Digital Processing and Reconstruction of Complex Signals

Predrag Petrovic, Ph. D. Milorad Stevanovic, Ph. D.

Digital Processing and Reconstruction of Complex AC Signals

~ Springer ACADEMIC MIND

Predrag Petrovic, Ph. D. University of Kragujevac, Technical Faculty Cacak

Milorad Stevanovic, Ph. D. University of Kragujevac, Technical Faculty Cacak

DIGITAL PROCESSING AND RECONSTRUCTION OF COMPLEX AC SIGNALS

Reviewers Srdan Stankovic, Ph. D. full professor, University of Belgrade, Faculty of Electrical Engineering

Slavoljub Marjanovic, Ph. D. full professor, University of Belgrade, Faculty of Electrical Engineering

(c) 2009 ACADEMIC MIND, Belgrade, Serbia SPRINGER-VERLAG, Berlin Heidelberg, Germany

Design of cover page Zorica Markovic, Academic Painter

Printed in Serbia by Planeta print, Belgrade

Circulation 400 copies

ISBN 978-86-7466-363-9 ISBN 978-3-642-03842-6

Library of Congress Control Number: assigned

NO TICE: No part of this publication may be reprodused, stored in a retreival system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers. All rights reserved by the publishers.

Digital Processing and Reconstruction of Complex AC Signals

CONTENTS

1. INTRODUCTION .....................................................................................................................•....•........5 6 1.1 Basic Principles of the Suggested Measuring Method 1.2 Mathematical Proof of Justification of the Suggested Measuring Method 11 14 1.3 Mathematical Proof of Correctness in Active Power Processing 18 1.3.1. Analysis of the Numerical Procedures Suggested in the Processing of Periodic Signals 1.4 Adaptability of the Suggested Algorithm Used for Measuring of Electrical Values in Electric Utilities .........•..........•.................................................•..•.....•...••................................................................... 19 1.5 Analysis of Possible Sources of Errors in Digital Processing With a Suggested Measurement Concept ..................•....•............................................................ 21 1.5.1 Simulation Results .....•.......•...•..............................................•......•.•...•...•..•..•..............................•26 1.5.2 Analysis of the Error Caused by Imprecision in Determining Sampling Interval ............•... 27 33 1.6 Simulation of the Suggested Measuring Method in the Matlab Program Package 1.6.1 Software Testing of the Suggested Measuring Concept of Electrical Values Based on Measurement Results in Real Electric Utilities........••....•.....•..•...................................................•...•..38 41 1.7 Practical Realization of the Suggested Digital Measuring System 49 1.8 Results of Practical Measurements with Realized Digital Wattmeter References 55 2. DIGITAL PROCESSING OF SYNCHRONOUSLY SAMPLED AC SIGNALS IN PRESENCE OF INTERHARMONICS AND SUBHARMONICS

58

59 2.1 Synchronous Sampling in the Presence of Subharmonics and Interharmonics 2.1.1 Derived Conditions for Precisely Processing..............•......•........•..•.............•....................•...••...•. 67 2.1.2 Asynchronous Sampling .••..•..•...•..•.••..•.••..•..•..•..•..•.••....•..••..•..•.•..•..•.••.•..•..•.•.••......•..........•..••..•••.. 69 2.2 Simulation Results 70 2.3 Calculation of the Truncation Errors in Case of Asynchronous Sampling of Complex AC Signals 73 2.3.1 Analysis of Worst Case Errors 73 2.3.1.1 Average Method ............•............................................................................................................ 75 2.3.1.2 Trapezoidal Method ...................................................................................................•..•..•....••... 76 2.3.1.3 Stenbakken's Compensation..............................•.•..•...•.................•........•...•.••..•.....••.................. 77 2.3.1.4 Zu-Liang Compensation ...•............•.....................................................................•..•.................. 77 2.3.1.5 Average Method- approximate expression ...•........................................•...•..••.......••.•..•..••........• 78 2.3.1.6 Trapezoidal Method- approximate expression...................................................................•....•. 78 2.3.1.7 Stenbakken's Compensation- approximate expression ....................................................•......• 78 2.3.1.8 Zu-Liang Compensation- approximate expression 79 2.3.2 Simulation Results 79 References .............................................................................•.•..•.......................................•.•.................83 Appendix A 84 3. RECONSTRUCTION OF NONUNIFORMLY SAMPLED AC SIGNALS

