly, while penalizing any design that allows a nonzero error to remain for a long time. Neither criterion allows highly underdamped or highly overdamped systems to be optimum. The ITAE criterion usually results in a system whose step response has a slight overshoot and well-damped oscillations. The integral squared-error (ISE) and integral-oftime-multiplied squared-error (ITSE) criteria are analogous to the IAE and ITAE criteria, except that the square of the error is employed for three reasons: (a) in some applications, the squared error represents the system’s power consumption; (b) squaring the error weights large errors much more heavily than small errors; (c) the squared error is much easier to handle analytically. The derivative of a squared term is easier to compute than that of an absolute value and does not have a discontinuity at e = 0. These differences are important when the system is of high order with multiple error terms. The closed-form solution for the response is not required to evaluate a performance index. For a given set of parameter values, the response and the resulting index value can be computed numerically. The optimum solution can be obtained using systematic computer search procedures; this makes this approach suitable for use with nonlinear systems. 7.2 Optimal-Control Methods Optimal-control theory includes a number of algorithms for systematic design of a control law to minimize a performance index, such as the following generalization of the ISE index, called the quadratic index:

J =

∞

(xT Qx + uT Ru) dt

(29)

0

where x and u are the deviations of the state and control vectors from the desired reference values. For example, in a servomechanism, the state vector might consist of the position and velocity, and the control vector might be a scalar—the force or torque produced by the actuator. The matrices Q and R are chosen by the designer to provide relative weighting for the elements of x and u. If the plant can be described by the linear state-variable model x˙ = Ax + Bu

(30)

y = Cx + Du

(31)

where y is the vector of outputs—for example, position and velocity—then the solution of this linearquadratic control problem is the linear control law: u = Ky

(32)

where K is a matrix of gains that can be found by several algorithms.1,6,7 A valid solution is guaranteed

to yield a stable closed-loop system, a major benefit of this method. Even if it is possible to formulate the control problem in this way, several practical difficulties arise. Some of the terms in (29) might be beyond the influence of the control vector u; the system is then uncontrollable. Also, there might not be enough information in the output equation (31) to achieve control, and the system is then unobservable. Several tests are available to check controllability and observability. Not all of the necessary state variables might be available for feedback or the feedback measurements might be noisy or biased. Algorithms known as observers, state reconstructors, estimators, and digital filters are available to compensate for the missing information. Another source of error is the uncertainty in the values of the coefficient matrices A, B, C, and D. Identification schemes can be used to compare the predicted and the actual system performance and to adjust the coefficient values “online.” 7.3 Ziegler–Nichols Rules The difficulty of obtaining accurate transfer function models for some processes has led to the development of empirically based rules of thumb for computing the optimum gain values for a controller. Commonly used guidelines are the Ziegler–Nichols rules, which have proved so helpful that they are still in use 50 years after their development. The rules actually consist of two separate methods. The first method requires the open-loop step response of the plant, while the second uses the results of experiments performed with the controller already installed. While primarily intended for use with systems for which no analytical model is available, the rules are also helpful even when a model can be developed. Ziegler and Nichols developed their rules from experiments and analysis of various industrial processes. Using the IAE criterion with a unit step response, they found that controllers adjusted according to the following rules usually had a step response that was oscillatory but with enough damping so that the second overshoot was less than 25% of the first (peak) overshoot. This is the quarter-decay criterion and is sometimes used as a specification. The first method is the process reaction method and relies on the fact that many processes have an open-loop step response like that shown in Fig. 33. This is the process signature and is characterized by two parameters, R and L, where R is the slope of a line tangent to the steepest part of the response curve and L is the time at which this line intersects the time axis. First- and second-order linear systems do not yield positive values for L, and so the method cannot be applied to such systems. However, third- and higher order linear systems with sufficient damping do yield such a response. If so, the Ziegler–Nichols rules recommend the controller settings given in Table 2. The ultimate-cycle method uses experiments with the controller in place. All control modes except


Fig. 33

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Process signature for unit step input.1

proportional are turned off, and the process is started with the proportional gain KP set at a low value. The gain is slowly increased until the process begins to exhibit sustained oscillations. Denote the period of this oscillation by Pu and the corresponding ultimate gain by KP u . The Ziegler–Nichols recommendations are given in Table 2 in terms of these parameters. The proportional gain is lower for PI control than for proportional control and is higher for PID control because integral action increases the order of the system and thus tends to destabilize it; thus, a lower gain is needed. On the other hand, derivative action tends to stabilize the system; hence, the proportional gain can be increased without degrading the stability characteristics. Because the rules were developed for a typical case out of many types of processes, final tuning of the gains in the field is usually necessary. 7.4 Nonlinearities and Controller Performance

All physical systems have nonlinear characteristics of some sort, although they can often be modeled as linear systems provided the deviations from the linearization reference condition are not too great. Under certain

conditions, however, the nonlinearities have significant effects on the system’s performance. One such situation can occur during the startup of a controller if the initial conditions are much different from the reference condition for linearization. The linearized model is then not accurate, and nonlinearities govern the behavior. If the nonlinearities are mild, there might not be much of a problem. Where the nonlinearities are severe, such as in process control, special consideration must be given to startup. Usually, in such cases, the control signal sent to the final control elements is manually adjusted until the system variables are within the linear range of the controller. Then the system is switched into automatic mode. Digital computers are often used to replace the manual adjustment process because they can be readily coded to produce complicated functions for the startup signals. Care must also be taken when switching from manual to automatic. For example, the integrators in electronic controllers must be provided with the proper initial conditions. 7.5

Reset Windup

In practice, all actuators and final control elements have a limited operating range. For example, a motor–amplifier combination can produce a torque proportional to the input voltage over only a limited range. No amplifier can supply an infinite current; there is a maximum current and thus a maximum torque that the system can produce. The final control elements are said to be overdriven when they are commanded by the controller to do something they cannot do. Since the limitations of the final control elements are ultimately due to the limited rate at which they can supply energy, it is important that all system performance specifications and controller designs be consistent with the energy delivery capabilities of the elements to be used. Controllers using integral action can exhibit the phenomenon called reset windup or integrator buildup when overdriven, if they are not properly designed. For a step change in set point, the proportional term

Table 2 Ziegler–Nichols Rules

1 + TD s Controller transfer function G(s) = Kp 1 + TI s Control Mode P control PI control

Process Reaction Method 1 RL 0.9 Kp = RL Kp =

TI = 3.3L PID control

Kp =

1.2 RL

TI = 2L TD = 0.5L

Ultimate-Cycle Method Kp = 0.5Kpu Kp = 0.45Kpu TI = 0.83Pu Kp = 0.6Kpu TI = 0.5Pu TD = 0.125Pu

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responds instantly and saturates immediately if the setpoint change is large enough. On the other hand, the integral term does not respond as fast. It integrates the error signal and saturates some time later if the error remains large for a long enough time. As the error decreases, the proportional term no longer causes saturation. However, the integral term continues to increase as long as the error has not changed sign, and thus the manipulated variable remains saturated. Even though the output is very near its desired value, the manipulated variable remains saturated until after the error has reversed sign. The result can be an undesirable overshoot in the response of the controlled variable. Limits on the controller prevent the voltages from exceeding the value required to saturate the actuator and thus protect the actuator, but they do not prevent the integral buildup that causes the overshoot. One way to prevent integrator buildup is to select the gains so that saturation will never occur. This requires knowledge of the maximum input magnitude that the

system will encounter. General algorithms for doing this are not available; some methods for low-order systems are presented in Ref. 1, Chapter 7; Ref. 2, Chapter 7, and Ref. 4, Chapter 11. Integrator buildup is easier to prevent when using digital control; this is discussed in Section 10. 8 COMPENSATION AND ALTERNATIVE CONTROL STRUCTURES

A common design technique is to insert a compensator into the system when the PID control algorithm can be made to satisfy most but not all of the design specifications. A compensator is a device that alters the response of the controller so that the overall system will have satisfactory performance. The three categories of compensation techniques generally recognized are series compensation, parallel (or feedback ) compensation, and feedforward compensation. The three structures are loosely illustrated in Fig. 34,

Fig. 34 General structures of three compensation types: (a) series; (b) parallel (or feedback); (c) feedforward. Compensator transfer function is Gc (s).1


where we assume the final control elements have a unity transfer function. The transfer function of the controller is G1 (s). The feedback elements are represented by H (s), and the compensator by Gc (s). We assume that the plant is unalterable, as is usually the case in control system design. The choice of compensation structure depends on what type of specifications must be satisfied. The physical devices used as compensators are similar to the pneumatic, hydraulic, and electrical devices treated previously. Compensators can be implemented in software for digital control applications. 8.1 Series Compensation

The most commonly used series compensators are the lead, the lag, and the lead–lag compensators. Electrical implementations of these are shown in Fig. 35. Other physical implementations are available. Generally, the lead compensator improves the speed of response; the lag compensator decreases the steadystate error; and the lead–lag affects both. Graphical aids, such as the root-locus and frequency response plots, are usually needed to design these compensators

Fig. 35

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(Ref. 1, Chapter 8; Ref. 2, Chapter 9; and Ref. 4, Chapter 11). 8.2 Feedback Compensation and Cascade Control The use of a tachometer to obtain velocity feedback, as in Fig. 24, is a case of feedback compensation. The feedback compensation principle of Fig. 3 is another. Another form is cascade control, in which another controller is inserted within the loop of the original control system (Fig. 36). The new controller can be used to achieve better control of variables within the forward path of the system. Its set point is manipulated by the first controller. Cascade control is frequently used when the plant cannot be satisfactorily approximated with a model of second order or lower. This is because the difficulty of analysis and control increases rapidly with system order. The characteristic roots of a second-order system can easily be expressed in analytical form. This is not so for third order or higher, and few general design rules are available. When faced with the problem of controlling a high-order system, the designer should first see if the performance requirements can be

Passive electrical compensators: (a) lead; (b) lag; (c) lead–lag.

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Fig. 36 Cascade control structure.

relaxed so that the system can be approximated with a low-order model. If this is not possible, the designer should attempt to divide the plant into subsystems, each of which is second order or lower. A controller is then designed for each subsystem. An application using cascade control is given in Section 11. 8.3 Feedforward Compensation The control algorithms considered thus far have counteracted disturbances by using measurements of the output. One difficulty with this approach is that the effects of the disturbance must show up in the output of the plant before the controller can begin to take action. On the other hand, if we can measure the disturbance, the response of the controller can be improved by using the measurement to augment the control signal sent from the controller to the final control elements. This is the essence of feedforward compensation of the disturbance, as shown in Fig. 34c. Feedforward compensation modified the output of the main controller. Instead of doing this by measuring the disturbance, another form of feedforward compensation utilizes the command input. Figure 37 is an example of this approach. The closed-loop transfer function is

Kf + K (s) = r (s) Is + c + K

(33)

For a unit step input, the steady-state output is ωss = (Kf + K)/(c + K). Thus, if we choose the feedforward gain Kf to be Kf = c, then ωss = 1 as desired,

Fig. 37

and the error is zero. Note that this form of feedforward compensation does not affect the disturbance response. Its effectiveness depends on how accurately we know the value of c. A digital application of feedforward compensation is presented in Section 11. 8.4 State-Variable Feedback

There are techniques for improving system performance that do not fall entirely into one of the three compensation categories considered previously. In some forms these techniques can be viewed as a type of feedback compensation, while in other forms they constitute a modification of the control law. Statevariable feedback (SVFB) is a technique that uses information about all the system’s state variables to modify either the control signal or the actuating signal. These two forms are illustrated in Fig. 38. Both forms require that the state vector x be measurable or at least derivable from other information. Devices or algorithms used to obtain state-variable information other than directly from measurements are variously termed state reconstructors, estimators, observers, or filters in the literature. 8.5 Pseudoderivative Feedback

Pseudoderivative feedback (PDF) is an extension of the velocity feedback compensation concept of Fig. 24.1,2 It uses integral action in the forward path plus an internal feedback loop whose operator H (s) depends on the plant (Fig. 39). For G(s) = 1/(I s + c), H (s) = K1 . For G(s) = 1/I s 2 , H (s) = K1 + K2 s.

Feedforward compensation of command input to augment proportional control.2


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Fig. 38 Two forms of state-variable feedback: (a) internal compensation of the control signal; (b) modification of the actuating signal.1

Fig. 39 Structure of pseudoderivative feedback.

The primary advantage of PDF is that it does not need derivative action in the forward path to achieve the desired stability and damping characteristics. 9 GRAPHICAL DESIGN METHODS

Higher order models commonly arise in control systems design. For example, integral action is often used with a second-order plant, and this produces a third-order system to be designed. Although algebraic solutions are available for third- and fourth-order polynomials, these solutions are cumbersome for design purposes. Fortunately, there exist graphical techniques to aid the designer. Frequency response plots of both the open- and closed-loop transfer functions are useful. The Bode plot and the Nyquist plot present the frequency response information in different forms. Each form has its own advantages. The root-locus plot shows the location of the characteristic roots for a range of values of some parameters, such as a controller gain. The design of two-position and other nonlinear control systems is facilitated by the describing function, which is a linearized approximation based on the frequency response of the controller. Graphical design methods are discussed in more detail in Refs. 1–4.

attention on the region around the point −1 + i0 on the polar plot of the open-loop transfer function. Figure 40 shows the polar plot of the open-loop transfer function of an arbitrary system that is assumed to be openloop stable. The Nyquist stability theorem is stated as follows: A system is closed-loop stable if and only if the point −1 + i0 lies to the left of the open-loop Nyquist plot relative to an observer traveling along the plot in the direction of increasing frequency ω. Therefore, the system described by Fig. 39 is closedloop stable. The Nyquist theorem provides a convenient measure of the relative stability of a system. A measure of the proximity of the plot to the −1 + i0 point is given by the angle between the negative real axis and a line from the origin to the point where the plot crosses the unit circle (see Fig. 39). The frequency corresponding to this intersection is denoted ωg . This angle is the phase margin (PM) and is positive when measured down from the negative real axis. The phase margin is the phase at the frequency ωg where the magnitude ratio or “gain” of G(iω)H (iω) is unity, or 0 decibels (dB). The frequency ωp , the phase crossover frequency, is the frequency at which the phase angle is −180◦ . The gain margin (GM) is

9.1 Nyquist Stability Theorem

The Nyquist stability theorem is a powerful tool for linear system analysis. If the open-loop system has no poles with positive real parts, we can concentrate our

Fig. 40 Nyquist plot for a stable system.1

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the difference in decibels between the unity gain condition (0 dB) and the value of |G(ωp )H (ωp )| decibels at the phase crossover frequency ωp . Thus, Gain margin = −|G(ωp )H (ωp )|

(dB)

(34)

Its magnitude and phase angle are |G(iω)H (iω)| = |P (iω)||e−iωD | = |P (iω)| ∠G(iω)H (iω) = ∠P (iω) + ∠e−iωD = ∠P (iω) − ωD

A system is stable only if the phase and gain margins are both positive. The phase and gain margins can be illustrated on the Bode plots shown in Fig. 41. The phase and gain margins can be stated as safety margins in the design specifications. A typical set of such specifications is as follows: Gain margin ≥ 8 dB

Phase margin ≥ 30

◦

Gain margin ≥ 6 dB

Phase margin ≥ 40

◦

(35) In common design situations, only one of these equalities can be met, and the other margin is allowed to be greater than its minimum value. It is not desirable to make the margins too large because this results in a low gain, which might produce sluggish response and a large steady-state error. Another commonly used set of specifications is

(36) The 6-dB limit corresponds to the quarter amplitude decay response obtained with the gain settings given by the Ziegler–Nichols ultimate-cycle method (Table 2). 9.2 Systems with Dead-Time Elements

The Nyquist theorem is particularly useful for systems with dead-time elements, especially when the plant is of an order high enough to make the root-locus method cumbersome. A delay D in either the manipulated variable or the measurement will result in an open-loop transfer function of the form G(s)H (s) = e−Ds P (s)

(37)

Fig. 41 Bode plot showing definitions of phase and gain margin.1

(38)

(39)

Thus, the dead time decreases the phase angle proportionally to the frequency ω, but it does not change the gain curve. This makes the analysis of its effects easier to accomplish with the open-loop frequency response plot. 9.3 Open-Loop Design for PID Control Some general comments can be made about the effects of proportional, integral, and derivative control actions on the phase and gain margins. Proportional action does not affect the phase curve at all and thus can be used to raise or lower the open-loop gain curve until the specifications for the gain and phase margins are satisfied. If integral action or derivative action is included, the proportional gain is selected last. Therefore, when using this approach to the design, it is best to write the PID algorithm with the proportional gain factored out, as

F (s) = KP

1 1+ + TD s E(s) TI s

(40)

Derivative action affects both the phase and gain curves. Therefore, the selection of the derivative gain is more difficult than the proportional gain. The increase in phase margin due to the positive phase angle introduced by Derivative action is partly negated by the derivative gain, which reduces the gain margin. Increasing the derivative gain increases the speed of response, makes the system more stable, and allows a larger proportional gain to be used to improve the system’s accuracy. However, if the phase curve is too steep near −180◦ , it is difficult to use Derivative action to improve the performance. Integral action also affects both the gain and phase curves. It can be used to increase the open-loop gain at low frequencies. However, it lowers the phase crossover frequency ωp and thus reduces some of the benefits provided by derivative action. If required, the derivative action term is usually designed first, followed by integral action and proportional action, respectively. The classical design methods based on the Bode plots obviously have a large component of trial and error because usually both the phase and gain curves must be manipulated to achieve an acceptable design. Given the same set of specifications, two designers can use these methods and arrive at substantially different designs. Many rules of thumb and ad hoc procedures have been developed, but a general foolproof procedure does not exist. However, an experienced designer can often obtain a good design quickly with these


techniques. The use of a computer plotting routine greatly speeds up the design process.

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As another example, let the plant transfer function be

9.4 Design with Root Locus

The effect of derivative action as a series compensator can be seen with the root locus. The term 1 + TD s in Fig. 32 can be considered as a series compensator to the proportional controller. The derivative action adds an open-loop zero at s = −1/TD . For example, a plant with the transfer function 1/s(s + 1)(s + 2), when subjected to proportional control, has the root locus shown in Fig. 42a. If the proportional gain is too high, the system will be unstable. The smallest achievable time constant corresponds to the root s = −0.42 and is τ = 1/0.42 = 2.4. If derivative action is used to put an open-loop zero at s = −1.5, the resulting root locus is given by Fig. 42b. The derivative action prevents the system from becoming unstable and allows a smaller time constant to be achieved (τ can be made close to 1/0.75 = 1.3 by using a high proportional gain). The integral action in PI control can be considered to add an open-loop pole at s = 0 and a zero at s = −1/TI . Proportional control of the plant 1/(s + 1)(s + 2) gives a root locus like that shown in Fig. 43, with a = 1 and b = 2. A steady-state error will exist for a step input. With the PI compensator applied to this plant, the root locus is given by Fig. 42b, with TI = 23 . The steady-state error is eliminated, but the response of the system has been slowed because the dominant paths of the root locus of the compensated system lie closer to the imaginary axis than those of the uncompensated system.

Fig. 42

GP (s) =

1 s 2 + a2 s + a1

(41)

where a1 > 0 and a2 > 0. PI control applied to this plant gives the closed-loop command transfer function T1 (s) =

KP s + KI s 3 + a2 s 2 + (a1 + KP )s + KI

(42)

Note that the Ziegler–Nichols rules cannot be used to set the gains KP and KI . The second-order plant, Eq. (41), does not have the S-shaped signature of Fig. 33, so the process reaction method does not apply. The ultimatecycle method requires KI to be set to zero and the ultimate gain KP u determined. With KI = 0 in Eq. (42) the resulting system is stable for all KP > 0, and thus a positive ultimate gain does not exist. Take the form of the PI control law given by Eq. (42) with TD = 0, and assume that the characteristic roots of the plant (Fig. 44) are real values −r1 and −r2 such that −r2 < −r1 . In this case the open-loop transfer function of the control system is G(s)H (s) =

KP (s + 1/TI ) s(s + r1 )(s + r2 )

(43)

One design approach is to select TI and plot the locus with KP as the parameter. If the zero at s = −1/TI

(a) Root-locus plot for s(s + 1)(s + 2) + K = 0, for K ≥ 0. (b) The effect of PD control with TD =

2 3.

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Fig. 43


Root-locus plot for (s + a)(s + b) + K = 0.

is located to the right of s = −r1 , the dominant time constant cannot be made as small as is possible with the zero located between the poles at s = −r1 and s = −r2 (Fig. 44). A large integral gain (small TI and/or large KP ) is desirable for reducing the overshoot due to a disturbance, but the zero should not be placed to the left of s = −r2 because the dominant time constant will be larger than that obtainable with the placement shown in Fig. 44 for large values of KP . Sketch the root-locus plots to see this. A similar situation exists if the poles of the plant are complex. The effects of the lead compensator in terms of time domain specifications (characteristic roots) can be shown with the root-locus plot. Consider the second-order plant with the real distinct roots s = −α,

Fig. 44 plant.

Root-locus plot for PI control of a second-order

s = −β. The root locus for this system with proportional control is shown in Fig. 45a. The smallest dominant time constant obtainable is τ1 , marked in the figure. A lead compensator introduces a pole at s = −1/T and a zero at s = −1/aT , and the root locus becomes that shown in Fig. 45b. The pole and zero introduced by the compensator reshape the locus so that a smaller dominant time constant can be obtained. This is done by choosing the proportional gain high enough to place the roots close to the asymptotes.

Fig. 45 Effects of series lead and lag compensators: (a) uncompensated system’s root locus; (b) root locus with lead compensation; (c) root locus with lag compensation.1


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With reference to the proportional control system whose root locus is shown in Fig. 45a, suppose that the desired damping ratio ζ1 and desired time constant τ1 are obtainable with a proportional gain of KP 1 , but the resulting steady-state error αβ/(αβ + KP 1 ) due to a step input is too large. We need to increase the gain while preserving the desired damping ratio and time constant. With the lag compensator, the root locus is as shown in Fig. 45c. By considering specific numerical values, one can show that for the compensated system, roots with a damping ratio ζ1 correspond to a high value of the proportional gain. Call this value KP 2 . Thus KP 2 > KP 1 , and the steady-state error will be reduced. If the value of T is chosen large enough, the pole at s = −1/T is approximately canceled by the zero at s = −1/aT , and the open-loop transfer function is given approximately by G(s)H (s) =

aKP (s + α)(s + β)

(44)

Thus, the system’s response is governed approximately by the complex roots corresponding to the gain value KP 2 . By comparing Fig. 45a with 45c, we see that the compensation leaves the time constant relatively unchanged. From Eq. (44) it can be seen that since a < 1, KP can be selected as the larger value KP 2 . The ratio of KP 1 to KP 2 is approximately given by the parameter a. Design by pole–zero cancellation can be difficult to accomplish because a response pattern of the system is essentially ignored. The pattern corresponds to the behavior generated by the canceled pole and zero, and this response can be shown to be beyond the influence of the controller. In this example, the canceled pole gives a stable response because it lies in the left-hand plane. However, another input not modeled here, such as a disturbance, might excite the response and cause unexpected behavior. The designer should therefore proceed with caution. None of the physical parameters of the system are known exactly, so exact pole–zero cancellation is not possible. A root-locus study of the effects of parameter uncertainty and a simulation study of the response are often advised before the design is accepted as final.

10 PRINCIPLES OF DIGITAL CONTROL Digital control has several advantages over analog devices. A greater variety of control algorithms is possible, including nonlinear algorithms and ones with time-varying coefficients. Also, greater accuracy is possible with digital systems. However, their additional hardware complexity can result in lower reliability, and their application is limited to signals whose time variation is slow enough to be handled by the samplers and the logic circuitry. This is now less of a problem because of the large increase in the speed of digital systems. 10.1 Digital Controller Structure The basic structure of a single-loop controller is shown in Fig. 46. The computer with its internal clock drives the digital-to-analog (D/A) and analogto-digital (A/D) converters. It compares the command signals with the feedback signals and generates the control signals to be sent to the final control elements. These control signals are computed from the control algorithm stored in the memory. Slightly different structures exist, but Fig. 46 shows the important aspects. For example, the comparison between the command and feedback signals can be done with analog elements, and the A/D conversion made on the resulting error signal. The software must also provide for interrupts, which are conditions that call for the computer’s attention to do something other than computing the control algorithm. The time required for the control system to complete one loop of the algorithm is the time T , the sampling time of the control system. It depends on the time required for the computer to calculate the control algorithm and on the time required for the interfaces to convert data. Modern systems are capable of very high rates, with sample times under 1 µs. In most digital control applications, the plant is an analog system, but the controller is a discrete-time system. Thus, to design a digital control system, we must either model the controller as an analog system or model the plant as a discrete-time system. Each approach has its own merits, and we will examine both. If we model the controller as an analog system, we use methods based on differential equations to

Fig. 46 Structure of digital control system.1

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compute the gains. However, a digital control system requires difference equations to describe its behavior. Thus, from a strictly mathematical point of view, the gain values we will compute will not give the predicted response exactly. However, if the sampling time is small compared to the smallest time constant in the system, then the digital system will act like an analog system, and our designs will work properly. Because most physical systems of interest have time constants greater than 1 ms and controllers can now achieve sampling times less than 1µs, controllers designed with analog methods will often be adequate. 10.2

and subtracting this from (46) to obtain f (tk ) = f (tk−1 ) + KP [e(tk ) − e(tk−1 )] + KI T e(tk ) (47) This form—called the incremental or velocity algorithm—is well suited for incremental output devices such as stepper motors. Its use also avoids the problem of integrator buildup, the condition in which the actuator saturates but the control algorithm continues to integrate the error. The simplest approximation to the derivative is the first-order difference approximation e(tk ) − e(tk−1 ) de ≈ dt T

Digital Forms of PID Control

There are a number of ways that PID control can be implemented in software in a digital control system because the integral and derivative terms must be approximated with formulas chosen from a variety of available algorithms. The simplest integral approximation is to replace the integral with a sum of rectangular areas. With this rectangular approximation, the error integral is calculated as

(k+1)T

The corresponding PID approximation using the rectangular integral approximation is f (tk ) = KP e(tk ) + KI T

e(t) dt ≈ T e(0) + T e(t1 ) + T e(t2 )

+ · · · + T e(tk ) = T

k

e(ti )

(45)

i =0

f (tk ) = KP e(tk ) + KI T

k

KD [e(tk ) − e(tk−1 )] T

(49)

(k+1)T

e(t) dt ≈ T

0

k 1 i=0

2

[e(ti+1 + e(ti )]

(50)

The accuracy of the derivative approximation can be improved by using values of the sampled error signal at more instants. Using the four-point centraldifference method (Refs. 1 and 2), the derivative term is approximated by 1 de ≈ [e(tk ) + 3e(tk−1 ) − 3e(tk−2 ) − e(tk−3 )] dt 6T The derivative action is sensitive to the resulting rapid change in the error samples that follows a step input. This effect can be eliminated by reformulating the control algorithm as follows (Refs. 1 and 2): f (tk ) = f (tk−1 ) + KP [c(tk−1 ) − c(tk )]

e(ti )

(46)

i =0

k−1 i=0

+ KI T [r(tk ) − c(tk )] +

This can be written in a more efficient form by noting that f (tk−1 ) = KP e(tk−1 ) + KI T

e(ti )

The accuracy of the integral approximation can be improved by substituting a more sophisticated algorithm, such as the following trapezoidal rule:

where tk = kT and the width of each rectangle is the sampling time T = ti+1 − ti . The times ti are the times at which the computer updates its calculation of the control algorithm after receiving an updated command signal and an updated measurement from the sensor through the A/D interfaces. If the time T is small, then the value of the sum in (45) is close to the value of the integral. After the control algorithm calculation is made, the calculated value of the control signal f (tk ) is sent to the actuator via the output interface. This interface includes a D/A converter and a hold circuit that “holds” or keeps the analog voltage corresponding to the control signal applied to the actuator until the next updated value is passed along from the computer. The simplest digital form of PI control uses (45) for the integral term. It is

k i=0

+

0

(48)

e(ti )

KD [−c(tk ) + 2c(tk−1 ) − c(tk−2 )] T

(51)

where r(tk ) is the command input and c(tk ) is the variable being controlled. Because the command input r(tk ) appears in this algorithm only in the integral term, we cannot apply this algorithm to PD control; that is, the integral gain KI must be nonzero.


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11 UNIQUELY DIGITAL ALGORITHMS

Development of analog control algorithms was constrained by the need to design physical devices that could implement the algorithm. However, digital control algorithms simply need to be programmable and are thus less constrained than analog algorithms. 11.1 Digital Feedforward Compensation

Classical control system design methods depend on linear models of the plant. With linearization we can obtain an approximately linear model, which is valid only over a limited operating range. Digital control now allows us to deal with nonlinear models more directly using the concepts of feedforward compensation discussed in Section 8. Computed Torque Method Figure 47 illustrates a variation of feedforward compensation of the disturbance called the computed torque method. It is used to control the motion of robots. A simple model of a robot arm is the following nonlinear equation:

algorithm to handle. The nonlinear torque calculations required to control multi-degree-of-freedom robots are very complicated and can be done only with a digital controller. Feedforward Command Compensation Computers can store lookup tables, which can be used to control systems that are difficult to model entirely with differential equations and analytical functions. Figure 48 shows a speed control system for an internal combustion engine. The fuel flow rate required to achieve a desired speed depends in a complicated way on many variables not shown in the figure, such as temperature, humidity, and so on. This dependence can be summarized in tables stored in the control computer and can be used to estimate the required fuel flow rate. A PID algorithm can be used to adjust the estimate based on the speed error. This application is an example of feedforward compensation of the command input, and it requires a digital computer.

(52)

11.2 Control Design in z Plane There are two common approaches to designing a digital controller:

where θ is the arm angle, I is its inertia, mg is its weight, and L is the distance from its mass center to the arm joint where the motor acts. The motor supplies the torque T . To position the arm at some desired angle θr , we can use PID control on the angle error θr − θ . This works well if the arm angle θ is never far from the desired angle θr so that we can linearize the plant model about θr . However, the controller will work for large-angle excursions if we compute the nonlinear gravity torque term mgL sin θ and add it to the PID output. That is, part of the motor torque will be computed specifically to cancel the gravity torque, in effect producing a linear system for the PID

1. The performance is specified in terms of the desired continuous-time response, and the controller design is done entirely in the s plane, as with an analog controller. The resulting control law is then converted to discrete-time form, using approximations for the integral and derivative terms. This method can be successfully applied if the sampling time is small. The technique is widely used for two reasons. When existing analog controllers are converted to digital control, the form of the control law and the values of its associated gains are known to have been satisfactory. Therefore, the digital version

I θ¨ = T − mgL sin θ

Fig. 47 Computed torque method applied to robot arm control.

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Fig. 48 Feedforward compensation applied to engine control.

can use the same control law and gain values. Second, because analog design methods are well established, many engineers prefer to take this route and then convert the design into a discretetime equivalent. 2. The performance specifications are given in terms of the desired continuous-time response and/or desired root locations in the s plane. From these the corresponding root locations in the z plane are found and a discrete control law is designed. This method avoids the derivative and integral approximation errors that are inherent in the first method and is the preferred method when the sampling time T is large. However, the algebraic manipulations are more cumbersome. The second approach uses the z transform and pulse transfer functions. If we have an analog model of the plant, with its transfer function G(s), we can obtain its pulse transfer function G(z) by finding the z transform of the impulse response g(t) = L −1 [G(s)]; that is, G(z) = Z [g(t)]. A table of transforms facilitates this process; see Refs. 1 and 2. Figure 49a shows the basic elements of a digital control system. Figure 49b is an equivalent diagram with the analog transfer functions inserted. Figure 49c represents the same system in terms of pulse transfer functions. From the diagram we can find the closed-loop pulse transfer function. It is G(z)P (z) C(z) = R(z) 1 + G(z)P (z)

(53)

The variable z is related to the Laplace variable s by z = esT

(54)

If we know the desired root locations and the sampling time T , we can compute the z roots from this equation. Digital PI Control Design For example, the firstorder plant 1/(2s + 1) with a zero-order hold has the following pulse transfer function (Refs. 1 and 2):

P (z) =

1 − e−0.5T z − e−0.5T

(55)

Suppose we use a control algorithm described by the following pulse transfer function: G(z) =

F (z) K1 z + K2 K1 + K2 z−1 = = E(z) z−1 1 − z−1

(56)

The corresponding difference equation that the control computer must implement is f (tk ) = f (tk−1 ) + K1 e(tk ) + K2 e(tk−1 )

(57)

where e(tk ) = r(tk ) − c(tk ). By comparing (57) with (47), it can be seen that this is the digital equivalent of PI control, where KP = −K2 and KI = (K1 + K2 )/T . Using the form of G(z) given by (56), the closed-loop transfer function is (1 − b)(K1 z + K2 ) C(z) = 2 R(z) z + (K1 − 1 − b − bK1 )z + b + K2 − bK2 (58) where b = e−0.5T . If the design specifications call for τ = 1 and ζ = 1, then the desired s roots are s = −1, −1, and the analog PI gains required to achieve these roots are KP = 3 and KI = 2. Using a sampling time of


793

Fig. 49 Block diagrams of typical digital controller. (a) Diagram showing the components. (b) Diagram of the s-plane relations. (c) Diagram of the z-plane relations.

T = 0.1, the z roots must be z = e−0.1 , e−0.1 . To achieve these roots, the denominator of the transfer function (58) must be z2 − 2e−0.1 z + e−0.2 . Thus the control gains must be K1 = 2.903 and K2 = −2.717. These values of K1 and K2 correspond to KP = 2.72 and KI = 1.86, which are close to the PI gains computed for an analog controller. If we had used a sampling time smaller than 0.1, say T = 0.01, the values of KP and KI computed from K1 and K2 would be KP = 2.97 and KI = 1.98, which are even closer to the analog gain values. This illustrates the earlier claim that analog design methods can be used when the sampling time is small enough. Digital Series Compensation Series compensation can be implemented digitally by applying suitable discrete-time approximations for the derivative and

integral to the model represented by the compensator’s transfer function Gc (s). For example, the form of a lead or a lag compensator’s transfer function is Gc (s) =

s+c M(s) =K F (s) s+d

(59)

where m(t) is the actuator command and f (t) is the control signal produced by the main (PID) controller. The differential equation corresponding to (59) is m ˙ + dm = K(f˙ + cf )

(60)

Using the simplest approximation for the derivative, Eq. (48), we obtain the following difference equation

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which has the form

that the digital compensator must implement:

F (z) z2 − az + b = Kc E(z) z−1

m(tk ) − m(tk−1 ) + dm(tk ) T

f (tk ) − f (tk−1 ) =K + cf (tk ) T

where Kc , a, and b can be expressed in terms of KP , KI , KD , and T . Note that the algorithm has two zeros and one pole, which is fixed at z = 1. Sometimes the algorithm is expressed in the more general form

In the z plane, the equation becomes

1 − z−1 1 − z−1 M(z) + dM(z) = K F (z) + cF (z) T T (61) The compensator’s pulse transfer function is thus seen to be Gc (z) =

K(1 − z−1 ) + cT M(z) = F (z) 1 − z−1 + dT

which has the form Gc (z) = Kc

z+a z+b

(62)

where Kc , a, and b can be expressed in terms of K, c, d, and T if we wish to use analog design methods to design the compensator. When using commercial controllers, the user might be required to enter the values of the gain, the pole, and the zero of the compensator. The user must ascertain whether these values should be entered as s-plane values (i.e., K, c, and d) or as z-plane values (Kc , a, and b). Note that the digital compensator has the same number of poles and zeros as the analog compensator. This is a result of the simple approximation used for the derivative. Note that Eq. (61) shows that when we use this approximation, we can simply replace s in the analog transfer function with 1 − z−1 . Because the integration operation is the inverse of differentiation, we can replace 1/s with 1/(1 − z−1 ) when integration is used. [This is equivalent to using the rectangular approximation for the integral and can be verified by finding the pulse transfer function of the incremental algorithm (47) with KP = 0.] Some commercial controllers treat the PID algorithm as a series compensator, and the user is expected to enter the controller’s values, not as PID gains, but as pole and zero locations in the z plane. The PID transfer function is KI F (s) = KP + + KD s E(s) s

(63)

Making the indicated replacements for the s terms, we obtain KI F (z) = KP + + KD (1 − z−1 ) E(z) 1 − z−1

(64)

z2 − az + b F (z) = Kc F (z) z−c

(65)

to allow the user to select the pole as well. Digital compensator design can be done with frequency response methods or with the root-locus plot applied to the z plane rather than the s plane. However, when better approximations are used for the derivative and integral, the digital series compensator will have more poles and zeros than its analog counterpart. This means that the root-locus plot will have more root paths, and the analysis will be more difficult. This topic is discussed in more detail in Refs. 1–3 and 8. 11.3 Direct Design of Digital Algorithms Because almost any algorithm can be implemented digitally, we can specify the desired response and work backward to find the required control algorithm. This is the direct-design method. If we let D(z) be the desired form of the closed-loop transfer function C(z) / R(z) and solve for the controller transfer function G(z), we obtain

G(z) =

D(z) P (z)[1 − D(z)]

(66)

We can pick D(z) directly or obtain it from the specified input transform R(z) and the desired output transform C(z), because D(z) = C(z)/R(z). Finite-Settling-Time Algorithm This method can be used to design a controller to compensate for the effects of process dead time. A plant having such a response can often be approximately described by a first-order model with a dead-time element; that is,

GP (s) = K

e−Ds τs + 1

(67)

where D is the dead time. This model also approximately describes the S-shaped response curve used with the Ziegler–Nichols method (Fig. 33). When combined with a zero-order hold, this plant has the following pulse transfer function: P (z) = Kz−n

1−a z−a

(68)


795

where a = exp(−T /τ ) and n = D/T . If we choose D(z) = z−(n+1) , then with a step command input, the output c(k) will reach its desired value in n + 1 sample times, one more than is in the dead time D. This is the fastest response possible. From (66) the required controller transfer function is G(z) =

1 1 − az−1 K(1 − a) 1 − z−(n+1)

(69)

The corresponding difference equation that the control computer must implement is 1 [e(tk ) − ae(tk−1 )] K(1 − a) (70) This algorithm is called a finite-settling-time algorithm because the response reaches its desired value in a finite, prescribed time. The maximum value of the manipulated variable required by this algorithm occurs at t = 0 and is 1/K(1 − a). If this value saturates the actuator, this method will not work as predicted. Its success depends also on the accuracy of the plant model. f (tk ) = f (tk−n−1 ) +

Dahlin’s Algorithm This sensitivity to plant modeling errors can be reduced by relaxing the minimum response time requirement. For example, choosing D(z) to have the same form as P (z), namely,

D(z) = Kd z−n

1 − ad z − ad

(71)

we obtain from (66) the following controller transfer function: Kd (1 − ad ) 1 − az−1 −1 K(1 − a) 1 − ad z − Kd (1 − ad )z−(n+1) (72) This is Dahlin’s algorithm.3 The corresponding difference equation that the control computer must implement is G(z) =

f (tk ) = ad f (tk−1 ) + Kd (1 − ad )f (tk−n−1 ) +

Kd (1 − ad ) [e(tk ) − ae(tk−1 )] K(1 − a)

(73)

Normally we would first try setting Kd = K and ad = a, but since we might not have good estimates of K and a, we can use Kd and ad as tuning parameters to adjust the controller’s performance. The constant ad is related to the time constant τd of the desired response: ad = exp(−T /τd ). Choosing τd smaller gives faster response. Algorithms such as these are often used for system startup, after which the control mode is switched to PID, which is more capable of handling disturbances.

12 HARDWARE AND SOFTWARE FOR DIGITAL CONTROL

This section provides an overview of the general categories of digital controllers that are commercially available. This is followed by a summary of the software currently available for digital control and for control system design. 12.1

Digital Control Hardware Commercially available controllers have different capabilities, such as different speeds and operator interfaces, depending on their targeted application. Programmable Logic Controllers (PLCs) These are controllers that are programmed with relay ladder logic, which is based on Boolean algebra. Now designed around microprocessors, they are the successors to the large relay panels, mechanical counters, and drum programmers used up to the 1960s for sequencing control and control applications requiring only a finite set of output values (e.g., opening and closing of valves). Some models now have the ability to perform advanced mathematical calculations required for PID control, thus allowing them to be used for modulated control as well as finite-state control. There are numerous manufacturers of PLCs. Digital Signal Processors (DSPs) A modern development is the digital signal processor (DSP), which has proved useful for feedback control as well as signal processing.9 This special type of processor chip has separate buses for moving data and instructions and is constructed to perform rapidly the kind of mathematical operations required for digital filtering and signal processing. The separate buses allow the data and the instructions to move in parallel rather than sequentially. Because the PID control algorithm can be written in the form of a digital filter, DSPs can also be used as controllers. The DSP architecture was developed to handle the types of calculations required for digital filters and discrete Fourier transforms, which form the basis of most signal-processing operations. DSPs usually lack the extensive memory management capabilities of generalpurpose computers because they need not store large programs or large amounts of data. Some DSPs contain A/D and D/A converters, serial ports, timers, and other features. They are programmed with specialized software that runs on popular personal computers. Low-cost DSPs are now widely used in consumer electronics and automotive applications, with Texas Instruments being a major supplier. Motion Controllers Motion controllers are specialized control systems that provide feedback control for one or more motors. They also provide a convenient operator interface for generating the commanded trajectories. Motion controllers are particularly well suited for applications requiring coordinated motion of

796


two or more axes and for applications where the commanded trajectory is complicated. A higher level host computer might transmit required distance, speed, and acceleration rates to the motion controller, which then constructs and implements the continuous position profile required for each motor. For example, the host computer would supply the required total displacement, the acceleration and deceleration times, and the desired slew speed (the speed during the zero acceleration phase). The motion controller would generate the commanded position versus time for each motor. The motion controller also has the task of providing feedback control for each motor to ensure that the system follows the required position profile. Figure 50 shows the functional elements of a typical motion controller, such as those built by Galil Motion Control, Inc. Provision for both analog and digital input signals allows these controllers to perform other control tasks besides motion control. Compared to DSPs, such controllers generally have greater capabilities for motion control and have operator interfaces that are better suited for such applications. Motion controllers are available as plug-in cards for most computer bus types. Some are available as stand-alone units. Motion controllers use a PID control algorithm to provide feedback control for each motor (some manufacturers call this algorithm a “filter”). The user enters the values of the PID gains (some manufacturers provide preset gain values, which can be changed; others provide tuning software that assists in selecting the proper gain values). Such controllers also have

their own language for programming a variety of motion profiles and other applications. For example, they provide for linear and circular interpolation for two-dimensional coordinated motion, motion smoothing (to eliminate jerk), contouring, helical motion, and electronic gearing. The latter is a control mode that emulates mechanical gearing in software, in which one motor (the slave) is driven in proportion to the position of another motor (the master) or an encoder. Process Controllers Process controllers are designed to handle inputs from sensors, such as thermocouples, and outputs to actuators, such as valve positioners, that are commonly found in process control applications. Figure 51 illustrates the input–output capabilities of a typical process controller such as those manufactured by Honeywell, which is a major supplier of such devices. This device is a stand-alone unit designed to be mounted in an instrumentation panel. The voltage and current ranges of the analog inputs are those normally found with thermocouple-based temperature sensors. The current outputs are designed for devices like valve positioners, which usually require 4–20-mA signals. The controller contains a microcomputer with builtin math functions normally required for process control, such as thermocouple linearization, weighted averaging, square roots, ratio/bias calculations, and the PID control algorithm. These controllers do not have the same software and memory capabilities as desktop computers, but they are less expensive. Their operator interface consists of a small keypad with typically

Fig. 50 Functional diagram of motion controller.


Fig. 51

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Functional diagram of digital process controller.

fewer than 10 keys, a small graphical display for displaying bar graphs of the set points and the process variables, indicator lights, and an alphanumeric display for programming the controller. The PID gains are entered by the user. Some units allow multiple sets of gains to be stored; the unit can be programmed to switch between gain settings when certain conditions occur. Some controllers have an adaptive tuning feature that is supposed to adjust

Fig. 52

the gains to prevent overshoot in startup mode, to adapt to changing process dynamics, and to adapt to disturbances. However, at this time, adaptive tuning cannot claim a 100% success rate, and further research and development in adaptive control is needed. Some process controllers have more than one PID control loop for controlling several variables. Figure 52 illustrates a boiler feedwater control application for a controller with two PID loops arranged

Application of two-loop process controller for feedwater control.

798


in a cascade control structure. Loop 1 is the main or outer loop controller for maintaining the desired water volume in the boiler. It uses sensing of the steam flow rate to implement feedforward compensation. Loop 2 is the inner loop controller that directly controls the feedwater control valve. 12.2

challenging, partly because of the need to provide adequately for interrupts. Software packages are now available that provide real-time control capability, usually a form of the PID algorithm, that can be programmed through user-friendly graphical interfaces. Examples include the Galil motion controllers and the add-on modules for Labview and MATLAB.

Software for Digital Control

The software available to the modern control engineer is quite varied and powerful and can be categorized according to the following tasks: 1. Control algorithm design, gain selection, and simulation 2. Tuning 3. Motion programming 4. Instrumentation configuration 5. Read-time control functions Many analysis and simulation packages now contain algorithms of specific interest to control system designers. MATLAB is one such package that is widely used. It contains built-in functions for generating rootlocus and frequency response plots, system simulation, digital filtering, calculation of control gains, and data analysis. It can accept model descriptions in the form of transfer functions or as state-variable equations.1,4,10 Some manufacturers provide software to assist the engineer in sizing and selecting components. An example is the Motion Component Selector (MCS) sold by Galil Motion Control, Inc. It assists the engineer in computing the load inertia, including the effects of the mechanical drive, and then selects the proper motor and amplifier based on the user’s description of the desired motion profile. Some hardware manufacturers supply software to assist the engineer in selecting control gains and modifying (tuning) them to achieve good response. This might require that the system to be controlled be available for experiments prior to installation. Some controllers, such as some Honeywell process controllers, have an autotuning feature that adjusts the gains in real time to improve performance. Motion programming software supplied with motion controllers was mentioned previously. Some packages, such as Galil’s, allow the user to simulate a multiaxis system having more than one motor and to display the resulting trajectory. Instrumentation configuration software, such as LabView, provides specialized programming languages for interacting with instruments and for creating graphical real-time displays of instrument outputs. Until recently, development of real-time digital control software involved tedious programming, often in assembly language. Even when implemented in a higher level language, such as Fortran or C, programming real-time control algorithms can be very

12.3 Embedded Control Systems and Hardware-in-the Loop Testing

An embedded control system is a microprocessor and sensor suite designed to be an integral part of a product. The aerospace and automotive industries have used embedded controllers for some time, but the decreased cost of components now makes embedded controllers feasible for more consumer and biomedical applications. For example, embedded controllers can greatly increase the performance of orthopedic devices. One model of an artificial leg now uses sensors to measure in real time the walking speed, the knee joint angle, and the loading due to the foot and ankle. These measurements are used by the controller to adjust the hydraulic resistance of a piston to produce a stable, natural, and efficient gait. The controller algorithms are adaptive in that they can be tuned to an individual’s characteristics and their settings changed to accommodate different physical activities. Engines incorporate embedded controllers to improve efficiency. Embedded controllers in new active suspensions use actuators to improve on the performance of traditional passive systems consisting only of springs and dampers. One design phase of such systems is hardware-in-the-loop testing, in which the controlled object (the engine or vehicle suspension) is replaced with a real-time simulation of its behavior. This enables the embedded system hardware and software to be tested faster and less expensively than with the physical prototype and perhaps even before the prototype is available. Simulink, which is built on top of MATLAB and requires MATLAB to run, is often used to create the simulation model for hardware-in-the-loop testing. Some of the toolboxes available for MATLAB, such as the control systems toolbox, the signal-processing toolbox, and the DSP and fixed-point blocksets, are also useful for such applications. 13 SOFTWARE SUPPORT FOR CONTROL SYSTEM DESIGN

Software packages are available for graphical control system design methods and control system simulation. These greatly reduce the tedious manual computation, plotting, and programming formerly required for control system design and simulation. 13.1 Software for Graphical Design Methods

Several software packages are available to support graphical control system design methods. The most


799

popular of these is MATLAB, which has extensive capabilities for generation and interactive analysis of root-locus plots and frequency response plots. Some of these capabilities are discussed in Refs. 1 and 4. 13.2 Software for Control Systems Simulation

It is difficult to obtain closed-form expressions for system response when the model contains dead time or nonlinear elements that represent realistic control system behavior. Dead time (also called transport delay), rate limiters, and actuator saturation are effects that often occur in real control systems, and simulation is often the only way to analyze their response. Several software packages are available to support system simulation. One of the most popular is Simulink. Systems having dead-time elements are easily simulated in Simulink. Figure 53 shows a Simulink model for PID control of the plant 53/(3.44s 2 + 2.61s + 1), with a dead time between the output of the controller and the plant. The block implementing the dead-time transfer function e−Ds is called the transport delay block. When you run this model, you will see the response in the scope block. In addition to being limited by saturation, some actuators have limits on how fast they can react. This limitation is independent of the time constant of the actuator and might be due to deliberate restrictions placed on the unit by its manufacturer. An example is a flow control valve whose rate of opening and closing is controlled by a rate limiter. Simulink has such a block, and it can be used in series with the saturation block to

Fig. 53

model the valve behavior. Consider the model of the height h of liquid in a tank whose input is a flow rate qi . For specific parameter values, such a model has the form H (s)/Qi (s) = 2/(5s + 1). A Simulink model is shown in Figure 54 for a specific PI controller whose gains are KP = 4 and KI = 54 . The saturation block models the fact that the valve opening must be between 0 and 100%. The model enables us to experiment with the lower and upper limits of the rate limiter block to see its effect on the system performance. An introduction to Simulink is given in Refs. 4 and 10. Applications of Simulink to control system simulation are given in Ref. 4. 14

FUTURE TRENDS IN CONTROL SYSTEMS

Microprocessors have rejuvenated the development of controllers for mechanical systems. Currently, there are several applications areas in which new control systems are indispensable to the product’s success: 1. 2. 3. 4. 5. 6.

Active vibration control Noise cancellation Adaptive optics Robotics Micromachines Precision engineering

Most of the design techniques presented here comprise “classical” control methods. These methods are

Simulink model of system with transport delay.

Fig. 54 Simulink model of system with actuator saturation and rate limiter.

800


widely used because when they are combined with some testing and computer simulation, an experienced engineer can rapidly achieve an acceptable design. Modern control algorithms, such as state-variable feedback and the linear–quadratic optimal controller, have had some significant mechanical engineering applications—for example, in the control of aerospace vehicles. The current approach to multivariable systems like the one shown in Fig. 55 is to use classical methods to design a controller for each subsystem because they can often be modeled with low-order linearized models. The coordination of the various low-level controllers is a nonlinear problem. High-order, nonlinear, multivariable systems that cannot be controlled with classical methods cannot yet be handled by modern control theory in a general way, and further research is needed. In addition to the improvements, such as lower cost, brought on by digital hardware, microprocessors have allowed designers to incorporate algorithms of much greater complexity into control systems. The following is a summary of the areas currently receiving much attention in the control systems community. 14.1 Fuzzy Logic Control In classical set theory, an object’s membership in a set is clearly defined and unambiguous. Fuzzy logic control is based on a generalization of classical set theory to allow objects to belong to several sets with various degrees of membership. Fuzzy logic can be used to describe processes that defy precise definition or precise measurement, and thus it can be used to model

the inexact and subjective aspects of human reasoning. For example, room temperature can be described as cold, cool, just right, warm, or hot. Development of a fuzzy logic temperature controller would require the designer to specify the membership functions that describe “warm” as a function of temperature, and so on. The control logic would then be developed as a linguistic algorithm that models a human operator’s decision process (e.g., if the room temperature is “cold,” then “greatly” increase the heater output; if the temperature is “cool,” then increase the heater output “slightly”). Fuzzy logic controllers have been implemented in a number of applications. Proponents of fuzzy logic control point to its ability to convert a human operator’s reasoning process into computer code. Its critics argue that because all the controller’s fuzzy calculations must eventually reduce to a specific output that must be given to the actuator (e.g., a specific voltage value or a specific valve position), why not be unambiguous from the start, and define a “cool” temperature to be the range between 65◦ and 68◦ , for example? Perhaps the proper role of fuzzy logic is at the human operator interface. Research is active in this area, and the issue is not yet settled.11,12 14.2 Nonlinear Control Most real systems are nonlinear, which means that they must be described by nonlinear differential equations. Control systems designed with the linear control theory described in this chapter depend on a linearized approximation to the original nonlinear model. This

Fig. 55 Computer control system for a boiler-generator. Each important variable requires its own controller. Interaction between variables calls for coordinated control of all loops.1


linearization can be explicitly performed, or implicitly made, as when we use the small-angle approximation: sin θ ≈ θ . This approach has been enormously successful because a well-designed controller will keep the system in the operating range where the linearization was done, thus preserving the accuracy of the linear model. However, it is difficult to control some systems accurately in this way because their operating range is too large. Robot arms are a good example.13,14 Their equations of motion are very nonlinear, due primarily to the fact that their inertia varies greatly as their configuration changes. Nonlinear systems encompass everything that is “not linear,” and thus there is no general theory for nonlinear systems. There have been many nonlinear control methods proposed—too many to summarize here.15 Lyapunov’s stability theory and Popov’s method play a central role in many such schemes. Adaptive control is a subcase of nonlinear control (see below). The high speeds of modern digital computers now allow us to implement nonlinear control algorithms not possible with earlier hardware. An example is the computed-torque method for controlling robot arms, which was discussed in Section 11 (see Fig. 47).

801

are (a) the determination of the minimum-fuel (or minimum-time) trajectory and the open-loop control outputs (e.g., rocket thrust as a function of time) required to achieve the trajectory and (b) the design of a feedback controller to keep the system near the optimal trajectory. Many optimal control problems are nonlinear, and thus no general theory is available. Two classes of problems that have achieved some practical successes are the bang-bang control problem in which the control variable switches between two fixed values (e.g., on and off or open and closed),6 and the linearquadratic regulator (LQG), discussed in Section 7, which has proven useful for high-order systems.1,6 Closely related to optimal control theory are methods based on stochastic process theory, including stochastic control theory,17 estimators, Kalman filters, and observers.1,6,17 REFERENCES 1. 2. 3.

14.3 Adaptive Control

The term adaptive control, which unfortunately has been loosely used, describes control systems that can change the form of the control algorithm or the values of the control gains in real time, as the controller improves its internal model of the process dynamics or in response to unmodeled disturbances.16 Constant control gains do not provide adequate response for some systems that exhibit large changes in their dynamics over their entire operating range, and some adaptive controllers use several models of the process, each of which is accurate within a certain operating range. The adaptive controller switches between gain settings that are appropriate for each operating range. Adaptive controllers are difficult to design and are prone to instability. Most existing adaptive controllers change only the gain values, not the form of the control algorithm. Many problems remain to be solved before adaptive control theory becomes widely implemented. 14.4 Optimal Control

A rocket might be required to reach orbit using minimum fuel or it might need to reach a given intercept point in minimum time. These are examples of potential applications of optimal control theory. Optimal control problems often consist of two subproblems. For the rocket example, these subproblems

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Palm III, W. J., Modeling, Analysis, and Control of Dynamic Systems, 2nd ed., Wiley, New York, 2000. Palm III, W. J., Control Systems Engineering, Wiley, New York, 1986. Seborg, D. E., Edgar, T. F., and Mellichamp, D. A., Process Dynamics and Control, Wiley, New York, 1989. Palm III, W. J., System Dynamics, McGraw-Hill, New York, 2005. McCloy, D., and Martin, H., The Control of Fluid Power, 2nd ed., Halsted, London, 1980. Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Blaisdell, Waltham, MA, 1969. Lewis, F., Optimal Control, Wiley, New York, 1986. Astrom, K. J., and Wittenmark, B., Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, NJ, 1984. Dote, Y., Servo Motor and Motion Control Using Digital Signal Processors, Prentice-Hall, Englewood Cliffs, NJ, 1990. Palm III, W. J., Introduction to MATLAB 7 for Engineers, McGraw-Hill, New York, 2005. Klir, G., and Yuan, B., Fuzzy Sets and Fuzzy Logic, Prentice-Hall, Englewood Cliffs, NJ, 1995. Kosko, B., Neural Networks and Fuzzy Systems, Prentice-Hall, Englewood Cliffs, NJ, 1992. Craig, J., Introduction to Robotics, 3rd ed., AddisonWesley, Reading, MA, 2005. Spong, M. W., and Vidyasagar, M., Robot Dynamics and Control, Wiley, New York, 1989. Slotine, J., and Li, W., Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. Astrom, K. J., Adaptive Control, Addison-Wesley, Reading, MA, 1989. Stengel, R., Stochastic Optimal Control, Wiley, New York, 1986.

CHAPTER 14 THERMODYNAMICS FUNDAMENTALS Adrian Bejan Department of Mechanical Engineering and Materials Science Duke University Durham, North Carolina

1

INTRODUCTION

2

FIRST LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS

3 4

802 803

SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS

805

ENERGY-MINIMUM PRINCIPLE

807

5 6 7

LAWS OF THERMODYNAMICS FOR OPEN SYSTEMS

807

RELATIONS AMONG THERMODYNAMIC PROPERTIES

808

ANALYSIS OF ENGINEERING SYSTEM COMPONENTS

815

REFERENCES

1 INTRODUCTION Thermodynamics describes the relationship between mechanical work and other forms of energy. There are two facets of contemporary thermodynamics that must be stressed in a review such as this. The first is the equivalence of work and heat as two possible forms of energy exchange. This facet is expressed by the first law of thermodynamics. The second aspect is the one-way character, or irreversibility, of all flows that occur in nature. As expressed by the second law of thermodynamics, irreversibility or entropy generation is what prevents us from extracting the most possible work from various sources; it is also what prevents us from doing the most with the work that is already at our disposal. The objective of this chapter is to review the first and second laws of thermodynamics and their implications in mechanical engineering, particularly with respect to such issues as energy conversion and conservation. The analytical aspects (the formulas) of engineering thermodynamics are reviewed primarily in terms of the behavior of a pure substance, as would be the case of the working fluid in a heat engine or in a refrigeration machine. Symbols and Units

c

specific heat of incompressible substance, J/(kg · K)

Reprinted from Mechanical Engineers’ Handbook, Vol. 4, Wiley, New York, 2006, with permission of the publisher. 802

cP cT cv COP E f F g g h K m m ˙ mi M M n N0 P δQ ˙ Q r R s S Sgen S˙gen T

817

specific heat at constant pressure, J/(kg · K) constant temperature coefficient, m3 / kg specific heat at constant volume, J/(kg · K) coefficient of performance energy, J specific Helmholtz free energy (u − T s), J/kg force vector, N gravitational acceleration, m/s2 specific Gibbs free energy (h − T s), J/kg specific enthalpy (u + P v), J/kg isothermal compressibility, m2 /N mass of closed system, kg mass flow rate, kg/s mass of component in a mixture, kg mass inventory of control volume, kg molar mass, g/mol or kg/kmol number of moles, mol Avogadro’s constant pressure infinitesimal heat interaction, J heat transfer rate, W position vector, m ideal gas constant, J/(kg · K) specific entropy, J/(kg · K) entropy, J/K entropy generation, J/K entropy generation rate, W/K absolute temperature, K


THERMODYNAMICS FUNDAMENTALS

u U v v V V δW W˙ lost W˙ sh x x Z β γ η ηI ηII θ

specific internal energy, J/kg internal energy, J specific volume, m3 / kg specific volume of incompressible substance, m3 / kg volume, m3 velocity, m/s infinitesimal work interaction, J rate of lost available work, W rate of shaft (shear) work transfer, W linear coordinate, m quality of liquid and vapor mixture vertical coordinate, m coefficient of thermal expansion, 1/K ratio of specific heats, cP /cv “efficiency” ratio first-law efficiency second-law efficiency relative temperature,◦ C

Subscripts

( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

)f )g )s )in )out )rev )H )L )max )T )C )N )D )0 )1 )2 )− )+

saturated liquid state (f = “fluid”) saturated vapor state (g = “gas”) saturated solid state (s = “solid”) inlet port outlet port reversible path high-temperature reservoir low-temperature reservoir maximum turbine compressor nozzle diffuser reference state initial state final state moderately compressed liquid state slightly superheated vapor state

Definitions

Boundary: The real or imaginary surface delineating the thermodynamic system. The boundary separates the system from its environment. The boundary is an unambiguously defined surface. The boundary has zero thickness and zero volume. Closed System: A thermodynamic system whose boundary is not crossed by mass flow. Cycle: The special process in which the final state coincides with the initial state. Environment: The thermodynamic system external to the thermodynamic system.

803

Extensive Properties: Properties whose values depend on the size of the system (e.g., mass, volume, energy, enthalpy, entropy). Intensive Properties: Properties whose values do not depend on the system size (e.g., pressure, temperature). The collection of all intensive properties constitutes the intensive state. Open System: A thermodynamic system whose boundary is permeable to mass flow. Open systems (flow systems) have their own nomenclature: The thermodynamic system is usually referred to as the control volume, the boundary of the open system is the control surface, and the particular regions of the boundary that are crossed by mass flows are the inlet and outlet ports. Phase: The collection of all system elements that have the same intensive state (e.g., the liquid droplets dispersed in a liquid–vapor mixture have the same intensive state, that is, the same pressure, temperature, specific volume, specific entropy, etc.). Process: The change of state from one initial state to a final state. In addition to the end states, knowledge of the process implies knowledge of the interactions experienced by the system while in communication with its environment (e.g., work transfer, heat transfer, mass transfer, and entropy transfer). To know the process also means to know the path (the history, or the succession of states) followed by the system from the initial to the final state. State: The condition (the being) of a thermodynamic system at a particular point in time, as described by an ensemble of quantities called thermodynamic properties (e.g., pressure, volume, temperature, energy, enthalpy, entropy). Thermodynamic properties are only those quantities that do not depend on the “history” of the system between two different states. Quantities that depend on the system evolution (path) between states are not thermodynamic properties (examples of nonproperties are the work, heat, and mass transfer; the entropy transfer; the entropy generation; and the destroyed exergy—see also the definition of process). Thermodynamic System: The region or the collection of matter in space selected for analysis. 2 FIRST LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS The first law of thermodynamics is a statement that brings together three concepts in thermodynamics: work transfer, heat transfer, and energy change. Of these concepts, only energy change, or simply energy, is a thermodynamic property. We begin with a review1 of the concepts of work transfer, heat transfer, and energy change. Consider the force Fx experienced by a system at a point on its boundary. The infinitesimal work transfer between system and environment is

δW = −Fx dx

804


where the boundary displacement dx is defined as positive in the direction of the force Fx . When the force F and the displacement of its point of application dr are not collinear, the general definition of infinitesimal work transfer is

Fahrenheit, θ (◦ F); these alternative temperature readings are related through the conversion formulas ◦

θ ( C) = 59 [θ (◦ F) − 32] ◦

θ ( F) = 59 θ (◦ C) + 32

δW = −F · dr The work transfer interaction is considered positive when the system does work on its environment—in other words, when F and dr are oriented in opposite directions. This sign convention has its origin in heat engine engineering, because the purpose of heat engines as thermodynamic systems is to deliver work while receiving heat. For a system to experience work transfer, two things must occur: (1) a force must be present on the boundary and (2) the point of application of this force (hence, the boundary) must move. The mere presence of forces on the boundary, without the displacement or the deformation of the boundary, does not mean work transfer. Likewise, the mere presence of boundary displacement without a force opposing or driving this motion does not mean work transfer. For example, in the free expansion of a gas into an evacuated space, the gas system does not experience work transfer because throughout the expansion the pressure at the imaginary system–environment interface is zero. If a closed system can interact with its environment only via work transfer (i.e., in the absence of heat transfer δQ discussed later), then measurements show that the work transfer during a change of state from state 1 to state 2 is the same for all processes linking states 1 and 2, −

2

= E2 − E1

δW 1

δQ=0

In this special case the work transfer interaction (W1 – 2 )δQ=0 is a property of the system because its value depends solely on the end states. This thermodynamic property is the energy change of the system, E2 − E1 . The statement that preceded the last equation is the first law of thermodynamics for closed systems that do not experience heat transfer. Heat transfer is, like work transfer, an energy interaction that can take place between a system and its environment. The distinction between δQ and δW is made by the second law of thermodynamics discussed in the next section: Heat transfer is the energy interaction accompanied by entropy transfer, whereas work transfer is the energy interaction taking place in the absence of entropy transfer. The transfer of heat is driven by the temperature difference established between the system and its environment.2 The system temperature is measured by placing the system in thermal communication with a test system called thermometer. The result of this measurement is the relative temperature θ expressed in degrees Celsius, θ (◦ C), or

◦

1 F=

5◦ 9 C

The boundary that prevents the transfer of heat, regardless of the magnitude of the system–environment temperature difference, is termed adiabatic. Conversely, the boundary that is crossed by heat even in the limit of a vanishingly small system–environment temperature difference is termed diathermal. Measurements also show that a closed system undergoing a change of state 1 → 2 in the absence of work transfer experiences a heat interaction whose magnitude depends solely on the end states:

2

δQ δW =0

1

= E2 − E1

In the special case of zero work transfer, the heat transfer interaction is a thermodynamic property of the system, which is by definition equal to the energy change experienced by the system in going from state 1 to state 2. The last equation is the first law of thermodynamics for closed systems incapable of experiencing work transfer. Note that, unlike work transfer, the heat transfer is considered positive when it increases the energy of the system. Most thermodynamic systems do not manifest purely mechanical (δQ = 0) or purely thermal (δW = 0) behavior. Most systems manifest a coupled mechanical and thermal behavior. The preceding firstlaw statements can be used to show that the first law of thermodynamics for a process executed by a closed system experiencing both work transfer and heat transfer is 2 δQ − δW = E2 − E1 1 1

2

heat transfer

work transfer

energy interaction (nonproperties)

energy change

(property)

The first law means that the net heat transfer into the system equals the work done by the system on the environment plus the increase in the energy of the system. The first law of thermodynamics for a cycle or for an integral number of cycles executed by a closed system is

δQ =

δW = 0


805

Note that the net change in the thermodynamic property energy is zero during a cycle or an integral number of cycles. The energy change term E2 − E1 appearing on the right-hand side of the first law can be replaced by a more general notation that distinguishes between macroscopically identifiable forms of energy storage (kinetic, gravitational) and energy stored internally, E2 − E1 = U2 − U1 + energy change

internal energy change

mV22

−

2

mV12 2

kinetic energy change

+ mgZ2 − mgZ1 gravitational energy change

If the closed system expands or contracts quasistatically (i.e., slowly enough, in mechanical equilibrium internally and with the environment) so that at every point in time the pressure P is uniform throughout the system, then the work transfer term can be calculated as being equal to the work done by all the boundary pressure forces as they move with their respective points of application,

2

δW =

1

distinction between reversible and irreversible cycles executed by closed systems in communication with no more than one temperature reservoir is δW = 0 (reversible) δW < 0 (irreversible) To summarize, the first and second laws for closed systems operating cyclically in contact with no more than one temperature reservoir are (Fig. 1)

δW =

δQ≤0

This statement of the second law can be used to show1 that in the case of a closed system executing one or an integral number of cycles while in communication with two temperature reservoirs the following inequality holds (Fig. 1): QL QH + ≤0 TH TL

2

P dV 1

The work transfer integral can be evaluated provided the path of the quasistatic process, P (V ), is known; this is another reminder that the work transfer is path dependent (i.e., not a thermodynamic property). 3 SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS A temperature reservoir is a thermodynamic system that experiences only heat transfer and whose temperature remains constant during such interactions. Consider first a closed system executing a cycle or an integral number of cycles while in thermal communication with no more than one temperature reservoir. To state the second law for this case is to observe that the net work transfer during each cycle cannot be positive,

δW = 0 In other words, a closed system cannot deliver work during one cycle while in communication with one temperature reservoir or with no temperature reservoir at all. Examples of such cyclic operation are the vibration of a spring–mass system or a ball bouncing on the pavement: For these systems to return to their respective initial heights, that is, for them to execute cycles, the environment (e.g., humans) must perform work on them. The limiting case of frictionless cyclic operation is termed reversible because in this limit the system returns to its initial state without intervention (work transfer) from the environment. Therefore, the

where H and L denote the high-temperature and the low-temperature reservoirs, respectively. Symbols QH and QL stand for the value of the cyclic integral δQ, where δQ is in one case exchanged only with the H reservoir and in the other with the L reservoir. In the reversible limit, the second law reduces to TH /TL = −QH /QL , which serves as definition for the absolute thermodynamic temperature scale denoted by symbol T . Absolute temperatures are expressed either in kelvins, T (K), or in degrees Rankine, T (◦ R); the relationships between absolute and relative temperatures are T (K) = θ (◦ C) + 273.15K T (◦ R) = θ (◦ F) + 459.67◦ R 1◦ R = 1◦ F 1K = 1◦ C

A heat engine is a special case of a closed system operating cyclically while in thermal communication with two temperature reservoirs, a system that during each cycle receives heat and delivers work:

δQ = QH + QL > 0

δW =

The goodness of the heat engine can be described in terms of the heat engine efficiency or the first-law efficiency η=

δW TL ≤1− QH TH

806


Fig. 1 First and second laws of thermodynamics for a closed system operating cyclically while in communication with one or two heat reservoirs.

Alternatively, the second-law efficiency of the heat engine is defined as1,3,4

Qi

i

δW ηI ηII = = 1 − TL /TH ( δW )maximum (reversiblecase) A refrigerating machine or a heat pump operates cyclically between two temperature reservoirs in such a way that during each cycle it receives work and delivers net heat to the environment,

δW =

δQ = QH + QL < 0

The goodness of such machines can be expressed in terms of a coefficient of performance (COP) COPrefrigerator =

1 QL ≤ TH /TL − 1 − δW

=

1 −QH ≤ 1 − TL /TH − δW

COPheat

pump

Generalizing the second law for closed systems operating cyclically, one can show that, if during each cycle the system experiences any number of heat interactions Qi with any number of temperature reservoirs whose respective absolute temperatures are Ti , then

Ti

≤0

Note that Ti is the absolute temperature of the boundary region crossed by Qi . Another way to write the second law in this case is δQ ≤0 T where, again, T is the temperature of the boundary pierced by δQ. Of special interest is the reversible cycle limit, in which the second law states ( δQ/T )rev = 0. According to the definition of thermodynamic property, the second law implies that during a reversible process the quantity δQ/T is the infinitesimal change in a property of the system: By definition, that property is the entropy change 2 δQ δQ or S2 − S1 = dS = T rev T rev 1 Combining this definition with the second law for a cycle, δQ/T ≤ 0, yields the second law of thermodynamics for any process executed by a closed system,

S2 − S1 − entropy change (property)

2

δQ ≥0 T 1 entropy transfer (nonproperty)


The entire left-hand side in this inequality is by definition the entropy generated by the process,

2

Sgen = S2 − S1 − 1

δQ T

The entropy generation is a measure of the inequality sign in the second law and hence a measure of the irreversibility of the process. The entropy generation is proportional to the useful work destroyed during the process.1,3,4 Note again that any heat interaction (δQ) is accompanied by entropy transfer (δQ/T ), whereas the work transfer δW is not. 4 ENERGY-MINIMUM PRINCIPLE Consider now a closed system that executes an infinitesimally small change of state, which means that its state changes from (U, S, . . .) to (U + dU, S + dS, . . .). The first- and second-law statements are

δQ − δW = dU

dS −

δQ ≥0 T

If the system is isolated from its environment, then δW = 0 and δQ = 0, and the two laws dictate that during any such process the energy inventory stays constant (dU = 0) and the entropy inventory cannot decrease, dS ≥ 0

from the outside, for example, the removal of one or more of the internal constraints plotted qualitatively in the vertical direction in Fig. 2. When all the constraints are removed, changes cease, and, according to dS ≥ 0, the entropy inventory reaches its highest possible level. This entropy-maximum principle is a consequence of the first and second laws. When all the internal constraints have disappeared, the system has reached the unconstrained equilibrium state. Alternatively, if changes occur in the absence of work transfer and at constant S, the first law and the second law require, respectively, dU = δQ and δQ ≤ 0, and hence dU ≤ 0 The energy inventory cannot increase, and when the unconstrained equilibrium state is reached, the system energy inventory is minimum. This energy-minimum principle is also a consequence of the first and second laws for closed systems. The interest in this classical formulation of the laws (e.g., Fig. 2) has been renewed by the emergence of an analogous principle of performance increase (the constructal law) in the search for optimal configurations in the design of open (flow) systems.5 This analogy is based on the constructal law of maximization of flow access.1,6 5 LAWS OF THERMODYNAMICS FOR OPEN SYSTEMS

Isolated systems undergo processes when they experience internal changes that do not require intervention

Fig. 2

807

If m ˙ represents the mass flow rate through a port in the control surface, the principle of mass conservation

Energy-minimum principle or entropy-maximum principle.

808


in the control volume is

in

m ˙ −

∂M ∂t

m ˙ =

out

W˙ ≤

mass change

mass transfer

Subscripts in and out refer to summation over all the inlet and outlet ports, respectively, while M stands for the instantaneous mass inventory of the control volume. The first law of thermodynamics is more general than the statement encountered earlier for closed systems because this time we must account for the flow of energy associated with the m ˙ streams:

˙ in m

h+

V2 2

+ gZ − out m ˙ h+ ˙ i − W˙ + iQ

V2 2

+ gZ

V2 + gZ − T0 s m ˙ h+ 2 in

V2 ∂ + gZ − T0 s − (E − T0 s) m ˙ h+ − 2 ∂t out

The right-hand side in this inequality is the maximum work transfer rate W˙ sh,max , which would exist only in the ideal limit of reversible operation. The rate of lost work, or the rate of exergy (availability) destruction, is defined as W˙ lost = W˙ max − W˙

Again, using both laws, one can show that lost work is directly proportional to entropy generation,

W˙ lost = T0 S˙gen

energy transfer

∂E ∂t = energy change On the left-hand side we have the energy interactions: heat, work, and the energy transfer associated with mass flow across the control surface. The specific enthalpy h, fluid velocity V , and height Z are evaluated right at the boundary. On the right-hand side, E is the instantaneous system energy integrated over the control volume. The second law of thermodynamics for an open system assumes the form

in

ms ˙ −

ms ˙ +

out

Q ˙i i

Ti

entropy transfer

≤

∂S ∂t entropy change

The specific entropy s is representative of the thermodynamic state of each stream right at the system boundary. The entropy generation rate is defined by

Q ˙i ∂S

S˙gen = + ms ˙ − ms ˙ − ∂t Ti out in

i

and is a measure of the irreversibility of open-system operation. The engineering importance of S˙gen stems from its proportionality to the rate of destruction of available work. If the following parameters are fixed—all the mass flows (m), ˙ the peripheral conditions (h, s, V, Z ), and the heat interactions (Qi , Ti ) except (Q0 , T0 )—then one can use the first law and the second law to show that the work transfer rate cannot exceed a theoretical maximum1,3,4 :

This result is known as the Gouy–Stodola theorem.1,3,4 Conservation of useful work (exergy) in thermodynamic systems can only be achieved based on the systematic minimization of entropy generation in all the components of the system. Engineering applications of entropy generation minimization as a design optimization philosophy may be found in Refs. 1, 3, and 4. 6 RELATIONS AMONG THERMODYNAMIC PROPERTIES

The analytical forms of the first and second laws of thermodynamics contain properties such as internal energy, enthalpy, and entropy, which cannot be measured directly. The values of these properties are derived from measurements that can be carried out in the laboratory (e.g., pressure, volume, temperature, specific heat); the formulas connecting the derived properties to the measurable properties are reviewed in this section. Consider an infinitesimal change of state experienced by a closed system. If kinetic and gravitational energy changes can be neglected, the first law reads δQany

path

− δWany

path

= dU

which emphasizes that dU is path independent. In particular, for a reversible path (rev), the same dU is given by δQrev − δWrev = dU Note that from the second law for closed systems we have δQrev = T dS. Reversibility (or zeroentropy generation) also requires internal mechanical equilibrium at every stage during the process; hence,


δWrev = P dV , as for a quasistatic change in volume. The infinitesimal change experienced by U is therefore T dS − P dV = dU Note that this formula holds for an infinitesimal change of state along any path (because dU is path independent); however, T dS matches δQ and P dV matches δW only if the path is reversible. In general, δQ < T dS and δW < P dV . The formula derived above for dU can be written for a unit mass: T ds − P dv = du. Additional identities implied by this relation are ∂u ∂u T = −P = ∂s v ∂v s 2 ∂ u ∂T ∂P = =− ∂s ∂v ∂v s ∂s v where the subscript indicates which variable is held constant during partial differentiation. Similar relations and partial derivative identities exist in conjunction with other derived functions such as enthalpy, Gibbs free energy, and Helmholtz free energy: • Enthalpy (defined as h = u + P v): dh = T ds + v dP ∂h ∂h T = v= ∂s P ∂P s ∂2h ∂T ∂v = = ∂s ∂P ∂P s ∂s P • Gibbs free energy (defined as g = h − T s): dg = −s dT + v dP ∂g ∂g −s = v= ∂T P ∂P T ∂2g ∂s ∂v =− = ∂T ∂P ∂P T ∂T P • Helmholtz free energy (defined as f = u − T s): df = −s dT − P dv ∂f ∂f −s = −P = ∂T v ∂v T ∂2f ∂s ∂P =− =− ∂T ∂v ∂v T ∂T v

809

In addition to the (P, v, T ) surface, which can be determined based on measurements (Fig. 3), the following partial derivatives are furnished by special experiments1 : • The specific heat at constant volume, cv = (∂u/∂T )v , follows directly from the constantvolume (∂W = 0) heating of a unit mass of pure substance. • The specific heat at constant pressure, cP = (∂h/∂T )P , is determined during the constantpressure heating of a unit mass of pure substance. • The Joule–Thompson coefficient, µ= (∂T /∂P )h , is measured during a throttling process, that is, during the flow of a stream through an adiabatic duct with friction (see the first law for an open system in the steady state). • The coefficient of thermal expansion β = (1/v)(∂v/∂T )P . • The isothermal compressibility K = (−1/v) (∂v/∂P )T . • The constant-temperature coefficient cT = (∂h/∂P )T . Two noteworthy relationships between some of the partial-derivative measurements are cP − cv =

T vβ 2 K

µ=

1 cP

∂v T −v ∂T P

The general equations relating the derived properties (u, h, s) to measurable quantities are ∂P − P dv du = cv dT + T ∂T v ∂v + v dP dh = cP dT + −T ∂T P cv ∂v dT + ds = dv or T ∂T v cP ∂v dT − dP ds = T ∂T P These relations also suggest the following identities:

∂u ∂T

∂h ∂T

=T

v

=T

P

∂s ∂T ∂s ∂T

= cv = cP P

The relationships between thermodynamic properties and the analyses associated with applying the laws of thermodynamics are simplified considerably in cases

810


Fig. 3 The (P, v, T) surface for a pure substance that contracts upon freezing, showing regions of ideal-gas and incompressible fluid behavior: S = solid, V = vapor, L = liquid, TP = triple point.

where the pure substance exhibits ideal-gas behavior. As shown in Fig. 3, this behavior sets in at sufficiently high temperatures and low pressures; in this limit, the (P, v, T ) surface is fitted closely by the simple expression Pv =R T

(constant)

where R is the ideal-gas constant of the substance of interest (Table 1). The formulas for internal energy,

enthalpy, and entropy, which concluded the preceding section, assume the following form in the ideal-gas limit: du = cv dT

cv = cv (T )

dh = cP dT

cP = cP (T ) = cv + R

R cv dT + dv or T v cP cv dP + dv ds = P v

ds =

ds =

cP R dT − dP T P

or


811

Table 1 Ideal-Gas Constants and Specific Heats at Constant Volume for Gases Encountered in Mechanical Engineering Ideal Gas Air Argon, Ar Butane, C4 H10 Carbon dioxide, CO2 Carbon monoxide, CO Ethane, C2 H6 Ethylene, C2 H4 Helium, He2 Hydrogen, H Methane, CH4 Neon, Ne Nitrogen, N2 Octane, C8 H18 Oxygen, O2 Propane, C3 H8 Steam, H2 O

R[J/(kg · K)]

cP [J/(kg · K)]

286.8 208.1 143.2 188.8 296.8 276.3 296.4 2076.7 4123.6 518.3 412.0 296.8 72.85 259.6 188.4 461.4

715.9 316.5 1595.2 661.5 745.3 1511.4 1423.5 3152.7 10216.0 1687.3 618.4 741.1 1641.2 657.3 1515.6 1402.6

Source: From Ref. 1.

2

δQ = 0

1

1

2

P2 V 2 δW = γ −1

V2 V1

γ −1

−1

where γ = cP /cv . • Path: γ

γ

P V γ = P1 V 1 = P2 V 2

(constant)

• Entropy change: S2 − S1 = 0 Hence the name isoentropic or isentropic for this process. • Entropy generation:

If the coefficients cv and cP are constant in the temperature domain of interest, then the changes in specific internal energy, enthalpy, and entropy relative to a reference state ( )0 are given by the formulas u − u0 = cv (T − T0 ) h − h0 = cP (T − T0 ) (where h0 = u0 + RT0 )  T v   + R ln cv ln   T0 v0    T P − R ln s − s0 = cP ln  T0 P0     P v  cv ln + cP ln P0 v0 The ideal-gas model rests on two empirical constants, cv and cP , or cv and R, or cP and R. The ideal-gas limit is also characterized by µ=0

β=

1 P

K=

1 P

cT = 0

The extent to which a thermodynamic system destroys available work is intimately tied to the system’s entropy generation, that is, to the system’s departure from the theoretical limit of reversible operation. Idealized processes that can be modeled as reversible occupy a central role in engineering thermodynamics because they can serve as standard in assessing the goodness of real processes. Two benchmark reversible processes executed by closed ideal-gas systems are particularly simple and useful. A quasistatic adiabatic process 1 → 2 executed by a closed ideal-gas system has the following characteristics:

Sgen1→2 = S2 − S1 −

2

1

δQ = 0 (reversible) T

A quasistatic isothermal process 1 → 2 executed by a closed ideal-gas system in communication with a single temperature reservoir T is characterized by: • Energy interactions:

2

δQ =

1

2

δW = mRT ln

1

V2 V1

• Path: T = T1 = T 2

(constant)

P V = P1 V 1 = P2 V 2

or

(constant)

• Entropy change: S2 − S1 = mR ln

V2 V1

• Entropy generation: Sgen1→2 = S2 − S1 −

1

2

δQ = 0 (reversible) T

Mixtures of ideal gases also behave as ideal gases in the high-temperature, low-pressure limit. If a certain mixture of mass m contains ideal gases mixed in mass proportions mi , and if the ideal-gas constants of each

812


component are (cvi , cP i , Ri ), then the equivalent idealgas constants of the mixture are cv =

1

mi cvi m

cp =

i

1

mi cP i m i

1

R= mi R i m i

where m = i mi . One mole is the amount of substance of a system that contains as many elementary entities (e.g., molecules) as there are in 12 g of carbon 12; the number of such entities is Avogadro’s constant, N0 ∼ = 6.022 × 1023 . The mole is not a mass unit because the mass of 1 mol is not the same for all substances. The molar mass M of a given molecular species is the mass of 1 mol of that species, so that the total mass m is equal to M times the number of moles n, m = nM Thus, the ideal-gas equation of state can be written as P V = nMRT where the product MR is the universal gas constant R = MR = 8.314 J/(mol · K) The equivalent molar mass of a mixture of ideal gases with individual molar masses Mi is M=

1

ni Mi n

where n = ni . The molar mass of air, as a mixture of nitrogen, oxygen, and traces of other gases, is 28.966 g/mol (or 28.966 kg/kmol). A more useful model of the air gas mixture relies on only nitrogen and oxygen as constituents, in the proportion 3.76 mol of nitrogen to every mole of oxygen; this simple model is used frequently in the field of combustion.1 At the opposite end of the spectrum is the incompressible substance model. At sufficiently high pressures and low temperatures in Fig. 3, solids and liquids behave so that their density or specific volume is practically constant. In this limit the (P, v, T ) surface is adequately represented by the equation v=v

(constant)

The formulas for calculating changes in internal energy, enthalpy, and entropy become (see the end of the section on relations among thermodynamic properties) du = c dT

dh = c dT + v dP

ds =

c dT T

where c is the sole specific heat of the incompressible substance, c = cv = cP The specific heat c is a function of temperature only. In a sufficiently narrow temperature range where c can be regarded as constant, the finite changes in internal energy, enthalpy, and entropy relative to a reference state denoted by ( )0 are u − u0 = c (T − T0 ) h − h0 = c (T − T0 ) + v (P − P0 ) (where h0 = u0 + P0 v) s − s0 = c ln

T T0

The incompressible substance model rests on two empirical constants, c and v. As shown in Fig. 3, the domains in which the pure substance behaves either as an ideal gas or as an incompressible substance intersect over regions where the substance exists as a mixture of two phases, liquid and vapor, solid and liquid, or solid and vapor. The two-phase regions themselves intersect along the triple-point line labeled TP-TP on the middle sketch of Fig. 3. In engineering cycle calculations, the projections of the (P, v, T ) surface on the P –v plane or, through the relations reviewed earlier, on the T –s plane are useful. The terminology associated with twophase equilibrium states is defined on the P –v diagram of Fig. 4a, where we imagine the isothermal compression of a unit mass of substance (a closed system). As the specific volume v decreases, the substance ceases to be a pure vapor at state g, where the first droplets of liquid are formed. State g is a saturated vapor state. It is observed that isothermal compression beyond g proceeds at constant pressure up to state f , where the last bubble (immersed in liquid) is suppressed. State f is a saturated liquid state. Isothermal compression beyond f is accompanied by a steep rise in pressure, depending on the compressibility of the liquid phase. The critical state is the intersection of the locus of saturated vapor states with the locus of saturated liquid states (Fig. 4a). The temperature and pressure corresponding to the critical state are the critical temperature and critical pressure. Table 2 contains a compilation of criticalstate properties of some of the more common substances. Figure 4b shows the projection of the liquid and vapor domain on the T –s plane. On the same drawing is shown the relative positioning (the relative slopes) of the traces of various constant-property cuts through the three-dimensional surface on which all the equilibrium states are positioned. In the two-phase region, the temperature is a unique function of pressure. This


Fig. 4

Locus of two-phase (liquid and vapor) states, projected on (a) P–v plane and (b) T –s plane.

813

814 Table 2

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Critical-State Properties Critical Temperature [K (◦ C)]

Fluid Air Alcohol (methyl) Alcohol (ethyl) Ammonia Argon Butane Carbon dioxide Carbon monoxide Carbon tetrachloride Chlorine Ethane Ethylene Helium Hexane Hydrogen Methane Methyl chloride Neon Nitric oxide Nitrogen Octane Oxygen Propane Sulfur dioxide Water

133.2(-140) 513.2(240) 516.5(243.3) 405.4(132.2) 150.9(-122.2) 425.9(152.8) 304.3(31.1) 134.3(-138.9) 555.9(282.8) 417 (143.9) 305.4(32.2) 282.6(9.4) 5.2(-268) 508.2(235) 33.2(-240) 190.9(-82.2) 416.5(143.3) 44.2(-288.9) 179.2(-93.9) 125.9(-147.2) 569.3(296.1) 154.3(-118.9) 368.7(95.6) 430.4(157.2) 647 (373.9)

Critical Pressure [MPa (atm)] 3.77(37.2) 7.98(78.7) 6.39(63.1) 11.3(111.6) 4.86(48) 3.65(36) 7.4(73) 3.54(35) 4.56(45) 7.72(76.14) 4.94(48.8) 5.85(57.7) 0.228(2.25) 2.99(29.5) 1.30(12.79) 4.64(45.8) 6.67(65.8) 2.7(26.6) 6.58(65) 3.39(33.5) 2.5(24.63) 5.03(49.7) 4.36(43) 7.87(77.7) 22.1(218.2)

Critical Specific Volume (cm3 /g) 2.9 3.7 3.6 4.25 1.88 4.4 2.2 3.2 1.81 1.75 4.75 4.6 14.4 4.25 32.3 6.2 2.7 2.1 1.94 3.25 4.25 2.3 4.4 1.94 3.1

Source: From Ref. 1.

one-to-one relationship is indicated also by the Clapeyron relation sg − sf hg − hf dP = = dT sat T (vg − vf ) vg − vf where the subscript sat is a reminder that the relation holds for saturated states (such as g and f ) and for mixtures of two saturated phases. Subscripts g and f indicate properties corresponding to the saturated vapor and liquid states found at temperature Tsat (and pressure Psat ). Built into the last equation is the identity hg − hf = T (sg − sf ) which is equivalent to the statement that the Gibbs free energy is the same for the saturated states and their mixtures found at the same temperature, gg = gf . The properties of a two-phase mixture depend on the proportion in which saturated vapor, mg , and saturated liquid, mf , enter the mixture. The composition of the mixture is described by the property called quality, x=

mg mf + mg

The quality varies between 0 at state f and 1 at state g. Other properties of the mixture can be calculated in

terms of the properties of the saturated states found at the same temperature, u = uf + xuf g

s = sf + xsf g

h = hf + xhf g

v = vf + xvf g

with the notation ( )f g = ( )g − ( )f . Similar relations can be used to calculate the properties of two-phase states other than liquid and vapor, namely, solid and vapor or solid and liquid. For example, the enthalpy of a solid and liquid mixture is given by h = hs + xhsf , where subscript s stands for the saturated solid state found at the same temperature as for the twophase state and hsf is the latent heat of melting or solidification. In general, the states situated immediately outside the two-phase dome sketched in Figs. 3 and 4 do not follow very well the limiting models discussed earlier in this section (ideal gas, incompressible substance). Because the properties of closely neighboring states are usually not available in tabular form, the following approximate calculation proves useful. For a moderately compressed liquid state, which is indicated by the subscript ()∗ , that is, for a state situated close to the left of the dome in Fig. 4, the properties may be calculated as slight deviations from those of the saturated liquid state found at the same temperature as the


815

compressed liquid state of interest, h∗ ∼ = (hf )T ∗ + (vf )T ∗ [P∗ − (Pf )T ∗ ]

s∼ = (sf )T ∗

For a slightly superheated vapor state, that is, a state situated close to the right of the dome in Fig. 4, the properties may be estimated in terms of those of the saturated vapor state found at the same temperature: h+ ∼ = (hg )T +

s+ ∼ = (sg )T + +

Pg v g Tg

ln T+

(Pg )T + P+

In these expressions, subscript ()+ indicates the properties of the slightly superheated vapor state.

7 ANALYSIS OF ENGINEERING SYSTEM COMPONENTS

This section contains a summary1 of the equations obtained by applying the first and second laws of thermodynamics to the components encountered in most engineering systems, such as power plants and refrigeration plants. It is assumed that each component operates in steady flow : • Valve (throttle) or adiabatic duct with friction (Fig. 5a): First law:

h1 = h2

Second law: S˙gen = m(s ˙ 2 − s1 ) > 0

Fig. 5 Engineering system components and their inlet and outlet states on the T –s plane: PH = high pressure; PL = low pressure.

816


Fig. 5 (Continued)


• Expander or turbine with negligible heat transfer to the ambient (Fig. 5b): ˙ 1 − h2 ) First law: W˙ T = m(h Second law: S˙gen = m(s ˙ 2 − s1 ) ≥ 0 Efficiency:

ηT =

h1 − h2 ≤1 h1 − h2,rev

• Compressor or pump with negligible heat transfer to the ambient (Fig. 5c):

817

First law: h2 − h1 = 12 (V12 − V22 ) ˙ 2 − s1 ) ≥ 0 Second law: S˙gen = m(s Efficiency: ηD =

• Heat exchangers with negligible heat transfer to the ambient (Figs. 5f and g): ˙ cold (h4 − h3 ) First law: m ˙ hot (h1 − h2 ) = m Second law: S˙gen − m ˙ hot (s2 − s1 )

˙ 2 − h1 ) First law: W˙ c = m(h Second law: S˙gen = m(s ˙ 2 − s1 ) ≥ 0 Efficiency:

h2,rev − h1 ≤1 ηc = h2 − h1

• Nozzle with negligible heat transfer to the ambient (Fig. 5d): First law:

1 2 2 (V2

− V12 ) = h1 − h2

Second law: S˙gen = m(s ˙ 2 − s1 ) ≥ 0 Efficiency:

ηN =

V22 − V12 ≤1 2 V2,rev − V12

• Diffuser with negligible heat transfer to the ambient (Fig. 5e):

h2,rev − h1 ≤1 h2 − h1

+m ˙ cold (s4 − s3 ) ≥ 0 Figures 5f and g show that a pressure drop always occurs in the direction of flow in any heat exchanger flow passage. REFERENCES 1. Bejan, A., Advanced Engineering Thermodynamics, 2nd ed., Wiley, New York, 1997. 2. Bejan, A., Heat Transfer, Wiley, New York, 1993. 3. Bejan, A., Entropy Generation through Heat and Fluid Flow, Wiley, New York, 1982. 4. Bejan, A., Entropy Generation Minimization, CRC Press, Boca Raton, FL, 1996. 5. Bejan, A., and Lorente, S., “The Constructal Law and the Thermodynamics of Flow Systems with Configuration,” Int. J. Heat Mass Transfer, 47, 3203–3214 (2004). 6. Bejan, A., Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000.

CHAPTER 15 HEAT TRANSFER FUNDAMENTALS G. P. Peterson Rensselaer Polytechnic Institute Troy, New York

1

CONDUCTION HEAT TRANSFER

819

1.1

Thermal Conductivity

820

1.2

One-Dimensional Steady-State Heat Conduction

1.3

820

RADIATION HEAT TRANSFER

844

3.1

Blackbody Radiation

846

3.2

Radiation Properties

846

3.3

Configuration Factor

849

Radiative Exchange among Diffuse Gray Surfaces in Enclosure

854

Thermal Radiation Properties of Gases

856

Two-Dimensional Steady-State Heat Conduction

822

3.4

Heat Conduction with Convection Heat Transfer on Boundaries

825

3.5

1.5

Transient Heat Conduction

830

1.6

Conduction at Microscale

830

CONVECTION HEAT TRANSFER

834

2.1

Forced Convection—Internal Flow

837

2.2

Forced Convection—External Flow

838

2.3

Free Convection

841

2.4

Log-Mean Temperature Difference

843

1.4

2

3

SYMBOLS AND UNITS A Area of heat transfer Bi Biot number, hL/k, dimensionless C Circumference, m, constant defined in text Cp Specific heat under constant pressure, J/kg·K D Diameter, m e Emissive power, W/m2 f Drag coefficient, dimensionless F Cross-flow correction factor, dimensionless Fi−j Configuration factor from surface i to surface j , dimensionless Fo Fourier number, αtA2 /V 2 , dimensionless Fo−λT Radiation function, dimensionless G Irradiation, W/m2 ; mass velocity, kg/m2 ·s g Local gravitational acceleration, 9.8 m/s2 gc Proportionality constant, 1 kg·m/N·s2 Gr Grashof number, gL3 β T /v 2 dimensionless h Convection heat transfer coefficient, equals q/A T , W/m2 ·K

4

BOILING AND CONDENSATION HEAT TRANSFER 4.1

Boiling

860

4.2

Condensation

862

4.3

Heat Pipes

864

REFERENCES

868

BIBLIOGRAPHY

869

hfg J k K L Ma N Nu Nu P Pe Pr q q R r Ra

Heat of vaporization, J/kg Radiosity, W/m2 Thermal conductivity, W/m·K Wick permeability, m2 Length, m Mach number, dimensionless Screen mesh number, m−1 Nusselt number, NuL = hL/k, NuD = hD/k, dimensionless Nusselt number averaged over length, dimensionless Pressure, N/m2 , perimeter, m Peclet number, RePr, dimensionless Prandtl number, Cp µ/k, dimensionless Rate of heat transfer, W Rate of heat transfer per unit area, W/m2 Distance, m; thermal resistance, K/W Radial coordinate, m; recovery factor, dimensionless Rayleigh number, GrPr; RaL = GrL Pr, dimensionless

Reprinted from Mechanical Engineers’ Handbook, Vol. 4, Wiley, New York, 2006, with permission of the publisher. 818

858


HEAT TRANSFER FUNDAMENTALS

Re S T t Tαs Tsat Tb Te Tf Ti T0 Ts T∞ U V w We x

Reynolds Number, ReL = ρV L/µ, ReD = ρV D/µ, dimensionless Conduction shape factor, m Temperature, K or ◦ C Time, s Adiabatic surface temperature, K Saturation temperature, K Fluid bulk temperature or base temperature of fins, K Excessive temperature, Ts − Tsat , K or ◦ C Film temperature, (T∞ + Ts )/2, K Initial temperature; at t = 0, K Stagnation temperature, K Surface temperature, K Free-stream fluid temperature, K Overall heat transfer coefficient, W/m2 ·K Fluid velocity, m/s; volume, m3 Groove width, m; or wire spacing, m Weber number, dimensionless One of the axes of Cartesian reference frame, m

GREEK SYMBOLS α Thermal diffusivity, k/ρCp , m2 /s; absorptivity, dimensionless β Coefficient of volume expansion, 1/K Mass flow rate of condensate per unit width, kg/m·s γ Specific heat ratio, dimensionless T Temperature difference, K δ Thickness of cavity space, groove depth, m ∈ Emissivity, dimensionless ε Wick porosity, dimensionless λ Wavelength, µm Fin efficiency, dimensionless ηf µ Viscosity, kg/m·s ν Kinematic viscosity, m2 /s ρ Reflectivity, dimensionless; density, kg/m3 σ Surface tension, N/m; Stefan–Boltzmann constant, 5.729 × 10−8 W/m2 ·K4 τ Transmissivity, dimensionless, shear stress, N/m2 Angle of inclination, degrees or radians

SUBSCRIPTS a Adiabatic section, air b Boiling, blackbody c Convection, capillary, capillary limitation, condenser e Entrainment, evaporator section eff Effective f Fin i Inner l Liquid

819

Mean, maximum Nucleation Outer Stagnation condition Pipe Radiation Surface, sonic or sphere Wire spacing, wick Vapor Spectral Free stream Axial hydrostatic pressure Normal hydrostatic pressure

m n o O p r s w v λ ∞ − +

Transport phenomena represents the overall field of study and encompasses a number of subfields. One of these is heat transfer, which focuses primarily on the energy transfer occurring as a result of an energy gradient that manifests itself as a temperature difference. This form of energy transfer can occur as a result of a number of different mechanisms, including conduction, which focuses on the transfer of energy through the direct impact of molecules; convection, which results from the energy transferred through the motion of a fluid; and radiation, which focuses on the transmission of energy through electromagnetic waves. In the following review, as is the case with most texts on heat transfer, phase change heat transfer, that is, boiling and condensation, will be treated as a subset of convection heat transfer. 1 CONDUCTION HEAT TRANSFER The exchange of energy or heat resulting from the kinetic energy transferred through the direct impact of molecules is referred to as conduction, and takes place from a region of high energy (or temperature) to a region of lower energy (or temperature). The fundamental relationship that governs this form of heat transfer is Fourier’s law of heat conduction, which states that in a one-dimensional system with no fluid motion, the rate of heat flow in a given direction is proportional to the product of the temperature gradient in that direction and the area normal to the direction of heat flow. For conduction heat transfer in the x direction this expression takes the form

qx = −kA

∂T ∂x

where qx is the heat transfer in the x direction, A is the area normal to the heat flow, ∂T /∂x is the temperature gradient, and k is the thermal conductivity of the substance. Writing an energy balance for a three-dimensional body and utilizing Fourier’s law of heat conduction yields an expression for the transient diffusion occurring within a body or substance: ∂ ∂x

k

∂T ∂x

+

∂ ∂y

k

∂T ∂y

+

∂ ∂T ∂ ∂T k + q˙ = ρcp ∂z ∂z ∂x ∂t

820


This expression, usually referred to as the heat diffusion equation or heat equation, provides a basis for most types of heat conduction analyses. Specialized cases of this equation can be used to solve many steady-state or transient problems. Some of these specialized cases are as follows: Thermal conductivity is a constant: ρcp ∂T ∂ 2T ∂ 2T q˙ ∂2T + + = ∂x 2 ∂y 2 ∂z2 k k ∂t Steady state with heat generation: ∂ ∂x

∂T ∂ ∂T ∂ ∂T k + k + k + q˙ = 0 ∂x ∂y ∂y ∂z ∂z

Steady-state, one-dimensional heat transfer with no heat sink (i.e., a fin): ∂ ∂x

∂T ∂x

+

q˙ =0 k

One-dimensional heat transfer with no internal heat generation: ρcp ∂T ∂ ∂T = ∂x ∂x k ∂t In the following sections, the heat diffusion equation will be utilized for several specific cases. However, in general, for a three-dimensional body of constant thermal properties without heat generation under steadystate heat conduction the temperature field satisfies the expression

lower, increases with increasing temperature and decreases with increasing molecular weight. The thermal conductivities of a number of commonly used metals and nonmetals are tabulated in Tables 1 and 2, respectively. Insulating materials, which are used to prevent or reduce the transfer of heat between two substance or a substance and the surroundings, are listed in Tables 3 and 4, along with the thermal properties. The thermal conductivities for liquids, molten metals, and gasses are given in Tables 5, 6 and 7, respectively. 1.2 One-Dimensional Steady-State Heat Conduction The steady-state rate of heat transfer resulting from heat conduction through a homogeneous material can be expressed in terms of the rate of heat transfer, q, or q = T /R, where T is the temperature difference and R is the thermal resistance. This thermal resistance is the reciprocal of the thermal conductance (C = 1/R) and is related to the thermal conductivity by the cross-sectional area. Expressions for the thermal resistance, the temperature distribution, and the rate of heat transfer are given in Table 8 for a plane wall, a cylinder, and a sphere. For a plane wall, the heat transfer is typically assumed to be one dimensional (i.e., heat is conducted in only the x direction) and for a cylinder and sphere, only in the radial direction. Aside from the heat transfer in these simple geometric configurations, other common problems encountered in practical applications is that of heat transfer through layers or composite walls consisting of N layers, where the thickness of each layer is represented by xn and the thermal conductivity by kn for n = 1, 2, . . . , N. Assuming that the interfacial resistance is negligible (i.e., there is no thermal resistance at the contacting surfaces), the overall thermal resistance can be expressed as

∇ 2T = 0 1.1 Thermal Conductivity The ability of a substance to transfer heat through conduction can be represented by the constant of proportionality, k, referred to as the thermal conductivity. Figure 1 illustrates the characteristics of the thermal conductivity as a function of temperature for several solids, liquids, and gases. As shown, the thermal conductivity of solids is higher than liquids, and liquids higher than gases. Metals typically have higher thermal conductivities than nonmetals, with pure metals having thermal conductivities that decrease with increasing temperature, while the thermal conductivity of nonmetallic solids generally increases with increasing temperature and density. The addition of other metals to create alloys, or the presence of impurities, usually decreases the thermal conductivity of a pure metal. In general, the thermal conductivity of liquids decreases with increasing temperature. Alternatively, the thermal conductivity of gases and vapors, while

R=

N xn n=1

kn A

Similarly, for conduction heat transfer in the radial direction through a number of N concentric cylinders with negligible interfacial resistance, the overall thermal resistance can be expressed as R=

N ln(rn+1 /rn ) n=1

2πkn L

where r1 = inner radius, rN +1 = outer radius. For N concentric spheres with negligible interfacial resistance, the thermal resistance can be expressed as R=

N 1/rn − 1/rn+1 n=1

4πk

where r1 = inner radius, rN +1 = outer radius.


821

Fig. 1 Temperature dependence of thermal conductivity of (a) selected solids, (b) selected nonmetallic liquids under saturated conditions, and (c) selected gases at normal pressures.1

822 Table 1

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Thermal Properties of Metallic Solidsa Properties at Various Temperatures (K) k(W/m·K); Cp (J/kg·K)

Properties at 300 K

Composition

Melting Point (K)

ρ (kg/m3 )

Cp (J/kg·K)

k (W/m·K)

α × 106 (m2 /s)

100

Aluminum Copper Gold Iron Lead Magnesium Molybdenum Nickel Platinum Silicon Silver Tin Titanium Tungsten Zinc

933 1358 1336 1810 601 923 2894 1728 2045 1685 1235 505 1953 3660 693

2702 8933 19300 7870 11340 1740 10240 8900 21450 2330 10500 7310 4500 19300 7140

903 385 129 447 129 1024 251 444 133 712 235 227 522 132 389

237 401 317 80.2 35.3 156 138 90.7 71.6 148 429 66.6 21.9 174 116

97.1 117 127 23.1 24.1 87.6 53.7 23.0 25.1 89.2 174 40.1 9.32 68.3 41.8

302; 482 482; 252 327; 109 134; 216 39.7; 118 169; 649 179; 141 164; 232 77.5; 100 884; 259 444; 187 85.2; 188 30.5; 300 208; 87 117; 297

a

600 231; 1033 379; 417 298; 135 54.7; 574 31.4; 142 149; 1170 126; 275 65.6; 592 73.2; 141 61.9; 867 412; 250 19.4; 591 137; 142 103; 436

1200 339; 480 255; 155 28.3; 609 105; 308 76.2; 594 82.6; 157 25.7; 967 361; 292 22.0; 620 113; 152

Adapted from Ref. 1.

Table 2

Thermal Properties of Nonmetals

Description/Composition Bakelite Brick, refractory Carborundum Chrome-brick Fire clay brick Clay Coal, anthracite Concrete (stone mix) Cotton Glass, window Rock, limestone Rubber, hard Soil, dry Teflon

Temperature (K)

Density, ρ(kg/m3 )

Thermal Conductivity, k (W/m·K)

Specific Heat, Cp (J/kg·K)

α × 106 (m2 /s)

300

1300

0.232

1465

0.122

872 473 478 300 300 300 300 300 300 300 300 300 400

— 3010 2645 1460 1350 2300 80 2700 2320 1190 2050 2200 —

18.5 2.32 1.0 1.3 0.26 1.4 0.059 0.78 2.15 0.160 0.52 0.35 0.45

— 835 960 880 1260 880 1300 840 810 — 1840 — —

— 0.915 0.394 1.01 0.153 0.692 0.567 0.344 1.14 — 0.138 — —

1.3 Two-Dimensional Steady-State Heat Conduction

Two-dimensional heat transfer in an isotropic, homogeneous material with no internal heat generation requires solution of the heat diffusion equation of the form ∂ 2 T /∂X 2 + ∂T /∂y 2 = 0, referred to as the Laplace equation. For certain geometries and a limited number of fairly simple combinations of boundary conditions, exact solutions can be obtained analytically. However, for anything but simple geometries or for simple geometries with complicated boundary conditions, development of an appropriate analytical solution can be difficult and other methods are

usually employed. Among these are solution procedures involving the use of graphical or numerical approaches. In the first of these, the rate of heat transfer between two isotherms, T1 and T2 , is expressed in terms of the conduction shape factor, defined by q = kS(T1 − T2 ) Table 9 illustrates the shape factor for a number of common geometric configurations. By combining these shape factors, the heat transfer characteristics for a wide variety of geometric configurations can be obtained.


823

Table 3 Thermal Properties of Building and Insulating Materials (at 300 K)a

Description/Composition Building boards Plywood Acoustic tile Hardboard, siding Woods Hardwoods (oak, maple) Softwoods (fir, pine) Masonry materials Cement mortor Brick, common Plastering materials Cement plaster, sand aggregate Gypsum plaster, sand aggregate Blanket and batt Glass fiber, paper faced Glass fiber, coated; duct liner Board and slab Cellular glass Wood, shredded/cemented Cork Loose fill Glass fiber, poured or blown Vermiculite, flakes a Adapted

Density ρ(kg/m3 )

Thermal Conductivity, k (W/m·K)

Specific Heat, Cp (J/kg·K)

α × 106 (m2 /s)

545 290 640

0.12 0.058 0.094

1215 1340 1170

0.181 0.149 0.126

720 510

0.16 0.12

1255 1380

0.177 0.171

1860 1920

0.72 0.72

780 835

0.496 0.449

1860 1680

0.72 0.22

— 1085

— 0.121

16 32

0.046 0.038

— 835

— 1.422

145 350 120

0.058 0.087 0.039

1000 1590 1800

0.400 0.156 0.181

16 80

0.043 0.068

835 835

3.219 1.018

from Ref. 1.

Table 4 Thermal Conductivities for Some Industrial Insulating Materialsa

Description/Composition Blankets Blanket, mineral fiber, glass; fine fiber organic bonded Blanket, alumina-silica fiber Felt, semirigid; organic bonded Felt, laminated; no binder Blocks, boards, and pipe insulations Asbestos paper, laminated and corruagated, 4-ply Calcium silicate Polystyrene, rigid Extruded (R-12) Molded beads Rubber, rigid foamed Insulating cement Mineral fiber (rock, slag, or glass) With clay binder With hydraulic setting binder Loose fill Cellulose, wood, or paper pulp Perlite, expanded Vermiculite, expanded a Adapted

from Ref. 1.

Maximum Service Temperature (K) 450

Typical Density (kg/m3 )

Typical Thermal Conductivity, k (W/m·K), at Various Temperature (K) 200

300 0.048 0.033

1530 480 920

10 48 48 50–125 120

420 920

190 190

0.078

350 350 340

56 16 70

1255 922

430 560

— — —

45 105 122

420

645

0.105 0.038

0.023 0.026

0.036

0.063 0.051

0.087

0.063

0.089

0.088 0.123

0.123

0.027 0.040 0.032

0.039 0.053 0.068

824 Table 5

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Thermal Properties of Saturated Liquidsa

Liquid Ammonia, NH3 Carbon dioxide, CO2 Engine oil (unused) Ethylene glycol, C2 H4 (OH)2 Clycerin, C3 H5 (OH)3 Freon (Refrigerant-12), CCI2 F2

T (K)

ρ (kg/m3 )

Cp (kJ/kg·K)

223 323 223 303 273 430 273 373 273 320 230 320

703.7 564.3 1156.3 597.8 899.1 806.5 1130.8 1058.5 1276.0 1247.2 1528.4 1228.6

4.463 5.116 1.84 36.4 1.796 2.471 2.294 2.742 2.261 2.564 0.8816 1.0155

a Adapted

from Ref. 2. See Table 22 for H2 O.

Table 6

Thermal Properties of Liquid Metalsa

Composition

Melting Point (K)

Bismuth

544

Lead

600

Mercury

234

Potassium

337

Sodium

371

NaK (56%/44%)

292

PbBi (44.5%/55.5%)

398

a Adapted

T (K)

ρ (kg/m3 )

589 1033 644 755 273 600 422 977 366 977 366 977 422 644

10,011 9,467 10,540 10,412 13,595 12,809 807.3 674.4 929.1 778.5 887.4 740.1 10,524 10,236

v × 106 (m2 /s) 0.435 0.330 0.119 0.080 4280 5.83 57.6 2.03 8310 168 0.299 0.190

k × 103 (W/m·K) 547 476 85.5 70.3 147 132 242 263 282 287 68 68

α × 107 (m2 /s)

Pr

β × 103 (K−1 )

1.742 1.654 0.402 0.028 0.910 0.662 0.933 0.906 0.977 0.897 0.505 0.545

2.60 1.99 2.96 28.7 47,000 88 617.0 22.4 85,000 1,870 5.9 3.5

2.45 2.45 14.0 14.0 0.70 0.70 0.65 0.65 0.47 0.50 1.85 3.50

Cp (kJ/kg·K)

v × 107 (m2 /s)

k (W/m·K)

0.1444 0.1645 0.159 0.155 0.140 0.136 0.80 0.75 1.38 1.26 1.130 1.043 0.147 0.147

1.617 0.8343 2.276 1.849 1.240 0.711 4.608 1.905 7.516 2.285 6.522 2.174 — 1.496

16.4 15.6 16.1 15.6 8.180 11.95 45.0 33.1 86.2 59.7 25.6 28.9 9.05 11.86

α × 105 (m2 /s)

Pr

0.138 1.001 1.084 1.223 0.429 0.688 6.99 6.55 6.71 6.12 2.55 3.74 0.586 0.790

0.0142 0.0083 0.024 0.017 0.0290 0.0103 0.0066 0.0029 0.011 0.0037 0.026 0.0058 — 0.189

from Liquid Metals Handbook, The Atomic Energy Commission, Department of the Navy, Washington, DC, 1952.

