Industrial Mathematics : The 1998 CRSC Workshop

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INDUSTRIAL MATHEMATICS

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INDUSTRIAL MATHEMATICS The 1998 CRSC Workshop

Edited by Pierre A. Gremaud Zhilin Li Ralph c Smith Hien T. Iran

Center for Research in Scientific Computation North Carolina State University Raleigh, North Carolina

Society for Industrial and Applied Mathematics Philadelphia

Copyright ©2000 by the Society for Industrial and Applied Mathematics. 109 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Industrial mathematics : the 1998 CRSC workshop / edited by Pierre A. Gremaud ... [et al.]. p. cm. "Proceedings of the Industrial Mathematics Modeling Workshop for Graduate Students sponsored by the Center for Research in Scientific Computation of North Carolina State University (NCSU), Raleigh, NC, from July 27 to August 4, 1998" -CIP pref. Includes bibliographical references and index. ISBN 0-89871-467-2 (pbk.) 1. Engineering mathematics-Industrial applications-Congresses. 2. Manufacturing processes-Mathematical models-Congresses. I. Gremaud, Pierre A. II. North Carolina State University. Center for Research in Scientific Computation. III. Industrial Mathematics Modeling Workshop for Graduate Students (4th : 1998 : Raleigh, N.C.) TA331 .153 2000 670'.1'51-dc21

This book resulted from the workshop held at North Carolina State University July 27-AugUSt 4, 1998.

00-058792

V

Contents Preface

vii

Participants

ix

Maximizing the Deflection to Voltage Ratio in a Circular Piezoelectric Actuator Design (Battelle Memorial Institute)

1

Modeling of the Radius of Curvature for THUNDER Actuators (FACE International Corporation)

11

Mathematical Modeling of High Power Amplifier (HPA) and HPA Linearization Through a Predistorter (The Aerospace Corporation) 23 Time Dependent Consolidation of Fine Powders (Jenike & Johanson, Inc.)

37

High Precision, High Accuracy Control of a Fluid Microdispenser Under Variable Load (D.H. Wagner Associates)

49

Mass Transport and Surface Reaction of Immunoassay (Ortho-Clinical Diagnostics, Inc., Johnson & Johnson Company)

59


vii

Preface This volume contains the proceedings of the Industrial Mathematics Modeling Workshop for Graduate Students sponsored by the Center for Research in Scientific Computation at North Carolina State University (NCSU), Raleigh, NC, from July 27 to August 4, 1998. The workshop, which was the fourth one of this type held at NCSU, brought together 35 graduate students. As noted in the participants list, these students represented a large number of graduate programs across the country including the University of Missouri-Rolla, University of New Orleans, University of Colorado, University of New Hampshire, Virginia Tech, University of California (Santa Barbara, Irvine, Berkeley, and Los Angeles), Texas AfeM, Louisiana State University, University of Texas, University of Tennessee, Worcester Polytech, University of Delaware, University of Massachusetts, Kansas State University, University of Alabama at Birmingham, University of Tulsa, Middle Tennessee State University, Washington University, University of Maryland, North Carolina State University, University of Pittsburgh and Ohio State University. The students were divided into six teams to work on "industrial mathematics" problems presented by industrial scientists. These were not the neat, wellposed academic exercises typically found in coursework, but were challenging real world problems from industry or applied science. The problems, which were explained to the students on the first day of the workshop, required fresh insights for their formulation and solution. Each group spent the first nine days of the workshop investigating their problem and then reported their findings in half-hour public seminars on the last day of the workshop. The following is a list of the presenters and the projects which they presented. • Chad Bouton (Battelle Memorial Institute): Maximizing the Deflection to Voltage Ratio of a Circular Piezoelectric Actuator Design. • Bryan Cunningham, Karla Mossi and Norvell Rose (FACE International Corporation): Modeling of the Radius of Curvature for THUNDER Actuators. • Tien M. Nguyen (The Aerospace Corporation): Mathematical Modeling of High Power Amplifier (HPA) and HPA Linearization Through a Predistorter. • T. Antony Royal (Jenike & Johanson, Inc.): Time Dependent Consolidation of Fine Powders. • Jeff Sachs (D.H. Wagner Associates): High Precision, High Accuracy Control of a Fluid Microdispenser Under Variable Load. • Zhong Ding (Ortho-Clinical Diagnostics, Inc., Johnson & Johnson Co.): Mass Transport and Surface Reaction of Immunoassay. These projects represent a broad spectrum of mathematical topics and applications. The problems presented by the Battelle and FACE scientists involved the modeling of novel high performance smart materials and required elements of elasticity theory, material science, engineering mechanics and thermodynamics. From a mathematical perspective, these projects required the formulation, analysis and solution of certain ODE and steady state equations and the consideration of commensurate numerical and optimization techniques. The problem presented by the Aerospace Corporation arises in a Global Broadcasting Service (GBS) application and entailed the investigation of the mathematics and physics associated with signal processing. Additionally, the students on this team developed simulation routines to numerically implement filters which they had designed. The problems investigated by Teams 4 and 5 involved the development and numerical solution of PDE models based on Newtonian principles and physical conservation laws. This included the investigation and

viii development of numerical methods utilizing state of the art ODE and optimization algorithms. The Johnson fe Johnson problem provided Team 6 with an exposure to the development of PDE models and associated numerical methods for a biomedical application. From discussions throughout the workshop as well as the formal presentations on the final day, the students observed that a large number of mathematical and physical disciplines play an important role in the solution of industrial problems. Students in previous and subsequent workshops have had similar exposure to industrial problems in which statistics and probability, number theory, algebra and combinatorics played a fundamental role. We, the organizers, strongly believe that workshops of this type provide substantive non-academic research related experiences for graduate students while contributing to the research efforts of industrial participants. In addition, these experiences facilitate the development of graduate students' ability to communicate and interact with scientists who are not traditional mathematicians but require and employ various mathematical tools in their research. Finally, the experience gained in the workshop will help to improve graduate student's vision about the type of mathematics and science that should be pursued by students who aspire to both non-academic and academic postgraduate careers. The success of the workshop was greatly enhanced by active participation in a very friendly atmosphere and almost uninterrupted work during the nine days of attendance, and the organizers are most grateful to participants for their contributions. The organizers would like to thank the National Security Agency, the National Science Foundation, the Center for Research in Scientific Computation, and the Department of Mathematics at North Carolina State University for their generous financial support of this and previous workshops. Special thanks and appreciation are extended to the faculty and staff of the Center for Research in Scientific Computation, the Department of Mathematics and North Carolina State University for the provision of excellent facilities and services. Pierre Gremaud Zhilin Li Ralph C. Smith Hien T. Tran Raleigh, 2000

ix

Participants A. Graduate Students 1. Karen Bliss, University of Missouri-Rolla. [email protected] 2. Philippe Bremond, University of New Orleans, [email protected] 3. Jim Buchanan, University of California, Santa Barbara, [email protected] 4. Danielle Bundy, University of Colorado, [email protected] 5. Chongsheng Cao, University of California, Irvine, [email protected] 6. Michelle Capozzoli, University of New Hampshire, [email protected] 7. Timothy Chevalier, University of New Hampshire, [email protected] 8. Jerawan Chudoung, Virginia Tech, [email protected] 9. Linda Connolly, University of New Orleans, [email protected] 10. Steven J. Coorpender, University of New Orleans, [email protected] 11. Daniel Finkel, North Carolina State University, [email protected] 12. Simone Flory, Louisiana State University, [email protected] 13. Jayadeep Gopalakrishnan, Texas A&M 14. Jeffrey Grant, North Carolina State University, [email protected] 15. Kevin Hogan, University of Texas, Austin, [email protected] 16. Henrik Kalisch, University of Texas, Austin, [email protected] 17. David Kan, University of California, [email protected] 18. Jennifer Kyzar, University of Tennessee, [email protected] 19. Brian Lewis, North Carolina State University, [email protected] 20. Anastasios Liakos, University of Pittsburgh, [email protected] 21. Renee Yong Ran Liu, University of California, Berkeley, [email protected] 22. Jordan E. Massad, North Carolina State University, [email protected] 23. John Matthews, North Carolina State University, [email protected] 24. Shailesh Naire, University of Delaware, [email protected] 25. Lakshmi Puthanveetil, University of Massachusetts, [email protected] 26. Sumithra Ramaswamy, Ohio State, [email protected] 27. Bob Sims, University of Alabama at Birmingham, [email protected] 28. Alexandra B. Smirnova, Kansas State University 29. Mohamed Tawhid, Univ. of Maryland Baltimore County, [email protected] 30. Troy Tokarchik, University of Tulsa, [email protected] 31. Ilker Tunay, Washington University, St. Louis, [email protected] 32. Shree Yvonne Whitaker, North Carolina State University, [email protected] 33. Steve Wilmarth, Texas A & M , [email protected] 34. Shannon Wynne, University of California, Irvine, 35. Igor Zhislin, Middle Tennessee State University, [email protected]