86

3. 2. Proposed Method of Processing 3.2.1 The Determinants ofthe Van der Monde Matrix 3.2.2 Reconstruction ofBand Limited Signals in Form ofFourier Series 3.3. Simulation Result and Error Analysis 2

88 89 90 95

3.4. Possible Hardware Realization of the Proposed Method of Processing 99 References ..•..••......••..••..........•.......•...••.•...•...••.••................•.•.•..........••..•..•••.••.•••..••..•••...•...••...........•.••.. 102 Appendix B ...•...•..•...•...•...•.•..............••.•..•..••..••..•...•...•..•..•.......•..........••.••..........••........•.....••.••..•..•.••.... 104 Appendix C •..••..••••••..•..•........•.....•.•....•.•..••.••...••.•••••..••••..•••••...........•.•..••.•..•••..••.•••..••......••••••••••••..•.••.•• 104 Appendix D •...•...••.....•...•.....••.•.•............•..••..••..••.•...........•.•....•......•.•....•.............•.....••...••••....•....•.•.••.••. 108 Appendix E ...........................................................................................................................•..•......••... 110 4. NEW METHOD FOR PROCESSING OF BASIC ELECTRICAL VALUES BASED ON DEFINITION FORMULA IN TIME DOMAIN ...•..•.........................................................•.....•....••. 112 4. 1. Suggested Method of Processing

112

4.1.1 EstimationofMeasuring Uncertainty

115

4.2. Simulation of the Suggested Measuring Method 115 4.3. Practical realization of the proposed algorithm 117 4.4. Experimental Results 119 References ..•.••...•................................•...•....................•.........•..•..........•......•..••....••..••............•....•..•.••.•. 121

3

PREFACE This monograph is the result of a long-term work of the authors on the problems of digital processing of complex periodic signals of voltage and current which can be found in the distribution network. This scientific field has been given special attention in the literature abroad through a great number of articles published in the leading journals, textbooks and the other publishing forms. Over the recent period, the authors of the monograph have published a large number of papers in a number of journals and have presented at the leading international conferences, which verifies the results they have achieved in this scientific field. Their patents which have been registered in their home country are the result of the work. The problem of the reconstruction of complex periodic signals, which is in the focus of Chapter three of this monograph, has been given special attention. A completely new protocol, which enables the development of much superior and more efficacious algorithms, has been developed, and the obtained results are unique in global practice. Chapter two deals with the processing of nonharmonic components of ac signals, i.e. interharmonics and subharmonics based on the principles of synchronous sampling. The conditions, required for the performance of such a processing, have necessarily been derived. Chapter four looks at a newly developed method for the calculation of basic parameters of the processed voltage and current signals. Having being practically verified, this method has demonstrated exceptionally favourable performance. The authors believe that the monograph will be beneficial to specialists and experts involved in problems of this scientific field, and hope that it will serve as a useful guideline to follow in further investigations. The authors wish to express their gratitude to their language editor Lidija Palurovic, M.A. Philo!'

4

1. INTRODUCTION The first chapter of this book is dedicated to the problem of measuring basic electric quantities in electric utilities (voltage, current, power, frequency), both from the aspect of accuracy of this type of measurements and the possibilities of simple and practical realization. The conventional algorithms used to this purpose (calculating the active power and the RMS value of the voltage and current signals that are the object of processing) are based on the use of integration or summation process on limit time interval. A characteristic of this approach is that it offers the correct result if the input signal is periodical in time. However, in real systems, voltage and current signals are not necessarily of a periodical quantity, due to the presence of nonharmonic components or/and possible stochastic variation. It is for this reason that, very often, the result of processing is not an instantaneous value of processing signal - it is rather a processed (observed) value in some time interval, both with conventional and certain new algorithms for processing [1-19]. The measuring equipment used to this purpose worldwide (as well as in our electric utilities) can be of different precision class (from class 0.1 to class 2), and measuring usually presupposes that the recording on the net takes place from a specified moment to another moment, specified with a particular protocol (these two moments can be some minutes or some hours away). This further implies that we deal with measurements not conducted on-line (continuously in time), and for which we can use a realized measurement system due to its accuracy and price. Many applications involve digital processing of periodic signals [1-19]. For example, both voltage and current in electric utilities are periodic signals containing harmonic components. The measuring method proposed in this chapter is based on selecting samples of the input variable in a large number of periods in which the system (electric utilities, in this case) is considered to be stationary. The stationary condition can be proved by the values obtained from measuring of the RMS (root mean square or effective) values of the voltage system. Stationarity of the system suggests that slowly changing quantities, such as current and voltage, and their harmonic content, are constant within the measuring interval. In this case, unlike the Nyquist criteria, undersampling is possible. The current makes this system nonlinear as the type of load which will be used and the time of its connection to the investigated system cannot be predicted. However, after a certain number of periods, the current can be considered as a slowly changing variable during processing. The utilities are inert systems in character, so that the sampling during several periods of the observed system variables can be performed. It is for this reason that very slow, low-cost, but very accurate AID converters, such as a dual-slope type, were used in the proposed measuring system. Voltage and current from real electric utilities were used as input variables. The sampling procedure is initiated arbitrarily. The distance between two consecutive samples is given by:

tdelay = N · T + L\t

(1.1)

where N is the number of periods between sampling, T is the period of the input voltage, and .~3

.~{siq(r+s)2Jr+f//r+¢r]-sir(f//r +¢r)}=0 (1.50)

=0 M

=>p=II~ +IR~' Lk,lrco(f//r -r/Jr) r=1

The final expression (1.50) defines the power based on the definition including the harmonic voltage and currentcontent. On the other hand: p' =t

t {~ -iiv. tk,Sin~llj(NT +

s'

M

+ M)+

W

lfI.J}

+1

+ .[iIe ' tl,sin[sllj(NT + M)+ M

¢J} =

M

=IJVI + J2~VI LIsLsin[sO!i(NT+~tA,)+¢J+ J2~vRLkrLsin[rmj(NT+~t)+f//r]+ s=1 j=1 r=1 j=1 + IR;R

(1.51)

ffk,lsI{cos[(r-s)aJj(NT + Llt)+V/r -¢J-cos[(r + s)ad(NT + Llt)+V/r +¢sn r=1

s=l

j=l

Introducing the.following expression: w

w

j~

j~

B. =Lsirlsaj(NT+!!J)+¢S] =L{co~sirlsaj(NT+!!J)]+siIYAco~saj(NT+!!J)]}=co~Cz +siIYAG; while: 15

(1.52)

C1 =

w

W

L cos [sOJj(NT +!1t)lc = L sin [sOJj(NT +!1t)]

(1.53)

2

}=1

}=1

It yields: C + iC 2 1

=

Lw esmj(NT +~t)i = L \esmi(NT +~t))\i = esmi(NT +~t). L \esmi(NT +~t))\i j=1

W {

W -1 (

j=1

j=O

.( ) smi(NT +~t}w -1 = esm1 NT+~t . e esmi(NT +~t) -1

=

e

. S{j)i(NT+~t)W

.(

= esm1

) Tie

NT+~t .

. • sin

2

+~t)

smi (NT 2ie 2

SOJ

-(NT + ~t)v

2

s OJ

.sin-(NT+~t)

2

s:

SWi(NT+MXW+l) sin (NT + L1t)w 2 .--=------

sin sO) (NT + L1t) 2 sin sO) (NT + L1t)w

C1 = cos sO) (NT +L1tXW +1).

2 ,C 2 sin sO) (NT + L1t) 2

2

= sin

sin sO) (NT + L1t)w sO) (NT +L1tXW+1)._--=2::...-_ 2 sin sO) (NT + L1t) 2

(1.54)

That is: . W( N + S~)

C\ =cos[(W+lXsNJr+ S~I)]. Sl~ (s

Jr

~)

sin sNJr+ S~I

sOJflt

. sOJfltN

[~] cosWsNJr,sln-cos(W +1~NJr'cos (W +1)~ . 2 cossNJrsinsm t 2

1

. sOJfltW

WSN cos(W +1) ' - - ' S l n - =(_lyw+l)sN ( ) 2 2 . S OJflt (- l)sN

.-=i-.

[(

=

Sln--

2

[

{

C, =sin (W +\sN7r+

s~t

SOJ~tJ

J]

sinW ( sNJr+--

.

. (

s~ J

sin sN Jr +- -

)SOJflt]. sOJfltW 2 2 . S OJflt Sln-2

. ( ) sOJ~t . sOJ~tW ( l)wsN sin W +1 , - - , s l n - (_lYW+'~N. (~I)SN . ~OJflt 2 Sln--

2

2

(1.56)

. [(W + 1) .SOJflt] . sOJ~tW sin - - 'S ln--

2

(1.55)

cos W +1 - - ' S l n - -

2

sOJ~t

.