Prior to the development of high-speed digital computers, shape factor and analytical methods were the most prevalent methods utilized for evaluating steady-state and transient conduction problems. However, more recently, solution procedures for problems involving complicated geometries or boundary conditions utilize the finite-difference method (FDM). Using this approach, the solid object is divided into a number of distinct or discrete regions, referred to as nodes, each with a specified boundary condition. An energy balance is then written for each nodal region and these equations are solved simultaneously. For interior nodes in a two-dimensional system with no internal heat generation, the energy equation takes the form of the Laplace equation discussed earlier. However, because the system is characterized in terms of a nodal network, a finite-difference approximation must be used. This approximation is derived by substituting the following

equation for the x-direction rate of change expression Tm+1,n + Tm−1,n − 2Tm,n ∂ 2 T ≈ 2 ∂x m,n (x)2 and for the y-direction rate of change expression: ∂ 2 T Tm,n+1 + Tm,n−1 + Tm,n ∂y 2 m,n (y)2 Assuming x = y and substituting into the Laplace equation and results in the following expression: Tm,n+1 + Tm,n−1 + Tm+1,n + Tm−1,n − 4Tm,n = 0 which reduces the exact difference to an approximate algebraic expression.


825

Table 7 Thermal Properties of Gases at Atmospheric Pressurea Gas Air Ammonia, NH3 Carbon dioxide Carbon monoxide Helium Hydrogen Nitrogen Oxygen Steam (H2 O vapor) a

T (K)

ρ (kg/m3 )

Cp (kJ/kg·K)

v × 106 (m2 /s)

k (W/m·K)

α × 104 (m2 /s)

100 300 2500 220 473 220 600 220 600 33 900 30 300 1000 100 300 1200 100 300 600 380 850

3.6010 1.1774 0.1394 0.3828 0.4405 2.4733 0.8938 1.5536 0.5685 1.4657 0.05286 0.8472 0.0819 0.0819 3.4808 1.1421 0.2851 3.9918 1.3007 0.6504 0.5863 0.2579

1.0266 1.0057 1.688 2.198 2.395 0.783 1.076 1.0429 1.0877 5.200 5.200 10.840 14.314 14.314 1.0722 1.0408 1.2037 0.9479 0.9203 1.0044 2.060 2.186

1.923 16.84 543.0 19.0 37.4 4.490 30.02 8.903 52.06 3.42 781.3 1.895 109.5 109.5 1.971 15.63 156.1 1.946 15.86 52.15 21.6 115.2

0.009246 0.02624 0.175 0.0171 0.0467 0.01081 0.04311 0.01906 0.04446 0.0353 0.298 0.0228 0.182 0.182 0.009450 0.0262 0.07184 0.00903 0.02676 0.04832 0.0246 0.0637

0.0250 0.2216 7.437 0.2054 0.4421 0.0592 0.4483 0.1176 0.7190 0.04625 10.834 0.02493 1.554 1.554 0.02531 0.204 2.0932 0.02388 0.2235 0.7399 0.2036 1.130

Pr 0.768 0.708 0.730 0.93 0.84 0.818 0.668 0.758 0.724 0.74 0.72 0.759 0.706 0.706 0.786 0.713 0.748 0.815 0.709 0.704 1.060 1.019

Adapted from Ref. 2.

Combining this temperature difference with Fourier’s law yields an expression for each internal node Tm,n+1 + Tm,n+1 + Tm−1,n + Tm−1,n +

q˙ x y 1 k − 4Tm,n = 0

Similar equations for other geometries (i.e., corners) and boundary conditions (i.e., convection) and combinations of the two are listed in Table 10. These equations must then be solved using some form of matrix inversion technique, Gauss–Seidel iteration method or other method for solving large numbers of simultaneous equations. 1.4 Heat Conduction with Convection Heat Transfer on Boundaries In physical situations where a solid is immersed in a fluid, or a portion of the surface is exposed to a liquid or gas, heat transfer will occur by convection (or when there is a large temperature difference, through some combination of convection and/or radiation). In these situations, the heat transfer is governed by Newton’s law of cooling, which is expressed as

q = hA T where h is the convection heat transfer coefficient (Section 2), T is the temperature difference between

the solid surface and the fluid, and A is the surface area in contact with the fluid. The resistance occurring at the surface abounding the solid and fluid is referred to as the thermal resistance and is given by 1/hA, that is, the convection resistance. Combining this resistance term with the appropriate conduction resistance yields an overall heat transfer coefficient U. Usage of this term allows the overall heat transfer to be defined as q = U A T . Table 8 shows the overall heat transfer coefficients for some simple geometries. Note that U may be based either on the inner surface (U1 ) or on the outer surface (U2 ) for the cylinders and spheres. Critical Radius of Insulation for Cylinders A large number of practical applications involve the use of insulation materials to reduce the transfer of heat into or out of cylindrical surfaces. This is particularly true of steam or hot water pipes where concentric cylinders of insulation are typically added to the outside of the pipes to reduce the heat loss. Beyond a certain thickness, however, the continued addition of insulation may not result in continued reductions in the heat loss. To optimize the thickness of insulation required for these types of applications, a value typically referred to as the critical radius, defined as rcr = k/ h, is used. If the outer radius of the object to be insulated is less than rcr then the addition of insulation will increase the heat loss, while for cases where the outer radii is greater than rcr any additional increases in insulation thickness will result in a decrease in heat loss.

826 One-Dimensional Heat Conduction

Hollow sphere

Hollow cylinder

Plane wall

Geometry

Table 8

ln(r2 /r1 ) 2π kL

R=

R=

q=

1/r1 − 1/r2 4π k

T1 − T2 (1/r1 − 1/r2 )/4π k r1 1 r1 T= (T1 − T2 ) + T2 − T1 (1 − r1 /r2 ) r r2

T2 − T1 r ln ln(r2 /r1 ) r1

T=

T1 − T2 q= [ln(r2 /r1 )]/2π kL

R = (xx − x1 )/ kA

T2 − T1 (x − x1 ) xx − x1

T1 − T2 (x2 − x1 )/ kA

T = T1 +

q=

Heat Transfer Rate and Temperature Distribution

1 11 x2 − x2 + + h1 k h2

−1

U2 =

U1 =

r1 r2

2

r1 r2

2

1 h2 r 2 (1/r1 − 1/r2 ) 1 1 + + 2 h1 k h2

r 2 (1/r1 − 1/r2 ) 1 + + 1 h1 k

= 4π r22 U2 (T∞,1 − T∞,2 )

q = 4π r12 U1 (T∞,1 − T∞,2 )

1 r1 ln(r2 /r1 ) r1 1 −1 + + h1 k r2 h2 r2 1 r2 ln(r2 /r1 ) 1 −1 U2 = + + r1 h1 k h2 U1 =

= 2π r1 LU2 (T∞,1 − T∞,2 )

q = 2π r1 LU1 (T∞,1 − T∞,2 )

U=

q = UA(T∞,1 − T∞,2 )

Heat Transfer Rate and Overall Heat Transfer Coefficient with Convection at the Boundaries

−1

−1


827

Table 9 Conduction Shape Factors System

Schematic

Restrictions

Isothermal sphere buried in a semi-infinite medium having isothermal surface

z > D/2

Horizontal isothermal cylinder of length L buried in a semi-infinite medium having isothermal surface

LD LD z > 3D/2

Shape Factor 2π D 1 − D/4z

2π L cosh−1 (2z/D) 2π L ln(4z/D)

The cylinder of length L with eccentric bore

L D1 , D2

cosh−1

2π L D21 + D22 − 4ε2 2D1 D2

2π L

Conduction between two cylinders of length L in infinite medium

L D1 , D2

Circular cylinder of length L in a square solid

LW w>D

2π L ln(1.08 w/D)

Conduction through the edge of adjoining walls

D > L/5

0.54 D

Conduction through corner of three walls with inside and outside temperature, respectively, at T1 and T2

L length and width of wall

0.15 L

Extended Surfaces In examining Newton’s law of cooling, it is clear that the rate of heat transfer between a solid and the surrounding ambient fluid may be increased by increasing the surface area of the solid that is exposed to the fluid. This is typically done through the addition of extended surfaces or fins to the primary surface. Numerous examples often exist,

cosh

−1

4 W 2 − D21 − D22 2D1 D2

including the cooling fins on air-cooled engines, that is, motorcycles or lawn mowers or the fins attached to automobile radiators. Figure 2 illustrates a common uniform cross-section extended surface, fin, with a constant base temperature, Tb , a constant cross-sectional area, A, a circumference of C = 2W + 2t, and a length, L, which is much larger

828 Table 10

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Summary of Nodal Finite-Difference Equations

Configuration Case 1. Interior node

Finite-Difference Equation for x = y Tm,n+1 + Tm,n−1 + Tm−1,n − 4Tm,n = 0

Case 2. Node at an internal corner with convection

2(Tm−1,n + Tm,n+1 ) + (Tm+1,n + Tm,n−1 ) h x h x +2 T∞ − 2 3 + Tm,n = 0 k k

Case 3. Node at a plane surface with convection

2(Tm−1,n + Tm,n+1 + Tm,n−1) + h x + 2 Tm,n = 0 −2 k

Case 4. Node at an external corner with convection

Case 5. Node near a curved surface maintained at a nonuniform temperature

h x (Tm,n−1 + Tm−1,n ) + 2 T∞ k h x + 1 Tm,n = 0 −2 k

2 2 Tm+1,n + Tm,n−1 a+1 b+1 2 2 T1 + T2 + a(a + 1) b(b + 1) 2 2 Tm,n = 0 − + a b

2h x T∞ k


Fig. 2

829

Heat transfer by extended surfaces.

than the thickness, t. For these conditions, the temperature distribution in the fin must satisfy the following expression: hC d 2T (T − T∞ ) = 0 − dx 2 kA The solution of this equation depends on the boundary conditions existing at the tip, that is, at x = L. Table 11 shows the temperature distribution and heat transfer rate for fins of uniform cross section subjected to a number of different tip conditions, assuming a constant value for the heat transfer coefficient, h. Two terms are used to evaluate fins and their usefulness. The first of these is the fin effectiveness,

defined as the ratio of the heat transfer rate with the fin to the heat transfer rate that would exist if the fin were not used. For most practical applications, the use of a fin is justified only when the fin effectiveness is significantly greater than 2. A second term used to evaluate the usefulness of a fin is the fin efficiency, ηf , This term represents the ratio of actual heat transfer rate from a fin to the heat transfer rate that would occur if the entire fin surface could be maintained at a uniform temperature equal to the temperature of the base of the fin. For this case, Newton’s law of cooling can be written as q = ηf hAf (Tb − T∞ )

Table 11 Temperature Distribution and Heat Transfer Rate at the Fin Base (m = Condition at x = L h(Tx=L − T∞ ) = −k

(convection) dT =0 dx x=L (insulated)

hc/kA)

T − T∞ Tb − T∞

dT dx

x=L

Tx=L = TL (prescribed temperature) Tx=L = T∞ (infinitely long fin, L → ∞)

Heat Transfer Rate q/mkA (Tb − T∞ )

cosh m(L − x) + [h/(mk)] sinh m(L − x) cosh mL + [h/(mk)] sinh mL

sinh mL + [h/(mk)] cosh mL cosh mL + [h/(mk)] sinh mL

cosh m(L − x) cosh mL

tanh mL

(TL − T∞ )/(Tb − T∞ ) sinh mx + sinh m(L − x) sinh ml

cosh mL − (TL − T∞ )/(Tb − T∞ ) sinh ml

e−mx

1

830


where Af is the total surface area of the fin and Tb is the temperature of the fin at the base. The application of fins for heat removal can be applied to either forced or natural convection of gases, and while some advantages can be gained in terms of increasing the liquid–solid or solid–vapor surface area, fins as such are not normally utilized for situations involving phase change heat transfer, such as boiling or condensation. 1.5 Transient Heat Conduction

Given a solid body, at a uniform temperature, T∞i , immersed in a fluid of different temperature T∞ , the surface of the solid body will be subject to heat losses (or gains) through convection from the surface to the fluid. In this situation, the heat lost (or gained) at the surface results from the conduction of heat from inside the body. To determine the significance of these two heat transfer modes, a dimensionless parameter referred to as the Biot number is used. This dimensionless number is defined as Bi = hL/k, where L = V /A or the ratio of the volume of the solid to the surface area of the solid, and really represents a comparative relationship of the importance of convections from the outer surface to the conduction occurring inside. When this value is less than 0.1, the temperature of the solid may be assumed uniform and dependent on time alone. When this value is greater than 0.1, there is some spatial temperature variation that will affect the solution procedure. For the first case, Bi < 0.1, an approximation referred to as the lumped heat capacity method may be used. In this method, the temperature of the solid is given by T − T∞ −t = exp(−Bi Fo) = exp Ti − T∞ τt where τt is the time constant and is equal to ρCp V/ hA. Increasing the value of the time constant, τt , will result in a decrease in the thermal response of the solid to the environment and hence, will increase the time required for it to reach thermal equilibrium (i.e., T = T∞ ). In this expression, Fo represents the dimensionless time and is called the Fourier number, the value of which is equal to αtA2 /V 2 . The Fourier number, along with the Biot number, can be used to characterize transient heat conduction problems. The total heat flow through the surface of the solid over the time interval from t = 0 to time t can be expressed as

−t Q = ρV Cp (Ti − T∞ ) 1 − exp τt

Transient Heat Transfer for Infinite Plate, Infinite Cylinder, and Sphere Subjected to Surface Convection Generalized analytical solutions to transient heat transfer problems involving infinite plates, cylinders, and finite diameter spheres subjected

to surface convection have been developed. These solutions can be presented in graphical form through the use of the Heisler charts,3 illustrated in Figs. 3–11 for plane walls, cylinders, and spheres. In this procedure, the solid is assumed to be at a uniform temperature, Ti , at time t = 0 and then is suddenly subjected to or immersed in a fluid at a uniform temperature T∞ . The convection heat transfer coefficient, h, is assumed to be constant, as is the temperature of the fluid. Combining Figs. 3 and 4 for plane walls, Figs. 6 and 7 for cylinders, and Figs. 9 and 10 for spheres allows the resulting time-dependent temperature of any point within the solid to be found. The total amount of energy, Q, transferred to or from the solid surface from time t = 0 to time t can be found from Figs. 5, 8, and 11. 1.6 Conduction at Microscale The mean free path of electrons and the size of the volume involved has long been recognized as having a pronounced effect on electron transport phenomena. This is particularly true in applications involving thin metallic films or wires where the characteristic length may be close to the same order of magnitude as the scattering mean free path of the electrons.4a The first notable work in this area was performed by Tien et al.,4b where the thermal conductivity of thin metallic films and wires were calculated at cryogenic temperatures. Because the length of the mean free path in these types of applications is shortened near the surface, due to termination at the boundary, a reduction in transport coefficients, such as electrical and thermal conductivities, was observed. Tests at cryogenic temperatures were first performed because the electron mean free path increases as temperature decreases, and the size effects were expected to become especially significant in this range. The primary purpose of this investigation was to outline in a systematic manner a method by which the thermal conductivity of such films and wires at cryogenic temperatures could be determined. The results indicated that, particularly in the case of thin metallic films, size effects may become an increasingly important part of the design and analysis required for application. Due to the increased use of thin films in optical components and solid-state devices and systems, there has been an increasing interest in the effect of decreasing size on the transport properties of thin solid films and wires. The most common method for calculating the thermal conductivities in thin films and wires consists of three essential steps:

1. Identifying the appropriate expression for the electrical conductivity size effect 2. Determining the mean free path for electrical conductivity, which is essential in calculations of all electron transport properties 3. Applying the electrical–thermal transport analogy for calculating the thermal conductivity size effect4a

831

Fig. 3

Midplane temperature as a function of time for a plane wall of thickness 2L. (Adapted from Ref. 3.)

832


Fig. 4 Temperature distribution in a plane wall of thickness 2L. (Adapted from Ref. 3.)

Fig. 5 Internal energy change as a function of time for a plane wall of thickness 2L.4 (Used with the permission of McGraw-Hill Book Company.)

833

Fig. 6

Centerline temperature as function of time for an infinite cylinder of radius ro . (Adapted from Ref. 3.)

834


Fig. 7 Temperature distribution in an infinite cylinder of radius ro . (Adapted from Ref. 3.)

Fig. 8 Internal energy change as function of time for an infinite cylinder of radius ro .4 (Used with the permission of McGraw-Hill Book Company.)

For domain thicknesses on the order of the carrier mean free path, jump boundary conditions significantly affect the solution of the conduction problem. This problem can be resolved through the solution of the hyperbolic heat equation-based analysis, which is generally justifiable engineering applications.4c

2 CONVECTION HEAT TRANSFER As discussed earlier, convection heat transfer is the mode of energy transport in which the energy is transferred by means of fluid motion. This transfer can be the result of the random molecular motion or bulk motion of the fluid. If the fluid motion is

835

Fig. 9 Center temperature as function of time in a sphere of radius ro . (Adapted from Ref. 3.)

836


Fig. 10 Temperature distribution in sphere of radius ro . (Adapted from Ref. 3.)

Fig. 11 Internal energy change as function of time for a sphere of radius ro .4 (Used with the permission of McGraw-Hill Book Company.)

caused by external forces, the energy transfer is called forced convection. If the fluid motion arises from a buoyancy effect caused by density differences, the energy transfer is called free convection or natural convection. For either case, the heat transfer rate, q, can be expressed in terms of the surface area, A, and the temperature difference, T , by Newton’s law of

cooling: q = hA T In this expression, h is referred to as the convection heat transfer coefficient or film coefficient and a function of the velocity and physical properties of the fluid, and the shape and nature of the surface. The


nondimensional heat transfer coefficient Nu = hL/k is called the Nusselt number, where L is a characteristic length and k is the thermal conductivity of the fluid. 2.1 Forced Convection—Internal Flow

For internal flow in a tube or pipe, the convection heat transfer coefficient is typically defined as a function of the temperature difference existing between the temperature at the surface of the tube and the bulk or mixing-cup temperature, Tb , that is, T = Ts − Tb can be defined as Cp T d m ˙ Tb = Cp d m ˙ where m ˙ is the axial flow rate. Using this value, heat transfer between the tube and the fluid can be written as q = hA(Ts − Tb ). In the entrance region of a tube or pipe, the flow is quite different from that occurring downstream from the entrance. The rate of heat transfer differs significantly, depending on whether the flow is laminar or turbulent. From fluid mechanics, the flow is considered to be turbulent when ReD = Vm D/v > 2300 for a smooth tube. This transition from laminar to turbulent, however, also depends on the roughness of tube wall and other factors. The generally accepted range for transition is 200 < ReD < 4000.

837

Laminar Fully Developed Flow For situations where both the thermal and velocity profiles are fully developed, the Nusselt number is constant and depends only on the thermal boundary conditions. For circular tubes with Pr ≥ 0.6, and x/D ReD Pr > 0.05, the Nusselt numbers have been shown to be NuD = 3.66 and 4.36, for constant temperature and constant heat flux conditions, respectively. Here, the fluid properties are based on the mean bulk temperature. For noncircular tubes, the hydraulic diameter, Dh = 4 × the flow cross-sectional area/wetted perimeter, is used to define the Nusselt number NuD and the Reynolds number ReD . Table 12 shows the Nusselt numbers based on hydraulic diameter for various crosssectional shapes. Laminar Flow for Short Tubes At the entrance of a tube, the Nusselt number is infinite, and decreases asymptotically to the value for fully developed flow as the flow progresses down the tube. The Sieder–Tate equation5 gives good correlation for the combined entry length, that is, that region where the thermal and velocity profiles are both developing or for short tubes:

NuD = hD = 1.86(Re D Pr)1/13 k

D L

1/3

NuH1

NuH2

Nur

3.608

3.091

2.976

4.123

3.017

3.391

5.099

4.35

3.66

6.490

2.904

5.597

8.235

8.235

7.541

5.385

—

4.861

4.364

4.364

3.657

NuH1 = average Nusselt number for uniform heat flux in flow direction and uniform wall temperature at particular flow cross section. NuH2 = average Nusselt number for uniform heat flux both in flow direction and around periphery. NuHrr = average Nusselt number for uniform wall temperature. a

0.14

for Ts = constant, 0.48 < Pr < 16,700, 0.0044 < µ/µs < 9.75, and (ReD Pr D/L)1/3 (µ/µs )0.14 > 2.

Table 12 Nusselt Numbers for Fully Developed Laminar Flow for Tubes of Various Cross Sectionsa Geometry (L/DH > 100)

µ µs

838


In this expression, all of the fluid properties are evaluated at the mean bulk temperature except for µs , which is evaluated at the wall surface temperature. The average convection heat transfer coefficient h is based on the arithmetic average of the inlet and outlet temperature differences. Turbulent Flow in Circular Tubes In turbulent flow, the velocity and thermal entry lengths are much shorter than for a laminar flow. As a result, with the exception of short tubes, the fully developed flow values of the Nusselt number are frequently used directly in the calculation of the heat transfer. In general, the Nusselt number obtained for the constant heat flux case is greater than the Nusselt number obtained for the constant temperature case. The one exception to this is the case of liquid metals, where the difference is smaller than for laminar flow and becomes negligible for Pr > 1.0. The Dittus–Boelter equation6 is typically used if the difference between the pipe surface temperature and the bulk fluid temperature is less than 6◦ C (10◦ F) for liquids or 56◦ C (100◦ F) for gases: n NuD = 0.023 Re0.8 D Pr

for 0.7 ≤ Pr ≤ 160, ReD ≥ 10,000, and L/D ≥ 60, where n = 0.4 for heating, Ts > Tb = 0.3 for cooling, Ts < Tb For temperature differences greater than specified above, use5 1/3 NuD = 0.027 Re0.8 D Pr

µ µs

0.14

for 0.7 ≤ Pr ≤ 16,700, ReD ≥ 10,000, and L/D ≥ 60. In this expression, the properties are all evaluated at the mean bulk fluid temperature with the exception of µs , which is again evaluated at the tube surface temperature. For concentric tube annuli, the hydraulic diameter Dh = Do − Di (outer diameter − inner diameter) must be used for NuD and ReD , and the coefficient h at either surface of the annulus must be evaluated from the Dittus–Boelter equation. Here, it should be noted that the foregoing equations apply for smooth surfaces and that the heat transfer rate will be larger for rough surfaces and are not applicable to liquid metals. Fully Developed Turbulent Flow of Liquid Metals in Circular Tubes Because the Prandtl number for liquid metals is on the order of 0.01, the Nusselt number is primarily dependent on a dimensionless

parameter number referred to as the Peclet number, which in general is defined as Pe = RePr: NuD = 5.0 + 0.025 Pe0.8 D which is valid for situations where Ts = a constant and PeD > 100 and L/D > 60. For q = constant and 3.6 × 103 < ReD < 9.05 × 105 , 102 < PeD < 104 , and L/D > 60, the Nusselt number can be expressed as NuD = 4.8 + 0.0185 Pe0.827 D 2.2 Forced Convection—External Flow

In forced convection heat transfer, the heat transfer coefficient, h, is based on the temperature difference between the wall surface temperature and the fluid temperature in the free stream outside the thermal boundary layer. The total heat transfer rate from the wall to the fluid is given by q = hA(Ts − T∞ ). The Reynolds numbers are based on the free-stream velocity. The fluid properties are evaluated either at the free-stream temperature T∞ or at the film temperature Tf = (Ts + T∞ )/2. Laminar Flow on a Flat Plate When the flow velocity along a constant temperature semi-infinite plate is uniform, the boundary layer originates from the leading edge and is laminar and the flow remains laminar until the local Reynolds number Rex = U∞ x/v reaches the critical Reynolds number, Rec . When the surface is smooth, the Reynolds number is generally assumed to be Rec = 5 × 105 , but the value will depend on several parameters, including the surface roughness. For a given distance x from the leading edge, the local Nusselt number and the average Nusselt number between x = 0 and x = L are given below (Rex and ReL ≤ 5 × 105 ):

For Pr ≥ 0.6: hx 1/3 = 0.332 Re0.5 x Pr k hL 1/3 = 0.664 Re0.5 NuL = L Pr k Nux =

For Pr ≤ 0.6: Nux = 0.565(Rex Pr)0.5

NuL = 1.13(ReL Pr)0.5

Here, all of the fluid properties are evaluated at the mean or average film temperature.


839

Turbulent Flow on Flat Plate When the flow over a flat plate is turbulent from the leading edge, expressions for the local Nusselt number can be written as 1/3 Nux = 0.0292 Re0.8 x Pr

Circular Cylinders in Cross Flow For circular cylinders in cross flow, the Nusselt number is based upon the diameter and can be expressed as

2/3 NuD = (0.4 Re0.5 )Pr0.4 D + 0.06 Re

1/3 NuL = 0.036 Re0.8 L Pr

µ∞ µs

0.25

where the fluid properties are all based on the mean film temperature and 5 × 105 ≤ Rex and ReL ≤ 108 and 0.6 ≤ Pr ≤ 60.

for 0.67 < Pr < 300, 10 < ReD < 105 , and 0.25 < 5.2. Here, the fluid properties are evaluated at the free stream temperature except µs , which is evaluated at the surface temperature.8

Average Nusselt Number between x = 0 and x = L with Transition For situations where transition occurs immediately once the critical Reynolds number Rec has been reached7

Cylinders of Noncircular Cross Section in Cross Flow of Gases For noncircular cylinders in cross flow, the Nusselt number is again based on the diameter, but is expressed as

0.8 0.5 NuL = 0.036 Pr1/3 [Re0.8 L − Rec + 18.44 Rec ]

NuD = C(ReD )m Pr1/3

provided that 5 × 105 ≤ ReL ≤ 108 and 0.6 ≤ Pr ≤ 60. Specialized cases exist for this situation, that is, NuL = 0.036 Pr

1/3

(Re0.8 L

where C and m are listed in Table 13, and the fluid properties are evaluated at the mean film temperature.9 Flow Past a Sphere For flow over a sphere, the Nusselt number is based on the sphere diameter and can be expressed as

− 18,700)

for Rec = 4 × 105 , or

2/3

0.4 NuD = 2 + (0.4 Re0.5 D + 0.06 ReD )Pr

NuL = 0.036 Pr1/3 (Re0.8 L − 23,000) for Rec = 5 × 105 . Again, all fluid properties are evaluated at the mean film temperature. Table 13

Square

Hexagon

Vertical plate

0.25

for the case of 3.5 < ReD < 8 × 104 , 0.7 < Pr < 380, and 1.0 < µ∞ /µs < 3.2. The fluid properties are calculated at the free-stream temperature except µs , which is evaluated at the surface temperature.8

Constants and m for Noncircular Cylinders in Cross Flow

Geometry

µ∞ µs

ReD

C

m

5 × 103 –105 5 × 103 –105

0.246 0.102

0.588 0.675

5 × 103 –1.95 × 104 1.95 × 104 –105

0.160 0.0385

0.538 0.782

5 × 103 –105 4 × 103 –1.5 × 104

0.153 0.228

0.638 0.731

840


Fig. 12

Tube arrangement.

Flow across Banks of Tubes For banks of tubes, the tube arrangement may be either staggered or aligned (Fig. 12), and the heat transfer coefficient for the first row is approximately equal to that for a single tube. In turbulent flow, the heat transfer coefficient for tubes in the first row is smaller than that of the subsequent rows. However, beyond the fourth or fifth row, the heat transfer coefficient becomes approximately constant. For tube banks with more than 20 rows, 0.7 < Pr < 500, and 1000 < ReD,max < 2 × 106 , the average Nusselt number for the entire tube bundle can be expressed as10

NuD = C(ReD,max )m Pr0.36

Pr∞ Prs

0.25

Table 14

ST + D SL2 + (ST /2)2 > 2 or as ST V Vmax = 2 2 SL + (ST /2)2 for staggered if

ST + D SL2 + (ST /2)2 < 2

where all fluid properties are evaluated at T∞ except Prs , which is evaluated at the surface temperature. The constants C and m used in this expression are listed in Table 14, and the Reynolds number is based on the maximum fluid velocity occurring at the minimum free flow area available for the fluid. Using the nomenclature shown in Fig. 12, the maximum fluid velocity can be determined by Vmax =

for the aligned or staggered configuration provided

ST V ST − D

Liquid Metals in Cross Flow over Banks of Tubes The average Nusselt number for tubes in the inner rows can be expressed as

NuD = 4.03 + 0.228(ReD,max Pr)0.67 which is valid for 2 × 104 < ReD,max < 8 × 104 and Pr < 0.03 and the fluid properties are evaluated at the mean film temperature.11

Constants C and m of Heat Transfer Coefficient for the Banks in Cross Flow

Configuration

ReD,max

C

m

Aligned Staggered (ST /SL < 2) Staggered (SG /SL > 2) Aligned Staggered

× 103 –2 × 105

0.27 0.35(ST /SL )1/5

0.63 0.60

103 –2 × 105

0.40

0.60

2 × 105 –2 × 106 2 × 105 –2 × 106

0.21 0.022

0.84 0.84

103 –2

105


High-Speed Flow over a Flat Plate When the free stream velocity is very high, the effects of viscous dissipation and fluid compressibility must be considered in the determination of the convection heat transfer. For these types of situations, the convection heat transfer can be described as q = hA(Ts − Tαs ), where Tαs is the adiabatic surface temperature or recovery temperature, and is related to the recovery factor by r = (Tαs − T∞ )/(T0 − T∞ ). The value of the stagnation temperature, T0 , is related to the free-stream static temperature, T∞ , by the expression

γ −1 2 T0 M∞ =1+ T∞ 2 where γ is the specific heat ratio of the fluid and M∞ is the ratio of the free-stream velocity and the acoustic velocity. For the case where 0.6 < Pr < 15, r=

Pr1/2

for laminar flow (Rex < 5 × 105 )

Pr1/3

for turbulent flow (Rex > 5 × 105 )

Here, all of the fluid properties are evaluated at the reference temperature Tref = T∞ + 0.5(Ts − T∞ )+ 0.22(Tαs − T∞ ). Expressions for the local heat transfer coefficients at a given distance x from the leading edge are given as2  1/3 0.332 Re0.5  x Pr   1/3 Nux = 0.0292 Re0.8 x Pr    0.185 Rex (log Rex )−2.584

for Rex < 5 × 105 for 5 × 105 < Rex < 107 for 107 < Rex < 109

In the case of gaseous fluids flowing at very high freestream velocities, dissociation of the gas may occur, and will cause large variations in the properties within the boundary layer. For these cases, the heat transfer coefficient must be defined in terms of the enthalpy difference, that is, q = hA(is − iαs ), and the recovery factor will be given by r = (is − iαs )/(i0 − i∞ ), where iαs represents the enthalpy at the adiabatic wall conditions. Similar expressions to those shown above for Nux can be used by substituting the properties evaluated at a reference enthalpy defined as iref = i∞ + 0.5(is − i∞ ) + 0.22(iαs − i∞ ). High-Speed Gas Flow Past Cones For the case of high-speed gaseous flows over conical-shaped objects the following expressions can be used:

Nux =

1/3 0.575 Re0.5 x Pr

for Rex < 105

1/3 0.0292 Re0.8 for Rex > 105 x Pr

where the fluid properties are evaluated at Tref as in the plate.12

841

Stagnation Point Heating for Gases When the conditions are such that the flow can be assumed to behave as incompressible, the Reynolds number is based on the free-stream velocity and h is defined as q = hA(Ts − T∞ ).13 Estimations of the Nusselt can be made using the following relationship: 0.4 NuD = C Re0.5 D Pr

where C = 1.14 for cylinders and 1.32 for spheres, and the fluid properties are evaluated at the mean film temperature. When the flow becomes supersonic, a bow shock wave will occur just off the front of the body. In this situation, the fluid properties must be evaluated at the stagnation state occurring behind the bow shock and the Nusselt number can be written as 0.4 NuD = C Re0.5 D Pr

ρ∞ ρ0

0.25

where C = 0.95 for cylinders and 1.28 for spheres; ρ∞ is the free-stream gas density and ρ0 is the stagnation density of stream behind the bow shock. The heat transfer rate for this case, is given by q = hA (Ts − T0 ). 2.3

Free Convection

In free convection the fluid motion is caused by the buoyant force resulting from the density difference near the body surface, which is at a temperature different from that of the free fluid far removed from the surface where velocity is zero. In all free convection correlations, except for the enclosed cavities, the fluid properties are usually evaluated at the mean film temperature Tf = (T1 + T∞ )/2. The thermal expansion coefficient β, however, is evaluated at the free fluid temperature T∞ . The convection heat transfer coefficient h is based on the temperature difference between the surface and the free fluid. Free Convection from Flat Plates and Cylinders For free convection from flat plates and cylinders, the average Nusselt number NuL can be expressed as4

NuL = C(GrL Pr)m where the constants C and m are given as shown in Table 15. The Grashof Prandtl number product, (GrL Pr) is called the Rayleigh number (RaL ) and for certain ranges of this value, Figs. 13 and 14 are used instead of the above equation. Reasonable approximations for other types of three-dimensional shapes, such as short cylinders and blocks, can be made for 104 < RaL < 109 , by using this expression and C = 0.6, m = 1/4, provided that the characteristic length, L, is determined from 1/L = 1/Lhor + 1/Lver , where

842 Table 15

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Constants for Free Convection from Flat Plates and Cylinders

Geometry Vertical flat plates and cylinders

Horizontal cylinders

Upper surface of heated plates or lower surface of cooled plates Lower surface of heated plates or upper surface of cooled plates

C

m

10−1 –104

GrK Pr

Use Fig. 12

Use Fig. 12

104 –109 109 –1013 0–10−5 10−5 –104 104 –109 109 –1013 2 × 104 –8 × 106

0.59 0.10 0.4 Use Fig. 13 0.53 0.13 0.54

1/ 4 1/ 3

0 Use Fig. 13 1/ 4 1/ 3 1/ 4

8 × 106 –1011 105 –1011

0.15 0.58

1/ 3 1/ 5

Fig. 13 Free-convection heat transfer correlation for heated vertical plates and cylinders. (Adapted from Ref. 14. Used with permission of McGraw-Hill Book Company.)

Lver is the height and Lhor is the horizontal dimension of the object in question. For unsymmetrical horizontal square, rectangular, or circular surfaces, the characteristic length L can be calculated from the expression L = A/P , where A is the area and P is the wetted perimeter of the surface. Free Convection from Spheres For free convection from spheres, the following correlation has been developed:

NuD = 2 + 0.43(GrD Pr)0.25

for 1 < GrD < 105

L Height of plates and cylinders; 1/4 restricted to D/L ≥ 35/GrL for cylinders

Diameter D

Length of a side for square plates, the average length of the two sides for rectangular plates 0.9D for circular disks

Fig. 14 Free-convection heat transfer correlation from heated horizontal cylinders. (Adapted from Ref. 14. Used with permission of McGraw-Hill Book Company.)

Although this expression was designed primarily for gases, Pr ≈ 1, it may be used to approximate the values for liquids as well.15 Free Convection in Enclosed Spaces Heat transfer in an enclosure occurs in a number of different situations and with a variety of configurations. Then a temperature difference is imposed on two opposing walls that enclose a space filled with a fluid, convective heat transfer will occur. For small values of the Rayleigh number, the heat transfer may be dominated by conduction, but as the Rayleigh number increases, the contribution made by free convection will increase. Following are a number of correlations, each designed


for a specific geometry. For all of these, the fluid properties are evaluated at the average temperature of the two walls. Cavities between Two Horizontal Walls at Temperatures T1 and T2 Separated by Distance δ (T1 for Lower Wall, T1 > T2 )

Nuδ =

q = h(T1 − T2 ) 0.069

1/3 Raδ

1.0

Pr0.074

843

for 2 < H /δ < 10, Pr < 105 Raδ < 1010 ;

Pr Raδ Nuδ = 0.18 0.2 + Pr

0.29

for 1 < H /δ < 2, 103 < Pr < 105 , and 103 < Raδ Pr/ (0.2 + Pr); and Nuδ = 0.42 Ra0.25 Pr0.012 (δ/H )0.3 δ

for 3 × 105 < Raδ < 7 × 109 for Raδ < 1700

where Raδ = gβ (T1 − T2 ) δ 3 /αv; δ is the thickness of the space.16 Cavities between Two Vertical Walls of Height H at Temperature by Distance T and T Separated by Distance δ 17,18

q = h(T1 − T2 ) 0.28 0.25 δ Pr Raδ Nuδ = 0.22 0.2 + Pr H

Fig. 15

for 10 < H /δ < 40, 1 < Pr < 2 × 104 , and 104 < Raδ < 107 . 2.4

Log-Mean Temperature Difference

The simplest and most common type of heat exchanger is the double-pipe heat exchanger illustrated in Fig. 15. For this type of heat exchanger, the heat transfer between the two fluids can be found by assuming a constant overall heat transfer coefficient found from Table 8 and a constant fluid specific heat. For this type, the heat transfer is given by q = U A Tm

Temperature profiles for parallel flow and counterflow in double-pipe heat exchanger.

844


Fig. 16 Correction factor for shell-and-tube heat exchanger with one shell and any multiple of two tube passes (two, four, etc., tube passes.)

heat transfer, q, can be determined by

where Tm =

T2 − T1 ln(T2 /T1 )

q = U AF Tm

In this expression, the temperature difference, Tm , is referred to as the log-mean temperature difference (LMTD); T1 represents the temperature difference between the two fluids at one end and T2 at the other end. For the case where the ratio T2 /T1 is less than two, the arithmetic mean temperature difference, (T2 + T1 )/2, may be used to calculate heat transfer rate without introducing any significant error. As shown in Fig. 15, T1 = Th,i − Tc,i

T2 = Th,o − Tc,o

for parallel flow

T1 = Th,i − Tc,o

T2 = Th,o − Tc,i

for counterflow

Cross-Flow Coefficient In other types of heat exchangers, where the values of the overall heat transfer coefficient, U , may vary over the area of the surface, the LMTD may not be representative of the actual average temperature difference. In these cases, it is necessary to utilize a correction factor such that the

Here the value of Tm is computed assuming counterflow conditions, that is, T1 = Th,i − Tc,i and T2 = Th,o − Tc,o . Figures 16 and 17 illustrate some examples of the correction factor F for various multiple-pass heat exchangers. 3 RADIATION HEAT TRANSFER Heat transfer can occur in the absence of a participating medium through the transmission of energy by electromagnetic waves, characterized by a wavelength, λ, and frequency, v, which are related by c = λv. The parameter c represents the velocity of light, which in a vacuum is co = 2.9979 × 108 m/s. Energy transmitted in this fashion is referred to as radiant energy and the heat transfer process that occurs is called radiation heat transfer or simply radiation. In this mode of heat transfer, the energy is transferred through electromagnetic waves or through photons, with the energy of a photon being given by hv, where h represents Planck’s constant. In nature, every substance has a characteristic wave velocity that is smaller than that occurring in a vacuum.


845

Fig. 17 Correction factor for shell-and-tube heat exchanger with two shell passes and any multiple of four tubes passes (four, eight, etc., tube passes.)

Fig. 18 Electromagnetic radiation spectrum.

These velocities can be related to co by c = co /n, where n indicates the refractive index. The value of the refractive index, n, for air is approximately equal to 1. The wavelength of the energy given or for the radiation that comes from a surface depends

on the nature of the source and various wavelengths sensed in different ways. For example, as shown in Fig. 18, the electromagnetic spectrum consists of a number of different types of radiation. Radiation in the visible spectrum occurs in the range λ = 0.4–0.74 µm,

846


while radiation in the wavelength range 0.1–100 µm is classified as thermal radiation and is sensed as heat. For radiant energy in this range, the amount of energy given off is governed by the temperature of the emitting body. 3.1 Blackbody Radiation

All objects in space are continuously being bombarded by radiant energy of one form or another and all of this energy is either absorbed, reflected, or transmitted. An ideal body that absorbs all the radiant energy falling upon it, regardless of the wavelength and direction, is referred to as a blackbody. Such a body emits maximum energy for a prescribed temperature and wavelength. Radiation from a blackbody is independent of direction and is referred to as a diffuse emitter. Stefan–Boltzmann Law The Stefan–Boltzmann law describes the rate at which energy is radiated from a blackbody and states that this radiation is proportional to the fourth power of the absolute temperature of the body,

eb = σ T 4 where eb is the total emissive power and σ is the Stefan–Boltzmann constant, which has the value 5.729 × 10−8 W/m2 ·K4 (0.173 × 10−8 Btu/h·ft2 ·◦ R4 ). Planck’s Distribution Law The temperature amount of energy leaving a blackbody is described as the spectral emissive power, eλb , and is a function of wavelength. This function, which was derived from quantum theory by Planck, is

eλb =

2πC1 λ5 [exp(C2 /λT ) − 1]

where eλb has a unit W/m ·µm (Btu/h·ft ·µm). Values of the constants C1 and C2 are 0.59544 × 10−16 W·m2 (0.18892 × 108 Btu·µm4 /h·ft2 ) and 14, 388 µm·K (25, 898 µm ·◦ R), respectively. The distribution of the spectral emissive power from a blackbody at various temperatures is shown in Fig. 19, which shows that the energy emitted at all wavelengths increases as the temperature increases. The maximum or peak values of the constant temperature curves illustrated in Fig. 20 shift to the left for shorter wavelengths as the temperatures increase. The fraction of the emissive power of a blackbody at a given temperature and in the wavelength interval between λ1 and λ2 can be described by 2

Fλ1 T −λ2 T =

1 σT 4

λ1

2

eλb dλ −

0

= Fo−λ1 T − Fo−λ2 T

0

λ2

eλb dλ

λ where the function Fo−λT = (1/σ T 4 ) o eλb dλ is given in Table 16. This function is useful for the evaluation of total properties involving integration on the wavelength in which the spectral properties are piecewise constant. Wien’s Displacement Law The relationship between these peak or maximum temperatures can be described by Wien’s displacement law,

λmax T = 2897.8 µm·K or ◦

λmax T = 5216.0 µm· R 3.2 Radiation Properties While to some degree all surfaces follow the general trends described by the Stefan–Boltzmann and Planck laws, the behavior of real surfaces deviates somewhat from these. In fact, because blackbodies are ideal, all real surfaces emit and absorb less radiant energy than a blackbody. The amount of energy a body emits can be described in terms of the emissivity and is, in general, a function of the type of material, the temperature, and the surface conditions, such as roughness, oxide layer thickness, and chemical contamination. The emissivity is, in fact, a measure of how well a real body radiates energy as compared with a blackbody of the same temperature. The radiant energy emitted into the entire hemispherical space above a real surface element, including all wavelengths is given by e = εσ T 4 , where ε is less than 1.0 and is called the hemispherical emissivity (or total hemispherical emissivity to indicate averaging over the total wavelength spectrum). For a given wavelength the spectral hemispherical emissivity ελ of a real surface is defined as

ελ =

eλ eλb

where eλ is the hemispherical emissive power of the real surface and eλb is that of a blackbody at the same temperature. Spectral irradiation, Gλ , (W/m2 ·µm), is defined as the rate at which radiation is incident upon a surface per unit area of the surface, per unit wavelength about the wavelength λ, and encompasses the incident radiation from all directions. Spectral hemispherical reflectivity, ρλ , is defined as the radiant energy reflected per unit time, per unit area of the surface, per unit wavelength per Gλ . Spectral hemispherical absorptivity, αλ , is defined as the radiant energy absorbed per unit area of the surface per unit wavelength about the wavelength per Gλ . Spectral hemispherical transmissivity is defined as the radiant energy transmitted per unit area of the surface, per unit wavelength about the wavelength per Gλ .


847

Fig. 19 Hemispherical spectral emissive power of a blackbody for various temperatures.

For any surface, the sum of the reflectivity, absorptivity, and transmissivity must equal unit, that is,

∞

ρ λ Gλ 0

dλ G

∞

τ λ Gλ 0

When these values are averaged over the entire wavelength from λ = 0 to ∞, they are referred to as total values. Hence, the total hemispherical reflectivity, total hemispherical absorptivity, and total hemispherical transmissivity can be written as

τ=

αλ − ρλ τ λ = 1

ρ=

and

α=

∞

α λ Gλ 0

dλ G

dλ G

respectively, where G=

∞

Gλ dλ 0

As was the case for the wavelength-dependent parameters, the sum of the total reflectivity, total absorptivity,

848


Fig. 20 Table 16

Configuration factor for radiation exchange between surfaces of area dAi and dAj .

Radiation Function Fo−λT

λT µm·K 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300

λT µm· ◦ R 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700 2880 3060 3240 3420 3600 3780 3960 4140 4320 4500 4680 4860 5040 5220 5400 5580 5760 5940

Fo−λT 10−11

0.1864 × 0.1298 × 10−8 0.9290 × 10−7 0.1838 × 10−5 0.1643 × 10−4 0.8701 × 10−4 0.3207 × 10−3 0.9111 × 10−3 0.2134 × 10−2 0.4316 × 10−2 0.7789 × 10−2 0.1285 × 10−1 0.1972 × 10−1 0.2853 × 10−1 0.3934 × 10−1 0.5210 × 10−1 0.6673 × 10−1 0.8305 × 10−1 0.1009 0.1200 0.1402 0.1613 0.1831 0.2053 0.2279 0.2505 0.2732 0.2958 0.3181 0.3401

λT

µm·K

µm· ◦ R

Fo−λT

µm·K

µm· ◦ R

Fo−λT

3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300

6120 6300 6480 6660 6840 7020 7200 7380 7560 7740 7920 8100 8280 8460 8640 8820 9000 9180 9360 9540 9720 9900 10,080 10,260 10,440 10,620 10,800 10,980 11,160 11,340

0.3617 0.3829 0.4036 0.4238 0.4434 0.4624 0.4809 0.4987 0.5160 0.5327 0.5488 0.5643 0.5793 0.5937 0.6075 0.6209 0.6337 0.6461 0.6579 0.6694 0.6803 0.6909 0.7010 0.7108 0.7201 0.7291 0.7378 0.7461 0.7541 0.7618

6400 6500 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 10,000 11,000 12,000 13,000 14,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000 60,000

11,520 11,700 11,880 12,240 12,600 12,960 13,320 13,680 14,040 14,400 14,760 15,120 15,480 15,840 16,200 18,000 19,800 21,600 23,400 25,200 27,000 36,000 45,000 54,000 63,000 72,000 81,000 90,000 99,000 108,000

0.7692 0.7763 0.7832 0.7961 0.8081 0.8192 0.8295 0.8391 0.8480 0.8562 0.8640 0.8712 0.8779 0.8841 0.8900 0.9142 0.9318 0.9451 0.9551 0.9628 0.9689 0.9856 0.9922 0.9953 0.9970 0.9979 0.9985 0.9989 0.9992 0.9994


and total transmissivity must be equal to unity, that is, α+ρ+τ =1 It is important to note that while the emissivity is a function of the material, temperature, and surface conditions, the absorptivity and reflectivity depend on both the surface characteristics and the nature of the incident radiation. The terms reflectance, absorptance, and transmittance are used by some authors for the real surfaces and the terms reflectivity, absorptivity, and transmissivity are reserved for the properties of the ideal surfaces (i.e., those optically smooth and pure substances perfectly uncontaminated). Surfaces that allow no radiation to pass through are referred to as opaque, that is, τλ = 0, and all of the incident energy will be either reflected or absorbed. For such a surface, αλ + ρλ = 1

α+ρ =1

Light rays reflected from a surface can be reflected in such a manner that the incident and reflected rays are symmetric with respect to the surface normal at the point of incidence. This type of radiation is referred to as specular. The radiation is referred to as diffuse if the intensity of the reflected radiation is uniform over all angles of reflection and is independent of the incident direction, and the surface is called a diffuse surface if the radiation properties are independent of the direction. If they are independent of the wavelength, the surface is called a gray surface, and a diffuse-gray surface absorbs a fixed fraction of incident radiation from any direction and at any wavelength, and αλ = ελ = α = ε. Kirchhoff’s Law of Radiation The directional characteristics can be specified by the addition of a prime to the value; for example, the spectral emissivity for radiation in a particular direction would be denoted by αλ . For radiation in a particular direction, the spectral emissivity is equal to the directional spectral absorptivity for the surface irradiated by a blackbody at the same temperature. The most general form of this expression states that αλ = ελ . If the incident radiation is independent of angle or if the surface is diffuse, then αλ = ελ for the hemispherical properties. This relationship can have various conditions imposed on it, depending on whether spectral, total, directional, or hemispherical quantities are being considered.19 Emissivity of Metallic Surfaces The properties of pure smooth metallic surfaces are often characterized by low emissivity and absorptivity values and high values of reflectivity. The spectral emissivity of metals tends to increase with decreasing wavelength, and exhibits a peak near the visible region. At wavelengths λ > ∼5 µm the spectral emissivity increases with increasing temperature, but this trend reverses at

849

shorter wavelengths (λ < ∼1.27 µm). Surface roughness has a pronounced effect on both the hemispherical emissivity and absorptivity, and large optical roughnesses, defined as the mean square roughness of the surface divided by the wavelength, will increase the hemispherical emissivity. For cases where the optical roughness is small, the directional properties will approach the values obtained for smooth surfaces. The presence of impurities, such as oxides or other nonmetallic contaminants, will change the properties significantly and increase the emissivity of an otherwise pure metallic body. A summary of the normal total emissivities for metals are given in Table 17. It should be noted that the hemispherical emissivity for metals is typically 10–30% higher than the values normally encountered for normal emissivity. Emissivity of Nonmetallic Materials Large values of total hemispherical emissivity and absorptivity are typical for nonmetallic surfaces at moderate temperatures and, as shown in Table 18, which lists the normal total emissivity of some nonmetals, the temperature dependence is small. Absorptivity for Solar Incident Radiation The spectral distribution of solar radiation can be approximated by blackbody radiation at a temperature of approximately 5800 K (10, 000◦ R) and yields an average solar irradiation at the outer limit of the atmosphere of approximately 1353 W/m2 (429 Btu/ft2 ·h). This solar irradiation is called the solar constant and is greater than the solar irradiation received at the surface of the earth, due to the radiation scattering by air molecules, water vapor, and dust, and the absorption by O3 , H2 O, and CO2 in the atmosphere. The absorptivity of a substance depends not only on the surface properties but also on the sources of incident radiation. Since solar radiation is concentrated at a shorter wavelength, due to the high source temperature, the absorptivity for certain materials when exposed to solar radiation may be quite different from that which occurs for low-temperature radiation, where the radiation is concentrated in the longer wavelength range. A comparison of absorptivities for a number of different materials is given in Table 19 for both solar and low-temperature radiation. 3.3 Configuration Factor The magnitude of the radiant energy exchanged between any two given surfaces is a function of the emisssivity, absorptivity, and transmissivity. In addition, the energy exchange is a strong function of how one surface is viewed from the other. This aspect can be defined in terms of the configuration factor (sometimes called the radiation shape factor, view factor, angle factor, or interception factor ). As shown in Fig. 20, the configuration factor, Fi – j , is defined as that fraction of the radiation leaving a black surface, i, that is intercepted by a black or gray surface, j , and is based on the relative geometry, position, and shape

850


Normal Total Emissivity of Metalsa Surface Temperature (K)

Materials Aluminum Highly polished plate Polished plate Heavily oxidized Bismuth, bright Chromium, polished Copper Highly polished Slightly polished Black oxidized Gold, highly polished Iron Highly polished, electrolytic Polished Wrought iron, polished Cast iron, rough, strongly oxidized Lead Polished Rough unoxidized Mercury, unoxidized Molybdenum, polished Nickel Electrolytic Electroplated on iron, not polished Nickel oxide Platinum, electrolytic Silver, polished Steel Polished sheet Mild steel, polished Sheet with rough oxide layer Tin, polished sheet Tungsten, clean Zinc Polished Gray oxidized a Adapted

Normal Total Emissivity

480–870 373 370–810 350 310–1370

0.038–0.06 0.095 0.20–0.33 0.34 0.08–0.40

310 310 310 370–870

0.02 0.15 0.78 0.018–0.035

310–530 700–760 310–530 310–530

0.05–0.07 0.14–0.38 0.28 0.95

310–530 310 280–370 310–3030

0.06–0.08 0.43 0.09–0.12 0.05–0.29

310–530 293 920–1530 530–810 310–810

0.04–0.06 0.11 0.59–0.86 0.06–0.10 0.01–0.03

90–420 530–920 295 310 310–810

0.07–0.14 0.27–0.31 0.81 0.05 0.03–0.08

310–810 295

0.02–0.05 0.23–0.28

from Ref. 19.

of the two surfaces. The configuration factor can also be expressed in terms of the differential fraction of the energy or dFi – dj , which indicates the differential fraction of energy from a finite area Ai that is intercepted by an infinitesimal area dAj . Expressions for a number of different cases are given below for several common geometries: Infinitesimal area dAj to infinitesimal area dAj : dFdi – dj =

cos θi cos θj dAj πR 2

Infinitesimal area dAj to finite area Aj : Fdi – j = Aj

cos θi cos θj dAj πR 2

Finite area Ai to finite area Aj : Fi – j =

1 Ai

Aj

Aj

cos θi cos θj dAi dAj πR 2

Analytical expressions of other configuration factors have been found for a wide variety of simple geometries, and a number of these are presented in Figs. 21–24 for surfaces that emit and reflect diffusely. Reciprocity Relations The configuration factors can be combined and manipulated using algebraic rules referred to as configuration factor geometry. These expressions take several forms, one of which is the reciprocal properties between different configuration


851

Table 18

Normal Total Emissivity of Nonmetalsa

Materials

Surface Temperature (K)

Normal Total Emissivity

310

0.96

Asbestos, board Brick White refractory Rough red Carbon, lampsoot Concrete, rough Ice, smooth Magnesium oxide, refractory Paint Oil, all colors Lacquer, flat black Paper, white Plaster Porcelain, glazed Rubber, hard Sandstone Silicon carbide Snow Water, deep Wood, sawdust a Adapted

1370 310 310 310 273 420–760

0.29 0.93 0.95 0.94 0.966 0.69–0.55

373 310–370 310 310 295 293 310–530 420–920 270 273–373 310

0.92–0.96 0.96–0.98 0.95 0.91 0.92 0.92 0.83–0.90 0.83–0.96 0.82 0.96 0.75

from Ref. 19.

factors, which allow one configuration factor to be determined from knowledge of the others:

of an infinite number of complex shapes and geometries form a few select, known geometries. These are summarized in the following sections.

dAi dFdi – dj = dAj dFdj – di

Additive Property For a surface Ai subdivided into N parts (Ai1 , Ai2 , . . . , AiN ) and a surface Aj subdivided into M parts (Aj1 , Aj2 , . . . , AjM ),

dAi dFdi – j = Aj dFj – di Ai Fi – j = Aj Fj – i

Ai Fi – j =

These relationships can be combined with other basic rules to allow the determination of the configuration

N M

Ain Fin – jm

n=1 m=1

Table 19 Comparison of Absorptivities of Various Surfaces to Solar and Low-Temperature Thermal Radiationa Absorptivity For Solar Radiation

Surface Aluminum, highly polished Copper, highly polished Tarnished Cast iron Stainless steel, No. 301, polished White marble Asphalt Brick, red Gravel Flat black lacquer White paints, various types of pigments a Adapted

from Ref. 20 after J. P. Holman.27

0.15 0.18 0.65 0.94 0.37 0.46 0.90 0.75 0.29 0.96 0.12–0.16

For Low-Temperature Radiation (∼300 K) 0.04 0.03 0.75 0.21 0.60 0.95 0.90 0.93 0.85 0.95 0.90–0.95

852


Fig. 21 Configuration factors for some simple geometries.19


Fig. 22

Fig. 23

853

Configuration factor for coaxial parallel circular disks.

Configuration factor for aligned parallel rectangles.

854


Fig. 24 Configuration factor for rectangles with common edge.

Relation in an Enclosure When a surface is completely enclosed, the surface can be subdivided into N parts having areas A1 , A2 , . . . , AN , respectively, and N

Fi−j = 1

Blackbody Radiation Exchange For black surfaces Ai , and Aj at temperatures Ti and Tj , respectively, the net radiative exchange, qij , can be expressed as

qij = Ai Fi – j σ (Ti4 − Tj4 ) and for a surface completely enclosed and subdivided into N surfaces maintained at temperatures T1 , T2 , . . . , TN , the net radiative heat transfer, qi , to surface area Ai is N j =1

Ai Fi – j σ (Ti4 − Tj4 ) =

J = εσ T 4 + (1 − ε)G For an enclosure consisting of N surfaces, the irradiation on a given surface i can be expressed as

j =1

qi =

that leaves a surface per unit time and per unit area. For an opaque surface, this term is defined as

N

qij

j =1

3.4 Radiative Exchange among Diffuse Gray Surfaces in Enclosure One method for solving for the radiation exchange between a number of surfaces or bodies is through the use of the radiocity, J, defined as the total radiation

Gi =

N

Jj Fi−j

j =1

and the net radiative heat transfer rate at given surface i is εi Ai qi = Ai (Ji − Gi ) = (σ Ti4 − Ji ) 1 − εi For every surface in the enclosure, a uniform temperature or a constant heat transfer rate can be specified. If the surface temperature is given, the heat transfer rate can be determined for that surface and vice versa. Shown below are several specific cases that are commonly encountered. Case I. The temperatures of the surfaces, Ti (i = 1, 2, . . . , N), are known for each of the N surfaces and the values of the radiocity, Ji , are solved from the expression N {δij − (1 − εi )Fi−j }Ji = εi σ Ti4 j =1

1≤i≤N


855

The net heat transfer rate to surface i can then be determined from the fundamental relationship qi = Ai

εi (σ Ti4 − Ji ) 1 − εi

1≤i≤N

where δij = 0 for i = j and δij = 1 for i = j . Case II. The heat transfer rates, qi (i = 1, 2, . . . , N), to each of the N surfaces are known and the values of the radiocity, Ji , are determined from N qi {δij − Fi−j }Jj = Ai

1≤i≤N

j =1

The surface temperature can then be determined from Ti =

1 σ

1 − εi qi + Ji εi Ai

1/4 1≤i≤N

Case III. The temperatures, Ti (i = 1, . . . , N1 ), for Ni surfaces and heat transfer rates qi (i = N1 + 1, . . . , N) for (N − Ni ) surfaces are known and

the radiocities are determined by N {δij − (1 − εi )Fi−j }Jj = εi αTi4

1 ≤ i ≤ N1

j =1 N qi {δij − Fi−j }Jj = Ai

N1 + 1 ≤ i ≤ N

j =1

The net heat transfer rates and temperatures van be found as εi (σ Ti4 − Ji ) 1 − εi 1/4 1 1 − εi qi Ti = + Ji σ εi Ai qi = Ai

1 ≤ i ≤ N1 N1 + 1 ≤ i ≤ N

Two Diffuse Gray Surfaces Forming an Enclosure The net radiative exchange, q12 , for two diffuse gray surfaces forming an enclosure are shown in Table 20 for several simple geometries. Radiation Shields Often in practice, it is desirable to reduce the radiation heat transfer between two surfaces. This can be accomplished by placing a highly reflective surface between the two surfaces. For this configuration, the ratio of the net radiative exchange

Table 20 Net Radiative Exchange between Two Surfaces Forming an Enclosure Large (infinite) parallel planes

A1 = A2 = A

q12 =

Long (infinite) concentric cylinders

A1 r1 = A2 r2

q12 =

Concentric sphere

r2 A1 = 12 A2 r2

q12 =

small convex object in a large cavity

A1 ≈0 A2

Aσ (T14 − T24 ) 1 1 + −1 ε1 ε2

σ A1 (T14 − T24 ) 1 1 − ε2 r1 + ε1 ε2 r2

σ A1 (T14 − T24 ) 1 1 − ε2 r1 2 + ε1 ε2 r2

q12 = σ A1 ε1 (T14 − T24 )

856


with the shield to that without the shield can be expressed by the relationship 1 q12 with shield = q12 without shield 1+χ Values for this ratio, χ, for shields between parallel plates, concentric cylinders, and concentric spheres are summarized in Table 21. For the special case of parallel plates involving more than one or N shields, where all of the emissivities are equal, the value of χ equals N. Radiation Heat Transfer Coefficient The rate at which radiation heat transfer occurs can be expressed in a form similar to Fourier’s law or Newton’s law of cooling, by expressing it in terms of the temperature difference T1 − T2 , or as

q = hr A(T1 − T2 ) where hr is the radiation heat transfer coefficient or radiation film coefficient. For the case of radiation between two large parallel plates with emissivities, respectively, of ε1 and ε2 ,

hr

Table 21

σ (T14 − T24 ) T1 − T2 (1/ε1 + 1/ε2 − 1)

All of the previous expressions assumed that the medium present between the surfaces did not affect the radiation exchange. In reality, gases such as air, oxygen (O2 ), hydrogen (H2 ), and nitrogen (N2 ) have a symmetrical molecular structure and neither emit nor absorb radiation at low to moderate temperatures. Hence, for most engineering applications, such nonparticipating gases can be ignored. However, polyatomic gases such as water vapor (H2 O), carbon dioxide (CO2 ), carbon monoxide (CO), sulfur dioxide (SO2 ), and various hydrocarbons emit and absorb significant amounts of radiation. These participating gases absorb and emit radiation in limited spectral ranges, referred to as spectral bands. In calculating the emitted or absorbed radiation for a gas layer, its thickness, shape, surface area, pressure, and temperature distribution must be considered. Although a precise method for calculating the effect of these participating media is quite complex, an approximate method developed by Hottel21 will yield results that are reasonably accurate. The effective total emissivities of carbon dioxide and water vapor are a function of the temperature and the product of the partial pressure and the mean beam length of the substance as indicated in Figs. 25 and 26, respectively. The mean beam length, Le , is the characteristic length that corresponds to the radius of a hemisphere of gas, such that the energy flux radiated to the center of the base is equal to the average flux radiated to the area of interest by the actual gas volume. Table 22 lists the mean beam lengths of several simple shapes. For a geometry for which Le has not been determined, it is generally approximated

Values of X for Radiative Shields

Geometry 1 1 + −1 εs1 εs2 1 1 + −1 ε1 ε2

Shield

3.5 Thermal Radiation Properties of Gases

2

1 1 + −1 εs1 εs2 2 1 1 r1 + −1 ε1 ε2 r2 r1 r2

X Infinitely long parallel plates

n = 1 for infinitely long concentric cylinders n = 2 for concentric spheres


857

Fig. 25 Total emissivity of CO2 in a mixture having a total pressure of 1 atm. (From Ref. 21. Used with the permission of McGraw-Hill Book Company.)

by Le = 3.6V /A for an entire gas volume V radiating to its entire boundary surface A. The data in Figs. 25 and 26 were obtained for a total pressure of 1 atm and zero partial pressure of the water vapor. For other total and partial pressures the emissivities are corrected by multiplying CCO2 (Fig. 27) and CH2 O (Fig. 28), respectively, to εCO2 and εH2 O which are found from Figs. 25 and 26. These results can be applied when water vapor or carbon dioxide appear separately or in a mixture with other nonparticipating gases. For mixtures of CO2 and water vapor in a nonparticipating gas, the total emissivity of the mixture, εg , can be estimated from the expression εg = CCO2 εCO2 + CH2 O εH2 O − ε where ε is a correction factor given in Fig. 29. Radiative Exchange between Gas Volume and Black Enclosure of Uniform Temperature When radiative energy is exchanged between a gas volume and a black enclosure, the exchange per unit area, q , for a gas volume at uniform temperature, Tg , and a

uniform wall temperature, Tw , is given by q = εg (Tg )σ Tg4 − αg (Tw )σ Tw4 where εg (Tg ) is the gas emissivity at a temperature Tg and αg (Tw ) is the absorptivity of gas for the radiation from the black enclosure at Tw . As a result of the nature of the band structure of the gas, the absorptivity, αg , for black radiation at a temperature Tw is different from the emissivity, εg , at a gas temperature of Tg . When a mixture of carbon dioxide and water vapor is present, the empirical expression for αg is αg = αCO2 + αH2 O − α where αCO2 = CCO2 εCO 2

αH2 O = CH2 O εH 2 O

Tg Tw Tg Tw

0.65 0.45

where α = ε and all properties are evaluated at Tw .

858


Fig. 26 Total emissivity of H2 O at 1 atm total pressure and zero partial pressure (From Ref. 21. Used with the permission of McGraw-Hill Book Company.)

In this expression, the values of εCO and εH 2 O 2 can be found from Figs. 25 and 26 using an abscissa of Tw , but substituting the parameters pCO2 Le Tw /Tg and pH2 O Le Tw /Tg for pCO2 Le and pH2 O Le , respectively.

Radiative Exchange between a Gray Enclosure and a Gas Volume When the emissivity of the enclosure, εw , is larger than 0.8, the rate of heat transfer may be approximated by

qgray =

εw + 1 qblack 2

where qgray is the heat transfer rate for gray enclosure and qblack is that for black enclosure. For values of εw < 0.8, the band structures of the participating gas must be taken into account for heat transfer calculations.

4 BOILING AND CONDENSATION HEAT TRANSFER

Boiling and condensation are both forms of convection in which the fluid medium is undergoing a change of phase. When a liquid comes into contact with a solid surface maintained at a temperature above the saturation temperature of the liquid, the liquid may vaporize, resulting in boiling. This process is always accompanied by a change of phase, from the liquid to the vapor state, and results in large rates of heat transfer from the solid surface, due to the latent heat of vaporization of the liquid. The process of condensation is usually accomplished by allowing the vapor to come into contact with a surface at a temperature below the saturation temperature of the vapor, in which case the liquid undergoes a change in state from the vapor state to the liquid state, giving up the latent heat of vaporization. The heat transfer coefficients for condensation and boiling are generally larger than that for convection


859

Table 22 Mean Beam Lengtha Geometry of Gas Volume Hemisphere radiating to element at center of base Sphere radiating to its surface Circular cylinder of infinite height radiating to concave bounding surface Circular cylinder of semi-infinite height radiating to: Element at center of base Entire base Circular cylinder of height equal to diameter radiating to: Element at center of base Entire surface Circular cylinder of height equal to two diameters radiating to: Plane end Concave surface Entire surface Infinite slab of gas radiating to: Element on one face Both bounding planes Cube radiating to a face Gas volume surrounding an infinite tube bundle and radiating to a single tube: Equilateral triangular array: S = 2D S = 3D Square array: S = 2D a Adapted

Characteristic Length

Le

Radius R

R

Diameter D Diameter D

0.65D 0.95D


0.90D 0.65D


0.71D 0.60D

Diameter D Diameter D Diameter D

0.60D 0.76D 0.73D

Slab thickness D Slab thickness D Edge X

1.8D 1.8D 0.6X

Tube diameter D and spacing between tube centers, S

3.0(S − D) 3.8(S − D) 3.5(S − D)

from Ref. 19.

Fig. 27 Pressure correction for CO2 total emissivity for values of P other than 1 atm. (Adapted from Ref. 21. Used with the permission of McGraw-Hill Book Company.)

without phase change, sometimes by as much as several orders of magnitude. Application of boiling and condensation heat transfer may be seen in a closedloop power cycle or in a device referred to as a heat pipe, which will be discussed in the following section.

In power cycles, the liquid is vaporized in a boiler at high pressure and temperature. After producing work by means of expansion through a turbine, the vapor is condensed to the liquid state in a condenser and then returned to the boiler where the cycle is repeated.

860


Fig. 28 Pressure correction for water vapor total emissivity for values of pH2 O and P other than 0 and 1 atm. (Adapted from Ref. 21. Used with the permission of McGraw-Hill Book Company.)

Fig. 29 Correction on total emissivity for band overlap when both CO2 and water vapor are present: (a) gas temperature Tg = 400 K (720◦ R); (b) gas temperature Tg = 810 K (1460◦ R); (c) gas temperature Tg = 1200 K (2160◦ R). (Adapted from Ref. 21. Used with the permission of McGraw-Hill Book Company.)

4.1 Boiling The formation of vapor bubbles on a hot surface in contact with a quiescent liquid without external agitation is called pool boiling. This differs from forcedconvection boiling in which forced convection occurs simultaneously with boiling. When the temperature of the liquid is below the saturation temperature, the process is referred to as subcooled boiling. When the liquid temperature is maintained or exceeds the saturation temperature, the process is referred to as saturated or saturation boiling. Figure 30 depicts the surface heat flux, q , as a function of the excess temperature, Te = Ts − Tsat , for typical pool boiling of water using an electrically heated wire. In the region 0 < Te < Te,A bubbles occur only on selected spots of

the heating surface, and the heat transfer occurs primarily through free convection. This process is called free-convection boiling. When Te,A < Te < Te,C , the heated surface is densely populated with bubbles, and the bubble separation and eventual rise due to buoyancy induce a considerable stirring action in the fluid near the surface. This stirring action substantially increases the heat transfer from the solid surface. This process or region of the curve is referred to as nucleate boiling. When the excess temperature is raised to Te,C , the heat flux reaches a maximum value, and further increases in the temperature will result in a decrease in the heat flux. The point at which the heat flux is at a maximum value, is called the critical heat flux.


861

where the subscripts l and v denote saturated liquid and vapor, respectively. The surface tension of the liquid is σ (N/m). The quantity gc is the proportionality constant equal to 1 kg·m/N·s2 . The quantity g is the local gravitational acceleration in m/s2 . The values of C are given in Table 23. The above equation may be applied to different geometries, such as plates, wire, or cylinders. The critical heat flux (point C of Fig. 30) is given by28 ρv 0.5 π σggc (ρl − ρv ) 0.25 h f g ρv 1 + qc = 24 ρv2 ρl

Fig. 30 Typical boiling curve for a wire in a pool of water at atmospheric pressure.

For a water–steel combination, qc ≈ 1290 kW/m2 and Te,c ≈ 30◦ C. For water–chrome-plated copper, qc ≈ 940–1260 KW/m2 and Te,c ≈ 23–28◦ C.

Film boiling occurs in the region where Te > Te,D , and the entire heating surface is covered by a vapor film. In this region the heat transfer to the liquid is caused by conduction and radiation through the vapor. Between points C and D, the heat flux decreases with increasing Te . In this region, part of the surface is covered by bubbles and part by a film. The vaporization in this region is called transition boiling or partial film boiling. The point of maximum heat flux, point C, is called the burnout point or the Linden frost point. Although it is desirable to operate the vapor generators at heat fluxes close to qc , to permit the maximum use of the surface area, in most engineering applications it is necessary to control the heat flux and great care is taken to avoid reaching this point. The primary reason for this is that, as illustrated, when the heat flux is increased gradually, the temperature rises steadily until point C is reached. Any increase of heat flux beyond the value of qc , however, will dramatically change the surface temperature to Ts = Tsat + Te,E , typically exceeding the solid melting point and leading to failure of the material in which the liquid is held or from which the heater is fabricated.

Film Pool Boiling The heat transfer from the surface to the liquid is due to both convection and radiation. A total heat transfer coefficient is defined by the combination of convection and radiation heat transfer coefficients of the following form29 for the outside surfaces of horizontal tubes:

Nucleate Pool Boiling The heat flux data are best correlated by26

Nucleate Boiling in Forced Convection The total heat transfer rate can be obtained by simply superimposing the heat transfer due to nucleate boiling and forced convection:

q = µl hf g

g(ρl − ρv ) gc σ

1/2

cp,l Te Chf g Pr1.7 l

3

1/3 h4/3 = h4/3 c + hr h

where

k 3 ρv (ρl − ρv )g(hf g + 0.4cp,v Te ) hc = 0.62 v µv D Te

1/4

and hr =

r 5.73 × 10−8 ε(Ts4 − Tsat ) Ts − Tsat

The vapor properties are evaluated at the film temperature Tf = (Ts + Tsat )/2. The temperatures Ts and Tsat are in kelvins for the evaluation of hr . The emissivity of the metallic solids can be found from Table 17. Note that q = hA (Ts − Tsat ).

+ qforced q = qboiling

convection

Table 23 Values of the Constant C for Various Liquid–Surface Combinations Fluid-Heating Surface Combinations Water with polished copper, platinum, or mechanically polished stainless steel Water with brass or nickel Water with ground and polished stainless steel Water with Teflon-plated stainless steel

C 0.0130 0.006 0.008 0.008

862


For forced convection, it is recommended that the coefficient 0.023 be replaced by 0.014 in the Dittus–Boelter equation (Section 2.1). The above equation is generally applicable to forced convection where the bulk liquid temperature is subcooled (local forced convection boiling). Simplified Relations for Boiling in Water For nucleate boiling,30

h = C(Te )

n

p pa

0.4

where p and pa are, respectively, the system pressure and standard atmospheric pressure. The constants C and n are listed in Table 24. For local forced convection boiling inside vertical tubes, valid over a pressure range of 5–170 atm,31 h = 2.54(Te )3 ep/1.551 where h has the unit W/m2 · ◦ C, Te is in◦ C, and p is the pressure in 106 N/m3 . 4.2 Condensation

Depending on the surface conditions, the condensation may be a film condensation or a dropwise condensation. Film condensation usually occurs when a vapor, relatively free of impurities, is allowed to condense on a clean, uncontaminated surface. Dropwise condensation occurs on highly polished surfaces or on surfaces coated with substances that inhibit wetting. The condensate provides a resistance to heat transfer between the vapor and the surface. Therefore, it is desirable to use short vertical surfaces or horizontal cylinders to prevent the condensate from growing too thick. The heat transfer rate for dropwise condensation is usually an order of magnitude larger than that for film condensation under similar conditions. Silicones, Teflon, and certain fatty acids can be used to coat the surfaces to promote dropwise condensation. However, such coatings may lose their effectiveness owing to oxidation or outright removal. Thus, except under carefully controlled conditions, film condensation may be expected to occur in most instances, and the condenser design calculations are often based on the assumption of film condensation. Table 24 Surface Horizontal Vertical

For condensation on the surface at temperature Ts the total heat transfer rate to the surface is given by q = hL A (Tsat − Ts ), where Tsat is the saturation temperature of the vapor. The mass flow rate is determined by m ˙ = q/ hf g ; hf g is the latent heat of vaporization of the fluid (see Table 25 for saturated water). Correlations are based on the evaluation of liquid properties at Tf = (Ts + Tsat )/2, except hf g , which is to be taken at Tsat . Film Condensation on a Vertical Plate The Reynolds number for condensate flow is defined by Re = ρl Vm Dh /µl , where ρl and µl are the density and viscosity of the liquid, Vm is the average velocity of condensate, and Dh is the hydraulic diameter defined by Dh = 4 × condensate film cross-sectional area/wetted perimeter. For the condensation on a vertical plate Re = 4/µl , where is the mass flow rate of condensate per unit width evaluated at the lowest point on the condensing surface. The condensate flow is generally considered to be laminar for Re < 1800, and turbulent for Re > 1800. The average Nusselt number is given by22  0.25 gρl (ρl − ρv )hf g L3    1.13   µl kl (Tsat − Ts )  NuL = 1/3   3  0.0077 gρl (ρl − ρv )L  Re0.4  µ2l

for Re < 1800

for Re > 1800

Film Condensation on the Outside of Horizontal Tubes and Tube Banks

gρl (ρl − ρv )hf g D 3 NuD = 0.725 Nµl kl (Tsat − Ts )

0.25

where N is the number of horizontal tubes placed one above the other; N = 1 for a single tube.23 Film Condensation Inside Horizontal Tubes For low vapor velocities such that ReD based on the vapor velocities at the pipe inlet is less than 350024

NuD = 0.555

gρl (ρl − ρl )hf g D 3

0.25

µl kl (Tsat − Ts )

Values of C and n for Simplified Relations for Boiling in Water q (kW/m2 )

q < 16 16 < q < 240 q < 3 3 < q < 63

C 1042 5.56 5.7 7.96

n 1/3 3 1/7 3

863

Pressure, P (bar) 0.00611 0.03531 0.1053 0.2713 0.6209 1.2869 2.455 9.319 26.40 61.19 123.5 221.2

Temperature, T (K)

273.15 300 320 340 360 380 400 450 500 550 600 647.3

1.000 1.003 1.011 1.021 1.034 1.049 1.067 1.123 1.203 1.323 1.541 3.170

vf × 103 206.3 39.13 13.98 5.74 2.645 1.337 0.731 0.208 0.0766 0.0317 0.0137 0.0032

vu

Specific Volume (m3 / kg)

2502 2438 2390 2342 2291 2239 2183 2024 1825 1564 1176 0

Heat of Vaporization, hfg (KJ/kg)

Table 25 Thermophysical Properties of Saturated Water

4.217 4.179 4.180 4.188 4.203 4.226 4.256 4.40 4.66 5.24 7.00 ∞

Cp,l 1.854 1.872 1.895 1.930 1.983 2.057 2.158 2.56 3.27 4.64 8.75 ∞

Cp,u

Specific Heat (kJ/kg·K)

1750 855 577 420 324 260 217 152 118 97 81 45

µl × 106

Thermal Conductivity (W/m·K)

8.02 9.09 9.89 10.69 11.49 12.29 13.05 14.85 16.59 18.6 22.7 45

659 613 640 660 674 683 688 678 642 580 497 238

18.2 19.6 21.0 22.3 23.7 25.4 27.2 33.1 42.3 58.3 92.9 238

µv × 103 kl × 103 kv × 103

Viscosity (N·s/m2 )

12.99 5.83 3.77 2.66 2.02 1.61 1.34 0.99 0.86 0.87 1.14 ∞

Prl 0.815 0.857 0.894 0.925 0.960 0.999 1.033 1.14 1.28 1.47 2.15 ∞

Prv

Prandtl Number

75.5 71.7 68.3 64.9 61.4 57.6 63.6 42.9 31.6 19.7 8.4 0.0

−68.05 276.1 436.7 566.0 697.9 788 896

Surface Expansion Tension Coefficient, σ1 × 103 β1 × 106 (N/m) (K−1 )

864


where hf g + 3/8Cp,l (Tsat − Ts ). For higher flow rate,25 ReG > 5 × 104 , 1/3 NuD = 0.0265 Re0.8 G Pr

where the Reynolds number ReG = GD/µl is based on the equivalent mass velocity G = Gl + Gv (ρl / ρv )0.5 . The mass velocity for the liquid Gl and that for vapor Gv are calculated as if each occupied the entire flow area. Effect of Noncondensable Gases If noncondensable gas such as air is present in a vapor, even in a small amount, the heat transfer coefficient for condensation may be greatly reduced. It has been found that the presence of a few percent of air by volume in steam reduces the coefficient by 50% or more. Therefore, it is desirable in the condenser design to vent the noncondensable gases as much as possible. 4.3 Heat Pipes

Heat pipes are two-phase heat transfer devices that operate on a closed two-phase cycle32 and come in a wide variety of sizes and shapes.33,34 As shown in Fig. 31, they typically consist of three distinct regions: the evaporator or heat addition region, the condenser or heat rejection region, and the adiabatic or isothermal region. Heat added to the evaporator region of the container causes the working fluid in the evaporator wicking structure to be vaporized. The

Fig. 31

high temperature and corresponding high pressure in this region result in flow of the vapor to the other, cooler end of the container where the vapor condenses, giving up its latent heat of vaporization. The capillary forces existing in the wicking structure then pump the liquid back to the evaporator section. Other similar devices, referred to as two-phase thermosyphons have no wick, and utilize gravitational forces to provide the liquid return. Thus, the heat pipe functions as a nearly isothermal device, adjusting the evaporation rate to accommodate a wide range of power inputs, while maintaining a relatively constant source temperature. Transport Limitations The transport capacity of a heat pipe is limited by several important mechanisms. Among these are the capillary wicking limit, viscous limit, sonic limit, entrainment, and boiling limits. The capillary wicking limit and viscous limits deal with the pressure drops occurring in the liquid and vapor phases, respectively. The sonic limit results from the occurrence of choked flow in the vapor passage, while the entrainment limit is due to the high liquid vapor shear forces developed when the vapor passes in counterflow over the liquid saturated wick. The boiling limit is reached when the heat flux applied in the evaporator portion is high enough that nucleate boiling occurs in the evaporator wick, creating vapor bubbles that partially block the return of fluid. To function properly, the net capillary pressure difference between the condenser and the evaporator in a heat pipe must be greater than the pressure losses throughout the liquid and vapor flow paths. This

Typical heat pipe construction and operation.35


relationship can be expressed as

865

where Leff is the effective heat pipe length defined as Leff = 0.5Le + La + 0.5Lc

Pc ≥ P+ + P− + Pl + Pυ where Pc P+ P− Pl

= = = =

net capillary pressure difference normal hydrostatic pressure drop axial hydrostatic pressure drop viscous pressure drop occurring in the liquid phase Pv = viscous pressure drop occurring in the vapor phase

If these conditions are not met, the heat pipe is said to have reached the capillary limitation. Expressions for each of these terms have been developed for steady-state operation, and are summarized below. 2σ Capillary pressure: Pc,m = rc,e

and K is the liquid permeability as shown in Table 27. V apor pressure drop

C(fv Rev )µv Leff q 2(rh,v )2 Av ρv hf g

Although during steady-state operation the liquid flow regime is always laminar, the vapor flow may be either laminar or turbulent. It is therefore necessary to determine the vapor flow regime as a function of the heat flux. This can be accomplished by evaluating the local axial Reynolds and Mach numbers and substituting the values as shown below: Rev < 2300

Mav < 0.2

(fv Rev ) = 16 C = 1.00

Values for the effective capillary radius, rc , can be found theoretically for simple geometries or experimentally for pores or structures of more complex geometry. Table 26 gives values for some common wicking structures.

Rev < 2300 Mav > 0.2 (fv Rev ) = 16 1/2 γv − 1 C = 1+ Ma2v 2

Normal and axial hydrostatic pressure drop P+ + ρl gdυ cos ψ

Rev > 2300 Mav < 0.2 2(rh,v )q 3/4 (fv Rev ) = 0.038 Av µv hf g

P− = ρl gL sin ψ In a gravitational environment, the axial hydrostatic pressure term may either assist or hinder the capillary pumping process, depending on whether the tilt of the heat pipe promotes or hinders the flow of liquid back to the evaporator (i.e., the evaporator lies either below or above the condenser). In a zero-g environment, both this term and the normal hydrostatic pressure drop term can be neglected because of the absence of body forces. µl Leff q Liquid pressure drop Pl = KAw hf g ρl Table 26

Pv =

C = 1.00 Rev > 2300

Mav > 0.2 2(rh,v )q 3/4 (fv Rev ) = 0.038 Av µv hf g −1/2 γv − 1 Ma2v C = 1+ 2

Expressions for the Effective Capillary Radius for Several Wick Structures

Structure

rc

Data

Circular cylinder (artery or tunnel wick)

r

r = radius of liquid flow passage ω = groove width ω = groove width β = half-included angle ω = wire spacing d = wire diameter N = screen mesh number ω = wire spacing rs = sphere radius

Rectangular groove Triangular groove Parallel wires Wire screens

Packed spheres

ω ω/ cos β ω (ω + dω )/2 = 12 N

0.41rs

866


Wick Permeability for Several Wick Structures

Structure

K r 2 /8

Circular cylinder (artery or tunnel wick)

2ε(rh,l )2 /(fl Rel ) = ω/s

Open rectangular grooves

Circular annular wick Wrapped screen wick

2(rh,l )2 /(fl Rel ) 1/22 dω2 ε3 /(1 − ε)2

Packed sphere

1/37.5rs2 ε3 /(1 − ε)2

Since the equations used to evaluate both the Reynolds number and the Mach number are functions of the heat transport capacity, it is necessary to first assume the conditions of the vapor flow. Using these assumptions, the maximum heat capacity, qc,m , can be determined by substituting the values of the individual pressure drops into Eq. (1) and solving for qc,m . Once the value of qc,m is known, it can then be substituted into the expressions for the vapor Reynolds number and Mach number to determine the accuracy of the original assumption. Using this iterative approach, accurate values for the capillary limitation as a function of the operating temperature can be determined in units of W-m or watts for (qL)c,m and qc,m , respectively. The viscous limitation in heat pipes occurs when the viscous forces within the vapor region are dominant and limit the heat pipe operation: Pv < 0.1 Pv for determining when this limit might be of a concern. Due to the operating temperature range, this limitation will normally be of little consequence in the design of heat pipes for use in the thermal control of electronic components and devices. The sonic limitation in heat pipes is analogous to the sonic limitation in a converging–diverging nozzle and can be determined from qs,m = Av ρv hf g

γv Rv Tv 2(γv + 1)

1/2

where Tv is the mean vapor temperature within the heat pipe. Since the liquid and vapor flow in opposite directions in a heat pipe, at high enough vapor velocities, liquid droplets may be picked up or entrained in the vapor flow. This entrainment results in excess liquid

Data r = radius of liquid flow passage ε = wick porosity ω = groove width s = groove pitch δ = groove depth (rh,l ) = 2ωδ/(ω + 2δ) (rh,l ) = r1 − r2 dω = wire diameter ε = 1 − (1.05π Ndω/4) N = mesh number rs = sphere radius ε = porosity (dependent on packing mode)

accumulation in the condenser and, hence, dryout of the evaporator wick. Using the Weber number, We, defined as the ratio of the viscous shear force to the force resulting from the liquid surface tension, an expression for the entrainment limit can be found as qe,m = Av hf g

σρv 2(rh,w )

1/2

where (rh,w ) is the hydraulic radius of the wick structure, defined as twice the area of the wick pore at the wick–vapor interface divided by the wetted perimeter at the wick–vapor interface. The boiling limit occurs when the input heat flux is so high that nucleate boiling occurs in the wicking structure and bubbles may become trapped in the wick, blocking the liquid return and resulting in evaporator dryout. This phenomenon, referred to as the boiling limit, differs from the other limitations previously discussed in that it depends on the evaporator heat flux as opposed to the axial heat flux. This expression, which is a function of the fluid properties, can be written as 2σ 2πLeff keff Tv − Pc,m qb,m = hf g ρv ln(ri /rv ) rn where keff is the effective thermal conductivity of the liquid–wick combination, given in Table 28, ri is the inner radius of the heat pipe wall, and rn is the nucleation site radius. After the power level associated with each of the four limitations is established, determination of the maximum heat transport capacity is only a matter of selecting the lowest limitation for any given operating temperature. Heat Pipe Thermal Resistance The heat pipe thermal resistance can be found using an analogous electrothermal network. Figure 32 illustrates the

HEAT TRANSFER FUNDAMENTALS Table 28

867

Effective Thermal Conductivity for Liquid-Saturated Wick Structures

Wick Structures

keff

Wick and liquid in series

kl kw εkw + kl (1 − ε)

Wick and liquid in parallel

εkl + kw (1 − ε)

Wrapped screen

kl [(kl + kw ) − (1 − ε)(kl − kw )] (kl + kw ) + (1 − ε)(kl − kw )]

Packed spheres

kl [(2kl + kw ) − 2(1 − ε)(kl − kw )] (2kl + kw ) + (1 − ε)(kl − kw ) (wf kl kw δ) + wkl (0.185 wf kw + δkl ) (w + wf )(0.185 wf kf + δkl )

Rectangular grooves

Fig. 32 Equivalent thermal resistance of heat pipe.

electrothermal analog for the heat pipe illustrated in Fig. 31. As shown, the overall thermal resistance is composed of nine different resistances arranged in a series/parallel combination, which can be summarized as follows: Rpe Rwe Rie Rya Rpa Rwa Ric

Radial resistance of pipe wall at evaporator Resistance of liquid–wick combination at evaporator Resistance of liquid–vapor interface at evaporator Resistance of adiabatic vapor section Axial resistance of pipe wall Axial resistance of liquid–wick combination Resistance of liquid–vapor interface at condenser

Rwc Rpc

Resistance of liquid–wick combination at condenser Radial resistance of pipe wall at condenser

Because of the comparative magnitudes of the resistance of the vapor space and the axial resistances of the pipe wall and liquid–wick combinations, the axial resistance of both the pipe wall and the liquid–wick combination may be treated as open circuits and neglected. Also, because of the comparative resistances, the liquid–vapor interface resistances and the axial vapor resistance can, in most situations, be assumed to be negligible. This leaves only the pipe wall radial resistances and the liquid–wick resistances at both the evaporator and condenser. The radial resistances at the pipe wall can be computed from Fourier’s

868

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS 7.

law as Rpe =

δ kp Ae

for flat plates, where δ is the plate thickness and Ae is the evaporator area, or Rpe =

ln(Do /Di ) 2πLe kp

8. 9. 10.

11. 12.

for cylindrical pipes, where Le is the evaporator length. An expression for the equivalent thermal resistance of the liquid–wick combination in circular pipes is Rwe =

ln(Do /Di ) 2πLe keff

13. 14. 15.

where values for the effective conductivity, keff , can be found in Table 28. The adiabatic vapor resistance, although usually negligible, can be found as Rva =

Tv (Pv,e − Pv,c ) ρv h f g q

where Pv,e and Pv,c are the vapor pressures at the evaporator and condenser. Combining these individual resistances provides a mechanism by which the overall thermal resistance can be computed and hence the temperature drop associated with various axial heat fluxes can be computed.

16. 17. 18.

19. 20.

REFERENCES 1. 2. 3. 4. 4a. 4b.

4c. 5. 6.

Incropera, F. P., and Dewitt, D. P., Fundamentals of Heat Transfer, Wiley, New York, 1981. Eckert, E. R. G., and Drake, R. M., Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972. Heisler, M. P., “Temperature Charts for Induction and Constant Temperature Heating,” Trans. ASME, 69, 227 (1947). Grober, H., and Erk, S., Fundamentals of Heat Transfer, McGraw-Hill, New York, 1961. Duncan, A. B., and Peterson, G. P., “A Review of Microscale Heat Transfer,” invited review article, Appl. Mechan. Rev., 47(9), 397–428 (1994). Tien, C. L., Armaly, B. F., and Jagannathan, P. S., “Thermal Conductivity of Thin Metallic Films,” in Proc. 8th Conference on Thermal Conductivity, October 7–10, 1968. Bai, C., and Lavine, A. S., “Thermal Boundary Conditions for Hyperbolic Heat Conduction,” ASME HTD, 253, 37–44 (1993). Sieder, E. N., and Tate, C. E., “Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind. Eng. Chem., 28, 1429 (1936). Dittus, F. W., and Boelter, L. M. K., Univ. Calif., Berkeley, Engineering Publication 2, 443 (1930).

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Chapman, A. J., Heat Transfer, Macmillan, New York, 1974. Whitaker, S., “Forced Convection Heat Transfer Correlations,” AICHE J., 18, 361 (1972). Jakob, M., Heat Transfer, Vol. 1, Wiley, New York, 1949. Zhukauska, A., “Heat Transfer from Tubes in Cross Flow,” in Advances in Heat Transfer, Vol. 8, J. P. Hartnett and T. F. Irvine, Jr. (Eds.), Academic, New York, 1972. Kreith, F., Principles of Heat Transfer, Harper & Row, New York, 1973. Johnson, H. A., and Rubesin, M. W., “Aerodynamic Heating and Convective Heat Transfer,” Trans. ASME, 71, 447 (1949). Lin, C. C. (Ed.), Turbulent Flows and Heat Transfer, High Speed Aerodynamics and Jet Propulsion, Vol. V, Princeton University Press, Princeton, NJ, 1959. McAdams, W. H., Heat Transmission, McGraw-Hill, New York, 1954. Yuge, T., “Experiments on Heat Transfer from Spheres Including Combined Natural and Forced Convection,” J. Heat Transfer, 82, 214 (1960). Globe, S., and Dropkin, D., “Natural Convection Heat Transfer in Liquids Confined between Two Horizontal Plates,” J. Heat Transfer, 81C, 24 (1959). Catton, I., “Natural Convection in Enclosures,” in Proc. 6th International Heat Transfer Conference, Vol. 6, Toronto, Canada, 1978. MacGregor, R. K., and Emery, A. P., “Free Convection through Vertical Plane Layers: Moderate and High Prandtl Number Fluids,” J. Heat Transfer, 91, 391(1969). Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer, McGraw-Hill, New York, 1981. Gubareff, G. G., Janssen, J. E., and Torborg, R. H., Thermal Radiation Properties Survey, 2nd ed., Minneapolis Honeywell Regulator Co., Minneapolis, MN, 1960. Hottel, H. C., in Heat Transmission, W. C. McAdams (Ed.), McGraw-Hill, New York, 1954, Chapter 2. McAdams, W. H., Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954. Rohsenow, W. M., “Film Condensation” in Handbook of Heat Transfer, W. M. Rohsenow and J. P. Hartnett (Eds.), McGraw-Hill, New York, 1973. Chato, J. C., “Laminar Condensation inside Horizontal and Inclined Tubes,” ASHRAE J., 4, 52 (1962). Akers, W. W., Deans, H. A., and Crosser, O. K., “Condensing Heat Transfer within Horizontal Tubes,” Chem. Eng. Prog., Sym. Ser., 55(29), 171 (1958). Rohsenow, W. M., “A Method of Correlating Heat Transfer Data for Surface Boiling Liquids,” Trans. ASME, 74, 969 (1952). Holman, J. P., Heat Transfer, McGraw-Hill, New York, 1981. Zuber, N., “On the Stability of Boiling Heat Transfer,” Trans. ASME, 80, 711 (1958). Bromley, L. A., “Heat Transfer in Stable Film Boiling,” Chem. Eng. Prog., 46, 221 (1950). Jacob, M., and Hawkins, G. A., Elements of Heat Transfer, Wiley, New York, 1957.

HEAT TRANSFER FUNDAMENTALS 31.

Jacob, M., Heat Transfer, Vol. 2, Wiley, New York, 1957, p. 584. 32. Peterson, G. P., An Introduction to Heat Pipes: Modeling, Testing and Applications, Wiley, New York, 1994. 33. Peterson, G. P., Duncan, A. B., and Weichold, M. H., “Experimental Investigation of Micro Heat Pipes Fabricated in Silicon Wafers,” ASME J. Heat Transfer, 115(3), 751 (1993). 34. Peterson, G. P., “Capillary Priming Characteristics of a High Capacity Dual Passage Heat Pipe,” Chem. Eng. Commun., 27, 1, 119 (1984). 35. Peterson, G. P., and Fletcher, L. S., “Effective Thermal Conductivity of Sintered Heat Pipe Wicks,” AIAA J. Thermophys. Heat Transfer, 1(3), 36 (1987).

BIBLIOGRAPHY American Society of Heating, Refrigerating and Air Conditioning Engineering, ASHRAE Handbook of Fundamentals, 1972. Arpaci, V. S., Conduction Heat Transfer, Addison-Wesley, Reading, MA, 1966. Carslaw, H. S., and Jager, J. C., Conduction of Heat in Solid, Oxford University Press, London, 1959. Chi, S. W., Heat Pipe Theory and Practice, McGraw-Hill, New York, 1976. Duffie, J. A., and Beckman, W. A., Solar Engineering of Thermal Process, Wiley, New York, 1980. Dunn. P. D., and Reay, D. A., Heat Pipes, 3rd ed., Pergamon, New York, 1983. Gebhart, B., Heat Transfer, McGraw-Hill, New York, 1971.

869 Hottel, H. C., and Saroffin, A. F., Radiative Transfer, McGraw-Hill, New York, 1967. Kays, W. M., Convective Heat and Mass Transfer, McGrawHill, New York, 1966. Knudsen, J. G., and Katz, D. L., Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. Ozisik, M. N., Radiative Transfer and Interaction with Conduction and Convection, Wiley, New York, 1973. Ozisik, M. N., Heat Conduction, Wiley, New York, 1980. Peterson, G. P., An Introduction to Heat Pipes: Modeling, Testing and Applications, Wiley, New York, 1994. Planck, M., The Theory of Heat Radiation, Dover, New York, 1959. Rohsenow, W. M., and Choi, H. Y., Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, NJ, 1961. Rohsenow, W. M., and Hartnett, J. P., Handbook of Heat Transfer, McGraw-Hill, New York, 1973. Schlichting, H., Boundary-Layer Theory, McGraw-Hill, New York, 1979. Schneider, P. J., Conduction Heat Transfer, Addison-Wesley, Reading, MA, 1955. Sparrow, E. M., and Cess, R. D., Radiation Heat Transfer, Wadsworth, Belmont, CA, 1966. Tien, C. L., “Fluid Mechanics of Heat Pipes,” Annu. Rev. Fluid Mechan., 7, 167 (1975). Turner, W. C., and Malloy, J. F., Thermal Insulation Handbook, McGraw-Hill, New York, 1981. Vargafik, N. B., Table of Thermophysical Properties of Liquids and Gases, Hemisphere, Washington, DC, 1975. Wiebelt, J. A., Engineering Radiation Heat Transfer, Holt, Rinehart & Winston, New York, 1966.

CHAPTER 16 ELECTRIC CIRCUITS Albert J. Rosa Professor Emeritus University of Denver Denver, Colorado

1

INTRODUCTION

870

4.2

Energy Storage Devices

910

1.1

Overview

870

4.3

1.2

Fundamentals

873

Phasor Analysis of Alternating Current Circuits

916

Power in Sinusoidal Steady State

924

TRANSIENT RESPONSE OF CIRCUITS

928

4.4 2

DIRECT-CURRENT (DC) CIRCUITS

879

2.1

Node Voltage Analysis

879

2.2

Mesh Current Analysis

881

4

1

First-Order Circuits

928

5.2

Second-Order Circuits

932

Linearity Properties

884

2.4

Thevenin and Norton Equivalent Circuits

885

2.5

Maximum Signal Transfer

887

6.1

Transfer Functions and Input Impedance

935

889

6.2

Cascade Connection and Chain Rule

938

6.3

Frequency Response Descriptors

939

6.4

First-Order Frequency Response and Filter Design

941

Interface Circuit Design

LINEAR ACTIVE CIRCUITS

891

6

FREQUENCY RESPONSE

935

3.1

Dependent Sources

891

3.2

Operational Amplifier

895

6.5

Second-Order RLC Filters

946

AC CIRCUITS

905

6.6

Compound Filters

948

4.1

905

Signals

INTRODUCTION

1.1 Overview

The purpose of this chapter is to introduce the analysis and design of linear circuits. Circuits are important in electrical engineering because they process electrical signals that carry energy and information. For the present a circuit is defined as an interconnection of electrical devices and a signal as a time-varying electrical quantity. A modern technological society is intimately dependent on the generation, transfer, and conversion of electrical energy. Recording media like CDs, DVDs, thumb drives, hard drives, and tapebased products; communication systems like radar, cell phones, radio, television, and the Internet; information systems like computers and the world wide web; instrumentation and control systems; and the national electrical power grid X all involve circuits that process 870

5.1

2.3

2.6 3

5

REFERENCES

948

and transfer signals carrying either energy or information or both. This chapter will focus on linear circuits. An important feature of a linear circuit is that the amplitude of the output signal is proportional to the input signal amplitude. The proportionality property of linear circuits greatly simplifies the process of circuit analysis and design. Most circuits are only linear within a restricted range of signal levels. When driven outside this range, they become nonlinear and proportionality no longer applies. Hence only circuits operating within their linear range will be studied. An important aspect of this study involves interface circuits. An interface is defined as a pair of accessible terminals at which signals may be observed or specified. The interface concept is especially important with integrated circuit (IC) technology. Integrated circuits involve many thousands of interconnections, but only


ELECTRIC CIRCUITS Table 1

871

Some Important Quantities, Symbols, and Unit Abbreviations

Quantity Time Frequency Radian frequency Phase angle Energy Power Charge Current Electric field Voltage Impedance Admittance Resistance Conductance Reactance Susceptance Inductance, self Inductance, mutual Capacitance Magnetic flux Flux linkages Power ratio

Symbol

Unit

t f ω θ, ϕ w p q i E v Z Y R G X B L M C n λ log10 (p2 /p1 )

Second Hertz Radians per second Degree or radian Joule Watt Coulomb Ampere Volt per meter Volt Ohm Siemen Ohm Siemens Ohm Siemen Henry Henry Farad Weber Weber-turns Bel

a small number are accessible to the user. Creating systems using ICs involves interconnecting large circuits at a few accessible terminals in such a way that the circuits are compatible. Ensuring compatibility often involves relatively small circuits whose purpose is to change signal levels or formats. Such interface circuits are intentionally introduced to ensure that the appropriate signal conditions exist at the connections between two larger circuits. In terms of signal processing, analysis involves determining the output signals of a given circuit with known input signals. Analysis has the compelling feature that a unique solution exists in linear circuits. Circuit analysis will occupy the bulk of the study of linear circuits, since it provides the foundation for understanding the interaction of signals and circuits. Design involves devising circuits that perform a prescribed signal-processing function. In contrast to analysis, a design problem may have no solution or several solutions. The latter possibility leads to evaluation. Given several circuits that perform the same basic function, the alternative designs are rank ordered using factors such as cost, power consumption, and part counts. In reality the engineer’s role involves analysis, design, and evaluation, and the boundaries between these functions are often blurred. There are some worked examples to help the reader understand how to apply the concepts needed to master the concepts covered. These examples describe in detail the steps needed to obtain the final answer. They usually treat analysis problems, although design examples and application notes are included where appropriate.

Unit Abbreviation s Hz rad/s ◦ or rad J W C A V/m V S S S H H F Wb Wb-t B

Symbols and Units This chapter uses the International System (SI) of units. The SI units include six fundamental units: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), and candela (cd). All the other units can be derived from these six. Table 1 contains the quantities important to this chapter. Numerical values encountered in electrical engineering range over many orders of magnitude. Consequently, the system of standard decimal prefixes in Table 2 is used. These prefixes on the unit abbreviation of a quantity indicate the power of 10 that is applied to the numerical value of the quantity. Circuit Variables The underlying physical quantities in the study of electronic systems are two basic Table 2 Multiplier 1018 1015 1012 109 106 103 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18

Standard Decimal Prefixes Prefix

Abbreviation

Exa Peta Tera Giga Mega Kilo Deci Centi Milli Micro Nano Pico Femto Atto

E P T G M k d c m µ n p f a

872


variables, charge and energy. The concept of electrical charge explains the very strong electrical forces that occur in nature. To explain both attraction and repulsion, we say there are two kinds of charge—positive and negative. Like charges repel while unlike charges attract. The symbol q is used to represent charge. If the amount of charge is varying with time, we emphasize the fact by writing q(t). In SI charge is measured in coulombs (abbreviated C). The smallest quantity of charge in nature is an electron’s charge (qE = 1.6 × 10−19 C). There are 6.24 × 1018 electrons in 1 C. Electrical charge is a rather cumbersome variable to work with in practice. Moreover, in many situations the charges are moving, and so it is more convenient to measure the amount of charge passing a given point per unit time. To do this in differential form, a signal variable i called current is defined as follows: dq (1) i= dt Current is a measure of the flow of electrical charge. It is the time rate of change of charge passing a given point. The physical dimensions of current are coulombs per second. The unit of current is the ampere (abbreviated A). That is, 1 coulomb per second = 1 ampere In electrical engineering it is customary to define the direction of current as the direction of the net flow of positive charges, that is, the opposite of electron flow. A second signal variable called voltage is related to the change in energy that would be experienced by a charge as it passes through a circuit. The symbol w is commonly used to represent energy. Energy carries the units of joules (abbreviated J). If a small charge dq were to experience a change in energy dw in passing from point A to point B, then the voltage v between A and B is defined as the change in energy per unit charge. One can express this definition in differential form as dw (2) v= dq Voltage does not depend on the path followed by the charge dq in moving from point A to point B. Furthermore, there can be a voltage between two points even if there is no charge motion (i.e., no current), since voltage is a measure of how much energy dw would be involved if a charge dq were moved. The dimensions of voltage are joules per coulomb. The unit of voltage is the volt (abbreviated V). That is,

The dimensions of power are joules per second, which is called a watt (abbreviated W). In electrical situations, it is useful to have power expressed in terms of current and voltage. Using the chain rule, Eq. (3) and Eqs. (1) and (2) can be combined as p=

dw dq

dq dt

=v·i

(4)

This shows that the electrical power associated with a situation is determined by the product of voltage and current. Signal References The three signal variables (current, voltage, and power) are defined in terms of two basic variables (charge and energy). Charge and energy, like mass, length, and time, are basic concepts of physics that provide the scientific foundation for electrical engineering. However, engineering problems rarely involve charge and energy directly but are usually stated in terms of the signal variables because current and voltage are much easier to measure. A signal can be either a current or a voltage, but it is essential that the reader recognize that current and voltage, while interrelated, are quite different variables. Current is a measure of the time rate of charge passing a point. Since current indicates the direction of the flow of electrical charge, one thinks of current as a through variable. Voltage is best thought as an across variable because it inherently involves two points. Voltage is a measure of the net change in energy involved in moving a charge from one point to another. Voltage is measured not at a single point but rather between two points or across an element. Figure 1 shows the notation used for assigning reference directions to current and voltage. The reference mark for current [the arrow below i(t)] does not indicate the actual direction of the current. The actual direction may be reversing a million times per second. However, when the actual direction coincides with the reference direction, the current is positive. When the opposite occurs, the current is negative. If the net flow of positive charge in Fig. 1 is to the right, the current i(t) is positive. Conversely, if the current i(t) is

1 joule per coulomb = 1 volt A third signal variable, power, is defined as the time rate of change of energy: p=

dw dt

(3)

Fig. 1 Voltage and current reference marks for twoterminal device.1

ELECTRIC CIRCUITS

positive, then the net flow of positive charge is to the right. Similarly, the voltage reference marks (+ and B symbols) in Fig. 1 do not imply that the potential at the positive terminal is always higher than the potential at the B terminal. However, when this is true, the voltage across the device is positive. When the opposite is true, the voltage is negative. The importance of relating the reference directions (the plus and minus voltage signs and the current arrows) to the actual direction of the current (in and out) and voltage (high and low) can be used to determine the power associated with a device. That is, if the actual direction of the current is the same as the reference arrow drawn on the device, the current goes “in” and comes “out” of the device in the same direction as the reference arrow. Also, the voltage is “high” at the positive reference and “low” at the negative reference. If the actual and reference directions agree and i and v have the same sign, the power associated with this device is positive since the product of the current and voltage is positive. A positive sign for the associated power indicates that the device absorbs or consumes power. If the actual and reference direction disagrees for either voltage or current so i and v have opposite signs, p = i · v is negative and the device provides power. This definition of reference marks is called the passive-sign convention. Certain devices such as heaters (e.g., a toaster) can only absorb power. On the other hand, the power associated with a battery is positive when it is charging (absorbing power) and negative when it is discharging (delivering power). The passive-sign convention is used throughout electrical engineering. It is also the convention used by computer circuit simulation programs. Ground Voltage as an across variable is defined and measured between two points. It is convenient to identify one of the points as a reference point commonly called ground. This is similar to measuring elevation with respect to mean sea level. The heights of mountains, cities, and so on, are given relative to sea level. Similarly, the voltages at all other points in a circuit are defined with respect to ground. Circuit references are denoted using one of the “ground” symbols shown in Fig. 2. The voltage at the ground point is always taken to be 0 V .

873

1.2

Fundamentals A circuit is a collection of interconnected electrical devices that performs a useful function. An electrical device is a component that is treated as a distinct entity. Element Constraints A two-terminal device is described by its i–v characteristic, that is, the relationship between the voltage across and current through the device. In most cases the relationship is complicated and nonlinear so we use simpler linear models which adequately approximate the dominant features of a device. Resistor A resistor is a linear device described by a simple i–v characteristic as follows:

v = Ri

or

i = Gv

(5)

where R and G are positive constants related as G = 1/R. The power rating of the resistor determines the range over which the i–v characteristic can be represented by this linear relation. Equations (5) are collectively known as Ohm’s law. The parameter R is called resistance and has the unit ohms (). The parameter G is called conductance with the unit siemens (S). The power associated with the resistor can be found from p = v · i. Using Eqs. (5) to eliminate v or i from this relationship yields p = i 2 R = v2 G =

v2 R

(6)

Since the parameter R is positive, these equations state that the power is always nonnegative. Under the passive-sign convention this means the resistor always absorbs power. Example 1. A resistor functions as a linear element as long as the voltage and current are within the limits defined by its power rating. Determine the maximum current and voltage that can be applied to a 47-k resistor with a power rating of 0.25 W and remain within its linear operating range. Solution. Using Eq. (6) to relate power and current, we obtain PMAX 0.25 = = 2.31 mA IMAX = R 47 × 103

Similarly, using Eq. (6) to relate power and voltage, VMAX = Fig. 2 Ground symbol indicates a common voltage reference point.1

RPMAX =

47 × 103 × 0.25 = 108 V

A resistor with infinite resistance, that is, R = 4 , is called an open circuit. By Ohm’s law no current can flow through such a device. Similarly, a resistor with

874


Fig. 3 (a) Resistor symbol; (b) open circuit; (c) short circuit.

no resistance, that is, R = 0 , is called a short circuit. The voltage across a short circuit is always zero. In circuit analysis the devices in a circuit are assumed to be interconnected by zero-resistance wire, that is, by short circuits. Figure 3 shows the circuit symbols for a resistor and open and short circuits. Ideal Sources The signal and power sources required to operate electronic circuits are modeled using two elements: voltage sources and current sources. These sources can produce either constant or timevarying signals. The circuit symbols of an ideal voltage source and an ideal current source are shown in Fig. 4. The i–v characteristic of an ideal voltage source in Fig. 4 is described by the element equations

v = vS

and

i = any value

(7)

The element equations mean the ideal voltage source produces vS volts across its terminals and will supply whatever current may be required by the circuit to which it is connected. The i–v characteristic of an ideal current source in Fig. 4 is described by the element equations i = iS

and

v = any value

(8)

The ideal current source supplies iS amperes in the direction of its arrow symbol and will furnish whatever voltage is required by the circuit to which it is connected. In practice, circuit analysis involves selecting an appropriate model for the actual device. Figure 5 shows the practical models for the voltage and current sources. These models are called practical because they more accurately represent the properties of real-world

Fig. 4 (a) Voltage source; (b) battery (traditional symbol); (c) current source.

Fig. 5 Circuit symbols for practical independent sources: (a) practical voltage source; (b) practical current source.1

sources than do the ideal models. The resistances RS in the practical source models in Fig. 5 do not represent physical resistors but represent circuit elements used to account for resistive effects within the devices being modeled. Connection Constraints The previous section dealt with individual devices and models while this section considers the constraints introduced by interconnections of devices to form circuits. Kirchhoff’s laws are derived from conservation laws as applied to circuits and are called connection constraints because they are based only on the circuit connections and not on the specific devices in the circuit. The treatment of Kirchhoff’s laws uses the following definitions.

A circuit is any collection of devices connected at their terminals. A node is an electrical juncture of two or more devices. A loop is a closed path formed by tracing through a sequence of devices without passing through any node more than once. While it is customary to designate a juncture of two or more elements as a node, it is important to realize that a node is not confined to a point but includes all the wire from the point to each element. Kirchhoff’s Current Law Kirchhoff’s first law is based on the principle of conservation of charge. Kirchhoff’s current law (KCL) states that the algebraic sum of the currents entering a node is zero at every instant. In forming the algebraic sum of currents, one must take into account the current reference directions associated with the devices. If the current reference direction is into the node, a positive sign is assigned to the algebraic sum of the corresponding current. If the reference direction is away from the node, a negative sign is assigned. There are two signs associated with each current in the application of KCL. The first is the sign given to a current in writing a KCL connection equation. This sign is determined by the orientation of the current reference direction relative to a node. The second sign is determined by the actual direction of the current relative to the reference direction.