X

B. Problem Presenters and Faculty 1. H.T. Banks, North Carolina State University, [email protected] 2. Negash Begashaw, Clark Atlanta University 3. Chad Bouton, Battelle Memorial Institute, [email protected] 4. Bryan Cunningham, FACE International Corp., [email protected] 5. Zhong Ding, Ortho-Clinical Diag, Johnson & Johnson, [email protected] 6. Rich Fabiano, University of North Carolina, Greensboro, [email protected] 7. Pierre Gremaud, North Carolina State University, [email protected] 8. Kazi Ito, North Carolina State University, [email protected] 9. Zhilin Li, North Carolina State University, [email protected] 10. Hung Ly, North Carolina State University, [email protected] 11. Negash Medhin, Clark Atlanta University, [email protected] 12. Karla Mossi, FACE International Corp., [email protected] 13. Tien Nguyen, Aerospace Corp., [email protected] 14. Norvell Rose, FACE International Corp., [email protected] 15. Tony Royal, Jenike & Johanson, Inc., [email protected] 16. Jeff Sachs, D.H. Wagner Assoc., [email protected] 17. Ralph C. Smith, North Carolina State University, [email protected] 18. Hien T. Iran, North Carolina State University, [email protected]

Chapter 1

Maximizing the Deflection to Voltage Ratio in a Circular Piezoelectric Actuator Design Student Team: Steven Coorpender1, Daniel Finkel2, Jennifer Kyzar 3 , Bob Sims4, Alexandra Smirnova5, Mohamed Taw hid6 Problem Presenter: Chad Bouton, Battelle Memorial Institute Abstract Piezoelectric actuators are employed in numerous applications, particularly when small, accurate deflections are required. The amount of deflection that can be attained is dependent on many factors including the amount of applied voltage and the geometry of the design. Efficient actuators are especially important when power is limited. We developed an ODE model of a thin circular piezoelectric patch attached to a single layer circular mounting surface (a unimorph-type actuator). All material properties and boundary conditions were specified in the design. The model was used to optimize the design of the two-layer structure by varying the patch radius and mounting layer thickness.

1.1

Introduction

Battelle Memorial Institute, based in Columbus, Ohio, is a non-profit research and development organization with laboratories throughout the world. The institute develops innovative technology for government and industry that is integrated into a variety of products. Examples of these products range from hologram technology to drug developments for health needs. Chad Bouton, a research engineer at Battelle, was invited by the workshop organizers to present a current problem of interest to Battelle concerning modeling and design issues for a thin piezoceramic actuator bonded to an University of New Orleans 2 North Carolina State University 3 University of Tennessee 4 University of Alabama at Birmingham 5 Kansas State University 6 University of Maryland, Baltimore County 1

2

CHAPTER 1. MAXIMIZING

DEFLECTION IN A PIEZOELECTRIC

ACTUATOR

underlying substrate material. Specifically, we consider a circular piezoceramic patch bonded to a circular metallic plate of larger radius. The edges of the plate are assumed to be clamped. Unimorph actuators of this type are commonly employed in applications ranging from micropositioners to precision valves, and in all cases, the accurate characterization of the displacements in the underlying plate due to applied voltages to the patch are crucial to the performance of the device. The characterization of patch inputs to underlying structures, including plates, beams and shells, has been investigated from several perspectives. In all cases, the active contribution of the patches is incorporated through the strains produced by the elongation of dipole moments in response to an applied field. The models differ in the manner through which these input strains are coupled with the dynamics of the underlying structures. As detailed in [3], PDE models quantifying the structural dynamics can be derived by incorporating the generated strains in appropriate energy formulations for the structures. The success of this approach for obtaining a highly accurate model for a circular plate with surface-mounted piezoceramic patches is illustrated in [1] with the experimental implementation of a corresponding model-based controller described in [2]. While this approach provides a highly accurate model, the numerical solution of the modeling PDE can inhibit certain design considerations. In this paper, we consider a simplified model which characterizes the displacements of a fixed piezoceramic plate with a surface-mounted piezoceramic actuator in terms of an ODE which can be solved analytically. This analytic solution can then be employed to design actuators which provide optimal performance with minimal power input. We note that aspects of this approach are analogous to those employed by Kim and Jones [5] in their analysis of patch contributions to thin plates. The approaches differ, however, in the geometries, solution techniques, and resulting optimality relations. In Section 1.2, we derive the model for the composite structure in two steps. In the first, general ODE for the central circular plate/patch composite and outer plate annulus are derived for unspecified boundary conditions. In the second step, appropriate boundary conditions are derived and used to determine the unspecified coefficients in the inner and outer components, and hence characterize the displacement at any point in the plate. Attributes of the model are illustrated in Section 1.3 through consideration of design criteria such as the determination of an optimal patch radius and thickness. Finally, we note that aspects of this investigation are summarized in [4].

1.2

Structural Model

We consider a thin circular plate of radius RQ and thickness h having clamped edges. A circular patch having radius Rpe and thickness hpe is bonded to one side of the plate as depicted in Figure 1.1. To simplify the discussion, we assume that the bonding layer is negligible and direct the reader to [3] for details regarding the inclusion of the bonding layer. The Young's modulus, flexural rigidity and Poisson ratio for the plate are respectively denoted by E, D and v while the notation Epe,Dpe and vpe is used to denoted commensurate characteristics of the patch. The transverse displacement of the plate is denoted by w. Finally, we assume that the plate is subjected to a uniform pressure q. The model is developed by quantifying separately the displacements of the circular region composed of both the piezoceramic patch and plate and the displacement of the outer annular plate region. These components are then combined through the choice of appropriate boundary and interface conditions.

1.2.1

Plate Model

We consider first the development of a model for the annular region J7a comprised solely of the underlying plate. Under standard assumptions for linear, thin, uniform plates, force and moment

1.2. STRUCTURAL MODEL

3

Figure 1.1: Thin circular plate with a surface mounted piezoceramic actuator. balancing yield the expression (1.1)

where Q r ,M r and Mg respectively denote the radial shear resultant, radial moment result and tangential moment resultant ([3,6,7]). Furthermore, it is illustrated in [6] that the moments can be expressed as (1.2)

Finally, the force resultant and pressure can be related through the expression (1.3)

To quantify the displacement due to the pressure, we combine (1.!)-(!.3) to obtain the differential equation

or equivalent ly, (1.4)

The relation (1.4) can then be integrated to obtain (1.5)

where Ci,C2 and cs are constants which will ultimately be determined from the boundary conditions and interface conditions with the circular component SliThe model for the composite material, comprised of the plate and piezoceramic patch and denoted by fii in Figure 1.1, differs only in the expression of the flexural rigidity Dc. To specify Dc, it is necessary to first compute the height zns of the neutral surface for the composite. This is accomplished by balancing forces and noting that the neutral surface denotes the locus of points which

4



ACTUATOR

exhibit no strain under the action of a pure applied bending moment. If we let ar(z) denote the radial stress at height z, then summation of forces over a differential element of the plate yields

where A is the cross-sectional area for the composite. Under the assumption that the plate is axisymmetric, and letting 7£ denote the radius of curvature for the neutral surface, the formulation in terms of strains yields

Integration then yields the expression

for the neutral surface. A homogenized flexural rigidity for the composite material is then given by

(1.6)

It is noted that Dc depends on both the material properties and dimensions of the component materials. It is also observed that for the region of the plate devoid of patch material (hpe = 0 and z ns — h/2), the expression (1.6) reduces to the familiar relation

for a homogeneous plate.