Sln--

2

To be able to equalize the definition and the processed signal it can be noticed that: sin s~M :f. 0, sin sw~tW

= 0, (1 ~ 'is ~ M) OJ = 100Jr rad / s,!1t = 0,5.10-3 s, sin ~:f. 0 ~M ~ 39;

(1.57)

sin ~ W = 0 ~W 2 40k, kEN The derived conditions are arising as a result of the freely-assumed value of the L1t delay, which is limiting the possible values for the highest harmonic of the processing signal. It is clear that for the obtained value for M~39, in accordance with the Nyquist criterion, we must adopt the delay of L1t~0,5xl0­ 3s. If we introduce the following abbreviations in the calculation under the proposed method: M

M

s=1

r=}

~B; = J2I;~ lls( eoss~ +sinsC1) =0, '\Is,andB; = J2I: Dr( cos'Vr~ +sin'VrCl) =0, '\Is M

B3

=

IR;R. L «.t., Wcos{'Vr

- ~r) ~ B 3

r=1

M

= I R • VR

•

L krlrcos{'Vr - q,r) r=1

16

(1.58) (1.59)

W

W

W

j~

j~

j~

B4= Lco~((r-s)oj(NT+&)+~r -¢s)]= Lco(~r -¢s)co~(r-s)oj(NT +&)]- Lsin(~r -¢s)·

(1.60)

.sin[(r-s)oj(NT +&)]=co(~r -¢S)c3 -sin(~r -¢s)c4 Introducing the following shortens writing: w

w

C3

= L cos [(r - s )mj(NT

C4

= Lsin[(r-s)mj(NT +~t)]= Lsin(r-s)mj~t;

j=l

W

j=l

j=l

C + i- C

4

=

(1.62)

f e(r-s)wjMi = f (e(r-s)wMi Y= e(r-s)wMi . I (e(r-s)wMiY= 1

j=l

j=l

j=O

L

=

(1.61 )

j=l

W

We obtain: 3

L cos (r - s)mj~t;

+ ~t )]=

e(r-s)wMi . e(r-sJWMiW -1 e(r-s)wMi. e(r-s )(l)Mi _ 1

2i . e

(r-s )wMWi 2

(r-s )(OM

Ti- e -2-

. sin

• sin

(

r-

S

(

r-

S

i.. jW!:J.tW

2 \~_,

~(

e

2

.

( \~-'!:J.tW ), sin r - s jW

W+l1 .

jW!:J.t

(1.63)

--'---~2-sin (r - s )Wilt 2

2

To obtain complete equality of the suggested method and the definition, it is necessary that: ~t = 0,5.10-3 s, OJ = 100Jr rad / s, sin ri: Jr:;i: 0, (r:;i: s), M ~ 40; sin (r-:o)w Jr = 0, W ~ 40k ~B: = 0, kEN

(1.64)

The derived conditions are arising as the result of the same assumption as shown in the calculation of the relation (1.57). Taking into account that: M M

W

r=1 s=l

j=1

B; =_l~R 'LL k)sLco~(r+s)oj(NT+M)+f/lr+¢J W W W (1.65) B4 = Ico~(r+s)oj(NT+M)+f/lr +¢J= Ico~(r+s)j2mV +(r+s)utM+f/lr +¢J= Ico~(r+s)utM+f/lr +¢J= ~

~

~

W

W

j=1

j=l

= CO~f/lr + ¢s)I co~r + s)ojM- sin{f/lr + ¢s )Isin(r + s)ojM = CO~f/lr + ¢s)·C, - sin{f/lr + ¢s)·C6

Introducing the following abbreviations: W

C, =

W

Lj=l cos (r + s )mj!!,.t;C = Lj=l sin (r + s )mj!!"t 6

It follows that: . _ (r+s{llJM(w +1)i sin (r+s)t MW Cs + I . C 6 - e . . 0G:iliIii sm 2

(1.66)

(1.67)

To attain equality it is necessary to have: ~t = 0,5.10-3 S,

sin (r+s)wMW 2

OJ =

100Jr rad / s, sin r;os Jr :;i: 0, M ~ 19

= sin (r+s)W1i =0 ~ W > 40k kEN 40 ,

(1.68)

Based on stated above, we conclude that the demand (1.45) will be satisfied provided that the strictest of the three derived demands - (1.57), (1.64) and (1.68) - are satisfied: We can conclude that: p=p * (1.69) for ~t = 0,5 .10- 3 s, OJ = 100Jr rad / s, T = -to, M ~ 19, W ~ 40k, kEN

17

We can observe that the last equation is in accordance with the Nyquist criterion (as regards the delay in accordance with the harmonic content), and, consequently, in accordance with the theory of synchronous sampling. As a new criterion, issuing from the suggested measuring concept, a very complex dependency between the delay time and possible harmonic content of the signal, the object of processing, has been evidenced. When a system works at 60 Hz, the described conditions (1.57), (1.64) and (1.68) can take the following form (with points 1,2 and 3 in the next equations): ~t

= 0,5.10-3 s, OJ = 120trrad / s, T = 1;os,

I°M~49,W~100,

2° M

~

50, W ~ 100,

3°M

~

24, W ~ 100.