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The following general principle applies to writing KCL equations: In a circuit containing N nodes there are only N − 1 independent KCL connection equations. In general, to write these equations, we select one node as the reference or ground node and then write KCL equations at the remaining N − 1 nonreference nodes. Kirchhoff’s Voltage Law The second of Kirchhoff’s circuit laws is based on the principle of conservation of energy. Kirchhoff’s voltage law (KVL) states that the algebraic sum of all of the voltages around a loop is zero at every instant. There are two signs associated with each voltage. The first is the sign given the voltage when writing the KVL connection equation. The second is the sign determined by the actual polarity of a voltage relative to its assigned reference polarity. The following general principle applies to writing KVL equations: In a circuit containing E two-terminal elements and N nodes there are only E − N + 1 independent KVL connection equations. Voltage equations written around E − N + 1 different loops contain all of the independent connection constraints that can be derived from KVL. A sufficient condition for loops to be different is that each contains at least one element that is not contained in any other loop. Parallel and Series Connections Two types of connections occur so frequently in circuit analysis that they deserve special attention. Elements are said to be connected in parallel when they share two common nodes. In a parallel connection KVL forces equal voltages across the elements. The parallel connection is not restricted to two elements. Two elements are said to be connected in series when they have one common node to which no other current-drawing element is connected. A series connection results in equal current through each element. Any number of elements can be connected in series. Combined Constraints The usual goal of circuit analysis is to determine the currents or voltages at various places in a circuit. This analysis is based on constraints of two distinctly different types. The element constraints are based on the models of the specific devices connected in the circuit. The connection constraints are based on Kirchhoff’s laws and the circuit connections. The element equations are independent of the circuit in which the device is connected. Likewise, the connection equations are independent of the specific devices in the circuit. But taken together, the combined constraints from the element and connection equations provide the data needed to analyze a circuit. The study of combined constraints begins by considering the simple but important example in Fig. 6. This circuit is driven by the current source iS and the resulting responses are current/voltage pairs (iX , vX ) and (iO , vO ). The reference marks for the response pairs have been assigned using the passive-sign convention.

Fig. 6 Circuit used to demonstrate combined constraints.1

To solve for all four responses, four equations are required. The first two are the element equations: Current source: iX = iS Resistor: vO = R · iO

(9)

The first element equation states that the response current iX and the input driving force iS are equal in magnitude and direction. The second element equation is Ohm’s law relating vO and iO under the passive-sign convention. The connection equations are obtained by applying Kirchhoff’s laws. The circuit in Fig. 6 has two elements (E = 2) and two nodes (N = 2); hence for a total solution E − N + 1 = 1 KVL equation and N − 1 = 1 KCL equation are required. Selecting node B as the reference or ground node, a KCL at node A and a KVL around the loop yield KCL: − iX − iO = 0 KVL: − vX + vO = 0

(10)

With four equations and four unknowns all four responses can be found. Combining the KCL connection equation and the first element equations yields iO = −iX = −iS . Substituting this result into the second element equations (Ohm’s law) produces vO = −RiS . The minus sign in this equation does not mean vO is always negative. Nor does it mean the resistance is negative since resistance is always positive. It means that when the input driving force iS is positive, the response vO is negative, and vice versa. Example 2. Find all of the element currents and voltages in Fig. 7 for VO = 10 V, R1 = 2 k, and R2 = 3 k. Solution. Substituting the element constraints into the KVL connection constraint produces

−VO + R1 i1 + R2 i2 = 0 This equation can be used to solve for i1 since the second KCL connection equation requires that i2 = i1 . Hence i1 =

VO 10 = 2 mA = R1 + R2 2000 + 3000

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Equivalent Resistance Resistances connected in series simply add, while conductances connected in parallel also simply add. Since conductance is not normally used to describe a resistor, two resistors R1 and R2 connected in parallel result in the expression

R1 ||R2 = REQ = = Fig. 7 (From Ref. 1.)

By finding the current i1 , all currents can be found from −iA = i1 = i2 since all three elements are connected in series. Substituting all of the known values into the element equations gives vA = 10 V

v1 = R1 i1 = 4 V

v2 = R2 i2 = 6 V

Assigning Reference Marks In all previous examples and exercises the reference marks for the element currents (arrows) and voltages (+ and −) were given. When reference marks are not shown on a circuit diagram, they must be assigned by the person solving the problem. Beginners sometimes wonder how to assign reference marks when the actual voltage polarities and current directions are as yet unknown. It is important to remember that reference marks do not indicate the actual polarities and directions. They are benchmarks assigned in an arbitrary way at the beginning of the analysis. If it turns out the actual direction and reference direction agree, then the numerical value of the response will be positive. If they disagree, the numerical value will be negative. In other words, the sign of the answer together with arbitrarily assigned reference marks tells us the actual voltage polarity or current direction. When assigning reference marks in this chapter the passive-sign convention will always be used. By always following the passive-sign convention any confusion about the direction of power flow in a device will be avoided. In addition, Ohm’s law and other device i–v characteristics assume the voltage and current reference marks follow the passivesign convention. Always using this convention follows the practice used in all SPICE-based computer circuit analysis programs. Equivalent Circuits The analysis of a circuit can often be simplified by replacing part of the circuit with one which is equivalent but simpler. The underlying basis for two circuits to be equivalent is contained in their i–v relationships: Two circuits are said to be equivalent if they have identical i–v characteristics at a specified pair of terminals.

1 1 1 = = GEQ G1 + G2 1/R1 + 1/R2 R1 R2 R1 + R2

(11)

where the symbol || is shorthand for “in parallel.” The expression on the far right in Eq. (11) is called the product over the sum rule for two resistors in parallel. The product-over-the-sum rule only applies to two resistors connected in parallel. When more than two resistors are in parallel, the following must be used to obtain the equivalent resistance: REQ =

1 1 = GEQ 1/R1 + 1/R2 + 1/R3 + · · ·

Example 3.

(12)

Given the circuit in Fig. 8:

(a) Find the equivalent resistance REQ1 connected between terminals A and B. (b) Find the equivalent resistance REQ2 connected between terminals C and D. Solution. First resistors R2 and R3 are connected in parallel. Applying Eq. (11) results in

R2 ||R3 =

R2 R3 R2 + R3

(a) The equivalent resistance between terminals A and B equals R1 and the equivalent resistance

Fig. 8

(From Ref. 1.)

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R2 ||R3 connected in series. The total equivalent resistance REQ1 between terminals A and B thus is REQ1 = R1 + (R2 ||R3 ) REQ1 = R1 + REQ1 =

R2 R3 R2 + R3

R1 R2 + R1 R3 + R2 R3 R2 + R3

(b) Looking into terminals C and D yields a different result. In this case R1 is not involved, since there is an open circuit (an infinite resistance) between terminals A and B. Therefore only R2 ||R3 affects the resistance between terminals C and D. As a result REQ2 is simply REQ2

R2 R3 = R2 ||R3 = R2 + R3

This example shows that equivalent resistance depends upon the pair of terminals involved. Equivalent Sources The practical source models shown in Fig. 9 consist of an ideal voltage source in series with a resistance and an ideal current source in parallel with a resistance. If R1 = R2 = R and vS = iS R, the two practical sources have the same i–v relationship, making the two sources equivalent. When equivalency conditions are met, the rest of the circuit is unaffected regardless if driven by a practical voltage source or a practical current source. The source transformation equivalency means that either model will deliver the same voltage and current

Fig. 9

Equivalent practical source models.1

to the rest of the circuit. It does not mean the two models are identical in every way. For example, when the rest of the circuit is an open circuit, there is no current in the resistance of the practical voltage source and hence no i 2 R power loss. But the practical current source model has a power loss because the open-circuit voltage is produced by the source current in the parallel resistance. Y– Transformations The Y– connections shown in Fig. 10 occasionally occur in circuits and are especially prevalent in three-phase power circuits. One can transform from one configuration to the other by the following set of transformations:

RA =

R1 R2 +R2 R3 +R1 R3 R1

R1 =

RB RC RA +RB +RC

RB =

R1 R2 +R2 R3 +R1 R3 R2

R2 =

RA RC RA +RB +RC

R1 R2 +R2 R3 +R1 R3 R3

R3 =

RB RA RA +RB +RC (13) Solving Eqs. (13) for R1 , R2 , and R3 yields the equations for a -to-Y transformation while solving Eqs. (13) for RA , RB , and RC yields the equations for a Y-to- transformation. The Y and subcircuits are said to be balanced when R1 = R2 = R3 = RY and RA = RB = RC = R . Under balanced conditions the transformation equations reduce to RY = R /3 and R = 3RY . RC =

Voltage and Current Division These two analysis tools find wide application in circuit analysis and design. Voltage Division Voltage division allows us to solve for the voltage across each element in a series circuit. Figure 11 shows a circuit that lends itself to solution by voltage division. Applying KVL around the loop in Fig. 11 yields vS = v1 + v2 + v3 . Since all resistors are connected in series, the same current i exists in all three. Using Ohm’s law yields vS = R1 i + R2 i + R3 i. Solving for i yields i = vS /(R1 + R2 + R3 ). Once the current in the series circuit is

Fig. 10 Y– transformation.2

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Moving the movable wiper arm all the way to the top makes R1 zero, and voltage division yields vS . In other words, 100% of the applied voltage is delivered to the rest of the circuit. Moving the wiper to the other extreme delivers zero voltage. By adjusting the wiper arm position we can obtain an output voltage anywhere between zero and the applied voltage vS . Applications of a potentiometer include volume control, voltage balancing, and fine-tuning adjustment. Fig. 11

Voltage divider circuit.1

found, the voltage across each resistor is found using Ohm’s law: R1 vS v1 = R 1 i = R1 + R2 + R3 R2 vS v2 = R 2 i = (14) R1 + R2 + R3 R3 vS v3 = R 3 i = R1 + R2 + R3 In each case the element voltage is equal to its resistance divided by the equivalent series resistance in the circuit times the total voltage across the series circuit. Thus, the general expression of the voltage division rule is Rk vtotal (15) vk = REQ The operation of a potentiometer is based on the voltage division rule. The device is a three-terminal element which uses voltage (potential) division to meter out a fraction of the applied voltage. Figure 12 shows the circuit symbol of a potentiometer. Simply stated, a potentiometer is an adjustable voltage divider. The voltage vO in Fig. 12 can be adjusted by turning the shaft on the potentiometer to move the wiper arm contact. Using the voltage division rule, vO is found as Rtotal − R1 vO = vS (16) Rtotal

Fig. 12 tion.

Current Division Current division is the dual of voltage division. By duality current division allows for the solution of the current through each element in a parallel circuit. Figure 13 shows a parallel circuit that lends itself to solution by current division. Applying KCL at node A yields iS = i1 + i2 + i3 . The voltage v appears across all three conductances since they are connected in parallel. So using Ohm’s law we can write iS = vG1 + vG2 + vG3 and solve for v as v = iS /(G1 + G2 + G3 ). Given the voltage v, the current through any element is found using Ohm’s law as

i1 = vG1 = i2 = vG2 = i3 = vG3 =

G1 G1 + G2 + G3 G2 G1 + G2 + G3 G3 G1 + G2 + G3

iS iS iS

These results show that the source current divides among the parallel resistors in proportion to their conductances divided by the equivalent conductances in the parallel connection. Thus, the general expression for the current division rule is 1/Rk Gk itotal = itotal (17) ik = GEQ 1/REQ Circuit Reduction The concepts of series/parallel equivalence, voltage/current division, and source transformations can be used to analyze ladder circuits of the type shown in Fig. 14. The basic analysis strategy is to reduce the circuit to a simpler equivalent in which the desired voltage or current is easily found using voltage and/or current division and/or source transformation and/or Ohm’s law. There is no fixed pattern to

Potentiometer: (a) circuit symbol; (b) an applicaFig. 13

Current divider circuit.1

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equivalent resistance in Fig. 14c. The output voltage is found using Ohm’s law: vO =

v S

R

×

2R 5

=

2 vS 5

Several other analysis approaches are possible. 2

DIRECT-CURRENT (DC) CIRCUITS This section reviews basic DC analysis using traditional circuits theorems with application to circuit analysis and design. 2.1

Fig. 14

(From Ref. 1.)

the reduction process, and much depends on the insight of the analyst. When using circuit reduction it is important to remember that the unknown voltage exists between two nodes and the unknown current exists in a branch. The reduction process must not eliminate the required node pair or branch; otherwise the unknown voltage or current cannot be found. The next example will illustrate circuit reduction.

Node Voltage Analysis Using node voltage instead of element voltages as circuit variables can greatly reduce the number of equations that must be treated simultaneously. To define a set of node voltages, a reference node or ground is first selected. The node voltages are then defined as the voltages between the remaining nodes and the reference node. Figure 15 shows a reference node indicated by the ground symbol as well as the notation defining the three nonreference node voltages. The node voltages are identified by a voltage symbol adjacent to the nonreference nodes. This notation means that the positive reference mark for the node voltage is located at the node in question while the negative mark is at the reference node. Any circuit with N nodes involves N − 1 node voltages. The following is a fundamental property of node voltages: If the Kth two-terminal element is connected

Example 4. Find the output voltage vO and the input current iS in the ladder circuit shown in Fig. 14a. Solution. Breaking the circuit at points X and Y produces voltage source vS in series with a resistor R: Using source transformation this combination can be replace by an equivalent current source in parallel with the same resistor, as shown in Fig. 14b. Using current division the input current iS is

iS =

vS vS 3 vS R × = = , (2/3)R + R R (5/3)R 5R

The three parallel resistances in Fig. 14b can be combined into a single equivalent resistance without eliminating the node pair used to define the output voltage vF : REQ =

2R 1 = 1/R + 1/(2R) + 1/R 5

which yields the equivalent circuit in Fig. 14c. The current source vS /R determines the current through the

Fig. 15 Node voltage definition and notation.1

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between nodes X and Y, then the element voltage can be expressed in terms of the two node voltages as vK = vX − vY

(18)

where X is the node connected to the positive reference for element voltage vK . Equation (18) is a KVL constraint at the element level. If node Y is the reference node, then by definition vY = 0 and Eq. (18) reduces to vK = vX . On the other hand, if node X is the reference node, then vX = 0 and therefore vK = −vY . The minus sign here comes from the fact that the positive reference for the element is connected to the reference node. In any case, the important fact is that the voltage across any twoterminal element can be expressed as the difference of two node voltages, one of which may be zero. To formulate a circuit description using node voltages, device and connection analysis is used, except that the KVL connection equations are not explicitly written down. Instead the fundamental property of node analysis is used to express the element voltages in terms of the node voltages. The circuit in Fig. 16 will demonstrate the formulation of node voltage equations. The ground symbol identifies the reference node, six element currents (i0 , i1 , i2 , i3 , i4 , and i5 ), and three node voltages (vA , vB , and vC ). The KCL constraints at the three nonreference nodes are Node A:

− i0 + i1 + i2 = 0

Node B:

− i1 + i3 − i5 = 0

Node C:

− i2 + i5 + i4 = 0

Using the fundamental property of node analysis, device equations are used to relate the element currents to the node voltages: R1 : i1 =

R3 : i3 =

vA − vB vA − vC R2 : i2 = R1 R2 Voltage source: vS = vA

Substituting the element currents into the KCL equations yields Node A: −i0 + Node B: Node C:

Bridge circuit for node voltage example.1

vA − vB vA − vC + =0 R1 R2 vB vB − vA + − iS2 = 0 R1 R3 vC vC − vA + + iS2 = 0 R2 R4

But since the reference is connected to the negative side of the voltage source, vA = vS . Thus at node A the voltage is already known and this reduces the number of equations that need to be solved. The equation written above can be used to solve for the current through the voltage source if that is desired. Writing these equations in standard form with all of the unknown node voltages grouped on one side and the independent sources on the other yields Node A: Node B: Node C:

vA = vS 1 1 vS = iS + vB + R1 R3 R1 1 vS 1 = −iS + vC + R2 R4 R2

Using node voltage analysis there are only two equations and two unknowns (vB and vC ) to be solved. The coefficients in the equations on the left side depend only on circuit parameters, while the right side contains the known input driving forces. Supernodes When neither node of a voltage source can be selected as the reference, a supernode must be used. The fact that KCL applies to the currents penetrating a boundary can be used to write a node equation at the supernode. Then node equations at the remaining nonreference nodes are written in the usual way. This process reduces the number of available node equations to N − 3 plus one supernode equation, leaving us one equation short of the N − 1 required. The voltage source inside the supernode constrains the difference between the node voltages to be the value of the voltage source. The voltage source constraint provides the additional relationship needed to write N − 1 independent equations in the N − 1 node voltages. Example 5.

Fig. 16

vB − 0 vC − 0 R4 : i4 = R3 R4 Current source: i5 = iS2

For the circuit in Fig. 17:

(a) Formulate node voltage equations. (b) Solve for the output voltage vO using R1 = R4 = 2 k and R2 = R3 = 4 k.

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To find the output vO , we need to solve these equations for vC . The second equation yields vA = vC + vS1 , which when substituted into the first equation yields the required output: vO = vC = 13 vS2 − 12 vS1 Node voltage equations are very useful in the analysis of a variety of electronic circuits. These equations can always be formulated using KCL, the element constraints, and the fundamental property of node voltages. The following guidelines summarize this approach: Fig. 17

(From Ref. 1.)

Solution. (a) The voltage sources in Fig. 17 do not have a common node so a reference node that includes both sources cannot be selected. Choosing node D as the reference forces the condition vB = vS2 but leaves the other source vS1 ungrounded. The ungrounded source and all wires leading to it are encompassed by a supernode boundary as shown in the figure. Kirchhoff’s current law applies to the four-element currents that penetrate the supernode boundary and we can write i1 + i2 + i3 + i4 = 0

These currents can easily be expressed in terms of the node voltages: vA − vB vC − vB vC vA + + + =0 R1 R2 R3 R4 But since vB = vS2 , the standard form of this equation is vA

1 1 + R1 R2

+ vC

1 1 + R3 R4

= vS2

1 1 + R2 R3

We have one equation in the two unknown node voltages vA and vC . Applying the fundamental property of node voltages inside the supernode, we can write vA − vC = vS1 That is, the ungrounded voltage source constrains the difference between the two unknown node voltages inside the supernode and thereby supplies the relationship needed to obtain two equations in two unknowns. (b) Substituting in the given numerical values yields

7.5 × 10−4 vA + 7.5 × 10−4 vC = 5 × 10−4 vS2 vA − vC = vS1

1. Simplify the circuit by combining elements in series and parallel wherever possible. 2. If not specified, select a reference node so that as many dependent and independent voltage sources as possible are directly connected to the reference. 3. Label a node voltage adjacent to each nonreference node. 4. Create supernodes for dependent and independent voltage sources that are not directly connected to the reference node. 5. Node equations are required at supernodes and all other nonreference nodes except op amp outputs and nodes directly connected to the reference by a voltage source. 6. Write symmetrical node equations by treating dependent sources as independent sources and using the inspection method. 7. Write expressions relating the node and source voltages for voltage sources included in supernodes. 8. Substitute the expressions from step 7 into the node equations from step 6 and place the result in standard form. 2.2

Mesh Current Analysis Mesh currents are an alternative set of analysis variables that are useful in circuits containing many elements connected in series. To review terminology, a loop is a sequence of circuit elements that forms a closed path that passes through each element just once. A mesh is a special type of loop that does not enclose any elements. The following development of mesh analysis is restricted to planar circuits. A planar circuit can be drawn on a flat surface without crossovers in a “window pane” fashion. To define a set of variables, a mesh current (iA , iB , iC , . . .) is associated with each window pane and a reference direction assigned customarily in a clockwise sense. There is no momentous reason for this except perhaps tradition. Mesh currents are thought of as circulating through the elements in their respective meshes; however, this

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viewpoint is not based on the physics of circuit behavior. There are not different types of electrons flowing that somehow get assigned to mesh currents iA or iB . Mesh currents are variables used in circuit analysis. They are only somewhat abstractly related to the physical operation of a circuit and may be impossible to measure directly. Mesh currents have a unique feature that is the dual of the fundamental property of node voltages. In a planar circuit any given element is contained in at most two meshes. When an element is in two meshes, the two mesh currents circulate through the element in opposite directions. In such cases KCL declares that the net element current through the element is the difference of the two mesh currents. These observations lead to the fundamental property of mesh currents: If the Kth two-terminal element is contained in meshes X and Y, then the element current can be expressed in terms of the two mesh currents as (19) iK = iX − iY where X is the mesh whose reference direction agrees with the reference direction of iK . Equation (19) is a KCL constraint at the element level. If the element is contained in only one mesh, then iK = iX or iK = −iY depending on whether the reference direction for the element current agrees or disagrees with the mesh current. The key fact is that the current through every two-terminal element in a planar circuit can be expressed as the difference of at most two mesh currents. Mesh currents allow circuit equations to be formulated using device and connection constraints, except that the KCL constraints are not explicitly written down. Instead, the fundamental property of mesh currents is used to express the device constraints in terms of the mesh currents, thereby avoiding using the element currents and working only with the element voltages and mesh currents. For example, the planar circuit in Fig. 18 can be analyzed using the mesh current method. In the figure two mesh currents are shown as well the voltages across each of the five elements. The KVL constraints around each mesh using the element voltages yield Mesh A:

− v0 + v1 + v3 = 0

Mesh B:

− v3 + v2 + v4 = 0

Using the fundamental property of mesh currents, the element voltages in terms of the mesh currents and input voltages are written as v1 = R1 iA

v0 = vS1

v2 = R2 iB

v4 = vS2

v3 = R3 (iA − iB ) Substituting these element equations into the KVL connection equations and arranging the result in standard form yield (R1 + R3 ) iA − R3 iB = vS1 −R3 iA + (R2 + R3 ) iB = −vS2 This results in two equations in two unknown mesh currents. The KCL equations i1 = iA , i2 = iB and i3 = iA − iB are implicitly used to write mesh equations. In effect, the fundamental property of mesh currents ensures that the KCL constraints are satisfied. Any general method of circuit analysis must satisfy KCL, KVL, and the device i–v relationships. Mesh current analysis appears to focus on the latter two but implicitly satisfies KCL when the device constraints are expressed in terms of the mesh currents. Solving for the mesh currents yields iA =

(R2 + R3 ) vS1 − R3 vS2 R1 R2 + R1 R3 + R2 R3

iB =

R3 vS1 − (R1 + R3 ) vS2 R1 R2 + R1 R3 + R2 R3

and

The results for iA and iB can now be substituted into the device constraints to solve for every voltage in the circuit. For instance, the voltage across R3 is vA = v3 = R3 (iA − iB ) =

R2 R3 vS1 + R1 R3 vS2 R1 R2 + R1 R3 + R2 R3

Example 6. Use mesh current equations to find iO in the circuit in Fig. 19a. Solution. The current source in this circuit can be handled by a source transformation. The 2-mA source in parallel with the 4-k resistor in Fig. 19a can be replaced by an equivalent 8-V source in series with the same resistor as shown in Fig. 19b. In this circuit the total resistance in mesh A is 6 k, the total resistance in mesh B is 11 k, and the resistance contained in both meshes is 2 k. By inspection the mesh equations for this circuit are

(6000) iA − (2000) iB = 5 Fig. 18 Circuit demonstrating mesh current analysis.1

− (2000) iA + (11000) iB = −8

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Fig. 20 Example of supermesh.1 Fig. 19

(From Ref. 1.)

Solving for the two mesh currents yields iA = 0.6290 mA

and

iB = 0.6129 mA

The supermesh voltage constraint yields one equation in the three unknown mesh currents. Applying KCL to each of the current sources yields iA = iS1

iB − iC = iS2

By KCL the required current is i0 = iA − iB = 1.2419 mA Supermesh If a current source is contained in two meshes and is not connected in parallel with a resistance, then a supermesh is created by excluding the current source and any elements connected in series with it. One mesh equation is written around the supermesh using the currents iA and iB . Then mesh equations of the remaining meshes are written in the usual way. This leaves the solution one equation short because parts of meshes A and B are included in the supermesh. However, the fundamental property of mesh currents relates the currents iS , iA , and iB as

iS = iA − iB This equation supplies the additional relationship needed to get the requisite number of equations in the unknown mesh currents. This approach is obviously the dual of the supernode method for modified node analysis. The following example demonstrates the use of a supermesh. Example 7. Use mesh current equations to find the vO in Fig. 20. Solution. The current source iS2 is in both mesh B and mesh C, so we exclude this element and create the supermesh shown in the figure. The sum of voltages around the supermesh is

R1 (iB − iA ) + R2 (iB ) + R4 (iC ) + R3 (iC − iA ) = 0

Because of KCL, the two current sources force constraints that supply two more equations. Using these two KCL constraints to eliminate iA and iB from the supermesh KVL constraint yields (R1 + R2 + R3 + R4 ) iC = (R1 + R3 ) iS1 − (R1 + R2 ) iS2 Hence, the required output voltage is vO = R4 iC = R4 ×

(R1 + R3 ) iS1 − (R1 + R2 ) iS2 R1 + R2 + R3 + R4

Mesh current equations can always be formulated from KVL, the element constraints, and the fundamental property of mesh currents. The following guidelines summarize an approach to formulating mesh equations for resistance circuits: 1. Simplify the circuit by combining elements in series or parallel wherever possible. 2. Assign a clockwise mesh current to each mesh. 3. Create a supermesh for dependent and independent current sources that are contained in two meshes. 4. Write symmetrical mesh equations for all meshes by treating dependent sources as independent sources and using the inspection method. 5. Write expressions relating the mesh and source currents for current sources contained in only one mesh.

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6. Write expressions relating the mesh and source currents for current sources included in supermeshes. 7. Substitute the expressions from steps 5 and 6 into the mesh equations from step 4 and place the result in standard form. 2.3 Linearity Properties

This chapter treats the analysis and design of linear circuits. A circuit is said to be linear if it can be adequately modeled using only linear elements and independent sources. The hallmark feature of a linear circuit is that outputs are linear functions of the inputs. Circuit inputs are the signals produced by independent sources and outputs are any other designated signals. Mathematically a function is said to be linear if it possesses two properties—homogeneity and additivity. In terms of circuit responses, homogeneity means the output of a linear circuit is proportional to the input. Additivity means the output due to two or more inputs can be found by adding the outputs obtained when each input is applied separately. Mathematically these properties are written as follows:

Fig. 22 Example of circuit exhibiting proportionality.1

which means x = vS K=

and

f (x1 + x2 ) = f (x1 ) + f (x2 )

(20)

where K is a scalar constant. In circuit analysis the homogeneity property is called proportionality while the additivity property is called superposition. Proportionality Property The proportionality property applies to linear circuits with one input. For linear resistive circuits proportionality states that every input–output relationship can be written as

y =K ·x where x is the input current or voltage, y is an output current or voltage, and K is a constant. The block diagram in Fig. 21 describes a relationship in which the input x is multiplied by the scalar constant K to produce the output y. Examples of proportionality abound. For instance, using voltage division in Fig. 22 produces R2 vS vO = R1 + R2

Fig. 21 Block diagram representation of proportionality property.1

R2 R1 + R2

In this example the proportionality constant K is dimensionless because the input and output have the same units. In other situations K could carry the units of ohms or siemens when the input or output does not have the same units. Example 8.

f (Kx) = Kf (x)

y = vO

Given the bridge circuit of Fig. 23:

(a) Find the proportionality constant K in the input–output relationship vO = KvS . (b) Find the sign of K when R2 R3 > R1 R4 , R2 R3 = R1 R4 , and R2 R3 < R1 R4 . Solution. (a) Note that the circuit consists of two voltage dividers. Applying the voltage division rule to each side of the bridge circuit yields

vA =

R3 vS R1 + R3

vB =

R4 vS R2 + R4

The fundamental property of node voltages allows us to write vO = vA − vB . Substituting the equations for vA and vB into this KVL equation yields  R4 R3   vS −   R2 + R4  R1 + R3 R2 R3 − R1 R4 vo =  vS     (R1 + R3 )(R2 + R4 ) (K)vS

Fig. 23

(From Ref. 1.)

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(b) The proportionality constant K can be positive, negative, or zero. Specifically: If R2 R3 > R1 R2 , then K > 0. If R2 R3 = R1 R2 , then K = 0. If R2 R3 < R1 R2 , then K < 0. When the product of the resistances in opposite legs of the bridge are equal, K = 0 and the bridge is said to be balanced. Superposition Property The additivity property, or superposition, states that any output current or voltage of a linear resistive circuit with multiple inputs can be expressed as a linear combination of several inputs: y = K 1 x1 + K 2 x2 + K 3 x3 + · · ·

where x1 , x2 , x3 , . . . are current or voltage inputs and K1 , K2 , K3 . . . are constants that depend on the circuit parameters. Since the output y above is a linear combination, the contribution of each input source is independent of all other inputs. This means that the output can be found by finding the contribution from each source acting alone and then adding the individual response to obtain the total response. This suggests that the output of a multiple-input linear circuit can be found by the following steps: Step 1: “Turn off” all independent input signal sources except one and find the output of the circuit due to that source acting alone. Step 2: Repeat the process in step 1 until each independent input source has been turned on and the output due to that source found. Step 3: The total output with all sources turned on is then a linear combination (algebraic sum) of the contributions of the individual independent sources. A voltage source is turned off by setting its voltage to zero (vS = 0) and replacing it with a short circuit. Similarly, turning off a current source (iS = 0) entails replacing it with an open circuit. Figure 24a shows that the circuit has two input sources. Figure 24b shows the circuit with the current source set to zero. The output of the circuit vO1 represents that part of the total output caused by the voltage source. Using voltage division yields R2 vS vO1 = R1 + R2 Next the voltage source is turned off and the current source is turned on, as shown in Fig. 24c. Using Ohm’s law, vO2 = iO2 R2 . Then using current division to express iO2 in terms of iS yields

R1 R2 R1 iS R2 = iS vO2 = iO2 × R2 = R1 + R2 R1 + R2

Fig. 24 Circuit analysis using superposition: (a) current source off; (b) voltage source off.1

Applying the superposition theorem, the response with both sources “turned on” is found by adding the two responses vO1 and vO2 : vO = vO1 + vO2

R2 R1 R2 vS + iS vO = R1 + R2 R1 + R2 Superposition is an important property of linear circuits and is used primarily as a conceptual tool to develop other circuit analysis and design techniques. It is useful, for example, to determine the contribution to a circuit by a certain source. 2.4

Thevenin and Norton Equivalent Circuits

An interface is a connection between circuits that perform different functions. Circuit interfaces occur frequently in electrical and electronic systems so special analysis methods are used to handle them. For the twoterminal interface shown in Fig. 25, one circuit can be considered as the source S and the other as the load L. Signals are produced by the source circuit and delivered to the load. The source–load interaction at an interface is one of the central problems of circuit analysis and design. The Thevenin and Norton equivalent circuits shown in Fig. 25 are valuable tools for dealing with circuit interfaces. The conditions under which these equivalent circuits exist can be stated as a theorem: If the source circuit in a two-terminal interface is linear, then the interface signals v and i do not change when the source circuit is replaced by its Thevenin or Norton equivalent circuit. The equivalence requires the source circuit to be linear but places no restriction on the linearity of

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Fig. 25 Equivalent circuits for source circuit: (a) Thevenin equivalent; (b) Norton equivalent.1

the load circuit. The Thevenin equivalent circuit consists of a voltage source (vT ) in series with a resistance (RT ). The Norton equivalent circuit is a current source (iN ) in parallel with a resistance (RN ). The Thevenin and Norton equivalent circuits are equivalent to each other since replacing one by the other leaves the interface signals unchanged. In essence the Thevenin and Norton equivalent circuits are related by the source transformation covered earlier under equivalent circuits. The two parameters can often be obtained using open-circuit and short-circuit loads. If the actual load is disconnected from the source, an open-circuit voltage vOC appears between terminals A and B. Connecting an open-circuit load to the Thevenin equivalent produces vOC = vT since the open circuit causes the current to be zero, resulting in no voltage drop across RT . Similarly, disconnecting the load and connecting a short circuit as shown produce a current iSC . Connecting a short-circuit load to the Norton equivalent produces iSC = iN since all of the source current iN is diverted through the short-circuit load. In summary, the parameters of the Thevenin and Norton equivalent circuits at a given interface can be found by determining the open-circuit voltage and the short-circuit current: vT = vOC

iN = iSC

RN = RT =

vOC iSC

(21)

General Applications Since even complex linear circuits can be replaced by their Thevenin or Norton equivalent, the chore of designing circuits that interface with these complex circuits is greatly simplified. Suppose a load resistance in Fig. 26a needs to be chosen so the source circuit to the left of the interface A–B delivers 4 V to the load. The Thevenin and Norton equivalents vOC and iSC are first found. The open-circuit voltage vOC is found by disconnecting the load at terminals A–B as shown in Fig. 26b. The voltage across the 15- resistor is zero because the current through it is zero due to the open circuit. The open-circuit voltage at the interface is the same as the voltage across the 10- resistor.

Using voltage division, this voltage is vT = vOC =

10 × 5 = 10 V 10 + 5

Then the short-circuit current iSC is calculated using the circuit in Fig. 26c. The total current iX delivered by the 15-V source is ix = 15/REQ , where REQ is the equivalent resistance seen by the voltage source with a short circuit at the interface: REQ = 5 +

10 × 15 = 11 10 + 15

The source current iX can now be found: iX = 15/11 = 1.36 A. Given iX , current division is used to obtain the short-circuit current, iN = iSC =

10 × iX = 0.545 A 10 + 15

Finally, we compute the Thevenin and Norton resistances: vOC = 18.3 RT = RN = iSC The resulting Thevenin and Norton equivalent circuits are shown in Figs. 26d,e. It now is an easy matter to select a load RL so 4 V is supplied to the load. Using the Thevenin equivalent circuit, the problem reduces to a voltage divider, 4V=

RL RL × 10 × VT = RL + RT RL + 18.3

Solving for RL yields RL = 12.2 . The Thevenin or Norton equivalent can always found from the open-circuit voltage and short-circuit current at the interface. Often they can be measured using a multimeter to measure the open-circuit voltage and the short-circuit current. Application to Nonlinear Loads An important use of Thevenin and Norton equivalent circuits is

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Fig. 26 Example of finding Thevenin and Norton equivalent circuits: (a) given circuit; (b) open circuit yields Thevenin voltage; (c) short circuit yields Norton current; (d) Thevenin equivalent circuit; (e) Norton equivalent circuit.1

finding the voltage across, current through, and power dissipated in a two-terminal nonlinear element (NLE). The method of analysis is a straightforward application of device i–v characteristics. An interface is defined at the terminals of the nonlinear element and the linear part of the circuit is reduced to the Thevenin equivalent in Fig. 27a. Treating the interface current i as the dependent variable, the i–v relationship of the Thevenin equivalent is written in the form 1 vT v+ i= − RT RT This is the equation of a straight line in the i–v plane shown in Fig. 27b. The line intersects the i axis (v = 0) at i = iSC = vT /RT and intersects the v axis (i = 0) at v = vOC = vT . This line is called the load line. The nonlinear element has the i–v characteristic shown in Fig. 27c. Mathematically this nonlinear characteristic has the form i = f (v). Both the nonlinear equation and the load line equation must be solved simultaneously. This can be done by numerical methods when f(v) is known explicitly, but often a graphical solution is adequate. By superimposing the load line on the i–v characteristic curve of the nonlinear element in Fig. 27d, the point or points of intersection represent the values of i and v that satisfy the source constraints given in the form of the Thevenin equivalent above,

Fig. 27 Graphical analysis of nonlinear circuit: (a) given circuit; (b) load line; (c) nonlinear device i–v characteristics; (d) Q point.1

and nonlinear element constraints. In the terminology of electronics the point of intersection is called the operating point, or Q point, or the quiescent point. 2.5 Maximum Signal Transfer Circuit interfacing involves interconnecting circuits in such a way that they are compatible. In this regard an

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compared to RT . Ideally RL should be zero (a short circuit), in which case iMAX =

Fig. 28 Two-terminal interface for deriving maximum signal transfer conditions.1

important consideration is the maximum signal levels that can be transferred across a given interface. This section defines the maximum voltage, current, and power available at an interface between a fixed source and an adjustable load. The source can be represented by its Thevenin equivalent and the load by an equivalent resistance RL , as shown in Fig. 28. For a fixed source the parameters vT and RT are given and the interface signal levels are functions of the load resistance RL . By voltage division, the interface voltage is v=

RL vT RL + RT

For a fixed source and a variable load, the voltage will be a maximum if RL is made very large compared to RT . Ideally RL should be made infinite (an open circuit), in which case vMAX = vT = vOC Therefore, the maximum voltage available at the interface is the source open-circuit voltage vOC . The current delivered at the interface is i=

vT RL + RT

Again, for a fixed source and a variable load, the current will be a maximum if RL is made very small

Fig. 29

vT = iN = iSC RT

Therefore, the maximum current available at the interface is the source short-circuit current iSC . The power delivered at the interface is equal to the product vi. Using interface voltage, and interface current results found above, the power is p =v×i =

RL vT2 (RT + RL )2

For a given source, the parameters vT and RT are fixed and the delivered power is a function of a single variable RL . The conditions for obtaining maximum voltage (RL → ∞) or maximum current (RL = 0) both produce zero power. The value of RL that maximizes the power lies somewhere between these two extremes. The value can be found by differentiating the power expression with respect to RL and solving for the value of RL that makes dp/dRL = 0. This occurs when RL = RT . Therefore, maximum power transfer occurs when the load resistance equals the Thevenin resistance of the source. When RL = RT the source and load are said to be matched. Substituting the condition RL = RT back into the power equation above shows the maximum power to be pMAX =

i 2 RT vT2 = N 4RT 4

These results are consequences of what is known as the maximum power transfer theorem: A fixed source with a Thevenin resistance RT delivers maximum power to an adjustable load RL when RL = RT . Figure 29 shows plots of the interface voltage, current, and power as functions of RL /RT . The plots of v/vOC , i/iSC , and p/pMAX are normalized to the maximum available signal levels so the ordinates in Fig. 29 range from 0 to 1.

Normalized plots of current, voltage, and power versus RL /RT .1

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The plot of the normalized power p/pMAX in the neighborhood of the maximum is not a particularly strong function of RL /RT . Changing the ratio RL /RT by a factor of 2 in either direction from the maximum reduces p/pMAX by less than 20%. The normalized voltage v/vOC is within 20% of its maximum when RL /RT = 4. Similarly, the normalized current is within 20% of its maximum when RL /RT = 3. In other words, for engineering purposes maximum signal levels can be approached with load resistances that only approximate the theoretical requirements. 2.6 Interface Circuit Design The maximum signal levels discussed in the previous section place bounds on what is achievable at an interface. However, those bounds are based on a fixed source and an adjustable load. In practice there are circumstances in which the source or the load or both can be adjusted to produce prescribed interface signal levels. Sometimes it is necessary to insert an interface circuit between the source and load. Figure 30 shows the general situations and some examples of resistive interface circuits. By its very nature the inserted circuit has two terminal pairs, or interfaces, at which voltage and current can be observed or specified. These terminal pairs are also called ports, and the interface circuit is referred to as a two-port network. The port connected to the source is called the input and the port connected to the load the output. The purpose of this two-port

Fig. 30

network is to ensure that the source and load interact in a prescribed way. Basic Circuit Design Concepts This section introduces a limited form of circuit design, as contrasted with circuit analysis. Although circuit analysis tools are essential in design, there are important differences. A linear circuit analysis problem generally has a unique solution. A circuit design problem may have many solutions or even no solution. The maximum available signal levels found above provide bounds that help test for the existence of a solution. Generally there will be several ways to meet the interface constraints, and it then becomes necessary to evaluate the alternatives using other factors such as cost, power consumption, or reliability. Currently only the resistor will be used to demonstrate interface design. In subsequent sections other useful devices such as op amps and capacitors and inductors will be used to design suitable interfaces. In a design situation the engineer must choose the resistance values in a proposed circuit. This decision is influenced by a host of practical considerations such as standard values, standard tolerances, manufacturing methods, power limitations, and parasitic elements. Example 9. Select the load resistance in Fig. 31 so the interface signals are in the range defined by v ≥ 4 V and i ≥ 30 mA.

General interface circuit and some examples.1

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Fig. 31 (From Ref. 1.)

Solution. In this design problem the source circuit is given and a suitable load needs to be selected. For a fixed source the maximum signal levels available at the interface are

vMAX = vT = 10 V

iMAX =

vT = 100 mA RT

The bounds given as design requirements are below the maximum available signal levels a suitable resistor can be found. Using voltage division, the interface voltage constraint requires v=

RL × 10 ≥ 4 100 + RL

or v = 10RL ≥ 4RL + 400

This condition yields RL ≥ 400/6 = 66.7 . The interface current constraint can be written as i=

10 ≥ 0.03 or i = 10 ≥ 3 + 0.03RL 100 + RL

which requires RL ≤ 7/0.03 = 233 . In principle any value of RL between 66.7 and 233 will work. However, to allow for circuit parameter variations, choose RL = 150 because it lies at the arithmetic midpoint of allowable range and is a standard value. Example 10. Design the two-port interface circuit in Fig. 32 so that the 10-A source delivers 100 V to the 50- load. Solution. The problem requires that the current delivered to the load is i = 100/50 = 2 A, which is well below the maximum available from the source. In fact, if the 10-A source is connected directly to the load, the source current divides equally between two 50- resistors producing 5 A through the load. Therefore, an interface circuit is needed to reduce the load current to the specified 2-A level. Two possible design solutions are shown in Fig. 32. Applying current division to the parallel-resistor case yields the following constraint:

i=

1/50 × 10 1/50 + 1/50 + 1/RPAR

Fig. 32

(From Ref. 1.)

For the i = 2-A design requirement this equation becomes 10 2= 2 + 50/RPAR Solving for RPAR yields RPAR =

50 3

= 16.7

Applying the two-path current division rule to the series-resistor case yields the following constraint: i=

50 × 10 = 2 A 50 + (50 + RSER )

Solving for RSER yields RSER = 150 Both these two designs meet the basic i = 2-A requirement. In practice, engineers evaluate alternative designs using additional criteria. One such consideration is the required power ratings of the resistors in each design. The voltage across the parallel resistor is v = 100 V, so the power loss is pPAR =

1002 = 600 W 50/3

The current through the series-resistor interface is i = 2 A so the power loss is pSER = 22 × 150 = 600 W In either design the resistors must have a power rating of at least 600 W. The series resistor is a standard

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value whereas the parallel resistor is not. Other factors besides power rating and standard size could determine which design should be selected. Example 11. Design the two-port interface circuit in Fig. 33 so the load is a match to 50 between terminals C and D, while simultaneously the source matches to a load resistance of 300 between A and B. Solution. No single-resistor interface circuit could work. Hence try an interface circuit containing two resistors. Since the load must see a smaller resistance than the source, it should “look” into a parallel resistor. Since the source must see a larger resistance than the load, it should look into a series resistor. A configuration that meets these conditions is the L circuit shown in Figs. 33b,c. The above discussion can be summarized mathematically. Using the L circuit, the design requirement at terminals C and D is

(R1 + 300) R2 = 50 R1 + 300 + R2 At terminals A and B the requirement is R11 +

50R2 = 300 R2 + 50

The design requirements yield two equations in two unknowns—what could be simpler? It turns out that

solving these nonlinear equations by hand analysis is a bit of a chore. They can easily be solved using a math solver such as MATLAB or MathCad. But a more heuristic approach might serve best. Given the L circuits in Fig. 33b, such an approach goes as follows. Let R2 = 50 . Then the requirement at terminals C and D will be met, at least approximately. Similarly, if R1 + R2 = 300 , the requirements at terminals A and B will be approximately satisfied. In other words, try R1 = 250 and R2 = 50 as a first cut. These values yield equivalent resistances of RCD = 50||550 = 45.8 and RAB = 250 + 50||50 = 275 . These equivalent resistances are not the exact values specified but are within ±10%. Since the tolerance on electrical components may be at least this high, a design using these values could be adequate. The exact values found by a math solver yields R1 = 273.861 and R2 = 54.772 . 3 LINEAR ACTIVE CIRCUITS This section treats the analysis and design of circuits containing active devices such as transistors or operational amplifiers (op amps). An active device is a component that requires an external power supply to operate correctly. An active circuit is one that contains one or more active devices. An important property of active circuits is that they are capable of providing signal amplification, one of the most important signal-processing functions in electrical engineering. Linear active circuits are governed by the proportionality property so their input–output relationships are of the form y = Kx. The term signal amplification means the proportionality factor K > 1 when the input x and output y have the same dimensions. Thus, active circuits can deliver more signal voltage, current, and power at their output than they receive from the input signal. The passive resistance circuits studied thus far cannot produce voltage, current, or power gains greater than unity. 3.1 Dependent Sources When active devices operate in a linear mode, they can be modeled using resistors and one or more of the four dependent source elements shown in Fig. 34.

Fig. 33

(From Ref. 1.)

Fig. 34 Dependent source circuit symbols: (a) currentcontrolled voltage source; (b) voltage-controlled voltage source; (c) current-controlled current source; (d) voltagecontrolled current source.1

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The dominant feature of a dependent source is that the strength or magnitude of the voltage source (VS) or current source (CS) is proportional to—that is, controlled by—a voltage (VC) or current (CC) appearing elsewhere in the circuit. For example, the dependent source model for a current-controlled current source (CCCS) is shown in Fig. 34c. The output current βi1 is dependent on the input current i1 and the dimensionless factor β. This dependency should be contrasted with the characteristics of the independent sources studied earlier. The voltage (current) delivered by an independent voltage (current) source does not depend on the circuit to which it is connected. Dependent sources are often but not always represented by the diamond symbol, in contrast to the circle symbol used for independent sources. Each dependent source is characterized by a single parameter, µ, β, r, or g. These parameters are often called simply the gain of the controlled source. Strictly speaking, the parameters µ and β are dimensionless quantities called the open-loop voltage gain and open-loop current gain, respectively. The parameter r has the dimensions of ohms and is called the transresistance, a contraction of transfer resistance. The parameter g is then called transconductance and has the dimensions of siemens. In every case the defining relationship for a dependent source has the form y = Kx, where x is the controlling variable, y is the controlled variable, and K is the element gain. It is this linear relationship between the controlling and controlled variables that make dependent sources linear elements. Although dependent sources are elements used in circuit analysis, they are conceptually different from the other circuit elements. The linear resistor and ideal switch are models of actual devices called resistors and switches. But dependent sources are not listed in catalogs. For this reason dependent sources are more abstract, since they are not models of identifiable physical devices. Dependent sources are used in combination with other resistive elements to create models of active devices. A voltage source acts like a short circuit when it is turned off. Likewise, a current source behaves like an open circuit when it is turned off. The same results apply to dependent sources, with one important difference. Dependent sources cannot be turned on and off individually because they depend on excitation supplied by independent sources. When applying the superposition principle or Thevenin’s theorem to active circuits, the state of a dependent source depends on excitation supplied by independent sources. In particular, for active circuits the superposition principle states that the response due to all independent sources acting simultaneously is equal to the sum of the responses due to each independent source acting one at a time. Analysis with Dependent Sources With certain modifications the circuit analysis tools developed for

passive circuits apply to active circuits as well. Circuit reduction applies to active circuits, but the control variable for a dependent source must not be eliminated. Applying a source transformation to a dependent source is sometimes helpful. Methods like node and mesh analysis can be adapted to handle dependent sources as well. But the main difference is that the properties of active circuits can be significantly different from those of the passive circuits. In the following example the objective is to determine the current, voltage, and power delivered to the 500- output load in Fig. 35. The control current iX is found using current division in the input circuit: 2 50 iS = iS iX = 50 + 25 3 Similarly the output current iO is found using current division in the output circuit: 3 300 iY = iY iO = 300 + 500 8 But at node A KCL requires that iY = −48iX . Combining this result with the equations for iX and iO yields the output current: (22) iO = 38 (−48) iX = −18 23 iS = −12iS The output voltage vO is found using Ohm’s law: vO = iO × 500 = −6000iS

(23)

The input–output relationships in Eqs. (22) and (23) are of the form y = Kx with K < 0. The proportionality constants are negative because the reference direction for iO in Fig. 35 is the opposite of the orientation of the dependent source reference mark. Active circuits often produce negative values of K. As a result the input and output signals have opposite signs, a result called signal inversion. In the analysis and design of active circuits it is important to keep track of signal inversions. The delivered output power is pO = vO iO = (−6000iS ) (−12iS ) = 72,000iS2

Fig. 35

Circuit with dependent source.1

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The input independent source delivers its power to the parallel combination of 50 and 25 . Hence, the power supplied by the independent source is pS = (50||25) iS2 =

50 3

iS2

Given the input power and output power, we find the power gain in the circuit: Power gain =

72, 000iS2 pO = = 432 pS (50/3) iS2

A power gain greater than unity means that the circuit delivers more power at its output than it receives from the input source. At first glance, this appears to be a violation of energy conservation, but dependent sources are models of active devices that require an external power supply to operate. In general, circuit designers do not show the external power supply in circuit diagrams. Control source models assume that the external supply and the active device can handle whatever power is required by the circuit. With real devices this is not the case, and in circuit design engineers must ensure that the power limits of the device and external supply are not exceeded. Node Voltage Analysis with Dependent Sources Node voltage analysis of active circuits follows the same process as for passive circuits except that the additional constraints implied by the dependent sources must be accounted for. For example, the circuit in Fig. 36 has five nodes. With node E as the reference both independent voltage sources are connected to ground and force the condition vA = vS1 and vB = vS2 . Node analysis involves expressing element currents in terms of the node voltages and applying KCL at each unknown node. The sum of the currents leaving node C is

vC − vS2 vC vC − vD vC − vS1 + + + =0 R1 R2 RB Rp

Fig. 36 Circuit used for node voltage analysis with dependent sources.1

Similarly, the sum of currents leaving node D is vD − vC vD + − βiB = 0 RP RE These two node equations can be rearranged into the forms 1 1 1 1 1 − Node C: vC + + + vD R1 R2 RB RP RP 1 1 = vS1 + vS2 R1 R2 1 1 1 vD Node D: − vC + + RP RP RE = βiB Applying the fundamental property of node voltages and Ohm’s law, the current iB can be written in terms of the node voltages as iB =

vC − vD RP

Substituting this expression for iB into the above node equation and putting the results in standard form yield Node C:

1 1 1 1 + + + R1 R2 RB RP

1 vD RP 1 1 = vS1+ vS2 R1 R2 vC −

1 Node D: − (β + 1) vC R P

1 1 vD = 0 + (β + 1) + RP RE

The final result involves two equations in the two unknown node voltages and includes the effect of the dependent source. This example illustrates a general approach to writing node voltage equations for circuits with dependent sources. Dependent sources are initially treated as if they are independent sources and node equations written for the resulting passive circuit. This step produces a set of symmetrical node equations with the independent and dependent source terms on the right side. Next the dependent source terms are expressed in terms of the unknown node voltages and moved to the left side with the other terms involving the unknowns. The last step destroys the coefficient symmetry but leads to a set of equations that can be solved for the active circuit response. Mesh Current Analysis with Dependent Sources Mesh current analysis of active circuits follows the same pattern noted for node voltage analysis. Treat the dependent sources initially as independent sources

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and write the mesh equations of the resulting passive circuit. Then account for the dependent sources by expressing their constraints in terms of unknown mesh currents. The following example illustrates the method.

(1.115 × 104 )iA − 102 iB = vS −(1.01 × 104 )iA + (5.1 × 103 )iB = 0

Example 12

(a) Formulate mesh current equations for the circuit in Fig. 37. (b) Use the mesh equations to find vO and RIN when R1 = 50 , R2 = 1 k, R3 = 100 , R4 = 5 k, and g = 100 mS. Solution. (a) Applying source transformation to the parallel combination of R3 and gvX in Fig. 37a produces the dependent voltage source R3gvX = µvX in Fig. 37b. In the modified circuit we have identified two mesh currents. Initially treating the dependent source (gR3 )vx as an independent source leads to two symmetrical mesh equations:

Mesh A: Mesh B:

(b) Substituting the numerical values into the mesh equations gives

(R1 + R2 + R3 )iA −R3 iB = vS − (gR3 )vX −R3 iA + (R3 + R4 )iB = (gR3 )vX

The control voltage vx can be written in terms of mesh currents as vX = R2 iA Substituting this equation for vx into the mesh equations and putting the equations in standard form yield (R1 + R2 + R3 + gR2 R3 ) iA − R3 iB = vS

Using Cramer’s rule the mesh currents are found to be iA = (0.9131 × 10−4 )vS

and

−4

iB = (1.808 × 10 )vS The output voltage and input resistance are found using Ohm’s law: vS RIN = = 10.95 k vO = R4 iB = 0.904vS iA Thevenin Equivalent Circuits with Dependent Sources To find the Thevenin equivalent of an active circuit, the independent sources are left on or else one must supply excitation from an external test source. This means that the Thevenin resistance can not be found by the “look-back” method, which requires that all independent sources be turned off. Turning off the independent sources deactivates the dependent sources as well and can result in a profound change in input and output characteristics of an active circuit. Thus, Thevenin equivalents of active circuits can be found using the open-circuit voltage and shortcircuit current at the interface.

− (R3 + gR2 R3 ) iA + (R3 + R4 ) iB = 0

Example 13. Find the Thevenin equivalent at the output interface of the circuit in Fig. 38.

The resulting mesh equations are not symmetrical because of the controlled source.

Solution. In this circuit the controlled voltage vX appears across an open circuit between nodes A and B. By the fundamental property of node voltages, vX = vS − vO . With the load disconnected and the input source turned off, vx = 0, the dependent voltage source µvX acts like a short circuit, and the Thevenin resistance looking back into the output port is RO . With the load connected and the input source turned on, the sum of currents leaving node B is

vO − µvX + iO = 0 RO


Fig. 38

(From Ref. 1.)

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Using the relationship vX = vS − vO to eliminate vX and then solving for vO produce the output i–v relationship of the circuit as vO =

µvS RO − iO µ+1 µ+1

The i–v relationship of a Thevenin circuit is v = vT iRT . By direct comparison, the Thevenin parameters of the active circuit are found to be vT =

µvS µ+1

and

RT =

RO µ+1

The circuit in Fig. 38 is a model of an op amp circuit called a voltage follower. The resistance RO for a general-purpose op amp is on the order of 100 , while the gain µ is about 105 . Thus, the active Thevenin resistance of the voltage follower is not 100 , as the look-back method suggests, but around a milliohm! 3.2 Operational Amplifier The operational amplifier is the premier linear active device made available by IC technology. John R. Ragazzini apparently first used the term operational amplifier in a 1947 paper and his colleagues who reported on work carried out for the National Defenses Research Council during World War II. The paper described high-gain dc amplifier circuits that perform mathematical operations (addition, subtraction, multiplication, division, integration, etc.)—hence the name “operational” amplifier. For more than a decade the most important applications were general- and specialpurpose analog computers using vacuum tube amplifiers. In the early 1960s general-purpose, discretetransistor, op amp became readily available and by the mid-1960s the first commercial IC op amps entered the market. The transition from vacuum tubes to ICs resulted in a decrease in size, power consumption, and cost of op amps by over three orders of magnitude. By the early 1970s the IC version became the dominant

Fig. 39

active device in analog circuits. The device itself is a complex array of transistors, resistors, diodes, and capacitors all fabricated and interconnected on a single silicon chip. In spite of its complexity, the op amp can be modeled by rather simple i–v characteristics. Op Amp Notation Certain matters of notation and nomenclature must be discussed before developing a circuit model for the op amp. The op amp is a five-terminal device, as shown in Fig. 39a. The “+” and “−” symbols identify the input terminals and are a shorthand notation for the noninverting and inverting input terminals, respectively. These “+” and “−” symbols identify the two input terminals and have nothing to do with the polarity of the voltages applied. The other terminals are the output and the positive and negative supply voltage, usually labeled +VCC and −VCC . While some op amps have more than five terminals, these five are always present. Figure 39b shows how these terminals are arranged in a common eight-pin IC package. While the two power supply terminals in Fig. 39 are not usually shown in circuit diagrams, they are always there because the external power supplies connected to these terminals make the op amp an active device. The power required for signal amplification comes through these terminals from an external power source. The +VCC and −VCC voltages applied to these terminals also determine the upper and lower limits on the op amp output voltage. Figure 40a shows a complete set of voltage and current variables for the op amp, while Fig. 40b shows the typical abbreviated set of signal variables. All voltages are defined with respect to a common reference node, usually ground. Voltage variables vP , vN , and vO are defined by writing a voltage symbol beside the corresponding terminals. This notation means the “+” reference mark is at the terminal in question and the “−” reference mark is at the reference or ground terminal. The reference directions for the currents are directed in at input terminals and out at the output. A global KCL equation for the complete set of variable in Fig. 40a is iO = IC+ + IC− + iP + iN , NOT iO = iN + iP , as might be inferred from Fig. 40b, since it does not

Op amp: (a) circuit symbol; (b) pin out diagram for eight-pin package.1

896


Fig. 40

Op amp voltage and current definitions: (a) complete set; (b) shorthand set.1

include all of the currents. More importantly, it implies that the output current comes from the inputs. In fact, this is wrong. The input currents are very small, ideally zero. The output current comes from the supply voltages even though these terminals are not shown on the abbreviated circuit diagram. Transfer Characteristics The dominant feature of the op amp is the transfer characteristic shown in Fig. 41. This characteristic provide the relationships between the noninverting input vP , the inverting input vN , and the output voltage vO . The transfer characteristic is divided into three regions or modes called +saturation, −saturation, and linear. In the linear region the op amp is a differential amplifier because the output is proportional to the difference between the two inputs. The slope of the line in the linear range is called the open-loop gain, denoted as µ. In the linear region the input–output relation is vO = µ(vP − vN ). The open-loop gain of an op amp is very large, usually greater than 105 . As long as the net input vP − vN is very small, the output will be proportional to the input. However, when µ|vP − vN | > VCC , the op amp is saturated and the output voltage is limited by the supply voltages (less some small internal losses).

Fig. 41

Op amp transfer characteristics.1

The op amp has three operating modes: 1. + Saturation mode when µ(vP − vN ) > +VCC and vO = +VCC . 2. − Saturation mode when µ(vP − vN ) < −VCC and vO = −VCC . 3. Linear mode when µ|vP − vN | < VCC and vO = µ(vP − vN ). Usually op amp circuits are analyzed and designed using the linear mode model. Ideal Op Amp Model A controlled source model of an op amp operating in its linear range is shown in Fig. 42. This model includes an input resistance (RI ), an output resistance (RO ), and a voltage-controlled voltage source whose gain is the open-loop gain µ. Some typical ranges for these op amp parameters are given in Table 3, along with the values for the ideal op amp. The high input and low output resistances and high open-loop gain are the key attributes of an op amp. The ideal model carries these traits to the extreme limiting values.

Fig. 42 Dependent source model of op amp operating in linear mode.1

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Table 3 Typical Op Amp Parameters Name

Parameter

Open-loop gain Input resistance Output resistance Supply voltages

µ RI RO VCC

Range 105 –108 106 –1013 10–100 ±5 to ±40 V

Ideal Values ∞ ∞ 0

The controlled source model can be used to develop the i–v relationships of the ideal model. This discussion is restricted to the linear region of operation. This means the output voltage is bounded as −VCC ≤ vo ≤ +VCC

−

VCC VCC ≤ (vP − vN ) ≤ + µ µ

The supply voltages VCC are most commonly ±15 V although other supply voltages are available, while µ is a very large number, usually 105 or greater. Consequently, linear operation requires that vP · vN . For the ideal op amp the open-loop gain is infinite (µ → ∞), in which case linear operation requires vP = vN . The input resistance RI of the ideal op amp is assumed to be infinite, so the currents at both input terminals are zero. In summary, the i–v relationships of the ideal model of the op amp are vP = vN

iP = iN = 0

(24)

At first glance the element constraints of the ideal op amp appear to be fairly useless. They actually look more like connection constraints and are totally silent about the output quantities (vO and iO ), which are usually the signals of interest. In fact, they seem to say that the op amp input terminals are simultaneously a short circuit (vP = vN ) and an open circuit (iP = iN = 0). The ideal model of the op amp is useful because in linear applications feedback is always present. That is, in order for the op amp to operate in a linear mode, it is necessary that there be feedback paths from the output to one or both of the inputs. These feedback paths ensure that vP = vN and allow for analysis of op amp circuits using the ideal op amp element constraints. Op Amp Circuit Analysis This section introduces op amp circuit analysis using circuits that are building blocks for analog signal-processing systems. The key to using the building block approach is to recognize the feedback pattern and to isolate the basic circuit as a building block. Noninverting Op Amp To illustrate the effects of feedback, consider the circuit in Fig. 43. This circuit has a feedback path from the output to the inverting input via a voltage divider. Since the ideal op amp draws no current at either input (iP = iN = 0),

Fig. 43

Noninverting amplifier circuit.1

voltage division determines the voltage at the inverting input as R2 vN = vO R1 + R2 The input source connection at the noninverting input requires the condition vP = vS . But the ideal op amp element constraints demand that vP = vN ; therefore, the input–output relationship of the overall circuit is vO =

R1 + R2 vS R2

(25)

The circuit in Fig. 43a is called a noninverting amplifier. The input–output relationship is of the form vO = KvS , a linear relationship. Figure 43b shows the functional building block for this circuit, where the proportionality constant K is K=

R1 + R2 R2

(26)

The constant K is called the closed-loop gain, since it includes the effect of the feedback path. When discussing op amp circuits, it is necessary to distinguish between two types of gains. The first is the open-loop gain µ provided by the op amp device. The gain µ is a large number with a large uncertainty tolerance. The second type is the closed-loop gain K of the op amp circuit with a feedback path. The gain K must be smaller than µ, typically no more than 1/100 of µ, and its value is determined by the resistance elements in the feedback path. For example, the closed-loop gain in Eq. (26) is really the voltage division rule upside down. The uncertainty tolerance assigned to K is determined by the quality of the resistors in the feedback path, and not the uncertainty in the actual value of the closedloop gain. In effect, feedback converts a very large but

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imprecisely known open-loop gain into a much smaller but precisely controllable closed-loop gain. Example 14. Design an amplifier with a closed-loop gain K = 10. Solution. Using a noninverting op amp circuit, the design problem is to select the values of the resistors in the feedback path. From Eq. (26) the design constraint is R1 + R2 10 = R2

This yields one constraint with two unknowns. Arbitrarily selecting R2 = 10 k makes R1 = 90 k. These resistors would normally have high precision (±1% or less) to produce a precisely controlled closed-loop gain. Comment: The problem of choosing resistance values in op amp circuit design problems deserves some discussion. Although resistances from a few ohms to several hundred megohms are commercially available, generally designers limit themselves to the range from about 1 k to perhaps 2.2 M . The lower limit of 1 k exists in part because of power dissipation in the resistors and to minimize the effects of loading (discussed later). Typically resistors with 3 W power ratings or less are used. The maximum voltages in op amp circuits are often around ±15 V although other values exist, including single-sided op amps, with a 0–5 V VCC for use in digital applications. The smallest 3-W resistance we can use is RMIN > (15)2 /0.25 = 900 , or about 1 k. The upper bound of 2.2 M exists because it is difficult to maintain precision in a highvalue resistor because of surface leakage caused by humidity. High-value resistors are also noisy, which leads to problems when they are in the feedback path. The range 1 k to 2.2 M should be used as a guideline and not an inviolate design rule. Actual design choices are influenced by system-specific factors and changes in technology. Voltage Follower The op amp in Fig. 44a is connected as voltage follower or buffer. In this case the feedback path is a direct connection from the output to the inverting input. The feedback connection forces the condition vN = vO . The input current iP = 0 so there is no voltage across the source resistance RS . Applying KVL results in vP = vS . The ideal op amp model requires vP = vN , so that vO = vS . By inspection the closed-loop gain is K = 1. The output exactly equals the input, that is, the output follows the input, and hence the name voltage follower. The voltage follower is used in interface circuits because it isolates the source and load—hence its other name, buffer. Note that the input–output relationship vO = vS does not depend on the source or load resistance. When the source is connected directly to the load as in Fig. 44b, the voltage delivered to the load depends on RS and RL . The source and load interaction limits the signals that can transfer across the interface.

Fig. 44 (a) Source–load interface with voltage follower; (b) interface without voltage follower.1

When the voltage follower is inserted between the source and load, the signal levels are limited by the capability of the op amp. Inverting Amplifier The circuit in Fig. 45 is called an inverting amplifier. The key feature of this circuit is that the input signal and the feedback are both applied at the inverting input. Note that the noninverting input is grounded, making vP = 0. Using the fundamental property of node voltages and KCL, the sum of currents entering node A can be written as

vO − vN vs − vN + − iN = 0 R1 R2

Fig. 45

Inverting amplifier circuit.1

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The element constraints for the op amp are vP = vN and iP = iN = 0. Since vP = 0, it follows that vN = 0. Substituting the op amp constraints and solving for the input–output relationship yield R2 vS (27) vO = − R1 This result is of the form vO = KvS , where K is the closed-loop gain. However, in this case the closedloop gain K = −R2 /R1 is negative, indicating a signal inversion—hence the name inverting amplifier. The block diagram symbol shown in Fig. 45b is used to indicate either the inverting or noninverting op amp configuration, since both circuits provide a gain of K. The op amp constraints mean that the input current i1 in Fig. 43a is i1 =

vS − vN vS = R1 R1

This in turn means that the input resistance seen by the source vS is RIN =

vS = R1 i1

(28)

grounded, vP = 0. This configuration is similar to the inverting amplifier; hence a similar analysis yields the circuit input–output relationship   − RF v + − RF v 1 2 R1 R2 (29) vO =  (K1 )v1 + (K2 )v2 The output is a weighted sum of the two inputs. The scale factors, or gains as they are called, are determined by the ratio of the feedback resistor RF to the input resistor for each input: that is, K1 = −RF /R1 and K2 = −RF /R2 . In the special case R1 = R2 = R, Eq. (29) reduces to vO = K (v1 + v2 ) where K = −RF /R. In this special case the output is proportional to the negative sum of the two inputs—hence the name inverting summing amplifier or simply adder. A block diagram representation of this circuit is shown in Fig. 46b. Example 15. Design an inverting summer that implements the input–output relationship vO = −(5v1 + 13v2 ).

In other words, the inverting amplifier has as finite input resistance determined by the external resistor R1 . This finite input resistance must be taken into account when analyzing circuits with op amps in the inverting amplifier configuration.

Solution. The design problem involves selecting the input and feedback resistors so that

Summing Amplifier The summing amplifier or adder circuit is shown in Fig. 46a. This circuit has two inputs connected at node A, which is called the summing point. Since the noninverting input is

One solution is to arbitrarily select RF = 65 k, which yields R1 = 13 k and R2 = 5 k. The resulting circuit is shown in Fig. 47a. The design can be modified to use standard resistance values for resistors with ±5% tolerance. Selecting the standard

Fig. 46

Inverting summer.1

RF =5 R1

and

RF = 13 R2


900


value RF = 56 k requires R1 = 11.2 k and R2 = 4.31 k. The nearest standard values are 11 and 4.3 k. The resulting circuit shown in Fig. 47b uses only standard value resistors and produces gains of K1 = 56/11 = 5.09 and K2 = 56/4.3 = 13.02. These nominal gains are within 2% of the values in the specified input–output relationship.

where K1 and K2 are the inverting and noninverting gains. Figure 48b shows how the differential amplifier is represented in a block diagram. In the special case of R1 = R2 = R3 = R4 , Eq. (30) reduces to vO = v2 − v1 . In this case the output is equal to the difference between the two inputs—hence the name differential amplifier or subtractor.

Differential Amplifier The circuit in Fig. 48a is called a differential amplifier or subtractor. Like the summer, this circuit has two inputs, but unlike the summer, one is applied at the inverting input and one at the noninverting input of the op amp. The input–output relationship can be obtained using the superposition principle. With source v2 off there is no excitation at the noninverting input and vP = 0. In effect, the noninverting input is grounded and the circuit acts like an inverting amplifier with the result that

Noninverting Summer The circuit in Fig. 49 is an example of a noninverting summer. The input–output relationship for a general noninverting summer is

vO1 = −

R2 v1 R1

Now turning v2 back on and turning v1 off, the circuit looks like a noninverting amplifier with a voltage divider connected at its input. Thus

R1 + R2 R4 vO2 = v2 R3 + R4 R1 Using superposition the two outputs are added to obtain the output with both sources on:  v +v O1 O2  

 R1 + R2 R2 R4 vO = − v1 + v2  R1 R3 + R4 R1   −[K1 ]v1 + [K2 ] v2 (30)

Fig. 48

Differential amplifier.1

REQ REQ v1 + v2 vO = K R1 R2

REQ vn + ··· + Rn

(31)

where REQ is the Thevenin resistance looking to the left at point P with all sources turned off (i.e., REQ = R1 R2 R3 · · · Rn ) and K is the gain of the noninverting amplifier circuit to the right of point P. Comparing this equation with the general inverting summer result in Eq. (29), we see several similarities. In both cases the weight assigned to an input voltage is proportional to a resistance ratio in which the denominator is its input resistance. In the inverting summer the numerator of the ratio is the feedback resistor RF and in the noninverting case the numerator is the equivalent of all input resistors REQ . Design with Op Amp Building Blocks The block diagram representation of the basic op amp circuit configurations were developed in the preceding section. The noninverting and inverting amplifiers are represented as gain blocks. The summing amplifier and differential amplifier require both gain blocks and the summing symbol. One should exercise care when translating from a block diagram to a circuit, or vice versa, since some gain blocks may involve negative gains. For example, the gain of the inverting amplifier is negative, as are the gains of the common inverting summing amplifier and the K1 gain of the differential amplifier. The minus sign is sometimes moved to the summing symbol and the gain within the block changed to a positive number. Since there is

Fig. 49

Noninverting summer.2

ELECTRIC CIRCUITS

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no standard convention for doing this, it is important to keep track of the signs associated with gain blocks and summing point symbol. Operational amplifier circuit design generally requires that a given equation or block diagram representation of a signal-processing function be created to implement that the function. Circuit design can often be accomplished by interconnecting the op amp, summer, and subtractor building blocks. The design process is greatly simplified by the near one-to-one correspondence between the op amp circuits and the elements in a block diagram. However, the design process is not unique since often there are several ways to use basic op amp circuits to meet the design objective. Some solutions are better than others are. The following example illustrates the design process. Example 16. Design an op amp circuit that implements the block diagram in Fig. 50. Solution. The input–output relationship represented by the block diagram is vO = 5v1 + 10v2 + 20v3 . An op amp adder can implement the summation required in this relationship. A three-input adder implements

Fig. 50

(From Ref. 2.)

Fig. 51

(From Ref. 2.)

the relationship vO = −

RF RF RF v1 + v2 + v3 R1 R2 R3

The required scale factors are realized by first selecting RF = 100 k and then choosing R1 = 20 k, R2 = 10 k, and R3 = 5 k. However, the adder involves a signal inversion. To correctly implement the block diagram, we must add an inverting amplifier (K = −R2 /R1 ) with R1 = R2 = 100 k. The final implementation is shown in Fig. 51a. An alternate solution avoiding the second inverting op amp by using a noninverting summer is shown in Fig. 51b. Digital-to-Analog Converters Operational amplifiers play an important role in the interface between digital systems and analog systems. The parallel 4-bit output in Fig. 52 is a digital representation of a signal. Each bit can only have two values: (a) a high or 1 (typically +5 V) and (b) a low or 0 (typically 0 V).

Fig. 52

A DAC.1

902


The bits have binary weights so that v1 is worth 23 = 8 times as much v4 , v2 is worth 22 = 4 times as much as v4 , v3 is worth 21 = 2 times as much v4 , and v4 is equal to 20 = 1 times itself. In a 4-bit DAC v4 is the least significant bit (LSB) and v1 the most significant bit (MSB). To convert the digital representation of the signal to analog form, each bit must be weighted so that the analog output vO is vO = ±K(8v1 + 4v2 + 2v3 + 1v4 )

(32)

where K is a scale factor or gain applied to the analog signal. Equation (32) is the input–output relationship of a 4-bit digital-to-analog converter (DAC). One way to implement Eq. (32) is to use an inverting summer with binary-weighted input resistors. Figure 53 shows the op amp circuit and a block diagram of the circuit input–output relationship. In either form, the output is seen to be a binary-weighted sum of the digital input scaled by −RF /R. That is, the output voltage is −RF vO = (8v1 + 4v2 + 2v3 + v4 ) R The R –2R ladder in Fig. 54a also implements a 4-bit DAC. The resistance seen looking back into the R –2R ladder at point A with all sources turned off is seen to be RT = R. A Thevenin equivalent circuit of the R –2R network is shown in Fig. 54b, where

Fig. 54 An R–2R ladder DAC.1

The output voltage is found using the inverting amplifier gain relationship: vO =

v2 v3 v4 −RF −RF v1 vT = + + + R R 2 4 8 16

Using RF = 16R yields vO = −(8v1 + 4v2 + 2v3 + v4 )

vT =

1 2 v1

+

1 4 v2

+

1 8 v3

+

1 16 v4

Fig. 53 Binary-weighted summer DAC.1

which shows the binary weights assigned to the digital inputs. In theory the circuits in Figs. 53 and 54 perform the same signal-processing function—4-bit digital-toanalog conversion. However, there are important practical differences between the two circuits. The inverting summer in Fig. 53 requires precision resistors with four different values spanning an 8 : 1 range. A more common 8-bit converter would require eight precision resistors spanning a 256 : 1 range. Moreover, the digital voltage sources in Fig. 53 see input resistances that span an 8 : 1 range; therefore, the source–load interface is not the same for each bit. On the other hand, the resistances in the R –2R ladder converter in Fig. 54 span only a 2 : 1 range regardless of the number of digital bits. The R –2R ladder also presents the same input resistance to each binary input. The R –2R ladder converters are readily made on integrated or thin-film circuits and are the preferred DAC type. Instrumentation Systems One of the most interesting and useful applications of op amp circuits is in instrumentation systems that collect and process data about physical phenomena. In such a system an input transducer (a device that converts some physical quantity, such as temperature, strain, light intensity,

ELECTRIC CIRCUITS

903

acceleration, wavelength, rotation, velocity, pressure, or whatever, into an electrical signal) generates an electrical signal that describes some ongoing physical process. In a simple system the transducer signal is processed by op amp circuits and displayed on an output transducer such as a meter or an oscilloscope or more commonly sent into a DAC for further processing or analysis by a microprocessor or digital computer. The output signal can also be used in a feedback control system to monitor and regulate the physical process itself or to control a robotic device. The block diagram in Fig. 55 shows an instrumentation system in its simplest form. The objective of the system is to deliver an output signal that is directly proportional to the physical quantity measured by the input transducer. The input transducer converts a physical variable x into an electrical voltage vTR . For many transducers this voltage is of the form vTR = mx + b, where m is a calibration constant and b is a constant

offset or bias. The transducer voltage is often quite small and must be amplified by the gain K, as indicated in Fig. 55. The amplified signal includes both a signal component K(mx) and a bias component K(b). The amplified bias K(b) is then removed by subtracting a constant electrical signal. The resulting output voltage K(mx) is directly proportional to the quantity measured and goes to an output transducer for display. The required gain K can be found from the relation K=

desired output range available input range

(33)

Example 17. Design a light intensity detector to detect 5–20 lm of incident light using a photocell serving as the input transducer. The system output is to be displayed on a 0–10-V voltmeter. The photocell characteristics are shown in Fig. 56a. The design

Fig. 55 Block diagram of instrumentation system.2


904



requirements are that 5 lm indicates 0 V and 20 lm indicates 10 V on the voltmeter. Solution. From the transducer’s characteristics the light intensity range L = 20 − 5 = 15 lm will produce an available range of v = (0.6 m − 0.2 m) = 0.4 mV at the system input. This 0.4-mV change must be translated into a 0–10-V range at the system output. To accomplish this, the transducer voltage must be amplified by a gain of

K= =

desired output range available input range 10 − 0 = 2.5 × 104 0.6 × 10−3 − 0.2 × 10−3

When the transducer’s output voltage range (0.2– 0.6 mV) is multiplied by the gain K found above, we obtain a voltage range of 5–15 V. This range is shifted to the required 0–10-V range by subtracting the 5-V bias from the amplified signal. A block diagram of the required signal-processing functions is shown in Fig. 56b. A cascade connection of op amp circuits is used to realize the signal-processing functions in the block diagram. Figure 57 shows one possible design using an inverting amplifier and an inverting adder. This design includes two inverting circuits in cascade so the signal inversions cancel in the output signal. Part of the overall gain of K = 2.5 × 104 is realized in the inverting amplifier (K1 = −200) and the remainder by the inverting summer (K2 = −125). Dividing the overall gain between the two stages avoids trying to produce too large of a gain in a single stage. A singlestage gain of K = 25,000 is not practical since the closed-loop gain is not small compared to the openloop gain µ of most op amps. The high gain would also require a very low input resistance that could load the input and an uncommonly large feedback resistance, for example, 100 and 2.5 M. Example 18. A strain gauge is a resistive device that measures the elongation (strain) of a solid material caused by applied forces (stress). A typical strain gauge consists of a thin film of conducting material deposited on an insulating substrate. When bonded to

a member under stress, the resistance of the gauge changes by an amount R = 2RG

L L

where RG is the resistance of the gage with no applied stress and L/L is the elongation of the material expressed as a fraction of the unstressed length L. The change in resistance R is only a few tenths of a milliohm, far too little to be measured with an ohmmeter. To detect such a small change, the strain gage is placed in a Wheatstone bridge circuit like the one shown in Fig. 58. The bridge contains fixed resistors RA and RB , two matched strain gages RG1 and RG2 , and a precisely controlled reference voltage vREF . The values of RA and RB are chosen so that the bridge is balanced (v1 = v2 ) when no stress is applied. When stress is applied, the resistance of the stressed gage changes to RG2 + R and the bridge is unbalanced (v1 = v2 ). The differential signal (v2 − v1 ) indicates the strain resulting from the applied stress. Design an op amp circuit to translate strains on the range 0 < L/L < 0.02% into an output voltage on the range 0 < vO < 4 for RG = 120 and vREF = 25 V. Solution. With external stress applied, the resistance RG2 changes to RG2 + R. Applying voltage division to each leg of the bridge yields

v2 =

RG2 + R VREF RG1 + RG2

Fig. 58

v1 =

RB VREF RA + RB

(From Ref. 2.)

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The differential voltage (v = v2 − v1 ) can be written as

RA RG1 + R v = v2 − v1 = VREF − RG1 + RG2 RA + RB By selecting RG1 = RG2 = RA = RB = RG , a balanced bridge is achieved in the unstressed state, in which case the differential voltage reduces to v = v2 − v1 = VREF

R 2RG

= VREF

L L

Thus, the differential voltage v is directly proportional to the strain L/L. However, for VREF = 25 V and L/L = 0.02% the differential voltage is only (VREF )(L/L) = 25 × 0.0002 = 5 mV. To obtain the required 4-V output, a voltage gain of K = 4/0.005 = 800 is required. The op amp subtractor is specifically designed to amplify differential signals. Selecting R1 = R3 = 10 k and R2 = R4 = 8 M produces an input– output relationship for the subtractor circuit of vO = 800 (v2 − v1 ) Figure 59 shows the selected design. The input resistance of the subtractor circuit must be large to avoid loading the bridge circuit. The Thevenin resistance look-back into the bridge circuit is RT = RG1 RG2 + RA RB = RG RG + RG RG = RG = 120 which is small compared to 10-k input resistance of the subtractor’s inverting input. Comment. The transducer in this example is the resistor RG2 . In the unstressed state the voltage across

this resistor is v2 = 12.5 V. In the stressed state the voltage is v2 = 12.5 V plus a 5-mV signal. In other words, the transducer’s 5-mV signal component is accompanied by a very large bias. It is important to amplify the 12.5-V bias component by K = 800 before subtracting it out. The bias is eliminated at the input by using a bridge circuit in which v1 = 12.5 V and then processing the differential signal v2 − v1 . The situation illustrated in this example is actually quite common. Consequently, the first amplifier stage in most instrumentation systems is a differential amplifier. 4

AC CIRCUITS

4.1

Signals Electrical engineers normally think of a signal as an electrical current i(t), voltage v(t), or power p(t). In any case, the time variation of the signal is called a waveform. More formally, a waveform is an equation or graph that defines the signal as a function of time. Waveforms that are constant for all time are called dc signals. The abbreviation dc stands for direct current, but it applies to either voltage or current. Mathematical expressions for a dc voltage v (t) or current i (t) take the form

v(t) = V0

i(t) = I0

for − ∞ < t < ∞

Although no physical signal can remain constant forever, it is still a useful model, however, because it approximates the signals produced by physical devices such as batteries. In a circuit diagram signal variables are normally accompanied by reference marks (+, −, → or ←). It is important to remember that these reference marks do not indicate the polarity of a voltage or the direction of current. The marks provide a baseline for determining the sign of the numerical value of the actual waveform. When the actual voltage polarity or current direction coincides with the reference directions, the signal has a positive value. When the opposite occurs, the value is negative. Since there are infinitely many different signals, it may seem that the study of signals involves the uninviting task of compiling a lengthy catalog of waveforms. Most of the waveforms of interest can be addressed using just three basic signal models: the step, exponential, and sinusoidal functions. Step Waveform The first basic signal in our catalog is the step waveform. The general step function is based on the unit step function defined as

u(t) ≡

Fig. 59

(From Ref. 2.)

0 for t < 0 1 for t ≥ 0

Mathematically, the function u(t) has a jump discontinuity at t = 0. While it is impossible to generate a true step function since signal variables like current and voltage cannot transition from one value to

906


another in zero time, it is possible to generate very good approximations to the step function. What is required is that the transition time be short compared with other response times in the circuit. The step waveform is a versatile signal used to construct a wide range of useful waveforms. It often is necessary to turn things on at a time other then zero and with an amplitude different from unity. Replacing t by t − TS produces a waveform VA u(t − TS ) which takes on the values VA u(t − TS ) =

0 VA

for t < TS for t ≥ TS

occur at t = 3 sec. Putting these observations together, we express the rectangular pulse as v(t) = 3u (t − 1) − 3u (t − 3) Figure 61b shows how the two step functions combine to produce the given rectangular pulse. Impulse Function The generalization of Example 19 is the waveform

v(t) = VA [u(t − T1 ) − u(t − T2 )]

(34)

The amplitude VA scales the size of the step discontinuity and the time shift parameter TS advances or delays the time at which the step occurs. Amplitude and time shift parameters are required to define the general step function. The amplitude VA carries the units of volts. The amplitude of the step function in an electric current is IA and carries the units of amperes. The constant TS carries the units of time, usually seconds. The parameters VA (or IA ) and TS can be positive, negative, or zero, as shown in Fig. 60. Example 19. Express the waveform in Fig. 61a in terms of step functions. Solution. The amplitude of the pulse jumps to a value of 3 V at t = 1 s; therefore, 3u(t − 1) is part of the equation for the waveform. The pulse returns to zero at t = 3 sec, so an equal and opposite step must

(35)

This waveform is a rectangular pulse of amplitude VA that turns on at t = T1 and off at t = T2 . Pulses that turn on at some time T1 and off at some later time T2 are sometimes called gating functions because they are used in conjunction with electronic switches to enable or inhibit the passage of another signal. A rectangular pulse centered on t = 0 is written in terms of step functions as

T T 1 u t+ −u t − (36) v1 (t) = T 2 2 The pulse in Eq. (36) is zero everywhere except in the range −T /2 ≤ t ≤ T /2, where its amplitude is 1/T . The area under the pulse is 1 because its amplitude is inversely proportional to its duration. As shown in Fig. 62a, the pulse becomes narrower and higher as T decreases but maintains its unit area. In the limit as T → 0 the amplitude approaches infinity but the area remains unity. The function obtained in the limit is called a unit impulse, symbolized as δ(t). The graphical representation of δ(t) is shown in Fig. 62b. The impulse is an idealized model of a large-amplitude, short-duration pulse. A formal definition of the unit impulse is t δ(t) = 0

for t = 0

δ(x) dx = u(t)

and −∞

Fig. 60 Effect time shifting on step function waveform.1


(37)

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t u(t) =

δ(x) dx

δ(t) =

du(t) dt

u(x) dx

u(t) =

dr(t) dt

−∞

t r(t) = −∞

Fig. 62

Rectangular pulse waveforms and impulse.1

The first condition says the impulse is zero everywhere except at t = 0. The second condition implies that the impulse is the derivative of a step function although it cannot be justified using elementary mathematics since the function u(t) has a discontinuity at t = 0 and its derivative at that point does not exist in the usual sense. However, the concept can be justified using limiting conditions on continuous functions as discussed in texts on signals and systems. The strength of an impulse is defined by its area since amplitude is infinite. An impulse of strength K is denoted Kδ(t), where K is the area under the impulse. In the graphical representation of the impulse the value of K is written in parentheses beside the arrow, as shown in Fig. 62b. Ramp Function The unit ramp is defined as the integral of a step function: t u(x) dx = tu(t) (38) r(t) = −∞

These signals are used to generate other waveforms and as test inputs to linear systems to characterize their responses. When applying the singularity functions in circuit analysis, it is important to remember that u(t) is a dimensionless function. But δ(t) carries the units of reciprocal seconds and r(t) carries units of seconds. Here, δ (t) is called a doublet and is included for completeness. It is the derivative of an impulse function and caries the units of reciprocal seconds squared. Exponential Waveform The exponential signal is a step function whose amplitude gradually decays to zero. The equation for this waveform is

v(t) = VA e−t/TC u(t)

Singularity Functions The impulse, step, and ramp form a triad of related signals that are referred to as singularity functions. They are related by integration or by differentiation as

t −∞

δ (x) dx

δ (t) =

dδ(t) dt

(40)

A graph of v(t) versus t/TC is shown in Fig. 63. The exponential starts out like a step function. It is zero for t < 0 and jumps to a maximum amplitude of VA at t = 0. Thereafter it monotonically decays toward zero versus time. The two parameters that define the waveform are the amplitude VA (in volts) and the time constant TC (in seconds). The amplitude of a current exponential would be written IA and carry the units of amperes. The time constant is of special interest, since it determines the rate at which the waveform decays to zero. An exponential decays to about 37% of its initial amplitude v(0) = VA in one time constant because, at t = TC , v(TC ) = VA e−1 or approximately 0.368VA . At t = 5TC , the value of the waveform is VA e−5

The unit-ramp waveform r(t) is zero for T ≤ 0 and is equal to t for t > 0. The slope of r(t) is unity. The general ramp waveform is written Kr(t − TS ). The general ramp is zero for t ≤ TS and equal to K(t − TS ) for t > 0. The scale factor K defines the slope of the ramp for t > 0. By adding a series of ramps the triangular and sawtooth waveforms can be created.

δ(t) =

(39)

Fig. 63

Exponential waveform.

908


or approximately 0.00674VA . An exponential signal decays to less than 1% of its initial amplitude in a time span of five time constants. In theory an exponential endures forever, but practically speaking after about 5TC the waveform amplitude becomes negligibly small. For this reason we define the duration of an exponential waveform to be 5TC . Decrement Property of Exponential Waveforms The decrement property describes the decay rate of an exponential signal. For t > 0 the exponential waveform is given by v(t) = VA e−t/TC . At time t + t the amplitude is

v (t + t) = VA e−(t+t)/TC = VA e−t/TC e−t/TC The ratio of these two amplitudes is VA e−t/TC e−t/TC v (t + t) = = e−t/TC v(t) VA e−t/TC

(41)

The decrement ratio is independent of amplitude and time. In any fixed time period t, the fractional decrease depends only on the time constant. The decrement property states that the same percentage decay occurs in equal time intervals. Slope Property of Exponential Waveforms The slope of the exponential waveform (for t > 0) is found by differentiating Eq. (40) with respect to time:

VA v(t) dv(t) = − e−t/TC = − dt TC TC

Fig. 64

Effect of time shifting on sinusoidal waveform.1

(42)

The slope property states that the time rate of change of the exponential waveform is inversely proportional to the time constant. Small time constants lead to large slopes or rapid decays, while large time constants produce shallow slopes and long decay times. Sinusoidal Waveform The cosine and sine functions are important in all branches of science and engineering. The corresponding time-varying waveform in Fig. 64 plays an especially prominent role in electrical engineering. In contrast with the step and exponential waveforms studied earlier, the sinusoid extends indefinitely in time in both the positive and negative directions. The sinusoid in Fig. 64 is an endless repetition of identical oscillations between positive and negative peaks. The amplitude VA defines the maximum and minimum values of the oscillations. The period T0 is the time required to complete one cycle of the oscillation. Using these two parameters, a voltage sinusoid can be expressed as 2πt V (43) v(t) = VA cos T0

The waveform v(t) carries the units of VA (volts in this case) and the period T0 carries the units of time t (usually seconds). Equation (43) produces the waveform in Fig. 64 which has a positive peak at t = 0 since v(0) = VA . As in the case of the step and exponential functions, the general sinusoid is obtained by replacing t by t − TS . Inserting this change in Eq. (43) yields a general expression for the sinusoid as v(t) = VA cos

2π (t − TS ) T0

(44)

where the constant TS is the time shift parameter. The sinusoid shifts to the right when TS > 0 and to the left when TS < 0. In effect, time shifting causes the positive peak nearest the origin to occur at t = TS . The time-shifting parameter can also be represented by an angle:

2πt +φ (45) v(t) = VA cos T0 The parameter φ is called the phase angle. The term phase angle is based on the circular interpretation of the cosine function where the period is divided into

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2π radians or 350◦ . In this sense the phase angle is the angle between t = 0 and the nearest positive peak. The relation between TS and φ is φ = −2π

TS T0

(46)

An alternative form of the general sinusoid is obtained by expanding Eq. (45) using the identity cos(x + y) = cos(x) cos(y) − sin(x) sin(y). This results in the general sinusoid being written as 2πt 2πt + b sin (47) v(t) = a cos T0 T0 The two amplitude-like parameters a and b have the same units as the waveform (volts in this case) and are called Fourier coefficients. By definition the Fourier coefficients are related to the amplitude and phase parameters by the equations √ VA = a 2 + b 2 a = VA cos φ −b (48) b = −VA sin φ φ = tan−1 a It is customary to describe the time variation of the sinusoid in terms of a frequency parameter. Cyclic frequency f0 is defined as the number of periods per unit time. By definition the period T0 is the number of seconds per cycle; consequently the number of cycles per second is f0 =

1 T0

(49)

where f0 is the cyclic frequency or simply the frequency. The unit of frequency (cycles per second) is the hertz (Hz). Because there are 2π radians per cycle, the angular frequency ω0 in radians per second is related to cyclic frequency by the relationship ω0 = 2πfO =

2π T0

(50)

In summary, there are several equivalent ways to describe the general sinusoid:

 2π(t − TS ) 2πt   = V cos cos + φ V A  A  T0 T0      2πt 2πt   + b sin   = a cos T T0 0 v(t) =  VA cos[2πf0 (t − TS )] = VA cos(2πf0 t + φ)     = a cos(2πf0 t + φ) + b sin(2πf0 t + φ)      VA cos[ω0 (t − TS )] = VA cos(ω0 t + φ)    = a cos(ω0 t) + b sin(ω0 t) (51)

Additive Property of Sinusoids The additive property of sinusoids states that summing two or more sinusoids with the same frequency yields a sinusoid with different amplitude and phase parameters but the same frequency. Derivative and Integral Property of Sinusoids The derivative and integral properties of the sinusoid state that a sinusoid maintains its wave shape when differentiated or integrated. These operations change the amplitude and phase angle but do not change the basic sinusoidal wave shape or frequency. The fact that the wave shape is unchanged by differentiation and integration is a key property of the sinusoid. No other periodic waveform has this shape-preserving property. Waveform Partial Descriptors An equation or graph defines a waveform for all time. The value of a waveform v(t), i(t), or p(t) at time t is called the instantaneous value of the waveform. Engineers often use numerical values or terminology that characterizes a waveform but do not give a complete description. These waveform partial descriptors fall into two categories: (a) those that describe temporal features and (b) those that describe amplitude features. Temporal Descriptors Temporal descriptors identify waveform attributes relative to the time axis. A signal v(t) is periodic if v(t + T0 ) = v(t) for all t, where the period T0 is the smallest value that meets this condition. Signals that are not periodic are called aperiodic. The fact that a waveform is periodic provides important information about the signal but does not specify all of its characteristics. The period and periodicity of a waveform are partial descriptors. A sine wave, square wave, and triangular wave are all periodic. Examples of aperiodic waveforms are the step function, exponential, and damped sine. Waveforms that are identically zero prior to some specified time are said to be causal. A signal v(t) is casual if v(t)/0 for t < T ; otherwise it is noncausal. It is usually assumed that a causal signal is zero for t < 0, since time shifting can always place the starting point of a waveform at t = 0. Examples of causal waveforms are the step function, exponential, and damped sine. An infinitely repeating periodic waveform is noncausal. Causal waveforms play a central role in circuit analysis. When the input driving force x(t) is causal, the circuit response y(t) must also be causal. That is, a physically realizable circuit cannot anticipate and respond to an input before it is applied. Causality is an important temporal feature but only a partial description of the waveform. Amplitude Descriptors Amplitude descriptors are generally positive scalars that identify size features of the waveform. Generally a waveform’s amplitude varies between two extreme values denoted as VMAX

910


and VMIN . The peak-to-peak value (Vpp ) describes the total excursion of v(t) and is defined as Vpp = VMAX − VMIN

(52)

Under this definition Vpp is always positive even if VMAX and VMIN are both negative. The peak value (vp ) is the maximum of the absolute value of the waveform. That is, VP = max {|VMAX | , |VMIN |} (53) The peak value is a positive number that indicates the maximum absolute excursion of the waveform from zero. Figure 65 shows examples of these two amplitude descriptors. The average value (vavg ) smoothes things out to reveal the underlying waveform baseline. Average value is the area under the waveform over some period of time T divided by that time period: Vavg

1 = T

t0 +T

v(x) dx

(54)

t0

For periodic signals the averaging interval T equals the period T0 . The average value measures the waveform’s baseline with respect to the v = 0 axis. In other words, it indicates whether the waveform contains a constant, non-time-varying component. The average value is also called the dc component of the waveform because dc signals are constant for all t.

Root-Mean-Square Value The root-mean-square value (vrms ) of a waveform is a measure of the average power carried by the signal. The instantaneous power delivered to a resistor R by a voltage v(t) is

p(t) =

1 [v(t)]2 R

The average power delivered to the resistor in time span T is defined as 1 T

Pavg =

t0 +T

p(t) dt t0

Combining the above equations yields Pavg

1 = R

1 T

t0 +T

2

[v(t)] dt t0

The quantity inside the large brackets is the average value of the square of the waveform. The units of the bracketed term are volts squared. The square root of this term defines the amplitude descriptor vrms : Vrms =

1 T

t0 +T

[v(t)]2 dt

(55)

t0

For periodic signals the averaging interval is one cycle since such a waveform repeats itself every T0 seconds. The average power delivered to a resistor in terms of vrms is 1 2 (56) Pavg = Vrms R The equation for average power in terms of vrms has the same form as the instantaneous power. For this reason the rms value is also called the effective value, since it determines the average power delivered to a resistor in the same way that a dc waveform v(t) = vdc determines the instantaneous power. If the waveform amplitude is doubled, its rms value is doubled, and the average power is quadrupled. Commercial electrical power systems use transmission voltages in the range of several hundred kilovolts (rms) to transfer large blocks of electrical power.

Fig. 65

Peak value (Vp ) and peak-to-peak value (Vpp ).1

4.2 Energy Storage Devices Capacitor A capacitor is a dynamic element involving the time variation of an electric field produced by a voltage. Figure 66a shows the parallel-plate capacitor, which is the simplest physical form of a capacitive device, and two common circuit symbols for the capacitor are shown in Fig. 66b. Electrostatics shows that a uniform electric field E (t) exists between the metal plates when a voltage exists across the capacitor. The electric field produces

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applied. The capacitor is a dynamic element because the current is zero unless the voltage is changing. However, a discontinuous change in voltage requires an infinite current, which is physically impossible. Therefore, the capacitor voltage must be a continuous function of time. Equation (59a) is the integral form of the capacitor i–v relationship where x is a dummy integration variable: 1 t vC (t) = vC (0) + iC (x) dx (59a) C 0 With the passive-sign convention the power associated with the capacitor is Fig. 66 Capacitor: (a) parallel-plate device; (b) circuit symbol.1

charge separation with equal and opposite charges appearing on the capacitor plates. When the separation d is small compared with the dimension of the plates, the electric field between the plates is

where ε is the permittivity of the dielectric, A is the area of the plates, and q(t) is the magnitude of the electric charge on each plate. The relationship between the electric field and the voltage across the capacitor vC (t) is given by vC (t) E (t) = d Setting both equations equal and solving for the charge q(t) yields

εA vC (t) = CvC (t) (57) q(t) = d The proportionality constant inside the bracket in this equation is the capacitance C. The unit of capacitance is the farad (F), a term that honors the British physicist Michael Faraday. Values of capacitance range from picofarads (10−12 F) in semiconductor devices to tens of millifarads (10−3 F) in industrial capacitor banks. Differentiating Eq. (57) with respect to time t and realizing that iC (t) is the time derivative of q(t) result in the capacitor i–v relationship d [CvC (t)] dvC (t) dq(t) = =C dt dt dt

(58)

The time derivative in Eq. (58) means the current is zero when the voltage across the capacitor is constant, and vice versa. In other words, the capacitor acts like an open circuit (iC = 0) when dc excitations are

(59b)

This equation shows that the power can be either positive or negative because the capacitor voltage and its time rate of change can have opposite signs. The ability to deliver power implies that the capacitor can store energy. Assuming that zero energy is stored at t = 0, the capacitor energy is expressed as wC (t) = 12 CvC2 (t)

q(t) E (t) = εA

iC (t) =

pC (t) = iC (t) × vC (t)

(60)

The stored energy is never negative, since it is proportional to the square of the voltage. The capacitor absorbs power from the circuit when storing energy and returns previously stored energy when delivering power to the circuit. Inductor The inductor is a dynamic circuit element involving the time variation of the magnetic field produced by a current. Magnetostatics shows that a magnetic flux ϕ surrounds a wire carrying an electric current. When the wire is wound into a coil the lines of flux concentrate along the axis of the coil as shown in Fig. 67a. In a linear magnetic medium the flux is proportional to both the current and the number of turns in the coil. Therefore, the total flux is

φ(t) = k1 NiL (t) where k1 is a constant of proportionality involving the permeability of the physical surroundings and dimensions of the wire. The magnetic flux intercepts or links the turns of the coil. The flux linkages in a coil is represent by the symbol λ, with units of webers (Wb), named after the German scientist Wilhelm Weber (1804–1891). The number of flux linkages is proportional to the number of turns in the coil and to the total magnetic flux, so λ is given as λ(t) = Nφ (t) Substituting for φ(t) gives λ(t) = k1 N 2 iL (t) = LiL (t)

(61)

912


variable: iL (t) = iL (0) +

1 L

t

vL (x) dx

(63)

0

With the passive-sign convention the inductor power is (64) pL (t) = iL (t) × vL (t) This expression shows that power can be positive or negative because the inductor current and its time derivative can have opposite signs. The ability to deliver power indicates that the inductor can store energy. Assuming that zero energy is stored at t = 0, the inductor energy is expressed as wL (t) = 12 LiL2 (t)

(65)

The energy stored in an inductor is never negative because it is proportional to the square of the current. The inductor stores energy when absorbing power and returns previously stored energy when delivering power.

Fig. 67 (a) Magnetic flux surrounding current-carrying coil. (b) Circuit symbol for inductor.1

The k1 N 2 inside the brackets in this equation is called the inductance L of the coil. The unit of inductance is the henry (H) (plural henrys), a name that honors American scientist Joseph Henry. Figure 67b shows the circuit symbol for an inductor. Equation (61) is the inductor element constraint in terms of current and flux linkages. Differentiating Eq. (61) with respect to time t and realizing that according to Faraday’s law vL (t) is the time derivative of λ(t) result in the inductor i–v relationship vL (t) =

d [Li L (t)] dil (t) d [λ(t)] = =L dt dt dt

Equivalent Capacitance and Inductance Resistors connected in series or parallel can be replaced by equivalent resistances. The same principle applies to connections of capacitors and inductors. N capacitors connected in parallel can be replaced by a single capacitor equal to the sum of the capacitance of the parallel capacitors, that is,

CEQ = C1 + C2 + · · · + CN

(parallel connection) (66) The initial voltage, if any, on the equivalent capacitance is v(0), the common voltage across all of the original N capacitors at t = 0. Likewise, N capacitors connected in series can be replaced by a single capacitor equal to CEQ =

1 1/C1 + 1/C2 + · · · + 1/CN (series connection) (67)

(62)

The time derivative in Eq. (62) means that the voltage across the inductor is zero unless the current is time varying. Under dc excitation the current is constant and vL = 0 so the inductor acts like a short circuit. The inductor is a dynamic element because only a changing current produces a nonzero voltage. However, a discontinuous change in current produces an infinite voltage, which is physically impossible. Therefore, the current iL (t) must be a continuous function of time t. Equation (63) is the integral form of the inductor i–v relationship where x is a dummy integration

The equivalent capacitance of a parallel connection is the sum of the individual capacitances. The reciprocal of the equivalent capacitance of a series connection is the sum of the reciprocals of the individual capacitances. Since the capacitor and inductor are dual elements, the corresponding results for inductors are found by interchanging the series and parallel equivalence rules for the capacitor. That is, in a series connection the equivalent inductance is the sum of the individual inductances: LEQ = L1 + L2 + · · · + LN

(series connection) (68)

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For the parallel connection the reciprocals add to produce the reciprocal of the equivalent inductance: LEQ =

1 1/L1 + 1/L2 + · · · + 1/LN (parallel connection) (69)

Example 20. Find the equivalent capacitance and inductance of the circuit in Fig. 68a. Solution. The circuit contains both inductors and capacitors. The inductors and the capacitors are combined separately. The 5-pF capacitor in parallel with the 0.1-µF capacitor yields an equivalent capacitance of 0.100005 µF. For all practical purposes the 5-pF capacitor can be ignored, leaving two 0.1-µF capacitors in series with equivalent capacitance of 0.05 µF. Combining this equivalent capacitance in parallel with the remaining 0.05-µF capacitor yields an overall equivalent capacitance of 0.1 µF. The parallel 700- and 300-µH inductors yield an equivalent inductance of 1/(1/700 + 1/300) = 210 µH. This equivalent inductance is effectively in series with the 1-mH inductor at the bottom, yielding 1000 + 210 = 1210 µH as the overall equivalent inductance. Figure 68b shows the simplified equivalent circuit. Mutual Inductance The i–v characteristics of the inductor result from the magnetic field produced by current in a coil of wire. The magnetic flux spreads out around the coil forming closed loops that cut or link with the turns in the coil. If the current is changing, then Faraday’s law states that voltage across the coil is equal to the time rate of change of the total flux linkages. Now suppose that a second coil is brought close to the first coil. The flux from the first coil will link with the turns of the second coil. If the current in the first

Fig. 69 (a) Inductors separated, only self-inductance present. (b) Inductors coupled, both self- and mutual inductance present.1

coil is changing, then these flux linkages will generate a voltage in the second coil. The coupling between a changing current in one coil and a voltage across a second coil results in mutual inductance. If there is coupling between the two coils in Fig. 69, there are two distinct effects occurring in the coils. First there is the self-inductance due to the current flowing in each individual coil and the voltage induced by that current in that coil. Second, there are the voltages occurring in the second coil caused by current flowing through the first coil and vice versa. A double-subscript notation is used because it clearly identifies the various cause-and-effect relationships. The first subscript indicates the coil in which the effect takes place and the second identifies the coil in which the cause occurs. For example, v11 (t) is the voltage across coil 1 due to causes occurring in coil 1 itself, while v12 (t) is the voltage across coil 1 due to causes occurring in coil 2. The selfinductance is Coil 1:

Coil 2:

Fig. 68

(From Ref. 1.)

dλ11 (t) dφ1 (t) = N1 dt dt (t) di 1 = k1 N12 dt dλ22 (t) dφ2 (t) = N2 v22 (t) = dt dt (t) di 2 = k2 N22 dt v11 (t) =

(70)

Equations (70) provide the i–v relationships for the coils when there is no mutual coupling. The mutual

914


inductance is dλ12 (t) dφ12 (t) = N1 dt dt di2 (t) = [k12 N1 N2 ] dt dλ21 (t) dφ21 (t)) = N2 Coil 2: v21 (t) = dt dt di1 (t) (71) = [k21 N1 N2 ] dt Coil 1: v12 (t) =

The quantity ϕ12 (t) is the flux intercepting coil 1 due to the current in coil 2 and ϕ21 (t) is the flux intercepting coil 2 due to the current in coil 1. The expressions in Eq. (71) are the i–v relationships describing the cross coupling between coils when there is mutual coupling. When the magnetic medium supporting the fluxes is linear, the superposition principle applies, and the total voltage across the coils is the sum of the results in Eqs. (70) and (71): Coil 1: v1 (t) = v11 (t) + v12 (t) Coil 2: v2 (t) = v21 (t) + v22 (t) There are four inductance parameters in these equations. Two self-inductance parameters L1 = k1 N12 and L2 = k2 N22 and two mutual inductances M12 = k12 N1 N2 and M21 = k21 N2 N1 . In a linear magnetic medium k12 = k21 = kM , there is a single mutual inductance parameter M defined as M = M12 = M21 = kM N1 N2 . Putting these all together yields di2 (t) di1 (t) ±M dt dt di1 (t) di2 (t) + L2 Coil 2: v2 (t) = ±M dt dt Coil 1: v1 (t) = L1

(72)

The coupling across coils can be additive or subtractive. This gives rise to the ± sign in front of the mutual inductance M. Additive (+) coupling means that a positive rate of change of current in coil 2 induces a positive voltage in coil 1, and vice versa for subtractive coupling (−). When applying these element equations, it is necessary to know when to use a plus sign and when to use a minus sign. Since the additive or subtractive nature of a coupled-coil set is predetermined by the manufacturer of the windings, a dot convention is used. The dots shown near one terminal of each coil are special reference marks indicating the relative orientation of the coils. Figure 70 shows the dot convention. The correct sign for the mutual inductance term hinges on how the reference marks for currents and voltages are assigned relative to the coil dots: Mutual

Fig. 70 Winding orientations and corresponding reference dots: (a) additive; (b) subtractive.

inductance is additive when both current reference directions point toward or both point away from dotted terminals; otherwise, it is subtractive. Ideal Transformer A transformer is an electrical device that utilizes mutual inductance coupling between two coils. Transformers find application in virtually every type of electrical system, especially in power supplies and commercial power grids. In Fig. 71 the transformer is shown as an interface device between a source and a load. The coil connected to the source is called the primary winding and the coil connected to the load the secondary winding. In most applications the transformer is a coupling device that transfers signals (especially power) from the source to the load. The basic purpose of the device is to change voltage and current levels so the signal conditions at the source and load are compatible. Transformer design involves two primary goals: (a) to maximize the magnetic coupling between the two windings and (b) to minimize the power loss in the windings. The first goal produces near-perfect coupling (k ∼ = 1) so that almost all of the flux in one winding

Fig. 71 Transformer interface.1

connected

at

source–load

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links the other. The second goal produces nearly zero power loss so that almost all of the power delivered to the primary winding transfers to the load. The ideal transformer is a circuit element in which coupled coils are assumed to have perfect coupling and zero power loss. Perfect coupling assumes that all the coupling coefficients are equal to each other, that is, k11 = k22 = k12 = k21 = kM ∼ = 1. Dividing the two equations in Eq. (72) and using the concept of perfect coupling result in the equation N2 v2 (t) =± = ±n v1 (t) N1

(73)

where n is the turns ratio. With perfect coupling the secondary voltage is proportional to the primary voltage so they have the same wave shape. For example, when the primary voltage is v1 (t) = VA sin ωt, the secondary voltage is v2 (t) = ±nVA sin ωt. When the turns ratio n > 1, the secondary-voltage amplitude is larger than the primary and the device is called a step-up transformer. Conversely, when n < 1, the secondary voltage is smaller than the primary and the device is called a step-down transformer. The ability to increase or decrease ac voltage levels is a basic feature of transformers. Commercial power systems use transmission voltages of several hundred kilovolts. For residential applications the transmission voltage is reduced to safer levels (typically 220/110 Vrms ) using step-down transformers. The ± sign in Eq. (73) depends on the reference marks given the primary and secondary currents relative to the dots indicating the relative coil orientations. The rule for the ideal transformer is a corollary of the rule for selecting the sign of the mutual inductance term in coupled-coil element equations. The ideal transformer model also assumes that there is no power loss in the transformer. With the passivesign convention, the power in the primary winding and secondary windings is v1 (t)i1 (t) and v2 (t)i2 (t), respectively. Zero power loss requires

The correct sign in this equation depends on the orientation of the current reference directions relative to the dots describing the transformer structure. With both perfect coupling and zero power loss, the secondary current is inversely proportional to the turns ratio. A step-up transformer (n > 1) increases the voltage and decreases the current, which improves transmission line efficiency because the i 2 R losses in the conductors are smaller. Using the ideal transformer model requires some caution. The relationships in Eqs. (73) and (74) state that the secondary signals are proportional to the primary signals. These element equations appear to apply to dc signals. This is of course wrong. The element equations are an idealization of mutual inductance, and mutual inductance requires time-varying signals to provide the coupling between two coils. Equivalent Input Resistance Because a transformer changes the voltage and current levels, it effectively changes the load resistance seen by a source in the primary circuit. Consider the circuit shown in Fig. 72. The device equations are

Resistor: v2 (t) = RL iL (t) Transformer: v2 (t) = nv1 (t) 1 i2 (t) = − i1 (t) n Dividing the first transformer equation by the second and inserting the load resistance constraint yield iL (t)RL v1 (t) v2 (t) = = −n2 i2 (t) i2 (t) i1 (t) Applying KCL at the output interface tells us iL (t) = −i2 (t). Therefore, the equivalent resistance seen on the primary side is REQ =

1 v1 (t) = 2 RL i1 (t) n

(75)

The equivalent load resistance seen on the primary side depends on the turns ratio and the load resistance.

v1 (t)i1 (t) + v2 (t)i2 (t) = 0 which can be rearranged in the form v1 (t) i2 (t) =− i1 (t) v2 (t) But under the perfect-coupling assumption v2 (t)/v1 (t) = ±n. With zero power loss and perfect coupling the primary and secondary currents are related as 1 i2 (t) =∓ i1 (t) n

(74)

Fig. 72

Equivalent resistance seen in primary winding.1

916


Adjusting the turns ratio can make REQ equal to the source resistance. Transformer coupling can produce the resistance match condition for maximum power transfer when the source and load resistances are not equal. 4.3 Phasor Analysis of Alternating Current Circuits Those ac circuits that are excited by a single frequency, for example, power systems, can be easily and effectively analyzed using sinusoidal steady-state techniques. Such a technique was first proposed by Charles Steinmetz (1865–1923) using a vector representation of sinusoids called phasors. Sinusoids and Phasors The phasor concept is the foundation for the analysis of linear circuits in the sinusoidal steady state. Simply put, a phasor is a complex number representing the amplitude and phase angle of a sinusoidal voltage or current. The connection between sine waves and complex numbers is provided by Euler’s relationship:

ej θ = cos θ + j sin θ To develop the phasor concept, it is necessary to adopt the point of view that the cosine and sine functions can be written in the form cos θ = Re{ej θ }

and

sin θ = Im{ej θ }

where Re stands for the “real part of” and Im for the “imaginary part of.” Development of the phasor concept begins with reference of phasors to the cosine function as v(t) = VA cos(ωt + φ) = VA Re ej (ωt+φ) (76) = VA Re ej ωt ej φ = Re VA ej φ ej ωt Moving the amplitude VA inside the real-part operation does not change the final result because it is real constant. By definition, the quantity VA ej ϕ in Eq. (76) is the phasor representation of the sinusoid v(t). The phasor V—a boldface V —or sometimes written with a tilde above the variable, V˜ , can be represented in polar or rectangular form as V=

V ej ϕ = VA (cos φ + j sin φ) A polar form rectangular form

(77)

Note that V is a complex number determined by the amplitude and phase angle of the sinusoid. Figure 73 shows a graphical representation commonly called a phasor diagram. An alternative way to write the polar form of a phasor is to replace the exponential ej ϕ by the shorthand notation ∠ϕ, that is, V = VA ∠ϕ,

Fig. 73

Complex exponential Vejωt .1

which is equivalent to the polar form in Eq. (77). It is important to realize that a phasor is determined by its amplitude and phase angle and does not contain any information about the frequency of the sinusoid. The first feature points out that signals can be described in different ways. Although the phasor V and waveform v(t) are related concepts, they have quite different physical interpretations and one must clearly distinguish between them. The absence of frequency information in the phasors results from the fact that in the sinusoidal steady state all currents and voltages are sinusoids with the same frequency. Carrying frequency information in the phasor would be redundant, since it is the same for all phasors in any given steady-state circuit problem. In summary, given a sinusoidal waveform v(t) = VA cos(ωt + ϕ), the corresponding phasor representation is V = VA ej ϕ . Conversely, given the phasor V = VA ej ϕ , the corresponding sinusoid waveform is found by multiplying the phasor by ej ωt and reversing the steps in Eq. (76) as follows: v(t) = Re Vej ωt = Re VA ej φ ej ωt = VA Re ej (ωt+φ) = VA cos (ωt + φ) The frequency ω in the complex exponential Vej ωt in Eq. (76) must be expressed or implied in a problem statement, since by definition it is not contained in the phasor. Figure 73 shows a geometric interpretation of the complex exponential Vej ωt as a vector in the complex plane of length vA , which rotates counterclockwise with a constant angular velocity ω. The real-part operation projects the rotating vector onto the horizontal (real) axis and thereby generates v(t) = VA cos (ωt + ϕ). The complex exponential is sometimes called a rotating phasor, and the phasor V is viewed as a snapshot of the situation at t = 0. Properties of Phasors Phasors have two properties. The additive property states that the phasor representing a sum of sinusoids of the same frequency is

ELECTRIC CIRCUITS

917

obtained by adding the phasor representations of the component sinusoids. To establish this property, we write the expression

Resistor:

v(t) = v1 (t) + v2 (t) + · · · + vN (t) v(t) = Re{V1 ej ωt } + Re{V2 ej ωt } + · · · + Re{VN ej ωt } (78) where v1 (t), v2 (t), . . . , vN (t) are sinusoids of the same frequency whose phasor representations are V1 , V2 , . . . , VN . The real-part operation is additive, so the sum of real parts equals the real part of the sum. Consequently, Eq. (78) can be written in the form v(t) = Re{V1 ej ωt + V2 ej ωt + · · · + VN ej ωt } = Re{(V1 + V2 + · · · + VN )ej ωt }

(79)

(80)

The result in Eq. (80) applies only if the component sinusoids all have the same frequency so that ej ωt can be factored out as shown in the last line in Eq. (79). The derivative property of phasors allows us to easily relate the phasor representing a sinusoid to the phasor representing its derivative. Differentiating Eq. (76) with respect to time t yields d d dv(t) = Re{Vej ωt } = Re V ej ωt dt dt dt = Re{(jωV)ej ωt }

Inductor: Capacitor:

vR (t) = RiR (t) diL (t) vL (t) = L dt dvC (t) iC (t) = C dt

(82)

Now in the sinusoidal steady state all of these currents and voltages are sinusoids. In the sinusoidal steady state the voltage and current of the resistor can be written in terms of phasors as vR (t) = Re{VR ej ωt } and iR (t) = Re{IR ej ωt }. Consequently, the resistor i–v relationship in Eq. (82) can be expressed in terms of phasors as follows: Re VR ej ωt = R × Re IR ej ωt

Hence the phasor V representing v(t) is V = V1 + V2 + · · · + VN

Device Constraints in Phasor Form The device constraints of the three passive elements are

Moving R inside the real-part operation on the right side of this equation does not change things because R is a real constant: Re VR ej ωt = Re RIR ej ωt This relationship holds only if the phasor voltage and current for a resistor are related as VR = RIR

(81)

From the definition of a phasor we see that the quantity j ωV on the right side of this equation is the phasor representation of the time derivative of the sinusoidal waveform. In summary, the additive property states that adding phasors is equivalent to adding sinusoidal waveforms of the same frequency. The derivative property states that multiplying a phasor by j ω is equivalent to differentiating the corresponding sinusoidal waveform. Phasor Circuit Analysis Phasor circuit analysis is a method of finding sinusoidal steady-state responses directly from the circuit without using differential equations. Connection Constraints in Phasor Form Kirchhoff’s laws in phasor form are as follows:

KVL: The algebraic sum of phasor voltages around a loop is zero. KCL: The algebraic sum of phasor currents at a node is zero.