The displacement w for the composite is then specified by the differential equation (1.4) with the flexural rigidity specified by (1.6). This then yields the general solution

(1.7) where the constants 04, c$ and eg are specified by the boundary and interface conditions.

1.2.2

Boundary and Interface Conditions

The relations (1.5) and (1.7) specify the displacements of the circular composite and annular plate region in terms of constants c\-c% which are dependent upon the boundary conditions and interface conditions between the regions. To ensure compatibility between the two components and enforce mechanical boundary conditions, we enforce the following constraints. To accommodate the clamp around the edge of the disk, we consider the fixed boundary conditions

(1.8)

1.2 STRUCTURAL MODEL

5

A third constraint is provided by the implicit condition witch is used to conforce axial symmetry at the center of the disk. The remaining three constraints arise when enforcing compatibility between the composite dis and outer annual region. We first enforce equal displacements and slopes through the constraints (1.10) The third interface condition is the moment constraint (1.11) which ensures that the radial moment in the annulus is equal and opposite when summed with the total moment for the composite. The former is defined in (1.2). The total moment for the composite is given by where Mpe denotes the moment generated by the piezoceramic patch when it strains in response to an applied voltage. To specify Mpe, we first note that the free strains generated by the patch, for an input voltage V, are approximately linear and given by

The piezoelectric constant dsi is typically estimated via a least squares fit to data, especially for patches which have aged or been repoled. The active moment generated by the patch is then

The enforcement of the constraints (1.8)-(1.11) yields the expressions

6



ACTUATOR

and

for the constants of integration. When employed in the displacement relations (1.5) and (1.7), this provides the capability for computing the displacement at any point which is due to the uniform pressure field and applied voltage to the patch. Representative results are provided in the next section.

1.3

Properties of the Model

The examples of this section illustrate various aspects about the model. The ultimate goal in these examples is the determination of parameters such the thickness of the underlying plate (mounting layer) h, the patch radius Rpe, and the patch thickness hpe. Example 1. Variable Plate Thickness and Patch Radius The radius of the plate was held constant at 4.0 mm arid the thickness of the piezoelectric patch was held constant at 200 microns. We then varied the thickness of the plate and the radius of the piezoceramic patch in order to find the maximum deflection to voltage ratio as illustrated in Figures 1.2 and 1.3. We observed the optimal thickness of the mounting layer to be 49.85 microns and the optimal radius of the piezoceramic layer to be 3.67 mm.

Figure 1.2: Relationship between plate thickness and patch radius.

1.3. PROPERTIES OF THE MODEL

7

Example 2. Constant Plate Thickness Once the optimal thickness of the plate was determined, we investigated the effect of varying the radius of the piezoceramic patch. Figure 1.4 shows a cross section of deflection to voltage ratio versus radius of the piezoceramic patch with the constant plate thickness set at the optimal value of 49.85 microns. Note that the deflection drastically decreases when the radius extends beyond the optimal value of 3.67 mm.

Figure 1.3: Relationship between plate thickness and patch radius.

Figure 1.4: Deflection as a function of radius at the optimal patch thickness.

8


DEFLECTION IN A PIEZOELECTRICC AACTUATOR

Example 3. Constant Radius of Piezoceramic Patch Figure 1.5 illustrates a cross section of deflection to voltage ratio versus plate thickness with the radius of the piezoceramic held constant at its optimal value of 3.67 mm. Note the optimal thickness is approximately 49 microns. The amount of deflection will decrease if the thickness is allowed to become too small. We found the deflection for the optimal design to be -22.54 nm as illustrated in Figure 1.6. This is within the range of values observed by Battelle Memorial Institute in their applications.

Figure 1.5: Deflection as a function of thickness at the optimal patch radius.

Figure 1.6: Optimal deflection at one volt.

1.4. CONCLUSIONS

1.4

9

Conclusions

We have presented a model for the specified piezoelectric actuator system and optimized the design with regard to certain aspects of the geometry. In the problem presented to us, the radius of the piezoceramic patch and the thickness of the mounting layer were the only variable parameters. With this model, it was also possible to investigate changes in deflection which would result from adjusting other parameters such as material stiffness, patch thickness, and mounting layer radius. In addition, the model could be easily generalized to include additional piezoceramic or mounting layers, and an improvement in accuracy could be achieved through a consideration of bonding materials. The original goal for this workshop was to maximize the deflection while minimizing the power consumption in a circular piezoelectric design. Unfortunately, we did not have time to incorporate power consumption into the model. This is a future consideration and a model similar to ours will most likely be used in this development. Acknowledgments

We would like to thank Chad Bouton, Rich Fabiano, and Ralph Smith for their invaluable assistance during the workshop. We would also like to thank everyone from North Carolina State University and the Center for Research in Scientific Computation for their generous hospitality and the organizers for giving us the opportunity to participate.

References [1] H.T. Banks, R.C. Smith, D.E. Brown, V.L. Metcalf and R.J. Silcox, "The Estimation of Material and Patch Parameters in a PDE-Based Circular Plate Model," Journal of Sound and Vibration, 199(5), pp. 777-799, 1997. [2] H.T. Banks, R.C. Smith, D.E. Brown, R.J. Silcox and V.L. Metcalf, "Experimental Confirmation of a PDE-Based Approach to Design of Feedback Controls," SIAM Journal on Control and Optimization, 35(4), pp. 1263-1296, 1997. [3] H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Masson/John Wiley, Paris/Chichester, 1996. [4] S. Coorpender, D. Finkel, J. Kyzar, R. Sims, A. Smirnova, M. Tawhid, C. Bouton and R.C. Smith, "Modeling and Optimization Issues Concerning a Circular Piezoelectric Actuator Design," Proceedings of the 1999 Winter Annual Meeting of the ASME. [5] S. J. Kim J.D. Jones, "Optimization of Piezo-Actuator/Substructure Coupling for Active Noise and Vibration Control," Proceedings of the Conference on Recent Advances in Active Control of Sound and Vibration, Eds., C.A. Rogers and C.R. Fuller, Blacksburg, VA, April 15-17, 1991, pp. 78-91, 1991. [6] A.W. Leissa, Vibration of Plates, NASA SP-160, 1969, Reprinted by the Acoustical Society of America through the American Institute of Physics, 1993. [7] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, Second Edition, McGrawHill Book Company, Inc., New York, 1987.


Chapter 2

Modeling of the Radius of Curvature for THUNDER Actuators Student Team: Michelle Capozzoli1, Jayadeep Gopalakrishnan2, Kevin Hogan3, Jordan Massad4, Troy Tokarchik5, Steve Wilmarth6 Problem Presenters: Bryan Cunningham, Karla Mossi and Norvell Rose, FACE International Corporation, http://www.faceco.com Abstract

THUNDER is a composite ferroelectric structure which, for our investigation, was comprised of an aluminum layer, a PZT-5A ceramic, a stainless steel layer, and a hot-melt adhesive sheet between each layer. This assembly is heated to an elevated temperature under pressure and then cooled to room temperature. The adhesive solidifies at a high temperature. Further cooling imparts a compressive pre-stress to the surfaces of the ceramic which significantly enhances the resulting actuator's performance and ruggedness. Due to the resulting internal stress-state, the actuator possess a radius of curvature which to some extent determines its characteristic behavior. The radius of curvature is the result of the properties of the constituent materials, the geometry of the constituent materials, the temperature at which the adhesive solidifies, and the reorientatiori of dipoles during repoling. An ideal modeling algorithm would predict a radius of curvature arid its impact on the actuator's properties. With this tool such variables as switching from stainless steel to another material, or changing the thickness of the substrate could be modeled and evaluated and THUNDER elements could be custom designed for specific applications.