(1.70)

Derived conditions have been assuming that delay is L1t=0,5xl0-3s, but apart from this they are of general type. They define the number of harmonics M needed for accurate processing of average power according to suggested algorithm, as well as the number of samples W, both of voltage and current, needed to satisfy the equation (1.45).

1.3.1. Analysis of the Numerical Procedures Suggested in the Processing of Periodic Signals The precision of each numerical method depends on the available processing potential, as well as the time in which it can be realized. In [23], with "practically" unlimited processing capacities and under ideal conditions post-processing of the described signals was realized, therefore these results cannot be identified with the results obtained within a real system, as in this monograph. The synchronization with the signal of fundamental frequency is a separate problem. By [23], we attempted to show that the error in the trapezoidal rule was a small one, since it is reduced to harmonics so high that they practically do not occur within the range of the real ac signal. This is true if you possess a strong processor and enough time for processing, i.e. if you do not restrict the number of significant digits in the record of calculated values - all of which is not applicable for a real-instrument situation. The statement that the precision of processing is not increased with the increase in the number of measurements (from 40 to 80), as explained in [23], is also a result of the above supposition. In the real system, however, the precision was increased, and this was verified during practical measurements performed in two authorized laboratories (one was the National Institute for Measuring), especially if the source of the system error is eliminated, within a system such as a circuit for sample and hold. By increasing the number of samples, we reduce step L1 t between two consecutive measurements so that the number of samples is effectively increased within the period of the processed signal. This will undoubtedly lead to an increased precision in processing, especially if the source of the system error is eliminated, within a system such as a circuit for sample and hold. Although the authors [23] tried to describe the error of the trapezoidal rule through equation (5) [23], the literature provides evidence [24] that the mistake in a function with a continuous second derivative at the interval in which integration is done is described as: b-a ( ~ b-a (1.71) ~::;-h2SUplf"x~ h = 12

[a,b]

n

where f(x) is the function whose integral is decided in the interval from a to b, with the step h at n points. This error decreases if, for the basic algorithm (depending on the calculation), a different method is chosen, e.g. Simpson's rule [24]. In such a case, the calculus is more complex as well as the interpretation of the derived conditions. For this reason, the authors of this monograph have started to derive the conditions from definition formulas for calculating the basic ac values, without favoring any of the numerical procedures. Under the derived condition (6) in [23], we are completely uncertain not only about what qJm is on the other side, but also about the real value of the error of the proposed procedure. 18

Stenbakken [25] derived an expression for the value of the error with this type of average power processing procedure, for simple sin signals and complex periodic signals. In the case of synchronized sampling, this error can be expressed as follows:

E = AfpqcosqJq Sin(2 1llJ iTs] + AfpqsinqJq COS(2 1llJ iTs] = T

q=l 00

T

q=l

(1.72)

00

=LXqPq cosqJq + LYqPq sinqJq q~

q~

where:

X q = ASine; iTs} Yq = ACos(2; tr,

J

(1.73)

where A represents the processing operator, Ts=T/W is the sampling interval, T is period of fundamental harmonic, qJq is the phase angle of the qth harmonic power signal, W is the number of samples. The Xq, Yq factors for the algorithm of the trapezoid formula are given as:

J~

X q = A Sin( 2; n; = 2 Yq = A cos( ;

( 0,5sin 0 +

~ Sin( 2m;,Ts i] + 0,5Sin( 2m;,T W]} s

rr, J= W1 ( 0,5cosO+ t;cos 2m;,Ts i ] +0,5cos( 2~ T W]J W -1

(

(1.74)

s

The same principle is applied in the calculation of the RMS values. All of this is directly related to the problem of synchronization of the processed signal. The above relation and conclusions are the key reason for using the original definition relation to obtain the value of the processed active power. In the first paragraph of Section 4 authors of [23] is said that, in case that the number of periods L=1 +nM over which is calculated the average value of processing signals (n is the number of samples, M is the natural number), then the generalized undersampling method is reduced to the method described here. This is incorrect for a simple reason that in its first equations define the step between two consecutive measurements as tstep=NT+Jt, where Nand Jt are not connected - Jt is directly connected to the Nyquist frequency. This is the reason why the procedure described here represents a general form of synchronized undersampling. Practically, the result obtained through [23] is the same as the one derived in this monograph. Similarly, it is in compliance with the theory of synchronized sampling defined by Stockton and Clarke [3].