(83a)

If the current through a resistor is iR (t) = IA cos(ωt + ϕ). Then the phasor current is IR = IA ej ϕ and, according to Eq. (83a), the phasor voltage across the resistor is VR = RIA ej φ (83b) This result shows that the voltage has the same phase angle (ϕ) as the current. Phasors with the same phase angle are said to be in phase; otherwise they are said to be out of phase. In the sinusoidal steady state the voltage and phasor current for the inductor can be written in terms of phasors as vL (t) = Re{VL ej ωt } and iL (t) = Re{IL ej ωt }. Using the derivative property of phasors, the inductor i–v relationship can be expressed as Re{VL ej ωt } = L × Re{jωIL ej ωt } = Re{j ωLIL ej ωt }

(84)

Moving the real constant L inside the real-part operation does not change things, leading to the conclusion that phasor voltage and current for an inductor are related as (85) VL = j ωLIL

918


When the current is iL (t) = iA cos(ωt + ϕ), the corresponding phasor is IL = IA ej ϕ and the i–v constraint in Eq. (85) yields ◦

VL = j ωLIL = (ωLej 90 )(IA ej φ ) ◦

= ωLIA ej (φ+90 ) The resulting phasor diagram in Fig. 74 shows that the inductor voltage and current are 90◦ out of phase. The voltage phasor is advanced by 90◦ counterclockwise, which is in the direction of rotation of the complex exponential eωt . When the voltage phasor is advanced counter clockwise, that is, ahead of the rotating current phasor, the voltage phasor leads the current phasor by 90◦ or equivalently the current lags the voltage by 90◦ . Finally, the capacitor voltage and current in the sinusoidal steady state can be written in terms of phasors as vC (t) = Re{VC ej ωt } and iC (t) = Re{IC ej ωt }. Using the derivative property of phasors, the i–v relationship of the capacitor becomes Re IC ejωt = C × Re j ωVC ej ωt (86) = Re j ωCVC ej ωt Moving the real constant C inside the real-part operation does not change the final results, so we conclude that the phasor voltage and current for a capacitor are related as IC = j ωCVC

or

VC =

1 IC j ωC

(87)

When iC (t) = IA cos(ωt + ϕ), then Eq. (87) the phasor voltage across the capacitor is 1 1 −j 90◦ IC = e (IA ej φ ) VC = j ωC ωC IA j (φ−90◦ ) e = ωC

Fig. 75 Phasor i–v characteristics of capacitor.1

The resulting phasor diagram in Fig. 75 shows that voltage and current are 90◦ out of phase. In this case the voltage phasor is retarded by 90◦ clockwise, which is in a direction opposite to the rotation of the complex exponential ej ωt . When the voltage is retarded clockwise, that is, behind the rotating current phasor, we say the voltage phasor lags the current phasor by 90◦ or equivalently the current leads the voltage by 90◦ . Impedance Concept The I–V constraints in Eqs. (83a), (85), and (87) are all of the form

V = ZI

(88)

where Z is called the impedance of the element. Equation (88) is analogous to Ohm’s law in resistive circuits. Impedance is the proportionality constant relating phasor voltage and phasor current in linear, two-terminal elements. The impedances of the three passive elements are ZR = R ZL = j ωL j 1 =− Capacitor: ZC = j ωC ωC

Resistor: Inductor:

(89)

Since impedance relates phasor voltage to phasor current, it is a complex quantity whose units are ohms. Although impedance can be a complex number, it is not a phasor. Phasors represent sinusoidal signals while impedances characterize circuit elements in the sinusoidal steady state.

Fig. 74

Phasor i–v characteristics of inductor.1

Basic Circuit Analysis in Phasor Domain The phasor constraints have the same format as the constraints for resistance circuits; therefore, familiar tools such as series and parallel equivalence, voltage and current division, proportionality and superposition, and Thevenin and Norton equivalent circuits are applicable to phasor circuit analysis. The major difference is that the circuit responses are complex numbers (phasors) and not waveforms.

ELECTRIC CIRCUITS

919

Series Equivalence and Voltage Division Consider a simple series circuit with several impedances connected to a phasor voltage. The same phasor responses V and I exist when the series-connected elements are replaced by equivalent impedance ZEQ :

(c) Solve for the phasor voltage across each element. (d) Find the waveforms corresponding to the phasors found in (b) and (c). Solution

ZEQ

V = Z1 + Z2 + · · · + ZN = I

In general, the equivalent impedance ZEQ is a complex quantity of the form ZEQ = R + j X

ZR = R = 50 ZL = j ωL = j 1000 × 25 × 10−3 = j 25

where R is the real part and X is the imaginary part. The real part of Z is called resistance and the imaginary part (X, not jX ) is called reactance. Both resistance and reactance are expressed in ohms. For passive circuits resistance is always positive while reactance X can be either positive or negative. A positive X is called an inductive reactance because the reactance of an inductor is ωL, which is always positive. A negative X is called a capacitive reactance because the reactance of a capacitor is −1/ωC, which is always negative. The phasor voltage across the kth element in the series connection is Zk V Vk = Z k I k = ZEQ

(a) The phasor representing the input source voltage is VS = 35∠0◦ . The impedances of the three passive elements are

(90)

Equation (90) is the phasor version of the voltage division principle. The phasor voltage across any element in a series connection equals the ratio of its impedance to the equivalent impedance of the connection times the total phasor voltage across the connection. Example 21. The circuit in Fig. 76a is operating in the sinusoidal steady state with vS (t) = 35 cos 1000t volts.

(a) Transform the circuit into the phasor domain. (b) Solve for the phasor current I.

ZC =

1 1 = = −j 100 j ωC j 1000 × 10−5

Using these, results we obtain the phasor domain circuit in Fig. 76b. (b) The equivalent impedance of the series connection is ZEQ = 50 + j 25 − j 100 = 50 − j 75 ◦

= 90.1∠ − 56.3 The current in the series circuit is I=

VS 35∠0◦ ◦ = = 0.388∠56.3 A ZEQ 90.1∠ − 56.3◦

(c) The current I exists in all three series elements so the voltage across each passive element is VR = ZR I = 50 × 0.388∠56.3

◦

◦

= 19.4∠56.3 V VL = ZL I = j 25 × 0.388∠56.3

◦

◦

= 9.70∠146.3 V VC = ZC I = −j 100 × 0.388∠56.3

◦

◦

= 38.8∠ − 33.7 V (d) The sinusoidal steady-state waveforms corresponding to the phasors in (b) and (c) are ◦

i(t) = Re{0.388ej 56.3 ej 1000t } ◦

= 0.388 cos(1000t + 56.3 ) A ◦

vR (t) = Re{19.4ej 56.3 ej 1000t } ◦

= 19.4 cos(1000t + 56.3 ) V ◦

vL (t) = Re{9.70ej 146.3 ej 1000t } Fig. 76

(From Ref. 1.)

◦

= 9.70 cos(1000t + 146.3 ) V

920

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS ◦

vC (t) = Re{38.8ej −33.7 ej 1000t } ◦

The equivalent admittance of the parallel connection is

= 38.8 cos(1000t − 33.7 ) V Parallel Equivalence and Current Division Consider a number of impedances connected in parallel so the same phasor voltage V appears across them. The same phasor responses V and I exist when the parallel-connected elements are replaced by equivalent impedance ZEQ :

YEQ = Y1 + Y2 = 4 × 10−4 + j 12 × 10−4 ◦

= 12.6 × 10−4 ∠71.6 S and the voltage across the parallel circuit is V=

1 I 1 1 1 = = + + ··· + ZEQ V Z1 Z2 ZN These results can also be written in terms of admittance Y , which is defined as the reciprocal of impedance: Y =

1 = G + jB Z

The real part of Y is called conductance and the imaginary part B is called susceptance, both of which are expressed in units of siemens. The phasor current through the kth element of the parallel connection is Yk I (91) I k = Y k Vk = YEQ Equation (91) is the phasor version of the current division principle. The phasor current through any element in a parallel connection equals the ratio of its admittance to the equivalent admittance of the connection times the total phasor current entering the connection. Example 22. For the circuit in Fig. 77 solve for the phasor voltage V and for the phasor current through each branch. Solution

(a) The admittances of the two parallel branches are Y1 =

1 = j 2 × 10−3 S −j 500

Y2 =

1 = 4 × 10−4 − j 8 × 10−4 S 500 + j 1000


0.05∠0◦ IS = YEQ 12.6 × 10−4 ∠71.6◦ ◦

= 39.7∠ − 71.6 V (b) The current through each parallel branch is I1 = Y1 V = j 2 × 10−3 × 39.7∠ − 71.6

◦

◦

= 79.4∠18.4 mA I2 = Y2 V = (4 × 10−4 − j 8 × 10−4 ) ◦

◦

× 39.7∠ − 71.6 = 35.5∠ − 135 mA Y– Transformations In section 1.2 in the discussion of equivalent circuits the equivalence of - and Y -connected resistors to simplify resistance circuits with no series- or parallel-connected branches was covered. The same basic concept applies to the - and Y -connected impedances (see Fig. 78). The equations for the –Y transformation are

Z1 =

ZB ZC ZA + ZB + ZC

Z3 =

ZA ZB ZA + ZB + ZC

Z2 =

ZC ZA ZA + ZB + ZC (92)

The equations for a Y– transformation are ZA =

Z1 Z2 + Z2 Z3 + Z1 Z3 Z1

ZB =

Z1 Z2 + Z2 Z3 + Z1 Z3 Z2

ZC =

Z1 Z2 + Z2 Z3 + Z1 Z3 Z3

Fig. 78

Y– impedance transformation.2

(93)

ELECTRIC CIRCUITS

921

The equations have the same form except that here they involve impedances rather than resistances. Example 23.

Find the phasor current IIN in Fig. 79a.

Solution. One cannot use basic reduction tools on the circuit because no elements are connected in series or parallel. However, by replacing either the upper (A, B, C) or lower (A, B, D) by an equivalent Y subcircuit, series and parallel reduction methods can be applied. Choosing the upper because it has two equal resistors simplifies the transformation equations. The sum of the impedance in the upper is 100 + j 200. This sum is the denominator in the expression in –Y transformation equations. The three Y impedances are found to be

Z1 =

(50)(j 200) = 40 + j 20 100 + j 200

Z2 =

(50)(j 200) = 40 + j 20 100 + j 200

Z3 =

(50)(50) = 5 − j 10 100 + j 200

Figure 79b shows the revised circuit with the equivalent Y inserted in place of the upper . Note that the transformation introduces a new node labeled N. The revised circuit can be reduced by series and parallel equivalence. The total impedance of the path NAD is 40 −j 100 . The total impedance of the path NBD is 100 + j 20 . These paths are connected in parallel so

the equivalent impedance between nodes N and D is ZND =

1 1/(40 − j 100) + 1/(100 + j 20)

= 60.6 − j 31.1 The impedance ZN D is connected in series with the remaining leg of the equivalent Y , so the equivalent impedance seen by the voltage source is ZEQ = 5 − j 10 + ZN D = 65.6 − j 41.1 The input current then is IIN =

VS 75∠0◦ = 0.891 + j 0.514 = ZEQ 65.6 − j 41.1 ◦

= 968∠32.0 mA Circuit Theorems in Phasor Domain Phasor analysis does not alter the linearity properties of circuits. Hence all of the theorems that are applied to resistive circuits can be applied to phasor analysis. These include proportionality, superposition, and Thevenin and Norton equivalence. Proportionality The proportionality property states that phasor output responses are proportional to the input phasor. Mathematically proportionality means that Y = KX, where X is the input phasor, Y the output phasor, and K the proportionality constant. In phasor circuit analysis the proportionality constant is generally a complex number. Superposition Care needs to be taken when applying superposition to phasor circuits. If the sources all have the same frequency, then one can transform the circuit into the phasor domain (impedances and phasors) and proceed as in dc circuits with the superposition theorem. If the sources have different frequencies, then superposition can still be used but its application is different. With different frequency sources each source must be treated in a separate steady-state analysis because the element impedances change with frequency. The phasor response for each source must be changed into waveforms and then superposition applied in the time domain. In other words, the superposition principle always applies in the time domain. It also applies in the phasor domain when all independent sources have the same frequency. The following example illustrates the latter case.

Fig. 79

(From Ref. 2.)

Example 24. Use superposition to find the steadystate current i(t) in Fig. 80 for R = 10 k, L = 200 mH, vS1 = 24 cos 20,000t V, and vS2 = 8 cos(60,000t + 30◦ ) V.

922



Solution. In this example the two sources operate at different frequencies. With source 2 off, the input phasor is VS1 = 24∠0◦ V at a frequency ω = 20 krad/sec. At this frequency the equivalent impedance of the inductor and resistor is

ZEQ1 = R + j ωL = (10 + j 4) k The phasor current due to source 1 is I1 =

VS1 24∠0◦ ◦ = 2.23∠ − 21.8 mA = ZEQ1 10,000 + j 4000

With source 1 off and source 2 on, the input phasor VS2 = 8∠30◦ V at a frequency ω = 60 krad/sec. At this frequency the equivalent impedance of the inductor and resistor is

elements and sources can be replaced by Thevenin or Norton equivalent circuits. The general concept of Thevenin’s and Norton’s theorems and their restrictions are the same as in the resistive circuit studied earlier. The important difference here is that the signals VT , IN , V, and I are phasors and ZT = 1/YN and ZL are complex numbers representing the source and load impedances. Thevenin equivalent circuits are useful to address the maximum power transfer problem. Consider the source–load interface as shown in Fig. 81. The source circuit is represented by a Thevenin equivalent circuit with source voltage VT and source impedance ZT = RT + j XT . The load circuit is represented by an equivalent impedance ZL = RL + j XL . In the maximumpower-transfer problem the source parameters VT , RT , and XT are given, and the objective is to adjust the load impedance RL and XL so that average power to the load is a maximum. The average power to the load is expressed in terms of the phasor current and load resistance: P = 12 RL |I|2 Then, using series equivalence, the magnitude of the interface current is VT |VT | = |I| = ZT + ZL |(RT + RL ) + j (XT + XL )| =

|VT | (RT + RL )2 + (XT + XL )2

Combining the last two equations yields the average power delivered across the interface:

ZEQ2 = R + j ωL = (10 + j 12) k P =

The phasor current due to source 2 is I2 =

VS2 8∠30◦ = ZEQ2 10,000 + j 12,000 ◦

= 0.512∠ − 20.2 mA The two input sources operate at different frequencies so the phasors responses I1 and I2 cannot be added to obtain the overall response. In this case the overall response is obtained by adding the corresponding time domain waveforms:

1 RL |VT |2 2 (RT + RL )2 + (XT + XL )2

Since the quantities |VT |, RT , and XT are fixed, P will be maximized when XL = −XT . This choice of XL always is possible because a reactance can be positive or negative. When the source Thevenin equivalent has an inductive reactance (XT > 0), the load is selected

i(t) = Re{I1 ej 20,000t } + Re{I2 ej 60,000t } ◦

i(t) = 2.23 cos(20,000t − 21.8 ) ◦

+ 0.512 cos(60,000t − 20.2 ) mA Thevenin and Norton Equivalent Circuits In the phasor domain a two-terminal circuit containing linear

Fig. 81 state.1

Source–load interface in the sinusoidal steady

ELECTRIC CIRCUITS

923

to have a capacitive reactance of the same magnitude and vice versa. This step reduces the net reactance of the series connection to zero, creating a condition in which the net impedance seen by the Thevenin voltage source is purely resistive. In summary, to obtain maximum power transfer in the sinusoidal steady state, we select the load resistance and reactance so that RL = RT and XL = −XT . The condition for maximum power transfer is called a conjugate match, since the load impedance is the conjugate of the source impedance ZL = ZT∗ . Under conjugate-match conditions the maximum average power available from the source circuit is |VT |2 PMAX = 8RT where |VT | is the peak amplitude of the Thevenin equivalent voltage. It is important to remember that conjugate matching applies when the source is fixed and the load is adjustable. These conditions arise frequently in powerlimited communication systems. However, conjugate matching does not apply to electrical power systems because the power transfer constraints are different. Node Voltage and Mesh Current Analysis in Phasor Domain The previous sections discuss basic analysis methods based on equivalence, reduction, and circuit theorems. These methods are valuable because they work directly with element impedances and thereby allow insight into steady-state circuit behavior. However, node and mesh analysis allows for solution of more complicated circuits than the basic methods can easily handle. There general methods use node voltage or mesh current variables to reduce the number of equations that must be solved simultaneously. These solution approaches are identical to those in resistive circuits except that phasors are used for signals and impedances in lieu of only resistors. The following are examples of node voltage and mesh current problems. Example 25. Use node analysis to find the node voltages VA and VB in Fig. 82a. Solution. The voltage source is connected in series with an impedance consisting of a resistor and inductor connected in parallel. The equivalent impedance of this parallel combination is

ZEQ =


Figure 82b shows the circuit produced by the source transformation. The node voltage equation at the remaining nonreference node in Fig. 82b is 1 1 1 ◦ + + VA = 0.1∠0 − (−0.1 − j 0.2) −j 50 j 100 50 Solving for VA yields VA =

0.2 + j 0.2 ◦ = 12 + j 4 = 12.6∠18.4 V 0.02 + j 0.01

Referring to Fig. 82a, KVL requires VB = VA + 10∠ − 90◦ . Therefore, VB is found to be ◦

VB = (12 + j 4) + 10∠ − 90 = 12 − j 6 ◦

= 13.4∠ − 26.6 V Example 26. The circuit in Fig. 83 is an equivalent circuit of an ac induction motor. The current IS is called the stator current, IR the rotor current, and IM the magnetizing current. Use the mesh current method to solve for the branch currents IS , IR , and IM .

1 = 40 + j 20 1/50 + 1/(j 100)

Applying a source transformation produces an equivalent current source of IEQ =

10∠ − 90◦ = −0.1 − j 0.2 A 40 + j 20


924


Solution. Applying KVL to the sum of voltages around each mesh yields

Mesh A:

◦

− 360∠0 + [0.1 + j 0.4]IA + j 10[IA − IB ] = 0

Mesh B:

j 10[IB − IA ] + [4 + j 0.4]IB = 0

Solving these equations for IA and IB produces IA = 79.0 − j 48.2 A

IB = 81.7 − j 14.9 A

The required stator, rotor, and magnetizing currents are related to these mesh currents as follows: ◦

IS = IA = 92.5∠31.4 A ◦

IR = −IB = −81.8 + j 14.9 = 83.0∠170 A ◦

IM = IA − IB = −2.68 − j 33.3 = 33.4∠ − 94.6 A 4.4 Power in Sinusoidal Steady State Average and Reactive Power In power applications it is normal to think of one circuit as the source and the other as the load. It is important to describe the flow of power across the interface between source and load when the circuit is operating in the sinusoidal steady state. The interface voltage and current in the time domain are sinusoids of the form

v(t) = VA cos(ωt + θ )

i(t) = IA cos ωt

where vA and iA are real, positive numbers representing the peak amplitudes of the voltage and current, respectively. The forms of v(t) and i(t) above are completely general. The positive maximum of the current i(t) occurs at t = 0 whereas v(t) contains a phase angle θ to account for the fact that the voltage maximum may not occur at the same time as the current’s. In the phasor domain the angle θ = ϕV − ϕI is the angle between the phasors V = VA ∠ϕV and I = iA ∠ϕI . In effect, choosing t = 0 at the current maximum shifts the phase reference by an amount −ϕI so that the voltage and current phasors become V = vA ∠θ and I = IA ∠0◦ . The instantaneous power in the time domain is p(t) = v(t) × i(t) = VA IA cos(ωt + θ ) cos ωtW This expression for instantaneous power contains both dc and ac components. Using the identities cos2 x = 2(1 + cos 2x) and cos x sin x = 2 sin 2x, p(t) can be written as

p(t) = 12 VA IA cos θ dc component + 12 VA IA cos θ cos 2ωt − 12 VA IA sin θ sin 2ωt ac component (94) The instantaneous power is the sum of a dc component and a double-frequency ac component. That is, the instantaneous power is the sum of a constant plus a sinusoid whose frequency is 2ω, which is twice the angular frequency of the voltage and current. The instantaneous power in Eq. (94) is periodic and its average value is 1 T p(t) dt P = T 0 where T = 2π/2ω is the period of p(t). Since the average value of a sinusoid is zero, the average value of p(t), denoted P , is equal to the constant or dc term in Eq. (94): (95) P = 12 VA IA cos θ The amplitude of the sin 2ωt term in Eq. (94) has a form much like the average power in Eq. (95), except it involves sin θ rather than cos θ . This amplitude factor is called the reactive power of p(t), where reactive power Q is defined as Q = 12 VA IA sin θ

(96)

The instantaneous power in terms of the average power and reactive power is p(t) = P (1 + cos 2ωt) − Q sin 2ωt unipolar bipolar

(97)

The first term in Eq. (97) is said to be unipolar because the factor 1 + cos 2ωt never changes sign. As a result, the first term is either always positive or always negative depending on the sign of P . The second term is said to be bipolar because the factor sin 2ωt alternates signs every half cycle. The energy transferred across the interface during one cycle T = 2π/2ω of p(t) is W =

T

p(t) dt 0

W =P

T 0

T (1 + cos2ωt) dt − Q sin 2ωt dt (98) 0 net energy no net energy

W =P ×T

−

0

ELECTRIC CIRCUITS

925

Only the unipolar term in Eq. (97) provides any net energy transfer and that energy is proportional to the average power P . With the passive-sign convention the energy flows from source to load when W > 0. Equation (98) shows that the net energy will be positive if the average power P > 0. Equation (95) points out that the average power P is positive when cos θ > 0, which in turn means |θ | < 90◦ . The bipolar term in Eq. (97) is a power oscillation which transfers no net energy across the interface. In the sinusoidal steady state the load borrows energy from the source circuit during part of a cycle and temporarily stores it in the load’s reactance, namely its inductance or capacitance. In another part of the cycle the borrowed energy is returned to the source unscathed. The amplitude of the power oscillation is called reactive power because it involves periodic energy storage and retrieval from the reactive elements of the load. The reactive power can be either positive or negative depending on the sign of sin θ . However, the sign of Q says nothing about the net energy transfer, which is controlled by the sign of P . Consumers are interested in average power since this component carries net energy from source to load. For most power system customers the basic cost of electrical service is proportional to the net energy delivered to the load. Large industrial users may also pay a service charge for their reactive power as well. This may seem unfair, since reactive power transfers no net energy. However, the electric energy borrowed and returned by the load is generated within a power system that has losses. From a power company’s viewpoint the reactive power is not free because there are losses in the system connecting the generators in the power plant to the source–load interface at which the lossless interchange of energy occurs. In ac power circuit analysis, it is necessary to keep track of both the average power and reactive power. These two components of power have the same dimensions, but because they represent quite different effects, they traditionally are given different units. The average power is expressed in watts while reactive power is expressed in volt-amperes reactive (VARs).

be easily converted to rms amplitudes, Eq. (95), as VA I A VA I A cos θ = √ √ cos θ 2 2 2 P = Vrms Irms cos θ P =

where θ = ϕV − ϕI is the angle between the voltage and current phasors. By similar reasoning, Eq. (96) becomes (100) Q = Vrms Irms sin θ Using rms phasors, we define the complex power (S) at a two-terminal interface as S = VI∗ = Vrms ej φV Irms e−j φI = [Vrms Irms ]ej (φV −φI ) (101) That is, the complex power at an interface is the product of the voltage phasor times the conjugate of the current phasor. Using Euler’s relationship and the fact that the angle θ = φV − φI , complex power can be written as S = [Vrms Irms ]ej θ = [Vrms Irms ] cos θ + j [Vrms Irms ] sin θ = P + j Q

and

pf =

Vrms Irms cos θ P = = cos θ |S| Vrms Irms

I = Irms ej φI

Equations (95) and (96) express average and reactive power in terms of peak amplitudes vA and iA . The peak and √ rms values of a sinusoid are related by Vrms = VA / 2. The expression for average power can

(102)

The real part of the complex power S is the average power while the imaginary part is the reactive power. Although S is a complex number, it is not a phasor. However, it is a convenient variable for keeping track of the two components of power when voltage and current are expressed as phasors. The power triangles in Fig. 84 provide a convenient way to remember complex power relationships and terminology. Considering those cases in which net energy is transferred from source to load, P > 0 and the power triangles fall in the first or fourth quadrant. The magnitude |S| = vrms Irms is called apparent power and is expressed using the unit volt-ampere (VA). The ratio of the average power to the apparent power is called the power factor (pf):

Complex Power It is important to relate average and reactive power to phasor quantities because ac circuit analysis is conveniently carried out using phasors. The magnitude of a phasor represents the peak amplitude of a sinusoid. However, in power circuit analysis it is convenient to express phasor magnitudes in rms values. In this chapter phasor voltages and currents are expressed as

V = Vrms ej φV

(99)

Fig. 84 Power triangles.1

926


Since pf = cos θ , the angle θ is called the power factor angle. When the power factor is unity, the phasors V and I are in phase (θ = 0◦ ) and the reactive power is zero since sin θ = 0. When the power factor is less than unity, the reactive power is not zero and its sign is indicated by the modifiers lagging or leading. The term lagging power factor means the current phasor lags the voltage phasor so that θ = φV − φI > 0. For a lagging power factor S falls in the first quadrant in Fig. 84 and the reactive power is positive since sin θ > 0. The term leading power factor means the current phasor leads the voltage phasor so that θ = ϕV − ϕI < 0. In this case S falls in the fourth quadrant in Fig. 84 and the reactive power is negative since sin θ < 0. Most industrial and residential loads have lagging power factors. The apparent power rating of electrical power equipment is an important design parameter. The ratings of generators, transfomers, and transmission lines are normally stated in kilovolt-amperes. The rating of most loads is stated in kilowatts and power factor. The wiring must be large enough to carry the required current and insulated well enough to withstand the rated voltage. However, only the average power is potentially available as useful output, since the reactive power represents a lossless interchange between the source and device. Because reactive power increases the apparent power rating without increasing the available output, it is desirable for electrical devices to operate as close as possible to unity power factor (zero reactive power). In many cases power circuit loads are described in terms of their power ratings at a specified voltage or current level. In order to find voltages and current elsewhere in the circuit, it is necessary to know the load impedance. In general, the load produces the element constraint V = ZI. Using this constraint in Eq. (101), we write the complex power of the load as S = V × I = ZI × I∗ = Z|I|2

is capacitive, since XC = −1/ωC is negative. The terms inductive load, lagging power factor, and positive reactive power are synonymous, as are the terms capacitive load, leading power factor, and negative reactive power. Example 27. At 440 V (rms) a two-terminal load draws 3 kVA of apparent power at a lagging power factor of 0.9. Find irms ,P , Q, and the load impedance. Solution

Irms =

|S| 3000 = 6.82 A (rms) = Vrms 440

P = Vrms Irms cos θ = 3000 × 0.9 = 2.7 kW. For cos θ = 0.9 lagging, sin θ = 0.436 and Q = vrms Irms sin θ = 1.31 kVAR. Z=

2700 + j 1310 P + jQ = 58.0 + j 28.2 = (Irms )2 46.5

Three-Phase Circuits The three-phase system shown in Fig. 85 is the predominant method of generating and distributing ac electrical power. The system uses four lines (A, B, C, N) to transmit power from the source to the loads. The symbols stand for the three phases A, B, and C and a neutral line labeled N. The three-phase generator in Fig. 85 is modeled as three independent sources, although the physical hardware is a single unit with three separate windings. Similarly, the loads are modeled as three separate impedances, although the actual equipment may be housed within a single container. The terminology Y connected and connected refers to the two ways the source and loads can be electrically connected. In a Y connection the three elements are connected from line to neutral, while in the connection they are connected from line to line.

2 = (R + j X)Irms

where R and X are the resistance and reactance of the load, respectively. Since S = P + j Q, we conclude that R=

P 2 Irms

and

X=

Q 2 Irms

(103)

The load resistance and reactance are proportional to the average and reactive power of the load, respectively. The first condition in Eq. (103) demonstrates that resistance cannot be negative, since P cannot be negative for a passive circuit. The second condition points out that when the reactive power is positive the load is inductive, since XL = ωL is positive. Conversely, when the reactive power is negative the load

Fig. 85 Three-phase source connected to three-phase Y connection and to three-phase connection.2

ELECTRIC CIRCUITS

927

In most systems the source is Y connected while the loads can be either Y or , although the latter is more common. Three-phase sources usually are Y connected because the connection involves a loop of voltage sources. Large currents may circulate in this loop if the three voltages do not exactly sum to zero. In analysis situations, a connection of ideal voltage sources is awkward because it is impossible to uniquely determine the current in each source. A double-subscript notation is used to identify voltages in the system. The reason is that there are at least six voltages to deal with: three line-to-line voltages and three line-to-neutral voltages. The two subscripts are used to define the points across which a voltage is defined. For example, VAB means the voltage between points A and B with an implied plus reference mark at the first subscript (A) and an implied minus at the second subscript (B). The three line-to-neutral voltages are called the phase voltages and are written in double-subscript notation as VAN , VBN , and VCN . Similarly, the three line-to-line voltages, called simply the line voltages, are identified as VAB , VBC , and VCA . From the definition of the double-subscript notation it follows that VXY = −VY X . Using this result and KVL we derive the relationships between the line voltages and phase voltages: VAB = VAN + VN B = VAN − VBN VBC = VBN + VN C = VBN − VCN

(104)

VCA = VCN + VN A = VCN − VAN A balanced three-phase source produces phase voltages that obey the following two constraints: |VAN | = |VBN | = |VCN | = VP VAN + VBN + VCN = 0 + j 0 That is, the phase voltages have equal amplitudes (vP ) and sum to zero. There are two ways to satisfy these constraints:

Positive Phase Sequence VAN = VP ∠0◦ VBN = VP ∠ − 120◦ VCN = VP ∠ − 240◦

Negative VAN = VBN = VCN =

Phase Sequence VP ∠0◦ VP ∠ − 240◦ VP ∠ − 120◦ (105) Figure 86 shows the phasor diagrams for the positive and negative phase sequences. It is apparent that both sequences involve three equal-length phasors that are separated by an angle of 120◦ . As a result, the sum of any two phasors cancels the third. In the positive sequence the phase B voltage lags the phase A voltage by 120◦ . In the negative sequence phase B lags by 240◦ . It also is apparent that one phase sequence can be converted into the other by simply interchanging the labels on lines B and C. From a circuit analysis viewpoint there is no conceptual difference between the two sequences. However, the reader is cautioned that “no conceptual difference” does not mean phase sequence is unimportant. It turns out that three-phase motors run in one direction when the positive sequence is applied and in the opposite direction for the negative sequence. In practice, it is essential that there be no confusion about which line A, B, and C is and whether the source phase sequence is positive or negative. A simple relationship between the line and phase voltages is obtained by substituting the positive-phasesequence voltages from Eq. (105) into the phasor sums in Eq. (104): √ ◦ VAB = VAN − VBN = 3VP ∠30 √ ◦ VBC = 3VP ∠ − 90 √ ◦ VCA = 3VP ∠ − 210 Figure 87 shows the phasor diagram of these results. The line voltage phasors have the same amplitude and are displaced from each other by 120◦ . Hence, they obey equal-amplitude and zero-sum constraints like the phase voltages. If the √ amplitude of the line voltages is vL , then VL = 3VP . In a balanced three-phase system the

Fig. 86 Two possible phase sequences: (a) positive; (b) negative.1

928


Substituting the i–v constraint into the source constraint produces the equation governing the RC series circuit: dv(t) + v(t) = vT (t) (106) RT C dt The unknown in Eq. (106) is the capacitor voltage v(t), which is called the state variable because it determines the amount or state of energy stored in the capacitive element. Writing a KCL equation for the RL circuit in Fig. 88b yields Fig. 87 Phasor diagram showing phase and line voltages for positive phase sequence.1

√ line voltage amplitude is 3 times the phase voltage amplitude. This ratio appears in equipment descriptions such as 277/480 V three-phase, where 277 is the phase voltage and 480 the line voltage. It is necessary to choose one of the phasors as the zero-phase reference when defining three-phase voltages and currents. Usually the reference is the line A phase voltage (i.e., VAN = VP ∠0◦ ), as illustrated in Figs. 86 and 87. 5 TRANSIENT RESPONSE OF CIRCUITS 5.1 First-Order Circuits First-order RC and RL circuits contain linear resistors and a single capacitor or a single inductor. Figure 88 shows RC and RL circuits divided into two parts: (a) the dynamic element and (b) the rest of the circuit containing only linear resistors and sources. Dealing first with the RC circuit in Fig. 88a, a KVL equation is RT i(t) + v(t) = vT (t)

The capacitor i–v constraint is i(t) = C

dv(t) dt

1 v(t) + i(t) = iN (t) RN The element constraint for the inductor is v(t) = L

di(t) dt

Combining the element and source constraints produces the differential equation for the RL circuit: L di(t) + i(t) = iN (t) RN dt

(107)

The unknown in Eq. (107) is the inductor current, also called the state variable because it determines the amount or state of energy stored in the inductive element. Note that Eqs. (106) and (107) have the same form. In fact, interchanging the following quantities converts one equation into the other: G↔R

L↔C

i↔v

iN ↔ vT

This interchange is an example of the principle of duality. Because of duality there is no need to study the RC and RL circuits as independent problems. Everything learned solving the RC circuit can be applied to the RL circuit as well. Step Response of RL and RC Circuits For the RC circuit the response v(t) must satisfy the differential equation (106) and the initial condition v(0). The initial energy can cause the circuit to have a nonzero response even when the input vT (t) = 0 for t ≥ 0. When the input to the RC circuit in Fig. 88 is a step function, the source can be written as vT (t) = vA u(t). The circuit differential equation (106) then becomes

RT C

Fig. 88

First-order circuits: (a) RC circuit; (b) RL circuit.1

dv(t) + v(t) = VA u(t) dt

The step response of this circuit is a function v(t) that satisfies this differential equation for t ≥ 0 and

ELECTRIC CIRCUITS

929

meets the initial condition v(0). Since u(t) = 1 for t ≥0 RT C

dv(t) + v(t) = VA dt

for t ≥ 0

(108)

The solution v(t) can be divided into two components: v(t) = vN (t) + vF (t) The first component vN (t) is the natural response and is the general solution equation (108) when the input is set to zero. The natural response has its origin in the physical characteristic of the circuit and does not depend on the form of the input. The component vF (t) is the forced response and is a particular solution of Eq. (108) when the input is the step function. Finding the natural response requires the general solution of Eq. (108) with the input set to zero:

But this is the homogeneous equation that produces the zero-input response. Therefore, the form of the natural response is vN (t) = Ke

t ≥0

dvF (t) + vF (t) = VA dt

for t ≥ 0

(109)

The equation requires that a linear combination of vF (t) and its derivative equal a constant vA for t ≥ 0. Setting vF (t) = KF meets this condition since dvF /dt = dVA /dt = 0. Substituting vF = KF into Eq. (109) results in KF = vA . Now combining the forced and natural responses yields v(t) = vN (t) + vF (t) v(t) = Ke−t/(RT C) + VA

t ≥0

This equation is the general solution for the step response because it satisfies Eq. (106) and contains an arbitrary constant K. This constant can now be evaluated using the initial condition, v(0) = V0 = Ke0 + VA = K + VA . The initial condition requires that K = V0 − VA . Substituting this conclusion into the

t ≥0

(110)

A typical plot of the waveform of v(t) is shown in Fig. 89. The RL circuit in Fig. 88 is the dual of the RC circuit, so the development of its step response is similar. The result is i(t) = (I0 − IA )e−RN t/L + IA

This is a general solution of the homogeneous equation because it contains an arbitrary constant K. To evaluate K from the initial condition the total response is needed since the initial condition applies to the total response (natural plus forced). Turning now to the forced response, a particular solution of the equation needs to be found: RT C

general solution yields the step response of the RC circuit: v(t) = (V0 − VA )e−t/(RT C) + VA

dvN (t) + vN (t) = 0 for t ≥ 0 RT C dt

−t/(RT C)

Fig. 89 Step response of first-order RC circuit.1

t ≥0

(111)

The RL circuit step response has the same form as the RC circuit step response in Eq. (110). At t = 0 the starting value of the response is i(0) = I0 as required by the initial condition. The final value is the forced response i(∞) = iF = iA , since the natural response decays to zero as time increases. Initial and Final Conditions The state variable responses can be written in the form

vc (t), iL (t) = [IC − FC]e−t/TC + FC

t ≥0 (112) where IC stands for the initial condition (t = 0) and FC for the final condition (t = 4). To determine the step response of any first-order circuit, only three quantities, IC, FC, and TC , are needed. The final condition can be calculated directly from the circuit by observing that for t > 5TC the step responses approach a constant, or dc, value. Under the dc condition a capacitor acts like an open circuit and an inductor acts like a short circuit, so the final value of the state variable can be calculated using resistance circuit analysis methods. Similarly the dc analysis method can be used to determine the initial condition in many practical situations. One common situation is a circuit containing a switch that remains in one state for a period of time that is long compared with the circuit time constant. If the switch is closed for a long period of time, then

930


the state variable approaches a final value determined by the dc input. If the switch is now opened at t = 0, a transient occurs in which the state variable is driven to a new final condition. The initial condition at t = 0 is the dc value of the state variable for the circuit configuration that existed before the switch was opened at t = 0. The switching action cannot cause an instantaneous change in the initial condition because capacitor voltage and inductor current are continuous functions of time. In other words, opening a switch at t = 0 marks the boundary between two eras. The dc condition of the state variable for the t < 0 era is the initial condition for the t > 0 era that follows. The parameters IC, FC, and TC in switched dynamic circuits are found using the following steps: Step 1: Find the initial condition IC by applying dc analysis to the circuit configuration for t < 0. Step 2: Find the final condition FC by applying dc analysis to the circuit configuration for t ≥ 0. Step 3: Find the time constant TC of the circuit with the switch in the position for t ≥ 0. Step 4: Write the step response directly using Eqs. (112) without formulating and solving the circuit differential equation. Example 28. For the circuit shown in Fig. 90a the switch has been closed for a long time. At t = 0 it

opens. Find the capacitor voltage v(t) and current i(t) for t ≥ 0. Solution

Step 1: The initial condition is found by dc analysis of the circuit configuration in Fig. 90b where the switch is closed. Using voltage division the initial capacitor voltage in found to be vC (0−) = IC =

R 2 V0 R1 + R2

Step 2: The final condition is found by dc analysis of the circuit configuration in Fig. 90c where the switch is open. Five time constants after the switch is opened the circuit has no practical dc excitation, so the final value of the capacitor voltage is zero. Step 3: The circuit in Fig. 90c also is used to calculate the time constant. Since R1 is connected in series with an open switch, the capacitor sees an equivalent resistance of only R2 . For t ≥ 0 the time constant is R2 C. Using Eq. (112) the capacitor voltage for t ≥ 0 is vC (t) = (IC − FC)e−t/TC + F C vC (t) =

t ≥0

R2 VA −t/(R2 C) e t ≥0 R1 + R2

This result is a zero-input response, since there is no excitation for t ≥ 0. To complete the analysis, the capacitor current is found by using its element constraint: iC (t) = C

V0 dvC =− e1/(R2 C) dt R1 + R2

t ≥0

For t < 0 the initial-condition circuit in Fig. 90b points out that iC (0−) = 0 since the capacitor acts like an open circuit. Example 29. The switch in Fig. 91a has been open for a “long time” and is closed at t = 0. Find the inductor current for t > 0. Solution. The initial condition is found using the circuit in Fig. 91b. By series equivalence the initial current is V0 i(0−) = IC = R1 + R2

The final condition and the time constant are determined from the circuit in Fig. 91c. Closing the switch shorts out R2 and the final condition and time constant for t > 0 are Fig. 90 Solving switched dynamic circuit using initial and final conditions.1

i(∞) = FC =

V0 R1

TC =

L L = RN R1

ELECTRIC CIRCUITS

931

switch. A solution function v(t) is needed that satisfies Eq. (113) for t ≥ 0, and that meets the prescribed initial condition v(0) = V0 . As with the step response, the solution is divided into two parts: natural response and forced response. The natural response is of the form vN (t) = Ke−t/(RT C) The natural response of a first-order circuit always has this form because it is a general solution of the homogeneous equation with input set to zero. The form of the natural response depends on physical characteristics of the circuit and is independent of the input. The forced response depends on both the circuit and the nature of the forcing function. The forced response is a particular solution of the equation Exponential input: RT C

dvF (t) + vF (t) dt

= VA e−αt

t ≥0

dvF (t) + vF (t) Sinusoidal input: RT C dt t ≥0 = VA cos ωt Fig. 91

(From Ref. 1.)

Using Eq. (112) the inductor current for t ≥ 0 is t ≥0 i(t) = (IC − FC)e−t/TC + FC

V0 −R1 t/L V0 V0 e − + A i(t) = R1 + R2 R1 R1

t ≥0

dv(t) + v(t) dt = VA e−αt u(t)

Exponential input: RT C

dv(t) + v(t) Sinusoidal input: RT C dt = VA cos ωtu(t)

This equation requires that vF (t) plus RT C times its first derivative add to produce either an exponential or a sinusoidal waveform for t ≥ 0. The only way this can happen is for vF (t) and its derivative to be either an exponential of the same decay or sinusoids of the same frequency. This requirement brings to mind the derivative property of the exponential or the sinusoid. Hence one chooses a solution in the form of Exponential: vF (t) = KF e−αt Sinusoidal:

First-Order Circuit Response to Other Than dc Signals The response of linear circuits to a variety of signal inputs is an important concept in electrical engineering. Of particular importance is the response to a step reviewed in the previous section to the exponential and sinusoid. If the input to the RC circuit in Fig. 88 is an exponential or a sinusoid, then the circuit differential equation is written as

(113)

The inputs on the right side of Eq. (113) are signals that start at t = 0 through some action such as closing a

(114)

vF (t) = KA cos ωt + KB sin ωt

In this expression the constant KF or the Fourier coefficients KA and KB are unknown. The approach we are using is called the method of undetermined coefficients. The unknown coefficients are found by inserting the forced solution vF (t) into the differential equation and equating the coefficients of the exponential in that case or of the sine and cosine terms. This yields the following: Exponential:

KF =

VA 1 − αRT C

Sinusoidal:

KA =

VA 1 + (ωRT C)2

KB =

ωRT CVA 1 + (ωRT C)2

The undetermined coefficients are now known, since these equations express constants in terms of known

932


circuit parameters (RT C) and known input signal parameters (ω and vA ). The forced and natural responses are combined and the initial condition used to find the remaining unknown constant K: Exponential: K = V0 + Sinusoidal:

K = V0 −

VA RT Cα − 1 VA 1 + (ωRT C)2

Combining these together yields the function v(t) that satisfies the differential equation and the initial conditions:     

 VA e−t/(RT C) v(t) = V0 +  RT Cα − 1     Natural response       VA −αt e u(t) V −  R Cα − 1  T    Forced response  Natural response    

 VA −t/(RT C) V0 − e =  1 + (ωRT C)2      Forced response      VA + (cos ωt + ωR C sin ωt) T  1 + (ωRT C)2     × u(t)

V

(115) Equations (115) are the complete responses of the RC circuit for an initial condition V0 and either an exponential or a sinusoidal input. 5.2 Second-Order Circuits

Second-order circuits contain two energy storage elements that cannot be replaced by a single equivalent element. They are called second-order circuits because the circuit differential equation involves the second derivative of the dependent variable. The series RLC circuit will illustrate almost all of the basic concepts of second-order circuits. The circuit in Fig. 92a has an inductor and a capacitor connected in series. The source–resistor circuit can be reduced to the Thevenin equivalent shown in Fig. 92b. Applying KVL around the loop on the right side of the interface and the two i–v characteristics of

Fig. 92 Series RLC circuit.1

the inductor and capacitor yields LC

d 2 vC (t) dvC (t) + vC (t) = vT (t) + RT C dt 2 dt vL (t) + vR (t) + vC (t) = vT (t) (116)

In effect, this is a KVL equation around the loop in Fig. 92b, where the inductor and resistor voltages have been expressed in terms of the capacitor voltage. The Thevenin voltage vT (t) is a known driving force. The initial conditions are determined by the values of the capacitor voltage and inductor current at t = 0, that is, V0 and I0 : vC (0) = V0

and

1 I0 dvC (0) = i(0) = dt C C

The circuit dynamic response for t ≥ 0 can be divided into two components: (1) the zero-input response caused by the initial conditions and (2) the zerostate response caused by driving forces applied after t = 0. With vT = 0 (zero input) Eq. (116) becomes LC

dvC (t) d 2 vC (t) + vC (t) = 0 + RT C dt 2 dt

This result is a second-order homogeneous differential equation in the capacitor voltage. Inserting a trial solution of vCN (t) = Kest into the above equation results in the following characteristic equation of the series RLC circuit: LCs 2 + RT Cs + 1 = 0 In general, the above quadratic characteristic equation has two roots: −RT C ± (RT C)2 − 4LC s1 , s2 = 2LC

ELECTRIC CIRCUITS

933

The roots can have three distinct possibilities: Case A: If (RT C)2 − 4LC > 0, the discriminant is positive and there are two real, unequal roots (s1 = −α1 = s2 = −α2 ). Case B: If (RT C)2 − 4LC = 0, the discriminant vanishes and there are two real, equal roots (s1 = s2 = −α). Case C: If (RT C)2 − 4LC < 0, the discriminant is negative and there are two complex conjugate roots (s1 = −α − jβ and s2 = −α + jβ). Second-Order Circuit Zero-Input Response Since the characteristic equation has two roots, there are two solutions to the homogeneous differential equation:

vC1 (t) = K1 es1 t

and

vC2 (t) = K2 es2 t

Therefore, the general solution for the zero-input response is of the form vC (t) = K1 es1 t + K2 es2 t

s2 V0 − I0 /C s1 t vc (t) = e s2 − s1 −s1 V0 + I0 /C s2 t e s2 − s1

t ≥0

vC (t) = V0 e−αt cos βt αV0 + I0 /C −αt e sin βt + β

(118)

Equation (118) is the general zero-input response of the series RLC circuit. The response depends on two initial conditions, V0 and I0 , and the circuit parameters RT , L, and C since s1 and s2 are the roots of the characteristic equation LCs 2 + RT Cs + 1 = 0. The response has different waveforms depending on whether the roots s1 and s2 fall under case A, B, or C. For case A the two roots are real and distinct. Using the notation s1 = −α1 and s2 = −α2 , the form of the zero-input response for t ≥ 0 is

α2 V0 + I0 /C −α1 t e vc (t) = α2 − α1

α1 V0 + I0 /C −α2 t e − t ≥0 α2 − α1 This form is called the overdamped response. The waveform has two time constants 1/α1 and 1/α2 . With case B the roots are real and equal. Using the notation s1 = s2 = −α, the general form becomes I0 te−αt t ≥0 vC (t) = V0 e−αt + αV0 + C

t ≥0

This form is called the underdamped response. The underdamped response contains a damped sinusoid waveform where the real part of the roots (α) provides the damping term in the exponential, while the imaginary part (β) defines the frequency of the sinusoidal oscillation. Second-Order Circuit Step Response The general second-order linear differential equation with a step function input has the form

(117)

The constants K1 and K2 can be found using the initial conditions:

+

This special form is called the critically damped response. The critically damped response includes an exponential and a damped ramp waveform. Case C produces complex-conjugate roots of the form s1 = −α − jβands2 = −α + jβ The form of case C is

a2

d 2 y(t) dy(t) + a0 y(t) = Au(t) + a1 2 dt dt

where y(t) is a voltage or current response, Au(t) is the step function input, and a2 , a1 , and a0 are constant coefficients. The step response is the general solution of this differential equation for t ≥ 0. The step response can be found by partitioning y(t) into forced and natural components: y(t) = yN (t) + yF (t) The natural response yN (t) is the general solution of the homogeneous equation (input set to zero), while the forced response yF (t) is a particular solution of the equation a2

d 2 yF (t) dyF (t) + a0 yF (t) = A + a1 dt 2 dt

t ≥0

The particular solution is simply yF = A/a0 . In a second-order circuit the zero-state and natural responses take one of the three possible forms: overdamped, critically damped, or underdamped. To describe the three possible forms, two parameters are used: ω0 , the undamped natural frequency, and ζ , the damping ratio. Using these two parameters, the general homogeneous equation is written in the form d 2 yN (t) dyN (t) + ω02 yN (t) = 0 + 2ζ ω0 dt 2 dt The above equation is written in standard form of the second-order linear differential equation. When a

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second-order equation is arranged in this format, its damping ratio and undamped natural frequency can be readily found by equating its coefficients with those in the standard form. For example, in the standard form the homogeneous equation for the series RLC circuit is

Combining the forced and natural responses yields the step response of the general second-order differential equation in the form y(t) = yN (t) +

2

1 d vC (t) RT dvc (t) + vC (t) = 0 + dt 2 L dt LC

1 LC

and

2ζ ω0 =

RT L

for the series RLC circuit. Note that the circuit elements determine the values of the parameters ω0 and ζ . The characteristic equation is s 2 + 2ζ ω0 s + ω02 = 0 and its roots are s1 , s2 = ω0 −ζ ± ζ 2 − 1 The expression under the radical defines the form of the roots and depends only on the damping ratio ζ : Case A: For ζ > 1 the discriminant is positive and there are two unequal, real roots s1 , s2 = −α1 , −α2 = ω0 −ζ ± ζ 2 − 1 and the natural response is of the form yN (t) = K1 e−α1 t + K2 e−α2 t

t ≥0

Solution. This is a series RLC circuit so the differential equation for the capacitor voltage is

d 2 vC (t) dvC (t) + 0.5 × 10−3 dt 2 dt t ≥0 + vC (t) = 10

10−6

By inspection the forced response is vCF (t) = 10 V. In standard format the homogeneous equation is dvCN (t) d 2 vCN (t) + 106 vCN (t) = 0 + 500 dt 2 dt

t ≥0

Comparing this format, the standard form yields

s1 = s2 = −α = −ζ ω0 and the natural response is of the form t ≥0

Example 30. The series RLC circuit in Fig. 93 is driven by a step function and is in the zero state at t = 0. Find the capacitor voltage for t ≥ 0.

(119)

Case B: For ζ = 1 the discriminant vanishes and there are two real, equal roots,

yN (t) = K1 e−αt + K2 t e−αt

t ≥0

The factor A/a0 is the forced response. The natural response yN (t) takes one of the forms in Eqs. (119)–(121) depending on the value of the damping ratio. The constants K1 and K2 in the natural response can be evaluated from the initial conditions.

Equating like terms yields ω02 =

A a0

(120)

Case C: For ζ < 1, the discriminant is negative leading to two complex, conjugate roots s1 , s2 = −α ± jβ, where α = ζ ω0 and β = ω0 1 − ζ 2 and the natural response is of the form

ω02 = 106

and

2ζ ω0 = 500

so that ω0 = 1000 and ζ = 0.25. Since ζ < 1, the natural response is underdamped (case C) and has the form α = ζ ω0 = 250 Np β = ω0 1 − ζ 2 = 968 rad/sec vCN (t) = K1 e−250t cos 968t + K2 e−250t sin 968t

yN (t) = e−αt (K1 cos βt + K2 sin βt)

t ≥0 (121) In other words, for ζ > 1 the natural response is overdamped, for ζ = 1 the natural response is critically damped, and for ζ < 1 the response is underdamped.

Fig. 93

(From Ref. 1.)

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6.1

Transfer Functions and Input Impedance The proportionality property of linear circuits states that the output is proportional to the input. In the phasor domain the proportionality factor is a rational function of j ω called a transfer function. More formally, in the phasor domain a transfer function is defined as the ratio of the output phasor to the input phasor with all initial conditions set to zero:

Transfer function = Fig. 94

(From Ref. 1.)

The general solution of the circuit differential equation is the sum of the forced and natural responses:

t ≥0

The constants K1 and K2 are determined by the initial conditions. The circuit is in the zero state at t = 0, so the initial conditions are vC (0) = 0 and iL (0) = 0. Applying the initial-condition constraints to the general solution yields two equations in the constants K1 and K2 : vC (0) = 10 + K1 = 0 dvC (0) = −250 K1 + 968K2 = 0 dt These equations yield K1 = −10 and K2 = −2.58. The step response of the capacitor voltage step response is vC (t) = 10 − 10e−250t cos 968t − 2.58e−250t sin 968t

V

To study the role of transfer functions in determining circuit responses is to write the phasor domain input–output relationship as Y (j ω) = H (j ω) · X(j ω)

vC (t) = 10 + K1 e−250t cos 968t + K2 e−250t sin 968t

Output phasor = H (j ω) Input phasor

t ≥0

A plot of vC (t) versus time is shown in Fig. 94. The waveform and its first derivative at t = 0 satisfy the initial conditions. The natural response decays to zero so the forced response determines the final value of vC (∞) = 10 V. Beginning at t = 0 the response climbs rapidly but overshoots the final value several times before eventually settling down. The damped sinusoidal behavior results from the fact that ζ < 1, producing an underdamped natural response. 6 FREQUENCY RESPONSE Linear circuits are often characterized by their behavior to sinusoids, in particular, how they process signals versus frequency. Audio, communication, instrumentation, and control systems all require signal processing that depends at least in part on their frequency response.

(122)

where H (j ω) is the transfer function, X(j ω) is the input signal transform (a voltage or a current phasor), and Y (j ω) is the output signal transform (also a voltage or current phasor). Figure 95 shows a block diagram representation of the phasor domain input–output relationship. In an analysis problem the circuit defined by H (j ω) and the input X(j ω) are known and the response Y (j ω) is sought. In a design problem the circuit is unknown. The input and the desired output or their ratio H (j ω) = Y (j ω)/X(j ω) are given, and the objective is to devise a circuit that realizes the specified input–output relationship. A linear circuit analysis problem has a unique solution, but a design problem may have one, many, or even no solution. Choosing the best of several solutions is referred to as an evaluation problem. There are two major types of functions that help define a circuit: input impedance and transfer functions. Input impedance relates the voltage and current at a pair of terminals called a port. The input impedance Z(j ω) of the one-port circuit in Fig. 96 is defined as V (j ω) Z(j ω) = (123) I (j ω) When the one port is driven by a current source, the response is V (j ω) = Z(j ω)I (j ω). On the other hand, when the one port is driven by a voltage source, the response is I (j ω) = [Z(j ω)]−1 V (j ω).

Fig. 95 Block diagram for phasor domain input–output relationship.1

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Determining Transfer Functions The divider circuits in Fig. 97 occur so frequently that it is worth taking time to develop their transfer functions in general terms. Using phasor domain analysis the voltage transfer function of a voltage divider circuit is

HV (j ω) =

Fig. 96 One-port circuit.1

The term input impedance means that the circuit is driven at one port and the response is observed at the same port. The impedances of the three basic circuit elements ZR (j ω), ZL (j ω), and ZC (j ω) are elementary examples of input impedances. The equivalent impedances found by combining elements in series and parallel are also effectively input impedances. The terms input impedance, driving-point impedance, and equivalent impedance are synonymous. Input impedance is useful in impedance-matching circuits at their interface and to help determine if loading will be an issue. Transfer functions are usually of greater interest in signal-processing applications than input impedances because they describe how a signal is modified by passing through a circuit. A transfer function relates an input and response (or output) at different ports in the circuit. Since the input and output signals can be either a current or a voltage, four kinds of transfer functions can be defined: V2 (j ω) HV (j ω) = voltage transfer function = V1 (j ω) I2 (j ω) HY (j ω) = transfer admittance = V1 (j ω) I2 (j ω) HI (j ω) = current transfer function = I1 (j ω) V2 (j ω) HZ (j ω) = transfer impedance = I1 (j ω) (124) The functions HV (j ω) and HI (j ω) are dimensionless since the input and output signals have the same units. The function HZ (j ω) has units of ohms and HY (j ω) has unit of siemens. Transfer functions always involve an input applied at one port and a response observed at a different port in the circuit. It is important to realize that a transfer function is only valid for a given input port and the specified output port. They cannot be turned upside down like the input impedance. For example, the voltage transfer function HV (j ω) relates the voltage V1 (j ω) applied at the input port to the voltage response V2 (j ω) observed at the output port in Fig. 95. The voltage transfer function for signal transmission in the opposite direction is usually not 1/HV (j ω).

Z2 (j ω) V2 (j ω) = V1 (j ω) Z1 (j ω) + Z2 (j ω)

Similarly, using phasor domain current division in Fig. 97b results in the current transfer function of a current divider circuit: HI (j ω) = =

1/[Z2 (j ω)] I2 (j ω) = I1 (j ω) 1/[Z1 (j ω)] + 1/[Z2 (j ω)] Z1 (j ω) Z1 (j ω) + Z2 (j ω)

By series equivalence the driving-point impedance at the input of the voltage divider is ZEQ (j ω) = Z1 (j ω) + Z2 (j ω). By parallel equivalence the drivingpoint impedance at the input of the current divider is ZEQ (j ω) = 1/[1/Z1 (j ω) + 1/Z2 (j ω)]. Two other useful circuits are the inverting and noninverting op amp configurations shown in Fig. 98. The voltage transfer function of the inverting circuit in Fig. 98a is HV (j ω) =

Z2 (j ω) V2 (j ω) =− V1 (j ω) Z1 (j ω)

The input impedance of this circuit is simply Z1 (j ω) since vB (j ω) = 0. The effect of Z1 (j ω) should be studied when connecting it to another circuit or a nonideal source since it can cause undesired loading. For the noninverting circuit in Fig. 98b the voltage transfer function is HV (j ω) =

Z1 (j ω) + Z2 (j ω) V2 (j ω) = V1 (j ω) Z1 (j ω)

The ideal op amp draws no current at its input terminals, so theoretically the input impedance of the noninverting circuit is infinite; in practice it is quite high, upward of 1010 .

Fig. 97 Basic divider circuits: (a) voltage divider; (b) current divider.1

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Fig. 98

Basic op amp circuits: (a) inverting amplifier; (b) noninverting amplifier.1

Example 31. For the circuit in Fig. 99, find (a) the input impedance seen by the voltage source and (b) the voltage transfer function HV (j ω) = V2 (j ω)/V1 (j ω).

HV (j ω) = =

Solution

(a) The circuit is a voltage divider. First find the equivalent impedances of the two legs of the divider. The two elements in parallel combine to produce the series leg impedance Z1 (j ω): Z1 (j ω) =

R1 1 = C1 j ω + 1/R1 R 1 C1 j ω + 1

The two elements in series combine to produce shunt (parallel) leg impedance Z2 (j ω): Z2 (j ω) = R2 +

R 2 C2 j ω + 1 1 = C2 j ω C2 j ω

Using series equivalence, the input impedance seen at the input is ZEQ (j ω) = Z1 (j ω) + Z2 (j ω)

Z2 (j ω) ZEQ (j ω) (R1 C1 j ω + 1)(R2 C2 j ω + 1) R1 C1 R2 C2 (j ω)2 + (R1 C1 + R2 C2 + R1 C2 )j ω + 1

Example 32

(a) Find the driving-point impedance seen by the voltage source in Fig. 100. (b) Find the voltage transfer function HV (j ω) = V2 (j ω)/V1 (j ω) of the circuit. (c) If R1 = 1 k, R2 = 10 k, C1 = 10 nF, and C2 = 1 µF, evaluate the driving-point impedance and the transfer function. Solution. The circuit is an inverting op amp configuration. The input impedance and voltage transfer function of this configuration are

ZIN (j ω) = Z1 (j ω)

HV (j ω) = −

and

Z2 (j ω) Z1 (j ω)

(a) The input impedance is

R1 C1 R2 C2 (j ω) + (R1 C1 + R2 C2 + R1 C2 )j ω + 1 = C2 j ω(R1 C1 j ω + 1) 2

Z1 (j ω) = R1 +

1 R 1 C1 j ω + 1 = j ωC1 j ωC1

(b) The feedback impedance is (b) Using voltage division, the voltage transfer function is

Fig. 99

(From Ref. 1.)

Z2 (j ω) =

R2 1 = j ωC2 + 1/R2 R 2 C2 j ω + 1


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and the voltage transfer function is HV (j ω) = −

Z2 (j ω) Z1 (j ω)

R 2 C1 j ω =− (R1 C1 j ω + 1) (R2 C2 j ω + 1) (c) For the values of R’s and C’s given 1000 j ω + 105 Z1 (j ω) = jω and HV (j ω) = −

1000j ω (j ω + 100) j ω + 105

6.2 Cascade Connection and Chain Rule Signal-processing circuits often involve a cascade connection in which the output voltage of one circuit serves as the input to the next stage. In some cases, the overall voltage transfer function of the cascade can be related to the transfer functions of the individual stages by a chain rule:

HV (j ω) = HV 1 (j ω) × HV 2 (j ω) × · · · × HV k (j ω) (125) where HV 1 , HV 2 , . . . , HV k are the voltage transfer functions of the individual stages when operated separately. It is important to understand when the chain rule applies since it greatly simplifies the analysis and design of cascade circuits. Figure 101 shows two RC circuits or stages connected in cascade at an interface. When disconnected

and operated separately, the transfer functions of each stage are easily found using voltage division as follows: HV 1 (j ω) =

Rj ωC R = R + 1/j ωC Rj ωC + 1

HV 2 (j ω) =

1 1/j ωC = R + 1/j ωC Rj ωC + 1

When connected in cascade the output of the first stage serves as the input to the second stage. If the chain rule applies, the overall transfer function would be expected to be V3 (j ω) V2 (j ω) V3 (j ω) = HV (j ω) = V1 (j ω) V1 (j ω) V2 (j ω) = HV 1 (j ω) × HV 2 (j ω) 1 Rj ωC = Rj ωC + 1 Rj ωC + 1 1st stage 2nd stage =

Rj ωC (Rj ωC)2 + 2Rj ωC + 1 combined

However, the overall transfer function of this circuit is actually found to be HV (j ω) =

Rj ωC (Rj ωC)2 + 3Rj ωC + 1

which disagrees with the chain rule result.

Fig. 101 (a) Two-port circuits connected in cascade. (b) Cascade connection with voltage follower isolation.1

ELECTRIC CIRCUITS

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The reason for the discrepancy is that when they are connected in cascade the second circuit “loads” the first circuit. That is, the voltage divider rule requires the current I2 (j ω) in Fig. 101a be zero. The no-load condition I2 (j ω) = 0 is valid when the stages operate separately, but when connected together the current is no longer zero. The chain rule does not apply here because loading caused by the second stage alters the transfer function of both stages. The loading problem goes away when an op amp voltage follower is inserted between the RC circuit stages (Fig. 101b). With this modification the chain rule in Eq. (125) applies because the voltage follower isolates the two RC circuits. Recall that ideally a voltage follower has infinite input resistance and zero output resistance. Therefore, the follower does not draw any current from the first RC circuit [I2 (j ω) = 0] and its transfer function of “1” allows V2 (j ω) to be applied directly across the input of the second RC circuit. The chain rule in Eq. (125) applies if connecting a stage does not change or load the output of the preceding stage. Loading can be avoided by connecting an op amp voltage follower between stages. More importantly, loading does not occur if the output of the preceding stage is the output of an op amp or controlled source unless the load resistance is very low. These elements act very close to ideal voltage sources whose outputs are unchanged by connecting the subsequent stage. For example, in the top representation the two circuits in Fig. 102 are connected in a cascade with circuit C1 appearing first in the cascade followed by circuit C2. The chain rule applies to this configuration

because the output of circuit C1 is an op amp that can handle the load presented by circuit C2. On the other hand, if the stages are interchanged so that the op amp circuit C1 follows the RC circuit C2 in the cascade, then the chain rule would not apply because the input impedance of circuit C1 would then load the output of circuit C2. 6.3

Frequency Response Descriptors

The relationships between the input and output sinusoids are important to frequency-sensitive circuits and can be summarized in the following statements. Realizing the circuit transfer function is usually a complex function of ω, its effect on the sinusoidal steady-state response can be found through its gain function |H (j ω)| and phase function ∠H (j ω) as follows: Magnitude of H (j ω) = |H (j ω)| =

output amplitude input amplitude

Angle of H (j ω) = ∠H (j ω) = output phase − input phase Taken together the gain and phase functions show how the circuit modifies the input amplitude and phase angle to produce the output sinusoid. These two functions define the frequency response of the circuit since they are frequency-dependent functions that relate the sinusoidal steady-state input and output. The gain and phase functions can be expressed mathematically or

Fig. 102 Effects of stage location on loading.1

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Fig. 103 Frequency response plots.1

presented graphically as in Fig. 103, which shows an example frequency response plot called a Bode diagram. These diagrams can be constructed by hand but are readily and more accurately produced by simulation and mathematical software products. The terminology used to describe the frequency response of circuits and systems is based on the form of the gain plot. For example, at high frequencies the gain in Fig. 103 falls off so that output signals in this frequency range are reduced in amplitude. The range of frequencies over which the output is significantly attenuated is called the stopband. At low frequencies the gain is essentially constant and there is relatively little attenuation. The frequency range over which there is little attenuation is called a passband. The frequency associated with the boundary between a passband and an adjacent stopband is called the cutoff frequency (ωC = 2πfC ). In general, the transition from the passband to the stopband is gradual so the precise location of the cutoff frequency is a matter of definition. The most widely used definition specifies the cutoff frequency to be the frequency √ at which the gain has decreased by a factor of 1/ 2 = 0.707 from its maximum value in the passband. Again this definition is arbitrary, since there is no sharp boundary between a passband and an adjacent stopband. However, the definition is motivated by the fact that the power delivered to a resistance by a sinusoidal current or voltage waveform is proportional to the square of its amplitude. At a√cutoff frequency the gain is reduced by a factor of 1/ 2 and the square of the output amplitude is reduced by a factor of 12 . For this reason the cutoff frequency is also called the halfpower frequency. In filter design the region from where the output amplitude is reduced by 0.707 and a second frequency wherein the output must have decayed to some specified value is called the transition region. This region is where much of the filter design attention is focused. How rapidly a filter transitions from the cutoff frequency to some necessary attenuation is what occupies much of the efforts of filter designers.

Additional frequency response descriptors are based on the four prototype gain characteristics shown in Fig. 104. A low-pass gain characteristic has a single passband extending from zero frequency (dc) to the cutoff frequency. A high-pass gain characteristic has a single passband extending from the cutoff frequency to infinite frequency. A bandpass gain has a single passband with two cutoff frequencies neither of which is zero or infinite. Finally, the bandstop gain has a single stopband with two cutoff frequencies neither of which is zero or infinite. The bandwidth of a gain characteristic is defined as the frequency range spanned by its passband. The bandwidth (BW) of a low-pass circuit is equal to its cutoff frequency (BW = ωC ). The bandwidth of a high-pass characteristic is infinite since passband extends to infinity. For the bandpass and bandstop cases in Fig. 104 the bandwidth is the difference in the two cutoff frequencies: BW = ωC2 − ωC1

(126)

For the bandstop case Eq. (126) defines the width of the stopband rather than the passband. The gain responses in Fig. 104 have different characteristics at zero and infinite frequency: Prototype

Gain at ω = 0

Gain at ω = 4

Low pass High pass Bandpass Bandstop

Finite 0 0 Finite

0 Finite 0 Finite

Since these extreme values form a unique pattern, the type of gain response can be inferred from the values of |H (0)| and |H (∞)|. These endpoint values in turn are usually determined by the impedance of capacitors and inductors in the circuit. In the sinusoidal steady state the impedances of these elements are ZC (j ω) =

1 j ωC

and

ZL (j ω) = j ωL

These impedances vary with respect to frequency. An inductor’s impedance increases linearly with increasing frequency, while that of a capacitor varies inversely with frequency. They form a unique pattern at zero and infinite frequency:

Element Capacitor (1/j ωC) Inductor (j ωL) Resistor (R)

Impedance () at ω = 0 (dc)

Impedance () at ω = 4

Infinite (open circuit) 0 (short circuit) R

0 (short circuit) Infinite (open circuit) R

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Fig. 104

Four basic gain responses.1

These observations often allow one to infer the type of gain response and hence the type of filter directly from the circuit itself without finding the transfer functions. Frequency response plots are almost always made using logarithmic scales for the frequency variable. The reason is that the frequency ranges of interest often span several orders of magnitude. A logarithmic frequency scale compresses the data range and highlights important features in the gain and phase responses. The use of a logarithmic frequency scale involves some special terminology. Any frequency range whose endpoints have a 2 : 1 ratio is called an octave. Any range whose endpoints have a 10 : 1 ratio is called a decade. For example, the frequency range from 10 to 20 Hz is one octave, as is the range from 20 to 40 MHz. The standard UHF (ultrahigh frequency) band spans one decade from 0.3 to 3 GHz. In frequency response plots the gain |H (j ω)| is often expressed in decibels (dB), defined as |H (j ω)|dB = 20 log10 |H (j ω)|

The gain in decibels can be positive, negative, or zero. A gain of 0 dB means that |H (j ω)| = 1; that is, the input and output amplitudes are equal. Positive decibel gains mean the output amplitude exceeds the input since |H (j ω)| > 1 and the circuit is said to amplify the signal. A negative decibel gain means the output amplitude is smaller than the input since |H (j ω)| < 1 and the circuit is said to attenuate the signal. A cutoff frequency occurs when the gain is reduced √ from its maximum passband value by a factor 1/ 2 or 3 dB. For this reason the cutoff is also called the 3-dB down frequency. 6.4 First-Order Frequency Response and Filter Design

Frequency-selective circuits are fundamental to all types of systems. First-order filters are simple to design and can be effective for many common applications. First-Order Low-Pass Response A first-order low-pass transfer function can be written as K H (j ω) = jω + α

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The constants K and α are real. The constant K can be positive or negative, but α must be positive so that the natural response of the circuit is stable. The gain and phase functions are given as |H (j ω)| = √

|K|

ω2 + α 2 ∠H (j ω) = ∠K − tan−1 (ω/α)

(127)

The gain function is a positive number. Since K is real, the angle of K(∠K) is either 0◦ when K > 0 or ± 180◦ when K < 0. An example of a negative K occurs in an inverting op amp configuration where H (j ω) = −Z2 (j ω)/Z1 (j ω). Figure 105 shows the gain and phase functions versus normalized frequency ωc /α. The maximum passband gain occurs at ω = 0 where |H (0)| = |K|/α. As frequency increases, the gain gradually decreases until at ω = α: |H (j α)| = √

|K| α2 + α2

=

|K| → α |H (0)| = √ √ 2 2

That is, the cutoff frequency of the first-order lowpass transfer function is ωC = α. The graph of the gain function in Fig. 105a displays a low-pass characteristic with a finite dc gain and zero infinite frequency gain. The low- and high-frequency gain asymptotes shown in Fig. 105a are especially important. The

low-frequency asymptote is the horizontal line and the high-frequency asymptote is the sloped line. At low frequencies (ω α) the gain approaches |H (j ω)| → |K|/α. At high frequencies (ω α) the gain approaches |H (j ω)| → |K|/ω. The intersection of the two asymptotes occurs when |K|/α = |K|/ω. The intersection forms a “corner” at ω = α, so the cutoff frequency is also called the corner frequency. The high-frequency gain asymptote decreases by a factor of 10 (−20 dB) whenever the frequency increases by a factor of 10 (one decade). As a result the high-frequency asymptote has a slope of −1 or −20 dB/decade and the low-frequency asymptote has a slope of 0 or 0 dB/decade. These two asymptotes provide a straight-line approximation to the gain response that differs from the true response by a maximum of 3 dB at the corner frequency. The semilog plot of the phase shift of the first-order low-pass transfer function is shown in Fig. 105b. At ω = α the phase angle in Eq. (127) is ∠K − 45◦ . At low frequency (ω < α) the phase angle approaches ∠K and at high frequencies (ω > α) the phase approaches ∠K − 90◦ . Almost all of the −90◦ phase change occurs in the two-decade range from ω/α = 0.1 to ω/α = 10. The straight-line segments in Fig. 105b provide an approximation of the phase response. The phase approximation below ω/α = 0.1 is θ = ∠K and above ω/α = 10 is H ∠θ = ∠K − 90◦ . Between these values the phase approximation is a straight line that begins at H ∠θ = ∠K, passes through H ∠θ = ∠K − 45◦ at the cutoff frequency, and reaches H ∠θ = ∠K − 90◦ at ω/α = 10. The slope of this line segment is −45◦ /decade since the total phase change is −90◦ over a two-decade range. To construct the straight-line approximations for a first-order low-pass transfer function, two parameters are needed, the value of H (0) and α. The parameter α defines the cutoff frequency and the value of H (0) defines the passband gain |H (0)| and the lowfrequency phase ∠H (0). The required quantities H (0) and α can be determined directly from the transfer function H (j ω) and can often be estimated by inspecting the circuit itself. Example 33. Design a low-pass filter with a passband gain of 4 and a cutoff frequency of 100 rad/sec. Solution. See Fig. 106. Start with an inverting amplifier configuration since a gain is required:

H (j ω) = −

Z2 (j ω) Z1 (j ω)

Z1 (j ω) = R1

and

1 R2 = Z2 (j ω) = j ωC2 + 1/R2 R 2 C2 j ω + 1 Fig. 105 First-order low-pass Bode

plots.1

H (j ω) = −

1 R2 × R1 R 2 C2 j ω + 1

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Rearrange the standard low-pass form as

ωC = α =

H (j ω) =

K/α j ω/α + 1

1 R 2 C2

and

H (0) = −

R2 R1

The design constraints require that ωC = 1/R2 C2 = 100 and |H (0)| = R2 /R1 = 4. Selecting R1 = 10 k requires R2 = 40 k and C = 250 nF. First-Order High-Pass Response A first-order high-pass transfer function is written as

H (j ω) =

Kj ω jω + α

The high-pass function differs from the low pass case by the introduction of a j ω in the numerator, resulting in the function becoming zero at ω = 0. Solving for the gain and phase functions yields |H (j ω)| = √

|K| ω ω2 + α 2 ◦

∠H (j ω) = ∠K + 90 − tan−1

ω α

(128)

Figure 107 shows the gain and phase functions versus normalized frequency ω/α. The maximum gain occurs at high frequency (ω > α) where |H (j ω)|6|K|. At low frequency (ω < α) the gain approaches |K|ω/α. At ω = α the gain is |H (j α)| = √

|K| α α2

+

α2

|K| = √ 2

which means the cutoff frequency is ωC = α. The gain response plot in Fig. 107a displays a high-pass characteristic with a passband extending from ω = α to infinity and a stopband between zero frequency and ω = α.

Fig. 107 First-order high-pass Bode plots.1

The low- and high-frequency gain asymptotes approximate the gain response in Fig. 107a. The highfrequency asymptote (ω > α) is the horizontal line whose ordinate is |K| (slope = 0 or 0 dB/decade). The low-frequency asymptote (ω < α) is a line of the form |K|ω/α (slope = +1 or +20 dB/decade). The intersection of these two asymptotes occurs when |K| = |K|ω/α, which defines a corner frequency at ω = α. The semilog plot of the phase shift of the first-order high-pass function is shown in Fig. 107b. The phase shift approaches ∠K at high frequency, passes through ∠K + 45◦ at the cutoff frequency, and approaches ∠K + 90◦ at low frequency. Most of the 90◦ phase change occurs over the two-decade range centered on the cutoff frequency. The phase shift can be approximated by the straight-line segments shown in the Fig. 107b. As in the low-pass case, ∠K is 0◦ when K is positive and ±180◦ when K is negative. Like the low-pass function, the first-order high-pass frequency response can be approximated by straightline segments. To construct these lines, we need two parameters, H (∞), and α. The parameter α defines the cutoff frequency and the quantity H (∞) gives the passband gain |H (∞)| and the high-frequency phase angle ∠H (∞). The quantities H (∞) and α can be determined directly from the transfer function or estimated directly from the circuit in some cases. The straight line shows the first-order high-pass response

944


Fig. 108 Cascade connection of high- and low-pass circuits.1

can be characterized by calculating the gain and phase over a two-decade band from one decade below to one decade above the cutoff frequency. Bandpass and Bandstop Responses Using FirstOrder Circuits The first-order high- and low-pass circuits can be used in a building block fashion to produce a circuit with bandpass and bandstop responses. Figure 108 shows a cascade connection of first-order high- and low-pass circuits. When the second stage does not load the first, the overall transfer function can be found by the chain rule:

H (j ω) = H1 (j ω) × H2 (j ω) K2 K1 j ω = j ω + α1 j ω + α2 high pass low pass

Fig. 109

input can reach the output via either a low- or a highpass path. The overall transfer function is the sum of the low- and high-pass transfer functions: 

Solving for the gain response yields 



Bandpass gain characteristic.1



 |K1 | ω   |K2 |  |H (j ω)| =  $ $  ω2 + α12 ω2 + α22 high pass low pass Note the gain of the cascade is zero at ω = 0 and at infinite frequency. When α1 < α2 the high-pass cutoff frequency is much lower than the low-pass cutoff frequency, and the overall transfer function has a bandpass characteristic. At low frequencies (ω < α1 < α2 ) the gain approaches |H (j ω)| → |K1 K2 |ω/α1 α2 . At midfrequencies (α1 < ω < α2 ) the gain approaches |H (j ω)| → |K1 K2 |/α2 . The low- and midfrequency asymptotes intersect when |K1 K2 |ω/α1 α2 = |K1 K2 |/α2 at ω = α1 , that is, at the cutoff frequency of the high-pass stage. At high frequencies (α1 < α2 < ω) the gain approaches |H (j ω)| → |K1 K2 |/ω. The high- and midfrequency asymptotes intersect when |K1 K2 |/ω = |K1 K2 |/α2 at ω = α2 , that is, at the cutoff frequency of the low-pass stage. The plot of these asymptotes in Fig. 109 shows that the asymptotic gain exhibits a passband between α1 and α2 . Input sinusoids whose frequencies are outside of this range fall in one of the two stopbands. In the bandpass cascade connection the input signal must pass both a low- and a high-pass stage to reach the output. In the parallel connection in Fig. 110 the







 |K1 | ω   |K2 |  |H (j ω)| =  $  + $  ω2 + α12 ω2 + α22 high pass low pass Any sinusoid whose frequency falls in either passband will find its way to the output unscathed. An input sinusoid whose frequency falls in both stopbands will be attenuated. When α1 > α2 , the high-pass cutoff frequency is much higher than the low-pass cutoff frequency, and the overall transfer function has a bandstop gain response as shown in Fig. 111. At low frequencies (ω < α2 < α1 ) the gain of the high-pass function is negligible and the overall gain approaches |H (j ω)| → |K2 |/α2 , which is the passband gain of the lowpass function. At high frequencies (α2 < α1 < ω) the low-pass function is negligible and the overall gain approaches |H (j ω)| → |K1 |, which is the passband gain of the high-pass function. With a bandstop function the two passbands normally have the same gain, hence |K1 | = |K2 |/α2 . Between these two passbands there is a stopband. For ω > α2 the low-pass asymptote is |K2 |/ω, and for ω < α1 the high-pass asymptote is |K1 |ω/α1 . The asymptotes intersect at ω2 = α1 |K2 |/|K1 |. But equal gains in the two passband frequencies requires |K √ 1 | = |K2 |/α2 , so the intersection frequency is ω = α1 α2 . Below this frequency the stopband attenuation is determined by the low-pass

ELECTRIC CIRCUITS

945

Fig. 110 Parallel connection of high- and low-pass circuits.1

ω

with the following constraints: α1 = = Upper cutoff frequency: α2 = = |K1 K2 | Midband gain: = α2 Lower cutoff frequency:

Fig. 111 Bandstop gain characteristic.1

function and above this frequency the attenuation is governed by the high-pass function. Analysis of the transfer functions illustrates that the asymptotic gain plots of the first-order functions can help one understand and describe other types of gain response. The asymptotic response in Figs. 109 and 111 are a reasonably good approximation as long as the two first-order cutoff frequencies are widely separated. The asymptotic analysis gives insight to see that to study the passband and stopband characteristics in greater detail one needs to calculate gain and phase responses on a frequency range from a decade below the lowest cutoff frequency to a decade above the highest. This frequency range could be very wide, since the two cutoff frequencies may be separated by several decades. Mathematical and simulation software packages can produce very accurate frequency response plots. Example 34. Design a first-order bandpass circuit with a passband gain of 10 and cutoff frequencies at 20 Hz and 20 kHz. Solution. A cascade connection of first-order lowand high-pass building blocks will satisfy the design. The required transfer function has the form

H (j ω) = H1 (j ω) × H2 (j ω) K2 K1 j ω = j ω + α1 j ω + α2 high pass low pass

2π(20) 40π rad/s 2π(20x103 ) 4π × 104 rad/s 10

With numerical values inserted, the required transfer function is

40π × 104 jω [10] j ω + 40π j ω + 40π × 104 high pass gain low pass

H (j ω) =

This transfer function can be realized using the highpass/low-pass cascade circuit in Fig. 112. The first stage is a passive RC high-pass circuit and the third stage is a passive RL low-pass circuit. The noninverting op amp second stage serves two purposes: (a) It isolates the first and third stages, so the chain rule applies, and (b) it supplies the midband gain. Using the chain rule, the transfer function of this circuit is

R1 + R2 R/L jω j ω + 1/RC R1 j ω + R/L high pass gain low pass

H (j ω) =

c


946


Comparing this to the required transfer function leads to the following design constraints:

where for a series RLC circuit ω0 =

High-Pass Stage: RC C = 1/40π. Let RC = 100 k. Then C = 79.6 nF. Gain Stage: (R1 + R2 )/R1 = 10. Let R1 = 10 k. Then R2 = 90 k. Low-Pass Stage: RL /L = 40000π. Let RL = 200 k. Then L = 0.628 H.

K −ω2 + 2ζ ω0 j ω + ω02

H (j ω)HP =

−Kω2 −ω2 + 2ζ ω0 j ω + ω02

H (j ω)BP =

Kj ω −ω2 + 2ζ ω0 j ω + ω02

R ζ = 2

and

LC

(129)

HP

BP

LP

L

R

C

10K ζ = 0.05

ζ = 0.5 K

ζ=1

– 40dB/dec

ω0

0.1ω0

Fig. 114

C . L

Fig. 113 Series RLC connected as low-pass (LP), highpass (HP), or bandpass (BP) filter.

|T(jω)|

0.1k

The undamped natural frequency ω0 is related to the cutoff frequency in the high- and low-pass cases and is the center frequency in the band pass case. Zeta (ζ ) is the damping ratio and determines the nature of the roots of the equation that translates to how quickly a transition is made from the passband to the stopband. In the band pass case ζ helps define the bandwidth of the circuit, that is, B = 2ζ ω0 . Figure 113 shows how a series RLC circuit can be connected to achieve the transfer functions given in Eq. (129). The gain |H (j ω)| plots of these circuits are shown in Figs. 114–116.

6.5 Second-Order RLC Filters Simple second-order low-pass, high-pass, or band pass filters can be made using series or parallel RLC circuits. Series or parallel RLC circuits can be connected to produce the following transfer functions:

H (j ω)LP =

√

Second-order low-pass gain responses.1

10 ω0

ω

ELECTRIC CIRCUITS

947 |T(jω)|

10K

ζ = 0.05

ζ = 0.5

K

+40 dB/dec ζ=1

0.1K 0.1ω0

ω0

Fig. 115

10ω0

ω

Second-order high-pass gain responses.1

|T(jω)| 10K

ζ = 0.05

ζ = 0.5 K

ζ=1 +20 dB/dec

0.1K 0.1ω0

−20 dB/dec

ω0

Fig. 116

Second-order bandpass gain responses.1

ω 10ω0

948


6.6 Compound Filters Compound filters are higher order filters obtained by cascading lower order designs. Ladder circuits are an important class of compound filters. Two of the more common passive ladder circuits are the constant-k and the m-derived filters, either of which can be configured using a T section, π section, L section (Fig. 117), or combinations thereof, the bridge-T network and parallel-T network. Active filters are generally designed using first-, second-, or third-order modules such as the Sallen–Key configurations shown in Fig. 118a and b, and the Delyannies–Friend configurations shown in Fig. 118c and d. The filters are then developed using an algorithmic approach following the Butterworth, elliptical, or Tchebycheff realizations.

Z1

1

Z1

Z1 Z2

Z2

2 Z2 2

1 (a)

1

Z1/2 Z1/2

Z1/2

Z1/2 Z1/2

Z2

Z1/2

2

Z2

Z2

2

1 (b)

1

Z1 2Z2 2Z2

Z1 2Z2 2Z2

Z1 2Z2 2Z2

Fig. 118 Second-order configurations: (a) LP; (b) HP; (c) tuned; (d) Notch.1 2 2Z2

2

1 (c)

Fig. 117 Passive cascaded filter sections: (a) L section; (b) T section; (c) π section.

REFERENCES 1. 2.

Thomas, R., and Rosa, A. J., The Analysis and Design of Linear Circuits, 5th ed., Wiley, 2005. Thomas, R., and Rosa, A. J., The Analysis and Design of Linear Circuits, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1998.

CHAPTER 17 ELECTRONICS∗ John D. Cressler

Halit Eren

Georgia Institute of Technology Atlantita, Georgia

Curtin University of Technology Bentley, Western Australia Australia

Kavita Nair, Chris Zillmer, Dennis Polla, and Ramesh Harjani

N. Ranganathan and Raju D. Venkataramana

University of Minnesota Minneapolis, Minnesota

University of South Florida Tampa, Florida

Arbee L. P. Chen and Yi-Hung Wu National Tsing Hua University Hsinchu, Taiwan Republic of China

Robert P. Colwell

Konstantinos Misiakos

Andrew Rusek

NCSR “Demokritos” Athens, Greece

Oakland University Rochester, Michigan

Clarence W. de Silva

Alex Q. Huang and Bo Zhang

Intel Corporation Hillsboro, Oregon

Virginia Polytechnic Institute and State University Blacksburg, Virginia

University of British Columbia Vancouver, British Columbia Canada

Georges Grinstein and Marjan Trutschl University of Massachusetts Lowell Lowell, Massachusetts

1

BIPOLAR TRANSISTORS

950

DATA ACQUISITION AND CONVERSION

964

1.1

Double-Polysilicon Bipolar Technology

951

2.1

Sensors

965

1.2

Theory of Operation

953

2.2

Data Converters

969

1.3

High-Injection Effects

959

2.3

System Design Examples

971

2.4

Conclusion

978

1.4

Scaling Issues

961

1.5

Future Directions

962

2

∗

Reprinted from Wiley Encyclopedia of Electrical and Electronics Engineering, Wiley, New York, 1999, with permission of the publisher. Eshbach’s Handbook of Engineering Fundamentals, Fifth Edition Edited by Myer Kutz Copyright © 2009 by John Wiley & Sons, Inc.

949

950 3

4

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS DATA ANALYSIS

979

INTEGRATED CIRCUITS

1042

3.1

980

8.1

Basic Technologies

1043 1044

6

7

8

3.2

Data Analysis on Internet Data

984

8.2

MOS Switch

3.3

Improvement of Data Analysis Methods

988

8.3

IC Design Methodology

1047

3.4

Summary

989

8.4

Circuit Design

1049

990

8.5

Simulation

1051

8.6

Layout

1052

8.7

Fabrication

1054

8.8

Testing

1058

DIODES 4.1

Fundamentals of p –n Junctions

991

4.2

Doping Carrier Profiles in Equilibrium and Quasi-Neutral Approximation

993

4.3

Forward- and Reverse-Bias Conditions

994

4.4

Approximate Analytical Expressions in Steady State Transient Response of Diodes

4.5

5

Data Analysis Methods

9

MICROPROCESSORS

1060

995

9.1

Microprocessors and Computers

1060

998

9.2

Moore’s Law

1062

Microprocessor Architectures

1063

4.6

Heavy Doping Effects in Emitter

1000

9.3

4.7

Diodes of Nonconventional Transport

1001

9.4

Evolution of ISAs

1065

9.5

Coprocessors and Multiple Processors

1065

9.6

High-End Microprocessor Systems

1065

9.7

Future Prospects for Microprocessors

1065

ELECTRONIC COMPONENTS

1003

5.1

Materials and Passive Components

1003

5.2

Active Components

1008

5.3

Light Emitters and Displays

1016

5.4

Light Sensors

1019

INPUT DEVICES

1022

6.1

Devices

1022

6.2

Commonly Used Input Devices

1022

6.3

Conclusions

1026

10

OSCILLOSCOPES

1066

10.1

Analog Scopes

1067

10.2

Sampling Scopes

1069

10.3

Digital Scopes

1070

10.4

Technical Parameters and Limitations

1072

10.5

Oscilloscope Probes

1073

10.6

Oscilloscope Measurements

1074

10.7

Programmability of Oscilloscopes

1076

INSTRUMENTS

1026

7.1

Design, Testing, and Use of Instruments

1027

7.2

Instrument Response and Drift

1030

7.3

Measurement Errors and Error Control Systems

1032

11.2

JBS Rectifiers

1078

11.3

Switches

1080

11

POWER DEVICES

1077

11.1

1077

Rectifiers

7.4

Standards and Reference Materials

1034

7.5

Calibration, Calibration Conditions, and Linear Calibration Model

11.4

Integrable Lateral Power Devices

1094

1036

11.5

Isolation Technologies for PICs

1096

7.6

Analog and Digital Instruments

1037

11.6

SiC-Based Power Devices

1100

7.7

Control of Instruments

1040

7.8

Industrial Measuring Instruments

1042

1 BIPOLAR TRANSISTORS John D. Cressler

The basic concept of the bipolar junction transistor (BJT) was patented by Shockley in 19471 , but

REFERENCES

1103

BIBLIOGRAPHY

1109

the BJT was not experimentally realized until 1951.2 Unlike the point contact transistor demonstrated earlier in 1947, the BJT can be completely formed inside the semiconductor crystal, and thus it proved to be more manufacturable and reliable and better suited

ELECTRONICS

for use in integrated circuits. In a real sense, the BJT was the device that launched the microelectronics revolution and, hence, spawned the Information Age. Until the widespread emergence of complementary metal–oxide–semiconductor (CMOS) technology in the 1980s, the BJT was the dominant semiconductor technology in microelectronics, and even today represents a significant fraction of the global semiconductor market. At its most basic level the BJT consists of two backto-back pn junctions (p–n–p or n–p–n depending on the doping polarity), in which the intermediate n or p region is made as thin as possible. In this configuration the resultant three-terminal (emitter–base–collector) device exhibits current amplification (current gain) and thus acts as a “transistor” that can be used to build a wide variety of electronic circuits. Modern applications of the BJT are varied and range from high-speed digital integrated circuits in mainframe computers, to precision analog circuits, to radio frequency (RF ) circuits found in radio communications systems. Compared to CMOS, the BJT exhibits higher output current per unit length, larger transconductance (gm ) per unit length, faster switching speeds (particularly under capacitive loading), and excellent properties for many analog and RF applications (e.g., lower 1/f and broadband noise). Today, frequency response above 50 GHz and circuit switching speeds below 20 ps are readily attainable using conventional fabrication techniques. The primary drawback of BJT circuits compared to CMOS circuits lies in their larger dc power dissipation and increased fabrication complexity, although in applications requiring the fastest possible switching speeds, the BJT remains the device of choice. Figure 1 shows unloaded emittercoupled logic (ECL) gate delay for today’s technology

Fig. 1 Unloaded ECL gate delay (as function of publication date) showing rapid decrease in delay with technology evolution.

951

and indicates that state-of-the-art BJT technology is rapidly approaching 10 ps switching times. In this section we review the essentials of modern bipolar technology, the operational principles of the BJT, second-order high-injection effects, issues associated with further technology advancements, and some future directions. Interested readers are referred to Refs. 3–5 for review articles on modern BJT technology, and to Ref. 6 for an interesting historical perspective on the development of the BJT. 1.1 Double-Polysilicon Bipolar Technology In contrast to the depictions commonly found in many standard electronics textbooks, BJT technology has evolved radically in the past 15 years, from doublediffused, large geometry, non-self-aligned structures to very compact, self-aligned, “double-polysilicon” structures. Figure 2 shows a schematic cross section of a modern double-polysilicon BJT. This device has deeptrench and shallow-trench isolation to separate one transistor from the next, a p + polysilicon extrinsic base contact, an n+ polysilicon emitter contact, and an ion-implanted intrinsic base region. The two polysilicon layers (hence the name double-polysilicon) act as both diffusion sources for the emitter and extrinsic base dopants as well as low-resistance contact layers. In addition, to form the active region of the transistor, a “hole” is etched into the p + polysilicon layer, and afterwards a thin dielectric “spacer” oxide is formed. In this manner, the emitter and extrinsic base regions are fabricated without the need of an additional lithography step (“self-aligned”), thereby dramatically reducing the size of the transistor and hence the associated parasitic resistances and capacitances of the structure. The first double-polysilicon BJT structures appeared in the early 1980s7,8 and today completely dominate the high-performance BJT technology market. The reader is referred to Refs. 9–15 for specific BJT technology examples in the recent literature. The doping profile from the intrinsic region of a state-of-the-art double-polysilicon BJT is shown in Fig. 3. The transistor from which this doping profile was measured has a peak cutoff frequency of about 40 GHz14 , and is typical of the state of the art. The emitter polysilicon layer is doped as heavily as possible with arsenic or phosphorus, and given a sort rapid-thermal-annealing (RTA) step to out-diffuse the dopants from the polysilicon layer. Typical metallurgical emitter–base junction depths range from 25 to 45 nm in modern BJT technologies. The collector region directly under the active region of the transistor is formed by local ion implantation of phosphorus. A collector doping of about 1 × 1017 cm−3 at the base–collector junction is adequate to obtain a peak cutoff frequency of 40 GHz at a collector-to-emitter breakdown voltage (BVCEO ) of about 3.5 V, consistent with the needs of digital ECL circuits. The intrinsic

952


Fig. 2

Schematic device cross section of modern double-polysilicon self-aligned bipolar transistor.

base region is also formed by low-energy ion implantation of boron. Resultant base widths range from about 60 to 150 nm at the state of the art, with peak base doping levels in the range of 3–5 × 1018 cm−3 . A traditional (measurable) metric describing the base profile in a BJT is the intrinsic base sheet resistance (Rbi ), which can be written in terms of the integrated base

doping (Nab ) according to Rbi = q

−1

Wb

µpb (x)Nab (x) dx

(1)

0

In Eq. (1), µpb is the position-dependent hole mobility in the base and Wb is the neutral base width. Typical

Fig. 3 Measured secondary ion mass spectroscopy (SIMS) doping profile from ion-implanted base bipolar technology with 40-GHz peak cutoff frequency.14

ELECTRONICS

953

Rbi values in modern BJT technologies range from 10 to 15 kW/f . 1.2 Theory of Operation Basic Physics The BJT is in essence a barriercontrolled device. A voltage bias is applied to the emitter–base junction such that we modulate the size of the potential barrier seen by the electrons moving from emitter to base, and thus can (exponentially) modulate the current flowing through the transistor. To best illustrate this process, we have used a one-dimensional device simulator called SCORPIO.16 SCORPIO is known as a “drift diffusion” simulator because it solves the electron and hole drift diffusion transport equations self-consistently with Poisson’s equation and the electron and hole current-continuity equations (see, e.g., Ref. 6 for a formulation of these equations and the inherent assumptions on their use). These five equations, together with the appropriate boundary conditions completely describe the BJT. Figure 4 depicts a “toy” doping profile of the ideal BJT being simulated. Both the layer thicknesses and doping levels are consistent with those found in modern BJTs, although the constancy of the doping profile in each region is idealized and hence unrealistic. Figure 5 shows the resultant electron energy band diagram of this device at zero bias (equilibrium). The base potential barrier seen by the electrons in the emitter is clearly evident. The equilibrium carrier concentrations for each region are shown in Fig. 6. The majority carrier densities are simply given by the doping level in each region, while the minority carrier densities are obtained by use of the “law of mass action” according to the following:

pe0 =

n2ie Nde

pb0 = Nab

ne0 = Nde nb0 =

n2ib Nab

(emitter)

(2)

(base)

(3)

Fig. 5 Simulated zero-bias energy band diagram of hypothetical bipolar transistor depicted in Fig. 4.

Fig. 6 Simulated electron and hole concentrations of hypothetical bipolar transistor depicted in Fig. 4. Also shown are analytical calculations.

pc0 =

n2i0 Ndc

nc0 = Ndc

(collector)

(4)

where ni0 is the intrinsic carrier density, the subscripts e, b, and c represent the emitter, base, and collector regions, respectively, N is the doping density, and, n2ie = n2i0 e n2ib = n2i0 e

app

Eg c /kT app E b /kT g

app

Fig. 4 Doping profile of hypothetical bipolar transistor used in one-dimensional SCORPIO simulations.

= NC NV e−Eg /kT e = NC NV e−Eg /kT e app

app

Eg c /kT app E b /kT g

(5) (6)

where Ege and Egb represent the heavy-dopinginduced apparent bandgap narrowing.17 The resultant

954


density can be expressed as20 Jc =

q[eqVBE /kT − 1] Wb Nab (x) dx Dnb (x)n2ib (x) 0

(7)

We see then that the collector current density in a BJT depends on the details of the base doping profile [more specifically the integrated base charge, and, hence, Rbi given in Eq. (1)]. The base current density can be obtained in a similar manner, except that the physics of the polysilicon emitter contact must be properly accounted for.21,22 For the “transparent emitter domain” in which the holes injected from the base to emitter do not recombine before reaching the emitter contact, the base current density can be written as Fig. 7 Simulated collector and base current densities as function of emitter–base bias. Also shown are analytical calculations.

collector current density (JC ) and base current density (JB ) from this structure are shown in Fig. 7. Observe that the BJT exhibits useful current gain (β = JC /JB ) over a wide operating range. The basic operational principles of the BJT can be described as follows. If we imagine forward biasing the emitter–base junction, and reverse biasing the base–collector junction (i.e., forward-active mode), electrons from the heavily doped emitter are injected into and diffuse across the base region and are collected at the collector contact, thereby giving rise to a useful collector current. At the same time, if the base region is thin enough, the base current consists primarily of the back-injected hole current from base to emitter. Because the emitter is doped heavily with respect to the base, the ratio of forward-injected (emitter-tobase) electron current to back-injected (base to emitter) hole current is large (roughly equal to the ratio of emitter-to-base doping), and the BJT exhibits useful current gain. It is critical that the intermediate base region be kept as thin as possible because (a) we do not want electrons traversing the base to have sufficient time to recombine with holes before they reach the collector contact, and (b) the transit time of the electrons through the base typically limits the frequency response and, hence, the speed of the transistor. In the forward-active mode, a schematic representation of the magnitude of the various currents flowing in an ideal BJT is illustrated in Fig. 8.6 Current–Voltage Characteristics For simplicity, we will limit this discussion to the currents flowing in the BJT under forward-active bias. Other bias regimes (e.g., saturation) are not typically encountered in highspeed circuits such as ECL. The reader is referred to Refs. 17–19 for a discussion of other operating regimes. In this case, for a BJT with a positiondependent base doping profile, the collector current

JB = 0

We

q[eqVBE /kT − 1] Nde (x) dx Nde (We ) + 2 Dpe (x)nie Spe n2ie (We )

(8)

where Spe is the “surface recombination velocity” characterizing the polysilicon emitter contact.21 More detailed base current density expressions can be found in Refs. 21 and 22. Observe that in this transparent domain, the base current density depends on the specifics of the emitter doping profile as well as the influence of the polysilicon emitter contact. For position-independent base and emitter doping profiles, with no polysilicon emitter contact, Eqs. (8) simplify to their familiar forms: qDnb n2ib qVBE /kT ∼ e (9) Jc = Wb Nab qDpe n2ie qVBE /kT ∼ JB = e (10) Lpe Nde from which the ideal BJT current gain can be obtained app app Dnb Lpe Nde (Egb Nde −Egb )/kT e ∝ β∼ = Dpe Wb Nab Nab

(11)

Thus, the current gain of the BJT depends on the ratio of emitter-to-base doping level. Given this fact, it is not surprising that the actual ratio of emitter-to-base doping level is typically found to be 100 (refer to Fig. 4), a common value for β in modern technologies. Note as well, however, from Eq. (11) that the ideal current gain in a BJT is reduced by the exponential dependence of the heavy-doping-induced bandgap narrowing parameters (the exponent is negative because the emitter is more heavily doped than the base). This latter dependence is also responsible for determining the temperature dependence of β in a BJT.

ELECTRONICS

955

Fig. 8

Schematic current flow distributions in realistic bipolar transistor.

If one compares the measured I–V characteristics of a BJT with those expected from Eqs. (9)–(11), substantial deviations are typically observed, as depicted schematically in Figs. 9 and 10 (the dashed lines represent the ideal results). Referring to Fig. 9, at low current levels, base current nonideality is the result of emitter–base space-charge region recombination effects; at high current levels, the deviations are the result of various “high-injection” effects (discussed in what follows). Only over an intermediate bias range are ideal characteristics usually observed. Figure 11 shows typical measured I–V characteristics (a socalled Gummel plot) from the same 40-GHz profile depicted in Fig. 3.14 The inset of Fig. 3 shows the linear “output characteristics” of the BJT. The shape and doping level of the collector profile controls the breakdown characteristics of the device. In this case, the collector-to-emitter breakdown voltage (BVCEO ) is approximately 3.3 V, typical for a high-performance digital BJT technology.

Fig. 9 Schematic Gummel characteristics for realistic bipolar transistor.

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fT =

Wb2 kT (Cbe + Cbc ) + qIc ηD˜ nb −1 Wbc 1 We2 We + + + + rc Cbc βac Spe 2Dpe 2vs (13)

and fmax =

Fig. 10 Schematic current gain versus bias for realistic bipolar transistor.

fT 8πCbc Rb

(14)

In Eqs. (12)–(14), gm is the transconductance (∂IC /∂VBE ), Cbe and Cbc are the base–emitter and base–collector capacitances, τb , τe , and τc are the base, emitter, and collector transit times, respectively, vs is the saturation velocity (1 × 107 cm/s), η accounts for any doping-gradient-induced electric fields in the base, and Rb is the base resistance; fT and, hence, fmax is typically limited by τb in conventional Si–BJT technologies. A major advantage of ion-implanted base, double-polysilicon BJT technology is that the base width can be made very small (typically < 150 nm), and thus the intrinsic frequency response quite large. Figure 12 shows measured fT data as a function of bias current for a variety of device sizes for the doping profile shown in Fig. 3.14 ECL Gate Delay Due to its nonsaturating properties and high logical functionality, the ECL is the highest speed bipolar logic family and is in widespread use in the high-speed digital bipolar world. Figure 13 shows a simplified two-phase ECL logic gate. A common large-signal performance figure of merit is the unloaded

Fig. 11 Measured Gummel characteristics for scaled 0.25-µm double-polysilicon bipolar technology.14 Inset shows common-emitter breakdown characteristics of transistor.

Frequency Response The frequency response of a BJT is determined by both the intrinsic speed of the carriers through the device (transit time), as well as the parasitic resistances and capacitances of the transistor. Two primary figures of merit are used to characterize the frequency response of a BJT, the unity gain cutoff frequency (fT ) and the maximum oscillation frequency (fmax ). Using a small-signal hybrid-π model both fT and fmax can be derived (17), yielding

−1 1 1 = (Cbe + Cbc ) + τb + τe + τc fT = 2πτec gm (12)

Fig. 12 Measured cutoff frequency as function of collector14 current for scaled 0.25-µm double-polysilicon bipolar technology. Shown are a variety of device geometries.

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Fig. 15 Measured output waveform from ECL ring oscillator.

VL [a1 Cbc + a2 Cbe ICS

=

α

+ a3 Ccs + a4 Cw + · · ·]

(16)

1 1 ∝ ICS power

(17)

Fig. 13 Circuit schematic of two-phase. ECL gate.

while under high current (or power) conditions the ECL gate delay can be written as ECL gate delay, which can be measured using a “ring oscillator.” A ring oscillator is essentially a delay chain of ECL inverters with output tied back to its input, thus rendering the resultant circuit unstable (Fig. 14). From the period of the oscillation (Fig. 15), the average gate delay can be determined for a given bias current. Multiple ring oscillators can then be configured to operate at various bias currents, and hence the “power delay” characteristics of the BJT technology determined (average gate delay is plotted as a function of average power dissipation—or current in this case, because the supply voltage is constant). Figure 16 shows a typical measured ECL power delay curve.14 A minimum ECL gate delay of 20.8 ps is achieved with this technology. Observe that the speed of the ECL gate becomes faster as the average switch current increases, until some minimum value of delay is reached. To better understand the functional shape of the power delay curve, asymptotic expressions can be developed using a weighted time constant approach.23 Under low-current (or low-power) conditions, the ECL gate delay is given by τECL (low power) ∼ = RCC

n k=1

a k Ck

(15)

τECL (high power) ∼ = Cdiff

n

bk R k

(18)

k=1

=

qτec ICS [b1 Rbi + b2 Bbx kT + b3 Re + b4 Rc + · · ·] (19)

α ICS α power

(20)

In Eqs. (15)–(20), RCC is the circuit pull-up resistor, VL is the logic swing, ak and bk are delay “weighting factors,” ICS is the switch current, and Cdiff is the transistor diffusion capacitance. We see then that at low currents, the parasitic capacitances dominate the ECL delay with a delay that is reciprocally proportional to the power dissipation, whereas at high currents, the parasitic resistances dominate the ECL delay, yielding a delay that is proportional to power dissipation. It is thus physically significant to plot the log of the ECL delay as a function of the log of the power (or current), as shown in Fig. 16. Also shown in Fig. 16 are large-signal circuit simulation results using the compact model depicted in Fig. 17, which confirm the stated dependence of delay on power.

Fig. 14 Schematic representation of ECL ring oscillator circuit configuration.

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Fig. 16 The ECL power delay characteristics for scaled 0.25-µm double-polysilicon bipolar technology.14 Minimum delay of 20.8 ps is achieved. The ECL circuits were operated on 3.6/2.1-V power supplies at 500-mV logic swing. Fan-in (FI) and fan-out (FO) of one was used. Impact of transistor scaling from 0.90/0.20-µm lithography to 0.45/0.06-µm lithography is indicated. Also shown are circuit simulations calibrated to data using compact circuit model implemented in ASTAP.

Fig. 17

Compact circuit model used in ASTAP circuit simulations.