2.1

Introduction

Headquartered in Norfolk, Virginia, FACE International Corporation is a developer and manufacturer of actuators and sensors. In 1997, FACE corporation obtained the manufacturing license for THUNDER (THin-layer composite UNimorph ferroelectric Driver and sEnsoR) developed by NASA. This '•University of New Hampshire Texas A &; M 3 University of Texas 4 North Carolina State University 5 University of Tulsa 6 Texas A & M 2

11

12

CHAPTER 2. MODELING THE CURVATURE FOR THUNDER ACTUATORS

technology allows the production of compact, adaptable and extremely rugged piezoelectric devices. Since 1997, FACE has introduced a variety of high performance actuators and sensors in a range of shapes and sizes. Companies representing a wide range of industries have expressed strong interest in using these devices. Practical applications exist in the automotive to the aerospace industries as well as in smart structures to industrial process control. Here we investigate the modeling of certain static properties of THUNDER actuators which result from the fabrication process. These actuators are typically constructed from layers of piezoceramic wafers, aluminum, stainless steel, brass or aluminum bonded with hot-melt adhesive. To construct the actuators, the assembly is heated under pressure to high temperature and then cooled to room temperature. During cooling, the adhesive solidifies and subsequent cooling leads to the production of internal stresses due to differing thermal properties of the component materials. This yields the characteristic curvature of the material. Once the material is cooled, it must be repoled since the temperatures required for bonding are in the proximity of the Curie temperature. The rotation of dipole moments in the piezoceramic provides a second source of internal stress which reduces the final curvature. This fabrication process significantly enhances the robustness of the actuator with respect to mechanical impacts and voltage levels. As a result of this construction, high voltages can be applied to the actuators and large displacements sustained without causing damage. This provides the actuators with significant strain and force capabilities. This investigation focused on the quantification of strains and subsequent curvature in the actuator produced during the manufacturing process. The strain model was constructed in two steps: (i) Model the strains due to thermal gradients and (ii) Characterize the strains due to repoling. The quantification of strains due to thermal gradients has been investigated for a variety of applications (e.g., see [1,5,6,7]) with certain aspects having been considered for THUNDER actuators [8]. We extend this analysis to obtain relations specifying the strains at the actuator surface and curvature for THUNDER actuators in terms of thermal properties of the components. The quantification of stresses and strains due to poling is based on domain theory for general ferroelectric materials [9,12] with certain analogies drawn from ferromagnetic domain theory [4,10]. The thermal and electrostatic stress are then combined with elasticity theory to provide a model for the induced curvature. Finally, we note that the results from this investigation are summarized in [3].

2.2

Manufacturing Process

A typical THUNDER actuator is comprised of a piezoceramic wafer, a metallic backing material, hot-melt adhesive layers, and optional metallic top layers as depicted in Figure 2.1. Some standard materials that are used for backing include aluminum, stainless steel, titanium and brass. The adhesive employed by FACE International Corporation is the high performance thermoplastic LaRCSI [2]. The design and performance of the actuator can be modified by varying the number of layers and/or raw materials. The composite materials are then bonded under the following conditions. The assembled THUNDER components are placed in a vacuum bag (100 kPa) and then heated in an autoclave from room temperature to 204° C at a rate of 5.6° C/min. The chamber is then pressurized with nitrogen to 241.3 kPa and the temperature is raised to 325° C. The composite is maintained under these conditions for 30 minutes and then cooled at a rate of 5.6° C/min to 52° C. At this point, both the autoclave pressure and vacuum are released at a gradual rate to minimize mechanical stresses which might crack or break the material. Details concerning the construction process can be found in [11].

13

2.3. STRAINS DUE TO THERMAL PROPERTIES

(a)

(b)

Figure 2.1: (a) Schematic of a THUNDER actuator; (b) Layers employed in the composite. The LaRC-SI adhesive solidifies at approximately 270° C and subsequent bending occurs as the composite is cooled due to differing thermal coefficients of the components. The modeling of the thermal effects on the final curvature is thus considered for temperatures from 270° C to 25° C. The Curie temperature for the employed piezoceramic compound PZT-5A is approximately 350° C so the previously poled material suffers some loss of polarization in the manufacture process. Hence it must be repoled after cooling through the application of a sustained DC voltage.

2.3

Strains Due to Thermal Properties

The first strain mechanism which we considered is that due to the differing thermal coefficients in the metallic backing and top layers, piezoceramic wafer, and adhesive layers. This produces differential contraction rates as the composite is cooled and yields the subsequent curvature. To model the thermal strains and corresponding stresses, we assume that the composite is comprised of N layers with the modulus of elasticity and coefficients of thermal expansion for each layer respectively denoted by Ej and oy. The coordinate system is oriented so that the backing material lies in the x, z-plane with y = 0 corresponding to the outer edge of the composite as depicted in Figure 2.2. The width of the jth layer is denoted by bj while hj indicates the y-coordinate of the top edge of the jth layer. Finally, the strain at the y = 0 position and curvature at the neutral axis are respectively denoted by £Q and k. The model is derived under the assumptions that linear elasticity and thermal relations can be employed in each layer and that the heating process is uniform. Furthermore, it is assumed that strains in the y and z-directions are negligible. We note that this latter assumption must be revaluated when considering the strains due to poling or actuation due to the magnitude of piezoelectric effect.

Figure 2.2: Orientation of the composite THUNDER actuator with five layers.

14

as

CHAPTER 2. MODELING THE CURVATURE FOR THUNDER ACTUATORS Under these assumptions, the total strain e(y) at a height y in the composite can be expressed

(2.1) where a(y) denotes the stress arid A£ is the change in temperature. The Youngs' modulus and thermal coefficient E(y) and a(y) are the piecewise constant functions given by E(y) = Ej and a(y) = Oj when y is in the jth layer. Furthermore, the assumption that strains are linear in the transverse direction yields the relation

(2.2) between the strain EQ at the outside edge of the composite and the strain at a height y. Combination of (2.1) and (2.2) then yields the stress expression

(2.3) Force and moment balancing provides the constraints necessary to solve for £Q and k. For a layer having cross sectional area Aj, the forces due to elastic and thermal stresses are

while the moments are Force balancing provides the relation

while moment balancing yields

To facilitate the solution of the strain EQ and curvature k, we reformulate the constraints as the linear system (2.4) where

whe

(2.5)

2.4. STRAINS DUE TO REPOLING

15

and

(2.6) It is noted that the matrix A and vector / are easily constructed once the geometrical and material properties of the layers in the composite have been specified. Once £ has been determined, the strains at a height y in the material are specified by (2.2) and the radius of curvature is given by R — 1/k. In the next section, we include the strains due to repoling. In combination, this forms the basis for determining the effect of manufacturing conditions on the final geometry of the actuator.

2.4

Strains Due to Repoling

The second component of the strains is due to the reorientation of domains which occurs when the materials are repoled following the bonding phase of the manufacturing process. To quantify these effects, it is necessary to consider certain aspects of ferroelectric domain theory. Details regarding this theory can be found in [9,12] with analogous theory for ferromagnetic materials provided in [4,10]. The piezoceramic wafers are in a poled state before the heating process with dipoles aligned in the transverse direction. In general^ when these materials are heated above the Curie temperature, the domain structure is lost and the properties of become paraelectric. In this state, the material acts as a normal dielectric. When the temperature is reduced below the Curie temperature, the domain structure reforms to minimize intergranular stresses. The material exhibits no net polarization, however, since the orientation of domains is random. Finally, repolarization through the application of a large applied DC voltage aligns the domains and yields a configuration which provides the material material with its actuator and sensor capabilities. The three stages are depicted in Figure 2.3. From this figure, it is observed that poling in the transverse direction produces positive strains in the y-direction and corresponding negative strains in the x-direction. To quantify the spontaneous strain ey in the material due to dipole ordering, we follow the approach in [4, pages 343-345] or [10, pages 99-104] where analogous relations are developed for ferromagnetic materials. It is first noted that if 0 is the angle between dipoles in a domain and the y-axis, then the strain varies as Under the assumption of a random orientation of domains, the strain due to the onset of domain formation is given by

Finally, it follows that the saturation electrostriction Xs due to dipole alignment during repoling is so that

(2.7) The relation (2.7) can be utilized in the following manner. The saturation electrostriction Xs can either be measured or estimated through a least squares fit to data for a given material. The spontaneous transverse strain ey due to domain alignment is then specified by (2.7). To determine the effect of polarization on curvature, it is necessary to quantify the resulting strain in the ^-direction.

16


Figure 2.3: Material behavior of the PZT wafer as it is cooled through the Curie temperature (after Jiles [10]). (a) Material in the paraelectric regime above the Curie temperature; (b) Random domain structure below the Curie temperature; (c) Saturation polarization due to an applied voltage. This can be accomplished through either a trigonometric analysis similar to that used to obtain (2.7) or by employing the relation where v denotes the Poisson ratio for the material. The longitudinal strains produced during repoling can then be expressed as

(2.8) where v and As are parameters which must be measured or estimated through a least squares fit to data for a given piezoceramic material. In the piezoceramic layer, the relation (2.8) can be combined with the thermal relation (2.1) to yield

For the composite structure, this yields the stress relation

where

isolates the electrostatic strains due to repoling to the piezoceramic layer. Force and moment balancing in this case yields the A matrix defined in (2.5) and the modified / vector (2.9)

2.5. MODEL VALIDATION

17

which incorporates the effects of repoling on the stresses in the piezoelectric layer. Solution of the system (2.4) then provides the strain SQ and curvature k which incorporate both thermal and electrostatic effects. The validity and limitations of the model are illustrated in the next section where the characterization of a THUNDER actuator through comparison with data is considered.

2.5

Model Validation

To illustrate the performance of the model, we compare the geometries determined by the model with data from a variety of THUNDER actuators designed and constructed by FACE International Corporation. The first step necessary for comparison with experimental data is the formulation of the geometry specified by the model with that measured for the actuators. The actuators are constructed with the bottom layer extending beyond the PZT and top layers as illustrated in Figure 2. la. The curvature in the region covered by the actuator layer is assumed uniform while the tabs on either side remain unbent since they are not subjected to either thermal or piezoelectric strains. This produces the configuration depicted in Figure 2.4. The model yields the radius of curvature R = I/A; for the actuator region while the height /i', which includes the tabs, is measured for the physical actuators. Furthermore, the actuator length s and length t of each tab are measured for a given actuator. It is thus necessary to quantify the relation between the radius of curvature R, the measured dome height h' and the lengths s and t. Given the radius of curvature R specified by the model and the measured length s, the subsumed angle 9 is given by Furthermore, the length t and height h of the tabs are related by the expressions

and

Since R = r + h', it follows directly that (2.10) Given the measured values of s and £, and the radius R specified by the model, the relation (2.10) provides the dome height h' which can be directly compared with measurements for various actuator configuration. Four actuators were considered with the dimensions and component materials summarized in Table 2.1. It is noted that the actuators differ in the metals employed for the bottom layer, the number of layers, and the thicknesses of the individual layers. The values for the modulus of elasticity E and thermal coefficients a reported by manufacturers for the component materials are compiled in Table 2.2. The first actuator had a bottom layer constructed from stainless steel and a thin aluminum top layer. The measured values for /i', which are reported in Table 2.3, indicate a 13.6% reduction in the dome height after repoling. This is in accordance with the explanation in Section 4 that the curvature is reduced when dipole moments align with the applied field in the y-direction. The performance of the model is indicated in Table 2.3 for two sets of parameters. For the actuator before repoling, the parameters specified by the manufacturers (Parameter Set 1) yields a dome height h' which differs from the measured value by approximately 12%. In the second case, values

18


Figure 2.4: Geometry of the actuator with tabs. of E and a were estimated through a least squares fit to the data to obtain the Parameter Set 2 which is summarized in columns 3 and 5 of Table 2.2. Due to the manner through which these parameters were determined, the corresponding dome height matches the experimental value for the actuator before poling. We note that the estimation of parameters for applications of this type is typically necessary to accommodate the bulk nature of certain mechanisms, introduced during the construction process, and the inherent uncertainties in reported values. To quantify the strains due to domain rotation during repoling, the Poisson ratio was taken to be v = .3 and the saturation electrostriction value \s = l.lle-3 was determined to quantify the height after poling. The feasibility of this value is indicated by comparing it with the strain SQ = -4.7e-3 computed at the bottom edge of the actuator. Actuator 2 differs both in the employed materials and the dimensions. It was constructed with a thin brass bottom layer and no top layer. As with Actuator 1, the use of the parameters specified by manufacturers yields a height h' which differs from the measured value by about 16% whereas the estimation of E and a for brass yields parameters which are within 6% of reported values and an exact fit to the height. The parameter values for the remaining layers in this second case were taken to be those estimated for the materials in Actuator 1. The value of h1 — .388 cm predicted by the model after repoling is obtained with the saturation electrostriction \s — l.lle-3 estimated for Actuator 1. The accuracy of the prediction attests to the validity of the model used to quantify strains due to domain rotation. Finally, we note that the physical height h' — .380 cm can be matched by the model with the parameter As = 1.245e-3. The third actuator differed from the second in that it had a titanium base layer as compared with the slightly thinner brass layer employed in Actuator 2. Because the thermal coefficient for titanium is significantly smaller than that of brass, the measured dome height is less than half that of Actuator 2. In this case, it is noted that the use of the manufacturer specified parameters (Parameter Set 1) yields a modeled dome height that is within 2.5% of the measured value. Moreover, when the previously estimated parameters for the LaRC-SI and PZT were employed with estimated parameters for titanium, the predicted dome height matches the measured value. Finally, when the saturation electrostriction Xs = l.lle-3, determined for Actuator 1, is employed for modeling the piezoelectric strains, the predicted height h1 after repoling is within 8.5% of the measured value. This further indicates the accuracy of the domain rotation model used to quantify strains due to repoling.

19

2.6. CONCLUDING REMARKS

To test the robustness of the model with parameters estimated for Actuator 1 when the dimensions are changed, the dome height h' of Actuator 4 was measured and compared to the modeled value obtained with Parameter Set 2. It is noted in Table 2.2 that Actuator 4 is constructed from the same material as Actuator 1 but with significantly different dimensions and layer thicknesses. The measured dome height for Actuator 4 was found to be .339 cm. The modeled height obtained with Parameter Set 1 (manufacturer specifications) was .248 cm while the height predicted with Parameter Set 2 (estimated for Actuator 1) was .304 cm. Hence it is observed that the model with parameters identified for a significantly different configuration predicts the dome height to within f 1% accuracy. It is hypothesized that the discrepancy is due to variations introduced in the manufacturing process.

Layer Layer Actuator 1 Layer Layer Layer Layer Actuator 2 Layer Layer Layer Layer Actuator 3 Layer Layer Layer Layer Layer Actuator 4 Layer Layer Layer

1 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 4 5

Materials Thickness (cm) Stainless Steel 304 2.540e-2 LaRC-SI 2.540e-3 PZT-3195Hd 3.810e-2 LaRC-SI 2.540e-3 Aluminum 3003 2.540e-3 Brass 7.620e-3 LaRC-SI 2.540e-3 PZT-3195Hd 2.032e-2 LaRC-SI 2.540e-3 Titanium 1.016e-2 LaRC-SI 2.540e-3 PZT-3195Hd 2.032e-2 LaRC-SI 2.540e-3 Stainless Steel 304 7.620e-3 LaRC-SI 2.540e-3 PZT-3195Hd 2.032e-2 LaRC-SI 2.540e-3 Aluminum 3003 2.540e-3

Length, Width and Tabs (cm) Length s = 5.08e-3 Width b = 5.08e-3 Tabs t = 1.27e-3 Length s = 3.81e-3 Width b = 1.27e-3 Tabs t = 6.35e-4 Length s = 3.81e-3 Width b = 1.27e-3 Tabs t = 6.35e-4

Length s = 3.81e-3 Width b = 1.27e-3 Tabs t = 6.35e-4

Table 2.1: Components and dimensions of THUNDER actuators.