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1.3 High-Injection Effects

Substantial deviations from ideal behavior occur for BJTs operating at high current densities (as a rule of thumb, for JC ∼ 1.0 mA/µm2 in a modern highperformance technology). This deviation from simple theory can be observed in the premature roll-off of both the current gain and the cutoff frequency at high current densities, as shown in Figs. 10–12. These socalled high-injection effects are particularly important because most high-performance BJT circuits will be biased at high current densities in order to achieve maximum transistor performance. High injection in a BJT can generally be defined as that current density at which the injected minority carrier density (e.g., electrons in the base) becomes comparable to the local doping density. High-injection effects are generally the result of a number of competing physical mechanisms in the collector, base, and emitter regions and are thus difficult to analyze together theoretically. In this work we will simply emphasize the physical origin of each high-injection phenomenon region by region, discuss their impact on device performance, and give some rule-of-thumb design guidelines. The interested reader is referred to Ref. 6 for a more indepth theoretical discussion. Collector Region Collector region high-injection effects in BJTs can be divided into two separate phenomena: (a) Kirk effect, sometimes referred to as “base push-out”24 ; and (b) quasi-saturation. The physical origin of the Kirk effect is as follows. As the collector current density continues to rise, the electron density in the base–collector space-charge region is no longer negligible and modifies the electric field distribution in the junction. At sufficiently high current density, the (positive) background space charge

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due to the donor doping in the collector (N+dc ) is compensated by the injected electrons, and the electric field in the junction collapses, thereby “pushing” the original base region deeper into the collector (Figs. 18 and 19). Because both β and fT depend reciprocally on Wb , this injection-induced increase in effective base width causes a strong degradation in both parameters. Approximate theoretical analysis can be used to determine the critical current density at which the Kirk effect is triggered, resulting in a BJT design equation JKirk

2εVBC ∼ = qvs Ndc 1 + 2 qWepi Ndc

(21)

From Eq. (21) it is apparent that increasing the collector doping level is the most efficient method of delaying the onset of the Kirk effect, although this will have a detrimental impact on the BVCEO and collector–base capacitance of the transistor. As the Kirk effect is typically the limiting high-injection phenomenon in modern high-performance BJTs, a fundamental tradeoff thus exists between peak fT and BVCEO . The second major collector region high-injection phenomenon is called “quasi-saturation.” At a basic level, quasi-saturation is the result of the finite collector resistance of the n-type epi layer separating the base from the heavily doped subcollector in a BJT. At sufficiently high current levels, the infrared (IR) drop associated with the collector epi becomes large enough to internally forward bias the base–collector junction, even though an external reverse bias on the collector is applied. For instance, for a collector resistance of 1 k and a collector current of 2 mA, an internal voltage drop of 2 V is obtained. If the BJT

Fig. 18 Simulated electron profile in bipolar transistor at both low injection (3.2 µA/µm2 ) and high injection (1.05 mA/µm2 ).

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Fig. 19 Simulated hole profile in bipolar transistor at both low injection (3.2 µA/µm2 ) and high injection (1.05 mA/µm2 ). Observe that at high-injection levels hole profile in base exceeds local doping level (as indicated by low-injection result), and holes are present in (n-type) collector region.

were biased at a base–collector reverse voltage of 1 V, then the internal base–collector junction would be forward-biased by 1 V, artificially saturating the transistor. With both base–emitter and base–collector junctions forward biased, the dc signature of quasisatuation is a strong increase in base current together with a “clipping” of the collector current. Dynamically, quasi-saturation has a strong negative impact on the fT and, hence, circuit speed because excess minority charge is injected into the base region under saturation. Theoretically, quasi-saturation is difficult to model because the resistance of the epi layer is strongly bias-dependent and the collector doping profile in real devices is highly position dependent. In a welldesigned high-performance BJT, the Kirk effect is much more important than quasi-saturation. Base Region High injection in the base region of a BJT leads to two major degradation mechanisms: (a) the Webster–Rittner effect,25,26 sometimes known as “base conductivity modulation,” and (b) emitter current crowding. In the Webster–Rittner effect, the large electron density in the base region under high injection is no longer small compared to the doping in the base. To maintain charge neutrality in the neutral base, the hole density must therefore rise (refer to Figs. 18 and 19), changing the (low-injection) Shockley boundary condition at the emitter–base junction, and effectively doubling the electron diffusivity in the base. The result is a different voltage dependence of the collector current, which changes to one-half the slope of the exponential low-injection collector current

according to q2Dnb nib (0) qVBE /2kT ∼ JC (Webster–Rittner) = e Wb (22) This slope change of JC has a detrimental impact on the current gain, although in practice for highperformance BJTs, the Kirk effect typically onsets before the Webster–Rittner effect because the base is much more heavily doped than the collector. Emitter current crowding is the result of the finite lateral resistance associated with the intrinsic base profile (i.e., Rbi ). Because the collector current depends on the actual base–emitter voltage applied at the junction itself, rather than that applied at the base and emitter terminals, large base currents flowing at highinjection levels can produce a lateral voltage drop across the base. This yields a lateral distribution in the actual base–emitter voltage at the junction, resulting in higher bias at the emitter periphery than in the center of the device. In essence, then, the collector current “crowds” to the emitter edge where the static and dynamic properties of the device are generally worse, and can even produce “thermal runaway” and catastrophic device burnout. This is typically only a problem in large geometry power transistors, not highspeed digital technologies. In addition, as the base current is a factor of β smaller than the collector current, emitter current crowding is not generally a problem unless there is very large base resistance in the device. Emitter Region Because it is very heavily doped, the emitter region in modern BJTs always operate in

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Fig. 20 The ECL power delay characteristics showing impact of idealized scaling.

low injection. Thus, the only significant emitter region high-injection effect is the result of the finite emitter resistance of the transistor. Because polysilicon emitter contacts in fact exhibit reasonably high specific contact resistance (e.g., 20–60 µm2 ), however, emitter resistance (RE ) can be a serious design constraint. Emitter resistance degrades the collector and base currents exponentially as it decreases the applied base–emitter voltage according to IC = IC0 eq(VBE −IE RE )/kT IB = IB0 e

q(VBE −IE RE )/kT

(23) (24)

For instance, for a 1.0-µm2 emitter area transistor operating at a collector current of 1.0 mA, a specific emitter contact resistance of 60 µm2 results in an emitter–base voltage loss of 60 mV, yielding a 10× decrease in collector current. Proper process optimization associated with the polysilicon emitter contact is key to obtaining a robust high-speed BJT technology, particularly as the emitter geometry shrinks. 1.4 Scaling Issues

Device miniaturization (“scaling”) has been a dominant theme in bipolar technology over the past 15 years, and has produced a monotonic decrease in circuit delay over that period (refer to Fig. 1). In general, optimized BJT scaling requires a coordinated reduction in both lateral and vertical transistor dimensions, as well as a change in circuit operating point.23 Unlike in CMOS technology, BJT circuit operating voltages (for conventional circuits such as ECL) cannot be scaled because the junction builtin voltage is only weakly dependent on doping. The

evolution of BJT technology from non-self-aligned, double-diffused transistor structures to self-aligned, ion-implanted, double-polysilicon transistor structures was the focus for BJT scaling in the 1980s. During the 1990s more emphasis has been placed on vertical profile scaling and a progression toward both forms of advanced lithography [e.g., deep ultraviolet (UV) or electron beam lithography], low-thermal budget processing, and structural innovation to continue the advances in circuit speed over time. Figure 20 represents an idealized ECL power delay curve and indicates the three principle regions that require attention during optimized scaling. In region (a), which is dominated by parasitic transistor capacitances [see Eqs. (15)–(17)], a reduction in lithography, and hence decrease in transistor size, is effective in reducing circuit delay at low current levels. Region (b) is dominated by the intrinsic speed of the transistor (i.e., τec ). Thinning the vertical profile, particularly the base width, is key to reducing the ECL delay at intermediate current levels. The evolution of ion implantation has proven key to realizing viable sub-150-nm metallurgical base widths in modern BJT technologies. In region (c), the ECL delay is dominated by base resistance and high-injection roll-off of the frequency response of the device [Eqs. (18)–(20)]. Doping the base and collector regions more heavily is successful in improving the delay at very high current levels, although tradeoffs exist. For instance, doping the base more heavily decreases the peak fT of the transistor (due to a lower electron mobility), and, hence, degrades the speed in region (b) at intemediate current levels. In addition, increasing the collector doping level to improve the high-injection performance in region (c) effectively increases the collector–base capacitance, degrading the ECL delay in region (a) at low-current levels. Optimized scaling is thus a complex tradeoff between many different profile design issues. Clever solutions to certain scaling tradeoffs have emerged over the years and include, for instance, the now pervasive use of the so-called self-aligned, implanted collector (SIC) process. In an SIC process (see Ref. 10), phosphorus is implanted through the emitter window in the base polysilicon layer (either before or after sidewall spacer formation) to increase the collector doping level locally under the intrinsic device without increasing the collector–base capacitance in the extrinsic transistor. Figure 21 and 22 show the results of a recent BJT lithographic scaling experiment.14 In this study a comparison was made between BJTs fabricated using three different lithographies (0.09-µm/0.20-µm lithographic linewidth/lithographic overlay, 0.45 µm/0.10 µm, and 0.45 µm/0.06 µm). The latter two processes used advanced electron-beam lithography. As can be seen, the impact of scaling on device parameters is dramatic, resulting in an expected improvement in ECL delay across the entire power delay characteristic, and a minimum ECL gate delay of 20.8 ps (Fig. 16).

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Fig. 21 Comparison of measured device parameters as function of scaling for: (1) 0.90/0.20 µm (lithographic image/overlay); (2) 0.45/0.10 µm; and (3) 0.45/0.06 µm transistors.14 Lumped ASTAP parameters are extracted from calibrated simulations of ECL ring oscillator data.

1.5 Future Directions

steadily eroded. This is due to both the improved performance of FET technology as gate lengths are scaled into the submicron domain, the widespread emergence of CMOS with its low power delay product, and the decreased cost associated with CMOS ICs compared to competing bipolar technologies. To confront this situation, many bipolar + CMOS (BiCMOS) technologies have been developed that seek to combine low-power CMOS with high-performance BJTs. The reader is referred to Ref. 4 for an examination of the process integration issues associated with modern BiCMOS technologies. In addition, there are several areas of current research with the potential to extend BJT technology well into the twenty-first century; they include: (a) complementary bipolar technology, (b) Silicon-oninsulator (SOI) bipolar technology, and (c) silicon– germanium (SiGe) bipolar technology. Each of these three research areas seeks to improve either the power dissipation associated with conventional BJT circuit families such as ECL, or improve the transistor performance to levels not possible in Si BJTs and thus to capture new and emerging IC markets.

Despite the continual improvements in speed that BJT technology has enjoyed over the past 15 years, and the inherent superiority of the analog and digital properties of BJTs compared to field-effect transistors (FETs), the world market for BJT integrated circuits (ICs) has

Complementary Bipolar Technology Complementary bipolar (C-bipolar) technology, which combines n–p –n and p –n–p transistors on the same chip, has been used for decades. In conventional usage, the n–p –n BJT is a standard, vertical high-performance

Nonetheless, practical limits do exist for conventional ion-implanted, double-polysilicon BJT technology. Obtaining metallurgical basewidths below 80–100 nm with reasonable base resistance using lowenergy ion implantation is very difficult and places a practical limit of about 40–50 GHz on the resultant fT of such transistors (see Fig. 12, which corresponds to the doping profile shown in Fig. 2). In addition, circuit operating voltages limit the useful BVCEO of the transistor to about 3.0 V, and thus place a practical limit on collector doping levels of about 1 × 1017 cm−3 and a consequent maximum operating current density of about 1–2 mA/µm2 . The emitter junction depth (and, hence, the thermal process associated with the polysilicon emitter) is limited to about 25–30 nm because the emitter–base space charge region must lie inside the single-crystal emitter region to avoid the generation/recombination centers associated with the heavily defective polysilicon region. More advanced profiles can be obtained using epitaxial growth techniques, as will be discussed in the next section.

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Fig. 22 Scaled comparison of (a) a 0.90/0.20-µm (lithographic image/overlay) transistor with (b) a 0.90/0.06-µm transistor.

transistor, while the p –n–p BJT is typically a slowspeed lateral device used only in analog circuits such as current sources where high speed is unnecessary. Modern implementations of C-bipolar technology, on the other hand, combine a high-performance vertical n–p –n BJT and a high-performance vertical p –n–p BJT (see, e.g., Refs. 27 and 28). The resulting IC technology, though inherently more complex than a traditional n–p –n only BJT technology, opens many new possibilities for novel high-speed, low-power circuit families. New C-bipolar circuit families such as accoupled push–pull emitter-coupled logic (ACPPECL) and nonthreshold logic with complementary emitter–follower (NTL-CEF) offer dramatic improvements in power delay product compared to conventional ECL (Fig. 23). Silicon-on-Insulator Bipolar Technology Siliconon-insulator IC technologies have existed since the 1960s but have emerged recently as a potential scaling path for advanced CMOS technologies. In SOI technology, a buried oxide dielectric layer is placed below the active Si region, either by ion implantation (SIMOX) or by wafer bonding (BESOI). For the

CMOS implementation, the active Si region is made thin, so that it is fully depleted during normal device operation, resulting in improved subthreshold slope, better leakage properties at elevated temperatures, and improved dynamic performance due primarily to the reduction in parasitic source/drain capacitance. Given this development, it is natural to implement a lateral BJT together with the SOI-CMOS to form an SOIBiCMOS technology. While lateral BJTs are not generally considered high-speed transistors, the reduction in parasitic capacitance in the lateral BJT, together with clever structural schemes which allow very aggressive base widths to be realized, have resulted in impressive performance.29 SiGe Bipolar Technology Attempts to reduce the base widths of modern BJT technologies below 100 nm typically rely on epitaxial growth techniques. A recent high-visibility avenue of research has been the incorporation of small amounts of germanium (Ge) into these epitaxial films to tailor the properties of the BJT selectively while maintaining compatibility with conventional Si fabrication techniques. The resultant

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Fig. 23 Measured power delay characteristics from an advanced complementary bipolar technology.27 Three circuit families are compared: (1) conventional (npn-only) ECL, (2) ACPP-ECL, and (3) nonthreshold logic with complementary emitter–(NTL-CEF). The NTL-CEF circuit achieved a minimum power delay product of 12 fJ.

device, called an SiGe heterojunction bipolar transistor (HBT), involves introducing strained epitaxial SiGe alloys into the base region of the transistor, and represents the first practical bandgap-engineered device in Si technology (refer to Refs. 30–32, and references contained within, for reviews of SiGe HBTs). Compared to an Si BJT with an identical doping profile, the SiGe HBT has significantly enhanced current gain, cutoff frequency, Early voltage (output conductance), and current gain Early voltage product, according to Refs. 31 and 32, βSiGe JCSiGe = JCSi βSi = γη

Eg,Ge (grade)/kT eEg,Ge (0)/kT 1 − e−Eg,Ge (grade)/kT

2 τb,SiGe fT ,Si α = τb,Si fT ,SiGe η

kT Eg,Ge (grade)

(25)

−Eg,Ge (grade)/kT

1−e Eg,Ge (grade)/kT (26) −Eg,Ge (grade)/kT 1−e Eg,Ge (grade)/kT (27)

1−

VA,SiGe = eEg,Ge (grade)/kT VA,Si

βVA |SiGe = γ ηeEg,Ge (0)/kT eEg,Ge grade/kT βVA |Si

(28)

where Eg,Ge (0) is the Ge-induced band offset at the emitter–base junction, Eg,Ge (grade) = Eg,Ge

(Wb ) − Eg,Ge (0) is the base bandgap grading factor, and γ , η are the strain-induced density-of-states reduction and mobility enhancement factors, respectively. With its improved transistor performance compared to Si BJTs and compatibility with standard Si fabrication processes, SiGe HBT technology is expected to pose a threat to more costly compound semiconductor technologies such as GaAs for emerging high-speed communications applications. Figure 24 shows a representative SiGe doping profile. Observe that the Ge is introduced only in the base region of the transistor. Experimental results comparing a SiGe HBT and a Si BJT having identical layout and doping profile are shown in Figs. 25 and 26 and indicate that significant enhancements compared to comparably designed Si devices are possible. It is now clear that cutoff frequencies well above 300 GHz are possible using SiGe HBT technology, and thus SiGe represents the next evolutionary step in Si BJT technology. 2

DATA ACQUISITION AND CONVERSION

Kavita Nair, Chris Zillmer, Dennis Polla, and Ramesh Hargani Data acquisition and conversion pertain to the generation of signals from sensors, their conditioning, and their conversion into a digital format. In this section we describe typical sensors that generate signals and examples of data converter topologies suitable for sensor interfaces. We restrict ourselves to integrated implementations of sensors and sensor interface circuits. In particular, we target sensors and sensor interfaces that are compatible with CMOS fabrication technologies.

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Fig. 24 Measured secondary ion mass spectroscopy (SIMS) doping profile comparing a 60-GHz cutoff frequency epitaxial SiGe base bipolar technology with an aggressive (40-GHz cutoff frequency) ion-implanted (I/I) base bipolar technology.

Fig. 25 Measured Gummel characteristics for SiGe and Si transistors with comparable doping profiles. Expected enhancement in collector current (4.51×) can be observed.

This section is organized as follows. First, we describe some examples of sensors and sensor interfaces; then we describe some sample data converter topologies. After that, we provide two complete design examples. 2.1 Sensors

Sensors are devices that respond to a physical or chemical stimulus and generate an output that can be used as a measure of the stimulus. The sensed inputs can be of many types: chemical, mechanical, electrical, magnetic, thermal, and so on. The input signal sensed by the sensor is then processed (amplified, converted from analog to digital, etc.) by some signal conditioning electronics, and the output transducer converts this

Fig. 26 Measured cutoff frequency as a function of collector current for SiGe and Si transistors with comparable doping profiles. The expected enhancement in collector current (1.71×) can be observed.

processed signal into the appropriate output form. The primary purpose of interface electronics is to convert the sensor’s signal into a format that is more compatible with the electronic system that controls the sensing system. The electric signals generated by sensors are usually small in amplitude. In addition to this, sensors often exhibit errors, such as offsets, drift, and nonlinearities that can be compensated for with the correct interface circuitry. Analog elements have been improved substantially to achieve high speed and high accuracy; however, for many applications digital is still the preferred format. The sensors yield a wide variety of electric output signals: voltages, currents, resistances, and capacitances. The signal conditioning

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ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Control parameters

Sensed parameters

two complete sensor systems that include an acoustic emission sensor and a temperature sensor.

Actuators

Sensors

Driver

Amp

DAC

ADC

Resistive Sensors Sensors based on the variation of electric resistance are called resistive sensors. They can be further classified according to the physical quantity that they measure: thermal, magnetic, optical, and so on. A potentiometer is a simple resistance measurement device in which the resistance is proportional to its length. However, the linearity of a potentiometer is limited because its resistance is not perfectly uniform. The resistance value also drifts with temperature. Applications of potentiometers are in the measurement of linear or rotary displacements. Another simple and commonly used resistive sensor is the strain gauge, which is based on the variation of the resistance of a conductor or semiconductor when subjected to a mechanical stress. The variation in the resistance of a metal is given by34

Microcomputer control

Higher-level control Fig. 27 Overall system architecture of a sensor–actuator control system.

R = R0 (1 + Gε) circuitry modifies the input signal into a format suitable for the follow-on data converter. Figure 27 shows the system architecture for a sensor–actuator-based control system. The sensor(s) senses the external physical and chemical parameters and converts them into an electrical format. The sensed data are processed and digitized using integrated circuitry and transmitted to the host controller. The host uses this information to make the appropriate decisions, and information is fed back to the external environment through a set of actuators.33 These microprocessor-based controllers have revolutionized the design and use of instrumentation systems by allowing system operation to be defined in software, thus permitting a substantial increase in signalprocessing and user–interface features. In general, a power supply is also connected to these blocks but is not explicitly shown in Fig. 27. If a sensor can provide a signal without a power supply, it is referred to as a self-generating sensor. Integrated sensors are used in many applications, including automotive, manufacturing, environmental monitoring, avionics, and defense. In the past few years, integrated sensors that monolithically combine the sensor structure and some signal-processing interface electronics on the same substrate have begun to emerge. By combining microsensors and circuits, integrated smart sensors increase accuracy, dynamic range, and reliability and at the same time reduce size and cost. Some examples of semiconductor sensors are pressure sensors used in pneumatic systems, magnetic sensors used in position control, temperature sensors used in automotive systems, chemical sensors used in biological diagnostic systems, and acoustic emission sensors used in structural diagnostics. We now illustrate the use of sensors and sensor interfaces with the two most common types of sensors: resistive and capacitive sensors. We then describe

(29)

where R0 is the resistance when there is no applied stress, G is the gauge factor, and ε is the strain. There are a number of limitations on strain gauges, such as temperature dependence, light dependence, and inaccuracies in the measurement of a nonuniform surface; but in spite of these limitations, they are among the most popular sensors because of their small size and linearity. Some of the applications of the strain gauge are in measuring force, torque, flow, acceleration, and pressure. Figure 28 shows a micromachined piezoresistive cantilever beam used as a strain gauge sensor. Strain gauges are capable of detecting deformations as small as 10 µm or lower. A resistance temperature detector (RTD) is a temperature detector based on the variation in electric resistance. An increase in temperature increases the vibrations of atoms around their equilibrium positions,

Fig. 28 Micromachined piezoresistive cantilever beam used as strain gauge sensor.

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and this increases the resistance in a metal: Thus there is a positive temperature coefficient of resistance. The complete temperature dependence can be expressed34 as R = R0 (1 + α1 T + α2 T 2 + · · · + αn T n )

(30)

where T is the temperature difference from the reference and R0 is the resistance at the reference temperature. The main advantages of these sensors are their high sensitivity, repeatability, and low cost. There are some limitations too. First, to avoid destruction through selfheating, the RTD cannot measure temperatures near the melting point of the metal. Second, the change in temperature may cause physical deformations in the sensor. Additionally, for each metal there is only a small range over which the RTD is linear. The most common metals used for RTDs are platinum, nickel, and copper. Thermistors are also temperature-dependent resistors but are made of semiconductors rather than metals. The temperature dependence of the resistance of a semiconductor is due to the variation in the available charge carriers. Semiconductors have a negative temperature coefficient, as the resistance is inversely proportional to the number of charge carriers. The temperature dependence of thermistors is given by34 1 1 − (31) RT = R0 exp B T T0 where T0 is the reference temperature, R0 is the resistance at T0 , and B is the characteristic temperature of the material, which itself is temperature dependent. The limitations and advantages of a thermistor are similar to those of an RTD, except that the thermistor is less stable. There are many types of thermistors available, and each type has its own applications. The foil and bead types are suitable for temperature measurement, whereas the disk and rod types are suitable for temperature control. Some of the applications of thermistors are in the measurement of temperature, flow, level, and time delay. Two simple applications of thermistors are discussed below. Light-dependent resistors, or LDRs, are devices whose resistance varies as a function of the illumination. LDRs are also known as photoconductors. The conductivity is primarily dependent on the number of carriers in the conduction band of the semiconductor material used. The basic working of the photoconductor is as follows. The valence and conduction bands in a semiconductor are quite close to each other. With increased illumination, electrons are excited from the valence to the conduction band, which increases the conductivity (reduces the resistance). The relation between resistance and optical radiation or illumination is given by34 R = AE −α

An important limitation of LDRs is their nonlinearity. Also, their sensitivity is limited by fluctuations caused by changes in temperature. Finally, the spectral response of LDRs is very narrow and primarily depends on the type of material used. Some of the most common LDRs are made of PbS, CdS, and PbSe. Some applications of LDRs are shutter control in cameras and contrast and brightness control in television receivers. Measurement Techniques for Resistive Sensors. Various measurement techniques can be used with resistive sensors. The basic requirement for any measurement circuitry is a power supply to convert the change in resistance into a measurable output signal. In addition, it is often necessary to custom-build interface circuits for some sensors. For example, we may be required to add a linearization circuit for thermistors. Resistance measurements can be made by either the deflection method or the nulling method. In the deflection method the actual current through the resistance or the voltage across the resistance is measured. In the nulling method a bridge is used. The two-readings method is a fundamental approach to resistance measurement. A known resistance is placed in series with the unknown resistance as shown in Fig. 29. The voltage is then measured across each of them. The two voltages can be written as

VK =

V RK RK + RU

(33)

VU =

V RU RK + RU

(34)

where V is the supply voltage, VK and RK are the known voltage and resistance, and VU and RU are the unknown voltage and resistance. Thus from the above equations RU can be written as follows: RU = RK

VU VK

(35)

RK

VK

RU

VU

V

(32)

where A and α are process constants, R is the resistance, and E is the illumination.

Fig. 29 Two-readings method for resistance measurement.

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R1 V

R2 G R3 = R0(1+x)

R4 Fig. 30 Simple method.

Wheatstone

bridge

measurement

A similar method is the voltage divider in which the unknown resistance is once again calculated from known voltages and resistances. It is easier to resolve small voltage changes for low voltages than it is for high voltages. Thus to measure small changes in resistance, another voltage divider is placed in parallel to the one with the sensor. The parallel voltage dividers are designed to give the same voltage for no input. Thus the signal obtained by taking the difference between their output signals is totally dependent on the measured signal. This method of measuring small changes using parallel voltage dividers is called the Wheatstone bridge method.34,34a,35,36 A simple Wheatstone bridge measurement method is shown in Fig. 30. The Wheatstone bridge is balanced with the help of a feedback system, which adjusts the value of the standard resistor until the current through the galvanometer is zero. Once this is done, the value for R3 is given by R3 = R4

R2 R1

(36)

Thus the resistance R3 is directly proportional to the change required in R4 in order to balance the circuit. The Wheatstone bridge can also be used for deflection measurement. In this case, instead of measuring the change needed to balance the bridge, the voltage difference between the bridge outputs is measured or the current through the center arm is measured. This method is shown in Fig. 30. When the bridge is completely balanced (i.e., x = 0), k is defined as follows: k=

R2 R1 = R4 R0

Capacitive Sensors Recently capacitive sensors have gained popularity. They generally exhibit lower temperature sensitivity, consume less power, and provide an overall higher sensor sensitivity with higher resolution than resistive sensors. For these reasons they have begun to show up in areas where resistive sensors were the norm. They are used in many applications such as pressure sensors and accelerometers. Capacitive sensors typically have one fixed plate and one moving plate that responds to the applied measurand. The capacitance between two plates separated by a distance d is given by C = εA/d, where ε is the dielectric constant and A is the area of the plate. It is easily seen that the capacitance is inversely proportional to the distance d. For capacitive sensors there are several possible interface schemes. Figure 31 shows one of the most common capacitive sensor interfaces. The circuit is simply a charge amplifier, which transfers the difference of the charges on the sensor capacitor Cs and the reference capacitor Cref to the integration capacitor CI . If this interface is used in a pressure sensor, the sensing capacitor Cs can be written as the sum of the sensor capacitor value Cs0 at zero pressure and the sensor capacitor variation Cs (p) with applied pressure: Cs = Cs0 + Cs (p). In many applications Cs0 can be 5 to 10 times larger than the full-scale sensor capacitance variation Cs (p)max ; the reference capacitor Cref is used to subtract the nominal value of the sensor capacitor at half the pressure range, which is Cref = Cs0 + Cs (p)max /2. This ensures that the transferred charge is the charge that results from the change in the capacitance. This results in a smaller integration capacitor and increased sensitivity. This type of capacitive interface is insensitive to the parasitic capacitance between the positive and negative terminals of the opamp, since the opamp maintains a virtual ground across the two terminals of the parasitic capacitor. This type of interface is also much faster than most other capacitive interfaces; its speed of operation is determined by the opamp’s settling time. This technique also allows for the amplifier’s offset and flicker noise to be removed very easily by using correlated double sampling or chopper stabilization. The resolution of this interface is in most cases limited by kT/C noise and charge injection due to the switches. q1

(37)

Thus the voltage difference between the outputs can be written as follows: kx R4 R3 =V − V0 = V R2 +R3 R1 +R4 (k+1)(k+1+x) (38) The maximum sensitivity for very small changes in x is obtained when k = 1.

CS Vref p

q1

q2 Cref

Vref n

q1

CI − +

CI

ADC

q2

Fig. 31

Capacitive sensor interface.

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digital logic circuit that performs the binary search. This logic circuit is called the successive approximation register (SAR). The output of the SAR is used to drive the digital-to-analog converter (DAC) that is connected to the positive input of the comparator. During the first clock period, the input is compared with the most significant bit (MSB). For this, the MSB is temporarily raised high. If the output of the comparator remains high, then the input lies somewhere between zero and Vref /2 and the MSB is reset to zero. However, if the comparator output is low, then the input signal is somewhere between Vref /2 and Vref and the MSB is set high. During the next clock period the MSB-1 bit is evaluated in the same manner. This procedure is repeated so that at the end of N clock periods all N bits have been resolved. The charge redistribution implementation of the successive approximation methodology is the most common topology in metal–oxide–semiconductor (MOS) technologies.38 The circuit diagram for a 4-bit charge redistribution converter is shown in Fig. 33. In this circuit the binary weighted capacitors {C, C/2, . . . , C/8} and the switches {S1 , S2 , . . . , S5 } form the 4-bit scaling DAC. For each conversion the circuit operates as a sequence of three phases. During the first phase (sample), switch S0 is closed and all the other switches S1 , S2 , . . . , S6 are connected so that the input voltage Vin is sampled onto all the capacitors. During the next phase (hold), S0 is open and the bottom plates of all the capacitors are connected to ground, that is, switches S1 , S2 , . . . , S5 are switched to ground. The voltage Vx at the top plate of the capacitors at this time is equal to −Vin , and the total charge in all the capacitors is equal to −2CVin . The final phase (redistribution) begins by testing the input voltage against the MSB. This is accomplished by keeping the switches S2 , S3 , . . . , S5 connected to ground and switching S1 and S6 so that the bottom plate of the largest capacitoris connected to Vref . The voltage at the top plate of the capacitor is equal to

There are a number of other sensor types, and two more will be discussed later in this section. However, we first describe the most common data converters that are used as part of sensor interfaces. 2.2 Data Converters

The analog signals generated and then conditioned by the signal conditioning circuit are usually converted into digital form via an analog-to-digital converter (ADC). In general, most of the signals generated by these sensors are in the low-frequency region. For this reason, certain data converter topologies are particularly well suited as sensor interface subblocks. These include the charge redistribution implementation of the successive approximation converter, along with incremental and sigma–delta converters.37 In the following we shall briefly describe successive approximation (incremental) and sigma–delta converters. Incremental and sigma–delta converters are very similar, and the details of the former are later described extensively as part of a sample system design. Successive Approximation Converter A block diagram for the successive approximation converter is shown in Fig. 32. The successive approximation topology requires N clock cycles to perform an N-bit conversion. For this reason, a sample-and-held (S/H) version of the input signal is provided to the negative input of the comparator. The comparator controls the

Vin

S/H

− Control

+

Successive approximation register

DAC

N bit output

Vx =

Fig. 32 Successive approximation converter: block diagram.

S0

Vref − Vin 2

+ C S1

C 2 S2

C 4 S3

C 8 S4

C 8

−

S5

Vin Vref

S6

Fig. 33 Charge redistribution implementation of successive approximation architecture.

(39)

970


If Vx > 0, then the comparator output goes high, signifying that Vin < Vref /2, and switch S1 is switched back to ground. If the comparator output is low, then Vin > Vref /2, and S1 is left connected to Vref and the MSB is set high. In a similar fashion the next bit, MSB-1, is evaluated. This procedure is continued until all N bits have been resolved. After the conversion process the voltage at the top plate is such that Vref Vref Vref Vref Vx = −Vin + b3 1 + b2 2 + b1 3 + b0 4 2 2 2 2 (40a) −

Vref < Vx < 0 24

(40b)

where bi is 0 or 1 depending on whether bit i was set to zero or one, and LSB is the least significant bit. One of the advantages of the charge redistribution topology is that the parasitic capacitance from the switches has little effect on its accuracy. Additionally, the clock feed-through from switch S0 only causes an offset, and those from switches S1 , S2 , . . . , S5 are independent of the input signal because the switches are always connected to either ground or Vref . However, any mismatch in the binary ratios of the capacitors in the array causes nonlinearity, which limits the accuracy to 10 or 12 bits. Self-calibrating39 techniques have been introduced that correct for errors in the binary ratios of the capacitors in charge redistribution topologies. However, these techniques are fairly complex, and for higher resolutions sigma–delta converters are the preferred topology. We now briefly describe sigma–delta converters. Sigma–Delta Data Converters Oversampling converters sample the input at a rate larger than the Nyquist frequency. If fs is the sampling rate, then fs /2f0 = OSR is called the oversampling ratio. Oversampling converters have the advantage over Nyquist rate converters that they do not require very tight tolerances from the analog components and that they simplify the design of the antialias filter. Sigma–delta converters40 are oversampling single-bit converters that use frequency shaping of the quantization noise to increase resolution without increasing the matching requirements for the analog components. Figure 34 shows a block diagram for a general noise-shaping oversampled converter. In a sigma–delta converter both the ADC and DAC shown in Fig. 34 are single-bit versions and as such provide perfect linearity. The ADC, a comparator in the case of a sigma–delta converter, quantizes the output of the loop filter, H1 . The quantization process approximates an analog value by a finite-resolution digital value. This step introduces a quantization error Qn . Further, if we assume that the quantization noise is not correlated to the input, then the system can be modeled as a linear

Vin

+

Σ

H1

ADC

V0

− DAC

Fig. 34 Figure for general noise-shaping oversampled converter.

system. The output voltage for this system can now be written as V0 =

Qn Vin H1 + 1 + H1 1 + H1

(41)

For most sigma–delta converters H1 has the characteristics of a low-pass filter and is usually implemented as a switched-capacitor integrator. For a first-order sigma–delta converter H1 is realized as a simple switched-capacitor integrator, H1 = z−1 /(1 − z−1 ). Making this substitution in Eq. (41), we can write the transfer function for the first-order sigma–delta converter as V0 = Vin z−1 + Qn (1 − z−1 )

(42)

As can be seen from Eq. (44) below, the output is a delayed version of the input plus the quantization noise multiplied by the factor 1 − z−1 . This function has a high-pass characteristic with the result that the quantization noise is reduced substantially at lower frequencies and increases slightly at higher frequencies. The analog modulator shown in Fig. 34 is followed by a low-pass filter in the digital domain that removes the out-of-band quantization noise. Thus, we are left with only the in-band (0 < f < f0 ) quantization noise. For simplicity the quantization noise is usually assumed √ to be white with a spectral density equal to erms 2/f s . Further, if the OSR is sufficiently large, then we can approximate the root-mean-square (rms) noise in the signal band by Nf0 ≈ erms

π 3

2f0 fs

3/2 (43)

As the oversampling ratio increases, the quantization noise in the signal band decreases; for a doubling of the oversampling ratio the quantization noise drops by 20(log 2)3/2 ≈ 9 dB. Therefore, for each doubling of the oversampling ratio we effectively increase the resolution of the converter by an additional 1.5 bits. Clearly, H1 can be replaced by other, higher order functions that have low-pass characteristics. For example, in Fig. 35 we show a second-order modulator. This modulator uses one forward delay integrator and one feedback delay integrator to avoid stability

ELECTRONICS

971 Qn

Vin

+ −

Σ

+

+

z−1

+

V0

+

z−1

Fig. 35

Modulator for second-order oversampled converter.

problems. The output voltage for this figure can be written as V0 = Vin z−1 + Qn (1 − z−1 )2

(44)

The quantization noise is shaped by a second-order difference equation. This serves to further reduce the quantization noise at low frequencies, with the result that the noise power in the signal bandwidth falls by 15 dB for every doubling of the oversampling ratio. Alternatively, the resolution increases by 2.5 bits for every doubling of the oversampling ratio. In general, increasing the order of the filter will reduce the necessary oversampling ratio for a given resolution. However, for stability reasons, topologies other than the simple Candy-style41 modulator discussed above are required for filter orders greater than two. Topologies that avoid this stability problem include the multistage delta–sigma (MASH) and interpolative topologies.37 For low-frequency inputs, the white-noise assumption for the quantization noise breaks down. This results in tones that reduce the effective resolution of lower order sigma–delta converters. Incremental converters utilize this observation to simplify the low-pass filter that follows the sigma–delta converter. Details for the incremental converter are discussed below. We now consider two system design examples. The first is an acoustic emission sensor system and the second is a temperature measurement system. 2.3 System Design Examples

We illustrate the sensor and sensor interface scenario with two examples. The first uses a piezoelectric acoustic emission sensor interfaced with a charge amplifier and a data converter. The second describes an integrated temperature sensor.

The piezoelectric effect is one of the most convenient ways to couple elastic waves to electrical circuits. Piezoelectricity is caused by the electric polarization produced by mechanical strain in certain crystals. Conversely, an electric polarization will induce a mechanical strain in piezoelectric crystals. As a consequence, when a voltage is applied to the electrodes of a piezoelectric film, it elongates or contracts depending on the polarity of the field. Conversely, when a mechanical force is applied to the film, a voltage develops across the film. Some properties of a good piezoelectric film are wide frequency range, high elastic compliance, high output voltage, high stability in wet and chemical environments, high dielectric strength, low acoustic impedance, and low fabrication costs. Piezoelectric materials are anisotropic, and hence their electrical and mechanical properties depend on the axis of the applied electric force. The choice of the piezoelectric material depends on the application. Crystalline quartz (SiO2 ) is a natural piezoelectric substance. Some other commonly used piezoelectric materials are ferroelectric single-crystal lithium niobate (LiNbO3 ) and thin films of ZnO and lead zirconium titanate (PZT). Recently, advances have been made in sensor technology with ultrasonic sensor configurations such as the surface acoustic wave (SAW) and acoustic plate mode (APM). In SAW devices the acoustic waves travel on the solid surface, and in an APM arrangement they bounce off at an acute angle between the bounding planes of a plate. The main types of acoustic wave sensors are shown in Fig. 36.42 Piezoelectric thin films are particularly well suited for microsensor applications that require high reliability and superior performance. When prepared under optimal conditions piezoelectric thin films have a dense

TSM

Acoustic Emission Sensing System Acoustic emission sensors are microsensors that are used for the detection of acoustic signals. These devices use elastic acoustic waves at high frequencies to measure physical, chemical, and biological quantities. Typically, integrated acoustic sensors can be made to be extremely sensitive and also to have a large dynamic range. The output of these sensors is usually a frequency, a charge, or a voltage.

SAW

FPW

APM

Top

Top

Side

Side Top

Side Bottom Side

End

Fig. 36 Types of acoustic wave sensors.

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microstructure without cracks and holes, good adherence, and good electrical properties. The three most popular materials used for thin films include ZnO (zinc oxide), AIN (aluminum nitride), and PZT (lead zirconium titalate). Deposition, sputtering, and sol–gel are some of the methods used for preparing piezo films; the choice depends on the material and substrate used. ZnO thin films are prepared using laser-assisted evaporation and are often doped with lithium. Such films have excellent orientation. AIN thin films maintain a high acoustic velocity and are able to withstand extremely high temperatures. PZT thin films have a much higher piezoelectric coefficient than ZnO and AIN. Recently, it has become possible to generate piezoelectric thin films with extremely good properties through the sol–gel process. This process consists of the following steps: synthesis of a metal–organic solution, deposition of this solution by spin coating, and a final heating that helps to crystallize the ceramic film. A cross-sectional view of a thin-film PZT sensor is shown in Fig. 37. The advantages of thin-film PZT sensors include their small size, which allows them to be positioned virtually anywhere, and their ability to operate at high frequencies. Measurement Techniques. The different modes of use for an acoustic sensor are summarized in Fig. 38. Using either a resonator-transducer or a delay line, measurements can be made on the device itself or incorporated into an oscillator circuit. There are

Gold PZT Ti/Pt electrode Poly Nitride Si

Fig. 37 Cross-sectional view of thin-film PZT sensor.

Elastic wave propagation Delay line Passive device Measure phase shift

basically two ways to implement this measurement technique: active or passive. In the case of passive bulk-wave resonators, we measure the resonant frequency to infer the wavelength and hence the velocity. Likewise, for passive delay lines the phase shift between the input and the output of the transducer, which are separated by a known distance, yields the velocity. On the other hand, for active resonators or delay-line oscillators, the frequency can be directly measured with the help of a digital counter. As an example, let us consider the complete design and implementation of an integrated acoustic emission sensor with low-power signal-conditioning circuitry for the detection of cracks and unusual wear in aircraft and submarines. Within a health and usage monitoring system, it is necessary by some means, either directly or indirectly, to monitor the condition of critical components, for example, airframe, gearboxes, and turbine blades. The overall aim is to replace the current practice of planned maintenance with a regime of required maintenance. Typical parameters used include stress (or strain), pressure, torque, temperature, vibration, and crack detection. In this example, acoustic emission sensors are used for crack detection. The thin-film piezoelectric sensor, coupled to an aircraft component, senses the outgoing ultrasonic waves from any acoustic emission event as shown in Fig. 39. The magnitude of the output signal is proportional to the magnitude of the acoustic emission event. For our example design, the acoustic emission signal bandwidth varies from 50 kHz to approximately 1 MHz. Mixed in with the desired acoustic emission signal is vibration noise due to fretting of the mechanical parts. However, this noise is limited to about 100 kHz and is easily filtered out. Due to the acoustic emission event, the piezoelectric sensor generates a charge on the top and bottom plates of the sensor. There are two basic methods of interfacing to this sensor. We can use either a voltage amplifier (Fig. 40) or a charge amplifier (Fig. 41). In general, the charge amplifier interface provides a number of advantages. First, it is not affected by parasitic capacitances at the input of the amplifier. Second, the output voltage at the piezoelectric sensor is very small. This is because the piezoelectric material, PZT, that is used for its high piezoelectric coefficient

Transducer

Active device

Active device

Measure oscillation frequency

Passive device

Output charge or voltage

0.3 mm to 1mm

Measure f, Q, Zin

Fig. 38 Different measurement techniques for acoustic sensors.

Ti−Pt PZT TiO2−Ti−Pt

Outgoing ultrasonic waves Acoustic emission event

Fig. 39

Acoustic emission sensor.

Si3N4 Si Package wear plate

ELECTRONICS

973

R2 R1

Cf

−

Vin

− V0

+

V0 Q

+

Cs V

Fig. 40 Voltage amplifier.

gm Fig. 42

Modified charge amplifier circuit.

Cf

−

Q

Cs

Fig. 41

+

V0

Charge amplifier.

also has a very high dielectric constant. As shown below, the output voltage is proportional to the charge and inversely proportional to the dielectric constant: V =

Q eSA eSd Q = = = C εA/d εA/d ε

(45)

[The output voltage can also be written in terms of the strain S, the distance d, the electron charge e, and the dielectric constant ε as shown in Eq. (47).] For these and other reasons the charge amplifier interface was selected for our design example. The charge amplifier circuit shown in Fig. 41 is in its simplest form. The charge Q and capacitance Cs are used to model the sensor charge and sensor capacitance. The inverting terminal of the operational amplifier is a virtual ground, and no charge flows into the operational amplifier inputs. Therefore, any charge that is generated across the sensor has to flow into the feedback capacitance Cf . The output voltage developed across the feedback capacitor is inversely proportional to the value of this capacitance. The voltage gain of the circuit is given by the ratio of Cs to Cf , and hence, to obtain high gain, Cf can be made much smaller than Cs . This basic topology has a number of limitations, including low-frequency flicker

noise of the amplifier, operational amplifier offset, and long-term drift. Traditionally, correlated double sampling and chopper stabilization are used to remove low-frequency noise and offset. However, as noted earlier, our signal band does not include the frequencies from dc to 50 kHz, and our maximum signal frequencies are fairly high. Therefore, an alternative design topology shown in Fig. 42 was selected to circumvent the problem. Here, low-frequency feedback is provided to reduce the effects of offset, long-term drift, and low-frequency noise. In the modified circuit, a transconductor is connected in negative feedback. The transfer function of the modified circuit is given by s(gma − gm − Cf s) V0 (s) =− Qin (s) Cs Cf s 2 + s(Cs gm + gma Cf ) + gma gm (46) In this equation, Cs is the sensor capacitance, Cf is the feedback capacitance of the operational amplifier, gma and gm are the transconductances of the operational amplifier and the transconductor. If the higher order terms are neglected, then Eq. (46) can be simplified to

s V0 (s) =− Qin (s) gm





  

 1  Cs s  1+ g ma

(47)

From Eq. (47) it is clear that the circuit has the characteristics of a high-pass filter, that is, none of the low-frequency noise or offsets affect the circuit performance. Next, we perform a power analysis to analyze the effects of different design tradeoffs. Both MOS and bipolar transistor technologies are considered, and power and noise analysis and design tradeoffs for both technologies are presented.

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Power Analysis. If MOS transistors in strong inversion (SI) are used to implement the operational amplifier, then the minimum power requirement is given by

P = VI =

V (2πBW C)2 2K(W/L)

2 VnT =

(48)

where BW is the signal bandwidth, C is the sensor capacitance, K is the transconductance factor, V is the output voltage, I is the supply current, and W/L is the aspect ratio of the transistor. From this equation it is clear that the power is proportional to the square of the signal bandwidth and sensor capacitance. If, however, bipolar transistors are used to implement the operational amplifier, the minimum power requirement is given by P = V I = V 2π BW UT C

Noise Analysis. The power spectral density for the wideband gate-referred noise voltage for MOS transistors is given by

(49)

Here, UT is the thermal voltage, which is equal to 26 mV at room temperature. From this equation it is clear that in the case of bipolar transistors, the power is linearly proportional to the signal bandwidth and sensor capacitance. This difference in the power consumption between bipolar and MOS implementations for a signal frequency of 1 MHz is shown in Fig. 43. Here we note that the power consumption for both MOS and bipolar implementations increases with increased sensor capacitance. However, for very low frequencies, the MOS devices can be operated in weak inversion (WI). In WI, MOS devices behave very similarly to bipolar devices, and hence the slopes for weak inversion and bipolar devices are initially very similar. However, at higher frequencies MOS devices are forced to operate in strong inversion and hence consume more power for the same performance. Next, we consider the design tradeoffs in connection with device noise.

8 kT 3 gm

(50)

Here, k is Boltzmann’s constant, T is the temperature, gm is the transconductance. Likewise, for bipolar transistors the power spectral density for the wide-band input-referred noise voltage is given by 2 = 2qIC VnT

(51)

For both MOS and bipolar implementations the total rms input-referred noise is independent of frequency and inversely proportional to the sensor capacitance as shown in Fig. 44. Here, we note that the ratio of the noise spectral density for the MOS and the bipolar implementations is a constant equal to 4. In summary we note that: For an MOS implementation the power consumption is proportional to the square of the sensor capacitance, whereas for a bipolar implementation it is linearly proportional to the sensor capacitance. On the other hand, the input-referred noise for both the MOS and bipolar implementations is inversely proportional to the sensor capacitance. Thus, there is a clear tradeoff between the minimum power consumption and the maximum input-referred noise. If the sensor capacitance is increased, then the inputreferred noise decreases, but the power increases, and vice versa. Using the equation above, we can calculate the minimum bound on the power requirements for our application. For 10 bits of accuracy and a signal bandwidth of 1 MHz, the minimum sensor capacitance size is 5 pF and the minimum power consumption is around 500 µW.

1000

1

0.01

SI

Bipolar

RMS noise (mV)

Power (mW)

MOS 0.1

WI

0.001 20 40 60 80 Sensor capacitance (pF)

10

1

100

Fig. 43 Minimum power requirements versus sensor capacitance for a MOS or bipolar design.

100

Fig. 44 tance.

1

10 Sensor capacitance (pF)

100

Noise power spectral density versus capaci-

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975

25 20

Gain (dB)

15 10 Simulated Measured

5 0 −5 −10 101

CS = 100 pF, Cf = 10 pF 102

103 104 105 Frequency (Hz)

106

10 7

Fig. 45 Small-signal frequency response of charge amplifier.

Next, we provide some simulation and measurement results for our acoustic emission sensor system. Results. Simulation and measurement results for the charge amplifier with a sensor capacitance of 100 pF and a feedback capacitance of 10 pF are shown in Fig. 45. For this measurement, discrete versions of the sensor and feedback capacitors were used. As expected, the signal band gain is given by the ratio of the sensor to the feedback capacitance, which is equal to 20 dB. Both measurement and simulation results agree fairly well with this value. The primary difference between the measurement and simulation results is in the low-frequency and high-frequency poles. It is expected that this is largely a result of parasitic capacitances and possibly a lower realized transconductance in comparison with the simulated value. The charge amplifier circuit design just described converts the sensor charge into a voltage. This amplified signal voltage is then converted to digital form using an ADC. For our implementation a 10-bit fourthorder sigma–delta implemented as a MASH topology was used. The fourth-order topology was used to keep the oversampling ratio low, as the signal frequency is fairly high. Details of this implementation are not included here; interested readers are referred to Ref. 37 for more information. Next, we describe a complete temperature sensor system.

most recent temperature measurement schemes concentrate on integrated silicon temperature sensors. The resulting smart sensor is extremely small and is also able to provide extremely high performance, as all the signal processing is done on chip before the data is transmitted. This avoids the usual signal corruption that results from data transmission. The disadvantage of the smart sensor is that since all the processing is done on chip, it is no longer possible to maintain the signal preprocessing circuits in an isothermal environment. The on-chip sensor interface electronics must therefore be temperature insensitive or be compensated to provide a temperature-insensitive output. A smart temperature sensor is a system that combines on the same chip all the functions needed for measurement and conversion into a digital output signal. A smart temperature sensor includes a temperature sensor, a voltage reference, an ADC, control circuitry, and calibration capabilities. A block diagram for a smart temperature sensor is shown in Fig. 46. The use of p –n junctions as temperature sensors and for the generation of the reference voltage signals has been reported extensively.43,44 A bandgap voltage reference can be generated with the help of a few p–n junctions. The basic principle for the operation of a bandgap voltage reference is illustrated in Fig. 47. The base–emitter voltage Vbe of a bipolar transistor decreases almost linearly with increasing temperature. The temperature coefficient varies with the applied current, but is approximately −2 mV/◦ C. It is also well known that the difference between the base–emitter voltages of two transistors, Vbe , operated at a constant ratio of their emitter current densities, possesses a positive temperature coefficient. At an emitter

Temp. sensor ADC Reference voltage

Digital filtering

Digital output

Control and calibration

Fig. 46

Smart temperature sensor.

V Vref = Vbe1 + G∆Vbe

Ic

Temperature-Sensing System In many control systems, temperature sensors are used as the primary sensor. Additionally, as most electronic components and circuits are affected by temperature fluctuations, temperature sensors are often needed in microsensor systems to compensate for the temperature variations of the primary sensor or sensors. Because integrated sensors can be manufactured on the same substrate as the signal-processing circuitry,

Bitstream

Vbe1 + PTAT cell Vbe generator

G

Vref

Vbe2 ∆Vbe Temperature

Fig. 47

Principle of bandgap reference.

G∆Vbe

976


R1

+

I1

The output voltage is the sum of the voltage across R1 and the voltage across Q1 . Since the voltage across R1 is equal to the voltage across R2 , the output voltage is equal to

R2 −

I2

Vref

R3

Q1

Fig. 48

Q2

Example bandgap voltage reference circuit.

current density ratio of 8, the temperature coefficient of this PTAT (proportional to absolute temperature) source is approximately 0.2 mV/◦ C. Amplifying this voltage (GVbe ) and adding it to a base–emitter voltage Vbe produces a voltage reference that is independent of temperature. Many circuits have been developed to realize bandgap voltage references using this principle.45,46 A circuit diagram for one of the early bandgap reference implementations is shown in Fig. 48.47 For an ideal operational amplifier, the differential input voltage is equal to zero, so that resistors R1 and R2 have equal voltages across them. Since the voltage across the resistors is the same, the two currents I1 and I2 must have a ratio that is determined solely by the ratio of the resistances R1 and R2 . The base–emitter voltage of a diode-connected bipolar transistor is given by Eq. (52), where T is the absolute temperature of the junction, Is is the reverse saturation current, Id is the current through the junction, k is Boltzmann’s constant, q is the electronic charge, and n is a constant that depends on the junction material and fabrication technique. To see this, we write Vbe =

Id + Is Id nkT nkT ln ln ≈ q Is q Is

(52)

Therefore, the difference between the two base–emitter voltages (Vbe ) is given by I1 Is2 R2 Is2 nkT nkT ln ln = q I2 Is1 q R1 Is1 (53) This voltage appears across R3 . Since the same current that flows through R3 also flows through R2 , the voltage across R2 is given by Vbe = Vbe1 − Vbe2 =

VR2 = as desired.

R2 Is2 R2 R2 nkT ln Vbe = R3 R3 q R1 Is1

R2 Is2 R2 nkT ln = Vbe1 + GVbe R3 q R1 Is1 (55) Therefore, this circuit behaves as a bandgap reference, where the gain factor G is set by the ratios R2 /R3 , R2 /R1 , and Is2 /Is1 . In many designs R2 = R1 and Is2 = 8Is1 . Since the reverse saturation current Is is proportional to the emitter area, to make Is2 = 8Is1 we let the emitter area of Q2 be 8 times as large as the emitter area of Q1 . The operational amplifier’s input-referred voltage offset is the largest error source in this type of voltage reference. This voltage offset is highly temperature dependent and nonlinear, making an accurate calibration of such a reference virtually impossible. It is therefore necessary to use some type of offset cancellation technique such as autozero or chopper stabilization.48 Another source of error is the nonzero temperature coefficient of the resistors. Usually, on-chip resistors are used in the form of polysilicon resistors or well resistors. Both of these resistor implementations tend to occupy very large amounts of chip area if low power is desired. Low-power implementations demand the use of large-value resistors, which unfortunately require large areas. Though well resistors have a much larger resistivity than polysilicon resistors, they also have a very nonlinear temperature coefficient, which makes for difficult calibration. A solution to these problems is to use switchedcapacitor circuits to implement the resistors in the voltage reference circuit. A switched-capacitor implementation makes offset removal simple and also reduces the power consumption, as the area occupied by large-value switched-capacitor resistors is significantly smaller than the area occupied by continuous-time resistors. In fact, the area occupied by switchedcapacitor resistors is inversely proportional to the value of the resistance desired. Another advantage is that the temperature coefficient of on-chip poly–poly capacitors is much smaller than that of on-chip resistors, making design and calibration easier. A switchedcapacitor implementation of the bandgap voltage reference is shown in Fig. 49. The structure of this voltage reference is similar to the one shown in Fig. 48, except that the continuous time resistors have been replaced by switchedcapacitor resistors, and capacitors CT and CF have been added. The switched capacitors emulate resistors with an effective resistance value given by Vout = Vbe1 +

(54)

Reff =

1 fC C

(56)

where fC is the clock frequency of the switch. The feedback capacitor CF is designed to be very small

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q1 q2

C2

q2

C2 +

q1 CT

Vref

−

q2

C3

CF 1×

8×

CT Gnd

Fig. 49 Switched-capacitor implementation of bandgap reference.

and is added to ensure the operational amplifier is never in an open-loop mode of operation. The capacitors located in parallel with the diodes act as tank capacitors to ensure that current is constantly supplied to the diodes. The output of this voltage reference can similarly be calculated and is given by C1 Is2 C3 nkT ln = Vbe1 + GVbe C2 q C2 Is1 (57) which is the desired bandgap voltage reference. Most temperature-sensing devices also use the difference between two diodes (Vbe ) as the sensing element of the system. Since the temperature coefficient of Vbe is small (≈0.2 mV/◦ C), it is almost always amplified to a much larger value (≈10 mV/◦ C) for increased sensitivity. Since we already have an amplified value of Vbe in the voltage reference (GVbe ), all that needs to be done is to subtract Vbe1 from the voltage reference to obtain an amplified value of Vbe . Vref = Vbe1 +

If more sensitivity is needed, the additional amplification can be incorporated in the ADC by simply adjusting the capacitor ratio of CA and CB as shown in Fig. 50. Additionally, the subtraction of Vbe1 from the voltage reference can be easily accomplished with the circuit shown in Fig. 51, where Vin1 is the output of the voltage reference, Vin2 is equal to Vbe1 , and VG is the negative input of the operational amplifier in the follow-on data converter. During clock cycle θ1 the capacitor C is charged to the input voltage Vin2 . During clock cycle θ2 , the charge (Vin1 − Vin2 )/C is transferred. This circuit effectively does the voltage subtraction that is needed to obtain the amplified temperature-dependent output voltage (GVbe ). Incorporating the voltage reference and temperaturesensing circuitry shown in Figs. 49 and 51 into a smart temperature sensor system involves some additional circuitry. Since switched capacitors are already being used for the voltage reference and the sensing circuitry, it makes sense to use switched-capacitor technology for the ADC. A simple ADC that utilizes oversampling techniques is the incremental converter.49 The advantage of this data converter topology, shown in Fig. 50, is its low power consumption, small area, and insensitivity to component mismatch. Additionally, in comparison with sigma–delta converters the postquantization digital low-pass filter is much simpler. It consists of just an up–down counter instead of a more complicated decimation filter. Unfortunately, the first-order incremental ADC has a relatively long conversion time, making this converter suitable only for very slow signals such as temperature.

Vin2

q1

Vin1

q2

C q1

q2

Fig. 51 Switched-capacitor subtraction circuit.

S6 CB Vin Vref

S1

CA

S2

S5 S4

S3

–

II

+

+ –

S1 to S1

Switch control logic Fig. 50

VG

Incremental ADC.

ai

Digital Up– output down counter

978

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS q1 q2 q3 q4 Integration period Fig. 52 Four-phase nonoverlapping clock.

The first-order incremental ADC shown in Fig. 50 is composed of a stray-insensitive switched-capacitor integrator, a comparator, switch control logic, and an up–down counter. A four-phase nonoverlapping clock as shown in Fig. 52 constitutes an integration period. The integrator output voltage is designated by VI [i, j ], where i corresponds to the current integration period and j to the clock cycle (1, 2, 3, or 4). During clock cycle θ1 , S1 and S4 are closed, charging CA to the input voltage Vin . During θ2 , S3 and S5 are closed, transferring the charge that was stored on CA to CB . At the end of the charge transfer from CA to CB the comparator output is denoted by ai =

VI [i, 2] > 0 VI [i, 2] < 0

1 if −1 if

During θ3 , S4 is closed, and if: ai =

1 S3 is closed −1 S2 is closed

During θ4 , S5 is closed, and if: ai =

1 S2 is closed −1 S3 is closed

Also during θ4 , the integrator output voltage VI [i, 4] is given by VI [i, 4] = VI [i, 1] +

CA (Vin − ai Vref ) CB

(58)

The final N-bit output code, denoted by Dout , that results from the up–down counter is obtained by evaluating the quantity 1 ai n n

Dout =

(59)

i=1

Here n is the number of integration periods, and is a function of the resolution that is required of the ADC.

The complete smart temperature sensor is shown in Fig. 53. The subtraction circuit of Fig. 51 is incorporated into the ADC by simply adding switch Ssub . The only difference in the operation of the incremental converter shown in Fig. 53 from the one shown in Fig. 50 is that now during θ2 , S3 is not closed but instead Ssub is closed. The calibration of this system is done in two steps. First the voltage reference is calibrated by adjusting the ratio of C3 and C2 ; next the amplified sensor voltage is calibrated by adjusting the ratio of CA and CB . Adjusting the ratios of the capacitors is done with the use of a capacitor array that is controlled digitally. The output is an N-bit digital word. In Fig. 54 we show measurement results for the voltage reference and final temperature output. For these results a first-pass design of the circuit in Fig. 53 was used. This design was not completely integrated and included external resistors to obtain gain. We expect final integrated results to behave similarly. Figure 54a shows the reference voltage obtained as a sum of a Vbe and an amplified Vbe as described in Eq. (57). The x axis shows the temperature in kelvin and the y axis shows the measured output reference voltage in volts. The measured value is fairly close to the expected value except for some small experimental variations. We suspect these variations are a result of the length of time used to stabilize the temperature between temperature output measurements. The graph in Fig. 54b shows the output voltage, which is Vref − Vbe . As expected, this voltage varies linearly with temperature. Figure 55 shows the expected 1-bit output stream (αi shown in Fig. 54) of the sigma–delta converter before the digital low-pass filter. This output corresponds to an input voltage equal to one-eighth of the reference voltage. We have provided detailed designs for two complete data acquisition systems, namely an acoustic emission sensor system and a smart temperature sensor system. We provide both measurement and simulation results to show their performance. 2.4 Conclusion

In this section we have provided brief descriptions of data acquisition and data conversion systems. In particular, we provided some general descriptions of integrated capacitive and resistive sensors. This was followed by descriptions of two of the most common data converter topologies used in sensor interface systems, namely successive approximation and sigma–delta. Finally, these were followed by detailed descriptions of two complete acquisition systems. The first system was based on a piezoelectric acoustic emission sensor interfaced to a charge amplifier and data converter. The second system was a smart temperature sensor. As feature sizes continue to decrease and integrated sensor technologies progress, it is likely that extremely smart and high-performance systems will be integrated on single chips. Additionally, significant

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979 q1

q1 q2

C2

q2

C2 V

+

q1

CF Q1

CT

Ref

−

q2

C3

Q1

CT

CB S1

CA

S2

S5 S4

Vbe1

S6

+

VI

−

+

ai

−

Up− down counter

Digital output

Ssub S1 −S6 Switch control logic

1.4

0.85

1.35

0.80 Vref − Vbel (V)

Vref (V)

Fig. 53 Smart temperature sensor circuit.

1.3 1.25 1.2 1.15

0.70 0.65 0.60 0.55

1.1 220 240 260 280 300 320 340 360 380 Temperature (K) (a) Fig. 54

0.75

0.50 220 240 260 280 300 320 340 360 380 Temperature (K) (b)

Measurement results for the (a) voltage reference and (b) temperature sensor.

reduction in power and area as a result of smaller feature sizes will make such systems ubiquitous. 3 DATA ANALYSIS Arbee L. P. Chen and Yi-Hung Wu

What is data analysis? Nolan50 gives a definition that is a way of making sense of the patterns that are in, or can be imposed on, sets of figures. In concrete terms, data analysis consists of an observation and an investigation

of the given data, and the derivation of characteristics from the data. Such characteristics, or features as they are sometimes called, contribute to the insight of the nature of data. Mathematically, the features can be regarded as some variables, and the data are modeled as a realization of these variables with some appropriate sets of values. In traditional data analysis,51 the values of the variables are usually numerical and may be transformed into symbolic representation. There are two general types of variables: discrete and continuous.

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3.1 Data Analysis Methods In data analysis, the goals are to find significant patterns in the data and apply this knowledge to some applications. Analysis is generally performed in the following stages:

1. Feature selection 2. Data classification 3. Conclusion evaluation

Fig. 55 Measurement converter.

results

for

analog-to-digital

Discrete variables vary in units, such as the number of words in a document or the population in a region. In contrast, continuous variables can vary in less than a unit to a certain degree of accuracy. The stock price and the height of people are examples of this type. The suitable method for collecting values of discrete variables is counting, and for continuous ones it is measurement. The task of data analysis is required among various application fields, such as agriculture, biology, economics, government, industry, medicine, military, psychology, and science. The source data provided for different purposes may be in various forms, such as text, image, or wave form. There are several basic types of purposes for data analysis: 1. Obtain the implicit structure of data 2. Derive the classification of data 3. Search particular objects in data For example, the stockbroker would like to get the future trend of the stock price, the biologist needs to divide animals into taxonomies, and the physician tries to find the related symptoms of a given disease. The techniques to accomplish these purposes are generally drawn from statistics that provide well-defined mathematical models and probability laws. In addition, some theories, such as fuzzy-set theory, are also useful for data analysis in particular. This section is an attempt to give a brief description of these techniques and concepts of data analysis. In Section 3.1, a variety of data analysis methods are introduced and illustrated by examples. We first give two categories of data analysis according to its initial conditions and resultant uses. Next, we show two well-known methods based on different mathematical models. In Section 3.2, an approach to data analysis for Internet applications is proposed. Some improvements of the data analysis methods are discussed in Section 3.3. Finally, we give a brief summary.

The first stage consists of the selection of the features in the data according to some criteria. For instance, features of people may include their height, skin color, and fingerprints. Considering the effectiveness of human recognition, the fingerprint, which is the least ambiguous, may get the highest priority for selection. In the second stage, the data are classified according to the selected features. If the data consist of at least two features, for example, the height and the weight of people, which can be plotted in a suitable coordinate system, we can inspect so-called scatter plots and detect clusters or contours for data grouping. Furthermore, we can investigate ways to express data similarity. In the final stage, the conclusions drawn from the data would be compared with the actual demands. A set of mathematical models has been developed for this evaluation. In the following sections, we first divide the study of data analysis into two categories according to different initial conditions and resultant uses. Then, we introduce two famous models for data analysis. Each method will be discussed first, followed by examples. Because the feature selection depends on the actual representations of data, we postpone the discussion about this stage until the next section. In this section, we focus on the classification procedure based on the given features. Categorization of Data Analysis There are a variety of ways to categorize the methods of data analysis. According to the initial conditions and the resultant uses, there are two categories, supervised data analysis and unsupervised data analysis. The term supervised means that human knowledge has to be provided for the process. In supervised data analysis, we specify a set of classes called a classification template and select some samples from the data for each class. These samples are then labeled by the names of the associated classes. Based on this initial condition, we can automatically classify the other data termed to-be-classified data. In unsupervised data analysis, there is no classification template, and the resultant classes depend on the samples. Following are descriptions of supervised and unsupervised data analysis with an emphasis on their differences. Supervised Data Analysis. The classification template and the well-chosen samples are given as an initial state and contribute to the high accuracy of

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data classification. Consider the K nearest-neighbor classifier, which is a typical example of supervised data analysis. The input to the classifier includes a set of labeled samples S, a constant value K, and a to-beclassified datum X. The output after the classfication is a label denoting a class to which X belongs. The classification procedure is as follows. 1. Find the K nearest neighbors (K NNs) of X from S. 2. Choose the dominant classes by K NNs. 3. If there exists only one dominant class, label X by this class; otherwise, label X by any dominant class. 4. Add X to S, and the process terminates. The first step selects K samples from S such that the values of the selected features (also called patterns) of these K samples are closest to those of X. Such a similarity may be expressed in a variety of ways. The measurement of distances among the patterns is one of the suitable instruments, for example, the Euclidean distance as shown in Eq. (60). Suppose the K samples belong to a set of classes; the second step is to find the set of dominant classes C . A dominant class is a class that contains the majority of the K samples. If there is only one element in C , say class Ci , we assign X to Ci . On the other hand, if C contains more than one element, X is assigned to an arbitrary class in C . After deciding on the class of X, we label it and add it into the set S. m (Xk −Yk )2 (60) δ(X , Y ) = k =1

where each datum is represented by m features. Example. Suppose there is a data set about the salaries and ages of people. Table 1 gives such a set of samples S and the corresponding labels. There are three labels that denote three classes: rich, fair, and poor. These classes are determined based on the assumption that richness depends on the values of the salary and age. In Table 1, we also append the rules for assigning labels for each age. From the above, we can get the set membership of each class.

Crich = {Y1 , Y4 , Y8 }

Cfair = {Y2 , Y5 , Y6 , Y10 }

Cpoor = {Y2 , Y7 , Y9 } If there is a to-be-classified datum X with age 26 and salary $35,000 (35k), we apply the classification procedure to classify it. Here we let the value of K be 4 and use the Euclidean distance as the similarity measure. 1. The set of 4 NNs is {Y4 , Y5 , Y6 , Y9 }.

Table 1

Set of Samples with Salary and Age Data

Sample

Age

Salary

Label

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

20 22 24 24 28 30 30 32 36 40

25k 15k 15k 40k 25k 40k 20k 60k 30k 70k

Rich Fair Poor Rich Fair Fair Poor Rich Poor Fair

rich, >20k; poor, 26k; poor, 35k; poor, 38k; poor, 44k; poor, 50k; poor, 56k; poor, 68k; poor, 80k; poor, 1. 2.M1 = (26.6, 21.6k), M2 = (31.5, 52.5k). 2. 3.4.There is no sample movement; the process terminates.

First iteration 1. 2.M1 = (24.6, 28.3k), M2 = (27.3, 33.3k), M3 = (32.5, 38.7k). 2. 3.Move Y4 from C1 to C2 , move Y2 and Y5 from C2 to C1 , move Y8 from C2 to C3 , move Y3 from C3 to C1 , move Y9 from C3 to C2 . 3. 4.For the new partition W: C1 = {Y1 , Y2 , Y3 , Y5 , Y7 }, C2 = {Y4 , Y9 }, C3 = {Y6 , Y8 , Y10 }. Second iteration 1. 2.M1 = (24.8, 20k), M2 = (30, 35k), M3 = (34, 56.6k). 2. 3.Move Y6 from C3 to C2 . 3. 4.For the new partition W: C1 = {Y1 , Y2 , Y3 , Y5 , Y7 }, C2 = {Y4 , Y6 , Y9 }, C3 = {Y8 , Y10 }. Third iteration 1. 2.M1 = (24.8, 20k), M2 = (30, 36.6k), M3 = (36, 65k). 2. 3.4.There is no sample movement; the process terminates. After three iterations, we have a stable partition and also conclude with the discriminant rule that all the samples with salaries lower than 30k belong to C1 , the other samples with salaries lower than 60k belong to C2 , and the remainder belongs to C3 . The total number of iterations depends on the initial partition, the number of clusters, the given features, and the similarity measure. Methods for Data Analysis In the following, we introduce two famous techniques for data analysis. One is Bayesian data analysis based on probability theory, and the other is fuzzy data analysis based on fuzzy-set theory.

We can easily find a simple discriminant rule behind this final partition. All the samples with salaries lower than 40k belong to C1 , and the others belong to C2 . Hence we may conclude with a discriminant rule that divides S into two clusters by checking the salary data. If we use another initial partition, say W , where C1 = {Y1 , Y3 , Y5 , Y7 , Y9 } and C2 = {Y2 , Y4 , Y6 , Y8 , Y10 }, the conclusion is the same. The following process yields another partition with three clusters.

Bayesian Data Analysis. Bayesian inference, as defined In Ref. 52, is the process of fitting a probability model to a set of samples, which results in a probability distribution to make predictions for to-be-classified data. In this environment, a set of samples is given in advance and labeled by their associated classes. Observing the patterns contained in these samples, we can obtain not only the distributions of samples for the classes but also the distributions of samples for the patterns. Therefore, we can compute a distribution of classes for these patterns and use this distribution to predict the classes for the to-be-classified data based on their patterns. A typical process of Bayesian data analysis contains the following stages:

1. For the initial partition W : C1 = {Y1 , Y4 , Y7 }, C2 = {Y2 , Y5 , Y8 }, C3 = {Y3 , Y6 , Y9 , Y10 }.

1. Compute the distributions from the set of labeled samples.

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2. Derive the distribution of classes for the patterns. 3. Evaluate the effectiveness of these distributions. Suppose a sample containing the pattern a on some features is labeled class Ci . First, we compute a set of probabilities P (Ci ) that denote a distribution of samples for different classes and let each P (a|Ci ) denote the conditional probability of a sample containing the pattern a, given that the sample belongs to the class Ci . In the second stage, the conditional probability of a sample belonging to class Ci , given that the sample contains the pattern a, can be formulated as follows: P (a|Ci )P (Ci ) P (a)

P (Ci |a) =

(61)

where P (a) =

P (a|Ci )P (Ci )

i

From Eq. (62), we can derive the probabilities of a sample belonging to classes according to the patterns contained in the sample. Finally, we can find a way to determine the class by using these probabilities. The following is a simple illustration of data analysis based on this probabilistic technique. Example. Consider the data in Table 1. We first gather the statistics and transform the continuous values into discrete ones as in Table 2. Here we have two discrete levels, young and old, representing the age data, and three levels, low, median, and high, referring to the salary data. We collect all the probabilities and derive the ones for prediction based on Eq. (62): P (young, low|Crich ) =

P (young, low|Cfair ) =

1 3

1 2

P (young, low|Cpoor ) = 13 , P (young, median|Crich ) =

P (young, median|Cfair ) = 0

1 3

P (young, median|Cpoor ) = 0, . . . P (young, low) =

P (young, median) =

4 10

1 10

P (young, high) = 0, . . . P (Crich ) =

3 10

P (Cfair ) =

P (Crich |young, low) =

1 4

P (Cpoor |young, low) =

1 4

2 5

P (Cpoor ) =

3 10

P (Cfair |young, low) =

1 2

P (Crich |young, median) = 1 P (Cfair |young, median) = 0 P (Cpoor |young, median) = 0, . . .

Because there are two features representing the data, we compute the joint probabilities instead of the individual probabilities. Here we assume that the two features have the same degree of significance. At this

Table 2 Summary of Probability Distribution for Data in Table 1 Sample

Rich

Fair

Poor

Young Old Low Median High

2 1 1 1 1

2 2 2 1 1

1 2 3 0 0

Expressions of New Condensed Features Age is lower than 30 Other ages Salary is lower than 36k Other salaries Salary is higher than 50k

point, we have constructed a model to express the data with their two features. The derived probabilities can be regarded as a set of rules to decide the class of any to-be-classified datum. If there is a to-be-classified datum X whose age is 26 and salary is 35k, we apply the derived rules to label X. We transform the pattern of X to indicate that the age is young and the salary is low. To find the suitable rules, we can define a penalty function λ(Ci |Cj ), which denotes the payment when a datum belonging to Cj is classified into Ci . Let the value of this function be 1 if Cj is not equal to Ci and 0 if two classes are the same. Furthermore, we can define a distance measure ι(X, Ci ) as in Eq. (64), which represents the total amount of payments when we classify X into Ci . We conclude that the lower the value of ι(X, Ci ), the higher the probability that X belongs to Ci . In this example, we label X by Cfair because ι(X, Cfair ) is the lowest. ι(X, Ci ) = λ(Ci |Cj )P (Cj |X) (62) j

ι(X, Crich ) = 0 × ι(X, Cfair ) =

1 2

1 4

+1×

1 2

+1×

ι(X, Cpoor ) =

1 4

=

2 4

3 4

Fuzzy Data Analysis. Fuzzy-set theory, established by Zadeh,53 allows a gradual membership MFA (X) for any datum X on a specified set A. Such an approach more adequately models the data uncertainty than using the common notion of set membership. Take cluster analysis as an example. Each datum belongs to exactly one cluster after the classification procedure. Often, however, the data cannot be assigned exactly to one cluster in the real world, such as the jobs of a busy person, the interests of a researcher, or the conditions of the weather. In the following, we replace the previous example for supervised data analysis with the fuzzy-set notion to show its characteristic. Consider a universe of data U and a subset A of U . Set theory allows to express the membership of A on U by the characteristic function FA (X) : U → {0, 1}. 1 X∈A FA (X) = (63) 0 X∈ /A

From the above, it can be clearly determined whether X is an element of A or not. However, many real-world

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phenomena make such a unique decision impossible. In this case, expressing membership is more suitable. A fuzzy set A on U can be represented by the set of pairs that describe the membership function MFA (X) : U → [0, 1] as defined in Ref. 54: A = {(X, MFA (X))|X ∈ U, MFA (X) ∈ [0, 1]} (64) Example. Table 3 contains a fuzzy-set representation of the data set in Table 1. The membership function of each sample is expressed in a form of possibility that stands for the degree of the acceptance that a sample belongs to a class. Under the case of supervised data analysis, the to-be-classified datum X needs to be labeled using an appropriate classification procedure. All the distances between each sample and X are calculated using the two features and Euclidean distance.

1. Find the K NNs of X from S. 2. Compute the membership function of X for each class. 3. Label X by the class with a maximal membership. 4. Add X to S and stop the process. The first stage in finding K samples with minimal distances is the same, so we have the same set of four nearest neighbors {Y4 , Y5 , Y6 , Y9 } when the value of K = 4. Let δ(X, Yj ) denote the distance between X and the sample Yj . In the next stage, we calculate the membership function MFCi (X) of X for each class Ci as follows:

MFCi (X) j MFCi (Yj )δ(X, Yj ) = j δ(X, Yj )

∀Yj ∈ k NNs of X

(65)

Table 3 Fuzzy-Set Membership Functions for Data in Table 1

Sample

Rich

Fair

Poor

Estimated Distances between the Sample and X

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

0.5 0.1 0 0.6 0.2 0.2 0 0.9 0 0.4

0.2 0.5 0.2 0.3 0.5 0.5 0 0.1 0.3 0.6

0.3 0.4 0.8 0.1 0.3 0.2 1 0 0.7 0

11.66 20.39 20.09 5.38 10.19 6.4 15.52 25.7 11.18 37.69

X

0.2

0.42

0.38

MFCrich (X) =

0.6 × 5.38 + 0.2 × 10.19 + 0.2 × 6.4 + 0 × 11.18 ≈ 0.2 5.38 + 10.19 + 6.4 + 11.18

MFCfair (X) =

0.3 × 5.38 + 0.5 × 10.19 + 0.6 × 6.4 + 0.3 × 11.18 ≈ 0.42 5.38 + 10.19 + 6.4 + 11.18

MFCpoor (X) =

0.1 × 5.38 + 0.3 × 10.19 + 0.2 × 6.4 + 0.7 × 11.18 ≈ 0.38 5.38 + 10.19 + 6.4 + 11.18

Because the membership of X for class Cfair is higher than all others, we label X by Cfair . The resultant membership directly gives a confidence measure of the classification. 3.2 Data Analysis on Internet Data The dramatic growth of information systems over the past years has brought about the rapid accumulation of data and an increasing need for information sharing. The World Wide Web (WWW) combines the technologies of the uniform resource locator (URL) and hypertext to organize the resources on the Internet into a distributed hypertext system.54 As more and more users and servers register on the WWW, data analysis on its rich content is expected to produce useful results for various applications. Many research communities such as network management, information retrieval, and database management have been working in this field.54 Many tools for Internet resource discovery55 use the results of data analysis on the WWW to help users find the correct positions of the desired resources. However, many of these tools essentially keep a keyword-based index of the available resources (Web pages). Owing to the imprecise relationship between the semantics of keywords and the Web pages,56 this approach clearly does not fit the user requests well. The goal of Internet data analysis is to derive a classification of a large amount of data, which can provide a valuable guide for the WWW users. Here the data are the Web pages produced by the information providers of the WWW. In some cases, data about the browsing behaviors of the WWW users are also interesting to the data analyzers, such as the most popular sites browsed or the relations among the sites in a sequence of browsing. Johnson and Fotouhi57 propose a technique to aid users to roam through the hypertext environment. They gather and analyze all the browsing paths of some users to generate a summary as a guide for other users. Many efforts have been made to apply the results of such data analysis.57 In this section, we focus on the Web pages that are the core data of the WWW. First, we present a study on the nature of Internet data. Then we show the feature selection stage and enforce a classification procedure to group the data at the end.

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Each site within the Web environment contains one or more Web pages. Under this environment, any WWW user can make a request to any site for any Web page in it. Moreover, the user can also roam through the Web by means of the anchor information provided in each Web page. Such an approach has resulted in several essential difficulties for data analysis. 1. Huge amounts of data 2. Frequent changes 3. Heterogeneous presentations Basically, Internet data originate from all over the world, and the amount of data is huge. As any WWW user can create, delete, and update the data, and change the locations of the data at any time, it is difficult to get a precise view of the data. Furthermore, the various forms of expressing the same data also reveal the status of the chaos on the WWW. As a whole, Internet data analysis should be able to handle the large amount of data and control the uncertainty factors in a practical way. The data analysis procedure consists of the following stages: 1. 2. 3. 4. 5.

Observe the data. Collect the samples. Select the features. Classify the data. Evaluate the results.

In the first stage, we observe the data and conclude with a set of features that may be effective for classifying the data. Next, we collect a set of samples based on a given scope. In the third stage, we estimate the fitness of each feature for the collected samples to determine a set of effective features. Then, we classify the to-beclassified data according to the similarity measure on the selected features. At last, we evaluate the classified results and find a way for further improvement. Data Observation In the following, we provide two directions for observing the data. Semantic Analysis. We may consider the semantics of a Web page as potential features. Keywords contained in a Web page can be analyzed to determine the semantics such as which fields it belongs to or what concepts it provides. There have been many efforts at developing techniques to derive the semantics of a Web page. The research results of information retrieval58,59 can also be applied for this purpose. Observing the data formats of Web pages, we can find several parts expressing the semantics of the Web pages to some extent. For example, the title of a Web page usually refers to a general concept of the Web page. An anchor, which is constructed by the home-page designer, provides a URL of another Web page and makes a connection between the two Web

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pages. As far as the home-page designer is concerned, the anchor texts must sufficiently express the semantics of the whole Web page to which the anchor points. As to the viewpoint of a WWW user, the motivation to follow an anchor is based on the fact that this anchor expresses desired semantics for the user. Therefore, we can make a proper connection between the user’s interests and those truly relevant Web pages. We can group the anchor texts to generate a corresponding classification of the Web pages pointed to by these anchor texts. Through this classification we can relieve the WWW users of the difficulties on Internet resource discovery through a query facility. Syntactic Analysis. Syntactic analysis is based on the syntax of the Web pages to derive a rough classification. Because the data formats of Web pages follow the standards provided on the WWW, for example, hypertext markup language (HTML) we can find potential features among the Web pages. Consider the features shown in Table 4. The white pages, which mean the Web pages with a list of URLs, can be distinguished from the ordinary Web pages by a large number of anchors and the short distances between two adjacent anchors within a Web page. Note that here the distance between two anchors means the number of characters between them. For publication, the set of the headings has to contain some specified keywords, such as “bibliography” or “Publications.” The average distance between two adjacent anchors has to be lower than a given threshold and the placement of anchors has to center to the bottom of the Web page. According to these features, some conclusions may be drawn in the form of classification rules. For instance, the Web page is designed for publication if it satisfies the requirements of the corresponding features. Obviously, this approach is effective only when the degree of support for such rules is high enough. Selection of effective features is a way to improve the precision of syntactic analysis. Sample Collection It is impossible to collect all the Web pages, and thus choosing a set of representative samples becomes a very important task. On the Internet, we have two approaches to gather these samples.

Table 4 Pages

Potential Features for Some Kinds of Web

Kind of Home Page White page Publication Person Resource

Potential Feature Number of anchors, average distance between two adjacent anchors Headings, average distance between two adjacent anchors, anchor position Title, URL directory Title, URL filename

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1. Supervised sampling 2. Unsupervised sampling

class of white pages. C0 = {Y1 , Y2 , Y3 , Y4 , Y5 }

Supervised sampling means the sampling process is based on human knowledge that specifies the scope of the samples. In supervised data analysis, there exists a classification template that consists of a set of classes. The sampling scope can be set based on the template. The sampling is more effective when all classes of the template contain at least one sample. On the other hand, we consider unsupervised sampling if there is not enough knowledge about the scope, as in the case of unsupervised data analysis. The most trivial way to get samples is to choose any subset of Web pages. However, this arbitrary sampling may not fit the requirement of random sampling well. We recommend the use of search engines that provide different kinds of Web pages in a form of directory. Feature Selection In addition to collecting enough samples, we have to select suitable features for the subsequent classification. No matter how good the classification scheme is, the accuracy of the results would not be satisfactory without effective features. A measure for the effectiveness of a feature is to estimate the degree of class separability. A better feature implies a higher class separability. This measure can be formulated as a criterion to select effective features.

C1 = {Y6 , Y7 , Y8 , Y9 , Y10 }

We can begin to formulate the class separability. In the following formula, we assume that the number of classes is c, the number of samples within class Cj is nj , and Yki denotes the kth sample in class Ci . First, we define the interclass separability Db , which represents the ability of a feature to distinguish the data between two classes. Next, we define the intraclass separability Dw , which expresses the power of a feature to separate the data within the same class. The two measures are formulated in Eqs. (69) and (67) based on the Euclidean distance defined in Eq. (60). Since a feature with larger Db and smaller Dw implies a better class separability, we define a simple criterion function DFj [Eq. (71)] as a composition of Db and Dw to evaluate the effectiveness of a feature Fj . Based on this criterion function, we get DF0 = 1.98 and DF1 = 8.78. Therefore, F1 is more effective than F0 due to its higher class separability. ni nj c 1 1 Db = Pi Pj δ(Yki , Ymj ) 2 ni nj i=1

j =i

(66)

k=1 m=1

where Example. Consider the samples shown in Table 5. From Table 4, there are two potential features for white pages, the number of anchors (F0 ) and the average distance between two adjacent anchors (F1 ). We assume that F0 ≥ 30 and F1 ≤ 3 when the sample is a white page. However, a sample may actually belong to the class of white pages although it does not satisfy the assumed conditions. For example, Y6 is a white page although its F0 < 30. Therefore, we need to find a way to select effective features. From the labels, the set membership of the two classes is as follows, where the class C1 refers to the

ni Pi = c

j =1 nj

nj ni c 1 1 Dw = Pi Pj δ(Yki , Ymj ) (67) 2 ni nj i=1

where

j =i

k=1 m=1

ni Pi = c

j =1

nj

and Table 5 Set of Samples with Two Features. The Labels Come from Human Knowledge Sample

F0a

F1b

White Page

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

8 15 25 35 50 20 25 40 50 80

5 3.5 2.5 4 10 2 1 2 2 8

No No No No No Yes Yes Yes Yes Yes

aF 0 b F1

denotes the number of anchors. denotes the average distance for two adjacent anchors.

DFj = Db − Dw

(68)

We have several ways to choose the most effective set of features: 1. 2. 3. 4.

Ranking approach Top-down approach Bottom-up approach Mixture approach

Ranking approach selects the features one by one according to the rank of their effectiveness. Each time we include a new feature from the rank, we compute the joint effectiveness of the features selected so far by Eqs. (69)–(71). When the effectiveness degenerates,

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the process terminates. Using a top-down approach, we consider all the features as the initial selection and drop the features one by one until the effectiveness degenerates. On the contrary, the bottom-up approach adds a feature at each iteration. The worse case of the above two approaches occurs if we choose the bad features earlier in the bottom-up approach or the good features earlier in the top-down approach. The last approach allows us to add and drop the features at each iteration by combining the above two approaches. After determining the set of effective features, we can start the classification process.

The first approach finds the sample in each class nearest to the to-be-classified page. Among these representative samples, we can choose the class containing the one with a minimal distance and assign the page to it. On the other hand, we can also find the farthest sample in each class from the page. Then we assign the page to the class that contains the representative sample with a minimal distance. The last approach is to take the mean of each class into consideration. As in the previous approaches, the mean of each class represents a whole class, and the one with a minimal distance from the page would be chosen. An example follows by using the mean approach.

Data Classification In the following, we only consider the anchor semantics as the feature, which is based on the dependency between an anchor and the Web page to which the anchor points. As mentioned previously, the semantics expressed by the anchor implies the semantics of the Web page to which the anchor points, and also describes the desired Web pages for the users. Therefore, grouping the semantics of the anchors is equivalent to classifying the Web pages into different classes. The classification procedure consists of the following stages:

Example. Inspect the data shown in Table 6. There are several Web pages and anchor texts contained in some anchors pointing to the Web pages. Here we consider six types of anchor texts, T1 , T2 , . . . , T6 . The value of an anchor text for a Web page stands for the number of the anchors pointing to the Web page, which contain the anchor text. The labeling is the same as in the previous example. We can calculate the means of the two classes:

1. Label all sample pages. 2. For each labeled pages, group the texts of the anchors pointing to it. 3. Record the texts of the anchors pointing to the to-be-classified page. 4. Classify the to-be-classified page based on the anchor information. 5. Refine the classification process. In the beginning, we label all the samples and record all the anchors pointing to them. Then we group together the anchor texts contained in the anchors pointing to the same sample. In the third stage, we group the anchor texts contained in the anchors pointing to the to-be-classified page. After the grouping, we determine the class of the to-be-classified page according to the corresponding anchor texts. At last, we can further improve the effectiveness of the classification process. There are two important measures during the classification process. One is the similarity measure of two data, and the other is the criterion for relevance feedback. Similarity Measure. After the grouping of samples, we have to measure the degree of membership between the to-be-classified page and each class. Considering the Euclidean distance again, there are three kinds of approaches for such measurement:

1. Nearest-neighbor approach 2. Farthest-neighbor approach 3. Mean approach

M0 = (0, 4, 1, 1, 1, 0.2, 1) M1 = (4.2, 3.4, 2.6, 1.4, 2, 1.4) Suppose there is a Web page X to be classified as shown in Table 6. We can compute the distances between X and the two means. They are δ(X, M0 ) = 6.94 and δ(X, M1 ) = 4.72. Thus we assign X to class C1 . Relevance Feedback. The set of samples may be enlarged after a successful classification by including the classified Web pages. However, the distance Table 6 Set of Web Pages with Corresponding Anchor Texts and Labels. The Labels Come from Human Knowledge Sample

T1a

T2b

T3c

T4d

T5e

T6f

White Page

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

0 0 0 0 2 1 3 4 5 8

0 1 2 0 2 3 3 2 5 4

0 2 0 3 0 0 1 5 3 4

1 0 4 0 0 0 6 0 0 1

1 0 0 0 0 2 3 1 0 4

2 2 0 1 0 3 0 0 2 2

No No No No No Yes Yes Yes Yes Yes

X

5

2

0

0

5

0

Yes

a T = ‘‘list.’’ 1 b T = ‘‘directory.’’ 2 c T = ‘‘classification.’’ 3 d T4 = ‘‘bookmark.’’ e T = ‘‘hot.’’ 5 f T6 = ‘‘resource.’’

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between a to-be-classified page and the nearest mean may be very large, which means that the current classification process does not work well on this Web page. In this case, we reject the classification of such a Web page and wait until more anchor texts for this Web page are accumulated. This kind of rejection not only expresses the extent of the current ability to classify Web pages, but also promotes the precision of the classified results. Furthermore, by the concept of class separability formulated in Eqs. (69)–(71), we can define a similar criterion function DS to evaluate the performance of the current set of samples. DS = DF (S)

(69)

where F is the set of all effective features and S is the current set of samples. Example. Reconsider the data shown in Table 6. Before we assign X to C1 , the initial DS = 0.75. When C1 contains X, DS∪{X} yields a smaller value 0.16. On the other hand, DS∪{X} becomes 1.26 if we assign X to C0 . Hence, although X is labeled C1 , it is not suitable to become a new sample for the subsequent classification. The set of samples can be enlarged only when such an addition of new samples gains a larger Ds value, which means the class separability is improved. 3.3 Improvement of Data Analysis Methods

Although the previous procedures are able to fit the requirements of data analysis well, there are still problems, such as speed or memory requirements and the complex nature of real-world data. We have to use some heuristic techniques to improve the classification performance. For example, the number of clusters given in unsupervised data analysis has significant impact on the time spent at each iteration and the quality of the final partition. Notice that the initial partition may contribute to a specific sequence of adjustments and then a particular solution. Therefore, we have to find an ideal number of clusters during the analysis according to the given initial partition. The bottomup approach for decreasing the number of clusters at each iteration is a way to determine the final partition. Given a threshold of similarity among the clusters, we can merge two clusters that are similar enough to become a new single cluster at each iteration. We can find a suitable number of clusters when there are no more similar clusters to be merged. In the following sections, we introduce two more techniques to improve the work of data analysis. Rough-Set-Based Data Analysis The approach to classifying Internet data by anchor semantics requires a large amount of anchor texts. These anchor texts may be contained in the anchors pointing to the Web pages in different classes. An anchor text is said to be indiscernible when it cannot be used to distinguish

the Web pages in different classes. We employ the rough-set theory60,61 to find the indiscernible anchor texts, which will then be removed. The remaining anchor texts will contribute to a higher degree of accuracy for the subsequent classification. In addition, the cost of distance computation can also be reduced. In the following, we introduce the basic idea of the rough-set theory and an example for the reduction of anchor texts. Rough-Set Theory. By the rough-set theory, an information system is modeled in the form of a 4-tuple (U,A, V,F ), where U represents a finite set of objects, A refers to a finite set of attributes, V is the union of all the domains of the attributes in A, and F is a binary function (U × A: → V ). The attribute set A often consists of two subsets, one refers to condition attributes C and the other stands for decision attributes D. In the approach of classification on Internet data, U stands for all the Web pages, A is the union of the anchor texts (C) and the class of Web pages (D)V is the union of all the domains of the attributes in A, and F handles the mappings. Let B be a subset of A. A binary relation called indiscernibility relation is defined as

INDB = {(Xi , Xj ) ∈ U × U |∀p ∈ B, p(Xi ) = p(Xj )} (70) That is, Xi and Xj are indiscernible by the set of attributes B if p(Xi ) is equal to p(Xj ) for every attribute p in B. INDB is an equivalence relation that produces an equivalence class denoted [Xi ]B for each sample Xi . With regard to the Internet data, two Web pages Xi and Xj , which have the same statistics for each anchor text in C belong to the same equivalence class [Xi ]C (or [Xj ]C). Let U be a subset of U . A lower approximation LOWB,U , which contains all the samples in each equivalence class [Xi ]B contained in U , is defined as LOWB,U = {Xi ∈ U |[Xi ]B ⊂ U }

(71)

Based on Eq. (75), LOW C, [Xi ]D contains the Web pages in the equivalence classes produced by IND C, and these equivalence classes are contained in [Xi ]D for a given Xi . A positive region POS C, D is defined as the union of LOW C, [Xi ]D for each equivalence class produced by IND D. POS D, D refers to the samples that belong to the same class when they have the same anchor texts. As defined in Ref. 62, C is independent of D if each subset C i in C satisfies the criterion that POS C, = POS C i , D; otherwise, C is said to be dependent on D. The degree of dependency γ C, D is defined as γC,D =

card(POSC,D ) card(U )

(72)

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989

where card denotes set cardinality; CONp,γC,D = γC,D − γC−|p|,D

(73)

From these equations, we define the contribution CONp,γ C, D of an anchor text p in C to the degree of dependency γ C, CIDbar; by using Eq. (73). According to Eq. (73), we say an anchor text p is dispensable if γ C − {p}, D = γ C, D. That is, the anchor text p makes no contribution to γ C, D and the value of CONp,γ C, D equals 0. The set of indispensable anchor texts is the core of the reduced set of anchor texts. The remaining task is to find a minimal subset of C called a reduct of C that satisfies Eq. (74) and the condition that the minimal subset is independent of D. POSC,D = POSminimal

subset of C,D

(74)

Reduction of Anchor Texts. To employ the concepts of the rough-set theory for the reduction of anchor texts, we transform the data shown in Table 6 into those in Table 7. The numerical value of each anchor text is transformed into a symbol according to the range in which the value falls. For instance, a value in the range between 0 and 2 is transformed into the symbol L. This process is a generalization technique usually used for a large database. By Eq. (73), we can compute CONp,γ C, D for each anchor text p and sort them in ascending order. In

Table 7 Set of Data in Symbolic Values Transformed from Table 6 Sample

T1

T2

T3

T4

T5

T6

White Page

Y1 Y2

La L

L L

L L

L L

L L

L L

No No

Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

L L L L M M M H

L L L M M L M M

L M L L L M M M

Mb L L L Hc L L L

L L L L M L L M

L L L M L L L L

No No No Yes Yes Yes Yes Yes

X

M

L

L

L

M

L

Yes

L = [0, 2]. b M = [3, 5]. c H = [6, 8]. a

this case, all CONp,γ C, D are 0 except CONT1 , γ C, D. That is, only the anchor text T1 is indispensable, which becomes the unique core of C Next, we use a heuristic method to find a reduct of C because such a task has been proved to be NP-complete in Ref. 63. Based on an arbitrary ordering of the dispensable anchor texts, we check the first anchor text to see whether it is dispensable. If it is, then remove it and continue to check the second anchor text. This process continues until no more anchor texts can be removed. Example. Suppose we sort the dispensable anchor texts as the sequence {T2 , T3 , T4 , T5 , T6 }, we then check one at a time to see whether it is dispensable. At last, we obtain the reduct {T1 , T6 }. During the classification process, we only consider these two anchor texts for similarity measure. Let the symbols used in each anchor text be transformed into three discrete values, 0, 1, and 2. The means of the two classes are M0 = (0, 0) and M1 = (1, 0.8). Therefore, we classify X into the class C1 due to its minimum distance. When we use the reduct {T1 , T6 } to classify data, the class separability D{T1 ,T6 } is 0.22. Different reducts may result in different values of class separability. For instance, the class separability becomes 0.27 if we choose the reduct {T1 , T2 }. Hierarchical Data Analysis Consider the 1-nearestneighbor classifier for supervised data analysis. We may not want to compute all the distances each time a to-be-classified datum X arrives. We can organize the set of samples into a hierarchy of subsets and record a mean Mi for each subset Si and the farthest distance di from Mi to any sample in Si . If there exists a nearest neighbor of X in a subset other than Si , we do not need to compute the distances between X and all the samples in Si as the triangular inequality [Eq. (75)] holds. Such techniques can reduce the computation time to find the nearest neighbor.

δ(X, Mi ) − di ≥ δ(X, Y )

(75)

where Y is the nearest neighbor of X. 3.4

Summary

In this section, we describe the techniques and concepts of data analysis. A variety of data analysis methods are introduced and illustrated by examples. Two categories, supervised data analysis and unsupervised data analysis, are presented according to their different initial conditions and resultant uses. Two methods for data analysis are also described, which are based on probability theory and fuzzy-set theory, respectively. An approach of data analysis Internet data is presented. Improvements for the data analysis methods are also discussed.

990


4 DIODES Konstantinos Misiakos

The word diode originates from the Greek word ιoδoζ meaning passage or way through. In electronics terminology, in fact, diode refers to a two-terminal device that allows current to flow in one direction while it blocks the flow of current in the opposite direction. Such devices usually employ semiconductor junctions or metal–semiconductor junctions. There are also diodes made of vacuum tubes or metal—purely ionic crystal contacts. This section deals with semiconductor p–n junction diodes because they are the most widely used in practice due to their versatility, reproducibility, stability, and compatibility with integrated circuit technology. Additionally, an insight into the operation of the p–n junction diode is the basis for understanding the device physics of other semiconductor devices, the majority of which use the p–n junction as the building block. The semiconductor of our choice will be silicon because almost all diodes, discrete or integrated, are made of this element. Extensions to other semiconductors will be made to generalize theoretical results or to set limits to the validity of certain equations. Figure 56 shows the electrical symbol of a diode. The arrow-type symbol indicates the conduction direction. For a diode to conduct an appreciable electric current, the voltage on the left side of the symbol must be a little higher than the voltage on the right side. If this polarity is reversed, the current drops to negligible values even for a large bias. The two previous polarity modes are known as forward and reverse bias, respectively. In Fig. 56b the very basic material structure of a diode is shown. The starting material is a high-purity silicon crystal, the properties of which are properly modified by selectively introducing dopants (elements) from either the third or the fifth column of the periodic table. The third-column elements, when introduced into the silicon lattice, behave as acceptors: They trap electrons from the valence band, thereby creating positively charged holes in the valence band and negatively charged immobile acceptor ions. The acceptor-doped part of the diode is called the p side. On the other hand, fifth-column elements behave as donors: They give up their fifth electron, creating a population of conduction band electrons and positively charged immobile donor ions. The donor-doped part of the device is the n side. The introduction of acceptor and donor dopants into silicon creates the two polarity sides of the diode, as shown in Fig. 56a. Schematically speaking, when applying a forward bias, the higher voltage on the p side makes the electron and hole gases move into each other. Thus, an electric current is created through electron–hole pair recombination. On the contrary, a lower voltage on the p side moves the charge carriers away, thus preventing recombination and eliminating the current. In terms of dopants, the previous account of how the p –n diode is formed also holds for germanium diodes, which also is a fourth-column elemental

p −

− −

−

n − −

+

−

(a)

+

+

−

+

+

+ +

(b)

Fig. 56 Electrical symbol of a diode (a) and illustration of a semiconductor p–n junction (b). In (b) the large circles with the minus and the plus signs are the acceptor and the donor ions, whereas the small circles are the holds (empty) and the electrons (dark).

I 1

2

V

Fig. 57 Current–voltage characteristics of an ideal diode, curve 1, and a realistic one, curve 2.

semiconductor. For compound semiconductors (e.g., GaAs, InP, CdTe), the chemical origin of donor and acceptor dopants is more complex in relation to the semiconductor elements themselves. In Fig. 57 the current–voltage (I –V ) characteristic of an ideal diode as well as of a realistic one is shown. The ideal diode would behave as a perfect switch when forward biased: Unlimited current flows without any voltage drop across the device. The same ideal diode would allow no current in the reverse direction, no matter what the magnitude of the reverse bias is. Now, a realistic semiconductor diode would exhibit a resistance to current flow in the forward direction, whereas in reverse bias a small current would always be present due to leakage mechanisms. The disagreement between the ideal and the actual electrical behavior is not restricted only to the static I –V characteristics shown in Fig. 57. It extends to the transient response obtained when applying a time-dependent terminal excitation. The response of a realistic diode cannot follow at exactly the same speed as the terminal excitation of an ideal diode would. When designing a diode to be used as a switching device, care is taken to bring the device electrical characteristics as close to the ideal ones as possible. This is done by choosing both the geometrical features and the fabrication process steps in a way to suppress the parasitic components of the diode. As a result of the semiconductor electronic band structure as well as technological constraints,

ELECTRONICS

material limitations impose certain basic restrictions on the device performance and create the subsequent deviation from the ideal performance. In the following sections, these restrictions will be investigated, and the deviation from the ideal performance will be analyzed in terms of the basic device physics, material constants, and geometry considerations. Before considering the device physics of the diode, we will briefly discuss the steps in the basic fabrication process employed when making a silicon diode. These steps determine its basic geometrical and technological characteristics, which in turn determine the device electrical behavior. Today, almost all silicon diodes are made through the standard planar process of the silicon integrated circuit technology. A silicon wafer is first oxidized at temperatures in the vicinity of 1000◦ C. Such oxidation creates a silicon dioxide (SiO2 ) cover layer with a thickness on the order of a micron. This layer is used as a mask for the subsequent technological steps. The SiO2 film is then patterned by lithographic techniques and through etching, which allows windows of exposed silicon to be opened. Then, either by diffusion or by ion implantation, dopants are introduced into the exposed areas. The dopants are of a type opposite to the one already existing in the original wafer. In this way, p –n junctions are created in the exposed areas. In the rest of the wafer, the SiO2 layer stops the ions and prevents diffusion into the silicon bulk. On the back surface, another diffusion or implantation of the same dopants as in the bulk is usually applied for reasons that will become clear in the next sections. At the end, metal contacts are evaporated on the front and the back. Lithography, again, on the front side defines the contacts of the individual diodes. The metal contacts are required for the diodes to interact with the external world in terms of terminal excitation (voltage or current) and terminal response (current or voltage, respectively). Similar methods are used for germanium diodes, whereas the compound semiconductor devices are usually made by epitaxial growth on proper substrates and by in situ doping. 4.1 Fundamentals of p–n Junctions

The basic p –n junction device physics was proposed by Shockley.64 He derived the current–voltage characteristics, considering the electron and hole current continuity equations and the relationship between the carrier quasi-Fermi levels and the externally applied potentials. Here, we rederive the general current–voltage relation of a p –n junction based on Shockley’s classic work64 and its later extension.65 Basic Equations and Assumptions To formulate the electron and hole transport in a semiconductor device mathematically, we can always start by expressing the carrier densities and currents in terms of the

991

carrier quasi-Fermi potentials under uniform temperature conditions: − eµn n ∇Fn (76a) Jn = eµn nE + eDn ∇n (76b) −eFn − Ei (76c) n = ni exp kT − eµp p ∇Fp (77a) Jp = (77b) eµp pE − eDp ∇p −(−eFp ) + Ei (77c) p = ni exp LT In Eqs. (76a) and (77a), Jn and Jp are the electric current densities of electrons and holes, respectively. Equations (76b) and (77b) express the currents in terms of drift and diffusion, where µn , Dn , n and µp , Dp , p are the mobilities, diffusivities, and volume densities of electrons and holes, respectively. Finally, Fn and Fp are the electron and hole quasiFermi potentials, Ei is the intrinsic energy level, and E the electric field density. Figure 58 shows the energy band diagram of a p–n junction under forward bias and illustrates the space dependence of the quasiFermi potentials, of the bottom of the conduction band Ec , and of the top of the valence band Ev . Equations (76c) and (77c) hold provided that the differences Ec − (eFn ) and (eFp ) − Ev are positive and at least several times the thermal energy kT. Equations (76a) and (77a) are borrowed from thermodynamics and hold provided that the bias is such that perturbations from equilibrium are small. Small, here, implies that the energy distribution of electrons and holes in the conduction and the valence band, respectively, continue (within a good approximation) to follow the Boltzmann statistics. Additionally, we assume that the mean free paths of the carriers are negligible compared to the physical dimensions of the device. Finally, Eqs. (76) and (77) hold provided quantum mechanical tunneling of carriers across potential barriers is not important. Such a constraint is relaxed in the last section of nonconventional transport diodes. The second set of equations to be considered is the electron and hole continuity equations: 1 ∂n = ∇Jn − U (n, p) + G ∂t e 1 ∂p = − ∇Jp − U (n, p) + G ∂t e

(78) (79)

where U is the electron–hole net recombination rate either by band-to-band transitions or through traps, whereas G is the band-to-band generation rate resulting from ionizing radiation or impact ionization processes. For the sake of simplicity, we assume that U

992

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS p

C1

Ec

Space charge region

Cj

n

−eFn −eV1

C2

−eFn

−eV2

−eFp

−eFp Ev

Ohmic contact

Ohmic contact

0

x

Fig. 58 Band diagram of forward-biased p–n junction. Boundaries C1 and C2 are the ohmic contacts, whereas Cj is the base-injecting boundary.

and G are the same for both carriers. Equations (78) and (79) are more general than the previous ones because they do the “bookkeeping” by equating the increase in the rate of carrier density to minus the carrier losses resulting from carrier out-fluxing (∇Jp /e and −∇Jn /e) and recombination. Next is the Poisson equation, which relates the electric field to the charge density caused by both mobile and immobile charges: ∇E =

e [−n + p + ND − NA ] ε

(80)

where ε is the semiconductor dielectric constant and ND and NA are the donor and acceptor densities, respectively. The charge density resulting from donors and acceptors is not carrier density dependent, unless the temperature drops to the cryogenic region. The final equation to be considered is the one that equates the electric field to the gradient of the electrostatic potential: E=

1 1 1 ∇Ec = ∇Ev = ∇Ei e e e

(81)

Equation (81) implies that the electrostatic potential is determined by conduction and the valence band edges because the carriers there have only potential energy. The last equation assumes that the separation in the energy scale of the three levels (Ec , Ev , and Ei ) is space independent. So, in Eq. (81), as well as in eqs. (76b), (76c), (77b), and (77c), we neglect band distortion resulting from heavy doping or other effects

(e.g., mechanical stain). This is discussed in a later section on heavy doping effects. Boundary Conditions. Equations (76)–(81) form a system of six relations with six unknown variables: Fn , Fp , Ei , Jn , Jp , and E. They apply, within the range of their validity, to any semiconductor device. In this sense, any semiconductor device understanding, design, operation, and performance is based on this set of six equations. What distinguishes a device of a particular kind is its boundary conditions, as well as the doping and trap density and type. Ohmic Contacts. For a diode, a two-terminal device, the boundary conditions necessarily include two ohmic contacts that will supply the charge to be transported through the device. The voltage across and the current through the two ohmic contacts are, interchangeably, the excitation or the response of the device. The ohmic contacts are realized by depositing metals (e.g., Ti or Al) on heavily doped regions of the semiconductor. An ideal ohmic contact should, by definition, establish thermodynamic equilibrium between the metal and the semiconductor at all the contact points. In analytical terms, this is expressed by equating the carrier quasi-Fermi potentials to the metal Fermi potential:

Fn (C1 ) = Fp (C1 ) = V1

(82a)

Fn (C2 ) = Fp (C2 ) = V2

(82b)

where V1 and V2 are the voltages of the metal contacts (V1 − V2 is the terminal voltage), whereas C1

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and C2 are the contact areas of the first and the second ohmic contact, respectively. The pinning of the Fermi potentials at the externally applied voltages is illustrated in Fig. 58. If V1 = V2 and G in Eqs. (78) and (79) were zero, the device would be in equilibrium. Then the solution of the previous system of six equations would be zero currents and equal and flat Fermi potentials throughout the diode. When an external voltage is applied, the splitting of the Fermi potential values between the two ohmic contacts drives the device out of equilibrium. Such a boundary value split enforces a separation of the electron and hole quasi-Fermi potentials through the device, as shown in Fig. 58. The separation of the two potentials implies that the nonequilibrium conditions mainly refer to the interaction between electrons and holes. At any point, excluding ohmic contacts, electrons are out of equilibrium with respect to holes because the relaxation time of interband transitions (recombination and generation mechanisms) required to bring them into equilibrium are too slow (milliseconds or microseconds for germanium and silicon and nanoseconds for most compound semiconductors). On the contrary, the intraband transitions resulting from scattering have short relaxation times (picoseconds) so that electrons or holes are nearly in equilibrium within their band. This is required for the carrier Fermi potentials to have a meaning, as mentioned in the discussion following Eqs. (76) and (77). Semiconductor–Insulator Interfaces. The surface that bounds the device includes, in addition to the ohmic contact, the semiconductor–vacuum or semiconductor–insulator interface. The exposed semiconductor surface is usually covered by an insulating film (SiO2 in silicon) to reduce recombination. If we assume that there is no injection in the insulator, then at the interface the boundary conditions for Eqs. (78) and (79) are

1 ∂ns = − Jnn − Us ∂t e 1 n ∂ps = Jp − Us ∂t e

(83) (84)

where the subscript s refers to surface densities and the superscript n refers to the normal component looking into the insulator. The boundary conditions for Eq. (80) are dictated by the lows of electrostatics. The discontinuity of the normal component of the dielectric displacement vector must be equal to the surfacecharge density, whereas the tangential component of the electric field must be continuous. Because the boundary conditions have been set, the system of six equations [Eqs. (76)–(81)] can be solved, in principle. As it turns out, the solution of such a nonlinear system of coupled equations can be found only numerically even for one-dimensional p –n junctions with uniform acceptor and donor densities. To

derive analytical approximations, we need to make certain assumptions regarding the physical makeup of the device and the degree of bias. These analytical expressions help predict the device response under reasonable bias, whereas the appreciation of their validity range provides an insight into the diode device physics. 4.2 Doping Carrier Profiles in Equilibrium and Quasi-Neutral Approximation As mentioned earlier, a p –n junction diode consists of an acceptor-doped p region in contact with a donor-doped n region. The two-dimensional area where the donor and acceptor densities are equal is called the metallurgical junction. Let us assume, for the moment, equilibrium conditions. In such a case, the currents are zero and the quasi-Fermi potentials are equal and spatially independent, Fn = Fp = F . Therefore, from Eqs. (76) and (77), pn = n2i , where ni is the intrinsic-carrier density. Now, the six equations reduce to the Poisson equation, which, with the help of Eqs. (76c), (77c), and (81), takes the form

e2 (−eF ) − Ei −ni exp [Ei − (−eE)] = ∈ kT Ei − (−eF ) + ND − NA (85) +ni exp kT 2

The last equation is known as the Boltzmann–Poisson equation. Approximate analytical solutions are possible when the donor and acceptor densities are uniform in the n and p regions, respectively. In this case, the field is zero, and the bands are flat everywhere except at and near the metallurgical junction. The finite-field region around the metallurgical junction is called the space-charge region, whereas the zero-field regions are called quasi-neutral regions, for reasons to be explained shortly. In the n and p quasi-neutral regions, electrons and holes are the majority carriers, respectively. The majority-carrier densities are equal to the respective doping densities. With reference to the metallurgical junction, the space-charge region extends WA and WD within the p and the n regions. At zero bias, and in one dimension, an approximate solution of Eq. (85) gives 2ε ND Vbi (86a) WA = e NA (NA + ND ) 2ε NA Vbi (86b) WD = e ND (NA + ND )

kT NA ND ln (86c) Vbi = e n2i In Eq. (86c) Vbi is the zero bias electrical potential difference, or barrier, between the p and the n side

994


reflected in the level differences of the flat bands of each side. Such a barrier prevents majority carriers from diffusing into the other side. These approximations result by assuming that the electron and hole densities are zero in the space-charge region. This is the depletion approximation, which reduces Eq. (85) to a linear second-order differential equation with constant terms and coefficients. The zero-field condition for the rest of the n and p sides, outside the space-charge region, apparently justifies the term quasi-neutral regions. This term also applies when the n and p regions have gradually changing dopant profiles in the sense that the net space charge is much less than the majority-carrier charge. Here, by gradually changing we mean that the doping profile N(x) in the quasi-neutral region is such that66 N(x) εkT 2 ln N(x) (87) ∇ e2 ni In such regions, the zero-bias majority-carrier density continues to be nearly the same as the net dopant density, but the electric field is not zero as in the uniform doping case. 4.3 Forward- and Reverse-Bias Conditions The quasi-neutrality condition of the n and p regions is preserved even under bias, but now the boundaries with the space-charge region move appropriately to accommodate the new boundary conditions. This neutrality condition can be expressed as

n ≈ p + ND − NA

(88)

Under a small forward bias, the applied voltage changes the electric field preferentially at the spacecharge region, because it is the region with the fewest carriers, has the highest resistance, and is in series with more conductive n and p regions. The equilibrium barrier height Vbi lowers under forward bias, and the majority electrons overcoming the repulsive field diffuse from the n side to the p side, whereas the holes are doing the opposite. The carrier quasi-Fermi potentials are no longer equal, as shown in Fig. 58. The diffusion process, through the space-charge region and inside the quasi-neutral regions, increases dramatically the minority-carrier population on either side and gives rise to an appreciable electric current. For forward voltages, the degree of bias defines three injection-level regimes distinguished by how the minority-carrier density compares to the majority one in the quasi-neutral regions. These regimes are the low-level, the moderate-level, and the high-level injection condition. In the low-level injection regime, the minority-carrier density is well below the majoritycarrier density, and the electric field in the quasineutral regions is practically unaffected by the bias. As a result, the applied voltage drops across the spacecharge region and reduces the barrier height from Vbi

to Vb = Vbi − V . Provided the depletion approximation still holds, Eq. (86) still applies with Vbi being replaced by Vb . In low-level injection, the majoritycarrier density is the same as at zero bias, as Eq. (88) points out, and is nearly equal to the net doping density. In the high-level injection regime, the minoritycarrier injection is so intense that the injected carriers have densities far exceeding the dopant densities. Now, both carrier densities are about the same, n = p, to preserve neutrality in the quasi-neutral region. In other words, there is no real distinction between minority and majority carriers in terms of concentrations, but we obtain an electron–hole plasma having densities well above that of the dopant densities instead. In the moderate injection, the minority-carrier density approaches the order of magnitude of the majority-carrier density causing the majority-carrier density to start to increase, as Eq. (88) implies. Under reverse bias, the built-in barrier increases in the space-charge region, the repulsive forces on the majority carriers coming from the quasi-neutral regions increase, and injection of minority carriers is not possible. The space-charge region is now totally depleted from both carriers, and a small leakage current exists as a result of thermal generation of electron and hole pairs in the depletion region. Recombination Currents in the Steady State Here, we will introduce the base and emitter terms as well as a general expression for the terminal current as the sum of recombination components. Between the two quasi-neutral regions, emitter is the one that is heavily doped, usually by diffusion or implantation, whereas the base is more lightly doped and occupies most of the substrate on which the device is made, at least in silicon. The heavy doping of the emitter excludes the possibility of moderate- or high-level injection conditions in this region. At forward bias, majority carriers from the emitter diffuse as minority carriers to the base where they recombine. Simultaneously, recombination occurs in the emitter because minority carriers are back-injected from the base, as well as in the space-charge region. At steady state, ∂n/∂t = ∂p/∂t = 0, and in the dark G = 0. Now, the continuity equations [Eqs. (78) and (79)] become after volume integration:

ID = Ie + Ib + ISCR

(89)

where ID is the terminal current and Ie , Ib , and ISCR are the net recombination currents in the emitter, the base, and the space-charge region, respectively. Equation (89) expresses the total current as the sum of the recombination currents in the three regions of the device. Therefore, excess carrier recombination along with diffusion are the two basic transport mechanisms that determine the diode current at a given bias. The carrier recombination occurs either at the ohmic contacts, at the surface, or in the bulk. The minority carriers that arrive at the ohmic contact are supposed

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to recombine simultaneously there, to preserve the boundary condition, Eq. (82b). The bulk recombination occurs either through traps or through band-toband transitions. In terms of trap-mediated recombination, the Shockley–Read–Hall mechanism67,68 is the most common: USRH

(pn − n2i )Nt = Et −Ei 1 σp vth n + ni exp kt Et − Ei 1 p + ni exp − + σn vth kt

−∇(Eµp p − Dp ∇p ) −

(90)

where Nt is the trap density, σn and σp are the electron and hole capture cross sections, respectively, Et , is the trap energy level in the gap, and νth is the carrier thermal velocity. The band-to-band recombination is discussed in the heavy-doping effects section. 4.4 Approximate Analytical Expressions in Steady State As previously mentioned, the set of Eqs. (76)–(81) has no analytical solution in the general case. Approximate closed-form expressions, though, are possible when low-level injection conditions dominate in the quasineutral base region. Without loss of generality, we assume that we are dealing with a p –n diode with a heavily doped p emitter and an n base. The steadystate situation results when a terminal bias, say a terminal voltage V , is steadily applied on the terminals, and we wait long enough for the initial transient to disappear. The steady-state version of the continuity equations [Eqs. (78) and (79)] is simplified because the time derivatives are set equal to zero. First, we will derive the expressions for the base current, and then extensions will be made for the recombination current in the emitter and the space-charge regions. If low-level injection conditions prevail in the base, then, to a good approximation, the original system of equations [Eqs. (76)–(81)] reduces to the minoritycarrier equations [Eqs. (77) and (79)], which are now decoupled from Eq. (80) (the Poisson equation). This decoupling results because, as previously mentioned, at low-level injection the electric field is practically bias independent. Any small field variations would affect only the drift current of the majority carriers because of their high density; the minority carriers would not be influenced. That is why we focus on the minority-carrier transport to exploit the Poisson equation decoupling. Another reason for focusing on the minority carriers is the fact that the recombination in low-level injection, where p n, can always be written as a linear function of their density:

U=

p − p0 τ

inverse of the derivative of the recombination rate with respect to the minority-carrier density. In the case of Shockley–Read–Hall recombination, τ = 1/σp νth Nt . Therefore, from Eqs. (77b), (79), and (91), we end up at

(91)

where p0 is the equilibrium carrier density, whereas the variable τ , called minority-carrier lifetime, is the

p +G=0 τ

(92)

where p = p − p0 is the excess minority-carrier density. Because of the field independence on p , Eq. (92) is linear and becomes homogenous if G = 0. In the later case, the solution is proportional to p (Cj ), the excess minority-carrier density at the injecting boundary (Fig. 58). Forward Bias and Low-Level Injection Under forward bias, a basic assumption will be made. This assumption allows the coupling of the minority-carrier density to the externally applied terminal voltage: the Fermi levels are flat in the regions where the carriers are a majority and also in the space-charge region. Under this condition and from Eqs. (76c), (77c), and (82),

p(Cj )n(Cj ) = n2i exp p (Cj ) =

eV kT

n2i [eV /kT ) − 1] ND (Cj )

(93) (94)

Equation (93) holds under any injection level, provided that the flat Fermi potential assumption holds, whereas Eq. (94) for the excess minority-carrier density holds only in low-level injection. The proportionality of the solution with respect to p (Cj ) forces all carrier densities and currents to become proportional to the term exp(eV /kT ) − 1. Here, we note that the surface recombination is also a linear function of the excess minority-carrier density when p n. More analytically, Eq. (84) becomes Jpn = eSp p

(95)

where Sp is called surface recombination velocity. Therefore, the total base recombination current in Eq. (89) is proportional to the term exp(eV /kT ) − 1. The same is true for the quasi-neutral emitter recombination. Thus, Eq. (89) becomes eV − 1 + ISCR ID = (I0e + I0b ) exp kT

(96)

where the preexponential factors I0b and I0e are called base and emitter saturation currents, respectively. Equation (94) points out that the saturation currents are proportional to n2i .