2.6

Concluding Remarks

The model developed in this investigation quantifies the strains produced in THUNDER actuators during the manufacturing process. The final strains produced in the material have two components: (i) Strains due to differences in the thermal properties of the constituent materials and (ii) Strains due to dipole rotation during the repoling process. The first component was modeled using linear thermoelastic relations while the second was modeled through the application of domain theory for ferroelectric materials. The resulting model was shown to accurately quantify the curvature of the final actuators and hence will be a viable design tool for determining the conditions necessary to construct actuators having prescribed performance capabilities.

20

CHAPTER 2. MODELING THE CURVATURE FOR THUNDER

Materials

Aluminum 3003 Stainless Steel 304 Brass Titanium PZT 3195HD LaRC-SI

Thermal Coefficient a (l(r 6 / 0 C) Manufacturer Specs. Estimated Param. (Parameter Set 1) (Parameter Set 2) 24.00 22.00 17.30 19.30 20.0 18.72 9.5 10.67 3.0 3.5 46 (23-150°C) 59.0 60 (150-200°C) 67.0

ACTUATORS

Modulus of Elasticity E (GPa) Manufacturer Specs. Estimated Param. (Parameter Set 1) (Parameter Set 2) 68.95 64.0 193.10 186.0 100.0 95.0 114.0 114.0 67.0 62.0 3.45 5.0

Table 2.2: Values of the thermal coefficients and moduli of elasticity reported by manufacturers and estimated through a fit to data.

Measured t i (cm) Modeled t i (cm) (Parameter Set 1) Modeled h ' (cm) (Parameter Set 2)

Actuator 1 Actuator 2 Actuator 3 Before Poling After Poling Before Poling After Poling Before Poling After Poling A45 ^ 3 8 5 A 5 2 ^ 3 8 0 A 9 4 !l41 ^392 ! 5 2 5 1 8 9 M5

J

&

5

A

5

2

1J88

194

129

Table 2.3: Measurements of the dome height h' for THUNDER actuators and model predictions with Parameter Set 1 (Manufacturer Specifications) and Parameter Set 2 (Estimated Parameters). Acknowledgments We would like to thank our faculty advisors Professors R.C. Smith and H.T. Banks for their expert advice and nearly infinite patience. We would also like to thank the problem presenters from FACE International Corporation, the workshop organizers and the staff at NCSU for their time and assistance.

References [1] B.A. Boley and J.H. Weiner, Theory of Thermal Stresses, John Wiley and sons, New York, 1960. [2] R.G. Bryant, "LaRC™-SI: a soluble aromatic polyimide," High Performance Polymers, 8, pp. 607-615, 1996. [3] M. Capozzoli, J. Gopalakrishnan, K. Hogan, J. Massad, T. Tokarchik, S. Wilmarth, H.T. Banks, K.M. Mossi and R.C. Smith, "Modeling Aspects Concerning THUNDER Actuators," Proceedings of the SPIE, Smart Structures and Materials 1999, Volume 3667, pp. 719-727, 1999. [4] S. Chikazumi, Physics of Ferromagnetism, Second Edition, Oxford University Press, New York, 1997.

2.6. CONCLUDING REMARKS

21

[5] D.L. Edberg, "Control of flexible structures by applied thermal gradients," AIAA Journal, 25(6), pp. 877-883, 1987. [6] A. Hamamoto and M.W. Hyer, "Nonlinear temperature-curvature relationships for unsymmetric graphite epoxy laminates," International Journal of Solids and Structures, 23(7), pp. 919-935, 1987. [7] M.W. Hyer, "The room-temperature shapes of four-layer unsymmetric cross-ply laminates," Journal of Composite Materials, 16(4), pp. 318-340, 1981. [8] M.W. Hyer and A. Jilani, "Predicting the deformation characteristics of rectangular unsymmetrically laminated piezoelectric materials," Smart Materials and Structures, 7, pp. 1-8, 1998. [9] B. Jaffe, W.R. Cook, Jr. and H. Jaffe, Piezoelectric Ceramics, Academic Press, New York, 1971. [10] D.C. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, New York, 1991. [11] K.M. Mossi, G.V. Selby and R.G. Bryant, "Thin-layer composite unimorph ferroelectric driver and sensor properties," Materials Letters, 35, pp. 39-49, 1998. [12] A.J. Moulson and J.M. Herbert, Electroceramics, Chapman and Hall, New York, 1990.


Chapter 3

Mathematical Modeling of High Power Amplifier (HPA) and HPA Linearization through a Predistorter Student Team: Philippe Bremond1, Danielle Bundy 2 , Timothy Chevalier3, Simone Flory 4 , Brian Lewis5, Sumithra Ramaswamy6 Problem Presenter: Tien M. Nguyen, The Aerospace Corporation Abstract

High Power Amplifiers (HPA's) are used by the Global Broadcasting Service (GBS) in order to boost the signal strength of transmitted data between satellite and ground-based stations. It is known that the use of these amplifiers induces distortion of the original signal before transmission. This is, in fact, a consequence of the nonlinear behavior of the amplifier in the saturation region (point of maximum power transfer). In particular, both the amplitude and phase of the signal are distorted, thereby compromising the integrity of the signal. This new signal exhibits both spectral re-growth (added signal spectra outside of the desired bandwidth) and increased Bit Error Rate (BER). To compensate for these undesired effects, it is necessary to predistort the signal so that the resultant signal, when amplified, has the same power spectrum as the original signal after baseband filtering. In effect, the predistorter 'linearizes' the output of the HPA. This simply means that there is a one-to-one correlation between the input and output power. This paper describes the implementation of two-types of predistorters. The first is a non-adaptive predistorter mechanism for an HPA with known characteristics. The second is an adaptive predistorter for an HPA with unknown characteristics.

3.1

Introduction

The Global Broadcasting Service (GBS) is involved with many aspects of our everyday lives from television broadcasts to military applications. Of major concern in the communications industry is 1

University of New Orleans University of Colorado 3 University of New Hampshire 4 Louisiana State University 5 North Carolina State University 6 Ohio State University 2

23

24

CHAPTER 3. MATHEMATICAL MODELING OF HIGH POWER AMPLIFIER (HPA)

the reliability in transferring information from one location to another. When dealing with satellite to ground-based communications, for example, high power amplifiers, such as Solid State Power Amplifiers (SSPA) or Traveling Wave Tube Amplifiers (TWTA), are needed to provide the transmitted signal with ample signal strength to account for signal attenuation due to path loss. Amplifying the signal to levels required for this type of long-range transmission induces distortion in the original signal. This distortion, which are of two types and are known as AM-to-AM (amplitude distortion) and AM-to-PM (phase distortion), is one of the major concerns to communication systems engineers. These effects can cause intermodulation (IM) components and spectral regrowth, which are undesirable to system designs. For example, the intermodulation components and the spectral regrowth can cause adjacent channel interference to other services in addition to a loss in power transmission. Finally, the effects of AM-AM and AM-PM can cause the signal distortion that can degrade the Bit Error Rate (BER) performance. Since cost is an issue in most engineering applications, a low cost mechanism for dealing with this distortion is desired. Working with the signal while it is on the satellite is simply impractical due to the logistics involved. Also, since the signal is transmitted at very high carrier frequencies (at least on the order of 106 Hertz) to keep antenna size down, any signal processing at the modulated frequency would be very costly. As a result, signal processing is performed at the ground-based stations using baseband frequencies. The actual technique used in correcting the nonlinear behavior of the HPA involves signal predistortion. Predistortion involves using known properties of the HPA in order to develop an algorithm for predistorting the original signal. This is done so that when the predistorted original signal is passed through the HPA and finally transmitted, its power spectrum should look identical to that of the original signal before amplification. In essence, the predistorter corrects for the nonlinear behavior of the HPA so that the resulting amplification is a linear process. Currently, there are a few predistorter models that correct for the nonlinear behavior exhibited by the HPA to varying degrees. The models studied by The Aerospace Corporation include Saleh's Model, the Extended Saleh's Model, the Linear-Log Model, and the Modified Linear-Log Model. It is important to note that the use of each of these models requires known characteristics of the HPA in order to design a successful predistorter. These schemes are deemed non-adaptive in nature because of this necessary prior knowledge of the HPA. The problem inherent with these non-adaptive schemes is that they have no way to account for changes in the characteristics of the HPA. In practical problems of interest, it is highly probable that the characteristics of the HPA may change with any number of known or unknown variables such as time or ambient temperature. As a consequence, these methods would provide erroneous results under varying HPA operating conditions. It was the goal of this research to develop a predistorter that would not require previous knowledge of the HPA characteristics. The predistorter would use the changing characteristics of the HPA as defined by previous signal measurements and adapt to the incoming signal. Hence, it would be an "adaptive predistorter." The group was faced with the challenge of modeling a basic communication channel incorporating the elements of baseband signal filtering, modulation and amplification and then incorporating a reliable adaptive pre-distortion algorithm to account for variable HPA operating conditions. The organization of this chapter is as follows. First, in Section 3.2, a brief background of the Saleh and the extended Saleh methods are presented. The adaptive predistorter, which is based on a variation of the Linear-Log model, is discussed in Section 3.3.1. Sections 3.3.2 and 3.4 discuss the implementation of the adaptive predistorter in the physical model, the Global Broadcasting Service, and in a simplified simulation model, respectively. Implementation issues and numerical results are presented in Section 3.5.

3.2. SALEH'S AND EXTENDED SALEH'S PREDISTORTER MODEL

3.2 3.2.1

25

Saleh's and Extended Saleh's Predistorter Model Saleh's Model

A simplified schematic diagram for the HPA with a predistorter is shown in Figure 3.1. The basis behind Saleh's model is that for an HPA of known characteristics (given by a set of discrete measured data), an interpolation technique is applied to curve-fit the data in an effort to develop a simple functional relationship between the predistorter input and output signals [1]. Let X ( t ) denote the input signal to the predistorter (PD), Y(t) denote the output of the PD, and Z(t) denote the output of the HPA. The baseband representations of these signals are (3.1a) (3.1b) (3.1c) The signal Z(t) may then be written in the form (3.2)

where M(py(t)) and inputs vi, V2 and output Q%. Combining these with an appropriate stepper motor dynamic equation with desired piston displacements as command inputs and piston displacements and velocities as outputs,

58 CHAPTER 5. HIGH PRECISION CONTROL OF A FLUID MICRODISPENSER a complete simulation model of the microdispenser pump is obtained. The model is transferred into Matlab Simulink and simulated using an algorithm designed for stiff systems. As an example, the behavior of the system to a typical input of triangular profiles in Figure 5.2 is shown in Figure 5.6. The profile aims to keep the output flow at 0.15 mL/s and corresponds to open-loop control. As seen in the plot of the flow across valve 2, backflow phenomenon is observed. This also creates a sharp dip in the output flow. Also notice the difference between the pressure rise times between the pistons. As piston 1 starts to deliver, it has to supply fluid to a larger volume, hence the slower rise compared to piston 2. Overall, the simplified model appears to approximate the reality sufficiently well for a first attempt. To gain more confidence, the model must be tested with several different control profiles employed in practice and the experimental measurements must be compared with the simulation results.

Figure 5.6: Pump pressure due to the triangular input depicted in Figure 5.2.

References [Ij Merritt, H. E., Hydraulic Control Systems, John Wiley and Sons, Inc. New York, 1967.

Chapter 6

Mass Transport and Surface Reaction of Immunoassay Student Team: Jim Buchanan1, Henrik Kalisch2, Anastasios Liakos3, Renee Yong Ran Liu 4 , Shree Yvonne Whitaker5, Igor Zhislin6 Problem Presenter: Zhong Ding, Johnson & Johnson Company Abstract

In clinical diagnostics product development and biochemical applications, the solid phase surface reactions and immunoassay technologies are widely used to determine the concentration of a protein in a fluid sample. In a sample, protein A with unknown concentration and protein B (a label) with known concentration bind competitively with protein C, which is coated on a solid surface. Any unbound proteins are washed out by the subsequent washing fluids. By measuring the concentration of B bound with C on the surface in a given time, the original concentration of A can be found. In this report, a fluid containing both A and B flows in a cylinder. The inner surface of a cylinder is uniformly coated with C. To assist the product design process, we need to predict the concentration of bound A on the surface as a function of time.

6.1

Introduction

When the human body is invaded by antigens (foreign substances that are usually protein), the immune system recognizes them and produces highly specific proteins called antibodies. Antibodies bind with antigens to form antibody-antigen complexes. The invading antigens thus become inactive and are removed from the organism later on. Although each type of antibody is highly specific for one particular type of antigen, antibodies share common structures and are different only in a region called the antigen binding site. Each antibody is a Y- shaped molecule. It has two identical antigen-binding sites, one at the end of each upper arm of the Y. University of California at Santa Barbara, buchananSmath.ucsb.edu University of Texas at Austin, kalischQfireeint.raa.utexas.edu 3 University of Pittsburgh, alistQmath.pitt.edu 4 University of California at Berkeley, reneeliuSmath.berkeley.edu 5 North Carolina State University, sywhitak8eos.ncsu.edu 6 Middle Tennessee State University, igorSfrank.mtsu.edu 2

59

60

CHAPTER 6. MASS TRANSPORT AND SURFACE REACTION OF IMMUNOASSAY

Antigens attached with labels are used in a variety of immunoassay techniques. The labeled antigen binds with the antibody and then is detected by using the label as a signal. In immunoassays, one can use the specificity of the antigen and antibody reactions to determine the existence and/or the concentration of a particular antigen. One typical immunoassay is called the Antigen Capture Immunoassay. In this assay, a known amount of labeled antigen (Lb) is added to the unlabeled sample antigen (Ag). Both antigens then compete in binding with the antibody (Ab) coated on the solid surface. The reactions taking place are as follows:

Any unbound proteins are washed out by the succeeding washing process. The greater the initial sample antigen concentration, the less labeled antigen is bound to antibody, and the smaller the final signal. Thus, the sample antigen's initial concentration can be determined by measuring the final signal and finding the corresponding initial concentration on the dose response curve. In this project, our final objective is to provide a good mathematical model for the surface reaction and fluid flow in the capillary. In the model, the inner surface of a cylinder is uniformly coated with antibody. A fluid containing antigen (with unknown concentration) and an antigen/label compound (with known concentration) flows inside the cylinder. The two bind competitively with the coated antibody in the surface of the cylinder. Our task is to predict the concentration of antigen (protein A) bound on the inner surface as a function of time along the axial direction of the cylinder. In order to get a precise result, we have to take into account the mass transport due to the movement of the fluid, the diffusion of antigen and antigen/label through the fluid, and the chemical reaction in the solid surface. In the first section we model the reaction as an isolated process, by assuming a homogeneous distribution of the reactions without mass transport. This will help us understand the kinetics of the antigen-antibody reactions and the effects of competitive antigen on the reaction. In the next section, we model the process with an infinite surface coated with antibody. The antigen and antigen/label diffuse to the surface and bind with the antibody competitively. The fluid remains stationary. This model is more realistic for the surface reactions in many applications. The effects of mass transport by diffusion can be investigated. The third model takes into account of steady state fluid flow in the axial direction of the capillary. Which can illustrate the effects of fluid flow on mass transport and surface reaction.