The space-charge region recombination is a current component that is hard to express in analytical terms. This difficulty arises because in this region the field depends on the bias and there is no such entity as a minority carrier. Consequently, the linearity conditions that allowed us to derive Eq. (96) no longer hold. To derive an approximate expression for the bias dependence of ISCR , certain simplifications must be made throughout the space-charge region regarding the integral of Eq. (90). These simplifications result in a bias dependence of the form exp(eV /nkT ) − 1, where n, the ideality or slope factor, takes values from 1 to 2.65 The specific value depends on the trap position in the energy gap, the doping profiles, and the cross section for hole capture relative to the cross section for electron capture. This range for n holds provided that the capture coefficients do not depend on the electric field. Now the preexponential factor is proportional to ni . Finally, the expression for ICSCR the forward current of a diode in the base of which low-level injection conditions prevail becomes eV −1 ID = (I0e + I0b ) exp kT eV exp −1 + ISCR nkT

(97)

For voltages higher than 3nkT/e, the unity can be dropped from Eqs. (96) and (97). Because of a better slope factor, the emitter and base recombination will dominate the diode current for voltages above a certain level. Below this level, the space-charge region recombination must be considered too. Such trends are seen in Fig. 59. Curve 1 is the I–V characteristic of a diode with a base doping of 5.5 × 1014 cm−3 and has an ideality factor of 1 in the bias range from 0.2 up to 0.4 V. For lower voltages, the space-charge region recombination slightly increases the ideality factor and makes the measured current deviate from the exp(eV /kT ) − 1 dependence. The ideality factor also increases for voltages above 0.4 V because of high-injection effects, which are discussed in the next subsection. The device of curve 2 has a very light doping density in the base, 4.5 × 1011 cm−3 and is driven in high injection at even smaller bias. As discussed in the next subsection, curve 2 exhibits unity slope factor even at very low voltages. Curve 3 shows what happens if the temperature is reduced to 78 K. The sharp reduction of the intrinsiccarrier density due to its ∼ = exp(−Eg /2kT ) dependence requires much higher voltages to reach the same current as at 300 K. In fact, to reach a current density of 10 mA/cm2 , a voltage in excess of 1 V is required. The reduction of ni reduces the recombination in the base and the emitter is much faster than in the spacecharge region because the proportionality constants are n2i and ni , respectively. Therefore, at low temperatures, the bias regions with higher than 1 ideality factor are

0.00

108

10.00

Reverse bias (−V) 20.00

30.00

107

40.00

n = 1.23

n=2

106 105 104 L (nA)

996

103

2 n=1

3

1

102

n = 3.5 n=1

10 1

1(−V)

10−1 10−2 10−3 0.00

0.20

0.40 0.60 Forward bias (V)

0.80

1.00

Fig. 59 Experimental I–V characteristics of two different diodes. Diode 1 has a base thickness of 250 µm, a base-doping density of 5.5 × 1014 cm−3 , and an area of 2.9 × 2.9 mm2 . Diode 2 has a base thickness of 300 µm, a base-doping density of 4.5 × 1011 cm−3 , and an area of 5 × 5 mm2 . The base in both devices is of n type. Plot 3 is the I–V characteristic of diode 2 at 78 K. The other plots are at 300 K. Curve 1(−V) is the reverse bias characteristic, with reversed sign, of diode 1 (top axis). The straight lines in curves 1, 2, and 3 are the exponential exp(eV/nkT) − 1 fits to the experimental points. The slope factor n is also shown.

expected to be wider. This is evidenced in curve 3 of Fig. 59, where the ideality factor is 3.5 for voltages below 950 mV. The increase of the ideality factor above 2 is a result of the Poole–Frenkel effect, which reduces the effective energy separation of the traps from the bands.69 The influence of small values of ni on the ideality factor is evident not only when the temperature drops but also when the bandgap increases. In several compound semiconductor devices, their large bandgap, compared to 1.1 eV of silicon, results in an intrinsic-carrier density, which is several orders of magnitude smaller than the 1010 cm−3 value for silicon at 300 K.70,71 Consequently, their I–V characteristics show slope factors substantially larger than 1 for the entire range of bias. On the contrary, germanium diodes have slope factors of 1 even at reduced temperatures because of the smaller gap, 0.66 eV, of the semiconductor. One-Dimensional Case. Equation (97) holds for any three-dimensional geometry and doping profiles because no assumption, except for low-level injection, was made so far regarding doping profiles and device topology. If, however, we want to express in close

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form the saturation values of the emitter and base recombination currents, then one-dimensional devices with uniform doping profiles must be considered. In such a case, the one-dimensional, homogenous, and constant-coefficient version of Eq. (92) becomes p d 2 p = 2 2 dx Lp

(98)

where Lp = Dp τ is the minority-carrier diffusion length. The first boundary condition for Eq. (98) is Eq. (94) applied at the injecting boundary. The other one refers to the ohmic contact. If it is an ideal ohmic contact deposited directly on the uniformly doped base, then the second boundary condition becomes, from Eq. (82), p (l) = 0. Here, l is the base length and the coordinate origin is at the injecting boundary, as shown in Fig. 58. In many cases, between the ohmic contact and the uniformly doped base, a thin and heavily doped region intervenes. This region has thickness on the order of a micrometer and a doping of the same type as the rest of the base. The purpose of such a layer, called back-surface field, is to provide a better ohmic contact and to isolate the contact from the lightly doped base so that carrier recombination generation is reduced.72 Such a backsurface field terminates the lightly doped base of diode 2 in Fig. 59 making it a p–i–n diode, where i stands for intrinsic. Therefore, in the presence of this contact layer, the base ends at a “low/high” n–n+ junction. In terms of minority-carrier recombination, this interface is characterized by an effective recombination velocity Spe , experienced by the minority carriers at the low side of the junction. The expression for Spe is Spe =

I0c ND en2i S

(99)

where I0c is the saturation value of the recombination current in the back-surface field and S is the device cross section. Equation (99) can be derived from Eq. (95), by applying Eq. (94) at the n–n+ junction and by equating the minority current at the low/high junction to the recombination in the heavily doped region. Under the previous boundary conditions, the solution of Eq. (98) yields for the base saturation current: Dp 1 + [Dp /(Spe Lp )] tanh(/Lp ) ND Lp tanh(/Lp ) + Dp /(Spe Lp ) (100) Equation (100) shows that, in terms of the onedimensional geometry, the quantity that matters is the ratio l/Lp . Values of this ratio much less than one define the short base, whereas values above 3 define the long base. In the long base case, Eq. (100) becomes I0b = Sen2i Dp /(ND Lp ). Similar equations hold for a uniform I0b = S

en2i

emitter, too, but now the heavy doping effects could modify the value of n2i , as will be discussed in the section on heavy doping effects. If the base doping is very light, as in a p–i–n diode, then the increased value of I0b will make the base recombination dominate the current components in Eq. (97). Accordingly, the influence of space-charge region recombination current on the slope factor will be suppressed even for voltages as low as a few kT/e, as shown in Fig. 59, curve 2. Also, by extrapolating the exp(eV/kT ) fit of curve 2 at zero voltage, a base recombination current of 0.7 nA is obtained. This corresponds to a 300-K saturation current density of 2.8nA/cm2 compared to emitter saturation current densities on the order of pA/cm2 . On the other hand, diode 1, with a base-doping density three orders of magnitude higher than that in diode 2, exhibits a saturation current of 24 pA/cm2 . This saturation current comes mainly from the base recombination as a result of its relatively light doping density and the absence of a backsurface field which gives Spe very high values. Equation (100) applies to uniformly doped regions. If the doping is nonuniform, close form expressions are not possible, in the general case. This is the case because the one-dimensional version of Eq. (92) is still an ordinary differential equation with nonconstant coefficients. However, analytical approximations can be derived based on iterative techniques.73 Diffusion in Three Dimensions. Equation (98) holds provided the cross-sectional dimensions of the diode are much larger than the diffusion length. Otherwise, lateral diffusion of minority carriers in the base becomes important. In such a case, the threedimensional version of Eq. (98) takes the form

∇p2 =

p L2p

(101)

The last equation can be solved very accurately by semianalytical techniques based on the two-dimensional Fourier transform.74 Simulation results are as shown in Fig. 60. As illustrated, in the case of a point contact diode having emitter dimensions of 0.1Lp , the base recombination is expected to increase by a factor of 25 as a result of the lateral carrier diffusion. High-Level Injection So far, our analytical approaches were based on the low-level injection assumption. In high-level injection, where n = p, an equation similar to Eq. (98) can be derived where now the hole diffusion length is replaced by the ambipolar diffusion length.75 The boundary conditions, however, are not linear and depend on the electric field, which, now, is a function of bias. If the quasi-Fermi potentials are flat in the quasi-neutral base, then the electron–hole plasma density p is space independent and equals ni exp (eV/2kT), as can be derived from Eq. (93). In such

998


such a case, the simulation is possible only by device simulators that solve the complete system of the transport equations.

Normalized 3-D current

10

(4) (2) (3) (1)

1

10

1 L /Ld

0.1

Fig. 60 Three-dimensional diffusion base saturation current of a planar p–n junction with a square emitter having a side length L. The top surface of the base is supposed to have zero recombination velocity. The current is normalized with respect to the one-dimensional diffusion current I0b = en2i L2 Dp /ND Lp . In curve 2, Spe = 0 and l = Lp /2. In curve 2, l is assumed to be infinite. In curve 3, Spe = Dp /Lp and l = Lp /2, whereas in curve 4, Spe is assumed to be infinite and l = Lp /2.

a case, by integrating the recombination current in the emitter, the base, and the back-surface field, we obtain ID = (I0e + I0c ) exp

eV kT

+

eSl τ

exp

eV 2kT

ni

(102) where τ is the high-injection lifetime defined as the ratio of the recombination rate divided by the plasma density. The flat Fermi level condition can easily be satisfied in p–i–n diodes where the light base doping density makes the high-level injection possible even at a bias of 0.4 V. In Fig. 59, curve 2 shows the exp(eV /2kT ) dependence, or slope factor of 2, for voltages of about 0.4 V, which drive the p–i–n device to high-level injection. For even higher voltages, the emitter and back-surface field recombination in Eq. (102) starts dominating the current, and the slope factor drops again. For higher base doping densities, as in curve 1 of diode 1, the required voltage for high-level injection conditions could exceed 0.5 V at 300 K. Now, the heavily doped region recombination in Eq. (102) competes with the bulk recombination, and the slope factor of 2 does not appear. The bent of both curves 1 and 2 at voltages near 0.6 V is a result of series resistance effects, which invalidate the assumption of flat Fermi levels across the base. In

Reverse Bias Under reverse bias where V < 0, the assumption of flat Fermi levels across the spacecharge region that led to Eq. (97) no longer hold. On the other hand, however, the space-charge region can be considered to be fully depleted from free carriers. In such a case, Eq. (86) holds with Vbi replaced by Vbi + |V |. Therefore, the depleted space-charge region will expand toward the base according to the square root of the bias for |V | > 5 V. In this region, the Shockley–Read–Hall Eq. (90) predicts a negative recombination or generation of electron–hole pairs. This generation current is the basic component of the leakage current in reverse bias. The contribution of the diffusion components from the base and the emitter, −I0b − I0e , is usually negligible unless the base is very lightly doped. The bottom line in Fig. 59 shows the reverse-bias current for diode 1. The square-root dependence on voltage is not exactly obeyed because of the Poole–Frenkel effect, which increases the generation rate at higher fields. 4.5 Transient Response of Diodes If a diode is subjected to a transient terminal bias, then in addition to currents due to carrier diffusion and recombination, we also have the dielectric displacement current resulting from the time dependence of the electric field. If low-level injection is observed in the quasi-neutral regions, the displacement current is restricted in the space-charge region. At the same time, low-level injection ensures that linearity holds in the base and the emitter, and Eq. (92) still applies with ∂p /∂t replacing zero in the righthand side of the relation. The solution of the timedependent edition of Eq. (92) provides the minoritycarrier currents at the injecting boundaries of the base and the emitter, Ib (t) and Ie (t), respectively. These currents have now two components: the minoritycarrier recombination and the minority-carrier storage current ∂Q /∂t, where Q is the total excess minority-carrier charge. To calculate the total transient current, reconsider Eq. (89) in its transient version. Therefore, in addition to Ib (t) and Ie (t), the transient space-charge region current is required. Unlike the base and the emitter, this current in addition to the recombination and storage component also includes the displacement current.76 Insofar as the displacement current is concerned, the space-charge region behaves as a parallel plate capacitor with a plate distance W = WA + WD , Eq. (86), a dielectric constant ε and a capacitance CSCR = εS/W . During transit, the dielectric displacement current is supplied by the majority carriers from either side of the junction. To calculate the transient currents in the base and the emitter, the boundary conditions must be defined. Boundary condition Eq. (95) holds because of

ELECTRONICS

ev eV exp −1 kT kT eV ev ∗ exp −1 + ISCR nkT nkT

∗ ∗ ID∗ = (I0e + I0b )

+ j ωCSCR v = v(G + j ωC)

(103)

The star exponents denote the complex values of the saturation currents as a result of the complex lifetime. In Eq. (103), G and C are the diode smallsignal parallel conductance and capacitance, respectively. These two components are of great importance because their frequency dependence can reveal minority-carrier properties, such as diffusivity and lifetime78 and allow the device circuit representation when the diode is part of a greater small-signal cir∗ cuit. For uniformly doped quasi-neutral regions, I0e ∗ and I0b can be obtained from Eq. (100) by replacing the diffusion length L = (Dτ )1/2 with the complex diffusion length L∗ = L/(1 + j ωτ )1/2 . For frequencies sufficiently high, the magnitude of the complex diffusion length will become much shorter than the base thickness. Then, the complex version of Eq. (100) predicts that the base current would change as 1/L∗ . If the base component in Eq (103) were to dominate, then beyond a certain frequency, C would change as ω−1/2 while G would change as ω1/2 . This frequency dependence is confirmed in Fig. 61, which shows the frequency response of diode 1, from

10 2

10

10

1

10 3

10 4 10 5 Frequency (Hz)

10 6

Conductance (mS)

Small-Signal Response In many cases, the device operates under sinusoidal small-signal excitation superimposed on a steady-state excitation. In such cases, Eq. (92) still holds, but now 1/τ will have to be replaced by 1/τ + j ω, where j is the imaginary unit and ω is the angular frequency of the excitation. This is the case because the time derivative of the small-signal carrier density is the carrier density amplitude times j ω. Having done the complex lifetime replacement, the analysis that followed Eq. (92) still holds. Now, however, the small-signal value of the excess minoritycarrier density at the injecting boundary will be the steady-state value in Eq. (94) times ev/kT. Here, v is the small-signal terminal voltage, which is supposed to be much less than kT/e. Under low-level injection and in view of the previous transient response discussion, the small-signal version of Eq. (97) will refer to a terminal current ID∗ having a real and an imaginary

component:

Capacitance (nF)

linearity. The other condition at the injecting boundary depends on the kind of transient to be considered (89). Here we will assume that the device is in equilibrium for t < 0, whereas at t = 0 a constant voltage V is applied. We can now assume that Eq. (94) applies with p (Cj ) replaced by p (Cj , t) for t > 0. This assumption has a validity range depending on how fast the flat quasi-Fermi potential condition can be established across the space-charge region. As a matter of fact, even in the absence of series resistance effects, it takes a short time for this condition to be established. This short time relates to the dielectric response time of the majority carriers and the minority-carrier diffusion time across the space-charge region.77 For almost all practical cases, the delay in establishing a fixed minority-carrier density at the edge of the quasineutral region will not exceed the limit of a few tens of a picosecond77 in the absence of series resistance effects. Therefore, if the time granularity used in solving the time-dependent version of Eq. (92) is restricted to about a nanosecond, then the solutions will be accurate. In practical cases, however, the very first part of the transient current, following the sudden application of a voltage, will be determined by charging CCSR through the series resistance of the majority carriers in the base and the emitter. The respective time constant could be on the order of a nanosecond. In such a case, the minority-carrier transport in the base will determine the transient only after several nanoseconds have elapsed since the application of the voltage. The transient base transport can be expressed in semianalytical forms using Laplace transform techniques,77 especially in the case of uniform and one-dimensional quasi-neutral regions. In a long-base diode, the transition will last for about a minority-carrier lifetime. In a short-base device with an ohmic contact at the base end, the transient will last approximately l 2 /2Dp , which is the minority-carrier diffusion time through the base.

999

1

Fig. 61 Experimentally measured capacitance (dots) and conductance (squares) at 300 K for diode 1. The bottom and the top curves correspond to two different bias points: 420 and 450 mV, respectively. The solid curves are the theoretical fits from the equivalent circuit of Fig. 62.

1000

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Cp Gp

Emitter contact

GSRC

RN

G +N

CNQNR

Base contact

CSRC

Fig. 62 The equivalent circuit model of a diode. The base injection currents in Eq. (103) corresponds to Cp , imaginary part, and Gp , real part. The injection in the emitter is represented by G+ N . The space-charge region recombination is represented by GSCR , whereas CSRC is the space-charge region capacitance. The rest of the components are accounted for in the text.

Fig. 59, at two bias points. The theoretical fit to the experimental results was obtained on the basis of the diode equivalent circuit shown in Fig. 62. This circuit includes all the components relating to carrier injection and storage in the device’s three regions in accordance with Eq. (103). It also includes the base resistance RN , which has been ignored in Eq. (103). In Fig. 61, the square root law is better obeyed at the higher bias point and for frequencies less than 1 MHz, especially for the capacitance. This is a combined result of the space-charge region capacitance, the relative contribution of which increases at lower bias, and the series resistance, the influence of which is stronger at high frequencies. The corner frequencies of the conductance and the capacitance depend on the base thickness and the lifetime. The fit shown in Fig. 61 gave a minority-hole lifetime in the n-type base of about 30 µs. Such a lifetime and Eq. (100) imply that the saturation current density of 24 pA/cm2 at 300 K, as shown in Fig. 59, is 90% due to base recombination. The emitter contribution of 10% is discussed in the section on heavy doping effects. The series resistance RN becomes the bulk majoritycarrier resistance under reverse bias or even under forward bias, provided that the frequency is high N in parallel with RN , (|L∗ | l). The capacitance CQNR as shown in Fig. 62, is the geometrical capacitance of the quasi-neutral base.79,80 For ordinary resistivity devices, it can be ignored unless the frequency is in the gigahertz range. However, for diodes made on high resistivity substrates, this capacitance must be considered especially at reverse bias and high frequencies.80 From Fig. 62 and in the limit of very high frequencies under forward bias, the parallel conductance saturates at 1/RN whereas the parallel capacN . This is because of the comitance does so at CQNR bination of the increasing injection conductancies and

the space-charge region capacitance. Then, the product N RN CQNR becomes the dielectric response time of the majority carriers in a uniform base. At high injection, the parallel conductance will saturate at the sum of the two carrier conductances.81 Under reverse bias, the circuit of Fig. 62 reduces to the space-charge region capacitance in series with the parallel combination of N CQNR and RN . Unlike the forward-bias case, where the circuit parameters depend roughly exponentially on the terminal voltage V , in reverse bias the voltage dependence would be restricted to V −1/2 . In the sense of the voltage dependence, the circuit of Fig. 62 is the circuit of a varactor. 4.6 Heavy Doping Effects in Emitter

In the previous subsection, the emitter saturation current density was estimated to be about 2 pA/cm2 . From Eq. (100) and by assuming microsecond lifetimes, we would expect saturation currents on the order of a fA/cm2 from an emitter doped in the range 1019 –1020 cm−3 . Such a discrepancy by three orders of magnitude is due to the heavy doping effects, namely the short lifetime resulting from Auger recombination and the effective increase of ni due to bandgap narrowing. In the Auger recombination process, a minority carrier recombines directly with a majority one, and the energy is transferred to another majority carrier. Because of such kinetics, the Auger minority-carrier lifetime is inversely proportional to the square of the majority-carrier density. The proportionality constant is ∼10−31 cm6 /s for minority electrons in p + emitters and 3 × 10−31 cm6 /s for minority holes in n+ emitters.82 In heavily doped regions, the Auger recombination rate is by far higher than the Shockley–Read–Hall rate and determines the lifetime. Therefore, nanosecond lifetimes are expected, especially for holes, in emitters doped in the vicinity of 1020 cm−3 . In a heavily doped region, every minority carrier interacts strongly with the majority carriers because of their high density. The minority-majority carrier attraction along with the carrier-dopant interaction and the semiconductor lattice random disruption by the dopant atoms reduces the banggap and changes the density of states in both bands.83,84 The net result is an effective shrinkage of the gap depending on the doping type and density.85 – 87 This shrinkage changes the intrinsiccarrier density ni to a much higher effective nie . The result of the band distortion is that the original system of transport equations [Eqs. (76)–(81)] no longer holds. More specifically, Eqs. (76b), (76c), (77b), and (77c) are not valid for the majority carriers even if nie substitutes ni because Boltzmann statistics must be replaced by Fermi–Dirac statistics. Also, Eq. (81) no longer holds in a nonuniform region because the band edges are not parallel any more and each carrier experiences a different field. However, the minority carriers still

ELECTRONICS 107 17 K 106 105

I (nA)

follow the Boltzmann statistics, and Eq. (92) holds for the minority carriers. Now E is the minority-carrier field (1/e∇Ec for electrons), and the boundary condition Eq. (94) is valid with ni replaced by nie . Therefore, Eq. (100) still applies for the minority-carrier recombination in a uniformly doped emitter. For an emitter doped at about 1020 cm−3 , a gap narrowing of about 100 meV is expected,15 – 87 which makes nie several tens higher than ni . If such an nie as well as nanosecond lifetimes replace ni and microsecond lifetimes in Eq. (100), an emitter saturation current on the order of pA/cm2 is predicted, in accordance with the experimental results of the previous section.

1001

104

13 K

103 102

4.7 Diodes of Nonconventional Transport

So far in this section, devices based on the drift and diffusion model of Eqs. (76) and (77) were studied. Charge carriers can be transported from one region to another by tunneling. Also, they can be temporarily trapped in energy-gap states, atom clusters, or crystallites imbedded in insulating films, thereby affecting the tunneling or the conventional transport of the free carriers. In this respect, the first device to be examined is the p–i–n diode 2 of Fig. 59, operating at cryogenic temperatures. Around 4.2 K, the equilibrium Fermi level in the lightly doped n− region is pinned at the donor level. These levels, now, are not ionized except for a fraction to compensate the charge of the unintentionally introduced acceptor ions. At such low temperatures, there are no free carriers in the base, and no measurable conduction is possible unless the voltage is raised enough to achieve the flat-band condition.88,89 For silicon, this voltage V0 would be about 1.1 V. For even higher voltages, conduction is possible only if electrons and holes can be injected in the frozen substrate from the n and p regions, respectively. In this sense, Eq. (99) based on the assumption of flat majority-carrier Fermi levels no longer holds. For T < 10 K, injection is possible by carrier tunneling through the small potential barrier existing at each of the p–i and i–n interfaces.59 These barriers exist because of the band distortion in the heavily doped regions and the smaller gap there, as outlined in the previous section. For V >V0 , electrons tunnel in the i layer, and the higher the forward bias, the higher the current due to a field-induced effective lowering of the barriers. As shown in Fig. 63, for temperatures below 10 K it takes at least several volts to establish a current of few nanoamps. The injected electrons in the i layer are trapped by the ionized donors and built a space charge and a subsequent potential barrier. For even higher voltages approaching 10 V, the barrier at the i–p interface lowers, holes now enter the i layer in large numbers. Their charge neutralizes the trapped electron charge and causes the voltage breakdown and the negative differential resistance that appears in Fig. 63 for T < 10 K. The negative resistance persists

7K 10 9K 1

1

10 V (V)

Fig. 63 Measured I–V characteristics of diode 2 at cryogenic temperatures. The square points correspond to 4.2 K. The T < 10 K plots exhibit a distinct voltage breakdown. Reprinted from K. Misiakos, D. Tsamakis, and E. Tsoi, Measurement and modeling of the anomalous dynamic response of high resistivity diodes at cryogenic temperatures, Solid State Electronics, 41: 1099–1103, 1997, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1 GB, UK.

and beyond breakdown as a result of new carrier generation by the impact ionization of occupied shallow donors by the injected carriers. The interaction of free and trapped carriers through impact ionization gives rise to a negative dynamic conductance and capacitance which for frequencies high enough change as ω−2 .90 For T > 10 K the injection mechanism changes to thermion emission over the interface potential barriers, whereas the space-charge effects are now less pronounced. Another example of tunneling injection mechanism is the breakdown effect in zener diodes. Here, the base is quite heavily doped (≈1018 cm−3 ), and the strong electric field in the space-charge region increases even further by applying a reverse bias. For fields approaching 106 V/cm, a valence band electron can tunnel to a conduction band state of the same energy. This way, electron–hole pairs are created, and the reverse current sharply increases. Another diode structure based on tunneling is a new metal–insulator–semiconductor device having silicon nanocrystals imbedded in the thin insulating film.91 One way to realize such diodes is by depositing an aluminum electrode on a thin (on the order of 10 nm) SiO2 layer containing silicon nanocrystals. The substrate is n-type crystalline silicon. The silicon nanocrystals can be created either by oxidizing deposited amorphous silicon layers91 or by

1002


low-energy silicon ion implantation in the SiO2 film.92 In the absence of the nanocrystals, by applying a negative voltage of a few volts on the aluminum electrode relative to the n-type silicon substrate, only a small tunneling current would be present. When the nanocrystals are introduced, much higher currents are observed while the conductance curve exhibits characteristic peaks. Such peaks are shown in Fig. 64 showing the reverse current and conductance of a quantum dot diode formed by low-energy implantation of silicon in a 10-nm SiO2 layer.92 The conductance peaks appear when the metal Fermi level is swept across the discrete energy states of the nanocrystals, thus enabling resonant tunneling from the metal to the semiconductor.91 The threedimensional confinement of electrons in the quantumbox crystallites creates a large separation between energy states, which along with the Coulomb blockade effect of the occupied states explains the large voltage separation of the three first conductance peaks in Fig. 64.91,92 Such quantum dot devices hold the

promise of single-electron transistors93 and siliconbased light-emitting diodes.94 As we end this section, we would like to mention the basic uses of the diode as a device. The most frequent use of the diode is the protection of CMOS integrated circuits from electrostatic discharges by clamping the output pads to the power-supply voltages through reverse-biased p –n junctions. In analog integrated circuits, forward-biased diodes are used for voltage shifting. Such diodes usually come from properly wired bipolar transistors (e.g., emitter–base diodes with base–collector short circuited). Diodes, as discrete devices, find applications mainly as rectifying elements in power circuits. The breakdown effect of zener diodes makes these devices useful as voltage reference sources in power supplies. Photodiodes are widely used for detecting photons or charge particles. Finally, large area diodes with exposed front surface and proper design and engineering can efficiently convert solar light into electricity and are used as solar cells.95

1.4

0.25

1.2 0.20

Current (mA)

0.15 0.8

0.6 0.10

Conductance (mS)

1.0

0.4 0.05 0.2

0.0

–16

–14

–12

–10 –8 –6 Gate voltage (V)

–4

–2

0

0.00

Fig. 64 Current and conductance plots of a reverse-biased quantum-dot diode. The conductance peaks correspond to steps in current curve. Reprinted from P. Normand et al., Silicon nanocrystal formation in thin thermal-oxide films by very low energy Si+ ion implantation, Microelectronic Engineering, 36(1–4): 79–82, 1997, with kind permission of Elsevier Science-NL, Sara Burgerharstraat 25, 1055 KV Amsterdam, The Netherlands.

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5 ELECTRONIC COMPONENTS Clarence W. de Silva

Table 8

Resistivities of Some Materials Resistivity ρ ( · m) at 20◦ C(68◦ F)

Material

5.1 Materials and Passive Components Conductive Material and Components Conductance and Resistance. When a voltage is applied across a conductor, a current will flow through the conductor. For a given voltage v (volts), the current i (amperes) will increase with the conductance G of the conductor. In the linear range of operation, this characteristic is expressed by Ohm’s law:

i = Gv Resistance R( ) is the inverse of conductance: R=

1 G

Silver, copper, gold, and aluminum are good conductors of electricity. Resistivity. For a conductor, resistance increases with the length (L) and decreases with the area of cross section (A). The corresponding relationship is

R=

ρL A

The constant of proportionality ρ is the resistivity of the conducting material. Hence, resistivity may be defined as the resistance of a conductor of unity length and unity cross-sectional area. It may be expressed in the units · cm2 / cm or · cm. A larger unit would be m2 /m or m. Alternatively, resistivity may be defined as the resistance of a conductor of unity length and unity diameter. According to this definition, R=

ρL d2

where d represents the wire diameter. If the wire diameter is 1 mil (or 1/1000 in), the wire area would be 1 circular mil (or cmil). Furthermore, if the wire length is 1 foot, the units of ρ would be · cmil/ft. Resistivities of several common materials are given in Table 8. Effect of Temperature. Electrical resistance of a material can change with many factors. For example, the resistance of a typical metal increases with temperature, and the resistance decreases with temperature for many nonmetals and semiconductors. Typically, temperature effects on hardware have to be minimized in precision equipment, and temperature compensation or calibration would be necessary. On the other hand, high-temperature sensitivity of resistance in

2.8×10−8 1.7×10−8 20.0 2.4×10−8 775.0×10−8 9.6×10−8 45.8×10−8 20.4×10−8 112.0×10−8 1×1010 1×1016 1×1016 1.6×10−8 15.9×10−8 11.5×10−8 5.5×10−8

Aluminum Copper Ferrite (manganese-zinc) Gold Graphite carbon Lead Magnesium Mercury Nichrome Polyester Polystyrene Porcelain Silver Steel Tin Tungsten

Note: Multiply by 6.0×108 to obtain the resistivity in · emil/ft.

some materials is exploited in temperature sensors such as RTDs and thermistors. The sensing element of an RTD is made of a metal such as nickel, copper, platinum, or silver. For not too large variations in temperature, the following linear relationship could be used: R = R0 (1 + α t) where R is the final resistance, R0 is the initial resistance, T is the change in temperature, and α is the temperature coefficient of resistance. Values of α for several common materials are given in Table 9. These values can be expressed in ppm/◦ C (parts per million per degree centigrade) by multiplying each value by 106 . Note that graphite has a negative temperature coefficient, and nichrome has

Table 9 Temperature Coefficients of Resistance for Several Materials Material Aluminum Brass Copper Gold Graphite carbon Iron Lead Nichrome Silver Steel Tin Tungsten

Temp. Coeff. Resistance α(per ◦ C) at 20◦ C(68◦ F) 0.0040 0.0015 0.0039 0.0034 −0.0005 0.0055 0.0039 0.0002 0.0038 0.0016 0.0042 0.0050


a very low temperature coefficient of resistance. A platinum RTD can operate accurately over a wide temperature range and possesses a high sensitivity (typically 0.4 /◦ C). Thermistors are made of semiconductor material such as oxides of cobalt, copper, manganese, and nickel. Their resistance decreases with temperature. The relationship is nonlinear and is given approximately by R = R0 e−β(1/T0 −1/T ) where the temperatures T and T0 are in absolute degrees (kelvins or rankines), and R and R0 are the corresponding resistances. The parameter β is a material constant. Effect of Strain. The property of resistance change with strain in materials, or piezoresistivity, is used in strain gauges. The foil strain gauges use metallic foils (e.g., a copper–nickel alloy called constantan) as their sensing elements. The semiconductor strain gauges use semiconductor elements (e.g., silicon with the trace impurity boron) in place of metal foils. An approximate relationship for a strain gauge is

R = Ss ε R where R is the change in resistance due to strain ε, R is initial resistance, and Ss is the sensitivity (gauge factor) of the strain gauge. The gauge factor is of the order of 4.0 for a metalfoil strain gauge and can range from 40.0 to 200.0 for a semiconductor strain gauge.96 Temperature effects have to be compensated for in high-precision measurements of strains. Compensation circuitry may be employed for this purpose. In semiconductor strain gauges, self-compensation for temperature effects can be achieved due to the fact that the temperature coefficient of resistance varies nonlinearly with the concentration of the dope material.96 The temperature coefficient curve of a p-type semiconductor strain gauge is shown in Fig. 65. Superconductivity. The resistivity of some materials drops virtually to zero when the temperature is decreased close to absolute zero, provided that the magnetic field strength of the environment is less than some critical value. Such materials are called superconducting materials. The superconducting temperature T (absolute) and the corresponding critical magnetic field strength H are related through

T 2 H = H0 1 − Tc where H0 is the critical magnetic field strength for a superconducting temperature of absolute zero, and Tc

x 10–4 Temperature coefficient (per °F)

1004

3

2

α = Temperature coefficient of resistance

α

1

0 1018

1019 1020 1021 Concentration of trace material (atoms/mL)

Fig. 65 Temperature coefficient of resistance of p-type semiconductor strain gauge.

is the superconducting temperature at zero magnetic field. The constants H0 and Tc for several materials are listed in Table 10. Superconducting elements can be used to produce high-frequency (e.g., 1 × 1011 Hz) switching elements (e.g., Josephson junctions) that can generate two stable states (e.g., zero voltage and a finite voltage, or zero magnetic field and a finite magnetic field). Hence, they are useful as computer memory elements. Other applications of superconductivity include powerful magnets with low dissipation (for medical imaging, magnetohydrodynamics, fusion reactors, particle accelerators, etc.), actuators (for motors, magnetically leviated vehicles, magnetic bearings, etc.), sensors, and in power systems. Color Code for Fixed Resistors. Carbon, wound metallic wire, and conductive plastics are commonly used as commercial resistors. A wire-wound resistor element is usually encapsulated in a casing made of an insulating material such as porcelain or bakelite. Axial

Table 10 Superconductivity Constants for Some Materials Material Aluminum Gallium Indium Lead Mercury Tin Vanadium Zinc

Tc (K) 1.2 1.1 3.4 7.2 4.0 3.7 5.3 0.9

H0 (A/m) 0.8×104 0.4×104 2.3×104 6.5×104 3.0×104 2.5×104 10.5×104 0.4×104

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Table 11 Color Code for Fixed Resistors

Color Silver Gold Black Brown Red Orange Yellow Green Blue Violet Gray White

First Stripe, First Digit

Second Stripe, Second Digit

Third Stripe, Multiplier

— — 0 1 2 3 4 5 6 7 8 9

— — 0 1 2 3 4 5 6 7 8 9

10−2 10−1 1 10 102 103 104 105 106 107 108 109

Fourth Stripe, Tolerance (%) ±10 ±5 — ±1 ±2

or radial leads are provided for external connection. The outer surface of a fixed resistor is color coded for the purpose of its specification. Four stripes are used for coding. The first stripe gives the first digit of a two-digit number, and the second stripe gives the second digit. The third stripe specifies a multiplier, which should be included with the two-digit number to give the resistance value in ohms. The fourth stripe gives the percentage tolerance of the resistance value. This color code is given in Table. 11 Dielectric Material and Components Dielectrics and Capacitors. Dielectric materials are insulators, having resistivities larger than 1 × 1012 · m and containing less than 1 × 106 mobile electrons per cubic meter. When a voltage is applied across a medium of dielectric material sandwiched between two electrode plates, a charge polarization takes place at the two electrodes. The resulting charge depends on the capacitance of the capacitor formed in this manner. In the linear range, the following relationship holds:

q = Cv where v is applied voltage (in volts), q is stored charge (in coulombs), and C is capacitance (farads). Since current (i) is the rate of change of charge (dq/dt), we can write i=C

dv dt

Hence, in the frequency domain (substitute j ω for the rate of change operator), we have i = Cj ωv

and the electrical impedance (v/i in the frequency domain) of a capacitor is given by 1 j ωC where ω is the frequency variable, and j =

√

−1.

Permittivity. Consider a capacitor made of a dielectric plate of thickness d sandwiched between two conducting plates (electrodes) of common (facing) area A. Neglecting the fringe effect, its capacitance is given by

C=

εA d

where ε is the permittivity of the dielectric material. The relative permittivity (or dielectric constant) εr is defined as εr =

ε ε0

where ε0 = permittivity of vacuum (approx. 8.85 × 10−12 F/m). Relative permittivities of some materials are given in Table 12. Capacitor Types. The capacitance of a capacitor is increased by increasing the common surface area of the electrode plates. This increase can be achieved, without excessively increasing the size of the capacitor, by employing a rolled-tube construction. Here, a dielectric sheet (e.g., paper or a polyester film) is placed between two metal foils, and the composite is rolled into a tube. Axial or radial leads are provided

Table 12

Dielectric Constants of Some Materials

Material Air Carbon dioxide gas Ceramic (high permittivity) Cloth Common salt Diamond Glass Hydrogen (liquid) Mica Oil (mineral) Paper (dry) Paraffin wax Polythene PVC Porcelain Quartz (SiO2 ) Vacuum Water Wood

Relative Permittivity εr 1.0006 1.001 8000.0 5.0 5.9 5.7 6.0 1.2 6.0 3.0 3.0 2.2 2.3 6.0 6.0 4.0 1.0 80.0 4.0

1006


for external connection. If the dielectric material is not flexible (e.g., mica), a stacked-plate construction may be employed in place of the rolled construction to obtain compact capacitors having high capacitance. High-permittivity ceramic disks are used as the dielectric plates in miniature, single-plate, high-capacitance capacitors. Electrolytic capacitors can be constructed using the rolled-tube method, using a paper soaked in an electrolyte in place of the dielectric sheet. When a voltage is applied across the capacitor, the paper becomes coated with a deposit of dielectric oxide that is formed through electrolysis. This becomes the dielectric medium of the capacitor. Capacitors having low capacitances of the order of 1 × 10−12 F (1 pF), and high capacitances of the order of 4 × 10−3 F are commercially available. An important specification for a capacitor is the breakdown voltage, which is the voltage at which discharge will occur through the dielectric medium (i.e., the dielectric medium ceases to function as an insulator). This is measured in terms of the dielectric strength, which is defined as the breakdown voltage for a dielectric element of thickness 1 mil (1 × 10−3 in). Approximate dielectric strengths of several useful materials are given in Table 13. Color Code for Fixed Capacitors. Color codes are used to indicate the specifications of a paper or ceramic capacitor. The code consists of a colored end followed by a series of four dots printed on the outer surface of the capacitor. The end color gives the temperature coefficient of the capacitance in parts per million per degree centigrade (ppm/◦ C). The first two dots specify a two-digit number. The third dot specifies a multiplier which, together with the two-digit number, gives the capacitance value of the capacitor in picofarads. The fourth dot gives the tolerance of the capacitance. This code is shown in Table 14. Piezoelectricity. Some materials, when subjected to a stress (strain), produce an electric charge. These are termed piezoelectric materials, and the effect is called piezoelectricity. Most materials that posses a nonsymmetric crystal structure are known to exhibit the piezoelectric effect. Examples are barium titanate, cadmium sulfide, lead zirconate titanate, quartz, and rochelle salt. The reverse piezoelectric effect (the

Table 13 Approximate Dielectric Strengths of Several Materials Material Air Ceramics Glass Mica Oil Paper

Dielectric Strength (V/mil) 25 1000 2000 3000 400 1500

Table 14 Color Code for Ceramic and Paper Capacitors End Color, Temp. Coeff. (ppm/◦ C)

Color Black Brown Red Orange Yellow Green Blue Violet Gray White

0 −30 −80 −150 −220 −330 −470 −750 30 100

First Second Third Fourth Dot Dot, Dot, Dot, Tolerance First Second For For Digit Digit Multiplier ≤10 pF >10 pF 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

1 10 1×102 1×103 1×104 — — — 0.01 0.1

±2 pF ±0.1 pF — — — ±0.5 pF — — ±0.25 pF ±1 pF

±20% ±1% ±2% ±2.5% — ±5% — — — ±10%

material deforms in an electric field) is also useful in practice. The piezoelectric characteristic of a material may be represented by its piezoelectric coefficient, kp , which is defined as kp =

change in strain(m/m) change in electric field strength(V/m)

with no applied stress. Piezoelectric coefficients of some common materials are given in Table 15. Applications of piezoelectric materials include actuators for ink-jet printers, miniature step motors, force sensors, precision shakers, high-frequency oscillators, and acoustic amplifiers. Note that large kp values are desirable in piezoelectric actuators. For instance, PZT is used in microminiature step motors.96 On the other hand, small kp values are desirable in piezoelectric sensors (e.g., quartz accelerometers). Magnetic Material and Components Magnetism and Permeability. When electrons move (or spin), a magnetic field is generated. The combined effect of such electron movements is the cause of magnetic properties of material.

Table 15 Piezoelectric Coefficients of Some Materials Material Barium titanate PZT Quartz Rochelle salt

Piezoelectric Coefficient kp (m/V) 2.5×10−10 6.0×10−10 0.02×10−10 3.5×10−10

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1007

In the linear range of operation of a magnetic element, we can write

Material

B = µH where B is the magnetic flux density (webers per meter squared or teslas), H is magnetic field strength (amperes per meter), and µ is the permeability of the magnetic material. The relative permeability µr of a magnetic material is defined as µ=

µ µ0

where µ0 is the permeability of a vacuum (approx. 4π × 10−7 H/m). (Note: 1 T = 1 Wb/m2 ; 1 H = 1 Wb/A.) Hysteresis Loop. The B versus H curve of a magnetic material is not linear and exhibits a hysteresis loop as shown in Fig. 66. It follows that µ is not a constant. Initial values (when magnetization is started at the demagnetized state of H = 0 and B = 0) are usually specified. Some representative values are given in Table 16. Properties of magnetic materials can be specified in terms of parameters of the hysteresis curve. Some important parameters are shown in Fig. 66:

Hc = coercive field or coercive force (A/m) Br = remnant flux density (Wb/m2 or T) Bsat = saturation flux density (T)

Magnetic flux density (T)

Magnetic parameters of a few permanent-magnetic materials are given in Table 17. Note that high values

B

Bsat Br

Hc

Table 16 Initial Relative Permeability (Approximate) of Some Materials

H

Magnetic field (A/m)

Fig. 66 Hysteresis curve (magnetization curve) of magnetic material.

Relative Permeability µr

Alnico (Fe2 Ni Al) Carbon steel Cobalt steel (35% Co) Ferrite (manganese-zinc) Iron Permalloy (78% Ni, 22% Fe) Silicon iron (grain oriented)

6.5 20 12 800–10,000 200 3000 500–1500

Table 17

Parameters of Some Magnetic Materials

Material

Hc (A/m)

Br (Wb/m2 )

Alnico Ferrites Steel (carbon) Steel (35% Co)

4.6×104 14.0×104 0.4×104 2.0×104

1.25 0.65 0.9 1.4

of Hc and Br are desirable for high-strength permanent magnets. Furthermore, high values of µ are desirable for core materials that are used to concentrate magnetic flux. Magnetic Materials. Magnetic characteristics of a material can be imagined as if contributed by a matrix of microminiature magnetic dipoles. Paramagnetic materials (e.g., platinum and tungsten) have their magnetic dipoles arranged in a somewhat random manner. These materials have a µr value approximately equal to 1 (i.e., no magnetization). Ferromagnetic materials (e.g., iron, cobalt, nickel, and some manganese alloys) have their magnetic dipoles aligned in one direction (parallel) with virtually no cancellation of polarity. These materials have a high µr (of the order of 1000) in general. At low H values, µr will be correspondingly low. Antiferromagnetic materials (e.g., chromium and manganese) have their magnetic dipoles arranged in parallel, but in an alternately opposing manner thereby virtually canceling the magnetization (µr = 1). Ferrites have parallel magnetic dipoles arranged alternately opposing, as in antiferromagnetic materials, but the adjacent dipoles have unequal strengths. Hence, there is a resultant magnetization (µr is of the order of 1000). Applications of magnets and magnetic materials include actuators (e.g., motors, magnetically leviated vehicles, tools, magnetic bearings), sensors and transducers, relays, resonators, and cores of inductors and transformers. Also, see the applications of superconductivity. Piezomagnetism. When a stress (strain) is applied to a piezomagnetic material, the degree of magnetization of the material changes. Conversely, a piezomagnetic material undergoes deformation when the magnetic field in which the material is situated is changed.

1008

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Hall sensor output v0 +

This induced voltage (v) is given by – Semiconductor element

v=

d(nφ) dφ =n dt dt

If the change in magnetic flux is brought about by a change in current (i), we can write v=L Magnetic field

di dt

where L is the inductance of the coil (in henries). In the frequency domain, we have v = Lj ωi

N Magnetic source

Supply vref Fig. 67 Hall effect sensor.

Hall Effect Sensors. Suppose that a dc voltage vref is applied to a semiconductor element that is placed in a magnetic field in an orthogonal direction, as shown in Fig. 67. A voltage v0 is generated in the third orthogonal direction, as indicated in the figure.96 This is known as the Hall effect. Hall effect sensors use this phenomenon. For example, the motion of a ferromagnetic element can be detected in this manner since the magnetic field in which the sensor is mounted would vary as a result of the motion of the ferromagnetic element. Hall effect sensors are useful as position sensors, speed sensors, commutation devices for motors, and instrument transformers for power transmission systems. Magnetic Bubble Memories. Consider a film of magnetic material such as gadolinium gallium oxide (Gd3 Ga5 O12 ) deposited on a nonmagnetic garnet layer (substrate). The direction of magnetization will be perpendicular to the surface of the film. Initially, some regions of the film will be N poles, and the remaining regions will be S poles. An external magnetic field can shrink either the N regions or the S regions, depending on the direction of the field. The size of the individual magnetic regions can be reduced to the order of 1 µm in this manner. These tiny magnetic bubbles are the means with which information is stored in a magnetic bubble memory. Inductance. Suppose that a conducting coil having n turns is placed in a magnetic field of flux φ (in webers). The resulting flux linkage is nφ. If the flux linkage is changed, a voltage is induced in the coil.

√ where ω = frequency and j = −1. It follows that the electrical impedance of an inductor is given bv j ωL. 5.2 Active Components Active components made of semiconductor junctions and field effect components are considered in this section. Junction diodes, bipolar junction transistors, and field-effect transistors are of particular interest here. Active components are widely used in the monolithic (integrated-circuit) form as well as in the form of discrete elements. pn Junctions A pure semiconductor can be doped to form either a p-type semiconductor or an n-type semiconductor. A pn junction is formed by joining a p-type semiconductor element and an n-type semiconductor element. Semiconductors. Semiconductor materials have resistivities that are several million times larger than those of conductors and several billion times smaller than those of insulators. Crystalline materials such as silicon and germanium are semiconductors. For example, the resistivity of pure silicon is about 5 × 1010 times that of silver. Similarly, the resistivity of pure germanium is about 5 × 107 times that of silver. Typically, semiconductors have resistivities ranging from 10−4 to 107 · m. Other examples of semiconductor materials are gallium arsenide, cadmium sulfide, and selenium. A pure (intrinsic) semiconductor material has some free electrons (negative charge carriers) and holes (positive charge carriers). Note that a hole is formed in an atom when an electron is removed. Strictly, the holes cannot move. But suppose that an electron shared by two atoms (a covalent electron) enters an existing hole in an atom, leaving behind a hole at the point of origin. The resulting movement of the electron is interpreted as a movement of a hole in the direction opposite to the actual movement of the covalent electron. The number of free electrons in a pure semiconductor is roughly equal to the number of holes.

ELECTRONICS

The number of free electrons or holes in a pure semiconductor can be drastically increased by adding traces of impurities in a controlled manner (doping) into the semiconductor during crystal growth (e.g., by alloying in a molten form, and by solid or gaseous diffusion of the trace). An atom of a pure semiconductor that has four electrons in its outer shell will need four more atoms to share in order to form a stable covalent bond. These covalent bonds are necessary to form a crystalline lattice structure of atoms that is typical of semiconductor materials. If the trace impurity is a material such as arsenic, phosphorus, or antimony whose atoms have five electrons in the outer shell (a donor impurity), a free electron will be left over after the formation of a bond with an impurity atom. The result will be an n-type semiconductor having a very large number of free electrons. If, on the other hand, the trace impurity is a material such as boron, gallium, aluminum, or indium whose atoms have only three electrons in the outer shell (an acceptor impurity), a hole will result on formation of a bond. In this case, a p-type semiconductor, consisting of a very large number of holes, will result. Doped semiconductors are termed extrinsic. Depletion Region. When a p-type semiconductor is joined with an n-type semiconductor, a pn junction is formed. A pn junction exhibits the diode effect, much larger resistance to current flow in one direction than in the opposite direction across the junction. As a pn junction is formed, electrons in the n-type material in the neighborhood of the common layer will diffuse across into the p-type material. Similarly, the holes in the p-type material near the junction will diffuse into the opposite side (strictly, the covalent electrons will diffuse in the opposite direction). The diffusion will proceed until an equilibrium state is reached. But, as a result of the loss of electrons and the gain of holes on the n side and the opposite process on the p side, a potential difference is generated across the pn junction, with a negative potential on the p side and a positive potential on the n side. Due to the diffusion of carriers across the junction, the small region surrounding the common area will be virtually free of carriers (free electrons and holes). Hence, this region is called the depletion region. The potential difference that exists in the depletion region is mainly responsible for the diode effect of a pn junction. Biasing. The forward biasing and the reverse biasing of a pn junction are shown in Fig. 68. In the case of forward biasing, a positive potential is connected to the p side of the junction, and a negative potential is connected to the n side. The polarities are reversed for reverse biasing. Note that in forward biasing, the external voltage (bias voltage v) complements the potential difference of the depletion region (Fig. 68a). The free electrons that crossed over to the p side from the n side will continue to flow toward the positive terminal of

1009 Depletion region

p– type

n– type

p– type

n– type

– + Current i

Electron flow v (a)

(b) Junction current i

Breakdown voltage vb

Leakage current 0

Forward bias Bias voltage v

Reverse bias (c)

Fig. 68 A pn junction diode: (a) forward biasing; (b) reverse biasing; (c) characteristic curve.

the external supply, thereby generating a current (junction current i). The junction current increases with the bias voltage, as shown in Fig. 68c. In reverse biasing, the potential in the depletion region is opposed by the bias voltage (Fig. 68b). Hence, the diffusion of free electrons from the n side into the p side is resisted. Since there are some (very few) free electrons in the p side and some holes in the n side, the reverse bias will reinforce the flow of these minority electrons and holes. This will create a very small current (about 10−9 A for silicon and 10−6 A for germanium at room temperature), known as the leakage current, in the opposite direction to the forwardbias current. If the reverse bias is increased, at some voltage (breakdown voltage vb in Fig. 68c) the junction will break down, generating a sudden increase in the reverse current. There are two main causes of this breakdown. First, the intense electric field of the external voltage can cause electrons to break away from neutral atoms in large numbers. This is known as Zener breakdown. Second, the external voltage will accelerate the minority free electrons on the p side (and minority holes on the n side), creating collisions that will cause electrons on the outer shells of neutral atoms to break away in large numbers. This is known as the avalanche breakdown. In some applications (e.g., rectifier circuits), junction breakdown is detrimental. In some other types of applications (e.g., as constant voltage sources and in some digital circuits), the breakdown state of specially designed diodes is practically utilized. Typical breakdown voltages of pn junctions made of three common semiconductor materials are given in Table 18. Note that the breakdown voltage decreases with the concentration of the trace material.

1010


Table 18 Typical Breakdown Voltage of pn Junction at Room Temperature

Semiconductor Germanium Silicon Gallium arsenide

Breakdown Voltage (V) Dope Dope Concentration Concentration = 1015 atoms/cm3 = 1017 atoms/cm3 400 300 150

5.0 11.0 16.0

The current through a reverse-biased pn junction will increase exponentially with temperature. For a forward-biased pn junction, current will increase with temperature at low to moderate voltages and will decrease with temperature at high levels of voltage. Diodes A semiconductor diode is formed by joining a p-type semiconductor with an n-type semiconductor. A diode offers much less resistance to current flow in one direction (forward) than in the opposite direction (reverse). There are many varieties of diodes. Zener diodes, voltage variable capacitor (VVC) diodes, tunnel diodes, microwave power diodes, pin diodes, photodiodes, and light-emitting diodes (LED) are examples. The last two varieties will be discussed in separate sections. Zener Diodes. Zener diodes are a particular type of diodes that are designed to operate in the neighborhood of the reverse breakdown (both Zener and avalanche breakdowns). In this manner, a somewhat constant voltage output (the breakdown voltage) can be generated. This voltage depends on the concentration of the trace impurity. By varying the impurity concentration, output voltages in the range of 2–200 V may be realized from a Zener diode. Special circuits would be needed to divert large currents that are generated at the breakdown point of the diode. The rated power dissipation of a Zener diode should take into consideration the current levels that are possible in the breakdown region. Applications of Zener diodes include constant voltage sources, voltage clipper circuits, filter circuits for voltage transients, digital circuits, and two-state devices. VVC Diodes. VVC diodes use the property of a diode that, in reverse bias, the capacitance decreases (nonlinearly) with the bias voltage. The depletion region of a pn junction is practically free of carriers (free electrons and holes) and, hence, behaves like the dielectric medium of a capacitor. The adjoining p region and n region serve as the two plates of the capacitor. The width of the depletion region increases with the bias voltage. Consequently, the capacitance of a reverse-biased pn junction decreases as the bias voltage is increased. The obtainable range of capacitance can be varied by changing the dope concentration and also by distributing the dope concentration

nonuniformly along the diode. For example, a capacitance variation of 5–500 pF may be obtained in this manner (note: 1 pF = 1 × 10−12 F). VVC diodes are also known as varactor diodes and varicaps and are useful in voltage-controlled tuners and oscillators. Tunnel Diodes. The depletion of a pn junction can be made very thin by using very high dope concentrations (in both the p and n sides). The result is a tunnel diode. Since the depletion region is very narrow, charge carriers (free electrons and holes) in the n and p sides of the diode can tunnel through the region into the opposite side on application of a relatively small voltage. The voltage–current characteristic of a tunnel diode is quite linear at low (forward and reverse) voltages. When the forward bias is further increased, however, the behavior will become very nonlinear; the junction current will peak, then drop (a negative conductance) to a minimum (valley), and finally rise again, as the voltage is increased. Due to the linear behavior of the tunnel diode at low voltages, almost instantaneous current reversal (i.e., very low reverse recovery time) can be achieved by switching the bias voltage. Tunnel diodes are useful in high-frequency switching devices, sensors, and signal conditioning circuits. pin Diodes. The width of the depletion region of a conventional pn junction varies with many factors, primarily the applied (bias) voltage. The capacitance of a junction depends on this width and will vary due to such factors. A diode with practically a constant capacitance is obtained by adding a layer of silicon in between the p and n elements. The sandwiched silicon layer is called the intrinsic layer, and the diode is called a pin diode. The resistance of a pin diode varies inversely with junction current. Pin diodes are useful as current-controlled resistors at constant capacitance. Schottky Barrier Diodes. Most diodes consist of semiconductor–semiconductor junctions. An exception is a Schottky barrier diode, which consists of a metal–semiconductor (n-type) junction. A metal such as gold, silver, platinum, or palladium and a semiconductor such as silicon or gallium arsenide may be used in the construction. Since no holes exist in the metal, a depletion region cannot be formed at the metal–semiconductor junction. Instead, an electron barrier is formed by the free electrons from the ntype semiconductor. Consequently, the junction capacitance will be negligible, and the reverse recovery time will be very small. For this reason. Schottkv diodes can handle very high switching frequencies (109 Hz range). Since the electron barrier is easier to penetrate than a depletion region, by using a reverse bias, Schotky diodes exhibit much lower breakdown voltages. Operating noise is also lower than for semiconductor–semiconductor diodes. Thyristors. A thyristor, also known as a siliconcontrolled rectifier, a solid-state controlled rectifier, a

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Gate

p

n

Gate vg

p

n

i

Cathode

Anode

Cathode

Anode v

(a)

(b)

Table 19 Characteristic Variables and Parameters for Diodes Diode Variable/Parameter Forward bias (vf ) Reverse bias (vr ) Breakdown voltage (vb )

Thyristor current i

Junction current (if ) Gate off (vg = 0)

vb

vfb

0 Gate on

Leakage current (ir ) Supply voltage v

(c)

Fig. 69 Thyristor: (a) schematic representation; (b) circuit symbol; (c) characteristic curve.

semiconductor-controlled rectifier, or simply an SCR, possesses some of the characteristics of a semiconductor diode. It consists of four layers (pnpn) of semiconductor and has three terminals—the anode, the cathode, and the gate—as shown in Fig. 69a. The circuit symbol for a thyristor is shown in Fig. 69b. The thyristor current is denoted by i, the external voltage is v, and the gate potential is vg . The characteristic curve of a thyristor is shown in Fig. 69c. Note that a thyristor cannot conduct in either direction (i almost zero) until either the reverse voltage reaches the reverse breakdown voltage (vb ) or the forward voltage reaches the forward breakover voltage (vfb ). The forward breakover is a bistable state, and once this voltage is reached, the voltage drops significantly, and the thyristor begins to conduct like a forwardbiased diode. When vg is less than or equal to zero with respect to the cathode, vfb becomes quite high. When vg is made positive, vfb becomes small, and vfb will decrease as the gate current (ig ) is increased. A small positive vg can make vfb very small, and then the thyristor will conduct from anode to cathode but not in the opposite direction (i.e., it behaves like a diode). It follows that a thyristor behaves like a voltage-triggered switch; a positive firing signal (a positive vg ) will close the switch. The switch will be opened when both i and vg are made zero. When the supply voltage v is dc and nonzero, the thyristor will not be able to turn itself off. In this case a commutating circuit that can make the trigger voltage vg slightly negative has to be employed. Thyristors are commonly used in control circuits for dc and ac motors. Parameter values for diodes are given in data sheets provided by the manufacturer. Commonly used variables and characteristic parameters in association with

Transition capacitance (Ct ) Diffusion capacitance (Cd ) Forward resistance (Rf ) Reverse recovery time (trr )

Operating temperature range (TA ) Storage temperature range (Tsrg ) Power dissipation (P)

Description Positive external voltage at p with respect to n Positive external voltage at n with respect to p Minimum reverse bias that will break down the junction resistance Forward current through a forward-biased diode Reverse current through a reverse-biased diode Capacitance (in the depletion region) of a reverse-biased diode Capacitance exhibited while a forward-biased diode is switched off Resistance of a forward-biased diode Time needed for the reverse current to reach a specified level when the diode is switched from forward to reverse Allowable temperature range for a diode during operation Temperature that should be maintained during storage of a diode Maximum power dissipation allowed for a diode at a specified temperature

diodes are described in Table 19. For thyristors, as mentioned before, several other quantities such as vfb , vg , and ig should be included. The time required for a thyristor to be turned on by the trigger signal (turn-on time) and the time for it to be turned off through commutation (turn-off time) determine the maximum switching frequency (bandwidth) for a thyristor. Another variable that is important is the holding current or latching current, which denotes the small forward current that exists at the breakover voltage. Bipolar Junction Transistors A bipolar junction transistor (BJT) has two junctions that are formed by joining p regions and n regions. Two types of transistors, npn and pnp, are possible with this structure. A BJT has three terminals, as indicated in Fig. 70a. The middle (sandwiched) region of a BJT is thinner than the end regions, and this region is known as the base. The end regions are termed the emitter and the collector. Under normal conditions, the emitter–base junction is forward biased, and the collector–base junction is reverse biased, as shown in Fig. 70b.

1012


Emitter

Collector n

p

C

E

n

p

n

ib= 120mA

Base (B)

8

B (a)

ie

10

p

( C)

E

C

ic

ie

E

C

B

B

ib

ib

ic

Collector current ic (mA)

(E)

ib= 100mA ib= 80mA

6

ib= 60mA 4 ib= 40mA 2

(b)

Fig. 70 Bipolar junction transistors: (a) npn and pnp transistors; (b) circuit symbols and biasing.

To explain the behavior of a BJT, consider an npn transistor under normal biasing. The forward bias at the emitter–base junction will cause free electrons in the emitter to flow into the base region, thereby creating the emitter current (ie ). The reverse bias at the collector–base junction will increase the depletion region there. The associated potential difference at the collector–base junction will accelerate the free electrons in the base into the collector and will form the collector current (ic ). Holes that are created in the base, for recombination with some free electrons that entered the base, will form the base current (ib ). Usually, ic is slightly smaller than ie . Furthermore, ib is much smaller than ic . Transistor Characteristics. The common-emitter connection is widely used for transistors in amplifier applications. In this configuration, the emitter terminal will be common to the input side and the output side of the circuit. Transistor characteristics are usually specified for this configuration. Figure 71 shows typical characteristic curves for a junction transistor in the common-emitter connection. In this configuration, both voltage gain (output voltage/input voltage) and current gain (collector current/base current) will be greater than unity, thereby providing a voltage amplification as well as a current amplification. Note from Fig. 71 that the control signal is the base current (ib ), and the characteristic of the transistor depends on ib . This is generally true for any bipolar junction transistor; a BJT is a current-controlled transistor. In the common-base configuration, the base terminal is common to both input and output. Maximum frequency of operation and allowable switching rate for a transistor are determined by parameters such as rise time, storage time, and fall time. These and some other useful ratings and characteristic parameters for bipolar junction transistors are defined in Table 20. Values for these parameters are normally given in the manufacturer’s data sheet for a particular transistor.

Base-current ib= 20mA 0

Fig. 71

2 4 6 8 Collector-emitter voltage vce (V)

10

Characteristic curves of common emitter BJT.

Table 20 Transistor Parameter

Rating Parameters for Transistors Description

Voltage limit across collector and Collector-to-base base with emitter open voltage (vcb ) Collector-to-emitter Voltage limit across collector and voltage (vce ) emitter with base connected to emitter Emitter-to-base Voltage limit across emitter and voltage (veb ) base with collector open Collector cutoff Reverse saturation current at current (ico ) collector with either emitter open (icbo ) or base open (ico ) Transistor Power dissipated by the transistor dissipation (PT ) at rated conditions Input impedance Input voltage/input current with (hi ) output voltage = 0 (Defined for both common emitter and common base configurations, hie , hib ) Output admittance Output current/output voltage with (ho ) input current = 0 (hoe , hob are defined) Forward current Output current/input current with transfer ratio (hf ) output voltage = 0 (hfe , hfb are defined) Reverse voltage Input voltage/output voltage with transfer ratio (hr ) input current = 0 (hre , hrb are defined) Rise time (tr ) Time taken to reach the full current level for the first time when turned on Storage time (ts ) Time taken to reach the steady current level when turned on Time taken for the current to reach Fall time (tf ) zero when turned off

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Field-Effect Transistors An FET, unlike a BJT, is a voltage-controlled transistor. The electrostatic field generated by a voltage applied to the gate terminal of an FET controls the behavior of the FET. Since the device is voltage controlled at very low input current levels, the input impedance is very high, and the input power is very low. Other advantages of an FET over a BJT are that the former is cheaper and requires significantly less space on a chip in the monolithic form. FETs are somewhat slower (in terms of switching rates) and more nonlinear than BJTs, however. There are two primary types of FETs: metal–oxide– semiconductor field-effect transistor (MOSFET) and junction field-effect transistor (JFET). Even though the Silicon Base dioxide (insulator)

Emitter Collector Aluminum plate N P

N

Silicon wafer (substrate)

Fig. 72 An npn transistor manufactured by the planar diffusion method.

Platinum plate Gate G Drain D SiO2 + id insulator

vg

– Source (S)

n n+

n+

p substrate Substrate –

n + = Heavily doped n regions

Channel (n)

(a) 30

Drain current id (mA)

Fabrication Process. The actual manufacturing process for a transistor is complex and delicate. For example, an npn transistor can be fabricated by starting with a crystal of n-type silicon. This starting element is called the wafer or substrate. The npn transistor is formed, by using the planar diffusion method, in the top half of the substrate as follows: The substrate is heated to about 1000◦ C. A gas stream containing a donor-type impurity (which forms n-type regions) is impinged on the crystal surface. This produces an ntype layer on the crystal. Next the crystal is oxidized by heating to a high temperature. The resulting layer of silicon dioxide acts as an insulating surface. A small area of this layer is then dissolved off using hydrofluoric acid. The crystal is again heated to 1000◦ C, and a gas stream containing acceptor-type impurity (which forms p-type regions) is impinged on the window thus formed. This produces a p region under the window on top of the n region, which was formed earlier. Oxidation is repeated to cover the newly formed p region. Using hydrofluoric acid, a smaller window is cut on the latest silicon dioxide layer, and a new n region is formed, as before, on top of the p region. The entire manufacturing process has to be properly controlled so as to control the properties of the resulting transistor. Aluminum contacts have to be deposited on the uppermost n region, the second p region (in a suitable annular window cut on the silicon dioxide layer), and on the n region below it or on the crystal substrate. A pictorial representation of an npn transistor fabricated in this manner is shown in Fig. 72.

ugs = 2V ugs = 1V

20

ugs = 0V ugs = –1V 10

0

Gate-source voltage ugs = –2 V

10 Drain-source voltage uds (V)

20

(b)

Fig. 73 MOSFET: (a) an n-channel depletion-type MOSFET; (b) D-MOSFET characteristics.

physical structure of the two types is somewhat different, their characteristics are quite similar. Insulated-gate FET (or IGFET) is a general name given to MOSFETs. MOSFET. An n-channel MOSFET is produced using a p-type silicon substrate, and a p-channel MOSFET by an n-type substrate. An n-channel MOSFET is shown in Fig. 73a. During manufacture, two heavily doped n-type regions are formed on the substrate. One region is termed source (S) and the other region drain (D). The two regions are connected by a moderately doped and narrow n region called a channel. A metal coating deposited over an insulating layer of silicon dioxide, which is formed on the channel, is the gate (G). The source lead is usually joined with the substrate lead. This is a depletion-type MOSFET (or D-MOSFET). Another type is the enhancement-type MOSFET (or E-MOSFET). In this type, a channel linking the drain and the source is not physically present in the substrate but is induced during operation of the transistor.

1014


Consider the operation of the n-channel D-MOSFET shown in Fig. 73a. Under normal operation, the drain is positively biased with respect to the source. Drain current id is considered the output of a MOSFET (analogous to the collector current of a BJT). The control signal of a MOSFET is the gate voltage vgs with respect to the source (analogous to the base current of a BJT). It follows that a MOSFET is a voltagecontrolled device. Since the source terminal is used as the reference for both input (gate voltage) and output (drain), this connection is called the common-source configuration. Suppose that the gate voltage is negative with respect to the source. This will induce holes in the channel, thereby decreasing the free electrons there through recombination. This, in turn, will reduce the concentration of free electrons in the drain region and, hence, will reduce the drain current id . Clearly, if the magnitude of the negative voltage at the gate is decreased, the drain current will increase, as indicated by the characteristic curves in Fig. 73b. A positive bias at the gate will further increase the drain current of an n-channel MOSFET as shown. The opposite will be true for a p-channel MOSFET. The JFET. A junction field-effect transistor (JFET) is different in physical structure to a MOSFET but similar in characteristics. The structure of an n-channel JFET is shown in Fig. 74. It consists of two p-type regions formed inside an n-type region. The two p regions are separated by a narrow n region called a Table 21

Gate (G)

Source (S) –

p n

n p

Drain (D) +

Fig. 74 An n-channel JFET.

channel. The channel links two n-type regions called source (S) and drain (D). The two p regions are linked by a common terminal and form the gate (G). As for a MOSFET, drain current id is considered the output of the JFET, and gate voltage vgs , with respect to the source, is considered the control signal. For normal operation, the drain is positively biased with respect to the source, as for an n-channel MOSFET, and the common-source configuration is used. To explain the operation of a JFET, consider the n-channel JFET shown in Fig. 74. Depletion regions are present at the two pn junctions of the JFET (as

Common Transistor Types Transistor Type

Abbreviation

Name

BJT

Bipolar junction transistor

FET

Field-effect transistor

MOSFET D-MOSFET E-MOSFET VMOS DG-MOS

Metal–oxide–semiconductor FET Depletion-type MOSFET Enhancement-type MOSFET V-shaped Gate MOSFET or VFET Dual-gate MOSFET

D-MOS

Double-diffused MOSFET

CMOS

Complementary symmetry MOSFET

GaAs

Gallium arsenide MOSFET

JFET

Junction FET

Description Three-layer device (npn or pnp) Current controlled Control = base current Output = collector current Physical or induced channel (n-channel or p-channel) voltage controlled Control = gate voltage Output = drain current n channel or p channel Channel is physically present Channel is induced An E-MOSFET with increased power-handling capacity Secondary gate is present between main gate and drain (lower capacitance) Channel layer is formed on a high-resistivity substrate and then source and drain are formed (by diffusion). High breakdown voltage Uses two E-MOSFETs (n channel and p channel). Symmetry is used to save space on chip. Cheaper and lower power consumption. Uses gallium arsenide, aluminum gallium arsenide, (AlGaAs), indium gallium arsenide phosphide (InGaAsP), etc. in place of silicon substrate. Faster operation p channel or n channel. Has two (n or p) regions in a (p or n) region linked by a channel (p or n) Control = gate voltage Output = drain current

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1015

for a semiconductor diode). If the gate voltage is made negative, the resulting field will weaken the p regions. As a result, the depletion regions will shrink. Some of the free electrons from the drain will diffuse toward the channel to occupy the growing n regions due to the shrinking depletion regions. This will reduce the drain current. It follows that drain current decreases as the magnitude of the negative voltage at the gate is increased. This behavior is similar to that of a MOSFET.

Physical representation

A p-channel JFET has two n regions representing the gate and two p regions forming the source and the drain, which are linked by a p-channel. Its characteristic is the reverse of an n-channel JFET. Common types of transistor are summarized in Table 21. Semiconductor devices have numerous uses. A common use is as switching devices or as two-state elements. Typical two-state elements are schematically illustrated in Fig. 75.

Circuit symbol

Schematic diagram

Characteristic curve

Relay contact

Output Output circuit

voltage

v0

Solenoid Output

On

To output circuit Off

Insulator

Control circuit

Control signal

Control

0

v0

current

(a)

i Anode –

+

p

+

Reverse

Cathode

breakdown voltage

i

n

–

v Aluminum

v

0

contacts (b)

ib

ic vce

+ Collector

Emitter

Collector

n p

ic

ib

–

n

Emitter

ie

Base

Base

vce

0 (c)

Aluminum plate

vds

p = substrate

D

–

Drain (D)

+

id

Source (S) + Gate (G)

G

S

n n p = substrate (silicon)

vgs

id

vgs

Silicon dioxide (insulator)

Channel (n)

Substrate 0 (d)

vds

Fig. 75 Discrete switching (two-state) elements: (a) electromagnetic relay; (b) Zener diode; (c) BJT (npn); (d) n-channel MOSFET.

1016


5.3 Light Emitters and Displays Visible light is part of the electromagnetic spectrum Electromagnetic waves in the wave length range of 390–770 nm (Note: 1 nm = 1 × 10−9 m) form the visible light. Ultraviolet rays and X rays are also electromagnetic waves but have lower wavelengths (higher frequencies). Infrared rays, microwaves, and radio waves are electromagnetic waves having higher wavelengths. Table 22 lists wavelengths of several types of electromagnetic waves. Visible light occupies a broad range of wavelengths. For example, in optical coupling applications, the narrower the wave spectrum, the clearer (noise free) the coupling process. Consequently, it is advantageous to use special light sources in applications of that type. Furthermore, since visible light can be contaminated by environmental light, thereby introducing an error signal into the system, it is also useful to consider electromagnetic waves that are different from what is commonly present in operating environments in applications such as sensing, optical coupling, and processing. Incandescent Lamps Tungsten-filament incandescent lamps that are commonly used in household illumination emit visible light in a broad spectrum. Furthermore, they are not efficient because they emit more infrared radiation than useful visible light. Ionizing lamps filled with gases such as halogens, sodium vapor, neon, or mercury vapor have much narrower spectra, and they emit very pure visible light (with negligible infrared radiation). Hence, these types of incandescent lamps are more efficient for illumination purposes. Regular fluorescent lamps are known to create a line-frequency (60 or 50 Hz) flicker but are quite efficient and durable. All these types of light sources are usually not suitable in many applications primarily because of the following disadvantages:

1. They are bulky. 2. They cannot be operated at high switching rates (from both time constant and component life points of view). 3. Their spectral bandwidth can be very wide. Note that a finite time is needed for an incandescent lamp to emit light once it is energized. That is, it has a Table 22 Wavelengths of Several Selected Components of the Electromagnetic Spectrum Wave Type Radio waves Microwaves Infrared rays Visible light Ultraviolet rays X rays

Approximate Wavelength Range (µm) 1×106 − 5×106 1×103 − 1×106 0.8 − 1×103 0.4 − 0.8 1×10−2 − 0.4 1×10−6 − 5×10−2

External leads

Semiconductor element

Metal housing

Glass window

(a)

(b) Fig. 76

LED: (a) physical construction; (b) circuit symbol.

large time constant. This limits the switching speed to less than 100 Hz. Furthermore, lamp life will decrease rapidly with increasing switching frequency. Light-Emitting Diodes The basic components of an LED are shown in Fig. 76a. The element symbol that is commonly used in electrical circuits is shown in Fig. 76b. The main component of an LED is a semiconductor diode element, typically made of gallium compounds (e.g., gallium arsenide or GaAs and gallium arsenide phosphide or GaAsP). When a voltage is applied in the forward-bias direction to this semiconductor element, it emits visible light (and also other electromagnetic wave components, primarily infrared). In the forward-bias configuration, electrons are injected into the p region of the diode and recombined with holes. Radiation energy (including visible light) is released spontaneously in this process. This is the principle of operation of an LED. Suitable doping with trace elements such as nitrogen will produce the desired effect. The radiation energy generated at the junction of a diode has to be directly transmitted to a window of the diode in order to reduce absorption losses. Two types of construction are commonly used; edge emitters emit radiation along the edges of the pn junction, and surface emitters emit radiation normal to the junction surface. Infrared light-emitting diodes (IRED) are LEDs that emit infrared radiation at a reasonable level of power. Gallium arsenide (GaAs), gallium aluminum arsenide (GaAlAs), and indium gallium arsenide phosphide (InGaAsP) are the commonly used IRED material. Gallium compounds and not silicon or germanium are used in LEDs for reasons of efficiency and intensity characteristics. (Gallium compounds exhibit sharp peaks of spectral output in the desired frequency bands.) Table 23 gives wavelength characteristics of common LED and ˚ = 1 × 10−10 m = 0.1 nm). Note that IRED types (1 A ˚ denotes the unit angstrom. A

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Table 23 Wavelength Characteristics of Common ˚ = 1×10−10 m) LEDs (1 A

LED Type Gallium arsenide Gallium arsenide phosphide Gallium phosphide Gallium aluminum arsenide Indium gallium arsenide phosphide

High-voltage dc supply

Partially reflective lens

Wavelength at Peak Intensity ˚ (A) Color 5500 9300 5500 7000 5500 8000 8500 13000

Green Infrared Green Red Green Red Infrared Infrared

Glass tube

Filament lamp

He, Ne, gases Silvered surface ac

Light-emitting diodes are widely used in optical electronics because they can be constructed in miniature sizes, they have small time constants and low impedances, they can provide high switching rates (typically over 1000 Hz), and they have much longer component life than incandescent lamps. They are useful as both light sources and displays. Lasers Laser (light amplification by stimulated emission of radiation) is a light source that emits a concentrated beam of light that will propagate typically at one or two frequencies (wavelengths) and in phase. Usually, the frequency band is extremely narrow (i.e., monochromatic), and the waves in each frequency are in phase (i.e., coherent). Furthermore, the energy of a laser is highly concentrated (power densities of the order of one billion watts/cm2 ). Consequently, a laser beam can travel in a straight line over a long distance with very little dispersion. Hence, it is useful in gauging and aligning applications. Lasers can be used in a wide variety of sensors (e.g., motion sensors, tactile sensors, laser-doppler velocity sensors) that employ photosensing and fiber optics. Also, lasers are used in medical applications, microsurgery in particular. Lasers have been used in manufacturing and material removal applications such as precision welding, cutting, and drilling of different types of materials, including metals, glass, plastics, ceramics, leather, and cloth. Lasers are used in inspection (detection of faults and irregularities) and gauging (measurement of dimensions) of parts. Other applications of lasers include heat treatment of alloys, holographic methods of nondestructive testing, communication, information processing, and high-quality printing. Lasers may be classified as solid, liquid, gas, and semiconductor. In a solid laser (e.g., ruby laser, glass laser), a solid rod with reflecting ends is used as the laser medium. The laser medium of a liquid laser (e.g., dye laser, salt-solution laser) is a liquid such as an organic solvent with a dye or an inorganic solvent with dissolved salt compound. Very high peak power levels are possible with liquid lasers. Gas lasers (e.g., helium–neon or He–Ne laser, helium–cadmium or He–Cd laser, carbon dioxide or CO2 laser) use a gas as

Laser beam

Gravity resonator

Fig. 77 Helium–neon (He–Ne) laser.

the laser medium. Semiconductor lasers (e.g., gallium arsenide laser) use a semiconductor diode similar to an edge-emitting LED. Some lasers have their main radiation components outside the visible spectrum of light. For example, a CO2 laser (wavelength of about ˚ primarily emits infrared radiation. 110,000 A) In a conventional laser unit, the laser beam is generated by first originating an excitation to create a light flash. This will initiate a process of emitting photons from molecules within the laser medium. This light is then reflected back and forth between two reflecting surfaces before the light beam is finally emitted as a laser. These waves will be limited to a very narrow frequency band (monochromatic) and will be in phase (coherent). For example, consider the He–Ne laser unit schematically shown in Fig. 77. The helium and neon gas mixture in the cavity resonator is heated by a filament lamp and ionized using a high dc voltage (2000 V). Electrons released in the process will be accelerated by the high voltage and will collide with the atoms, thereby releasing photons (light). These photons will collide with other molecules, releasing more photons. This process is known as lasing. The light generated in this manner is reflected back and forth by the silvered surface and the partially reflective lens (beam splitter) in the cavity resonator, thereby stimulating it. This is somewhat similar to a resonant action. The stimulated light is concentrated into a narrow beam by a glass tube and emitted as a laser beam through the partially silvered lens. A semiconductor laser is somewhat similar to an LED. The laser element is typically made of a pn junction (diode) of semiconductor material such as gallium arsenide (GaAs) or indium gallium arsenide phosphide (InGaAsP). The edges of the junction are reflective (naturally or by depositing a film of silver). As a voltage is applied to the semiconductor laser, the ionic injection and spontaneous recombination that take place near the pn junction will emit light as in an LED. This light will be reflected back and forth between the reflective surfaces, passing along

1018


Table 24 Properties of Several Types of Lasers ˚ = 1×10−10 m) (1 A Laser Type

Wavelength

Output Power

˚ (A)

(W/cm2 )

Solid Ruby 7, 000 Glass 1, 000 Liquid Dye 4,000–10,000 Gas Helium–neon 6, 330 Helium–cadmium 4, 000 Carbondioxide 110, 000 Semiconductor: GaAs 9, 000 InGaAsP 13, 000

0.1–100 0.1–500 0.001–1 0.001–2 0.001–1 1–1×104 0.002–0.01 0.001–0.005

the depletion region many times and creating more photons. The stimulated light (laser) beam is emitted through an edge of the pn junction. Semiconductor lasers are often maintained at very low temperatures in order to obtain a reasonable component life. Semiconductor lasers can be manufactured in very small sizes. They are lower in cost and require less power in comparison to the conventional lasers. Wave length and power output characteristics of several types of lasers are given in Table 24. Liquid Crystal Displays A liquid crystal display (LCD) consists of a medium of liquid crystal material (e.g., organic compounds such as cholesteryl nonanote and p-azoxyanisole) trapped between a glass sheet and a mirrored surface, as shown in Fig. 78. Pairs of transparent electrodes (e.g., indium tin oxide), arranged in a planar matrix, are deposited on the inner surfaces of the sandwiching plates. In the absence of an electric field across an electrode pair, the atoms of liquid crystal medium in that region will have a parallel orientation. As a result, any light that falls on the glass sheet will first travel through the liquid crystal, then will be reflected back by the mirrored surface, and finally will return unscattered. Once an electrode pair is energized, the molecular alignment of the entrapped Transparent electrodes

medium will change, causing some scattering. As a result, a dark region in the shape of the electrode will be visible. Alphanumeric characters and other graphic images can be displayed in this manner by energizing a particular pattern of electrodes. Other types of LCD construction are available. In one type, polarized glass sheets are used to entrap the liquid crystal. In addition, a special coating is applied on the inner surfaces of the two sheets that will polarize the liquid crystal medium in different directions. This polarization structure is altered by an electric field (supplied by an electrode pair), thereby displaying an image element. LCDs require external light to function. But they need significantly low currents and power levels to operate. For example, an LED display might need a watt of power, whereas a comparable LCD might require just a small fraction of a milliwatt. Similarly, the current requirement for an LCD will be in the microampere range. LCDs usually need an ac biasing, however. An image resolution on the order of 5 lines/mm is possible with an LCD. Plasma Displays A plasma display is somewhat similar to an LCD in construction. The medium used in a plasma display is an ionizing gas (e.g., neon with traces of argon or xenon). A planar matrix of electrode pairs is used on the inner surfaces of entrapping glass. When a voltage above the ionizing voltage of the medium is applied to the electrode pair, the gas will break down, and a discharge will result. The electron impacts that are generated at the cathode as a result will cause further release of electrons to sustain the discharge. A characteristic orange glow will result. The pattern of energized electrodes will determine the graphic image. The electrodes could be either dc coupled or ac coupled. In the case of the latter, the electrodes are coated with a layer of dielectric material to introduce a capacitor at the gas interface. The power efficiency of a plasma display is higher than that of an LED display. A typical image resolution of 2 lines/mm is obtainable. Cathode Ray Tubes A schematic representation of a cathode ray tube (CRT) is given in Fig. 79. In a CRT,

Glass surface

Beam deflector (magnetic/electrostatic) Cathode

Heating coil Electron beam

Liquid crystal medium

Mirrored surface

Fig. 78 LCD element.

Screen grid window Focus electrodes/coils Phosphor coating

Fig. 79 Schematic of CRT.

CRT screen (glass)

ELECTRONICS

an electron beam is used to trace lines, characters, and other graphic images on the CRT screen. The electron beam is generated by an electron gun. A cathode made of a metal such as nickel coated with an oxide such as barium strontium calcium oxide forms the electron gun and is heated (say, using a tungsten coil heater) to generate electrons. Electrons are accelerated toward the inner surface of the CRT screen using a series of anodes, biased in increasing steps. The CRT screen is made of glass. Its inner surface is coated with a crystalline phosphor material. The electrons that impinge on the screen will excite the phosphor layer, which will result in the release of additional electrons and radiation. As a result, the point of impingement will be illuminated. The electron beam is focused using either electrostatic (a pair of electrode plates) or magnetic (a coil) means. The position of the luminous spot on the screen is controlled using a similar method. Two pairs of electrodes (or two coils) will be needed to deflect the electron to an arbitrary position on the screen. Different types of phosphor material will provide different colors (red, green, blue, white, etc.). The color of a monochrome display is determined by this. Color displays employ one of two common techniques. In one method (masking), three guns are used for the three basic colors (red, green, and blue). The three beams pass through a small masking window and fall on the faceplate. The faceplate has a matrix of miniature phosphor spots (e.g., at 0.1-mm spacing). The matrix consists of a regular pattern of R–G–B phosphor elements. The three electron beams fall on three adjacent spots of R–G–B phosphor. A particular color is obtained as a mixture of the three basic colors by properly adjusting the intensity of the three beams. In the second method (penetration), the faceplate has several layers of phosphor. The color emitted will depend on the depth of penetration of the electron beam into the phosphor. Flicker in a CRT display, at low frequencies, will strain the eye and also can deteriorate dynamic images. Usually, a minimum flicker frequency of 40 Hz will be satisfactory, and even higher frequencies can be achieved with most types of phosphor coatings. Flicker effect worsens with the brightness of an image. The efficiency of a phosphor screen is determined by the light flux density per unit power input (measured in lumens/watt). A typical value is 40 lm/W. Time constant determines the time of decay of an image when power is turned off. Common types of phosphor and their time constants are given in Table 25. CRTs have numerous uses. Computer display screens, television picture tubes, radar displays, and oscilloscope tubes are common applications. The rasterscan method is a common way of generating an image on a computer or television screen. In this method, the electron beam continuously sweeps the screen (say, starting from the top left corner of the screen and tracing horizontal lines up to the bottom right corner, continuously repeating the process). The spot is turned on or

1019 Table 25 Phosphor P1 P4 P22 RP20

Time Constants of CRT Phosphor Color

Time Constant (ms)

Green White Red Green Blue Yellow–green

30.0 0.1 2.0 8.0 6.0 5.0

off using a controller according to some logic that will determine the image that is generated on the screen. In another method used in computer screens, the beam is directly moved to trace the curves that form the image. In oscilloscopes, the horizontal deflection of the beam can be time sequenced and cycled in order to enable the display of time signals. 5.4 Light Sensors A light sensor (also known as a photodetector or photosensor) is a device that is sensitive to light. Usually, it is a part of an electrical circuit with associated signal conditioning (amplification, filtering, etc.) so that an electrical signal representative of the intensity of light falling on the photosensor is obtained. Some photosensors can serve as energy sources (cells) as well. A photosensor may be an integral component of an optoisolator or other optically coupled system. In particular, a commercial optical coupler typically has an LED source and a photosensor in the same package, with a pair of leads for connecting it to other circuits, and perhaps power leads. By definition, the purpose of a photodetector or photosensor is to sense visible light. But there are many applications where sensing of adjoining bands of the electromagnetic spectrum, namely infrared radiation and ultraviolet radiation, would be useful. For instance, since objects emit reasonable levels of infrared radiation even at low temperatures, infrared sensing can be used in applications where imaging of an object in the dark is needed. Applications include infrared photography, security systems, and missile guidance. Also, since infrared radiation is essentially thermal energy, infrared sensing can be effectively used in thermal control systems. Ultraviolet sensing is not as widely applied as infrared sensing. Typically, a photosensor is a resistor, diode, or transistor element that brings about a change (e.g., generation of a potential or a change in resistance) into an electrical circuit in response to light that is falling on the sensor element. The power of the output signal may be derived primarily from the power source that energizes the electrical circuit. Alternatively, a photocell can be used as a photosensor. In this latter case, the energy of the light falling on the cell is converted into electrical energy of the output signal. Typically, a photosensor is available as a tiny cylindrical element with a sensor head consisting of a circular window (lens). Several types of photosensors are described below.

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Photoresistors A photoresistor (or photoconductor) has the property of decreasing resistance (increasing conductivity) as the intensity of light falling on it increases. Typically, the resistance of a photoresistor could change from very high values (megohms) in the dark to reasonably low values (less than 100 ) in bright light. As a result, very high sensitivity to light is possible. Some photocells can function as photoresistors because their impedance decreases (output increases) as the light intensity increases. Photocells used in this manner are termed photoconductive cells. The circuit symbol of a photoresistor is given in Fig. 80a. A photoresistor may be formed by sandwiching a photoconductive crystalline material such as cadmium sulfide (CdS) or cadmium selenide (CdSe) between two electrodes. Lead sulfide (PbS) or lead selenide (PbSe) may be used in infrared photoresistors. Photodiodes A photodiode is a pn junction of semiconductor material that produces electron–hole

(a)

pn (b) Collector (C)

Base (B)

Emitter (E) (c)

Drain (D) Gate (G)

Source (S) (d) +–

(e) Fig. 80 Circuit symbols of some photosensors: (a) photoresistor; (b) photodiode; (c) phototransistor (npn); (d) photo-FET (n-channel); (e) photocell.

pairs in response to light. The symbol for a photodiode is shown in Fig. 80b. Two types of photodiodes are available. A photovoltaic diode generates a sufficient potential at its junction in response to light (photons) falling on it. Hence, an external bias source is not necessary for a photovoltaic diode. A photoconductive diode undergoes a resistance change at its junction in response to photons. This type of photodiode is usually operated in reverse-biased form; the p lead of the diode is connected to the negative lead of the circuit, and n lead is connected to the positive lead of the circuit. The breakdown condition may occur at about 10 V, and the corresponding current will be nearly proportional to the intensity of light falling on the photodiode. Hence, this current can be used as a measure of the light intensity. Since the current level is usually low (a fraction of a milliampere), amplification might be necessary before using it in the subsequent application (e.g., actuation, control, display). Semiconductor materials such as silicon, germanium, cadmium sulfide, and cadmium selenide are commonly used in photodiodes. A diode with an intrinsic layer (a pin diode) can provide faster response than with a regular pn diode. Phototransistor Any semiconductor photosensor with amplification circuitry built into the same package (chip) is popularly called a phototransistor. Hence, a photodiode with an amplifier circuit in a single unit might be called a phototransistor. Strictly, a phototransistor is manufactured in the form of a conventional bipolar junction transistor with base (B), collector (C) and emitter (E) leads. Symbolic representation of a phototransistor is shown in Fig. 80c. This is an npn transistor. The base is the central (p) region of the transistor element. The collector and the emitter are the two end regions (n) of the element. Under operating conditions of the phototransistor, the collector–base junction is reverse biased (i.e., a positive lead of the circuit is connected to the collector, and a negative lead of the circuit is connected to the base of an npn transistor). Alternatively, a phototransistor may be connected as a two-terminal device with its base terminal floated and the collector terminal properly biased (positive for an npn transistor). For a given level of source voltage (usually applied between the emitter lead of the transistor and load, the negative potential being at the emitter load), the collector current (current through the collector lead) ic is nearly proportional to the intensity of the light falling on the collector–base junction of the transistor. Hence, ic can be used as a measure of the light intensity. Germanium or silicon is the semiconductor material that is commonly used in phototransistors. Photo-FET A photo–field-effect transistor is similar to a conventional FET. The symbol shown in Fig. 80d is for an n-channel photo-FET. This consists of an n-type semiconductor element (e.g., silicon doped with boron), called channel. A much smaller element of p-type material is attached to the n-type

ELECTRONICS

1021

element. The lead on the p-type element forms the gate (G). The drain (D) and the source (S) are the two leads on the channel. The operation of an FET depends on the electrostatic fields created by the potentials applied to the leads of the FET. Under operating conditions of a photo-FET, the gate is reverse biased (i.e., a negative potential is applied to the gate of an n-channel photo-FET). When light is projected at the gate, the drain current id will increase. Hence, drain current (current at the D lead) can be used as a measure of light intensity.

(e.g., 4096 × 4096 pixels) are available for specialized applications. A charge injection device (CID) is similar to a CCD. In a CID, however, there is a matrix of semiconductor capacitor pairs. Each capacitor pair can be directly addressed through voltage pulses. When a particular element is addressed, the potential well there will shrink, thereby injecting minority carriers into the substrate. The corresponding signal, tapped from the substrate, forms the video signal. The signal level of a CID is substantially smaller than that of a CCD, as a result of higher capacitance.

Photocells Photocells are similar to photosensors except that a photocell is used as an electricity source rather than a sensor of radiation. Solar cells, which are more effective in sunlight are commonly available. A typical photocell is a semiconductor junction element made of a material such as single-crystal silicon, polycrystalline silicon, and cadmium sulfide. Cell arrays are used in moderate-power applications. Typical power output is 10 mW/cm2 of surface area, with a potential of about 1.0 V. The circuit symbol of a photocell is given in Fig. 80e.

Applications of Optically Coupled Devices One direct application is in the isolation of electric circuitry. When two circuits are directly connected through electrical connections (cables, wires, etc.), a two-way path is created at the interface for the electrical signals. In other words, signals in circuit A will affect circuit B and signals in circuit B, will affect circuit A. This interaction means that noise in one circuit will directly affect the other. Furthermore, there will be loading problems; the source will be affected by the load. Both these situations are undesirable. If the two circuits are optically coupled, however, there is only a oneway interaction between the two circuits (see Fig. 82). Variations in the output circuit (load circuit) will not affect the input circuit. Hence, the input circuit is isolated from the output circuit. The connecting cables in an electrical circuit can introduce noise components such as electromagnetic interference, line noise, and ground-loop noise. The likelihood of these noise components affecting the overall system is also reduced by using optical coupling. In summary, isolation between two circuits and isolation of a circuit from noise can be achieved by optical coupling. Optical coupling is widely used in communication networks (telephones, computers, etc.) and in circuitry for high-precision signal conditioning (e.g., for sophisticated sensors and control systems) for these reasons. The medium through which light passes from the light source to the photosensor can create noise problems, however. If the medium is open (see Fig. 82), then ambient lighting conditions will affect the output circuit, resulting in an error. Also, environmental impurities (dust, smoke, moisture, etc.) will affect the light received by the photosensor. Hence, a more controlled medium of transmission would be desirable. Linking the light source and the photosensor using

Charge-Coupled Device A charge-coupled device (CCD) is an integrated circuit (a monolith device) element of semiconductor material. A CCD made from silicon is schematically represented in Fig. 81. A silicon wafer (p type or n type) is oxidized to generate a layer of SiO2 on its surface. A matrix of metal electrodes is deposited on the oxide layer and is linked to the CCD output leads. When light falls onto the CCD element, charge packets are generated within the substrate silicon wafer. Now if an external potential is applied to a particular electrode of the CCD, a potential well is formed under the electrode, and a charge packet is deposited here. This charge packet can be moved across the CCD to an output circuit by sequentially energizing the electrodes using pulses of external voltage. Such a charge packet corresponds to a pixel (a picture element). The circuit output is the video signal. The pulsing rate could be higher than 10 MHz. CCDs are commonly used in imaging application, particularly in video cameras. A typical CCD element with a facial area of a few square centimeters may detect 576 × 485 pixels, but larger elements

Silicon dioxide layer

Electrodes Output circuit

Input circuit

Silic on subst rate ( p or n) A potential well (receives a charge packet) Fig. 81

A CCD.

Photo sensor Input signal

Electrical circuitry

Electrical circuitry Light source

Fig. 82 An optically coupled device.

Output (to load)

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optical fibers is a good way to reduce problems due to ambient conditions in optically coupled systems. Optical coupling may be used in relay circuits where a low-power circuit is used to operate a highpower circuit. If the relay that operates the high-power circuit is activated using an optical coupler, reaction effects (noise and loading) on the low-power circuit can be eliminated. Optical coupling is used in power electronics and control systems in this manner. Many types of sensors and transducers that are based on optical methods do, indeed, employ optical coupling (e.g., optical encoders, fiberoptic tactile sensors). Optical sensors are widely used in industry for parts counting, parts detection, and level detection. In these sensors, a light beam is projected from a source to a photodetector, both units being stationary. An interruption of the beam through the passage of a part will generate a pulse at the detector, and this pulse is read by a counter or a parts detector. Furthermore, if the light beam is located horizontally at a required height, its interruption when the material filled into a container reaches that level could be used for filling control in the packaging industry. Note that the light source and the sensor could be located within a single package if a mirror is used to reflect light from the source back onto the detector. Further applications are within computer disk drive systems, for example, to detect the write protect notch as well as the position of the recording head. 6

INPUT DEVICES George Grinstein and Marjan Trutschl

Human–computer interaction (HCI) is now a multidisciplinary area focusing on the interface and interactions between people and computer systems. Figure 83 presents a conceptual view of HCI: A user interacts with a system (typically a processor or device) using one or multiple input devices. Input devices convert some form of energy, most often kinetic or potential energy, to electric energy. In this section we consider analog and digital input devices. Analog input devices generate voltages that vary over a continuous range (R = Vmax − Vmin ) of values and are converted to binary values by an ADC. Digital input devices are based on binary digits. An input device that generates logical 0’s and 1’s, on and

Input device(s) Processor/ device Output device User

Fig. 83 model.

Fundamental

human–computer

interaction

off, respectively, is called a binary switch. A binary switch generates the binary digit 1 when the input voltage is equal to or greater than a specified threshold value and the binary digit 0 otherwise. A second type of digital input device approximates an analog signal and provides a binary stream. Thus, any device that produces an electrical signal or responds to an electrical signal can be used as an input device. Preprocessed analog (digitized) and digital signals generated by an input device are passed on to the processor/device for processing. Once processed, the processor/device may, and often does, generate a new signal or a series of signals. These signals can be used to trigger events on some attached output device. Figure 84 shows examples of a signal produced by an analog input device. To be used with a digital computer, the analog signal can be processed to mimic an on/off switch or it can be digitized using an ADC. The performance of an ADC depends on its architecture. The more bits the ADC operates with, the better the resolution of the signal approximation. Input devices can be further classified as acoustic, inertial, mechanical, magnetic, and optical input devices. 6.1 Devices Based on their basic operation, input devices can be classified as 2-D, 3-D, 6-D, or n-D (degrees of freedom) input devices. Table 26 lists some of the most popular input devices and degrees of freedom associated with each. Many devices can fit in several categories. Also, as any device can emulate another, this table is to be used simply as a guide. Finally, there are other forms of input technologies that are described elsewhere in this encyclopedia. Many of the aforementioned devices can be used in combinations with other input devices, thus providing the notion of either two-handed input or multimodal input. For example, the use of two data gloves is considered two-handed input, as is the use of a mouse along with a Spaceball, whereas the use of a mouse along with speech recognition is considered to be multimodal input. 6.2 Commonly Used Input Devices Keyboard The keyboard is now considered the most essential input device and is used with the majority of computers. Keyboards provide a number of keys (typically more than 100) labeled with a letter or a function that the key performs. Keyboards manufactured for use with notebooks and palm computers or those designed for users with special needs typically provide a reduced set of keys. Different alphabets require different characters to be mapped to each key on the keyboard (i.e., English QWERTY versus German QWERTZ keyboard). Such mappings are achieved by reprogramming the keyboard’s instruction set. Certain keys (e.g., ALT, CTRL, and SHIFT) can be used in conjunction with other keys, thus permitting one key to map to several different functions.

Voltage

Voltage

1023

Voltage

ELECTRONICS Logic 1

Thresho ld

Time Analog signal

Digital signal on/off switch Fig. 84

Table 26 Input Device Classes Input Device Data glove Digitizer Eye tracker Graphic tablet Trackpoint device Joystick Lightpen Monkey Mouse Position tracker Scanner Slider Spaceball Touch screen Touchpad Trackball

1-D

× ×

Logic 0

Time

2-D

3-D

× × × × × ×

×

× × × × × × ×

4-D

6-D

n-D ×

Time Digital signal approximation

Input signals.

the light emitted from the LED is diffracted, resulting in a slight drop of a voltage on the sensor’s side (Fig. 86). These drops of voltage are used to determine the direction and speed of movement of the mouse. Trackball A trackball can be described as an inverted mouse. To move a cursor on the screen, the user moves

×

×

× × ×

× ×

× × ×

×

×

Perforated disk

Mouse Since its creation at Xerox Palo Alto Research Center (PARC), the mouse has become the most popular 2-D input device and has a wide number of variants. Regardless of the variation, each mouse has one, two, or three buttons. For most mice, the motion of a ball, located underneath the mouse, is converted to planar motion—a set of x and y values—using a photoelectric switch as an input transducer. The photoelectric switch contains an LED as a source, a phototransistor as a sensor, and a circular perforated disk as a switch. When the light emitted from the diode reaches the sensor, a pulse (logic 1) is generated and passed on to the interface electronics. The frequency of pulses is interpreted as the velocity of the mouse. There are two such input transducers built in a mouse—one for the x and one for the y axis. Figure 85 shows the principle of motion-to-electric energy conversion. The majority of mice use this principle of motion conversion. Optical mice take advantage of the reflective properties of mouse pads that have a grid of thin lines printed on their smooth and reflective surface. As the mouse passes across the line of a grid, a portion of

Sensor

Source Fig. 85

Motion-to-energy conversion.

Source

Sensor

Reflective surface (mousepad) Grid Fig. 86 Optical mouse structure.

1024


the ball in the desired direction. The motion of the ball is translated to electric signals using a set of perforated disks (one for the x and the other for the y direction). Trackballs, like mice, are equipped with one or more buttons that are pressed to perform a desired operation. Many notebooks and portable computers provide built-in trackballs, as these require much less space than a mouse. Joystick The joystick made its first major appearance in arcade machines in the early 1980s. The basic joystick is a 2-D input device that allows users to move a cursor or an object in any direction on a plane. Typically, a joystick consists of two major parts—a vertical handle (the stick) and a base—each providing one or more buttons that can be used to trigger events. To move the cursor or an object, the stick is moved in the desired direction. Figure 87 shows a major limitation imposed by the resolution of a joystick. The resolution in this example joystick makes it impossible to move in the indicated direction (desired direction), making navigation a bit difficult. There are two major types of joysticks—isotonic and isometric. Isotonic joysticks are precision positionsensitive devices, used in animation, special-effects development, and games. These joysticks are equipped with a set of springs, which return the joystick to the center position when released. A stream of x and y values is generated based on and proportional to the angle between the initial and the current position of the control stick. Some implementations of isotonic joysticks are insensitive to the angle α. These use switches to provide information on direction. Isometric joysticks provide no spring action—the control stick does not move. The x and y values generated by the joystick are proportional to the force applied to the control stick. Some newer joysticks also have been provided with tactile and force feedback.

Desired direction

Forward d

c b a

Left

Right

Back Fig. 87 Directional limitation of Joysticks.

Fig. 88 Slider box. (Image courtesy of Simulation Special Effect, LLC).

Slider A slider is a 1-D input device (Fig. 88). Although sliders are usually implemented in software as part of a graphical user interface (GUI), slider boxes are available as input devices in applications requiring a large number of independent parameters to be controlled [as in musical instrument digital interface (MIDI) applications requiring multiple channels to be manipulated independently]. Most windowing systems incorporate sliders to support panning of the window’s content or for color scale value selections. Spaceball The Spaceball is a 6-D input device used primarily in computer-aided design and engineering, animation, virtual reality, and computer games. It enables users to manipulate a 3-D model with 6degrees-of-freedom control (simultaneous x, y, z, translations and rotations) and as easily as if they were holding it in their hands. A Spaceball is often used in conjuction with the mouse. Spaceballs made their appearances initially with high-end graphic workstations, but this is not the case anymore. As desktop computers have become more powerful, many applications make use of the Spaceball and its derivatives. Touchpad A touchpad is a 2-D input device developed for use in areas with limited space. Touchpads provide precise cursor control by using a fingertip moving on a rectangular area. Buttons located on the side of the rectangular input area can be programmed to perform specific operations as modifier keys on keyboards. Touchpads are usually located under the SPACE bar or the cursor keys, or they can be attached to a computer through a serial port. Input Tablet An input tablet is a variation of a touchpad. It is larger than a touchpad, and instead of a finger, a penlike device with a button to perform specific operations is used. A coil in the pen generates a magnetic field, and a wire grid in the tablet transmits the signal to the tablet’s microprocessor. The output data include the pen’s location, the pressure of the pen

ELECTRONICS

1025

on the tablet, and the tilt of the pen in relation to tablet. Input tablets are mostly used in the design arts and in mechanical and engineering computer-aided design.

draw a grid on the object to be digitized to enable the acquisition of coordinates for every point on that grid. This is both time consuming and error prone.

Integrated Pointing Device—Stick A 2-D integrated pointing device, called a stick, is a miniature isometric joystick embedded between the keys on the keyboard. To move the cursor, the user pushes or pulls the stick in desired direction. The buttons associated with the stick are located under the SPACE bar on the keyboard.

3-D Laser Digitizers Nonmanual digitizers can automate several parts of the digitization process. These are primarily laser-based scanners. An object is positioned on a podium and the scanner rotates the podium while the digitization takes place. Some digitizers revolve around the object when the object is too big or to heavy to be rotated easily around its axes. Such scanners project a beam of laser light onto the model. The intersection of the laser beam and the surface of the object creates a contour of the model captured by a camera and displayed on the screen. This can be done in real time, and a color camera can be used to generate a color model. Most laser scanners use laser triangulation to reconstruct the object.