6.2

Kinetics in the Solution Phase

We first consider a simple model in which we do not take into account the geometry of the physical space. We obtain a set of five ordinary differential equations for the chemical reactions. Simplifying the notation we can rewrite our system as

6.2. KINETICS IN THE SOLUTION PHASE

61

with initial conditions X(Q) = XQ, Y(Q) = YQ and Z(0) = ZQ. Here we have set

We can obtain a numerical approximation to the solution using a finite-difference scheme. To carry out those calculations during the preparation of this report, we used the Matlab routine ode23s. In Figures 6.1 and 6.2, we show some results for different initial values and reaction coefficients. Note that depending on the values used, it may take more than 300 seconds to get near the equilibrium. To judge how close our solution is to the equilibrium, we set the time derivatives equal to zero to obtain the following relations:

Figure 6.1: Low initial conditions and low coefficients.

62


Figure 6.2: High initial conditions and low coefficients. In fact, we can prove mathematically that every solution within a certain range of initial values and constants will approach the equilibrium over time. First observe that we have the following symmetries in our system.

We integrate those equations using AQ, YQ-> and ZQ as constants of integration. With this, the original system can be reduced to the following

6.3. SURFACE REACTION IN AN INFINITE PLANAR SURFACE

63

Using the reduced system of two differential equations, and setting dX/dt and dY/dt equal to zero, we obtain a set of algebraic equations for X and Y

Solving for Y gives

A solution of this system determines an equilibrium point of the system of ODE's in the X-Y plane. As seen in Figure 6.3, the first equation behaves like 1/X for small X, but like X for large X. The second equation behaves almost like a constant, because the values of the coefficients are so small. We can see directly that there is only one solution in the first quadrant, i.e., a solution for which both X and Y are positive. Moreover, we can prove that the equilibrium point so determined is stable, i.e., the real part of both eigenvalues of the Jacobian matrix of the derivative of the function associated to the differential equations are negative. The Jacobian of the system is given by

where

The eigenvalues are given by

It is obvious that a + d < 0. It can also be shown that ad —be > 0. This ensures that both eigenvalues are either negative or complex with the negative real part. Thus for any choice of coefficients and initial concentrations the equilibrium point is stable. We can also see this graphically in Figure 6.4 where the vector field associated with the system of differential equations for X and Y is plotted.

6.3

Surface Reaction in an Infinite Planar Surface

In this section, we consider the reaction in an idealized infinite plane coated with antibody. We assume that at any time, the distribution of the reactants in the fluid is homogeneous in any direction parallel to the plane, but not vertical to the plane. We thus have to solve a one-dimensional problem: We consider an imagined ray of fluid above a point in the surface. Because of the depletion of antigen

64


Figure 6.3: Nullclines for X and Y.

Figure 6.4: Stability of Equilibrium. and antigen/label in the fluid near the surface, there will be diffusion of the reactants through the fluid towards the boundary. This process is governed as usual by the diffusion equation. To derive the boundary conditions, we have to consider the transport of the reactants from the fluid into the boundary. To this end, suppose there is a gap of size h between the fluid and the surface. We then have the following equations for the concentrations of X and Y


65

Letting h tend to zero, we obtain

for the concentrations X and Y in the surface. The coefficients dx and dy appearing here are the coefficients of mass transport (see Christensen [2]). The values of dx and dy are actually equal to the values of DX and Dy, respectively, although with different units (see Prank and Kamenetskii [3]). The system of equations for the model described above is

with boundary conditions

u> i/

The algorithm for obtaining a numerical solution for the partial differential equation is summarized as follows. 1. Initialize concentrations of the proteins along the boundary:

2. While time steps remain -Calculate the normal derivative (via approximation):

66

CHAPTER 6. MASS TRANSPORT AND SURFACE REACTION OF IMMUNOASSAY -Solve (by ODE solver or by forward Euler):

-Solve:

using the following finite difference scheme:

-Similarly, solve the equation for Y. 3. Repeat As a first check on the numerical results, we replaced the boundary conditions with homogeneous conditions. This simplification should not alter the solution too much since we expect the concentrations near the surface to be close to zero. Moreover, an exact solution to the problem

with

can be found using Fourier transforms

The graph of C(z, t) at t = 5 sec. is pictured in Figure 6.5 together with the graph obtained from the numerical approximation with zero-boundary conditions. As can be seen, there is good agreement of the numerical solution and the exact solution. For the actual boundary conditions, our calculation also gave reasonable results. In particular, the depletion layer appears to be rather thin (see Figure 6.6). This will be important in the third part of our investigation. If we graph the concentrations of the reactants at the boundary over time, we observe that an equilibrium stage is reached much sooner than in the previous part (Figures 6.8 and 6.9).


67

Figure 6.5: Exact versus numerical solutions for homogeneous boundary conditions.

Figure 6.6: X and Y versus z at t=300 sec. The dose-response curve which shows the bound label/antigen in the surface after 300 seconds graphed against the initial concentration of antigen in log-scale shows a steep decline (see Figure 6.7), which warrants its practical usefulness. Inspecting the graphs of the concentrations against time during the reaction (Figures 6.8 and 6.9), we see that it would be possible to get a similar doseresponse curve after 30 to 60 seconds. Of course this also depends on the particular values of the constants and initial concentrations, so it would be nice to have an estimate valid for a range of values. Using the capillary flow model it will hardly be possible to improve the results regarding the dose-response curve. The primary reason for using it is the lower cost of production.

68


Figure 6.7: Dose-response curve (log scale).

Figure 6.8: (a) X versus t at surface, (b) Y versus t at surface.

6.4

Surface Reaction in a Cylinder with Steady Fluid Flow

We now consider steady state fluid flow in a cylinder whose inner surface coated with antibody. The fluid, containing the antigen and antigen/label, is fed into the cylinder from one side and flows out the other. The reaction takes place in the inner surface of the cylinder. We can see directly that we have advection-diffusion equations for the concentrations of antigen and antigen/label

6.4. SURFACE REACTION IN A CYLINDER WITH STEADY FLUID FLOW

(a)

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(b)

Figure 6.9: (a) W versus t at surface, (b) P versus t at surface. The Laplacian is given in cylindrical coordinates, and we assume that the concentrations of X and Y are axially symmetric, so that the terms involving angle-derivatives are zero. The equations are then

On the cylinder 0 < r < R and 0 < z < I . We assume the fluid to be moving strictly in the z-direction, i.e., along the axis of the cylinder. The velocity profile V(r) is assumed to be quadratic

Since we observed in Section 6.2 that the depletion layed near the surface is rather thin, we can simplify the problem by assuming that the concentrations X and Y are nearly constant in the center of the cylinder. This enables us to solve the problem on the square 0 < z < 1, ro < r < R instead of the cylinder. The boundary conditions at r — r0 are then of homogeneous Neumann type

Assuming steady flow of liquid through the cylinder, we also have

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The fourth boundary condition is again given by the system of five ordinary differential equations, modified to take into account the transport from the fluid into the surface as in the previous section. We tried to numerically approximate a solution using a forward Euler scheme with a second order accurate spatial discretization. Unlike in the previous case, there are significant stability problems due to the terms £^r and p ^r, and the fact that we have to approximate both first and second spatial derivatives. After about 40 seconds, our approximation started to blow up. We expect that by fine-tuning our code we might be able to integrate up to 60 seconds. This would probably be enough to get a decent dose-response curve. To go further in time an implicit time discretization would be advisable; however, the computation time necessary for such an approach seems prohibitive.

Acknowledgments We would like to thank Dr. Zhong Ding of Ortho-Clinical Diagnostics, Inc. and Professor Zhilin Li of North Carolina State University for their valuable contributions to this report.

References [1] J.R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, Reading, MA, 1984. [2] Lars L. H. Christensen, "Theoretical Analysis of Protein Concentration Determination Using Biosensor Technology under Conditions of Partial Mass Transfer Limitation," Analytical Biochemistry, 249, pp. 153-164, 1997. [3] D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969. [4] G. Zubay, Biochemistry, 2rd edition, Macmillan Publishing Company, 1988.

Industrial Mathematics : The 1998 CRSC Workshop

Industrial Mathematics : The 1998 CRSC Workshop

Industrial Mathematics : The 1998 CRSC Workshop