Lightpen A lightpen is a penlike 2-D device attached to a computer through one of the communications ports or through a dedicated controller board. It is used to draw or select objects directly on the screen. Lightpens may be optically or pressure driven. An optically driven lightpen receives light from the refresh update on the screen; the x, y position of the refreshed pixel is then available for processing. A pressure-driven lightpen is triggered by pressing the lightpen on the screen or by pushing a button. Touch Screen A touch screen is a special type of a 2-D hybrid device because it can both display and acquire information at the same time. On the input side, a touch screen contains a set of sensors in the x and y directions. These sensors may be magnetic, optical, or pressure. Users simply touch the screen, and the sensors in both x and y directions detect an event at some x and y coordinate. Since users tend to use a finger to interact with the touch screen, the resolution of the input device is not fully utilized. In fact, it is often limited to the size of a fingertip. Touch screens are very popular in menu-driven environments such as information booths, fast-food restaurants, and control rooms. Scanner A scanner is a 2-D input device used to capture pictures, drawings, or text. Images, color or black and white, can be captured and stored in digital form for analysis, manipulation, or future retrieval. Associated application software is typically bundled with scanners. This includes imaging software, photo manipulation software, vector graphics conversion software, or text creation (using optical character recognition) software. Three major scanners are available: handheld, flatbed, and sheet scanners. Handheld scanners are suitable for small-scale scanning, flatbed scanners usually handle up to legal-size documents, and sheet scanners usually handle documents of fixed width but arbitrary length. Some engineering firms and geographers use special large-scale scanners for digitizing blueprints and maps. Digitizer A digitizer can be considered either a 2-D or a 3-D input device. There are numerous kinds of digitizers available. Many older and less expensive systems require a great deal of manual work to acquire the data points. For example, the user may need to

Position Trackers Position trackers are used to detect motion and are often attached to objects or body parts. Trackers perform reasonably well. Newer trackers have removed the tethering limitation of older trackers. Newer technologies are also solving the lineof-sight problem (the receiver’s requiring an unobstructed view of the sensors). Some trackers need to be recalibrated often to maintain a high degree of accuracy. Mechanical. Mechanical position trackers use a rigid jointed structure with a known geometry. Such a structure has one fixed and one active end, with the position of the active and available in real time. Mechanical tracking devices are very fast (less than 5 ms response time) and very accurate. The accuracy depends on the accuracy of joint angle encoders. A tracker with a full-color head-coupled stereoscopic display can provide high-quality, full-color stereoscopic images and full 6 degrees of freedom (translation along x, y, and z as well as roll, pitch, yaw). Magnetic. Magnetic trackers use a source that generates three fields of known strength. Detectors are attached to the object to be tracked and measure the magnetic field strengths at a given point. These values are used to determine 6 degrees of freedom in space. Magnetic trackers do not experience any line-of-sight problems and are scalable to many detectors. However, the amount of wiring increases as the number of detectors increases. Magnetic trackers do not operate well around ferrous materials. Ultrasonic. Ultrasonic trackers are often attached to a virtual reality (VR) headset. The tracker consists of three receivers and three transmitters. The position and orientation of the object is calculated based on the time required for each transmitted signal to reach a receiver. Ferrous materials do not affect such trackers. However, ultrasonic trackers are affected by the lineof-sight problem and may be affected by other sources of ultrasonic harmonics.

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High-Speed Video. High-speed video along with fiducial markings on a tracked object is used to determine the location of an object in space. A single picture or a series of pictures are acquired and later processed using image-processing techniques. Fiducial markings can also be located in the space (i.e., scene or walls) and the camera can be attached to the object itself. Such devices can then be used to control the navigation of a robot between two given locations. High-speed video is used for work in a large space because no extra wiring is necessary. Video is unaffected by ferrous and other metals, ultrasonic sound, and light. However, the line-of-sight problem does affect video-tracking systems. Inertial. Inertial position trackers are used to measure orientation and velocity. They are untethered and are not limited by the range or the size of the volume they operate in. Inertial position trackers provide almost complete environmental immunity. Such trackers are sensitive to vibrations and can thus result in inaccurate readings. Biological. Eye tracking is a relatively old technology although not in common use. Eye tracking can be used for control or monitoring. For example, a pilot can control various instruments by simply looking at them. A low-powered infrared (IR) beam is used to illuminate the eye, which in turn is captured using a small camera. The image is processed to track pupil and corneal reflection. Today’s eye tracking devices operate at one degree of resolution. It takes approximately one-third of a second to select, acquire, and fix on an image. Modern applications of eye tracking include its use as an input device for the disabled. Digital Whiteboard A digital whiteboard is a 2-D input device designed to replace traditional blackboards and whiteboards. Everything written on the digital whiteboard with a standard dry-erase marker can be transmitted to a computer. That information can then be used by any application, such as e-mail, fax, or teleconferencing. Data Glove A data glove is an input device that uses properties of leaky fiber-optic cables or resistive strain gauges to determine the amount of movement of fingers and wrists. Leaky fiber-optic cables provide good data, but it is the resistive strain-based input gloves that provide more accurate data. Each data glove is often combined with a 3-D tracker and with 10 strain gauges—at least one for each finger joint—which provides a very high degree of freedom. The latest data gloves also have been extended to provide tactile/force feedback using pneumatic pistons and air bladders. Data gloves can be used along with gestures to manipulate virtual objects or to perform other tasks. Microphone/Speech Recognition and Understanding The microphone has proved to be one of

the most useful input devices for digitizing voice and sound input or for issuing short commands that need to be recognized by a computer. Longer commands cannot be handled by simple recognition. Most sophisticated systems available today still cannot guarantee 100% understanding of human speech. Monkeys or Mannequins The first monkeys were humanlike input devices with a skeleton and precision rheostats at the joints to provide joint angles. Monkeys can be used to set up and capture humanlike motions and offer much better degree-of-freedom match than other devices. Since the first monkeys, a series of animal-like input devices and building blocks have been created that allow users to create their own creatures. Game Input Devices There are a number of other specialized input devices designed to make playing games a more exciting and more realistic experience. Most of these input devices offer additional degrees of freedom and can be used along with other input devices. 6.3 Conclusions

There are a large number of input devices, and the technology is rapidly changing. It is expected that speech recognition and command interpretation, gesture recognition for highly interactive environments (game and virtual), and real-time imaging will become more prominent in the next decade. These will increase the level of human participation in applications and the bandwidth of the data transferred. 7

INSTRUMENTS

Halit Eren Measurement is essential for observing and testing scientific and technological investigations. It is so fundamental and important to science and engineering that the whole science can be said to be dependent on it. Instruments are developed for monitoring the conditions of physical variables and converting them into symbolic output forms. They are designed to maintain prescribed relationships between the parameters being measured and the physical variables under investigation. The physical parameter being measured is known as the measurand. The sensors and transducers are the primary sensing elements in the measuring systems that sense the physical parameters to produce an output. The energy output from the sensor is supplied to a transducer, which converts energy from one form to another. Therefore, a transducer is a device capable of transferring energy between two physical systems. Measurement is a process of gathering information from a physical world and comparing this information with agreed standards. Measurement is carried out with instruments that are designed and manufactured to fulfill given specifications. After the sensor generates

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the signals, the type of signal processing depends on the information required from it. A diverse range of sensors and transducers may be available to meet the measurement requirements of a physical system. The sensors and transducers can be categorized in a number of ways depending on the energy input and output, input variables, sensing elements, and electrical or physical principles. For example, from an energy input and output point of view, there are three fundamental types of transducers: modifiers, self-generators, and modulators. In modifiers, a particular form of energy is modified rather than converted; therefore, the same form of energy exists at the input and the output. In selfgenerators, electrical signals are produced from nonelectric inputs without the application of external energy. These transducers produce very small signals, which may need additional conditioning. Typical examples are piezoelectric transducers and photovoltaic cells. Modulators, on the other hand, produce electric outputs from nonelectric inputs, but they require an external source of energy. Strain gauges are typical examples of such devices. The functionality of an instrument can be broken into smaller elements, as illustrated in Fig. 89. Most measurement systems have a sensor or transducer stage, a signal-conditioning stage, and an output or termination stage. All instruments have some or all of these functional blocks. Generally, if the behavior of the physical system under investigation is known, its performance can be assessed by means of a suitable method of sensing, signal conditioning, and termination. In the applications of instruments, the information about a physical variable is collected, organized, interpreted, and generalized. Experiments are conceived, performed, and repeated; as we acquire confidence in the results, they are expressed as scientific laws. The application of instruments ranges from laboratory conditions to arduous environments such as inside nuclear reactors or on satellite systems and spaceships. In order to meet diverse application requirements of high complexity and capability, many manufacturers have developed a large arsenal of instruments. Some of these manufacturers are listed in Table 27. In recent years, rapid growth of integrated circuit (IC) electronics and the availability of cheap analogto-digital and microprocessors have led to progress in the instrumentation field, with the development of

Physical quantities

Sensor and/or transducer

instruments, measuring techniques, distributed architectures, and standards aimed to improve performance. Instruments are applied for static or dynamic measurements. The static measurements are relatively easy since the physical quantity (e.g., fixed dimensions and weights) does not change in time. If the physical quantity is changing in time, which is often the case, the measurement is said to be dynamic. In this case, steady-state and transient behavior of the physical variable must be analyzed so that it can be matched with the dynamic behavior of the instrument. 7.1 Design, Testing, and Use of Instruments Instruments are designed on the basis of existing knowledge, which is gained either from the experiences of people about the physical process or from our structured understanding of the process. In any case, ideas conceived about an instrument must be translated into hardware and/or software that can perform well within the expected standards and easily be accepted by the end users. Usually, the design of instruments requires many multidisciplinary activities. In the wake of rapidly changing technology, instruments are upgraded often to meet the demands of the marketplace. Depending on the complexity of the proposed instrument, it may take many years to produce an instrument for a relatively short commercial lifetime. In the design and production of instruments, we must consider such factors as simplicity, appearance, ease and flexibility of use, maintenance requirements, lower production costs, lead time to product, and positioning strategy in the marketplace. In order to design and produce instruments, a firm must consider many factors. These include sound business plans, suitable infrastructure, plant, equipment, understanding of technological changes, skilled and trained personnel, adequate finance, marketing and distribution channels, and a clear understanding about worldwide instrument and instrumentation system trends. It is important to choose the right product that is very likely to be in demand in the years to come. Here entrepreneurial management skills may be an important factor. The design process itself may follow well-ordered procedures from idea to marketing stages. The process may be broken down into smaller tasks such as identifying specifications, developing possible solutions for these specifications, modeling, prototyping,

Excitation

Signal conditioner Signal

Signal processing

Output

Transmission or display

Fig. 89 An instrument has a number of relatively independent components that can be described as functional elements. These functional elements are the sensors and transducers, signal conditioners, and output or terminations. In general, if the behavior of the physical system is known, its performance is measured by a suitable arrangement and design of these components.

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List of Manufacturers

ABB, Inc. 501 Merritt 7, P.O. Box 5308 Norwalk, CT 06856-5308 Tel: 800-626-4999 Fax: 203-750-2263

Keithley Instrument, Inc. 28775-T Aurora Road Cleveland, OH 44139-1891 Tel: 800-552-1115 Fax: 440-248-6168

Allied Signal, Inc. 101 Columbia Road Morristown, NY 07962 Tel: 800-707-4555 Fax: 608-497-1001

MCS Calibration, Inc. Engineering Division 1533 Lincoln Avenue Halbrook, NY 11741 Tel: 800-790-0512 Fax: 512-471-6902

Bailey-Fisher and Porter Company 125 E County Line Road Wanminster, PA 18974 Tel: 800-268-8520 Fax: 215-674-7183

MSC Industrial Supply Company 151-T Sunnyside Boulevard Plainview, NY 11803 Tel: 800-753-7937 Fax: 516-349-0265

Consolidated Instrument, Inc. 510 Industrial Avenue Teterboro, NC 07608 Tel: 800-240-3633 Fax: 201-288-8006

National Instruments 6504 Bridge Point Parkway Austin, TX 78730-7186 Tel: 512-794-0100; 888-217-7186 Fax: 512-794-8411

Davies Instrument Manufacturing Company, Inc. 4701 Mt. Hope Drive Baltimore, MD 21215 Tel: 800-548-0409 Fax: 410-358-0252

Omega Engineering, Inc. P.O. Box 4047 Stamford, CT 06907 Tel: 800-826-6342 Fax: 203-359-7700

Dwyer Instrument, Inc. P.O. Box 373-T Michigan City, IN 46361-0373 Tel: 219-879-8000 Fax: 219-872-9057 Fuji Corporation of America Park 80 West, Plaza Two Saddlebrook, NJ 07663 Tel: 201-712-0555 Fax: 201-368-8258 Hanna Instrument, Inc. Highland Industrial Park 584 Park East Drive Woonscocket, RI 02895-0849 Tel: 800-999-4144 Fax: 401-765-7575 Hewlett-Packard Company 5301 Stevens Creek Boulevard Santa Clara, CA 95052-8059 Fax: 303-756-6800 Industrial Instruments and Supply, Inc. P.O. Box 416 12 County Line Industrial Park Southampton, PA 18966 Tel: 800-523-6079 Fax: 215-396-0833 Instrument and Control Services Company 1351-T Cedar Lake Road Lake Villa, IL 60046 Tel: 800-747-8367 Fax: 847-356-9007

Rosemount Analytical 600 S. Harbor Boulevard, Dept TR La Habra, CA 90631-6166 Tel: 800-338-8099 Fax: 562-690-7127 Scientific Instruments, Inc. 518 W Cherry Street Milwaukee, WI 53212 Tel: 414-263-1600 Fax: 415-263-5506 Space Age Control, Inc. 38850 20th Street East Palmdale, CA 93550 Tel: 800-366-3408 Fax: 805-273-4240 Tektronix, Inc. P.O. Box 500 Beaverton, OR 97077 Tel: 503-627-7111 Texas Instrument, Inc. 34 Forest Street, MS 23-01 P.O. Box 2964 Attleboro, MA 02703 Tel: 508-236-3287 Fax: 508-236-1598 Warren-Knight Instrument Company 2045 Bennett Drive Philadelphia, PA 19116 Tel: 215-464-9300 Fax: 215-464-9303 Yokogawa Corporation of America 2 Dart Road Newnon, GA 30265-1040 Tel: 800-258-2552 Fax: 770-251-2088

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Market research

Alternative products

Select product

Design, prototype, and test

Production

Marketing and supply Fig. 90 Design process from the conception of ideas to marketing follows carefully considered stages. The proper identification and effective implementation of these stages is important in the success of a specific instrument in the marketplace.

installing and testing, making modifications, manufacturing, planning marketing and distribution, evaluating customer feedback, and making design and technological improvements. Figure 90 illustrates the stages for the design and marketing of an instrument. Each one of these stages can be viewed in detail in the form of subtasks. For example, many different specifications may be considered for a particular product. These specifications include but are not limited to operational requirements, functional and technological requirements, quality, installation and maintenance, documentation and servicing, and acceptance level determination by the customers. In recent years, computers have been used extensively in the instrument manufacturing industry in the form of computer-aided design (CAD), automated testing, and in other applications. The computer enables rapid access to knowledge-based information and makes design time considerably shorter thus enabling manufacturers to meet rapid demand. In CAD systems, mechanical drafting software, electronic circuit design tools, control analysis tools, and mathematical and word processing tools are integrated to assist the design procedure. Design software is available from various manufacturers listed in Table 27. Testing and Use of Instruments After the instrument is designed and prototyped, various evaluation tests may be conducted. These tests may be

made under reference conditions or under simulated environmental conditions. Some examples of reference condition tests are accuracy, response time, drift, and warmup time. Simulated environmental tests may be compulsory, being regulated by governments and other authorities. Some simulated environment tests include climatic test, drop test, dust test, insulation resistance test, vibration test, electromagnetic compatibility (EMC) tests, and safety and health hazard tests. Many of these tests are strictly regulated by national and international standards. Adequate testing and proper use of instruments is important to achieve the best results out of them. When the instruments are installed, a regular calibration is necessary to ensure the consistency of the performance over the time period of operation. Incorrect measurements can cost a considerable amount of money or even result in the loss of lives. For maximum efficiency, an appropriate instrument for the measurement must be selected. Users should be fully aware of their application requirements, since instruments that do not fit their purpose will deliver false data resulting in wasted time and effort. When selecting the instrument, users must evaluate many factors such as accuracy, frequency response, electrical and physical loading effects, sensitivity, response time, calibration intervals, power supply needs, spare parts, technology, and maintenance requirements. They must ensure compatibility with their existing equipment. Also, when selecting and implementing instruments, quality becomes an important issue from both quantitative and qualitative perspectives. The quality of an instrument may be viewed differently depending on the people involved. For example, quality as viewed by the designer may be an instrument designed on sound physical principles, whereas from the user’s point of view quality may be reliability, maintainability, cost, and availability. For the accuracy and validity of information collected from the instruments, correct installation and proper use become very important. The instruments must be fully integrated with the overall system. Sufficient background work must be conducted prior to installation to avoid a possible shutdown of the process that is longer than necessary. Once the system is installed, the reliability of the instrument must be assessed, and its performance must be checked regularly. The reliability of the system may be defined as the probability that it will operate at an agreed level of performance for a specified period of time. The reliability of instruments follows a bath tub shape against time. Instruments tend to be unreliable in the early and later stages of their lives. During normal operations, if the process conditions change (e.g., installation of large machinery nearby), calibrations must be conducted to avoid possible performance deterioration of the instrument. Therefore, the correct operations of the instruments must be assured at all times throughout the lifetime of the device.


Once the instruments are installed, they may be left alone and expected to operate reliably. They may be communicating with other devices, and their performance may affect the performance of the rest of the system, as in the case of the process industry. In some applications, the instruments may be part of a large instrumentation system, taking a critical role in monitoring and/or controlling the process and operations. However, in many applications, instruments are used on a stand-alone basis for laboratory and experimental work, and the success of the experiments may entirely depend on their correct performance. In these cases, the experiments must be designed and conducted carefully by identifying the primary variables, controlling, selecting the correct instruments, assessing the relative performances, validating the results, and using the data effectively by employing comprehensive data analysis techniques. Set procedures for experimental designs can be found in various sources (e.g., see Ref. 97 as well as the Bibliography). After having performed the experiments, the data must be analyzed appropriately. This can be done at various stages by examining the consistency of the data, performing appropriate statistical analyses, estimating the uncertainties of the results, relating the results to the theory, and correlating the data. Details of statistical data analysis can be found in many books; also many computer software programs are available for the purpose of analysis including common packages such as Microsoft Excel. 7.2 Instrument Response and Drift

Instruments respond to physical phenomena by sensing and generating signals. Depending on the type of instrument used and the physical phenomenon observed, the signals may be either slow or fast to change, and may also contain transients. The response of the instruments to the signals can be analyzed in a number of ways by establishing static and dynamic performance characteristics. Although, the static performances are relatively simple, the dynamic performances may be complex. Static Response Instruments are often described by their dynamic ranges and full-scale deflections (span). The dynamic range indicates the largest and smallest quantities that can be measured. The full-scale deflection of an instrument refers to the maximum permissible value of the input quoted in the units of the particular quantity to be measured. In instruments, the change in output amplitude resulting from a change in input amplitude is called the sensitivity. System sensitivity often is a function of external physical variables such as temperature and humidity. The relative ratio of the output signal to the input signal is the gain. Both, the gain and sensitivity are dependent on the amplitude of the signals and the frequency, which will be discussed in the section on dynamic response.

Output

1030

Time Fig. 91 Drift in the output of an instrument. The main causes of the drift are aging, temperature, ambient conditions, and component deterioration. The drift in an instrument may be predicted by performance analysis of components, past experience, environmental tests, and so on.

In the design stages or during manufacturing, there might be small differences between the input and output, which is called the offset. In other words, when the input is zero the output is not zero or vice versa. The signal output also may change in time, which is known as drift. The drift can occur for many reasons including temperature and aging. Fortunately, drift usually occurs in a predictable manner. A typical drift curve of an instrument against time is illustrated in Fig. 91. During practical applications, readings taken from an instrument under the same conditions may not be repeatable. In this case, a repeatability test may be conducted, and statistical techniques must be employed to evaluate the repeatability of the instrument. Dynamic Response The dynamic response of an instrument is characterized by its natural frequency, amplitude, frequency response, phase shift, linearity and distortions, rise and settling times, slew rate, and the like. These characteristics are a common theme in many instrumentation, control, and electronics books. Although sufficient analysis will be given here, the detailed treatment of the topic can be very lengthy and complex; Hence the full treatment of this tonic is not within the scope of this section. Interested readers should refer to the literature (e.g., Ref. 98). The dynamic response of an instrument can be linear or nonlinear. Fortunately, most instruments exhibit linear characteristics, leading to simple mathematical modeling by using differential equations such as

an

d ny d n−1 y + an−1 n−1 + · · · + a0 y = x(t) n dt dt

(104)

where x is the input variable or the forcing function, y is the output variable, and an , an−1 , . . . , a0 are the coefficients or the constants of the system. The dynamic response of instruments can be categorized as zero-order, first-order, or second-order responses. Although higher order instruments may exist, their behaviors can be understood adequately in

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the form of a second-order system. From Eq. (104)

a2

zero order

(105)

dy + a0 y = x(t) dt

first order

(106)

d 2y dy + a0 y = x(t) + a1 dt 2 dt

1/a0 X(s) = 2 Y (s) s / ωn2 + 2ζ s/ωn + 1

second order (107)

Equations (105)–(107) can be written as Laplace transforms, thus enabling analysis in the frequency domain,    1 (108)      1 Y (s) = (109) X(s)  τ1 s + 1   1   (110) 0  (τ s + 1)(τ s + 1) 1 2 where s is the Laplace operator and τ is the coefficient also called time constant. In zero-order instruments, there is no frequency dependence between the input and output. The amplitude change is uniform across the spectrum of all possible frequencies. In practice, such instruments are difficult to obtain, except in a limited range of operations. In first-order instruments, the relation between the input and the output is frequency dependent. Figure 92 illustrates the response of a first-order instrument for a unit step input in the time domain. Mathematically, the output may be written as y(t) = Ke−t/τ

(111)

where K and τ are constants determined by the system parameters. In many cases, the input signals may be a complex rather than a simple step input. In the analysis, we need to multiply the transfer function, the second member of Eq. (109), by the Laplace transform of the input signal and then transform it back to the time domain if we are to understand the nature of transient and steady-state responses. Also, if the first-order

(112)

where ωn is the natural or undamped frequency (radians per second) and ζ is the damping ratio. As can be seen, the performance of instruments become a function of natural frequency and the damping ratio of the system. The natural frequency and damping ratios are related to the physical parameters of the devices, such as mass and dimensions. In the design stages, these physical parameters may be selected, tested, and modified to obtain a desired response from the system. Typical time response of a second-order system to unit step inputs is illustrated in Fig. 93. The response here indicates that a second-order system can either resonate or be unstable. Furthermore, we can deduce that, since the second-order system is dependent on time, wrong readings can be made depending on the time that the results are taken. Clearly, recording the output when the instrument is still under transient conditions will give an inadequate representation of the physical variable. The frequency compensation, selection of appropriate damping, acceptable time responses, and rise time settling time of instruments may need careful attention in both the design and

2.0 z = 0.1 z = 0.3 z = 0.4 z = 0.6 z = 0.8 z = 1.5

1.8 1.6 1.4 1.2 y(t)

a1

a0 y = x(t)

systems are cascaded, the relative magnitudes of the time constants become important; some may be dominant, and others may be neglected. Second-order systems exhibit the laws of simple harmonic motion, which can be described by linear wave equations. Equation (110) may be rearranged as

1.0 0.8 0.6

Output

Input

0.4 0.2

Output

0

1

2

3

4

5

6

7

8

9

10

11

12

13

wnt

Time (s) Fig. 92 First-order-hold instrument responds to a step input in an exponential form. For a good response the time delay must be small. Drift is usually expressed in percentage of output.

Fig. 93 Unit step time responses of a second-order system with various damping ratios. The maximum overshoot, delay, rise, settling times, and frequency of oscillation depend on the damping ratio. A smaller damping ratio gives a faster response but larger over shot. In many applications, a damping ratio of 0.707 is prefered.


dB

1032 20 ζ = 0.1 ζ = 0.2 ζ = 0.5

10

ζ = 0.7 z = 1.0

0

–10

Angle

0° ζ = 0.1 –90°

–180° 0.1

ζ = 0.2 ζ = 0.5 ζ = 0.7 z = 1.0 0.2

0.4

0.6 0.8 1

2

4

6

8 10

w/wn

Fig. 94 Bode plots of gains and phase angles against frequency of a second-order system. Curves are functions of frequencies as well as damping ratios. These plots can be obtained theoretically or by practical tests conducted in the frequency range.

application stages of an instrument. In these systems, system analysis is essential to ensure that they can measure the input measurand adequately. A typical frequency dependence of gain and phase angle between input and output is illustrated in Fig. 94 in the form of Bode diagrams. Here, the bandwidth, which is the frequencies over which the gain is reasonably constant, is also shown. Usually, half power point (3 dB), which symbolizes 70.7% of the maximum value, is taken as the bandwidth. An important concept in instruments is response time, which can be described as the time required for the instrument to respond to an input signal change. For automatic measurements, the response time is an indication of how many readings can be done per second. Response time is affected by many factors such as analog-to-digital (A/D) conversion time, settling time, delays in electronic components, and delays in sensors. 7.3 Measurement Errors and Error Control Systems The performance of an instrument depends on its static and dynamic characteristics. The performance may be indicated by its accuracy, which may be described as the closeness of measured values to the real values of the variable. The total response is a combination of dynamic and static responses. If the signals generated by the physical variable are changing rapidly, then the dynamic properties of the instrument become important. For slowly varying systems the dynamic errors

may be neglected. In order to describe the full relationships between the inputs and outputs, differential equations can be used, as discussed previously. The performance of an instrument may also be decided by other factors, such as the magnitudes of errors; the repeatability, which indicates the closeness of sets of measurements made in the short term; and the reproducibility of the instrument. The reproducibility is the closeness of sets of measurements when repeated in similar conditions over a long period of time. The ideal or perfect instrument would have perfect sensitivity, reliability, and repeatability without any spread of values and would be within the applicable standards. However, in many measurements, there will be imprecise and inaccurate results as a result of internal and external factors. The departure from the expected perfection is called the error. Often, sensitivity analyses are conducted to evaluate the effect of individual components that are causing these errors. Sensitivity to the affecting parameter can be obtained by varying that one parameter and keeping the others constant. This can be done practically by using the developed instruments or mathematically by means of appropriate models. When determining the performance of an instrument, it is essential to appreciate how errors arise. There may be many sources of errors; therefore, it is important to identify these sources and draw up an error budget. In the error budget, there may be many factors, such as (a) imperfections in electrical and mechanical components (e.g., high tolerances and noise or offset voltages), (b) changes in component performances (e.g., shift in gains, changes in chemistry, aging, and drifts in offsets), (c) external and ambient influences (e.g., temperature, pressure, and humidity), and (d) inherent physical fundamental laws (e.g., thermal and other electrical noises, Brownian motion in materials, and radiation). In instrumentation systems, errors can be broadly classified as systematic, random, or gross. Systematic Errors Systematic errors remain constant with repeated measurements. They can be divided into two basic groups as instrumental errors and environmental errors. Instrumental errors are inherent within the instrument, arising because of the mechanical structure, electronic design, improper adjustments, wrong applications, and so on. They can also be subclassified as loading error, scale error, zero error, and response time error. The environmental errors are caused by environmental factors such as temperature and humidity. Systematic errors can also be viewed as static or dynamic errors. Systematic errors can be quantified by mathematical and graphical means. They can be caused by the nonlinear response of the instrument to different inputs as a result of hysteresis. They also emerge from wrong biasing, wear and aging, and other factors such as modifying the effects environment (e.g., interference). Typical systematic error curves are illustrated in Fig. 95. Because of the predictability of systematic errors, deterministic mathematics can be employed. In the

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Error band

(%) Output

100 75

Hysteresis

Offset

y ± y = f (x1 ± x1 , x2 ± x2 , . . . , xn ± xn ) (116) For an approximate solution, the Taylor series may be applied to Eq. (116). By neglecting the higher order terms of the series, the total absolute error y of the system may be written as xn δy x1 δy x2 δy (117) + + ··· + y = δx1 δx2 δxn

50

The absolute error is predicted by measuring or calculating the values of the errors of each contributing component. Slight modification of Eq. (116) leads to uncertainty analysis, where

25 Ideal 25

50

75 100 (%) Input

wy = [(w1 δy/δx1 )2 + (w2 δy/δx2 )2 + . . .

Fig. 95 Systematic errors are static errors and they can be quantified theoretically or experimentally. There are many different types, including hysteresis, linearity, and offset. They are contained within an error band typical to particular instrument.

simplest form, the error of a measurement may be expressed as x(t) = xm (t) − xr (t)

(113)

where x(t) is the absolute error, xτ (t) is the correct reference value, and xm (t) is the measured value. From Eq. (113), the relative error re (t) may be calculated as re (t) =

x(t) xr (t)

(114)

However, in complex situations, correction curves obtained either empirically or theoretically may be used. Manufacturers usually supply correction curves, especially if their products embrace wide ranging and different applications (e.g., slurries with changing characteristics in time). In many applications, the measurement system is made up of many components that have errors in their own rights. The deterministic approach may be adapted to calculate the overall propagated error of the system, as y = f (x1 , x2 , x3 , . . . , xn )

(115)

where y is the overall output and x1 , x2 , . . . are the components affecting the output. Each variable affecting the output will have its own absolute error of xi . The term xi indicates the mathematically or experimentally determined error of each component under specified operating conditions. The overall performance of the overall system with the errors may be expressed as

+ (wn δy/δxn )2 ]1/2

(118)

where wy is the uncertainty of the overall system and w1 , w2 , . . . , wn are the uncertainties of affecting the component. Uncertainty differs from error in that it involves such human judgmental factors as estimating the possible values of errors. In measurement systems, apart from the uncertainties imposed by the instruments, experimental uncertainties also exist. In evaluating the total uncertainty, several alternative measuring techniques should be considered and assessed, and estimated accuracies must be worked out with care. Random and Gross Errors Random errors appear as a result of rounding, noise and interference, backlash and ambient influences, and so on. In experiments, the random errors vary by small amounts around a mean value. Therefore, the future value of any individual measurement cannot be predicted in a deterministic manner. Random errors may not easily be offset electronically; therefore, in the analysis and compensation, stochastic approaches are adapted by using the laws of probability. Depending on the system, the random error analysis may be made by applying different probability distribution models. But, most instrumentation systems obey normal distribution laws; therefore, the Gaussian model can broadly be applied enabling the determination of the mean values, standard deviations, confidence intervals, and the like, depending on the number of samples being taken. A typical example of a Gaussian curve is given in Fig. 96. The mean value x and the standard deviation σ may be found by xi (119) x= n

and σ =

(xi − x)2 n−1

(120)

1034


P(x)

2σ

σ

σ

2σ

x

x Mean

Fig. 96 Random errors of instruments can be analyzed by using probability methods. In many instruments the errors can be described by a Gaussian distribution curve.

Discussions relating to the application of stochastic theory in error analysis are very lengthy and will not be repeated here. Interested readers should refer to the literature (e.g., Ref. 98a). Gross errors are the result of human mistakes, equipment fault, and the like. Human errors may occur in the process of observations or during the recording and interpretation of experimental results. A large number of errors can be attributed to carelessness, the improper adjustment of instruments, the lack of knowledge about the instrument and the process, and so on. These errors cannot be treated mathematically and eliminated completely, but they can be minimized by having different observers repeat the experiments. Error Reduction Techniques Controlling errors is an essential part of instruments and instrumentation systems. Various techniques are available to achieve this objective. The error control begins in the design stages by choosing the appropriate components, filtering, and bandwidth selection, by reducing the noise, and by eliminating the errors generated by the individual subunits of the complete system. In a good design, the errors of the previous group may be compensated adequately by the following groups. The accuracy of instruments can be increased by postmeasurement corrections. Various calibration methods may be employed to alter parameters slightly to give correct results. In many cases, calibration graphs, mathematical equations, tables, the experiences of the operators, and the like are used to reduce measurement errors. In recent years, with the application of digital techniques and intelligent instruments, error corrections are made automatically by computers or the devices themselves. In many instrumentation systems, the application of compensation strategy is used to increase static and dynamic performances. In the case of static characteristics, compensations can be made by many methods

including introducing opposing nonlinear elements in the system, using isolation and zero environmental sensitivity, opposing compensating environmental inputs, using differential systems, and employing feedback systems. On the other hand, the dynamic compensation can be achieved by applying these techniques as well as by reducing harmonics, using filters, adjusting bandwidth, using feedback compensation techniques, and the like. Open-loop and close-loop dynamic compensations are popular methods employed in both static and dynamic error corrections. For example, using highgain negative feedback can reduce the nonlinearity generated by the system. A recent and fast developing trend is the use of computers for estimating measured values and providing compensation during the operations if any deviations occur from the estimated values. 7.4 Standards and Reference Materials

Standards of fundamental units of length, time, weight, temperature, and electrical quantities have been developed for measurements to be consistent all over the world. The length and weight standards—the meter and the kilogram—are kept in the International Bureau of Weights and Measures in Sèvres, France. Nevertheless, in 1983 the meter was defined as the length of the path traveled by light in a vacuum in the fraction 1/299,792,458 of a second, which was adopted as the standard meter. The standard unit of time—second—is established in terms of known oscillation frequencies of certain devices, such as the radiation of the cesium-133 atom. The standards of electrical quantities are derived from mechanical units of force, mass, length, and time. Temperature standards are established as international scale by taking 11 primary fixed points. If different units are involved, the relationship between different units are defined in fixed terms. For example, 1 lbm = 453.59237 g. Based on these standards, primary international units, SI (Système International d’Unités), are established for mass, length, time, electric current, luminous intensity, and temperature, as illustrated in Table 28. From these units, SI units of all physical quantities can be derived as exemplified in Table 29. The standard multiplier prefixes are illustrated in Table 30.

Table 28

Basic SI Units

Quantity Length Mass Time Electric current Temperature Amount of substance Luminous intensity Plane angle Solid angle

Unit meter kilogram second ampere kelvin mole candela radian steradian

Symbol m kg s A K mol cd rad sr

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Table 29 Fundamental, Supplementary, and Derived Units Quantity

Symbol

Unit Name

Unit Symbol

Mechanical Units Acceleration Angular acceleration Angular frequency Angular velocity Area Energy Force Frequency Gravitational field strength Moment of force Plane angle Power Pressure Solid angle Torque Velocity Volume Volume density Wavelength Weight Weight density Work

a α ω ω A E F f g M α, β, θ , φ P p ω T v V ρ λ W γ w

Admittance Capacitance Conductance Conductivity Current density Electric potential Electric field intensity Electrical energy Electrical power Impedance Permittivity of free space Quantity of electricity Reactance Resistance Resistivity

Y C G γ J V E W P Z ε Q X R ρ

Magnetic field intensity Magnetic flux Magnetic flux density Magnetic permeability Mutual inductance Permeability of free space Permeance Relative permeability Reluctance Self-inductance

H B µ M µo P µτ R L

Meter/second2 Radian/second2 Radian/second Radian/second Square meter Joule Newton Hertz Newton/kilogram Newton · meter Radian Watt Newton/meter3 Steradian Newton meter Meter/second Cubic meter Kilogram/meter3 Meter Newton Newton/cubic meter Joule

m/s2 rad/s2 rad/s rad/s m2 J(kg · m2 /s2 ) N(kg · m/s2 ) Hz N/kg N·m Rad W(J/s) N/m3 Sr N·m m/s m3 kg/m3 M N N/m3 J

Electrical Units Mho (siemen) Farad Mho (siemen) Mho/meter Ampere/meter2 Volt Volt/meter Joule Watt Ohm Farad/meter Coulomb Ohm Ohm Ohm · meter

mho (S) F(A · s/V) mho(S) mho/m(S/m) A/m2 V V/m J W F/m C(A · s) ·m

Magnetic Units Ampere/meter Weber Tesla (weber/meter2 ) Henry/meter Henry Henry/meter Henry — Henry−1 Henry

A/m Wb T (Wb/m2 ) H/m H H/m H — H−1 H

Optical Units Illumination Luminous flux Luminance Radiance Radiant energy Radiant flux Radiant intensity

lx lm cd Le W P Ic

Lux Lumen Candela/meter2 Watt/steradian · meter3 Joule Watt Watt/steradian

cd · sr/m2 cd · sr cd/m2 W/sr · m3 J W W/sr

1036 Table 30

ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Decimal Multiples

Name

Symbol

Exa Peta Tera Giga Mega Kilo Hecto Deca Deci Centi Milli Micro Nano Pico Femto Atto

E P T g M k h da d c m µ n p f a

Equivalent 1018 1015 1012 109 106 103 102 10 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18

In addition to primary international standards, standard instruments are available having stable and precisely defined characteristics that are used as references for other instruments that are performing the same function. Hence, the performance of an instrument can be cross-checked against a known device. At a global level, checking is done by using an international network of national and international laboratories, such as the National Bureau of Standards (NBS), the National Physical Laboratory (NPL), and the PhysikalischTechnische Bundesanstalt of Germany. A treaty between the world’s national laboratories regulates the international activity and coordinates development, acceptance, and intercomparisons. Basically, standards are kept in four stages: 1. International standards represent certain units of measurement with maximum accuracy possible within today’s available technology. These standards are under the responsibility of an international advisory committee and are not available to ordinary users for comparison or calibration purposes. 2. Primary standards are the national standards maintained by national laboratories in different parts of the world for verification of secondary standards. These standards are independently calibrated by absolute measurements that are periodically made against the international standards. The primary standards are compared against each other. 3. Secondary standards are maintained in the laboratories of industry and other organizations. They are periodically checked against primary standards and certified. 4. Working standards are used to calibrate general laboratory and field instruments.

Another type of standard is published and maintained by the Institute of Electrical and Electronics Engineer (IEEE) in New York. These standards are for test procedures, safety rules, definitions, nomenclature, and so on. The IEEE standards are adopted by many organizations around the world. Many nations also have their own standards for test procedures, instrument usage procedures, safety, and the like. 7.5 Calibration, Calibration Conditions, and Linear Calibration Model The calibration of all instruments is essential for checking their performances against known standards. This provides consistency in readings and reduces errors, thus validating the measurements to be valid universally. After an instrument is calibrated, future operation is deemed to be error bounded for a given period of time for similar operational conditions. The calibration procedure involves comparison of the instrument against primary or secondary standards. In some cases, it may be sufficient to calibrate a device against another one with a known accuracy. Many nations and organizations maintain laboratories with the primary functions of calibrating instruments and field measuring systems that are used in everyday operations. Examples of these laboratories are National Association of Testing Authorities (NATA) of Australia and the British Calibration Services (BCS). Calibrations may be made under static or dynamic conditions. A typical calibration procedure of a complex process involving many instruments is illustrated in Fig. 97. In an ideal situation, for an instrument that responds to a multitude of physical variables, a commonly employed method is to keep all inputs constant except one. The input is varied in increments in

Parameter 1 Standard instrument 1

Parameter 2 Standard instrument 2

Parameter n Standard instrument n

Element or system under calibration

Output 1 Output 2 Output k Standard instrument 1 Standard instrument 2 Standard instrument k Calibrated instrument 1 Calibrated instrument 2 Calibrated instrument k

Fig. 97 Instruments are frequently calibrated sequentially for all affected inputs. Calibrations are made under static or dynamic conditions, usually keeping all inputs constant and varying only one and observing the output. Calibration continues until all other inputs are covered.

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increasing and decreasing directions over a specified range. The observed output then becomes a function of that single input. The calibration is continued in a similar manner until all other inputs are covered. For better results, this procedure may be repeated by varying the sequences of inputs, thus developing a family of relationships between the inputs and outputs. As a result of these calibration readings, the input and output relation usually demonstrates statistical characteristics. From these characteristics, appropriate calibration curves can be obtained, and other statistical techniques can be applied. In many instruments, the effect of a single input may not represent the true output values when one input is varied and all others are kept constant. In these cases, calibration is conducted by varying several inputs simultaneously. Throughout the calibration procedure, the n number of variables of the system are monitored by appropriate standard instruments. The rule of thumb is that each calibrated variable must have a traceable ladder starting from laboratory standards and secondary standards leading to primary standards. This is known as the linear calibration model or traceability. Most instrument manufacturers supply calibrated instruments and reliable information about their products. But their claims of accuracy and reliability must be taken at face value. Therefore, in many cases, application-specific calibrations must be made periodically within the recommended calibration intervals. Usually, manufacturers supply calibration programs. In the absence of such programs, it is advisable to conduct frequent calibrations in the early stages of installation and lengthen the period between calibrations as the confidence builds based on satisfactory performance. Recently, with the wide applications of digital systems, computers can make automatic and self-calibrations as in the case of many intelligent instruments. In these cases, postmeasurement corrections are made, and the magnitudes of various errors are stored in the memory to be recalled and used in laboratory and field applications. 7.6 Analog and Digital Instruments Instruments can be analog or digital or a combination of the two. Nowadays, most instruments are produced to be digital because of the advantages that they offer. However, the front end of majority of instruments are still analog; that is, the majority of sensors and transducers generate analog signals. Initially, the signals are conditioned by analog circuits before they are put into digital form for signal processing. It is important

Physical variable


Input circuit

Preamplifier

Filter

to mention that digital instruments operating purely on digital principles are developing fast. For instance, today’s smart sensors contain the complete signal condition circuits in a single chip integrated with the sensor itself. The output of smart sensors can be interfaced directly with other digital devices. In analog instruments, the useful information is conveyed by changes in amplitudes, phases, or frequencies or a combination of the three. These signals can be deterministic or nondeterministic. In all analog or digital instruments, as in the case with all signalbearing systems, there are useful signals that respond to the physical phenomena and unwanted signal resulting from various forms of noise. In the case of digital instruments, additional noise is generated in the process of A/D conversion. Analog signals can also be nondeterministic; that is, the future state of the signal cannot be determined. If the signal varies in a probabilistic manner, its future can be foreseen only by statistical methods. The mathematical and practical treatment of analog and digital signals having deterministic, stochastic, and nondeterministic properties is a very lengthy subject and a vast body of information can be found in the literature; therefore, they will not be treated here. As is true of all instruments, when connecting electronic building blocks, it is necessary to minimize the loading effects of each block by ensuring that the signal is passed without attenuation, loss, or magnitude and phase alterations. It is also important to ensure maximum power transfer between blocks by appropriate impedance-matching techniques. Impedance matching is very important in all instruments but particularly at a frequency of 1 MHz and above. As a rule of thumb, output impedances of the blocks are usually kept low, and input impedances are kept high so that the loading effects can be minimized. Analog Instruments Analog instruments are characterized by continuous signals. A purely analog system measures, transmits, displays, and stores data in analog form. The signal conditioning is usually made by integrating many functional blocks such as bridges, amplifiers, filters, oscillators, modulators, offsets and level converters, buffers, and the like, as illustrated Fig. 98. Generally, in the initial stages, the signals produced by the sensors and transducers are conditioned mainly by analog electronics, even if they are configured as digital instruments later. Therefore, we pay more attention to analog instruments, keeping in mind

Amplifier

Transmission

Processing

Output display

Fig. 98 Analog instruments measure, transmit, display, and store data in analog form. Signal conditioning usually involves such components as bridges, amplifiers, filters, oscillators, modulators, offsets and level converters, buffers, and so on. These components are designed and tested carefully to suit the characteristics of particular instruments.

1038


that much of the information given here also may be used in various stages of the digital instruments. Instrument bridges are commonly used to measure such basic electrical quantities as resistance, capacitance, inductance, impedance, and admittance. Basically, they are two-port networks in which the component to be measured is connected to one of the branches of the network. There are two basic groups, ac and dc bridges. Also, there are many different types in each group, such as Wheatstone and Kelvin dc bridges and Schering, Maxwell, Hay, and Owen ac bridges. In a particular instrument, the selection of the bridge to be employed and the determination of values and tolerances of its components is very important. It is not our intent to cover all bridges here; however, as typical example of an ac bridge, a series RC bridge is given in Fig. 99. We also offer some analysis to illustrate briefly their typical operational principles. At balance, Z1 Z3 = Zx Zz

(121)

Substitution of impedance values gives j j R 3 R 1 − C1 = R x − Cx R 2 ω ω

(122)

Equating the real and imaginary terms gives the values of unknown components as Rx =

R1 R3 R2

and Cx =

1 Cf 1 Cx

C1

if dt = eex − eai − eex

(125)

ix dt = e0 − eai = e0

(126)

if + ix − iai = 0 = if + ix

R2

r1

D Zx Rx

(127)

Z2

Z1

Cx

(124)

In instruments, the selection and use of amplifiers and filters are also very important since many transducers generate extremely weak signals in comparison to the noise existing in the device. Today, operational amplifiers and high-precision instrumentation amplifiers are the building blocks of modern instruments. The operation amplifiers may be used as inverting and noninverting amplifiers, and by connecting suitable external components, they can be configured to perform many other functions, such as multipliers, adders, limiters, and filters. Instrumentation amplifiers are used in situations where operational amplifiers do not meet the requirements. They are essentially high-performance differential amplifiers consisting of several closed-loop operational amplifiers. The instrumentation amplifiers have improved common-mode rejection ratios (CMRR) (up to 160 dB), high input impedances (up to 500 M ), low output impedance, low offset currents and voltages, and better temperature characteristics. To illustrate amplifiers in instrumentation systems, a typical current amplifier used in charge amplification is illustrated in Fig. 100. In this circuit, if the input impedance of the operational amplifier is high, output is not saturated, and the differential input voltage is small, it is possible to write

(123)

R1

C1 R 2 R3

Cx Z3

if

R3

ix

– eex

Fig. 99 A series RC bridge wherein the unknown capacitance is compared with a known capacitance. The voltage drop across R1 balances the resistive voltage drop in branch Z2 . The bridge balance is achieved relatively easily when capacitive branches have substantial resistive components. The resistors R1 and either R2 or R3 are adjusted alternately to obtain the balance.

Cf

eai≈ 0 eai≈ 0

+

eo

Fig. 100 Using an operational amplifier signal processor is useful to eliminate the nonlinearity in the signals generated by capacitive sensors. With this type of arrangement, the output voltage can be made to be directly proportional to variations in the signal representing the nonlinear operation of the device.

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C1

R1 – +

C2

R5

R3 R4

R7 R6

–

– +

+ Vs

R2

Fig. 101 Filtering is used in various stages of signal processing to eliminate unwanted components of signals. They can be designed and constructed to eliminate or pass signals at certain frequency ranges. Suitable arrangements of components yield to bandpass, highpass, bandpass, bandstop, and notch filters. Filters can be classified as active and passive.

Manipulation of these equations gives e0 =

−Cf eex Cx

(128)

However, a practical circuit requires a resistance across Cf to limit output drift. The value of this resistance must be greater than the impedance of Cf at the lowest frequency of interest. Filtering is used to reject unwanted components of signals. For example, by using a filter that narrows the bandwidth, the broadband noise energy is reduced, and unwanted signals outside the passband are rejected. Analog filters can be designed by using various techniques, such as Butterworth, Chebyshev, and Bessel–Thomson filters. They can be low-pass, high-pass, bandpass, bandstop, and notch filters. Filters can be classified as active and passive. Active filters involve active components such as operational or instrumentation amplifiers, whereas passive filters are configured completely by inductive, capacitive, and resistive components. The choice of active or passive filters depends on the available components, the precision required, and the frequency of operations. A typical filter used in instrument is given in Fig. 101. Digital Instruments In modern instruments, the original data acquired from the physical variables are usually in analog form. This analog signal is converted to digital before being passed on to the other parts of the system. For conversion purposes, analogto-digital converters are used together with appropriate

Physical signal


Analog signal conditioner

Multiplexer

sample-and-hold devices. In addition, analog multiplexers enable the connection of a number of transducers to the same signal-processing media. The typical components of a digital instrument are illustrated in Fig. 102. The digital systems are particularly useful in performing mathematical operations and storing and transmitting data. Analog-to-digital conversion involves three stages: sampling, quantization, and encoding. The Nyquist sampling theorem must be observed during sampling; that is, “the number of samples per second must be at least twice the highest frequency present in the continuous signal.” As a rule of thumb, depending on the significance of the high frequencies, the sampling must be about 5–10 times the highest frequency of the signal. The next stage is the quantization, which determines the resolution of the sampled signals. The quantization error decreases as the number of bits increases. In the encoding stage, the quantized values are converted to binary numbers to be processed digitally. Figure 103 illustrates a typical A/D sampling process of an analog signal. After the signals are converted to digital form, the data can be further processed by employing such various techniques as fast Fourier transform (FFT) analysis, digital filtering, sequential or logical decision making, correlation methods, spectrum analysis, and so on. Virtual Instruments (VIs) Traditional instruments have three basic components—acquisition and control,

Analog signal

x(t)

Sample and hold

0

T

nT

2T

t

Fig. 103 Analog-to-digital converters involve three stages: sampling, quantization, and encoding. However, the digitization introduces a number of predictable errors. After the conversion, the data can be processed by techniques such as FFT analysis, discrete Fourier transform (DFT) analysis, digital filtering, sequential or logical decision making, correlation methods, spectrum analysis, and so on.

A/D converter

Computer or microprocessor

D/A converter

Fig. 102 Digital instruments have more signal-processing components than analog instruments. Usually, analog signals are converted to digital form by analog-to-digital (A/D) converters. The digital instruments have the advantage of processing, storing, and transmitting signals more easily than their analog counterparts.

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data analysis, and data presentation. In VIs, the use of digital techniques, software, and computers replace the display and processing capabilities of most traditional instruments. In this technology, plug-in data acquistion (DAQ) boards, PC cards (PCMCIA), and parallel-port input–output (I/O) devices are used to interface sensors and transducers of the system under investigation to computers. There are standard interface buses such as VXIbus, which stands for VMEbus Extensions for Instrumentation (also known as the IEEE Standard 1155–1992). Once the system is interfaced, the computer can be programmed to act just like a stand-alone instrument, but offering additional benefits of flexibility of the processing, display, and storage. In VIs, the data can be saved or loaded in memory to be processed in popular spreadsheet programs and word processors, and a report generation capability complements the raw data storage by adding timestamps, measurements, user names, and comments. VI technology allows the user to build test systems that fit specific applications. The VI software can be programmed to resemble familiar instrument panels, including buttons and dials. The user interface tools include knobs, meters, gauges, dials, tank displays, thermometers, graphs, strip charts, and the like to simulate the appearance of traditional instruments. Computer displays can show more colors and allow users to quickly change the way they display test data and controls as required. The software also contains analysis libraries with high-powered statistics, curve fitting, signal processing, and filtering to standard dynamic link libraries (DLLs). Designing a VI system is similar to designing a test system with stand-alone instruments. The first step is to determine what types of signals are needed to measure, including their frequencies, amplitudes, and other signal characteristics together with the level of accuracy expected from these signals. To develop the software for the test application, a programming language or test development software package needs to be selected such as C or Microsoft Visual Basic. Since the display is not fixed, as on a stand-alone instrument, it can be as complex or as simple as the application requires. Nowadays, users can configure their VIs to update front panels and display real-time, animated VIs over the Internet. The toolkits let applications be published over the web and viewed with a standard web browser with little additional programming. With these tools, developers can monitor VIs running in remote locations, publish experiment results on the Web, and automatically notify operators of alarm conditions or status information. 7.7 Control of Instruments Instruments can be manual, semiautomatic, or fully automatic. Manual instruments need human intervention for adjustment, parameter setting, and interpreting readings. Semiautomatic instruments need limited

intervention such as the selection of operating conditions and so on. In the fully automatic instruments, however, the variables are measured either periodically or continuously without human intervention. The information is either stored or transmitted to other devices automatically. Some of these instruments can also measure the values of process variables and regulate their deviations from preset points. It is often necessary to measure many parameters of a process by using two or more instruments. The resulting arrangement for performing the overall measurement function is called the measurement system. In measurement systems, instruments operate in an autonomously but coordinated manner. The information generated by each device is communicated between instruments themselves, or between the instrument and other devices such as recorders, display units, and computers. The coordination of instruments can be done in three ways: analog to analog, analog to digital, and digital to digital. Analog systems consist of instruments that generate continuous current and voltage waveforms in response to the physical variations. The signals are processed by using analog electronics; therefore, signal transmission between the instruments and other devices is also done in the analog form. In assembling these devices, the following characteristics must be considered: Signal transmission and conditioning Loading effects and buffering Proper grounding and shielding Inherent and imposed noises Ambient conditions Signal level compatibility Impedance matching Proper display units Proper data storage media Offset and level conversion is used to convert the output signal of an instrument from one level to another, compatible with the transmission medium in use. In analog systems, signals are usually transmitted at suitable current levels (4–20 mA). In this way, change in impedance does not affect the signal levels, and standard current signal levels can easily be exchanged. In digital instrumentation systems, analog data are converted and transmitted in digital form. The transmission of data between digital devices can be done relatively easily, by using serial or parallel transmission techniques. However, as the measurement system becomes large by the inclusion of many instruments, the communication becomes complex. To avoid this complexity, message interchange standards are used for digital signal transmission such as RS-232 and IEEE-488 VXIbus. Many instruments are manufactured with output ports to pass measurement data and various control signals. The IEEE-488 (also known as the GPIB) bus

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is one of the established industry standard instrumentation interfacings. It enables simultaneous measurements by interfacing up to 15 instruments together at the same time. It has 16 signal lines distributed as 8 data lines, 3 control lines, and 5 general interface management lines. The line configuration of an IEEE488 bus is given in Fig. 104. Once connected, any one device can transfer data to one or more other devices on the bus. All devices must be able to perform at least one of the following roles: talker, listener, controller. The minimum device consists of one talker and one listener without a controller. The length of cables connected to the bus cannot exceed 20 m, and the maximum data rate is restricted to 250 kbytes per second. RS-232 is issued by the Electronic Industries Association (EIA). It uses serial binary data interchange and applies specifically to the interconnection of data communication equipment (DCE) and data terminal equipment (DTM). Data communications equipment may include modems, which are the devices that convert digital signals suitable for transmission through telephone lines. The RS-232 uses standard DB-25 connectors, the pin connection is given in Table 31. Although 25 pins are assigned, a complete data transmission is possible by using only three pins—2, 3, and 7. The transmission speed can be set to certain baud rates such as 19,200 bits per second and can be used for synchronous or nonsynchronous communication purposes. The signal voltage levels are very flexible, with any voltage between −3 and −25 V representing logic 1 and any voltage between +3 and +25 V representing logic 0. In many industrial applications, the current loop digital communication is used. This communication is similar to analog current loop systems, but the signal is transmitted in digital form, with 20 mA signifying logic 1 and 0 mA representing logic 0. Depending on

Data bus

the external noise sources in the installation environment, the current loop can be extended up to 2 km. When data are transmitted distances greater than those permitted by the RS-232 or current loop, the modem, microwave, or RF transmissions are used. In this case, various signal modulation techniques are necessary to convert digital signals to suitable formats. For example, most modems, with medium-speed asynchronous data transmission, use frequency-shift keyed (FSK) modulation. The digital interface with modems uses various protocols such as MIL-STD-188C to transmit signals in simplex, half-duplex, or full-duplex forms depending on the directions of the data flow. The simplex interface transmits data in one direction, whereas full duplex transmits it in two directions simultaneously. Table 31

RS-232 Pin Connections

Pin Number

Direction

1 2 3 4 5 6 7 8 9 11 18 20 22 25

— Out In Out In In — In Out Out In Out In In

Function Frame ground Transmitted data (–TxD) Received data (–RxD) Request to send (RTS) Clear to send (CTS) Data set ready (DSR) Signal ground (SG) Received line signal detector (DCD) + Transmit current loop data − Transmit current loop data + Receive current loop data Data terminal ready (DTR) Ring indicator (RI) − Receive current loop return

DI01−I08

DAV (data valid) NRFD (not ready for data) NDAC (not data accepted)

Control bus

ATN (attention) IFC (interface clear) SRQ (service request) REN (remote enable) EOI (end or identify)

Interface management bus

1

15

Instruments

Fig. 104 The IEEE-488 or the GPIB bus is an industry standard for interface medium. It has 8 data lines, 3 control lines, and 5 general interface management lines. In noisy environments the maximum length of cable is recommended to be not more than 20 m.

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As far as industrial applications are concerned, several standards for digital data transmission are available, commonly known as field buses in the engineering literature. For example, WorldFIP and Profibus have been developed and Foundation Fieldbus is under development to increase the performance of the 20mA current loop. New devices allow for an increase in the data rates (e.g., National Instruments chips and boards operating with high-speed protocol HS488 for 8 Mbytes/s transfer rate). A new standard is under discussion at the IEEE by the working group for higher performance IEEE Std. 488.1, with a very high increase in the data rate. Concerning the design software, there are important tools that help implement control (application) software for automatic measuring equipment, such as LabWindows and LabVIEW from National Instruments and VEE from Hewlett-Packard. In many applications, many instruments (say over 1000) may be used to monitor and control the process as in the case of computer-integrated manufacturing (CIM). In these cases, instruments are networked either in groups or as whole via a center computer or group of computers. Appropriate network topologies (e.g., star, ring, field bus) may be employed to enable the signal flow between the instruments and computers, among the instruments themselves, or between instruments and control panels. 7.8 Industrial Measuring Instruments In industry, instruments are used to sense and maintain the functions of the process. Because the requirements of diverse industries are different, the instruments are made quite differently to suit applicational differences from one industry to another. Here, instruments specific to some industries will be discussed briefly. The process industry uses instruments extensively for on-line monitoring and off-line analysis. Specific instruments are used commonly for sensing variables such as temperature, pressure, volumetric and mass flow rate, density, weight, displacement, pH levels, color, absorbency, viscosity, material flow, dew point, organic and inorganic components, turbidity, solid and liquid level, humidity, and particle size distribution. The selection and use of these instruments constitute an important part of process engineering, which is a discipline in its own right. Additional information can be found in the Bibliography (e.g., Ref. 97). In medical technology, there are three basic types of instruments—imaging, physiological measurements, and laboratory analysis. In imaging and physiological measurements, the instruments are closely linked with patients. Some examples of these instruments are X-ray tomography, nuclear magnetic reasonance (NMR) and nuclear spin tomography, ultrasound imaging, thermography, brain and nervous system sensors, and respiratory sensors. Many instruments are based on the radiation and sound, force and tactile sensing, electromagnetic sensing, and chemical and bioanalytical sensors.

Power plants are instrumented for maximum availability, operational safety, and environmental planning. Therefore, their measurements must be as accurate as possible and reliable. Instruments are used for temperature, pressure, flow, level, vibration measurements, and water, steam, and gas analysis. For example, gas analysis requires instruments to measure carbon compounds, sulfur and nitrogen compounds, and dust and ash contents. Environmental monitoring requires a diverse range of instruments for air, water, and biological monitoring. Instruments are used for measuring various forms of radiation, chemicals hazards, air pollutants, and organic solvents. Many sophisticated instruments are also developed for remote monitoring via satellites, and they operate on optical, microwave, and RF electromagnetic radiation principles. In automobiles, instruments are used to assist drivers by sensing variables such as cooling, braking, fuel consumption, humidity control, speed, travel route monitoring, and position sensing. Instruments also find applications for safety and security purposes, such as passenger protection and locking and antitheft systems. Recently, with the advent of micromachined sensors, many diverse instruments such as engine control, fuel injection, air regulation, and torque sensing are developed. The manufacturing industry, especially automated manufacturing, requires a diverse range of instruments. Machine diagnosis and process parameters are made by instruments based on force, torque, pressure, speed, temperature, and electrical parameter-sensing instruments. Optics, tactile arrays, and acoustic scanning instruments are used for pattern recognition. Distance and displacement measurements are made by many methods (e.g., inductive, capacitive, optical, and acoustic techniques). Aerospace instrumentation requires an accurate indication of physical variables and the changes in these variables. Instruments are designed to suit specific conditions of operations. Some of the measurements are gas temperature and pressure, fluid flow, aircraft velocity, aircraft icing, thrust and acceleration, load, strain and force, position, altitude sensing, and direction finding. 8

INTEGRATED CIRCUITS

N. Ranganathan and Raju D. Venkataramana The invention of the transistor in 1947 by William Shockley and his colleagues John Bardeen and Walter Brattain at Bell Laboratories, Murray Hill, NJ, launched a new era of ICs. The transistor concept was based on the discovery that the flow of electric current through a solid semiconductor material like silicon can be controlled by adding impurities appropriately through the implantation processes. The transistor replaced the vacuum tube due to its better reliability, lesser power requirements, and, above all, its much

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smaller size. In the late 1950s, Jack Kilby of Texas Instruments developed the first integrated circuit. The ability to develop flat or planar ICs, which allowed the interconnection of circuits on a single substrate (due to Robert Noyce and Gordon Moore), began the microelectronics revolution. The substrate is the supporting semiconductor material on which the various devices that form the integrated circuit are attached. Researchers developed sophisticated photolithography techniques that helped in the reduction of the minimum feature size, leading to larger circuits being implemented on a chip. The miniaturization of the transistor led to the development of integrated circuit technology in which several hundreds and thousands of transistors could be integrated on a single silicon die. IC technology led to further developments, such as microprocessors, mainframe computers, and supercomputers. Since the first integrated circuit was designed following the invention of the transistor, several generations of integrated circuits have come into existence: SSI (small-scale integration) in the early 1960s, MSI (medium-scale integration) in the latter half of the 1960s, and LSI (large-scale integration) in the 1970s. The VLSI (very large scale integration) era began in the 1980s. While the SSI components consisted on the order of 10–100 transistors or devices per integrated circuit package, the MSI chips consisted of anywhere from 100 to 1000 devices per chip. The LSI components ranged from roughly 1000 to 20,000 transistors per chip, while the VLSI chips contain on the order of up to 3 million devices. When the chip density increases beyond a few million, the Japanese refer to the technology as ULSI (ultra large scale integration), but many in the rest of the world continue to call it VLSI. The driving factor behind integrated circuit technology was the scaling factor, which in turn affected the circuit density within a single packaged chip. In 1965, Gordon Moore predicted that the density of components per integrated circuit would continue to double at regular intervals. Amazingly, this has proved true, with a fair amount of accuracy.99 Another important factor used in measuring the advances in IC technology is the minimum feature size or the minimum line width within an integrated circuit (measured in microns). From about 8 µm in the early 1970s, the minimum feature size has decreased steadily, increasing the chip density or the number of devices that can be packed within a given die size. In the early 1990s, the minimum feature size decreased to about 0.5 µm, and currently 0.3, 0.25, and 0.1 µm technologies (also called deep submicron technologies) are becoming increasingly common. IC complexity refers, in general, to the increase in chip area (die size), the decrease in minimum feature size, and the increase in chip density. With the increase in IC complexity, the design time and the design automation complexity increase significantly. The advances in IC technology are the result of many factors, such as

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high-resolution lithography techniques, better processing capabilities, reliability and yield characteristics, sophisticated design automation tools, and accumulated architecture, circuit, and layout design experience. 8.1 Basic Technologies The field of integrated circuits is broad. The various basic technologies commonly known are shown in Fig. 105. The inert substrate processes, further divided as thin- and thick-film processes, yield devices with good resistive and temperature characteristics. However, they are mostly used in low-volume circuits and in hybrid ICs. The two most popular active substrate materials are silicon and gallium arsenide (GaAs). The silicon processes can be separated into two classes: MOS (the basic device is a metal–oxide–semiconductor fieldeffect transistor) and bipolar (the basic device is bipolar junction transistors). The bipolar process was commonly used in the 1960s and 1970s and yields high-speed circuits with the overhead of high-power dissipation and the disadvantage of low density. The transistor– transistor logic (TTL) family of circuits constitutes the most popular type of bipolar and is still used in many high-volume applications. The ECL devices are used for high-speed parts that form the critical path delay of the circuit. The MOS family of processes consists of PMOS, NMOS, CMOS, and BiCMOS. The term PMOS refers to a MOS process that uses only p-channel transistors, and NMOS refers to a MOS process that uses only n-channel transistors. PMOS is not used much due to its electrical characteristics, which are not as good as the n-channel FETs, primarily since the mobility of the n-channel material is almost twice compared to the mobility of the p-channel material. Also, the NMOS devices are smaller than the PMOS devices, and thus PMOS do not give good packing density. CMOS was introduced in the early 1960s; however, it was only used in limited applications, such as watches and calculators. This was primarily due to the fact that CMOS had slower speed, less packing density, and latchup problems although it had a high noise margin and lower power requirements. Thus, NMOS was

Basic technologies

Inert substrat e

Thin film

Thick film

MOS

NMOS

PMOS

Fig. 105

CMOS

Active substr ate

Silicon

Bipolar

BI-CMOS

Gallium arsenide

MESFET

TTL

I2 L

Overview of basic technologies.

Bipolar

ECL

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preferred over CMOS, in general, until the p-channel devices developed began to have similar characteristics as the nMOS and both the p-channel and n-channel transistors started delivering close to equal amounts of currents with similar transistor sizes. In the 1980s and the 1990s, the need for lower power consumption was the driving factor, and thus CMOS emerged as the leading IC technology.100 The BiCMOS technology combines both bipolar and CMOS devices in a single process. While CMOS is preferred for logic circuits, BiCMOS is preferred for input/output (I/O) and driver circuits due to its low input impedance and high current driving capability. Since the 1980s, efforts have been directed toward designing digital ICs using GaAs devices. In many high-resolution radar systems, space systems, highspeed communication circuits, and microwave circuits, the integrated circuits need to operate at speeds beyond several gigahertz. In silicon technology, it is possible to obtain speeds on the order of up to 10 GHz using ECL circuits, which is almost pushing the limits of the silicon technology. In GaAs technology, the basic device is the metal–semiconductor (Schottky gate) field-effect transistor, called the GaAs MESFET. Given similar conditions, the electrons in n-type GaAs material travel twice faster than in silicon. Thus, the GaAs circuits could function at twice the speed than the silicon ECL circuits for the same minimum feature size. The GaAs material has a larger bandgap and does not need gate oxide material, as in silicon, which makes it immune to radiation effects. Also, the GaAs

material has very high resistivity at room temperatures and lower parasitic capacitances, yielding high-quality transistor devices. However, the cost of fabricating large GaAs circuits is significantly high due to its low reliability and yield characteristics (primarily due to the presence of more defects in the material compared to silicon). The fabrication process is complex, expensive, and does not aid scaling. Also, the hole mobility is the same as in silicon, which means GaAs is not preferable for complementary circuits. Thus, the GaAs technology has not been as successful as initially promised. Since CMOS has been the most dominant technology for integrated circuits, we examine the MOS transistor and its characteristics as a switch in the next section. 8.2 MOS Switch The MOSFET is the basic building block of contemporary CMOS circuits, such as microprocessors and memories. A MOSFET is a unipolar device; that is, current is transported by means of only one type of polarity (electrons in an n type and holes in a p type). In this section, we describe the basic structure of MOSFETS and their operation and provide examples of gates built using MOS devices. Structure The basic structure of a MOSFET (n and p type) is shown in Fig. 106. We describe the structure of an n-type MOSFET.101,102 It consists of four terminals with a p-type substrate into which two n+ regions are implanted. The substrate is a silicon

G

Gate oxide Poly gate Field oxide

n+

n+

Field oxide

Channel

S

D

P-type substrate NMOS structure

NMOS symbol

G

Gate oxide Poly gate Field oxide

p+

p+

Field oxide

Channel

S

D

N-type substrate PMOS structure Fig. 106

Structure of n- and p-type MOSFET.

PMOS symbol

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wafer that provides stability and support. The region between the two n+ regions is covered by an insulator, typically polysilicon and a metal contact. This contact forms the gate of the transistor. The insulating layer is required to prevent the flow of current between the semiconductor and the gate. The two n+ regions form the source and the drain. Due to the symmetry of the structure, the source and the drain are equivalent. The gate input controls the operation of the MOSFET. A bias voltage on the gate causes the formation of a channel between the n+ regions. This channel causes a connection between the source and drain and is responsible for the flow of the current. The MOSFET is surrounded by a thick oxide, called the field oxide, which isolates it from neighboring devices. Reversal of n and p types in the discussion will result in a ptype MOSFET. Typical circuit symbols for n-type and p-type MOSFETS are also shown in Fig. 106. Operation When no gate bias is applied, the drain and the source behave as two pn junctions connected in series in the opposite direction. The only current that flows is the reverse leakage current from the source to the drain. When a positive voltage is applied to the gate, the electrons are attracted and the holes are repelled. This causes the formation of an inversion layer or a channel region. The source and the drain are connected by a conducting n channel through which the current can flow. This voltage-induced channel is formed only when the applied voltage is greater than the threshold voltage, Vt . MOS devices that do not conduct when no gate bias is applied are called enhancement mode or normally OFF transistors. In nMOS enhancement mode devices, a gate voltage greater than Vt should be applied for channel formation. In pMOS enhancement mode devices, a negative gate voltage whose magnitude is greater than Vt must be applied. MOS devices that conduct at zero gate bias are called normally ON or depletion mode devices. A gate voltage of appropriate polarity depletes the channel of majority carriers and hence turns it OFF. Considering an enhancement mode n-channel transistor, when the bias voltage is above the predefined threshold voltage, the gate acts as a closed switch between the source and drain, the terminals of which become electrically connected. When the gate voltage is cut off, the channel becomes absent, the transistor stops conducting, and the source and the drain channels get electrically disconnected. Similarly, the p-channel transistor conducts when the gate voltage is beneath the threshold voltage and stops conducting when the bias voltage is increased above the threshold. The behavior of the MOS transistor as a switch forms the fundamental basis for implementing digital Boolean circuits using MOS devices. Output Characteristics We describe the basis output characteristics103,104 of a MOS device in this subsection. There are three regions of operation for a MOS device:

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1. Cutoff region 2. Linear region 3. Saturation region In the cutoff region, no current flows and the device is said to be off. When a bias, Vgs , is applied to the gate such that Vg >Vt , the channel is formed. If a small drain voltage, Vds , is applied, drain current, Ids , flows from source to drain through the conducting channel. The channel acts like a resistance, and the drain current is proportional to the drain voltage. This is the linear region of operation. As the value of Vds is increased, the channel charge near the drain decreases. The channel is pinched off when Vds = Vgs − Vt . An increase in Vds beyond the pinchoff value causes little change in the drain current. This is the saturation region of operation of the MOS device. The output characteristics of n- and p-type devices is shown in Fig. 107. The equations that describe the regions of operation can be summarized as follows:

Ids

 0 if Vgs ≤ Vt    2  ] k/2[2(Vgs − Vt )Vds − Vds = if Vg > Vt , Vds ≤ (Vgs − Vt )   k/2(Vgs − Vt )2    if Vg > Vt , Vds > (Vgs − Vt )

(cutoff) (linear) (saturation)

where k is the transconductance parameter of the transistor. A detailed analysis of the structure and operation of MOS devices is described in Refs. 101, 103, 105, and 106. CMOS Inverter The basic structure of an inverter is shown in Fig. 108, and the process cross section is shown in Fig. 109. The gates of both the NMOS and the PMOS transistors are connected. The PMOs transistor is connected to the supply voltage Vdd , and the NMOS transistor is connected to Gnd . When a logical 0 is applied at the input Vin , then the PMOS

Vgs1 Linear region

Saturation region Vgs2

Ids

Vgs3 Vgs4 Vds Fig. 107 Output characteristics of MOS transistor.

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ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS Vdd

Vdd

In

Out

GND I

Fig. 108 Circuit schematic of inverter.

II

III

0 Vin

device is on and the output is pulled to Vdd . Hence the output is a logical 1. On the other hand, when a logical 1 is applied at the input, then the NMOS transistor is on and the output is pulled to the ground. Hence we have a logical 0. The operating regions of the transistor are shown in Fig. 110. In region I, the n device is off and the p device operates in the linear region. Hence the output is pulled to Vdd . In region II, the n and p devices operate in the linear and saturation region depending on the input voltage. In region III, the p device is cut off and the n device is operating in the linear region. The output is pulled to the ground. In region II, when both the transistors are on simultaneously, a short is produced between Vdd and Gnd . This accounts for the short circuit power dissipation in CMOS logic.

VDD

Fig. 110 Operating regions of transistor.

Transmission Gate Consider the device shown in Fig. 111, which represents an NMOS or a PMOS device. By suitably controlling the gate bias, the device can be made to turn on or off. It behaves as an electrical switch that either connects or disconnects the points s and d. An NMOS device is a good switch when it passes a logical 0, and a PMOS is a good switch when it passes a logical 1. In CMOS logic, both the NMOS and PMOS devices operate together. In general, the NMOS transistor pulls down the output

Input

Ground

Output

n well p substrate

Polysilicon

Field oxide

p diffusion

Metal (A1)

Gate oxide

n diffusion

Fig. 109 Process cross section of n-well inverter.

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MOS device

s

d

s

d

s

d

Vdd

Vdd

B

B

A

A

Appropriate gate bias

g

Out

Fig. 111

A MOS device as switch.

Out

A

A B

g

Gnd

Gnd

Two input NAND gate

Two input NOR gate

Fig. 113

a

B

Two-input NAND and NOR gate.

b

to Vdd . Hence this structure implements the operation f = (A + B) . The p structure is the logical dual of the n structure. An n input NAND and NOR gate can be constructed in a similar fashion. g Fig. 112

Transmission gate.

node to logical 0, and the PMOS device pulls up a node to logical 1. A transmission gate is obtained by connecting the two in parallel, as shown in Fig. 112. The control signal (say, g) applied to the n-type device is complemented and applied to the p-type device. When g is high, both the transistors are on and hence a good 1 or a 0 is passed. When g is low, both the devices are off. This is also called a complementary switch, or a C SWITCH.103 NAND and NOR Gates CMOS combinational gates are constructed by connecting the PMOS and NMOS devices in either series or parallel to generate different logical functions. The structures for a two-input NAND and NOR gate are shown in Fig. 113. NAND Gate. The p devices are connected in parallel, and the n devices are connected in series. When either of the inputs A or B is a logical 0, the output is pulled high to Vdd . When both A and B are high, then the output is pulled to the ground. Hence this structure implements the operation f = (AB) . NOR Gate. Similarly, in the NOR gate, the p devices are connected in series and the n devices are connected in parallel. When either of the inputs A or B is a logical 1, then the output is pulled to the ground. When both A and B are low, then the output is pulled

8.3 IC Design Methodology To design and realize VLSI circuits, several factors play key roles. The goal of an IC designer is to design a circuit that meets the given specifications and requirements while spending minimum design and development time avoiding design errors. The designed circuit should function correctly and meet the performance requirements, such as delay, timing, power, and size. A robust design methodology has been established over the years, and the design of complex integrated circuits has been made possible essentially due to advances in VLSI design automation. The various stages in the design flow are shown in Fig. 114. The design cycle ranges from the systemlevel specification and requirements to the end product of a fabricated, packaged, and tested integrated circuit. The basic design methodology is briefly described here, and the various stages are discussed in detail in the following sections using simple examples. The first step is to determine the system-level specifications, such as the overall functionality, size, power, performance, cost, application environment, IC fabrication process, technology, and chip-level and boardlevel interfaces required. There are several tradeoffs to be considered. The next step is the functional design and description, in which the system is partitioned into functional modules and the functionality of the different modules and their interfaces to each other are considered. The issues to be considered are regularity and modularity of structures, subsystem design, data flow organization, hierarchical design approach, cell types, geometric placements, and communication between the different blocks.

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ESHBACH’S HANDBOOK OF ENGINEERING FUNDAMENTALS System-level specification and requirements Functional design and description Architectur al design Logic design Circuit des ign Physical design Verification Fabrication Testing Fig. 114

IC design methodology.

Once the functionality of the various modules is determined, the architectural design of the modules is pursued. Many design alternatives are considered toward optimization. This stage also includes the design of any hardware algorithms to be mapped onto architectures. A behavioral-level description of the architecture is obtained and verified using extensive simulations, often with an iterative process. This stage is critical in obtaining an efficient circuit in the end and for simplifying the steps in some of the following stages. In the logic design stage, the architectural blocks are converted into corresponding gate-level logic designs, Boolean minimization is performed, and logic simulation is used to verify the design at this level. In some design flows, the circuit could be synthesized from the logic level by using gate-level libraries (this is referred to as logic synthesis). The logic design usually includes a conventioinal logic design approach and a nontraditional design, such as precharge logic. At this stage, gate delays are considered and timing diagrams are derived to verify the synchronization of the various logic modules. The next step is the circuit design stage, which essentially involves converting the logic design modules into a circuit representation. At this stage, the essential factors considered are clocking, switching speeds or delays, switching activity and power requirements, and other electrical characteristics (e.g., resistance, capacitance). The most complex step in VLSI design automation is the physical design, which includes floor planning, partitioning, placement, routing, layout, and compaction. This process converts the given circuit design or description into a physical layout that is a geometric representation of the entire circuit. Each step of the physical design by itself is complex and takes significant amounts of iterations and time. The various types

of transistors, the interconnecting wires, and contacts between different wires and transistors are represented as different geometric patterns consisting of many layers placed according to several design rules that govern a given fabrication technology and process. The floorplanning step involves higher level planning of the various components on the layout. The partitioning step converts the overall circuit into smaller blocks to help the other steps. It is usually impractical to synthesize the entire circuit in one step. Thus, logic partitioning is used to divide the given circuit into a smaller number of blocks, which can be individually synthesized and compacted. This step considers the size of the blocks, the number of blocks, and the interconnections between the blocks and yields a netlist for each block that can be used in the further design steps. During the next step, which is the placement of the blocks on the chip layout, the various blocks are placed such that the routing can be completed effectively and the blocks use minimum overall area, avoiding any white spaces. The placement task is iterative in that an initial placement is obtained first and evaluated for area minimization and effective routing possibility, and alternate arrangements are investigated until a good placement is obtained. The routing task completes the routing of the various interconnections, as specified by the netlists of the different blocks. The goal is to minimize the routing wire lengths and minimize the overall area needed for routing. The routing areas between the various blocks are referred to as channels or switchboxes. Initially, a global routing is performed in which a channel assignment is determined based on the routing requirements, and then a detailed routing step completes the actual point-to-point routing. The last step in the physical design is the compaction step, which tries to compact the layout in all directions to minimize the layout area. A compact layout leads to less wire lengths, lower capacitances, and more chip density since the chip area is used effectively. The compaction step is usually an interactive and iterative process in which the user can specify certain parameters and check if the compaction can be achieved. The goal of compaction, in general, is to achieve minimum layout area. The entire physical design process is iterative and is performed several times until an efficient layout for the given circuit is obtained. Once the layout is obtained, design verification needs to be done to ensure that the layout produced functions correctly and meets the specifications and requirements. In this stage, design rule checking is performed on the layout to make sure that the geometric placement and routing rules and the rules regarding the separation of the various layers, the dimensions of the transistors, and the width of the wires are followed correctly. Any design rule violations that occurred during the physical design steps are detected and removed. Then circuit extraction is performed to complete the functional verification of the layout. This step verifies

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the correctness of the layout produced by the physical design process. After layout verification, the circuit layout is ready to be submitted for fabrication, packaging, and testing. Usually, several dies are produced on a single wafer and the wafer is tested for faulty dies. The correct ones are diced out and packaged in the form of a pin grid array (PGA), dual in-line package (DIP), or any other packaging technology. The packaged chip is tested extensively for functionality, electrical and thermal characteristics, and performance. The process of designing and building an integrated circuit107 that meets the performance requirements and functions perfectly depends on the efficiency of the design automation tools. 8.4 Circuit Design

To create performance-optimized designs, two areas have to be addressed to achieve a prescribed behavior: (a) circuit or structural design, and (b) layout or physical design. While the layout design is discussed in a later section, this section focuses on the former. A logic circuit must function correctly and meet the timing requirements. There are several factors that can result in the incorrect functioning of a CMOS logic gate: (a) incorrect or insufficient power supplies, (b) noise on gate inputs, (c) faulty transistors, (d) faulty connections to transistors, (e) incorrect ratios in ratioed logic, and (f) charge sharing or incorrect clocking in dynamic gates. In any design, there are certain paths, called critical paths, that require attention to timing details since they determine the overall functional frequency. The critical paths are recognized and analyzed using timing analyzer tools and can be dealt with at four levels: 1. 2. 3. 4.

Architecture RTL/logic level Circuit level Layout level

Designing an efficient overall functional architecture helps to achieve good performance. To design an efficient architecture, it is important to understand the characteristics of the algorithm being implemented as the architecture. At the register transfer logic (RTL)/logic level, pipelining, the type of gates, and the fan-in and the fan-out of the gates are to be considered. Fan-in is the number of inputs to a logic gate, and fanout is the number of gate inputs that the output of a logic gate drives. Logic synthesis tools can be used to achieve the transformation of the RTL level. From the logic level, the circuit level can be designed to optimize a critical speed path. This is achieved by using different styles of CMOS logic, as explained later in this section. Finally, the speed of a set of logic can be affected by rearranging the physical layout. The following techniques can be used for specific design constraints.

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The various CMOS logic structures that can be used to implement circuit designs are as follows: 1. CMOS Complementary Logic. The CMOS complementary logic gates are designed as ratioless circuits. In these circuits, the output voltage is not a fraction of the Vdd (supply), and the gates are sized to meet the required electrical characteristics of the circuits. The gate consists of two blocks, and n block and a p block, that determine the function of the gate. The p block is a dual of the n block. Thus, an n-input gate will consist of 2n transistors. 2. Pseudo-NMOS Logic. In this logic, the load device is a single p transistor with the gate (103,108) . This is equivalent to connected to Vdd replacing the depletion NMOS load in a conventional NMOS gate by a p device. The design of this style of gate109,110 involves ratioed transistor sizes to ensure proper operation and is shown in Fig. 115. The static power dissipation that occurs whenever the pull-down chain is turned on is a major drawback of this logic style. 3. Dynamic CMOS Logic. In the dynamic CMOS logic style, an n-transistor logic structure’s output node is precharged to Vdd by a p transistor and conditionally discharged by an n transistor connected to Vss .103 The input capacitance of the gate is the same as the pseudo-NMOS gate. Here, the pull-up time is improved by virtue of the active switch, but the pull-down time is increased due to the ground. The disadvantage of this logic structure is that the inputs can only change during the precharge phase and must be stable during the evaluate portion of the cycle. Figure 116 depicts this logic style.

z a c b d

z = a⋅b + c⋅d Fig. 115 Pseudo-NMOS logic.

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Clk

–Clk

Clk to p block s

z Input s

n-logic block

Inputs

Clock

n-logic block

p-logi c block

Other p block s

n-logic block

Other n block s

Fig. 118 NP domino logic. Fig. 116 Dynamic CMOS logic.

4. Clocked CMOS Logic. To build CMOS logic gates with low-power dissipation,111 this logic structure was proposed. The reduced dynamic power dissipation is realized due to the metal gate CMOS layout considerations. The gates have larger rise and fall times because of the series clocking transistors, but the capacitance is similar to the CMOS complementary gates. This is a recommended strategy for “hot electron” effects because it places an additional n transistor in series with the logic transistors.112 5. CMOS Domino Logic. This is a modification of the clocked CMOS logic, in which a single clock is used to precharge and evaluate a cascaded set of dynamic logic blocks. This involves incorporating a static CMOS inverter into each logic gate,113 as shown in Fig. 117. During precharge, the output node is charged high and hence the output of the buffer is low. The transistors in the subsequent logic blocks

Weak p device Outputs

Inputs

n-logic block

Clock

Fig. 117 CMOS domino logic.

will be turned off since they are fed by the buffer. When the gate is evaluated, the output will conditionally go low (1–0), causing the buffer to conditionally go high (0–1). Hence, in a cascaded set of logic blocks, each state evaluates and causes the next stage to evaluate, provided the entire sequence can be evaluated in one clock cycle. Therefore, a single clock is used to precharge and evaluate all logic gates in a block. The disadvantages of this logic are that (a) every gate needs to be buffered, and (b) only noninverting structures can be used. 6. NP Domino Logic (Zipper CMOS). This is a further refinement of the domino CMOS.114 – 116 Here, the domino buffer is removed, and the cascading of logic blocks is achieved by alternately composed p and n blocks, as is shown in Fig. 118. When the clock is low, all the n-block stages are precharged high while all the p-block stages are precharged low. Some of the advantages of the dynamic logic styles include (a) smaller area than fully static gates, (b) smaller parasitic capacitances, and (c) glitchfree operation if designed carefully. 7. Cascade Voltage Switch Logic (CVSL). The CVSL is a differential style of logic requiring both true and complement signals to be routed to gates.117 Two complementary NMOS structures are constructed and then connected to a pair of cross-coupled p pull-up transistors. The gates here function similarly to the domino logic, but the advantage of this style is the ability to generate any logic expression involving both inverting and noninverting structures. Figure 119 gives a sketch of the CVSL logic style. 8. Pass Transistor Logic. Pass transistor logic is popular in NMOS-rich circuits. Formal methods for deriving pass transistor logic for NMOS are presented in Ref. 118. Here, a set of control signals and a set of pass signals are applied to the gates and sources of the n transistor,

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Output

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Clock

Output

n MOS combinational network

Differential inputs

categorized according to the way the delays are modeled in the circuit: (a) unit-delay simulators, in which each component is assumed to have a delay of one time unit, and (b) variable-delay simulators, which allow components to have arbitrary delays. While the former helps in simulating the functionality of the circuit, the latter allows for more accurate modeling of the fastchanging nodes. The timing is normally specified in terms of an inertial delay and a load-dependent delay, as follows: Tgate = Tintrinsic + Cload × Tload where Tgate Tintrinsic Cload Tload

Clock

Fig. 119

= = = =

delay of the gate intrinsic gate delay actual load in some units (pF) delay per load in some units (ns/pF)

Cascade voltage switch logic.

correspondingly. From these signals, the truth table for any logic equation can be realized. 9. Source Follower Pull-Up Logic (SFPL). This is similar to the pseudo-NMOS gate except that the pull-up is controlled by the inputs.119 In turn, this leads to the use of smaller pull-down circuits. The SFPL gate style reduces the selfloading of the output and improves the speed of the gate. Therefore, it shows a marked advantage in high fan-in gates. Using the various design styles, any circuit design can be built in a hierarchical fashion. The basic gates are first built, from which functional blocks like a multiplexer or an adder circuit can be realized. From these basic blocks, more complex circuits can be constructed. Once a design for a specific application has been designed, the functionality of the circuit needs to be verified. Also, other constraints, like the timing and electrical characteristics, have to be studied before the design can be manufactured. The techniques and tools to achieve this are the subject of the next section. 8.5 Simulation Simulation is required to verify if a design works the way it should. Simulation can be performed at various levels of abstraction. A circuit can be simulated at the logic level, the switch level, or with reference to the timing. Simulation is a critical procedure before committing the design to silicon. The simulators themselves are available in a wide variety of types.120 Logic-Level Simulation Logic-level simulation occurs at the highest level of abstraction. It uses primitive models of NOT, OR, AND, NOR, and NAMD gates. Virtually all digital logic simulators are event driven (i.e., a component is evaluated based on when an event occurs on its inputs). Logic simulators are

Earlier, logic simulators used preprogrammed models for the gates, which forced the user to describe the system in terms of these models. In modern simulators, programming primitives are provided that allow the user to write models for the components. The two most popular digital simulation systems in use today are VHDL and Verilog. Circuit-Level Simulation The most detailed and accurate simulation technique is referred to as circuit analysis. Circuit analysis simulators operate at the circuit level. Simulation programs typically solve a complex set of matrix equations that relate the circuit voltages, currents, and resistances. They provide accurate results but require long simulation times. If N is the number of nonlinear devices in the circuit, then the simulation time is typically proportional to N m , where m is between 1 and 2. Simulation programs are useful in verifying small circuits in detail but are unrealistic for verifying complex VLSI designs. They are based on transistor models and hence should not be assumed to predict accurately the performance of designs. The basic sources of error include (a) inaccuracies in the MOS model parameters, (b) an inappropriate MOS model, and (c) inaccuracies in parasitic capacitances and resistances. The circuit analysis programs widely used are the SPICE program, developed at the University of California at Berekely,121 and the ASTAP program from IBM.122 HSPICE 123 is the commercial variant of these programs. The SPICE program provides various levels of modeling. The simple models are optimized for speed, while the complex ones are used to get accurate solutions. As the feature size of the processes is reduced, the models used for the transistors are no longer valid and hence the simulators cannot predict the performance accurately unless new models are used. Switch-Level Simulation Switch-level simulation is simulation performed at the lowest level of abstraction. These simulators model transistors as switches

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to merge the logic-level and circuit-level simulation techniques. Although logic-level simulators also model transistors as switches, the unidirectional logic gate model cannot model charge sharing, which is a result of the bidirectionality of the MOS transistor. Hence, we assume that all wires have capacitance since we need to locate charge-sharing bugs. RSIM124 is an example of a switch-level simulator with timing. In RSIM, CMOS gates are modeled as either pull-up or pull-down structures, for which the program calculates a resistance to power or ground. The output capacitance of the gate is used with the resistance to predict the rise and the fall times of a gate. Timing Simulators Timing simulators allow simple nonmatrix calculations to be employed to solve for circuit behavior. This involves making approximations about the circuit, and hence accuracy is less than that of simulators like SPICE. The advantage is the execution time, which is over two orders of magnitude less than for SPICE. Timing simulator implementations typically use MOS-model equations or table look-up methods. Examples of these simulators are in Ref. 125. Mixed-Mode Simulators Mixed-mode simulators are available commercially today and merge the aforementioned different simulation techniques. Each circuit block can be simulated in the appropriate mode. The results of the simulation analysis are fed back to the design stage, where the design is tuned to incorporate the deviations. Once the circuit is perfected and the simulation results are satisfactory, the design can be fabricated. To do this, we need to generate a geometric layout of the transistors and the electrical connections between them. This has been a subject of intense research over the last decade and continues to be so. The following section introduces this problem and presents some of the well-known techniques for solving it. 8.6 Layout The layout design is considered a prescription for preparing the photomasks used in the fabrication of ICs.103 There is a set of rules, called the design rules, used for the layout; these serve as the link between the circuit and the process engineer. The physical design engineer, in addition to knowledge of the components and the rules of the layout, needs strategies to fit the layouts together with other circuits and provide good electrical properties. The main objective is to obtain circuits with optimum yield in as small an area as possible without sacrificing reliability. The starting point for the layout is a circuit schematic. Figure 106 depicts the schematic symbols for an n-type and p-type transistor. The circuit schematic is treated as a specification for which we must implement the transistors and connections between them in the layout. The circuit schematic of an inverter is shown in Fig. 108. We need to generate the exact layout of the transistors of this schematic, which can then be used to

Vdd In Vss Fig. 120

Out

Metal 2 Metal 1 Poly n diff p diff

Stick diagram of inverter.

build the photomask for the fabrication of the inverter. Generating a complete layout in terms of rectangles for a complex system can be overwhelming, although at some point we need to generate it. Hence designers use an abstraction between the traditional transistor schematic and the full layout to help organize the layout for complex systems. This abstraction is called a stick diagram. Figure 120 shows the stick diagram for the inverter schematic. As can be seen, the stick diagram represents the rectangles in the layout as lines, which represent wires and component symbols. Stick diagrams are not exact models of the layouts but let us evaluate a candidate design with relatively little effort. Area and aspect ratio are difficult to estimate from stick diagrams. Design Rules Design rules for a layout126 specify to the designer certain geometric constraints on the layout artwork so that the patterns on the processed wafer will preserve the topology and geometry of the designs. These help to prevent separate, isolated features from accidentally short circuiting, or thin features from opening, or contacts from slipping outside the area to be contacted. They represent a tolerance that ensures very high probability of correct fabrication and subsequent operation of the IC. The design rules address two issues primarily:

1. The geometrical reproduction of features that can be reproduced by the mask-making and lithographical process 2. The interaction among the different layers Several approaches can be used to descibe the design rules. These include the micron rules, stated at some micron resolution, and the lambda (λ)-based rules. The former are given as a list of minimum feature sizes and spacings for all masks in a process, which is the usual style for the industry. Mead– Conway127 popularized the λ-based approach, where λ is process dependent and is defined as the maximum distance by which a geometrical feature on any one layer can stray from another feature. The advantage of the λ-based approach is that by defining λ properly the design itself can be made independent of both the process and fabrication house, and the design can be rescaled. The goal is to devise rules that are simple, constant in time, applicable to many processes,

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Metal 1 -pdiff via p-type transistor Metal 1 Tub tie VDD ntub Poly

Metal 1 a

a′

Metal 1-poly via Metal 1-ndiff via n-type transisto r

ptub Metal 1

Tub tie

Fig. 121

Vss

Transistor layout of inverter.

standardized among many institutions, and have a small number of exceptions for specific processes. Figure 121 gives the layout of the inverter, with the design rules specified. To design and verify layouts, different CAD tools can be used. The most important of these are the layout editors,128 design rule checkers, and circuit extractors. The editor is an interactive graphic program that allows us to create and delete layout elements. Most editors work on hierarchical layouts, but some editors, like Berekely’s Magic tool,129 work on a symbolic layout. The latter include somewhat more detail than the stick diagram but are still more abstract than the pure layout. The design rule checker, or DRC, programs look for design rule violations in the layouts. Magic has an online design rule checking. The circuit extractor is an extension of the DRC programs. While the DRC must identify transistors and vias to ensure proper checks, the circuit extractor performs a complete job of component and wire extraction. It produces a netlist, which lists the transistors in the layouts and the electrical nets that connect their terminals. Physical Design From the circuit design of a certain application and the design rules of a specific process, the physical design problem is to generate a geometric layout of the transistors of the circuit design conforming to the specified design rules. From this layout, photomasks can be generated that will be used in the fabrication process. To achieve this, the different modules of the design need to be placed first and then electrical connections between them realized through the metal layers. For instance, a two-layer

metallization would allow the designer to lay out metal both vertically and horizontally on the floorplan. Whenever the wire changes direction, a via can be used to connect the two metal layers. Due to the complexity of this problem, most authors treat module placement and the routing between modules as two separate problems, although they are related critically. Also, in former days, when designs were less complex, design was done by hand. Now we require sophisticated tools for this process. Placement. Placement is the task of placing modules adjacent to each other to minimize area or cycle time. The literature consists of a number of different placement, algorithms that have been proposed.130 – 133 Most algorithms partition the problem into smaller parts and then combine them, or start with a random placement solution and then refine it to reach the optimal. The modules are usually considered as rectangular boxes with specified dimensions. The algorithms then use different approaches to fit these boxes in a minimal area or to optimize them to certain other constraints. For instance, consider a certain number of modules with specific dimensions and a given area in which to fit them. This is similar to the bin-packing algorithm. After the placement step, the different modules are placed in an optimal fashion and the electrical connections between them need to be realized. Routing. Once the modules have been placed, we need to create space for the electrical connections between them. To keep the area of the floorplan minimal, the first consideration is to determine the shortest path between nodes, although a cost-based approach may also be used. The cost is defined to include an estimate of the congestion, number of available wire tracks in a local area, individual or overall wire length, and so on. Since the problem is a complex one, the strategy is to split the problem into global or loose routing and local or detailed routing. Global routing is a preliminary step, in which each net is assigned to a particular routing area, and the goal is to make 100% assignment of nets to routing regions while minimizing the total wire length. Detailed routing then determines the exact route for each net within the global route. There are a number of approaches to both of these problems. Global Routing. Global routing134 is performed using a wire-length criterion because all timing critical nets must be routed with minimum wire length. The routing area itself can be divided into disjoint rectangular areas, which can be classified by their topology. A two-sided channel is a rectangular routing area with no obstruction inside and with pins on two parallel sides. A switch box is a rectangular routing area with no obstructions and signals entering and leaving through all four sides.135 The focus in this problem is only to create space between the modules for all the nets and not to determine the exact route. The algorithms

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proceed by routing one net at a time, choosing the shortest possible path. Since there is a lot of dependency among the nets, different heuristics are used to generate the least possible routing space in which to route the nets. Once space is created for all the nets, the exact route of each net can be determined. Detailed Routing. Detailed routing is usually done by either a maze-search or a line-search algorithm. The maze-running algorithm136,137 proposed by Lee–Moore finds the shortest path between any two points. For this, the layout is divided into a grid of nodes, in which each node is weighted by its distance from the source of the wire to be routed. The route that requires the smallest number of squares is then chosen. If a solution exists, this algorithm will find it, but an excessive amount of memory is required to achieve this. In the linesearch algorithm, vertical and horizontal lines are drawn from the source and the target, followed by horizontal or vertical lines that intersect the original lines. This is repeated until the source and target meet. There are also a number of other heuristic algorithms that exploit different characteristics of the design to generate optimal routing solutions. Genetic algorithms and simulated annealing approaches to this problem have gained importance in recent years. An introduction to the various algorithms that have been proposed for layouts can be found in Ref. 138. Once the layout has been determined and the photomasks made, the circuit can go to the fabrication plant for processing. 8.7 Fabrication

The section describes the approach used in building integrated circuits on monolithic pieces of silicon. The process involves the fabrication of successive layers of insulating, conducting, and semiconducting materials, which have to be patterned to perform specific functions. The fabrication therefore must be executed in a specific sequence, which constitutes an IC process flow. The manufacturing process itself is a complex interrelationship of chemistry, physics, material science, thermodynamics, and statistics. Semiconducting materials, as the name suggests, are neither good conductors nor good insulators. While there are many semiconducting elements, silicon is primarily chosen for manufacturing ICs because it exhibits few useful properties. Silicon devices can be built with a maximum operating temperature of about 150◦ C due to the smaller leakage currents as a result of the large bandgap of silicon (1.1 eV). IC planar processing requires the capability to fabricate a passivation layer on the semiconductor surface. The oxide of silicon, SiO2 , which could act as such a layer, is easy to form and is chemically stable. The controlled addition of specific impurities makes it possible to alter the characteristics of silicon. For these reasons, silicon is almost exclusively used for fabricating microelectronic components.

Silicon Material Technology Beach sand is first refined to obtain semiconductor-grade silicon. This is then reduced to obtain electronic-grade polysilicon in the form of long, slender rods. Single-crystal silicon is grown from this by the Czochralski (CZ) or floatzone (FZ) methods. In CZ growth, single-crystal ingots are pulled from molten silicon contained in a crucible. For VLSI applications, CZ silicon is preferred because it can better withstand thermal stresses139 and offers an internal gettering mechanism than can remove unwanted impurities from the device structures on wafer surfaces.140 FZ crystals are grown without any contact to a container or crucible and hence can attain higher purity and resistivity than CZ silicon. Most high-voltage, high-power devices are fabricated on FZ silicon. The single-crystal ingot is then subjected to a complex sequence of shaping and polishing, known as wafering, to produce starting material suitable for fabricating semiconductor devices. This involves the following steps:

1. The single-crystal ingot undergoes routine evaluation of resistivity, impurity content, crystal perfection size, and weight. 2. Since ingots are not perfectly round, they are shaped to the desired form and dimension. 3. The ingots are then sawed to produce silicon slices. The operation defines the surface orientation, thickness, taper, and bow of the slice. 4. To bring all the slices to within the specified thickness tolerance, lapping and grinding steps are employed. 5. The edges of the slices are then rounded to reduce substantially the incidence of chipping during normal wafer handling. 6. A chemical-mechanical polishing141 step is then used to produce the highly reflective and scratchand damage-free surface on one side of the wafer. 7. Most VLSI process technologies also require an epitaxial layer, which is grown by a chemical vapor deposition process. The most obvious trend in silicon material technology is the increasing size of the silicon wafers. The use of these larger-diameter wafers presents major challengers to semiconductor manufacturers. Several procedures have been investigated to increase axial impurity uniformity. These include the use of double crucibles, continuous liquid feed (CLF) systems,142 magnetic Czochralski growth (MCZ),142,143 and controlled evaporation from the melt. Epitaxial Layer The epitaxial growth process is a means of depositing a single-crystal film with the same crystal orientation as the underlying substrate. This can be achieved from the vapor phase, liquid phase, or solid phase. Vapor-phase epitaxy has the

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widest acceptance in silicon processing, since excellent control of the impurity concentration and crystalline perfection can be achieved. Epitaxial processes are used for the fabrication of advanced CMOS VLSI circuits because epitaxial processes minimize latch-up effects. Also in the epitaxial layer, doping concentration can be accurately controlled, and the layer can be made oxygen and carbon free. Epitaxial deposition is a chemical vapor deposition process.144 The four major chemical sources of silicon used commercially for this deposition are (a) silicon tetrachloride (SiCl4 ), (b) trichlorosilane (SiHCl3 ), (c) dichlorosilane (SiH2 Cl2 ), and (d) silane (SiH4 ). Depending on particular deposition conditions and film requirements, one of these sources can be used. Doping Silicon The active circuit elements of the IC are formed within the silicon substrate. To construct these elements, we need to create localized n-type and p-type regions by adding the appropriate dopant atoms. The process of introducing controlled amounts of impurities into the lattice of the monocrystalline silicon is known as doping. Dopants can be introduced selectively into the silicon using two techniques: diffusion and ion implantation. Diffusion. The process by which a species moves as a result of the presence of a chemical gradient is referred to as diffusion. Diffusion is a time- and temperature-dependent process. To achieve maximum control, most diffusions are performed in two steps. The first step is predeposition,145 which takes place at a furnace temperature and controls the amount of impurity that is introduced. The second step, the drivein step,145 controls the desired depth of diffusion. Predeposition. In predisposition, the impurity atoms are made available at the surface of the wafer. The atoms of the desired element in the form of a solution of controlled viscosity can be spun on the wafer, in the same manner as the photoresist. For these spin-on dopants, the viscosity and the spin rate are used to control the desired dopant film thickness. The wafer is then subjected to a selected high temperature to complete the predeposition diffusion. The impurity atoms can also be made available by employing a low-temperature chemical vapor deposition process in which the dopant is introduced as a gaseous compound—usually in the presence of nitrogen as a diluent. The oxygen concentration must be carefully controlled in this operation to prevent oxidation of the silicon surface of the wafer. Drive-In. After predeposition the dopant wafer is subjected to an elevated temperature. During this step, the atoms further diffuse into the silicon crystal lattice. The rate of diffusion is controlled by the temperature employed. The concentration of the dopant atoms is maximum at the wafer surface and reduces as the silicon substrate is penetrated further. As the atoms migrate

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during the diffusion, this concentration changes. Hence a specific dopant profile can be achieved by controlling the diffusion process. The dopant drive-in is usually performed in an oxidizing temperature to grow a protective layer of SiO2 over the newly diffused area. Ion Implantation. Ion implantation is a process in which energetic, charged atoms or molecules are directly introduced into the substrate. Ion implantation146,147 is superior to the chemical doping methods discussed previously. The most important advantage of this process is its ability to control more precisely the number of implanted dopant atoms into substrates. Using this method, the lateral diffusion is reduced considerably compared to the chemical doping methods. Ion implantation is a low-temperature process, and the parameters that control the ion implantation are amenable to automatic control. After this process the wafer is subjected to annealing to activate the dopant electrically. There are some limitations to this process. Since the wafer is bombarded with dopant atoms, the material structure of the target is damaged. The throughput is typically lower than diffusion doping process. Additionally, the equipment used causes safety hazards to operating personnel. Photolithography Photolithography is the most critical step in the fabrication sequence. It determines the minimum feature size that can be realized on silicon and is a photoengraving process that accurately transfers the circuit patterns to the wafer. Lithography148,149 involves the patterning of metals, dielectrics, and semiconductors. The photoresist material is first spin coated on the wafer substrate. It performs two important functions: (a) precise pattern formation and (b) protection of the substrate during etch. The most important property of the photoresist is that its solubility in certain solvents is greatly affected by exposure to ultraviolet radiation. The resist layer is then exposed to ultraviolet light. Patterns can be transferred to the wafer using either positive or negative masking techniques. The required pattern is formed when the wafer undergoes the development step. After development, the undesired material is removed by wet or dry etching. Resolution of the lithography process is important to this process step. It specifies the ability to print minimum size images consistently under conditions of reasonable manufacturing variation. Therefore, lithographic processes with submicron resolution must be available to build devices with submicron features. The resolution is limited by a number of factors, including (a) hardware, (b) optical properties of the resist material, and (c) process characteristics.150 Most IC processes require 5–10 patterns. Each one of them needs to be aligned precisely with those already on the wafer. Typically, the alignment distance between two patterns is less than 0.2 µm across the entire area of the wafer. The initial alignment is made with respect to the crystal lattice structure of the

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wafer, and subsequent patterns are aligned with the existing ones. Earlier, mask alignment was done using contact printing,151,152 in which the mask is held just off the wafer and visually aligned. The mask is then pressed into contact with the wafer and impinged with ultraviolet light. There is a variation of this technique called proximity printing, in which the mask is held slightly above the wafer during exposure. Hard contact printing was preferred because it reduced the diffraction of light, but it led to a number of yield and production problems. So the projection alignment and exposure system was developed, in which the mask and wafer never touch and an optical system projects and aligns the mask onto the wafer. Since there is no damage to the mask or photoresist, the mask life is virtually unlimited. VLSI devices use projection alignment as the standard production method. Junction Isolation When fabricating silicon ICs, it must be possible to isolate the devices from one another. These devices can then be connected through specific electrical paths to obtain the desired circuit configuration. From this perspective, the isolation technology is one of the critical aspects of IC fabrication. For different IC types, like NMOS, CMOS, and bipolar, a variety of techniques have been developed for device isolation. The most important technique developed was termed LOCOS, for LOCal Oxidation of Silicon. This involves the formation of semirecessed oxide in the nonactive or field areas of the substrate. With the advent of submicron-size device geometries, alternative approaches for isolation were needed. Modified LOCOS processes, trench isolation, and selective epitaxial isolation were among the newer approaches adopted. LOCOS. To isolate MOS transistors, it is necessary to prevent the formation of channels in the field regions. This implies that a large value of VT is required in the field region, in practice about 3–4 V above the supply voltage. Two ways to increase the field voltage are to increase the field oxide thickness and raise the doping beneath the field oxide. Thicker field oxide regions cause high enough threshold voltages but unfortunately lead to step coverage problems, and hence thinner field oxide regions are preferred. Therefore, the doping under the field oxide region is increased to realize higher threshold voltages. Nevertheless, the field oxide is made 7–10 times thicker than the gate oxide. Following this, in the channel-stop implant step, ion implantation is used to increase the doping under the field oxide. Until about 1970, the thick field oxide was grown using the grow-oxide-and-etch approach in which the oxide is grown over the entire wafer and then etched over the active regions. Two disadvantages of this approach prevented it from being used for VLSI applications: (a) Field-oxide steps have sharp upper corners, which poses a problem to the subsequent metallization steps: and (b) channel-stop implant must be performed before

the oxide is grown. In another approach, the oxide is selectively grown over the desired field regions. This process was introduced by Appels and Kooi in 1970153 and is widely used in the industry. This process is performed by preventing the oxidation of the active regions by covering them with a thin layer of silicon nitride. After etching the silicon nitride layer, the channel-stop dopant can be implanted selectively. The process has a number of limitations for submicron devices. The most important of these is the formation of the “bird’s beak,” which is a lateral extension of the field oxide into the active areas of the device. The LOCOS bird’s beak creates other problems as junctions become shallower in CMOS ICs. The LOCOS process was therefore modified in several ways to overcome these limitations: (a) etched-back LOCOS, (b) polybuffered LOCOS, and (c) sealedinterface local oxidation (SILO).154 Non-LOCOS Isolation Technologies. There have been non-LOCOS isolation technologies for VLSI and ultra-large-scale integration (ULSI) applications. The most prominent of these is trench isolation technology. Trench technologies are classified into three categories: (a) shallow trenches (3 µm deep, 1000 V) and high operating temperature (>200◦ C) due to the conductivity modulation. SIT Although the SiC SIT (static induction transistor) has a structure resembling that of the UMOSFET, as shown in Fig. 190,231 the operation mechanism is significantly different. The SiC SIT is a vertical device with an ohmic source contact on the top and an ohmic drain contact on the back of the wafer. Between the

Source Gate n+

n+

pp N− epitaxial layer

N+ substrate Drain Fig. 189 Structure of planar SiC MOSFET. An N− epitaxial drift layer is first grown on the N+ substrate. The DMOS structure is formed by using multiple high-energy boron (p region) and nitrogen implants (n region).

Source ohmic contact N+

Schottky gate

Schottky gate

_

N drift layer

N+ substrate Drain ohmic contact Fig. 190 Cross-sectional view of SiC SIT. An N− epitaxial drift layer is grown on the N+ substrate, and then an N+ layer is grown. Trenches are etched to define the channel region, and Schottky gate contacts are formed in the bottom and along the sidewalls of the trench.

N + source and N + drain regions is an N − epitaxial drift layer whose doping is one of the factors that determines the device breakdown voltage and pinchoff voltage. Trenches are etched to define the channel region, and Schottky gate contacts are formed in the bottom and along the side walls of the trench. Majority carriers flow from the source contact to the drain contact through the N-type channel region. By applying a negative voltage to the gate contact, the current flow can be modulated and even decreased to zero when depletion regions under each gate contact meet in the middle of the channel. The SiC SIT is ideally suited to high-power microwave devices owing to the remarkable transport properties, very high breakdown field strength, and thermal conductivity of SiC. The SiC SIT is being developed as a discrete power microwave transistor for operation at frequencies up to S-band. RF MESFET The cross-sectional view of an RF SiC MESFET is shown in Fig. 191.232 This device is a lateral device with both source and drain contacts on the top surface of the wafer. The MESFET epitaxial structure consists of an undoped P -buffer layer, Ntype channel layer, and N + contact layer. The majority of carriers flow laterally from source to drain, confined to the N-type channel by the P − buffer layer and controlled by the Schottky gate electrode. For RF Si LDMOS, GaAs MESFET, and SiC MESFET, the device parameters that are important in different power densities are low field electron mobility, breakdown electric field, and electron saturation velocity. At a doping density of 1 × 1017 cm−3 the electron mobility of 4H–SiC is 560 cm2 /V · s, which is slightly lower than that of Si (800 cm2 /V · s) and significantly lower than that of GaAs (4900 cm2 /V · s). On the

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and ion-implanted N + source and drain contact regions are used in place of the N + epitaxial region.

Ohmic contact

REFERENCES

Schottky gate Source

Drain

N+ Epi

N+ Epi N-type channel _

P buffer epitaxial layer N-type substrate Fig. 191 Cross-sectional view of RF SiC MESFET. In the SiC RF MESFET structure, all semiconductor layers are epitaxially grown. The RIE is used to define the Schottky gate.

other hand, the breakdown electric field of 4H–SiC is about 10 times that of Si and GaAs, and the saturated drift velocity is 2 times that of Si and GaAs. Consequently, at low voltages, GaAs MESFETs, which have the highest electron mobility, have the highest power density. The higher power density of SiC MESFETs is only achieved at drain voltages higher than those normally used with either Si or GaAs devices. RF JFET High-frequency SiC JFETs are of interest for high-temperature RF applications because a much lower gate leakage current can be obtained with a P –N junction at high temperature than with a Schottky gate.233 The cross-section of a SiC RF JFET (shown in Fig. 192) is similar to that of the RF MESFET, except a P + . SiC epitaxial region with an ohmic contact on top is used in place of a Schottky contact,

Ohmic contact

Gate Source N

P + Epi

+

N epitaxial layer

Drain N+

P − epitaxial layer N-type substrate Fig. 192 Cross-sectional view of SiC JFET. In the SiC RF JFET structure, a p− epitaxial layer is grown on the N-type substrate, and then an N-type epitaxial layer and high concentration P+ epitaxial layer are grown. The P+ mesa is formed by using RIE technology and N+ source and drain regions are formed by using ion implantation.

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BIBLIOGRAPHY Abele, M. G., Structures of Permanent Magnets, Wiley, New York, 1993. “About Oscilloscope,” http://www.hobbyprojects.com/ oscilloscope tutorial.html. “Advances in Oscilloscope Technology,” LeCroy, White Paper, http://www.lecroy.com. Annaratone, M., Digital CMOS Circuit Design, Kluwer, Norwell, MA, 1986. Baecker, R. M., and Buxton, W. S. (Eds.), Readings in Human-Computer Interaction: A Multidisciplinary Approach, Morgan Kaufmann, San Mateo, CA, 1987. Balakrishnan, et al., R., “The Rockin’ Mouse: Integral 3D Manipulation on a Plane,” in CHI97 Conf. Proc., Atlanta, GA, ACM, 1997. Barfield, W., and Furness III, T. A. (Eds.), Virtual Environments and Advanced Interface Design, Oxford University Press, Oxford, 1995. Belove, C. (Ed.), Handbook of Modern Electronics and Electrical Engineering, Wiley, New York, 1986. Bentley, J. P., Principles of Measurement Systems, 2nd ed., Longman Scientific and Technical, Burnt Mill, UK, 1988. Bogart, T. F., Electronic Devices and Circuits, 3rd ed., Macmillan, New York, 1993. Brey, B. B., Microprocessors and Peripherals: Hardware, Software, Interfacing, and Applications, 2nd ed., Macmillan, New York, 1988. Chang, C. Y., and Sze, S. M., ULSI Technology, McGrawHill, New York, 1996. Dahl, P. F., Superconductivity, American Institute of Physics, New York, 1992. Dix, A., et al., Human-Computer Interaction, Prentice-Hall, Englewood Cliffs, NJ, 1993. Esposito, C., User Interfaces for Virtual Reality Systems, Tutorial Notes, CHI’96, Vancouver, British Columbia, Canada, 1996. Fink, D. G., and Christiansen, D. (Eds.), Electronics Engineers’ Handbook, McGraw-Hill, New York, 1982. Fishbane, P. M., Gasiorowicz, S., and Thornton, S. T., Physics for Scientists and Engineers, Prentice-Hall, Upper Saddle River, NJ, 1996.

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Gallagher, R. S., Computer Visualization, CRC Press, Boca Raton, FL, 1995. Giancoli, D. C., Physics Principles and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1991. Gieger, R. L., Allen, P. E., and Strader, N. R., VLSI Design Techniques for Analog and Digital Circuits, McGraw-Hill, New York, 1990. Glasser, L. A., and Dobberpuhl, D. W., The Design and Analysis of VLSI Circuits, Addison-Wesley, Reading, MA, 1985. Gopel, W., Hesse, J., and Zemel, J. N., Sensors—A Comprehensive Survey, WCH, Weinheim, Germany, 1989. Hartson, H. R., and Hix, D., Advances in Human-Computer Interaction, Vol. 4, Ablex, Norwood, NJ, 1993. Herbst, L. J., Integrated Circuit Engineering, Oxford University Press, London, 1996. Holmes-Siedle, A., and Adams, L., Handbook of Radiation Effects, Oxford University Press, New York, 1993. Interrante, L. V., Casper, L. A., and Ellis, A. B. (Eds.), Materials Chemistry, American Chemical Society, Washington, DC, 1995. Kang, S. M., and Leblebici, Y., CMOS Digital Integrated Circuits, McGraw-Hill, New York, 1996. Kaufaman, M., and Seidman, A. H. (Eds.), Handbook for Electronic Engineering Technicians, McGraw-Hill, New York, 1984. Metzger, D., Electronic Components, Instruments, and Troubleshooting, Prentice-Hall, Englewood Cliffs, NJ, 1981. Mukherjee, A., Introduction to nMOS and CMOS VLSI Systems Design, Prentice-Hall, Englewood Cliffs, NJ, 1986. Pucknell, D. A., and Eshraghian, K., Basic VLSI Design: Systems and Circuits, Prentice-Hall of Australia, Sydney, 1988.

Rosenstein, M., and Morris, P., Modern Electronic Devices: Circuit Design and Application, Reston Publishing Company, Reston, VA, 1985. Sadiku, M. N. O., Elements of Electromagnetics, Saunders College Publishing, Orlando, FL, 1994. Schroeter, J., Surviving the ASIC Experience, Prentice-Hall, Englewood Cliffs, NJ, 1992. Seymour, J., Electronic Devices and Components, Pitman Publishing, London, 1981. Sherwani, N., Algorithms for VLSI Physical Design Automation, Kluwer, Boston, 1993. Shoji, M., CMOS Digital Circuit Technology, Prentice-Hall, Englewood Cliffs, NJ, 1988. Smith, M. S., Application Specific Integrated Circuits, Addison-Wesley, Reading, MA, 1997. Solymar, L., and Walsh, D., Lectures on the Electrical Properties of Materials, 3rd ed., Oxford University Press, Oxford, 1984. Thomson, C. M., Fundamentals of Electronics, Prentice-Hall, Englewood Cliffs, NJ, 1979. Tocci, R. J., Digital Systems Principles and Applications, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 1991. Tompkins, W. J., and Webster, J. G., Interfacing Sensors to the IBM PC, Prentice-Hall, Englewood Cliffs, NJ, 1988. “User’s and Service Guide, 3000 Series Oscilloscopes,” Agilent Technologies, http://www.agilent.com. Wolf, S., and Smith, R. F. M., Student Reference Manual for Electronic Instrumentation Laboratories, 2nd ed, PrenticeHall, Upper Saddle River, NJ, 2004. Wolf, W., Modern VLSI Design: A System Approach, Prentice-Hall, Englewood Cliffs, NJ, 1994.

CHAPTER 18 LIGHT AND RADIATION M. Parker Givens Institute of Optics University of Rochester Rochester, New York

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6.6

Colorimetry

1138

1113

6.7

Color Mixing

1138

6.8

Tristimulus Values and Trichromatic Coefficients

1140

6.9

Trichromatic Coefficients for Nonmonochromatic Light

1141

1119

6.10

Color of an Orange Skin

1142

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6.11

Chromaticity Diagram as Aid to Color Mixing

1143

4 LIGHT SOURCES

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6.12

Dominant Wavelength and Purity

1144

5 LASERS

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6.13

Average Reflectance

1145

1134

6.14

Subtractive Color Mixing

1145

1134

6.15

Munsell System

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6.16

Photometric Units

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1 INTRODUCTION 2 GEOMETRIC OPTICS 2.1

Aberrations

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2.2

Chromatic Aberration

1118

2.3

Spherical Aberration

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3 PHYSICAL OPTICS 3.1

Holography

6 THE EYE AND VISION 6.1

Structure of the Eye

6.2

Adaptation of Eye to Light

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6.3

Scotopic Vision

1136

7

6.4

Photopic Vision

1136

REFERENCES

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6.5

Color Vision

1136

BIBLIOGRAPHY

1149

1 INTRODUCTION Radiation is the transfer of energy through space without requiring any intervening medium; for example, the energy reaching Earth from the sun is classified as radiation. The majority of this is in the form of electromagnetic waves, which have a wide range of frequencies. Of this, a relatively narrow frequency band between 4 × 1014 and 8 × 1014 Hz is capable of stimulating the visual system; this is light. Our attention will be directed primarily toward the visible part of the spectrum, but most of the principles are valid in other parts of the spectrum. Electromagnetic radiation propagates through empty space with velocity c, which is one of the fundamental

DETECTORS OR OPTICAL TRANSDUCERS 1147

constants of nature; its approximate value is 2.998 . . . × 108 m/s. This velocity is independent of frequency. In any material medium, the velocity of propagation v is less than c; the ratio c/v ≡ n is called the index of refraction of the medium. The velocity v (and therefore n) depends upon the frequency of the radiation; this variation of velocity (or index) with frequency is known as dispersion. The simplest wave to discuss is one in which some physical quantity varies sinusoidally with time at any point in space and this variation propagates with velocity v. Such a wave is represented by an equation of the form


b = A sin

2π (x ± vt) λ

(1)

1111

1112


where b represents the value of the quantity at position x and at time t and A is the maximum value of b and is called the amplitude of the wave. This equation represents a plane wave, that is, the quantity b is constant over a plane surface perpendicular to the x axis. The minus sign gives a wave propagating in the positive x direction; the plus sign gives a wave propagating in the negative x direction. The wavelength λ represents the smallest, nonzero distance for which b(x + λ) = b(x) for all x; alternately, λ may be defined as the distance between adjacent crests of the wave. See Fig. 1, which is a plot of b as a function of x for some fixed t. Equation (1) is meaningful for all values of x and t; in this sense, it represents a wave of infinite extent in time and space. We can also define the period T of the wave as the time required to execute one cycle, or the smallest, nonzero, time for which b(t + T ) = b(t). Frequency ν is the reciprocal of the period; for period in seconds, the frequency unit is the hertz. Equation (1) is called a monochromatic wave since it contains only one frequency. These quantities are interrelated as follows: v = λν =

λ T

or

λ = vT

As the radiation passes from one medium to another, such as from glass to air, the frequency remains constant, but the velocity and wavelength change. No real wave extends indefinitely in time but must begin and end; real waves cannot be monochromatic in the strictest interpretation of the term. The methods of Fourier analysis enable us to construct a finite wave train as a sum of appropriately selected infinite wave trains of the proper phase. The amplitude as a function of time in the finite wave train and the amplitude as a function of frequency for the infinite components form a Fourier transform pair. Although the details in each case will depend upon the manner in which the wave builds up initially and dies away at the end, some “rule-of-thumb” statements are often helpful. These are: λ T λ ν ≈ ≈ ≈ ν t vt λ

where t is the duration of the wave train; ν is the frequency spread of the infinite, or monochromatic, components making up the wave train; ν, T , and λ are the average frequency, period, and wavelength of the finite wave. These are rule-of-thumb or orderof-magnitude statements. A wave for which ν ν is properly called quasi-monochromatic but is frequently called monochromatic. For many classical (i.e., prelaser) light sources, ν/ν ≈ 10−5 and the wave is, for most practical purposes, monochromatic. Sunlight, on the other hand, has a very broad spectral range (ν comparable to ν) and may be described equally well as a series of randomly spaced short pulses or a broad spectrum of randomly phased monochromatic waves. The two descriptions are equally valid and interchangeable. As already mentioned, light is electromagnetic radiation. The quantities described by Eq. (1) are electric and magnetic fields E and B. These two fields each obey Eq. (1) and are in phase with each other. They are perpendicular to each other in space and each is perpendicular to the direction of propagation. This is illustrated in Fig. 2, which shows E in the y direction and B in the z direction for a wave propagated in the positive x direction. Here, E, B, and v form a righthanded orthogonal system so that in order to have a wave propagated in the negative x direction, either E or B must be reversed. Within the constraint that it remain in a plane perpendicular to the direction of propagation, E may have any direction. Usually the direction of E changes in a random way, and the light is called unpolarized. If the direction of E remains constant, the light is called linearly polarized. Also, B is always perpendicular to E. Theoretical considerations indicate that in vacuum c = (µ0 ε0 )−1/2 , which experiment confirms. Theory also predicts that in a medium v = (µε)−1/2 or n = (µε/µ0 ε0 )1/2 . Here µ and ε are the permeability and permittivity of the medium; µ0 and ε0 are the corresponding quantities for vacuum. The prediction for the velocity in a real medium cannot be experimentally confirmed since µ and ε are frequency dependent and at optical frequencies the only available measurements are the measurements of n or v; there are no direct measurements of µ and ε. The velocity v in Eq. (1) is the phase velocity. For a finite wave train or pulse, the envelope of the pulse moves forward with the group velocity U. The value of U may be expressed in a variety of forms, including c λ dn c = 1+ U= (d/dν)(nν) n n dλ

Fig. 1 Simple harmonic wave of the form y = A sin(2π/λ) (x − vt). Plot is for fixed t = 0. Over time, disturbance moves to right.

(2)

For common transparent materials, dn/dλ < 0 and U < v. In nondispersive media (e.g., vacuum), U = v; λ is the wavelength in the medium. One common aspect of wave propagation, as observed with water waves, is the tendency of the

LIGHT AND RADIATION

1113

Fig. 2 Electromagnetic wave, where E, B, and v form right-handed orthogonal system as shown. Plot is for some fixed time; wave moves to right; Ey and Bz have maximum values for same value of x.

wave to spread into the shadow region behind barriers. This phenomenon is known as diffraction. Light also exhibits diffraction, but the effects are much smaller than for water waves because the wavelengths of light waves are so small (∼5 × 10−7 m). Diffraction effects are important if we attempt to pass light through openings only a few wavelengths wide or to focus the light into a very small spot. Otherwise, we may describe the light in terms of “rays” that represent the direction of energy flow and coincide with the direction of propagation. In a homogeneous isotropic medium, the rays are straight. That part of optics that may be treated by tracing rays is called geometric optics. 2 GEOMETRIC OPTICS If a ray of light strikes a boundary separating two homogeneous isotropic media such as air and glass (see Fig. 3), a simple wave calculation will show and experiment will confirm the following statements:

1. The incident ray will be partially reflected at the boundary and partially transmitted (refracted) into the second medium. 2. The incident ray, the reflected ray, the refracted ray, and the normal to the surface (erected at the point of incidence) are coplanar. 3. The angle of reflection θ is equal to the angle of incidence θ ; these angles are measured between the surface normal and the rays, as shown in Fig. 3. 4. The angle of refraction φ and the angle of incidence θ are related by the following equation, which is known as Snell’s law: v1 n2 sin θ = = sin φ v2 n1

(3)

where v1 and v2 are the velocities of propagation in medium 1 and medium 2 and n1 and n2 are the indices of refraction.

Fig. 3 Refraction at plane boundary separating two media with different indices of refraction. According to Snell’s law, n1 sin θ = n2 sin φ. In the special case for which θB satisfies condition tan θB = n2 /n1 , reflected light is linearly polarized.

These statements can be proven without postulating that the light wave is electromagnetic. However, one must recognize the electromagnetic nature of the wave in order to calculate the fraction of the light that is reflected or transmitted. The path of light is reversible; that is, light will travel from B to A along the same path BPA. It is a straightforward exercise in calculus to show that if A and B (of Fig. 3) are fixed points, one in each medium, and if P is an arbitrary point on the boundary, then the location of P that minimizes the propagation time from A to P to B is the same location of P for which Snell’s law is satisfied. Also, for A and C as fixed points in the same medium, the choice of P that produces a minimum of AP + P C is the same P for which θ = θ . These are examples of Fermat’s

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principle, which states that the path of an actual ray from one point to another is a path for which the transit time is stationary. By stationary we mean that the derivative of the transit time with respect to small changes in the path (such as small changes in the location of P ) must be zero. For a ray such as APB in Fig. 3, which passes through more than one medium, it is convenient to define the optical pathlength from A to B as n1 AP + passes through many media, n2 P B, or in case the ray the optical pathlength is i li ni , where li is the pathlength in medium i and ni is the index of medium i. The optical pathlength between two points A and B is the distance in vacuum that light could travel during the time required to propagate from A to B through the intervening media. Snell’s law and the law of reflection are sufficient to explain the image-forming properties of lenses and mirrors. We first turn our attention to lenses and to the case in which the same medium (air) is on both sides of the lens. The lens will have spherical surfaces and rotational symmetry about some line called the axis; distances are measured along and perpendicular to this axis. The following results are usually derived using small-angle approximations, sin α = α = tan α, and are called paraxial calculations. Consider first a ray that is parallel to the axis at a distance h above the axis (see Fig. 4). Upon passing through the lens, the ray will be refracted according to Snell’s law and cross the axis at the point F in the figure. For a good lens, the point F is independent of h; it is called the focal point of the lens. As the ray passes through F , it has a slope of u; in the figure, u is negative. The ratio h/(−u) is the focal length f . For a thin lens, f is the distance from the lens to F , where F is the second (or back) focal point. There is another point, F , in front

of the lens called the first (or front) focal point. Any ray that passes through F (with slope u) and strikes the lens will be refracted to be parallel to the axis (at height h). The front focal length, h/u ≡ f , will be equal to f provided there is the same medium on both sides of the lens. For a thin lens in air 1 1 1 1 = (n − 1) (4) = − f f r1 r2 where n is the index of refraction of the lens material and r1 and r2 are the radii of curvature of the first and second surfaces of the lens; r1 and r2 are considered positive (negative) if the center of curvature of the surface is to the right or downstream (left or upstream) relative to the surface; and 1/f is the power of the lens. In Fig. 4, r1 is positive and r2 is negative; for this lens, the focal length f is positive. This lens is convergent. Figure 5 shows the application of these definitions to a negative, or divergent, lens. In this case, r1 is negative and r2 is positive, making f negative. Notice that F , the second focal point, is to the left of the lens; the refracted ray does not pass through F but must be extended backward to intersect the axis (at F ). In Fig. 5b, the incident ray is headed toward F , the first focal point, but is refracted by the lens to be parallel to the axis. For both positive and negative lenses, a ray that crosses the axis at the center of the lens continues undeviated into the region beyond. This is called a chief ray. It follows from Eq. (4) that positive, or convergent, lenses are thicker on axis than at the edge, whereas negative, or divergent, lenses are thinner on axis than at the edge. They may have a variety of shapes, as illustrated in Fig. 6.

Fig. 4 Two focal points of positive lens. In (a) F is second (or back) focal point. In (b) F is first (or front) focal point. For lens shown, r1 (radius of curvature of front surface) is positive; r2 (radius of curvature of second surface) is negative.

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Fig. 5 Two focal points of negative lens. In (a) F, the second focal point, is in front of lens. In (b) F , first focal point, is behind lens. For lens shown, r1 is negative and r2 is positive.

image. The image is real since the rays actually arrive at point B. It is simple to calculate the image position by the equation xx = f 2 (5)

Fig. 6 Variety of positive lenses (upper group) and negative lenses (lower group).

where x is the distance from the object to the first focal point; it is taken as positive if (as shown in Fig. 7) the object-to-focal-point direction is the same as the direction of the light propagation. On the image side x is the distance from the second focal point to the image; it is positive in Fig. 7. From Eq. (5), we see that x and x always have the same sign. Since the product xx is constant, moving the object to the right (toward F ) moves the image to the right (away from F ). The lateral magnification m is given as m≡

Figure 7 shows a positive lens forming an image of point A at point B. All rays from A that pass through the lens converge to B, but only three are shown. It is assumed that the locations of F and F are known; rays 1 and 3 are drawn to satisfy the definitions of these points. Ray 2 passes undeviated through the center of the lens. The image is inverted and the lateral magnification m is defined as y /y or B B/A A; it is negative in the cases shown, indicating an inverted

f x y =− =− y x f

(6)

Another pair of equations may be used 1 1 1 + = p q f

and

m=−

q p

(7)

where p is the distance from the object to the lens and q is the distance from the lens to the image; they are considered positive if they are in the same

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Fig. 7 Image formation by positive lens, illustrating quantities that appear in Eqs. (5)– (7). Also shown are three rays easily used in graphical ray tracing.

direction as the light propagation. In Fig. 7, both p and q are positive. Equation (7) is very convenient for use with thin lenses. For thick lenses or lenses consisting of several elements, it is not obvious what point (or points) in the lens should be used for measuring p and q. By reversing the rays, A becomes the image of B, and A and B are said to be conjugate points. Figure 8 shows the corresponding situation for a negative lens. Here the first focal point F is to the right of the lens and the second focal point F is to the left of the lens. Ray 1 is parallel to the axis until it strikes the lens and is refracted along a line that appears to have come from F . Ray 3 is headed for F but is refracted to be parallel to the axis. Ray 2 passes straight through the center of the lens. These rays do not intersect anywhere to the right of the lens but if extended backward appear to have intersected at B. Only ray 2 actually passes through B. Point B is a virtual image of A (in contrast to the real image formed in Fig. 7). Equations (5) and (6) or Eq. (7) work for this case, but notice the following: f is negative; x is measured from A to F as before, but F is to the right of the lens; x is measured from F to B , but F is to the left of the lens; p is positive; Eq. (7) gives a negative value for q, indicating that B is to the left of the lens and therefore virtual; m is positive but less than 1, so the image is upright or erect and smaller than the object.

Equations (5)–(7) may be used to establish the information in Table 1. The focal length of a thin positive lens may be calculated from Eq. (4) if the curvatures and the index of the glass are known. It may also be measured in the laboratory by setting up on an optical bench an experiment similar to Fig. 7, measuring the appropriate distances and calculating f . A small luminous source, such as the filament of an unfrosted lightbulb, might serve as a suitable object. A ground glass screen is used to locate the image. Negative lenses cannot be measured in this way because the image in Fig. 8 is virtual and virtual images cannot be caught upon a screen. There are two ways around this problem. Two thin lenses of focal lengths f1 and f2 when placed in contact are equivalent to a single lens of focal length fc , given by the equation 1 1 1 = + fc f1 f2

(8)

If f1 is a negative lens under test, it may be combined with a positive lens of known focal length f2 . If f2 < |f1 |, then the combined focal length fc will be positive and can be measured by the experiment of Fig. 7; f1 is then calculated from Eq. (8). An alternate method is shown in Fig. 9. A positive lens is used to form a real image at A A; its position is determined and recorded by observing the image on

Fig. 8 Image formation by negative lens illustrating quantities from (5)–(7). Also shown are the rays easily used in graphical ray tracing.

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Table 1 Images Formed by Thin Lenses Lens

Object Position

Positive, or convex, f>0

Negative, or concave, fx>f x=f f >x>0 0 > x > −f

p=∞ ∞ > p > 2f p = 2f 2f > p > f f>p>0

x = 0 0 < x < f x = f f < x < ∞ −∞ < x < −f

q=f f < q < 2f q = 2f 2f < q < ∞ −∞ < q < 0

Real Real, inverted, |m| < 1 Real, inverted, |m| = 1 Real, inverted, |m| > 1 Virtual, erect, |m| > 1

x=∞ ∞ > x > −f

p=∞ ∞>p>0

x = 0 0 < x < −f

q = f = −|f| f L1 . The irradiance (incident power per unit area) onto S is E, given by E=

dφ = dA

L2 cos θ d

where the integration is over the solid angle subtended at S by the lens. This is the same irradiance that would be produced at S if the source S filled the exit pupil of the lens. The value of E depends upon the solid angle subtended. In photography, the solid angle is usually represented through the f number [see Eq. (26)] so that irradiance is proportional to (F # )−2 .

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Fig. 28 Neglecting losses in lens, L2 = n22 L1 /n21 . Assuming there is no diffusing screen at image position S , L2 remains the same to right of image; L2 cannot be greater than L1 if n2 = n1 .

If one were to place a small luminous source, such as the filament of a small lightbulb, at S and then place the pupil of his eye at S while looking back at the lens, he would see the exit pupil of the lens entirely filled with light. Only if the pupil of the eye is located at S will the exit pupil be entirely filled with light.

in the form of light and return to the ground state. The frequency and wavelength of the emitted light are given by the equation

5 LASERS In gases, the intermolecular spacing is generally so large that the molecules radiate independently of each other, and the spectrum produced is a line structure for monatomic molecules or a band structure of fine lines for polyatomic molecules. We comment briefly about the atomic (or monatomic) case. As an example, sodium vapor may be observed to emit light by introducing a little salt (NaCl) into a bunsen flame; alternatively, radiation from sodium vapor may be observed from the sodium arcs, which are widely used for highway lighting. In the heat of the bunsen flame, the salt dissociates into sodium and chlorine atoms. Most of these atoms are in their lowest energy (or “ground”) state. The atoms can, however, be put into more energetic (or “excited”) states; the energy of the excited states is specific, depending upon the atom. The number of atoms per unit volume, N1 , in the excited state and the number of atoms per unit volume, N0 , in the ground state will be in the ratio

This bundle of light energy is called a photon or a quantum. Since each atom (e.g., sodium, potassium, mercury, and hydrogen) is characterized by its own set of excited states, it is also characterized by its own set of spectral lines. In the case of the sodium arc lamp and other gaseous discharge lamps, the atomic excitations take place by inelastic collisions of the atoms with the moving electrons or ions that make up the electric current. Equation (45) is no longer an adequate description of the ratio N1 /N0 , but in most cases it remains true that N1 N0 ; that is, only a small fraction of the atoms are excited. If there are N1 excited atoms per unit volume, there will be a rate of spontaneous return to the ground state. That rate will be

N1 = e−E/kT N0

(45)

where E is the energy difference between an atom in the excited state and an atom in the ground state, T is the temperature (in Kelvin), and k is the Boltzmann constant. Clearly N1 < N0 . The atom will have several excited states, each with its own E, and there will be an equation similar to (45) for each state. The atoms in the more energetic states are said to be thermally excited. An excited atom may give up its excess energy

hν =

hc = E λ

dN1 = −A01 N1 dt

(46)

(47)

where A01 is a constant determined by the nature of the two states. Aside from the value of this constant, the rate of spontaneous return depends only upon N1 . Each atom that returns from the excited state to the ground state emits one quantum of light. If light of the resonant frequency [i.e., the frequency given by Eq. (46)] passes through the gas, some of the atoms in the ground state will absorb a quantum and be excited into the more energetic state. This process is called absorption, and the rate at which it takes place is given by dN0 = −B10 N0 L dt

(48)

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where L is the radiance at the resonant frequency and B10 is a constant determined by the nature of the two states. The rate at which atoms are excited (or quanta absorbed) is proportional to L. The atoms in the excited state are also affected by the light, causing some of them to emit a quantum and return to the ground state. This process is called stimulated emission. The rate at which it takes place is dN1 = −B01 N1 L dt

(49)

This is in addition to the spontaneous emission. It can be shown that B10 = B01 and A01 = (hν 3 /πc2 )B10 . The photon emitted by stimulated emission is indistinguishable from the photon that stimulated it. There are now two photons instead of one; they have the same direction of propagation, the same frequency, the same phase, and the same polarization. In most cases, N0 N1 , so that absorption predominates and stimulated emission is of little consequence. However, if it can be arranged so that N1 > N0 , then the stimulated emission will exceed the absorption and the light can increase in L as it propagates. This might be called negative absorption. This condition must exist in order to produce a laser. The condition in which N1 > N0 is called a population inversion. It may be created in a variety of ways. In the helium–neon laser, for example, the population inversion is produced as follows: The medium is a mixture of helium and neon atoms in the ratio of about 4 : 1; the gas pressure of the mixture is about 1.0 torr to obtain a stable discharge. Electrical discharge in this gas mixture has little direct effect upon the neon but serves to excite some of the helium atoms into metastable states known as the 21 S and the 23 S states. These states are metastable in the sense that there are no allowed radiative transitions by which the atoms can return to the ground state; these helium atoms remain excited long enough to experience inelastic collisions with neon atoms. Fortunately, the excitation energy of the 3s2 state of a neon atom is the same as the 21 S state of a helium atom; collision between an excited helium atom and an unexcited neon atom can result in energy transfer, producing a neon atom in the 3s2 state and a helium atom in the ground state. In the same way, a helium atom in the 23 S state can excite a neon atom to

the 2s2 state. In this way, a small but useful fraction of the neon atoms are excited into these two states even though the neon was not directly involved in the electrical discharge. At a lower energy than the two states we have been discussing is a state of the neon atom known as 2p4 . There are essentially no neon atoms in this state, so there is a population inversion between states 3s2 and 2p4 and between states 2s2 and 2p4 . These two population inversions can produce lasing at λ = 632.8 nm and λ = 1152.3 nm, respectively. For the process to run continuously, the neon atoms in the 2p4 state must return to the ground state. This involves a radiative transition to the 1s5 state and finally an inelastic collision of the neon atom with the walls of the tube. Since the rate of stimulated emission is proportional to the spectral irradiance (or to the spectral energy density) and the rate of spontaneous emission is independent of the spectral irradiance, it follows that stimulated emission will become the dominant process when the spectral irradiance is large. To bring this about, the lasing medium [in our case the helium–neon (He–Ne) gas mixture] is placed between two mirrors as indicated in Fig. 29. These mirrors should have high reflectivity; it is common for the reflectivities to exceed 99.5%. One of the mirrors should have a slight transmissivity (a few tenths of a percent) so that some of the light may escape from the space between the mirrors (called the cavity) into outside space. If the mirror separation l and the wavelength λm are such that a round-trip path equals an integral number m of wavelengths, that is, 2l = mλm

(50)

there will be constructive interference for light reflected from the mirrors upon successive round trips. The cavity is said to be resonant at wavelength λm and at the corresponding frequency νm = c/λm . The frequency spacing ν between νm and νm+1 is ν = c/2l. Cavity resonances are called modes. For the typical He–Ne laser, l may be about 30 cm so that m is a very large number and ν is about 500 MHz. If a resonance of the cavity exists at the same wavelength as the resonance of the neon states 3s2 and 2p4 and there is also a population inversion for these two states sufficiently large that the gain per pass exceeds

Fig. 29 Helium–neon laser with concave end mirrors and Brewster angle windows.

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the losses, then there will be lasing (oscillation at optical frequencies) within the cavity. The losses from the cavity include (a) the light that escapes through the partially transparent mirrors, (b) absorption by the mirrors, (c) diffraction and scattering from the beam inside the cavity, and (d) any reflection losses at the end windows of the discharge tube. The spectral lines of neon have some width, primarily because of the Doppler broadening due to thermal motion. For an atom in motion, its Doppler-shifted resonant frequency must match the incident photon frequency in order to produce stimulated emission; that is, the Doppler-shifted frequency of the atom must match the cavity-resonant frequency for the system to lase. The spectral width of the neon resonance due to Doppler broadening is temperature dependent, but in a typical laser it is roughly 1.0 GHz. This width is sufficient to cover two, or sometimes three, cavity modes of a 30-cm laser cavity. These two or three modes will lase simultaneously. In the first He–Ne lasers, the mirrors M1 and M2 were plane mirrors. It was necessary that they be accurately parallel to each other; if they were not, a ray reflected back and forth between them would not remain in the discharge tube to be amplified by the active medium. It was later discovered that curved mirrors could be used, in which case the alignment is less critical. It can be shown by diffraction calculations or by ray tracing that a ray from M1 to M2 that is slightly off axis will remain trapped near the axis if the center of curvature of M1 is to the right of M2 and the center of curvature of M2 is to the left of M1 . The axis is a line through the two centers of curvature; the axis should pass through the discharge tube. A limiting case is to use one plane mirror, M1 , and let the center of curvature of the other (M2 ) lie slightly to the left of M1 ; this is called a hemispherical cavity. There are other mirror arrangements that form stable cavities (stable in the sense that rays are trapped near the axis), but the ones just mentioned are commonly used. In contrast, there are unstable cavities in which a ray starting slightly off axis diverges from the axis and escapes from the cavity. As an example, we give a cavity for which the center of curvature of M1 lies slightly to the left of M2 and the center of curvature of M2 lies slightly to the left of M1 . (One center is inside the cavity and the other outside the cavity.) The rays are not trapped in this cavity but escape after a few round trips; the losses in this cavity are very large. It does not lase. In the earliest He–Ne lasers, the reflection losses at the windows on the ends of the discharge tube were so great that the system could be made to lase only by eliminating these windows and attaching the cavity mirrors directly onto the discharge tube. Later it was realized that by attaching the windows at the Brewster angle (see Fig. 3), one polarization (the polarization with E parallel to the plane of incidence) would experience no reflection loss and the cavity mirrors could

be mounted independently of the discharge tube. The light from such a laser is linearly polarized since the gain exceeds the losses only for the polarization, which experiences no loss at the Brewster angle windows. The mirrors M1 and M2 , which form the resonant cavity, are usually multilayer dielectric coatings since most metals do not have sufficiently high reflectivity. Also, with multilayer dielectric coatings the mirrors may be spectrally selective in their reflectivity. For example, the mirrors may be highly reflecting at λ = 632.8 nm with much lower reflectivity at λ = 1152.3 nm. In this case, the He–Ne laser will lase at λ = 632.8 nm but not at 1152.3 nm. By exchanging the mirrors for a pair with high reflectivity at 1152.3 nm, we can cause the system to lase at that wavelength. The modes considered a few paragraphs back are properly called longitudinal modes. A laser may also have several “transverse” modes, but the manufacturers of lasers usually suppress all the transverse modes except the TEM00 mode. In this mode the amplitude of the electric field at the output mirror as a function of distance off axis r is given by E = E 0 e−r

2 /w 2

(51)

where E 0 is the amplitude on axis, E is the amplitude at a distance r from the axis, and w is a constant depending upon the geometry of the cavity. The surface of the output mirror is a surface of constant phase (i.e., a wave front) for the emerging wave; the beam width w may be only a few millimeters. We have given our attention to the He–Ne laser because it is readily available and illustrates the principles involved. There are many other media in which population inversion can be produced and which can provide lasing if used in a suitable cavity. 6 THE EYE AND VISION The eye is important because most of the information obtained in a lifetime is brought to the brain through the eye. For the student of optics, the eye is important because many optical instruments, for example, microscopes and telescopes, are used in conjunction with the eye so that the eye becomes a part of the optical system. The pupil of the eye may become the aperture stop of the system or in some cases the eye may limit the spatial frequency response or resolution of the optical system. A geometrical or physical description is inadequate because the eye is a living, functioning organ that should be considered in terms of physiology and neurology, but these fields are beyond the scope of this chapter and can be considered only superficially. 6.1 Structure of the Eye The human eye is an almost spherical organ about an inch in diameter. It is shown in cross section in Fig. 30. Six muscles, two of which are shown in the figure as Z1 and Z2 , hold the eye in place and rotate

LIGHT AND RADIATION

Fig. 30

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Horizontal section of right human eye according to Helmholtz.

it relative to the head. These muscles are attached to the sclera S, which is a tough white skin covering most of the eye. At the front of the eye the sclera is replaced by the cornea C, which is a transparent membrane through which light enters the eye. After entering through the cornea, light passes through the aqueous humor AH, the crystalline lens L, the vitreous humor VH, and finally reaches the retina R. The aqueous humor is a weak salt solution; the vitreous humor is a soft jelly consisting primarily of water. The fluids of the eye are slightly (∼25 torr) above atmospheric pressure. This pressure helps to maintain the shape of the eyeball. The crystalline lens is a fibrous jelly contained in a thin membrane or sac; it is hard at the center and progressively softer toward the outside. The lens is held in place and attached to the ciliary muscle Y by the suspensory ligament G. When the ciliary muscle is relaxed, the second focal point is at the retina and distant objects are in focus. To view nearby objects, the ciliary muscle contracts, allowing the lens to become more nearly spherical. This is known as accommodation; with age, the lens becomes less elastic, and the ability to accommodate gradually decreases. The lens of the eye is not transparent to ultraviolet light. The retina is the interior lining for a large part of the eyeball. It consists of rods and cones that are lightsensitive nerve endings, along with a delicate network of nerve fibers connecting the rods and cones to the optic nerve O and a network of capillary blood vessels that supply the necessary oxygen and nutrients. The yellow spot, or macula lutea M, which contains many cones and relatively few rods, is a slight depression in the retina; the central region, called the fovea centralis, contains cones exclusively, no rods. The macula lutea is about 2 mm in diameter, and the fovea centralis is about 0.25 mm in diameter. Cones in the fovea centralis are about 1.5 µm in diameter, increasing in size to about 5.5 µm in the outer portion of the macula lutea and several times this size in other portions of the retina. See Fig. 31. In the outer portion of the retina, the rods outnumber the cones by 10 : 1.

Fig. 31 Rods and cones of retina: A, rod; B, cone from extrafoveal region; C, cone from central fovea.

Each human eye contains roughly 7 million cones and 120 million rods. Vision in the fovea centralis is so much more acute than in the extra foveal region that the muscles surrounding the eye involuntarily rotate the eyeball until the object of interest is imaged upon the fovea centralis. The angle in object space covered by the fovea centralis is less than 1◦ ; it is only a little more than sufficient to cover one letter of this printed page when the book is held at the usual reading distance of 25 cm. In reading or examining an extended object, the eye must move frequently. Extra foveal vision is not useful in observing details but enables one to be aware of objects around him. For a healthy eye, the total field is about 128◦ ; an early sympton of glaucoma is the shrinking of the field of view.

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The mosaic structure of the cones in the fovea centralis limits the resolution of the eye. Considering the size of the cones, this varies from 0.3 to 1.0 min for the angular resolution in object space. These numbers should be slightly larger because the cones are separated by a small amount of inactive tissue. It is interesting to observe that this is comparable to the resolution limit set by diffraction at the pupil and also comparable to the limits produced by aberrations, primarily spherical aberration of the optical system. One minute is a good round number representing the overall resolution of the eye. Part of the blood supply to the retina is provided by a network of blood vessels on the front of the retina. If one stares at a blue sky (or a white wall illuminated by blue or violet light), the red blood cells coursing through these blood vessels can be seen since they cast a shadow on the retina. Unlike the specks of dust that float upon the front of the eye or in the vitreous humor when one is tired, these shadows follow definite paths; that is, they are confined to the blood vessels. These shadows are called muscae volitantes, which means flying flies. Red blood cells are about 8 µm in diameter, so each one can cast a shadow over several cones of the fovea.

the atmosphere. However one describes it, the darkadapted eye is incredibly sensitive. Vision by the dark-adapted eye is called scotopic vision and takes place in the rods of the eye, not in the cones. Since there are no rods in the fovea, the dark-adapted eye has no central vision, and in order to see an object in subdued light, one must look not at the object of interest but to the side so that the object of interest will be imaged on the outer part of the retina, which contains rods. There is no color in scotopic vision. In the rods, the pigment that absorbs the light and somehow triggers the signal along the nerves to the brain is called rhodopsin. The chemical composition and structure is known to be a protein molecule combined with a molecule of retinal. Retinal is closely related to the compounds known as retinol (vitamin A) and carotene (the yellow pigment of carrots and many other yellow vegetables). The spectral sensitivity of rod vision is shown in curve B of Fig. 32. The ordinate at each wavelength is inversely proportional to the minimum amount of energy that is just perceptable (i.e., to the threshold of vision). The curve is normalized to 1 at its peak. This closely matches the absorption curve of rhodopsin.

6.2 Adaptation of Eye to Light

6.4 Photopic Vision

The iris diaphragm, I, is a ring-shaped involuntary muscle that controls the amount of light entering the eye. It is located just in front of the lens, and the diaphragm opening or pupil is the aperture stop of the eye. It varies in diameter from 2 to 8 mm. This is a factor of 42 , or 16, in the area of the entrance pupil. The eye functions under illumination conditions that vary by a factor of ∼109 . Variation in pupil size is certainly not sufficient to account for this wide range; most of the adaptation to light and dark is accomplished by changing the sensitivity of the retina. The photosensitive chemicals (or pigments) in the rods and cones are bleached or altered by light and must be constantly reconstituted. Due to the lower rate at which the pigment is consumed in low illumination, the steady-state concentration of the pigments is higher and the retina more sensitive in low illumination than in high illumination.

For conditions of ordinary illumination, the rhodopsin in the rods is almost completely bleached and vision is by the cones. This is called photopic vision, or cone vision. The spectral sensitivity for cone vision is shown by curve A of Fig. 32. Notice the shift of this curve toward the red relative to the scotopic curve B. Because of this shift, two nonluminous objects of different colors (e.g., yellow and blue-green) that appear “equally bright” in ordinary illumination will not appear equally bright in subdued illumination (e.g., twilight), the blue-green becoming much more conspicuous than the yellow. This shift in the spectral sensitivity and the resulting change in relative brightness of various colors is known as the Purkinje effect. It is a source of trouble in making visual comparisons of light sources of different colors. The level of illumination at which the eye changes from photopic to scotopic vision (or vice versa) with the attendant change in spectral sensitivity, loss of color discrimination, and foveal vision is about the illumination level produced by the full moon on a clear night, or 0.16 lux.

6.3 Scotopic Vision

When the eye has been dark adapted (i.e., kept for half an hour or more in darkness comparable to outdoor illumination by a moonless night sky), the eye becomes sufficiently sensitive to see a small source of 2 × 10−8 cd at a distance of 3 m. Neglecting atmospheric absorption, this is equivalent to seeing a standard candle at a distance of 13 miles. Astronomers observe that except under unusually good conditions, stars of sixth magnitude represent the limit of vision of the unaided eye. This corresponds to seeing a standard candle at a distance of about 6.6 miles through

6.5 Color Vision

Color vision takes place in the cones. There are three different types of cones in the eye; the three differ in that they contain different photosensitive pigments and have distinct spectral response curves. There is no observable physical structure that enables one to distinguish between the three types; the photosensitive pigments are present in such low concentrations that it is difficult to distinguish even on this basis.

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Fig. 32 Spectral sensitivity curves for normal human eye: A, light-adapted (photopic) eye; B, dark-adapted (scotopic) eye. Each curve is normalized to 1 at its maximum.

The three pigments are probably three different protein molecules, each in combination with a molecule of retinal. Because of the chemical similarity of the three dyes to each other and to rhodopsin (which is much more abundant), they are difficult to isolate and identify. The spectral sensitivity of the three cone types is given in Fig. 33. The output of each cone is determined by the intensity reaching it, the wavelength of the light, and the spectral sensitivity of the cone for that

Fig. 33

wavelength; the same output signal could be obtained by use of a lower intensity at a wavelength closer to the peak of the sensitivity curve. Each cone is color blind (just as the rods are color blind); the sense of color is derived from the relative response of the three types of cones. As shown in Fig. 33, the three cone types have peak sensitivities in the blue (∼440 nm), green (∼535 nm), and orange (∼565 nm); they are labeled C, B, and A,

Spectral sensitivity curves for three cone types of human eye. Each curve is normalized to 1 at its maximum.

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respectively. For each spectral wavelength, the relative response of these three cone types is unique and determines the color sensation. If several wavelengths are present, each wavelength evokes a response in each cone type, and the relative size of the total response in each of the three cones determines the color sensation. Curve A peaks at 565 nm, which is in the orange; at this wavelength, the other curves, in particular B, still have nonzero values and the sensation of orange is produced by a signal from A and a weaker signal from B. At longer wavelengths, cones of type A respond less than they did at 565 nm, but the response from cones of type B decreases even more rapidly so that the signal from A makes up a larger fraction of the total output and the color sensation changes from orange to red. Common forms of color blindness result from the absence of type A or type B cones. Protonopes are persons color blind due to lack of type A cones; duteranopes lack type B cones. In ordinary vision, the output of type A cones is added to the output of type B cones (with perhaps a weak contribution from type C cones), and this sum is transmitted to the brain along the optic nerves. This sum encoded as nerve pulses per second is interpreted by the brain as luminosity (white) without color information. Color information is transmitted in two channels as the difference (A − B) between the output of type A cones and type B cones and the difference (A − C) between the output of type A cones and type C cones. The data is processed into these sum and differences in or near the eye and then transmitted along nerve fibers to the brain in the form of an increase or a decrease of the pulse frequency from the spontaneous value of the pulse frequency that exists when the eye is in the dark. Type C cones have little effect upon the sensation of brightness but are effective in producing color discrimination. Although the system just described is believed by many to be the usual one, others are possible and sometimes effective because one can cover the right eye with a red filter and the left eye with a green filter (or vice versa) and obtain color vision. In this case, some of the data processing that usually takes place at the eye appears to be deferred to some later stage of the visual process, perhaps in the brain or perhaps at the optic chiasma, the point at which the two optic nerves (one from each eye) come together on their way to the brain. For a person with normal vision, the colors associated with various portions of the spectrum are as shown in Table 2.

6.6 Colorimetry The word color has several definitions. In one, it is associated with the properties of a dye; in another, it is a property of light; and in yet another, it is a physiological sensation produced in the brain by light entering the eye. In an earlier section, we have given a brief description of color vision. In this section, we use the word color as descriptive of the light entering the eye and present the methods used to give a quantitative description of the color. The branch of optics that deals with the quantitative specification of color is called colorimetry. 6.7 Color Mixing In order to understand colorimetry, we must first establish the basic facts of color mixing, which are illustrated by the following experiment. We attempt to match all possible colors by mixing three “primaries.” The selected primaries are monochromatic (or spectral) colors of wavelength 450 nm (blue), 550 nm (green), and 620 nm (red). We identify them as α, β, and γ , respectively. There is nothing unique about these particular wavelengths that entitle them to be primaries; we select them because the experimental data using these three primaries was carefully determined in early color-mixing experiments. We now allow the eye to look at a white diffusing card. Two adjacent areas of the card are illuminated (a) by light of arbitrary or unknown color and (b) by a mixture of the three primaries. Area 2 is illuminated by all three primaries, and the amount of each primary is adjusted to obtain a match with the unknown. Most colors can be matched by this mixing process; a few cannot. In cases for which the unknown cannot be matched by the preceding process, a match can be obtained by moving one (or very rarely two) of the primaries from area 2 to area 1 and then adjusting the amount of each primary; this is equivalent to subtracting or using a negative amount of the moved primary in area 2. The use of three primaries widely spaced in the spectrum, as are the ones suggested here, reduces the number of cases in which a negative amount of any of the primaries is required. Neither the unknown nor the primaries need be monochromatic (spectrally pure) colors; a match can always be made. If the unknown is represented by U and the amount of each primary by A, B, and C, respectively, the experimental results may be represented by the equation

U =A+B +C Table 2 Color Violet Blue Green Yellow Orange Red

(52)

Colors and Associated Wavelengths Wavelength (nm) 610

which is interpreted to mean that the sensation of light and color produced by the unknown may be duplicated by the mixture of the three primaries. The values of A, B, and C are unique if U is given. The eye sees the overall effect of the mixture; it is not aware of the individual primaries that make up the mixture. If we now restrict our unknown to spectrally pure (i.e., monochromatic) light and keep the power of the

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Fig. 34 Color mixture curves for matching spectrally pure colors by mixing primaries having wavelengths 450, 550, and 620 nm.

unknown constant but vary its wavelength, we can at each wavelength determine experimentally the power of each primary required to produce a match. The results of this experiment are given in Fig. 34, which gives the amount of each primary α, β, and γ required to match each spectral color. The curves have been normalized to β = 100 at 550 nm; α and γ are zero at this wavelength, which corresponds to the β primary. Each curve is normalized to 100 when the unknown wavelength is the same as that primary. For example, the curves indicate that a match is obtained for an unknown at 500 nm by combining 47.5 units of α (light at 450 nm) with 125 units of β (light at 550 nm) and subtracting (i.e., adding to the unknown) 30.0 units of γ (light at 620 nm). Notice that the primaries do not add to 100; this is because the spectral sensitivity of the eye for each of the primaries differs from its sensitivity at the unknown wavelength. In this case, the most significant difference is a factor of about 3 between the sensitivity of the eye to the β primary and its sensitivity to the 500 nm unknown. Similar curves for the mixing of other sets of monochromatic primaries could be determined experimentally, but it is unnecessary to do so because it

is possible to deduce them from the curves already given. The process is straightforward but tedious, and we shall not describe it. It is also possible to specify a new set of primaries by giving the curves α , β , and γ , which give the mixing data required to match spectral colors using the new primaries. As long as the new curves α , β , and γ (as functions of wavelength) are a linear combination of the experimental curves α, β, and γ (which were given in Fig. 34), the new system will give a satisfactory system of color specification. The requirement of algebraic linearity means that α = K11 α + K12 β + K13 γ β = K21 α + K22 β + K23 γ γ = K31 α + K32 β + K33 γ

(53)

where the Kij are real and independent of wavelength but are otherwise subject to no restriction except that the determinant K11 K12 K13 (54) K21 K22 K23 = 0 K31 K32 K33

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With such a wide choice of primaries and with the possibility of algebraic transformation from one set to another, color-mixing data could not provide any insight into the spectral sensitivity curves of Fig. 33, but this did not prevent the development of colorimetry in advance of a detailed understanding of color vision. With so much freedom in the choice of the Kij (or the curves α , β , and γ ), it is probably not surprising that some of the possible sets of primaries so described contain primaries that are not spectral colors (e.g., purples) or not spectrally pure (e.g., pinks). It is also true that many acceptable sets of primaries contain primary colors that are not real; by this we mean that they exist mathematically in terms of the mixing curves α , β , and γ , which produce real colors; in this sense, they are entirely satisfactory primaries, and yet they do not exist in the sense that the single primary alone cannot be seen as light and color. 6.8 Tristimulus Values and Trichromatic Coefficients

Since color specification is commercially important and since there is so much freedom in the choice of

primaries, it was inevitable that there should develop some agreement on what set of primaries would be used. In 1931, the International Commission on Illumination (ICI) [also known by its French name, Commission Internationale de l’Eclairage (CIE)] agreed to express all color specifications in terms of three primaries defined by the color-mixing curves of Fig. 35. The letters x, y, and z have become standard, replacing the α, β, and γ used by earlier workers. The ordinates, called tristimulus values, are in arbitrary units and have been adjusted so that the areas under the three curves are equal. The shape of curve y was arbitrarily chosen to be the same as curve A of Fig. 32. Curves x and z have shapes selected so that the three primaries satisfy Eq. (53). For computational convenience it was also required that none of the curves is ever negative. None of the primaries defined by this set of mixing curves is real; they form a satisfactory base for the quantitative specification of color, but only real colors can be produced and mixed in the laboratory. For monochromatic light with wavelength 500 nm (green), the tristimulus values are x = 0.0049

y = 0.3230

z = 0.2720

Fig. 35 Standard ICI (or CIE) tristimulus curves x, y, and z for unit power at indicated wavelength. Numerical values for these curves may be found in Refs. 1 and 2.

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Define three new quantities x, y, and z such that x≡

x x+y+z

y≡

y x+y+z

z≡

z x+y+z

These new quantities are called trichromatic coefficients and by definition have the property that x + y + z = 1; any two of the three quantities are sufficient to specify the color. For 500-nm light, the values are x = 0.0082

y = 0.5384

z = 0.4534

This system cannot contain any intensity information, only color information. In this system, any spectral color may be specified by giving any two of the trichromatic coefficients; the values of x and y are usually given. If we plot on ordinary graph paper the values of x and y for the spectral colors, we obtain the curve of Fig. 36, where the wavelength (in nanometers) is shown at various places along the curve. A diagram such as this in which color information is plotted using the trichromatic coefficients is called a chromaticity diagram; the curve is known as the spectrum locus.

6.9 Trichromatic Coefficients for Nonmonochromatic Light

In the previous section we defined the trichromatic coefficients of any monochromatic light using the ICI (or CIE) primaries. In very few cases is the light reaching the eye monochromatic; it is usually a mixture or distribution of spectral colors. If we represent the spectral distribution by the function f (λ) defined so that f (λ) dλ is the power (e.g., in watts) in the spectral interval between λ and λ + dλ, then we calculate the tristimulus values of the light by the equations X≡

∞

xf (λ) dλ

(55a)

yf (λ) dλ

(55b)

zf (λ) dλ

(55c)

0

Y ≡

∞

0

Z≡

∞

0

where x, y, and z are the functions represented in Fig. 35, the ICI color mixture curves. This process

Fig. 36 Chromaticity diagram: Horseshoe curve, spectrum locus; E, source for which f(λ) is constant; C, illuminant C, approximately daylight; A, illuminant A, illumination from tungsten filament lamp; S, light reflected from orange skin illuminated by illuminant C.

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amounts to treating each spectral color in the light by the methods of the previous section and then adding (integrating) all of these effects together. By definition, the integrals are from zero to infinity; but since the functions x, y, and z are zero outside of the visible range, the integration is effectively limited to the visible-wavelength interval. The functions involved cannot be integrated by elementary methods; the integration is carried out numerically. The numerical data represented by the curves x, y, and z may be found in the original ICI report1 or in any textbook on colorimetry.2 The three tristimulus values calculated by Eqs. (55) are converted to trichromatic coefficients by the equations X X+Y +Z Z z= X+Y +Z

x=

y=

Y X+Y +Z (56)

We again have the property that x+y+z=1

(57)

and any two of these may be used to specify the color of the light; x and y are usually used. Information about the total intensity of the light has been lost, but all color information is retained. As an example, c onsider light for which the spectral distribution f (λ) is a constant. This is light for

which, at any wavelength, a small wavelength interval dλ contains the same power as an equal interval dλ located at any other wavelength. Since the three curves x, y, and z of Fig. 35 have equal areas under them, the integrals of Eqs. (55) will, for this example, be equal, that is, X = Y = Z. When these are converted to the trichromatic coefficients x, y, and z, we obtain x = y = z = 0.3333 On the chromaticity diagram (Fig. 36), this is repre sented by the point E at 13 , 13 . Another important example is light from a source known as illuminant C. Illuminant C is intended to have the same spectral distribution as average daylight, at least in the visible. It consists of a gas-filled tungsten lamp operated at the color temperature 2848 K combined with filters designed to alter the spectral distribution of the lamp to that of daylight. The spectral distribution of illuminant C is in Fig. 37. From this distribution, one can evaluate numerically the integrals of Eqs. (55) and then the trichromatic coefficients x, y, and z of Eqs. (56). The results of these calculations give point C at (0.3101, 0.3163) on the chromaticity diagram in Fig. 37. Light from illuminant C is generally considered to be “white,” although the term white light has no universally accepted definition. 6.10 Color of an Orange Skin In Fig. 37, there is shown the reflectance of an orange skin as a function of wavelength, R(λ). This curve may be obtained by illuminating the orange skin successively at several different wavelengths; at each

Fig. 37 Dashed curve D, curve A, and curve C are spectral distributions f(λ) of average daylight, light from illuminant A, and light from illuminant C. (Vertical scale is arbitrary and not the same for the three curves.) Lower curve OS, spectral reflectance R(λ) of orange skin; magnesium carbonate powder is taken as 100%.

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wavelength, the reflected radiance is measured for the orange skin and for some white object. The reflectance R(λ) at each wavelength is the ratio of these two measurements. Since the orange skin is a diffuse reflector, the white comparison object should be a diffuse reflector also. Freshly fallen snow is a good white diffuse reflector, but in the laboratory a powder of magnesium carbonate is more practical. The observed color of the orange skin depends not only on its spectral reflectance but also on the spectral distribution of the illuminating light. Let us assume that illuminant C is used and we represent its spectral distribution by C(λ). The spectral distribution of the light reflected from the orange skin is the product C(λ)R(λ). We calculate the tristimulus values of this light from Eqs. (55): ∞ xC(λ)R(λ) dλ = 341 X≡ 0

Y ≡

∞

yC(λ)R(λ) dλ = 277

0

Z≡

∞

zC(λ)R(λ) dλ = 50

0

and when these are normalized to the trichromatic coefficients, we have x = 0.511

y = 0.414

z = 0.075

These locate a point (marked S) on the chromaticity diagram (see Fig. 36). This point is fairly close to the spectral locus for 586 nm. The light reflected from this orange skin is therefore close to the orange-yellow color of the sodium D lines. If we used another illuminant instead of illuminant C, the location of point S representing the chromaticity of the light reflected by the orange skin would have to be recalculated and would probably have changed. Two pieces of cloth that have the same spectral reflectance will always look alike, that is, have the same coordinates on the chromaticity diagram, as long as the same illuminant is used on each piece no matter what illuminant is used. It is possible, and sometimes happens, that two pieces of cloth that have different spectral reflectance curves may look alike when illuminant C is used but will be noticeably different when another illuminant, such as illuminant A, is used. Illuminant A is the gas-filled tungsten lamp operated at the color temperature 2848 K and used without filters; it is typical of the illumination produced by tungsten filament lamps. The chromaticity of illuminant A is represented by point A in Fig. 36. It more frequently happens that two pieces of cloth look alike under illuminant A but are noticeably different under illuminant C. Illuminant A is relatively weak in the short-wavelength region so that a match using this illuminant is relatively insensitive to the reflectance

of the cloth for blue light. Illuminant C is slightly stronger at the shorter wavelengths than it is at the longer wavelength (see Fig. 37). 6.11 Chromaticity Diagram as Aid to Color Mixing From the definitions of the trichromatic coefficients [Eqs. (55)], it follows that if we have two colors represented by points such as G and R (Fig. 38) of the chromaticity diagram, any additive mixture of these two colors will be represented by a point lying on the line GR. If each component (G and R) is assinged a weight proportional to the sum of its tristimulus values (X + Y + Z), the point representing the chromaticity of the mixture will lie at the center of gravity of these weights. For example, if the mixture contains more of light G than of light R such that the sum X + Y + Z for light G is twice the corresponding sum for light R, the mixture will have color represented by the point D on the line GR located so that the distance DR is twice the distance GD. Any color on the line from G to R may be obtained by additive mixing properly selected amounts of lights G and R. After obtaining light D in this way, light D may be mixed with some other light, such as that represented by B, to obtain any color along the line BD. It follows that by additive mixing properly selected amounts of the three lights represented by points G, R, and B, one can obtain any color within the triangle GRB. Colors outside this triangle cannot be produced by additive mixing of colors GRB. Since all the real colors are mixtures of the spectral colors, they must lie in the area enclosed by the horseshoe-shaped spectrum locus curve and the straight line connecting the violet and red ends of the horseshoe. If the triangle GRB is to enclose most of the real colors, the point G should lie close to the spectrum locus point for 520-nm (green) light, and the points R and B should be near the red and violet ends of the spectrum locus curve. In this sense, red, green, and blue are desirable primaries. Equations (55) and (56) give us a means of calculating the trichromatic coefficients (and therefore the location on the chromaticity diagram) for light with any given spectral distribution; the answer is unique. The reverse process is not unique. Given a light represented by a point such as G that has a specific set of trichromatic coefficients, this light may be matched by a mixture of two monochromatic colors with wavelengths 500 and 530 nm, by a pair with wavelengths 510 and 550 nm, or by several other pairs of monochromatic colors or a variety of continuously variable spectral distributions. These various matches are easily distinguished with the aid of a spectrometer, but to the unaided eye all look the same. There is no unique spectral distribution associated with a given point on the chromaticity diagram. Color television is an example of additive color mixing. The screen consists of a mosaic (dots) of three different phosphors that can be excited independently;

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Fig. 38 Chromaticity diagram as aid to color mixing. Any color along line GR may be produced by adding colors G and R. Any color within triangle GBR may be produced by adding properly selected amounts of light of colors G, B, and R. Color television uses primaries G , B , and R . Curve entering horseshoe near 610-nm locus gives color of blackbody radiation for several temperatures.

the three phosphors emit three different colors: red, green, and blue. At the customary viewing distance, the spacing of the dots is too small to be resolved by the eye and so the light from several dots is added together to give the sensation of color. The phosphors used give the colors represented by R , G , and B in Fig. 38. The coordinates of these points are R = (0.670, 0.330), G = (0.210, 0.710), and B = (0.140, 0.080). By exciting these three phosphors in the proper ratio, any color within the triangle R G B can be produced. In Fig. 38, we have added a curve representing the color of a blackbody at temperatures from 1000 to 4000 K. It is “cherry red” at 1000 K and progresses through orange toward white as the temperature rises. At very high temperatures (e.g., 15,000 K), the blackbody color is on the blue side of illuminant C. 6.12

Dominant Wavelength and Purity

Dominant wavelength and purity are physical properties of light that evoke the physiological sensations called hue and saturation.

As an illustration, consider again the orange skin. The chromaticity of the light scattered from it is represented by the point S in Fig. 36. Continuing to use C as the white point, we see that a line C to S may be extended to intersect the spectrum locus at λ = 587 nm, which is labeled D and has coordinates (0.56, 0.44). It follows that the light from the orange skin may be color matched by a mixture of white light (illuminant C) and monochromatic light of wavelength 587 nm. The light from the orange skin is said to have a dominant wavelength of 587 nm. The distance from C to S divided by the distance from C through S to D is 0.83; light from the orange skin is said to have spectral purity p of 83%. Similar procedures show that light represented by the point G has dominant wavelength of 519 nm and spectral purity of 80%. Specifying dominant wavelength and spectral purity is an alternate method of locating a point on the chromaticity diagram. For most people, these quantities are easier to interpret than the trichromatic coefficients x and y. One runs into trouble for colors in the lower part of the diagram, for example, the color represented by

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the point P . A line from C to P , if extended, does not intersect the spectrum locus but intersects at P , the straight line closing the bottom of the horseshoe. If the line is extended backward, it intersects the spectrum locus at P , or 550 nm. The color represented by P is said to have dominant wavelength of −550 nm, or complementary, 550 nm. The spectral purity is the ratio of the distances CP /CP , which is about 65% for this case. Colors commonly called “pastel colors” are of low spectral purity. The color pink is a red of low spectral purity; but in every-day language, low purity is often indicated by some adjectives (e.g., “baby” blue and “apple” green). Two colors are said to be complementary if they may be added to make white. In terms of the chromaticity diagram, two colors are complementary if the line joining the two points representing them passes through C. The negative, or complementary, greens are called magenta or purple; frequently they are incorrectly called red. 6.13 Average Reflectance We have seen that the dominant wavelength and purity of light reflected by an object depends upon the spectral distribution of the illuminant. The average reflectance also depends upon the illuminant; the average reflectance will be large if the spectral distribution of the illuminant is large at the wavelengths for which the reflectance of the object is also large. The average reflectance depends in this same way upon the spectral sensitivity of the detector. For the light-adapted (photopic) eye, the spectral sensitivity is represented by curve A of Fig. 32, which is the same as curve y of Fig. 35. Since the eye has its maximum sensitivity in the wavelength interval near 550 nm, the averaging process must be weighted in favor of these wavelengths. The average reflectance ra is calculated as

∞ ra =

0

r(λ)y(λ)C(λ) dλ

∞ 0 y(λ)C(λ) dλ

(58)

where r(λ) is the spectral reflectance of the object; r(λ) for an orange skin was given in Fig. 37. The spectral distribution of the illuminant is C(λ), and y(λ) is the spectral sensitivity of the photopic eye. If some other detector were used, its spectral sensitivity would replace y(λ) in the equation. For the orange skin, illuminant C, and the photopic eye we obtain an average reflectance ra = 0.26 or 26% 6.14 Subtractive Color Mixing If a white paper is used as a background for water colors, light must pass through the water color to get to the paper and after diffuse reflection from the paper again pass through the water color. Selective absorption by

the dye in the water color gives the scattered light its color. Consider a dye that is absorbent and transmits only a little (say, 10%) for wavelengths shorter than 500 nm but is only slightly absorbent, transmitting 80 or 90%, for wavelengths longer than 500 nm. If this dye is painted on white paper, it will produce a yellow color. Another dye may transmit well for wavelengths shorter than 550 nm and absorb most of the light of longer wavelength; this dye will produce a blue color. If these two dyes do not react chemically and are used one on top of the other (or mixed together) so that light must pass through both of them, then most of the light between 500 and 550 nm will emerge but only a little of the light outside this wavelength interval will emerge. The resultant is a green color. This is a subtractive process in which yellow and blue give green; it must not be confused with the additive processes discussed earlier. The principles in the water color experiment just described may be illustrated using a slide projector and two pieces of cellophane, one yellow and the other blue. Light projected onto a white wall through one piece of cellophane appears either yellow or blue, but when it passes through both pieces in series, it appears green. In Fig. 39 are the transmission curves of a yellow filter (A), a blue filter (B), and the two in series (G). At each wavelength, the transmission represented by curve G is the product of the transmission for A and the transmission for B. Color pictures and color slides (i.e., transparencies) use the subtractive method of producing colors. Three dyes are sufficient, and those that are most effective (i.e., produce the widest range of colors) are dyes that control the red, green, and blue. The dye that subtracts the red is blue in color and often is described as cyan. The dye that subtracts the green, leaving the red and blue, is magenta. The third dye subtracts the blue, leaving the green and longer wavelengths unaffected; it is yellow. By varying the concentration of each of these dyes, one can produce all real colors except the highly saturated ones. The available colors are sufficient for all ordinary use since spectrally pure colors are rare outside of the laboratory. In color printing, it is often necessary to include a black and white image in addition to the three subtractive colors described. The use of black controls the average reflectance of a given area of the picture, not its color. Good color reproduction in prints or in slides for projection necessitates prior knowledge of the illuminant used in viewing them. Slides are usually projected using a tungsten filament lamp so the dyes are adjusted on the assumption of illuminant A. In many cases, the color purity is increased deliberately and the blues emphasized because it is pleasing to have “bright colors” and “nice blue skys.” Color pictures are more likely to be viewed in daylight and therefore are processed for use with illuminant C. If the pictures were produced photographically, the illuminant used for the initial exposure affects the final color.

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Fig. 39 Subtractive combination of yellow filter, A, and blue filter, B, to produce green, G. At each wavelength, G is product of A and B (A and B are Wratten filters 9 and 47A).

6.15 Munsell System In matching or specifying paints, the average reflectance is as important as the dominant wavelength and spectral purity. In the Munsell system, paint samples are assigned a “value” from 0 to 10. Zero is black (i.e., nonreflecting) and 10 is white (100% diffuse reflecting); the intermediate shades of gray are equally spaced subjectively. At each value level is a plane polar arrangement in which the distance from the center indicates saturation or the sensation called chroma in subjectively equal steps, from zero or neutral at the center to 12 for a saturated color. Dominant waelength, or its subjective equivalent hue, is represented by the horizontal direction on the polar plot; five principal hues—red, yellow, green, blue, and purple—are recognized with the intermediate hues—yellow-red, green-yellow, and so on—making a total of 10 equally spaced hue segments. Each of these is divided into 10 numbered subdivisions (see Fig. 40). The Munsell quantities value, hue, and chroma are subjective, corresponding roughly to the physical quantities average reflectance, dominant wavelength, and purity. About a thousand distinguishable paints have been prepared and classified in this way. These samples preserved as an atlas are used to specify paints. The average reflectance and ICI color specification (under illuminant C) for these samples have been measured, but equating the subjective and the physical quantities is difficult. The Munsell system predates the ICI system.

Fig. 40 Munsell representation of color data.

6.16 Photometric Units

In our earlier discussions of sources, such as the blackbody, we measured the radiated energy in physical units (i.e., in watts). The units used are called radiometric units. Long before it became possible to make such measurements, light sources and levels of illumination

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were compared and measured using the eye as the detector. The eye is quite good at judging the equality of illumination on two adjacent areas. A whole set of units evolved around this process and are still in use; these are known as photometric units, often distinguished from the corresponding radiometric units by including the word luminous in the name and the subscript v on the symbol. The unit of luminous flux is the lumen. At the wavelength 555 nm, 1 W produces 683 lm; but since the sensitivity of the photopic eye follows curve A of Fig. 32 (or curve y of Fig. 35), the number of lumens per watt at other wavelengths is smaller, as indicated by this curve. For a nonmonochromatic source, the luminance (or luminous radiance) is Lv = 683

∞

L(λ)y(λ) dλ

(59)

0

and has units of nits. A nit is one lumen per square meter per steradian. A small source that radiates 1 lm into each steradian is said to have a luminous intensity Iv of 1 candela. (It was formerly called a “standard candle.”) Common units of illuminance are the foot-candle (1 lm/ft2 ), the lux (1 lm/m2 ), and the phot (1 lm/cm2 ). 7 DETECTORS OR OPTICAL TRANSDUCERS

Aside from the eye, there are numerous devices that are used to detect and measure radiant flux. These are usually separated into two groups: (a) thermal detectors and (b) quantum detectors. In thermal detectors, the radiation is absorbed and converted into heat, which raises the temperature of the detector. The temperature change causes a measurable change in some other physical property of the detector (e.g., its resistance). Thermal detectors are sensitive throughout the spectrum. For quantum detectors, the incident light (photons) affect the detector directly (i.e., without heating it); the best known of these is the photoelectric detector in which light causes electrons to be emitted from a surface. Two thermal detectors are in common use. (a) In the thermocouple, two different materials (usually metals) are connected to form a closed circuit. One junction is exposed to the radiation and thereby heated slightly while the other junction is shielded from the radiation and remains at ambient temperature. The temperature difference between the two junctions produces an electromotive force (emf) in the circuit that may be measured. (This is known as the Peltier effect.) (b) The bolometer depends upon the change of electrical resistance with temperature. Two identical small detectors are arranged in a Wheatstone bridge circuit; again one detector is exposed to the radiation and the other shielded from the radiation. Small changes in the resistance of the exposed detector are taken as a measure of the incident flux.

If the bolometer element is small and thermally well insulated (except for the thin connecting wires) from its surroundings, then a very small optical or radiant power will produce a relatively large temperature rise; in this sense, the detector is very sensitive. However, it will cool slowly and be unable to respond to rapid fluctuations of the incident flux. For rapid response, the thermal isolation of the bolometer should be reduced. For any given application, one must find the optimum compromise between good sensitivity and rapid response. Bolometer elements may be small flakes or ribbons of metal; nickel or platinum are commonly used. For metals, the resistance increases with increasing temperature, as represented by the equation R = R0 [1 + α(T − T0 )]

(60)

where R is the resistance at temperature T and R0 is the resistance at ambient temperature T0 . The constant α depends upon the metal used, but values of 0.003–0.004 per degree Kelvin are typical. Semiconductor bolometer elements (known as thermistors) are also available. For semiconductors, the resistance decreases with increasing temperature according to the equation eβ/T R(T ) = β/T R(T0 ) e 0

(61)

A typical value of β is 3600 K, which gives dR/dT equivalent to α = −0.04 per degree Kelvin in Eq. (60). In this sense, the thermistor is about 10 times as sensitive as a metal bolometer. Since the resistance decreases with increasing temperature, it must be used with a suitably large series resistance to prevent selfburnout. In detecting or measuring weak signals, the signalto-noise ratio becomes important. It is important to realize that a small object (such as a bolometer element) that is in thermal equilibrium with its surroundings is not at a constant temperature but is constantly exchanging energy with its surroundings and fluctuating in temperature. It will experience a root-meansquare random fluctuation of temperature T given by T 2 =

kT 2 C

(62)

where k is Boltzmann’s constant, T is the absolute temperature of the surroundings, and C is the thermal capacity of the small object. Even if all the amplifying and/or measuring circuits could be noise free, the random temperature fluctuations given by this equation represent unavoidable noise. If the incoming radiation in this ideal case produces a temperature rise equal to T , it is said to have noise equivalent power (NEP) and a signal-to-noise ratio of 1.0. In this respect, the

1148


thermistor has no advantage over the metal bolometer; it gives larger response to both signal and noise but does not improve the signal-to-noise ratio. The simplest quantum detector is an evacuated glass tube containing two electrodes. The anode (or positive electrode) collects electrons emitted from the cathode (or negative electrode). Light striking the cathode, called the photocathode, causes the emission of electrons from the cathode; these are collected by the anode and are measured as current in an external circuit. Not every photon causes the emission of an electron, and the term quantum efficiency is used to represent the ratio of the number of electrons emitted to the number of photons incident on the cathode. Quantum efficiencies of 10–15% are typical of good photocathode surfaces. Even in a light beam from a well-stabilized source, the photons do not arrive on the cathode at equally spaced times but at random time intervals. Also, the photons that produce electrons are randomly selected from those that arrive. The subject is usually treated by Poisson statistics, giving the result that if many measurements are made of n (the number of observed electrons in some constant time interval) and the average value of n is n, then the departures of the individual measurements from the average, n, will be given by (n − n)2 = n

(63)

This statistical fluctuation of n about its average value n is known as photon noise; it arises from the same statistical considerations as “shot noise” √ in electric circuits. The signal-to-noise ratio is n, which may be increased by making n larger by (a) increasing the rate at which photons arrive or (b) increasing the observation time for each measurement. The time allotted to each measurement is often built into the associated amplifier, that is, the reciprocal of its bandwidth. Quantum detectors are wavelength selective. The long-wavelength (low-frequency) limit is determined by the equation hc = hν ≥ φe λ where e is the electronic charge, φ is the “work function” of the photo cathode. Here, φe is the minimum energy required to remove an electron from the cathode into the vacuum, where φ is this energy expressed in electron volts and e is the electronic charge. The shortwavelength (high-frequency) limit is usually determined by the absorption of light by the glass walls of the vacuum tube. The sensitivity of photocathodes is wavelength dependent, and a variety of photocathodes are available having peak sensitivities at different regions of the spectrum. At room temperature, there will be some emission from the cathode even in the dark. This is known as “dark current” and is due mainly to thermionic emission; it may be reduced by refrigerating the detector.

Some photodetectors realize an amplification of about 10 in the current by having a few torrs of gas in the tube. The electrons are accelerated toward the anode and gain enough energy to ionize some of the gas; the ions then contribute to the current. The recommended cathode-to-anode potential difference must be maintained. Too little will not provide the specified gain; too much will result in a glow discharge independent of light input (and damaging to the cathode). Recommended potential differences are usually 50–100 V. The photomultiplier is a vacuum tube in which the photocathode is followed by several other electrodes called dynodes. Electrons emitted by the cathode are accelerated to the first dynode, which has a positive potential relative to the cathode on the order of 100 V. Each electron striking the dynode gives up its kinetic energy, thereby causing emission from the dynode of several (e.g., four) slow-moving electrons; this process is called secondary emission. These secondary electrons are accelerated to the next dynode where the process is repeated. It is repeated at each dynode until the electrons are finally collected on the anode and measured in some external circuit. If there is a gain of 4 electrons at each dynode and there are 10 dynodes, there will be ∼106 electrons at the anode for each electron that left the cathode; under these conditions, one can observe individual photoemissive events and count their number. The photon noise is determined by the number of electrons leaving the cathode; the large gain makes the individual events easier to count; it does not improve the signal-to-noise ratio. The anode is usually at or near ground potential; to get 100 V for each of 10 dynodes requires the cathode to be −1000 V. The gain is sensitive to this voltage. The photomultiplier is useful primarily at low levels of illumination. Advances in semiconductor science and technology have provided a number of solid quantum detectors that are more rugged and easier to use than the vacuum tube detectors of earlier years. The electrical behavior of solids is usually described in terms of allowed energy bands for the electrons. For intrinsic semiconductors, the valence band contains all of the valence electrons of the solid and is filled by these electrons. There is no room for any net motion of these electrons. Above the valence band there is an energy region in which no electrons can exist. The width of this forbidden region is called the bandgap φ (usually expressed in electron volts). Above the bandgap is an energy band in which electrons are permitted and in which they are free to move; this is called the conduction band. Normally there are no electrons in the conduction band except for a negligible few that may be thermally excited there from the valence band. Light of wavelength shorter than that given by Eq. (63) can be absorbed by the semiconductor. A photon so absorbed can excite an electron from the valence band to the conduction band. An empty space, or hole, is left in the valence band; this hole acts as a small positive charge and can move through the solid in the valence band.

LIGHT AND RADIATION

If the semiconductor just described is connected to a current meter and a source of small emf, the observed current is due to the motion of electrons and holes; it will depend upon the irradiance. This process is called photoconductivity, and a semiconductor used in this way is called a photoconductor. There are a number of photoconductors available, each having its characteristic bandgap. A semiconductor made by mixing mercury telluride and cadmium telluride will have a bandgap dependent upon the relative concentration of the components. In practice, one seeks a detector with bandgap a little less than the quantum energy hν of the radiation to be detected; in this way the detector becomes blind to undesired radiation at lower frequencies. The electron has a mean lifetime before it recombines with a hole, after which it no longer contributes to the current. The lifetime is a random variable. This randomness contributes to the noise and is known as generation recombination noise; it is larger than the photon noise of photoemissive detectors. An extrinsic semiconductor is an intrinsic semiconductor (such as silicon) into which a small concentration of “impurity” has been introduced. Silicon, as carbon, has four valence electrons, and these are just sufficient to fill the valence band. The bandgap for silicon is 1.14 eV. If a small concentration of an element with five valence electrons (e.g., phosphorus or arsenic) is included, each impurity atom will contribute four electrons to the valence band and the fifth electron will be loosely bound to its parent atom. A little energy, 0. The power reflection and transmission coefficients are given by R = |R|2 and T = 1 − R . For the case of a single pipe with a different cross-sectional area, for x > 0, Z0 = ρc/S2 . For branches in the pipe or for

pr

Zb

S1

pt

pi

S2

pr pi

Z2 pt

x=0 x=0 (a) Fig. 9

(b)

Reflection and transmission from geometric discontinuities: (a) pip expansion; (b) side branch with impedance Zb .

ACOUSTICS

1165

more complex impedances such as expansion chambers and constrictions, the measured or calculated acoustic impedance at x = 0 should be used.

Table 6 Daily Noise Exposure Limits (LpA ) in dBA Hours

(dBA)

Side Branch A special case of practical interest is where a side branch exists at x = 0. This could be a branching of the pipe or it could be a noise reduction device such as a Helmholtz resonator (see Section 9). Let the acoustic impedance of the side branch be given by Zb = Rb + j Xb and the impedance of the continuing pipe be ρc/S1 . Using these values leads to the power reflection coefficient, the power transmission coefficient for waves propagating further down the pipe, and the power transmission coefficient for waves propagating into the side branch:

8 6 4 3 2 1.5 1 0.5 1. High-Pass Filter An acoustic high-pass filter can be constructed using a side branch consisting of a short length of unflanged pipe (with radius a), such as is used for toneholes in musical instruments like a flute or clarinet, as shown in Fig. 19. The acoustic impedance of the side branch is given by

Zsb =

ρo ck 2 ρo Leff + jω 4π πa 2

(68)

ACOUSTICS

1171

Fig. 18 Power transmission coefficient for low-pass filter using enlarged section of pipe. The main pipe has a radius of 2.54 cm, and the enlarged section has a length of 30.5 cm and an area six times that of the main pipe.

S1

L

S Fig. 19 High-pass filter using open side branch.

The resulting power transmission coefficient is given by T =

1 1 + [S1 /(2SLeff k)]2

(69)

A typical response for this high-pass filter is shown in Fig. 20. It should again be remembered that this acoustic response is only valid for frequencies where the wavelength is significantly larger than the side branch. 9.6 Lined Ducts

Ducts lined with absorbing material are often used as dissipative muffling devices to muffle fans in heating and air conditioning systems. The liner material generally consists of a porous material such as fiberglass or rockwool, usually covered with a protective facing. The protective facing may be a thin layer of acoustically

transparent material, such as a lightweight plastic sheet, or it may be a perforated heavy-gage metal facing. If a perforated facing is used, it should have a minimum open area of 25% to ensure proper performance. The performance of a lined duct with liner thickness l and airway width 2h is shown in Fig. 21 for the case of zero mean flow. This figure shows some of the dependencies on the ratio of liner thickness to airway width as well as the flow resistivity. 9.7

Single- and Double-Leaf Partitions

Partitions (such as walls) are often used to separate a noise source from a receiving space. When partitions are used, flanking paths and leakage are important to check for. Small openings or flanking paths with low impedance can easily reduce the effectiveness of partitions significantly. Single-leaf partitions exist when there is a single surface or when both surfaces of the wall vibrate as a unit. Double-leaf partitions consist of two unconnected walls separated by a cavity. At low frequencies for single-leaf partitions, the transmission through the partition is governed by the mass of the partition, and the mass law governs this behavior. The intensity transmission coefficient is given in Eq. (56), and the resulting transmission loss can be expressed as TL = 20 log(fρ s ) − 47(dB)

(70)

1172


Fig. 20 Power transmission coefficient for high-pass filter using open side branch. The main pipe has a cross-sectional area of 28 cm2 , and the side branch has a length of 8 cm and a cross-sectional area of 7.5 cm2 .

where ρs is the surface density of the partition, given as the product of the density and thickness of the partition. Thus, at low frequencies, one must increase the density of the partition in order to increase the transmission loss; doubling the density of the partition increases the transmission loss by 6 dB. For a single-leaf partition, wave effects in the partition lead to a coincidence frequency, which corresponds to the condition where the flexural wavelength in the partition matches the acoustic wavelength along the direction of the partition. When this condition is met, the transmission loss drops significantly, with the decrease being governed by the damping in the partition. Above the coincidence frequency, the partition becomes stiffness controlled, and the transmission loss increases at a rate higher than the mass law (theoretically 18 dB/octave for a single angle of incidence or about 9 dB/octave for diffuse-field incidence). The coincidence frequency separates these two behaviors and is given by fco

1 = 2π

ρp h D

c sin φ

2 (71)

In this equation, c is the speed of sound in the fluid, ρp is the density of the partition, h is the thickness of the partition, and D is the bending rigidity of the partition, given by D = Eh3 /[12(1 − ν 2 )], where E is Young’s modulus of the partition and ν is Poisson’s ratio for the partition. The transmission loss through a double-leaf partition is noticeably higher than through a single-leaf partition with the same mass density. While the behavior is too

complex to be covered extensively here, the following characteristics are generally associated with double-leaf partitions. (a) At low frequencies, the transmission loss follows the mass law, with the combined mass of the two leaves being used to determine the surface density. (b) A mass–air–mass resonance exists where the air cavity between the two leaves behaves as a spring between two masses. (c) Above the mass–air–mass resonance, the transmission loss increases sharply (18 dB/octave) until resonance effects in the cavity become important. (d) The transmission loss oscillates as the cavity resonance effects become important. At resonances of the cavity (cavity depth ≈ nλ/2), the transmission loss drops to values consistent with the mass law, while the peaks in the transmission loss occur at antiresonances of the cavity and continue to rise at about 12 dB/octave. Experimental transmission loss measurements typically show some variation from these predicted trends, but the results are generally consistent with predicted behavior. 9.8 Enclosures For noisy equipment, one can install an enclosure around the piece of equipment. The insertion loss of the enclosure can be estimated using

IL = TL − C

(dB)

(72)

where TL is the transmission loss associated with the walls of the enclosure and SE (1 − α i ) dB (73) C = 10 log 0.3 + Si α i

ACOUSTICS

1173 3.5 3.0 2.5 2.0 1.5

Curve no

h

1 2 3 4 5

0.25 0.5 1 2 4

R1 ρc

=1

R1 ρc

2h

=2

1.0 3 2 1

Attenuation rate (dB per h duct length)

0.5

3

2

1

3

2

1

0.0 3.5 3.0

R1 ρc

=4

5

4

R1 ρc

=8

5

4

2.5 2.0 1.5 1.0 0.5

3

2

1

0.0 3.5 3.0

R1 ρc

R1 ρc

= 12

= 16

2.5 2.0 1.5

5

1.0

4 3 2

5

1

0.1

3 2 1

0.5 0.0 0.01

4

1

10 .01 2h/λ

0.1

1

10

Fig. 21 Predicted octave-band attenuations for rectangular duct lined on two opposite sides. Lined circular ducts or square ducts lined on all four sides give twice the attenuation shown here. The quantity ρ is the density of fluid flowing in the duct, c is the speed of sound in the duct, is the liner thickness, h is the half width of the airway, and R1 is the liner flow resistivity. For these results, a bulk reacting liner with no limp membrane covering and zero mean flow is assumed.14

αi is the mean Sabine absorption coefficient of the interior of the enclosure, Si is the interior surface area of the enclosure, and SE is the external surface area of the enclosure. 10 ACTIVE NOISE CONTROL

Over the last couple of decades, there has been considerable interest in the use of active noise control to address noise control applications. In many cases, it has been a technology that has not been well understood. There are many applications for which active noise control is not a good solution. In such cases, active noise control will be quite ineffective, and as a result, people can be easily disappointed. Thus, active noise control as a noise control solution should be

chosen with care. However, for applications where active noise control is appropriate, it works very well and can produce impressive results. While active noise control can be very effective for proper applications, it is generally not a straightforward “off-the-shelf” solution. Thus, the focus of this section is to give an overview of how active noise control works and the direction in identifying proper applications. If active noise control is a viable solution, expertise should be sought in implementing the solution. In deciding whether active noise control is a viable solution, there are several characteristics of active noise control that should be understood. First, active noise control is inherently a low-frequency solution. Implementation at high frequencies has several

1174


difficulties associated with it. First, effective control requires precise phase and amplitude matching. For example, if one wishes to achieve 20 dB of attenuation, the control signal must have a phase error of less than 4.7◦ (assuming perfect amplitude matching) or a magnitude error of less than 0.9 dB (assuming perfect phase matching). This tight tolerance in phase and magnitude matching is significantly easier to achieve at lower frequencies than at higher frequencies. It is also easier to achieve significant spatial control of the acoustic field at lower frequencies than it is at higher frequencies. In active noise control, one can achieve localized control or global control of the field, depending on the physical configuration of the problem. In those cases where local control is achieved, the spatial volume where significant attenuation occurs scales according to the wavelength. The diameter of the sphere where at least 10 dB of attenuation is achieved is about one-tenth of a wavelength. Thus, lower frequencies result in larger volumes of control. If global control is to be achieved, good spatial matching must be achieved, which is also easier to achieve at lower frequencies than at higher frequencies. In a number of applications, it is desirable to achieve global control of the acoustic field. In order to accomplish this, there must be a good acoustic coupling between the primary noise source and the secondary control source used to control the field. This can be achieved using one of two mechanisms. The first is to have the spacing between the primary noise source and the secondary control source be significantly less than an acoustic wavelength. For an extended noise source, this would require multiple control sources that will acoustically couple to the primary noise source. For an enclosed noise field (such as in rooms or cabs), it is also possible to achieve acoustic coupling without the control source being less than a wavelength away from the noise source. Instead, the coupling occurs through the acoustic modes of the enclosed field. The acoustic modes have a distinct spatial response, and by exciting the secondary control source properly, it is possible to achieve the same spatial (modal) response but with opposite phase, thus resulting in global attenuation of the field. However, it should be noted that this approach is effective only for low modal density fields. If too many modes are excited, global control can rarely be achieved and local control is the result. It is also important to understand that discrete tonal noise is significantly easier to control than broadband noise. When multiple frequencies must be controlled, it is necessary to control the precise phase and amplitude matching at all frequencies. This is easier to accomplish with a small number of discrete tones than it is with many frequencies or broadband noise. In addition, when controlling broadband noise, causality also becomes an important issue. When controlling tonal noise, if it is not possible to generate the control signal to arrive with the noise signal at the error sensor at exactly the same time, it is still possible to

Broadband Noise

(Global, broadband)

Increasing difficulty

Tonal Noise

(Local, tonal) Local Control

Increasing difficulty

Global Control

Fig. 22 Level of difficulty for various classes of noise control problems.

achieve effective control by delaying the control signal to match the noise signal one period later. However, with broadband noise, this approach is not possible. If broadband control is to be achieved, the control signal and the noise signal must be temporally aligned, which cannot occur if it takes longer for the control signal to get to the error sensor than the primary noise signal. To summarize, when considering a particular application, the lower the frequencies involved, the more effective the control can be, in general. In addition, the difficulty of the solution depends on the frequency content of the noise and the spatial extent of the control needed, as shown in Fig. 22. 10.1 Control Architectures

Active control can be implemented in either an adaptive mode or a nonadaptive mode. In a nonadaptive mode, the control filter is fixed such that if the filter is designed properly, good attenuation results. However, if the acoustic system changes, reduced effectiveness can result. In adaptive mode, the control filter has the ability to adjust itself to a changing acoustic environment, based on the response of one or more error sensors. Most active noise control solutions are based on an adaptive mode solution that is based on a digital signal processing (DSP) platform. There are two general architectures that can be implemented with an active noise control system. These are referred to as feedforward and feedback. While both adaptive control and nonadaptive control have been used with both architectures, adaptive control has generally been used for feedforward control systems. To understand the basic working of an adaptive feedforward control system, consider the control of a plane wave propagating in a duct, as shown in Fig. 23. This is a prototypical application and allows one to understand the basic configuration in a straightforward manner. The noise to be controlled is detected by a “reference sensor,” which could be a microphone in the duct or some other sensor whose output is correlated to the noise to be attenuated. It should be understood that the control system will only be capable of attenuating noise that is correlated with the signal

ACOUSTICS

1175 Reference sensor

Error sensor

x(t) Control System

x(t)

u(t)

e(t)

d(t)

Plant – P(z)

Control Filter – W(z)

Control actuator

u(t)

e(t)

Secondary Path – H(z)

Fig. 23 Adaptive feedforward control for a duct. The upper figure shows the physical layout of the system, while the lower figure shows a block diagram of the control system implementation.

from this reference sensor. This signal is then used as the input to an adaptive control filter that determines the control signal output. The output signal is passed to a control actuator (such as one or more loudspeakers), where it generates an acoustic response that combines with the uncontrolled field and is measured by one or more error sensors. The error sensor response is then used to update the adaptive control filter. For implementation of a feedforward control system, care must also be taken to account for any possible response from the control output at the reference sensor. In our example, not only does the control signal propagate “downstream” in the duct to provide the desired attenuation, but it also propagates “upstream” in the duct where it can alter the reference signal if a microphone is being used. This can be accounted for either by using a nonacoustic reference signal (such as a tachometer signal if the noise is being created by some piece of rotating equipment such as a fan) or by modeling the feedback contribution from the control output to the reference sensor and compensating for that feedback component in the control system. Causality is a potentially important issue in implementing feedforward control. There is an acoustic delay that exists as the noise to be controlled propagates from the reference sensor to the error sensor. If the reference signal can be processed and the response from the control actuator can arrive at the error sensor at the same time as the uncontrolled noise, the system will be causal. In this case, both random and periodic noise could be effectively controlled, since it will be possible to achieve the precise time alignment needed. If the noise to be controlled is periodic, the causality constraint can be relaxed. If the control signal does not arrive in time to be perfectly aligned in time, the DSP will adjust itself to effectively line up the signal properly one cycle later, thus still achieving the desired attenuation.

d(t) Control Filter – W(z)

u(t)

Plant – P(z) e(t)

Fig. 24 Block diagram of feedback control implementation. The uncontrolled signal is given by d(t).

Feedback control systems have used both adaptive and nonadaptive control configurations. Perhaps the most common example involving feedback control is with active headsets that are commercially available to reduce noise (although some headsets are now also implementing feedforward control). For a feedback control system (Fig. 24), the noise to be controlled is detected by a reference sensor that is used as the input to the control filter. The output of the control filter is again sent to a control actuator, and the sound generated combines with the uncontrolled field. The result is measured at the reference sensor and thus creates the “feedback loop.” Feedback control systems are generally more tolerant of model errors in the control system implementation. Thus, implementation of feedforward control in a nonadaptive mode is rarely effective in a practical application. However, feedback control systems can also easily become unstable if designed improperly. Staying within stability constraints often leads to a solution that does not achieve as much attenuation as with feedforward control. The stability constraint also often determines the frequency bandwidth that can be effectively controlled. It should also be understood that there is inherently an acoustic delay present in the feedback loop when

1176


implementing feedback control. This delay results from the finite response time of the control actuator, the acoustic propagation time from the control actuator to the reference sensor, and any delays associated with the electronics of the system. As a result, it is impossible to achieve perfect time alignment of the uncontrolled noise and the noise generated by the active control system. In general, the shorter the delay time in the feedback loop, the greater the attenuation that can be achieved and the wider the frequency bandwidth that can be controlled. 10.2

Attenuation Limits Good estimates of the maximum attenuation achievable can be determined for both feedforward and feedback control implementations. Because of the structure of feedforward systems, the control system will only attenuate noise that is correlated with the reference sensor signal. Thus, it is imperative that the error signal sensor and the reference signal sensor be correlated. This is embodied in the expression that gives the maximum obtainable attenuation, 2 (ω)] Lmax = 10 log[1 − γxd

T W = w0 , w1 , w2 , . . . , wI −1

(76)

Similarly, the secondary path transfer function (represented by H in Fig. 23) can be represented by a vector of filter coefficients: T H = h0 , h1 , h2 , . . . , hJ −1

(77)

The output of a digital filter is the convolution sum of the input signal with the filter response vector. Thus, if the reference input and control output signals are represented as vectors, X(t) = [x(t)x(t −1)x(t − 2) · · · x(t −I + 1)]T (78) U(t) = [u(t)u(t −1)u(t − 2) · · · u(t −J + 1)]T

(74)

where Lmax is the maximum obtainable attenuation 2 (ω) is the coherence (frequency dependent) and γxd between the reference signal, x(t), and the uncontrolled error signal, d(t). A quick method of determining if active noise control would be effective would be to measure the coherence between a proposed reference sensor and a proposed error sensor. It should be remembered that this gives a prediction of the attenuation at the error sensor. If local control is being achieved, this will give no indication as to how effective the control will be at locations removed from the error sensor location. For feedback control, the autocorrelation of the reference signal can be used to predict the attenuation that can be achieved if the time delay associated with the feedback loop is known. Conversely, if the desired attenuation is known, this autocorrelation can also be used to determine how short the time delay must be. The predicted attenuation is given by Ep Lmax = −10 log 1 − E0

architecture of most active noise control systems. In the DSP architecture, the control filter is implemented as a finite impulse response filter, whose response can be represented by a vector of the filter coefficients. Thus,

(75)

the control output signal is given by u(t) = WT X(t) and the error signal is given by e(t) = d(t) + HT U(t). Most active noise control systems are based on quadratic minimization techniques. For the filtered-x algorithm, the algorithm updates its coefficients according to the negative gradient (with respect to the control filter coefficients) of the squared instantaneous error signal. Calculating the gradient of the squared error signal leads to W(t + 1) = W(t) − µR(t)e(t)

(79)

which gives the control filter coefficients for the next iteration of the algorithm. In this expression, µ is a convergence parameter chosen to maintain stability and R(t) is the “filtered-x” signal vector, whose components are given by ˆ T X(t) r(t) = H

(80)

ˆ is a vector of filter coefficients that models where H the physical secondary path transfer function H. 10.4 System Identification

where E0 is the autocorrelation level at zero delay and Ep is the largest magnitude of the autocorrelation that exists in the autocorrelation at any time greater than the group delay of the feedback loop. 10.3 Filtered-x Algorithm The most common adaptive algorithm in current use for active noise control is the filtered-x algorithm or some variation of that algorithm. A brief review of the algorithm is helpful in understanding the general

In order to achieve stable, effective control, it is necessary to have a reasonable model of the secondary path transfer function H. It has been shown that the phase of the model is the primary concern in achieving good control. While phase errors of up to ±90◦ can be tolerated in order to maintain stability, the performance of the control system degrades seriously as the phase errors approach this limit. Thus, an accurate model with minimal phase errors will result in substantially improved control results.

ACOUSTICS

There are several methods that have been used to obtain a good model of H. The most straightforward method is to obtain a model of H a priori. This is done by injecting broadband noise into the secondary path (typically from the DSP used for the control) and measuring the response at the error signal with the primary noise source turned off. In this manner, a straightforward adaptive system identification routine can be used to obtain the coefficients of H. A second method implements an adaptive online secondary-path estimation technique by injecting lowlevel broadband noise, n(t), along with the control signal, u(t). This broadband signal is uncorrelated with the primary noise, d(t), and with the control signal, u(t). Thus, the error signal can be used as an output, with the noise signal n(t) as an input in a typical adaptive system identification routine. Since the primary noise and control signal are not correlated with n(t), they do not affect the system identification and the process proceeds similar to the offline approach. The difficulty with this approach is that the injected noise must be high enough in level to achieve good system identification and yet kept low enough in level to not affect the overall noise level at the error sensor. In many cases, this can involve adaptive gain control to maintain the correct balance between the control signal and the injected noise. A final method that has been used is also adaptive in nature. This method performs system identification not only for the secondary path but also for the plant, P. In other words, it implements a model of the entire system which is unlike the previous two methods. One of the results of this approach is that the model of the secondary path is not unique, unless the excitation of the system can be characterized as “persistent excitation,” which essentially means broadband excitation. For excitation signals that are narrowband in nature, although there is not a unique solution for the secondary path, it has been shown that the solution obtained leads to stable, effective control. For more information on both adaptive system identification methods, the reader is referred to Refs. 15 and 16.

1177

(ii)

(iii)

(iv)

10.5 Control Applications This section briefly outlines application areas where active noise control may be applicable:

(i) Active Control in Ducts. Active control of ducts has been implemented in a number of commercial applications, such as in exhaust stacks at industrial plants and in heating, ventilation, and air-conditioning (HVAC) ducts. The most successful applications have been at low frequencies, where only plane waves propagate in the duct. At higher frequencies, higher order modes propagate in the duct. While control of such fields can be effective, it requires sensing and actuating configurations that are able to sense and control those higher order modes. Another consideration is that even in

(v)

the frequency range where only plane waves propagate the control actuator will generate evanescent higher order modes. Thus, the control system must be configured so that all evanescent modes have effectively decayed by the time they reach the error sensor and/or the reference sensors. Active Control of Free-Field Radiation. Active control has been investigated for applications such as radiation from transformers and even as part of noise barriers for highways. For these applications, the control configuration significantly impacts the amount of control that can be achieved and whether the control is local or global. The control source configuration must be carefully selected for these applications if control in a desired direction or even global control is to be achieved. Active Control in Enclosures. The most successful application currently is for active headsets. These implement control in a small confined volume surrounding the ears and a number of active headsets are commercially available. Other applications include active control in automobile cabins, aircraft fuselages, other vehicles, and rooms. The active control will be more effective at lower frequencies and is dependent to a large extent on the modal density in the enclosure. In a number of applications, local control is achieved, although if the modal density is low, it can be possible to achieve global control, or at least control extended over a much broader portion of the volume. In general, if global control is desired, the number of control actuators used must be at least as great as the number of modes to be controlled. Active Vibration Isolation Mounts. This approach uses active vibration control to minimize the transmission of vibration energy through isolation mounts associated with engines, generators, and so forth. Active mounts have been investigated for automobile engine mounts and aircraft engine mounts, among others. Depending on the mount configuration, it may be necessary to control multiple degrees of freedom in the mount in order to achieve the desired isolation, and it may be necessary to use active mounts on most, if not all, of the engine mounts. Active Control of Transmission Loss. This approach is focused on increasing the transmission loss through a partition, such as an aircraft fuselage or a partition in a building. There are multiple possible approaches, including controlling the structural response of the partition using structural actuators or directly controlling the acoustic field (on either the source or receiver side) through the use

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of acoustic actuators. For these applications, a thorough understanding of the physics associated with the structural response of the partition and its coupling with the incident and transmitted acoustic fields is essential in developing an effective solution. With all of these applications, it is important to do a careful analysis of the noise reduction requirements in order to assess whether active noise control is an appropriate solution. Several applications using active noise control are currently commercially available and others are nearing commercialization. Nonetheless, if one does not carefully consider the application, it is easy to be disappointed in active noise control when it is not as effective as hoped. In review, active noise control is better suited for low frequencies. It is easier to achieve success for tonal noise than it is for broadband noise. It will generally be more effective for compact noise sources than for complex extended sources. If these criteria are met, active noise control could be a very effective and viable solution, although currently it would still generally require the involvement of someone knowledgeable in the field. 11 ARCHITECTURAL ACOUSTICS This section defines the four principal, physical measures used to determine performance hall listening quality. Several perceptual attributes correlated with a physical measure are also listed. The four main measures used to qualify concert halls are the binaural quality index (BQI), the early decay time (EDT), the strength (G), and the initial time delay gap (ITDG). The BQI is defined as BQI = 1 − IACC, where IACC is the average of the interaural cross correlation in octave bands of 500, 1000, and 2000 Hz. The BQI and IACC are measures of the relative sound reached at each ear. If a listener’s two

ears receive identical reflections such as from a ceiling, floor or back wall, the BQI would equal zero and the IACC would equal unity. If reflection at both ears are received from side walls or such that the reflections at both ears are not identical the BQI would be greater than zero and the IACC would be less than unity. BQI values of 0.65–0.71 are representative of the best concert halls. The second measure is EDT. This is a measure of the time required for a 10-dB decay to occur in the signal. This time is then multiplied by a factor of 6 that provides an extrapolated comparison to a 60-dB decay time that is a similar measure of the T reverberation time ( 60 ). Because of the 10-dB decay, the abbreviation of EDT10 is often used. EDT typically has a linear relationship in frequency between occupied and unoccupied halls. This can simplify the process of gathering data. The better concert halls have EDT values in the range from 1.7 to 2.1 sec. The third measure is G, defined as G = Lp − Lw + 31 dB, where Lp is the sound pressure level measured at the point of interest and Lw is the power level of the source. Typically, Lp will decrease as the room volume and room absorption increase. However, if the reverberation time (RT) also increases with room volume, the Lp can be held constant. The better concert halls have G values that range from 3–6 dB while relatively large concert halls with less sound quality have G values in the range of 0–3 dB. The fourth measure is the ITDG, which is defined as the time interval in milliseconds from the direct sound to the first reflected sound. ITDG values are functions of walls, balconies, and other obstacles which provide a reflective surface for the sound. ITDG values should not exceed 35 msec for best results. There are also several perceptual attributes which are used to describe concert halls. These attributes are listed and defined by providing a measure in Table 7.

Table 7 Perceptual Attributes, Physical Measures, and Optimal Values for Concert Halls Perceptual Attribute Spaciousness

Physical Measure

Reverberance

Binaural quality index, lateral fraction Early decay time

Dynamic loudness

Strength

Intimacy Clarity

Initial time delay gap Early/late ratio

Envelopment

Highly diffuse, reverberant sound Frequency dependence of EDT Early (15–35-msec) stage reflections, at frequencies above 500 Hz

Warmth Ensemble

Optimal Values BQI > 0.64 EDT and RT depend on type of music G > 3 dB, low background level ITDG (15–35 msec) Large early/late ratio for speech, depends on music Similarity of EDT and RT EDTlow > EDThigh Stage with ample reflecting surfaces

ACOUSTICS

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This table was produced from class notes provided by William Strong at Brigham Young University.17 It is important to note that halls that provide good speech intelligibility are not necessarily the best halls for music. 12 COMMUNITY AND ENVIRONMENTAL NOISE Measurement and analysis of the impact of noise on both individuals and communities represent a major subfield within acoustics. They are also topics inherently fraught with debate, because human perception of noise is ultimately a subjective phenomenon. To introduce this section, some of the basic principles of outdoor sound propagation are summarized. This is done to demonstrate the impact the propagation environment can have on the noise at a receiver. 12.1 Outdoor Sound Propagation Many phenomena can affect the propagation of noise from source to receiver and therefore have a direct impact on community noise issues. Some of these phenomena are:

• • • • • •

Geometric spreading Atmospheric absorption Ground effect Refraction Atmospheric turbulence Barriers

The effects that each of these can have on sound pressure level are now reviewed. Geometric spreading for distances much larger than the characteristic dimensions of the source will be spherical. For every doubling of distance, Lp will be −6 dB. There are situations, such as supersonic aircraft or steady traffic near a roadway, where the spreading will be cylindrical, which reduces Lp to −3 dB for every doubling of distance. Because of its frequency-dependent nature, absorption causes high-frequency energy to decay much more rapidly than low-frequency energy. In addition to overall level, absorption can play an important role in changing spectral shape and therefore community response to the noise. Another effect is that of reflections of nonplanar waves off a finite-impedance ground. If we consider the basic setup in Fig. 25, where the distance from the source to the receiver is r1 and the distance from the image source to the receiver is r2 , the complex pressure amplitude at the receiver may be expressed as p=

Ae−j kr1 Ae−j kr2 +Q r1 r2

(81)

where the quantity Q is the spherical wave reflection coefficient and may be calculated from Appendix D.4

Fig. 25 Direct and ground-reflected paths from source to receiver.

in (Ref. 18). This coefficient is generally complex and accounts for the amplitude and phase changes encountered when the nonplanar sound wave reflects off the finite-impedance ground. An example of the significant effect that the impedance of the ground can have is shown in Fig. 26, where the impedance values chosen are representative of grass, gravel, and asphalt. In this example, for which one-third octave bands are displayed, the source and receiver are both at a height of 6 ft (1.8 m) and are separated by a distance of 500 ft (152 m). At low frequencies, the wavelengths of sound are such that the direct and reflected sound waves arrive in phase, resulting in constructive interference and a doubling of pressure (+6 dB Lp ). The first interference null varies significantly in frequency for the three surfaces. Note that asphalt begins to approximate a rigid ground surface. Atmospheric refraction can also have a significant impact on the propagation of sound from source to receiver. Refraction is caused by variations in sound speed. The first cause of a variable sound speed is wind. For sound propagation upwind, upward refraction occurs. For sound propagation downwind, downward refraction occurs, as illustrated in Fig. 27. The second cause of sound speed variation is a temperature gradient. During the day, solar radiation causes the ground to warm up and a temperature lapse to occur, meaning that the temperature decreases as a function of height. This condition causes upward refraction to occur as the sound waves bend toward where the sound speed is slower and can create a “shadow region” near the ground where the sound (theoretically) does not reach. At night, however, temperature inversions can occur as the ground cools more quickly than the surrounding air. In this case, downward refraction occurs. This also occurs at the surface of a body of water, where the air temperature just above the water is cooler than the surrounding air. This condition makes

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Fig. 26 Change in sound pressure level relative to free-field propagation due to ground reflections. The source and receiver are both at a height of 6 ft and the distance between them is 500 ft.

Fig. 27 Effect of wind on direction of sound rays radiating from a source.

sound propagated over large distances near the ground more readily audible and can greatly impact community noise issues. Atmospheric turbulence may be viewed as smallscale refraction. Small-scale inhomogeneities in temperature or air velocity can cause sound to be scattered (diffracted). The main effect of turbulence is to generally lessen the impact of other propagation effects. For example, in Fig. 28, Lp has been calculated for propagation over grass with and without atmospheric turbulence. Turbulence minimizes the interference nulls at high frequencies because the scattered sound takes slightly different paths to the receiver. The effect for a refracting atmosphere is similar. Although the shadow zone can readily occur near the ground for upward refraction, turbulence causes some sound to be scattered into the region.

The final phenomenon in outdoor sound propagation that is discussed is the behavior of acoustic waves when a barrier is encountered. For the case of natural barriers, such as hills, acoustic propagation over a hill can often be treated as propagation through an upward-refracting atmosphere over a flat plane. For the case of man-made barriers, such as sound walls, analytical methods may be used to account for the sound that reaches a receiver. If the length of the barrier is much greater than the height, there are four basic paths that need to be accounted for, which are depicted in Fig. 29. The first path is a direct path from the source to barrier and then, due to diffraction, from the top of the barrier to the receiver. The other paths involve one or two ground reflections before reaching the receiver. Although more sophisticated analytical methods exist (e.g., Ref. 18), the basic effects of the multipath problem can be included in the following equation,19 which describes the insertion loss, in decibels, of a thin barrier for a point source and for ranges less than 100 m, where atmospheric effects are ignored: IL = −SPL = 10 log10 [3 + 10N] − Aground (82)

Fig. 28 Change in sound pressure level for propagation over grass with and without atmospheric turbulence.

ACOUSTICS

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Fig. 29 Four different paths that the barrier-diffracted sound can take between the source and receiver.

where Aground is the absorption due to the ground (in decibels) before the barrier is installed, JPL stands for sound pressure level, and N is the Fresnel number, which for sound of wavelength λ can be calculated as N=

2 (dSB + dBR − dSR ) λ

(83)

The distances are the distances from the source to the top of the barrier (SB), from the barrier to the receiver (BR), and the direct line-of-sight distance from the source to the receiver (SR). One point to make is that although the barrier effectiveness does generally increase as a function of frequency, the diffraction from the top of the barrier can play a significant role and result in diminished performance. This is especially true for pathlengths for which the interference is constructive. More general analytical techniques, applicable to thick barriers or diffraction due to gradual structures like hills, do exist and can be found in Refs. 14 and 18. However, explicit inclusion of atmospheric effects (e.g., refraction) is usually accomplished with numerical models.

12.2 Representations of Community Noise Data There are numerous ways of representing the noise to which communities are subjected. One way is to simply display the A-weighted sound pressure level (LA ) as a function of time. One example, displayed in Fig. 30, is the emptying of several trash dumpsters during the early morning at an apartment complex in Provo, Utah. Before the garbage truck arrived, major noise sources were due to intermittent traffic from the nearby street. The most significant noise events were due to the dumpsters being shaken by the hydraulic arms on the truck before being noisily set back down. Other representations of community noise are statistical in nature. Using the same garbage truck example, the estimated probability density function of the A-weighted level is displayed in Fig. 31. The broad tail of large values is caused primarily by the garbage truck noise events. Another statistical representation is a cumulative distribution, which displays the percentage of time that the noise levels exceed a given value. This is shown for the same garbage truck data in Fig. 32. Finally, statistical moments can also be calculated from the time series. For example, the mean level during the 15-min sampling period was 62.4 dBA and the skewness of the data was 1.1. This latter moment emphasizes the non-Gaussian characteristics of the noise distribution because skewness is zero for Gaussian distributions. In addition to A-weighted sound pressure level, there are many other single-number metrics that are used to describe community noise. These metrics have been the result of attempts to correlate subjective response with objective, albeit empirical measures. Some of the commonly used metrics are as follows:

• Equivalent Continuous Sound Level (Leq ). The A-weighted level of the steady sound that has the same time-averaged energy as the noise event. Common averaging times include hourly levels, day levels (7 am–10 pm), evening levels (7–10 pm), and night levels (10 pm–7 am).

Fig. 30 A-weighted Lp and Leq as function of time before, during, and after garbage truck arrival.

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Fig. 31 Probability density function (PDF) of Lp for time series shown in Fig. 30.

For the time interval T which runs between T1 and T2 , Leq is calculated as Leq = 10 log10

1 1 4 × 10−10 T

T2

T1

pA2 (t) dt

(84) where pA is the instantaneous A-weighted sound pressure. The Leq as a function of time was shown for the previous garbage truck example in Fig. 30. • Day–Night Level (DNL or Ldn ). The Leq obtained for a 24-hr period after a 10-dBA penalty is added to the night levels (10 pm–7 am). For individual Leq calculations carried out over 1-hr intervals (L1h ), Ldn may be expressed as Ldn = 10 log10

Fig. 32

1 24

0700

i=0100

100.1[L1h (i)+10]

+

2200 i=0800

10

0.1L1h (i)

+

2400

10

0.1[L1h (i)+10]

i=2300

(85) • Community Noise Equivalent Level (CNEL). The Leq obtained for a 24-hr period after 5 dBA is added to the evening levels (7–10 pm) and 10 dBA is added to the night levels. It can be calculated similar to Ldn , with the appropriate penalty given during the evening (between 2000 and 2200 hours). • X-Percentile-Exceeded Sound Level (LX ). Readily calculated from the cumulative distribution (e.g., see Fig. 32), LX is the level exceed X percent of the time. Common values are L10 , L50 , and L90 . In Fig. 33, Lx values are shown as bars for L99 , L10 , L50 , L90 , and L1 for representative noise environments. • C-Weighted Sound Pressure Level (LC ). Similar to A weighting, but designed to mimic the

Cumulative distribution of time series in Fig. 30 showing fraction of time sound level exceeds given Lp .

ACOUSTICS

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Third-floor apartment, next to freeway Second-floor tenement, new york Aircraft landing Urban residential, near a major airport Urban residential, six miles from a major airport Urban residential Aircraft takeoff

Urban residential, near a small airport

Aircraft overlight

Suburban residential, at city outskirts Small-town residential cul-de-sac

Main street traffic

Small-town residential, on main street Sightseeing aircraft

80% of data

Grand Canyon (north rim)

99%

10

20

30

40

50

60

70

80

50%

1%

90

A-weighted outdoor noise level (dB) Fig. 33

Community noise data showing L99 , L90 , L50 , L10 , and L1 data points for various noise events.

90-phon equal-loudness contour. Consequently, C weighting is more appropriate than A weighting for louder sounds. The equation for the Cweighting filter was given previously in Eq. (7). • D-Weighted Sound Pressure Level (LD ). Developed for assessing the auditory impact of aircraft noise. The weighting curve heavily penalizes high frequencies to which the ear is most sensitive (see Fig. 3). The equation for the D-weighting filter was given previously in Eq. (7). • Effective Perceived Noise Level (EPNL). This metric was designed for characterizing aircraft noise impact and is used by the Federal Aviation Administration (FAA; see FAR Part 36, Sec. A.36) in the certification of commercial aircraft. The metric accounts for (a) the nonuniform response of the human ear as a function of frequency (i.e., the perceived noise level), (b) the additional annoyance due to significant tonal components of the spectrum (the tone-corrected perceived noise level), and (c) the change in perceived noisiness due to the duration of the flyover event. Too involved to be repeated here, calculation procedures for EPNL may be found in FAR Part 36, Sec. A.36.4, or Ref. 20. 12.3 Community Noise Criteria

Because of increased awareness regarding community noise issues, city noise ordinances are becoming more

commonplace. Many of these ordinances are based on maximum allowable A-weighted sound pressure level, broken down into land usage and day or night. In addition, consideration can be given to the nature of the noise source (e.g., is it essential to commerce/industry) and its duration (e.g., is it intermittent or continuous). As an example, portions of the Provo, Utah, noise ordinance, which is representative of many cities, are summarized in Table 8. Continuous sounds are those that have a duration greater than 6 min, intermittent sounds last between 2 sec and 6 min, and impulse sounds last less than 2 sec. The level listed is not to be exceeded at the property line of interest. Noise levels are an important consideration when considering land use. Guidelines for outdoor DNL (for structures in 24-hr/day use) or Leq (for structures being used only part of the day) have been put forth in a land use compatibility report published by the FAA. If the appropriately measured outdoor levels for a yearly average are

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