1.4 Adaptability of the Suggested Algorithm Used for Measuring Electric Values in Electric Utilities The adaptability of the suggested algorithms look at the fact that we can in dependence of detected harmonic content of observed input signal to determine the necessary number of samples that we must take, in order to have accurate data about processing value. This is confirms the derived conditions (1.39), (1.40), and (1.41) while we processing the effective value of the net voltage, in other words demands (1.69) and (1.70) while we calculating the active power. Certainly, for the realization of this kind of processing concept we need a microprocessor capable of carrying out the DFT, a common option in modem microprocessors. It is a low-cost microprocessor type, acceptable in respect of the total price of the final digital system. Aiming at confirming that the presented mathematical proof is general in type, if we assume that the input signal is of general harmonic content, the following can be written: M

M

M

n=l

n=l

n=l

Lansn(nrot+'Vn) + Lbnco~nrot+'Vn) =LCnS~nrot+'Vn +8n)

(1.75)

While c, =~c{, +If. and oo;(}n =~, sitfln =~., which provides the same form of the input signal already considered.

19

In case that the assumed input signals are described in the following order: M

M

Lan si"'-nrof+ \fin) + Lbncos(nrof + <J>n) n~

(1.76)

n~

Ifwe introduce the following abbreviations:

(ancos\fln - bnsin<J>n) = an

(1.77)

(ansin\fln + bncos<J>n) = ~n It follows that: (an cosvn - b;sino,)sines+(an simpn +b;coso,)COSIDt = an sinox + ~ n cosox =

~a~ + ~~ sin( IDt+y n), while cosy n = ~ ~n siny n = ~ ~n by which the expression (1.76) comes to the known form. an+~n an+~n 2'

(1.78)

2 .,

Finally, we can conclude that the suggested method gives very accurate results when measuring the average power, and therefore can be accepted as reliable regardless of the harmonic content of the input signa1. In addition, this method can be defined in the form of an adaptive algorithm. The comparison of the suggested algorithm with those known provided by the literature suggests that its relative simplicity renders less processing time. Similarly, under very complex exploitation conditions, it ensures high level of performance and adaptability (depending on the harmonic content of the input signal). This is provided by the implementation of additional algorithms - these algorithms solve the synchronization problem and eliminate the quantization error. In addition, all the requirements imposed here can be satisfied using a wide range of low-cost microprocessors. The time needed for the processing by the suggested method (the time in which we require the input signal not to change its harmonic content) is very short - limited to I second. The dynamic characteristic of the suggested algorithm for measuring electrical values depends on the speed at which the processing signal changes (voltage or current). The speed of the used ADC does not allow us to follow sudden changes (as the jump function or similar). However, for all the changes for which the speed can be recorded with a sampling frequency that corresponds to a dual-slope ADC, it will be recorded with high accuracy. It cannot escape notice that the same measuring concept can be applied on other slowly-changing values such as pressure, temperature, etc. The realized mathematical analysis is general in type, and absolutely accurate from the mathematical point of view. The authors did not try to adapt the derived results to individual practical cases, which is certainly going to be the matter of consideration for those aiming at realizing the device practically. The expressions obtained put only a lower limit on the necessary calculations (necessary but insufficient condition). The delay used in the derivations of inequality (1.69) and (1.70), Lit=0,5ms, was taken as an example, not as an obligated assumption, for the purpose of illustrating the results obtained in nets working at the frequencies of 50 or 60 Hz. In the derivation we assumed that M was the highest harmonic component, whereas, for the reason of simplicity, we assumed that the signals of voltage and current possessed the same number of harmonics, by which the conclusions do not lose generality. In the real environment, it is common for the current signal to possess somewhat richer harmonic content. The claim made by authors of the monograph in their conclusions about the necessity of equidistant current and voltage samples has already been practically given by expressions in the above text. As shown by expressions (1.69) and (1.70), and according to the one with the highest demand, it can be concluded that it is necessary for this to be:

"* 0 :::::> (r+;)wA -< tt, I ~ 'tIr,s ~ M :::::> mA -< 11, (r+s ;WAW = kn, kEN (natural number) sin

(r+;)wA

(1.79)

The obtained conditions for the value of delay Lit are completely equal to the Nyquist condition. In other words, as it was described in the introduction, we are not able to carry out real spectral reconstruction of the processed signal by the proposed measurement method due to the extremely low speed of sampling. It is only possible to carry out "virtual" (delay in time) spectral reconstruction. Since

20

the delay L1t is responsible for the movement forward of the moment of sampling from period to period (or more periods, depending on the parameter N value), the delay must satisfy the Nyquist criterion. In other words, it must be in accordance with a basic postulate of synchronous sampling, where the proposed measurement method conditionally belongs. If the current and voltage signals have different harmonic content, the conditions for delay L1t would determine a signal with "richer" harmonic content (which is usually the current signal because of its greater dynamics): sin (r+slcoM 0 => (r+slmM -< 1!, 1 ~ "\Ir,s ~ M 1 , M 2 (1.80) => r.0,.t -< _Mj+M t r _ (r+s)m~tW = kn: k r S E N(natural number) 2 " ,

'*

t.v.

z'

where 0)=2 if is the angular frequency of the fundamental harmonic, M] is number of the highest harmonic of the voltage, and M2 is the number of the highest harmonic of the current signal. It is necessary to take the samples equidistantly on the interval of one period. In that way, on the basis of established limits in processing, it is possible to calculate accurately the observed electrical values in electric utilities. With the appearance of more modem microprocessors, we received hardware resources which enable very simple realization of the suggested measurement system. Namely, these microprocessors posses inbuilt ADC of integration type, which can activate 16-bit resolution in correlation with the input signal dynamic which is the object of processing. It can possess (internally) four different current sources, charged at a varied speed (in other words with different bias of changing), with the externally added capacitor. This will directly determine the speed of the conversion which, in the case of a 16-bit resolution, demands approximately 3ms for the realization of the complete process of converting the input analogue value into the digital output value. All of this completely satisfies the conditions suggested here for the digital measurement system.

1.5 Analysis of Possible Sources of Errors in DigitalProcessing With a Suggested Measurement Concept The synchronous sampling of alternating current (ac) signals enables a highly accurate recalculation of basic electrical values in a network with very low uncertainties (on the order of a few parts of 10-6 [28]). This is possible in cases when we have a modified signal that is spectrally limited and when we have a sufficient processing time and necessary recalculation capacities. For this method to be effective, it is necessary to precisely measure the period T, as well as to generate the sampling interval Ts = TIW, where T is the period of the processed signal, and W is the number of measurements necessary for exact calculation. This method is suitable for sinusoidal signal and complexperiodical signals with low harmonic content. There are various sources of error during the synchronous sampling of complex-periodical signals, such as the variable initial time of measurement to, the error of the sampling interval generator, which depends on the number of samples and the initial phase, the delay of the SIR circuit at a command signal, and the effect of the initial phase. Owing to the issues mentioned above and the non-ideal nature of the method, theoretically obtained discrete sampling moments are not in agreement with the experimentally obtained values. Therefore, an additional analysis under the designed conditions, based on the conclusions in above text, should be performed by considering the sensitivity of the procedure suggested in cases when sampling frequency does not correspond to the actual frequency of fundamental signals. It is only now that synchronous sampling is performed, this being the method most sensitive to this type of error. Stockton and Clarke [3] mathematically derived the value of the error in the case of sine signal sampling, when the signals are not spectrally limited. If we perform the sampling with W samples over M period, the error is a result of the effect of the harmonic of a q order which is bigger than W, which is considered as qMlW EZ. If we consider the initial time, at which measuring starts to be to=O (measurements are synchronized with the zero crossings of signals, O W

j==O

4;aiJj _

-

j==o

e41rai -I _ Zie21ra isin(Z7l"a) _ 21rai 4 ' 2 ' - e nca nca e W -1 2' 2Jl"a te sln

w '

W

;1sin(Z7l"a) 2

W

(1.97)

na sin-

W

where i is the imaginary number. We express the uncertainties in determining the effective value of the signal as:

= cos

M 1

4ift cos 0

2

21l'a(W -1) , ( ) sin 21l'a W . Zxa

sln-

W

, 21l'a(W -1) ,( ) . 4ift sin sin 21l'a + sin 0 _ _----'-'-W _ 2 ' 21l'a slnW

=-!sin(21l'a Xcos(4ifto + 2rp)sin(21l'a )+ sin(4ifto + 2rp) COS (21l'a)) 2

21l'a cosW (cos(4ifto + 2rp)cos(21l'a)- sin(4ifto+ 2rp)sin(21l'a)) -!Sin(21l'a)-2 2 ,1l'a slnW

In the case when phase angle qr=Q:

24

(1.98)

2·J IL

W-I ( 21ifto +~ =- W-I( I-cos( 41ifto Al = Lsin }=o W 2 }=o 2

4.JJ =-W +~ W

2

~

(1.99) cos-21Caj. W M31 =-~sin(21Ca sin(41ifto +21Ca)+cos(41ifto + 21Ca)-2 2 . 1Ca Sln-W If we consider the initial time, at which measuring starts to be 10=0 (measurements are synchronized with the zero crossings of signals):

~

cos2tra) M3l = -!sin(2tra sin(2tra )+cos(2tra ) - W 2 . 2 . tra SlnW

(1.100)

The absolute obtained relation must satisfy the next inequality: cos2tra]

lM3ll~!lsin(2tra~ ISin(2tra~+coS(2tra)-1 WI· 2 . 2tra [

(1.101)

Sln-

W

Using the conditionsI + 111 ~. I 1M31

1

(111 ~ 0) andsin x ~ x(x ~ 0), we obtain:

s ~cos~lsin(2tra~. 4tr

(1.102)

2W

With the introductionof the amplitude of the processed signal V, the definition formula for calculating the effectivevalue of the signal is:

VRMS

JJ

(10 +L. , = -1 W-I LV 2 sin"( 2;(

W}=o and with this equation we apply

WI

(1.103)

~ + Mil ~~1 + ~l ;1\(1 + t)~ 1+ ~ + [~}2 + [:}3 +.... Al

=

=

(1.104)

The error in calculatingthe effectivevalue can thus be presented as:

M=~Al +M1 -~ =fA:(M3 2A

1

1

I

2

-

3

J.

M31 + M31 +... 8A I

16AI

(1.105)

By neglectingthe higher members in the series, the following error in calculus is obtained: till '"

fA:

Mi2Al 2fA: Mil ~ ItillI:: ; 2MifA:l . =

(1.106)

I

By introducing the amplitude V in relation (1.106) and by using the definition formula for calculating the effectivevalue, the followingequation is obtained: IE/:S;

V ~cos~lsin(2i'Z'a) V 41C 2W =--cos~/sin(21Ca~, 2~.JW 4i'Z'Ji 2W

(1.107)

where E is the error (absolute) in the calculation of the effective value (it is easily reduced to an error in the calculation of average power) under a supposition that the initial moment of measuring is synchronizedwith the zero crossing of the signal (10=0). The errors in the AC voltage measurementwere comparedwith those given in [30], and were in good agreement. The followingconstraintsmust hold to attain minimumuncertainties: I) Ta=I/(Wf) must hold at all times for the multiple of two W. This is the condition for the synchronous samplingof the signal with the frequencyf generated from a common clock reference. 25

2) The number of sampled periods M must be an integer multiple of the number of power-line cycles in order to reduce the number of power line interferences. Conditions 1 and 2 prevent artificial spectral components (leakage) from appearing when performing the DFT of the sampled data. 3) The suppression ofhannonics of the power line frequency occurs when 1/(T!»> 1 and is an integer. In the case of complex input signals (with harmonic and nonhannonic components), the uncertainties probably are evaluated as the superposition's of harmonic errors (with the form defined in relation (1.107», and this is expected to be the theme of some future publications. The total uncertainty of the sampling method is approximately the same as that of the step calibration in the observed frequency range of 46-65 Hz. The presented result (1.107) enables a more accurate estimation of possible errors in calculating the RMS values of low frequency ac signals then those presented in [31].

1.5.1 Simulation Results The calculated results were further tested by simulation using the program package Matlab and module Simulink. In Figure 1.3, a block diagram of the suggested digital measuring system is shown. The system is made of ready-made Simulink models. The unique advantage of using such a program environment or surrounding is that we are able to provide an arbitrary input signal which is further processed. The signal (comprising two voltage signals or voltage and current signals) is introduced into the circuit for the sample and hold (unit delay), which is located in front of the actual ADC. Then the signal is transferred from the output sample-and-hold circuit into the D flip-flop as a delay element and clocked from the unique signal generator (rectangular series of impulses) for which an arbitrary duty ratio is given. In this manner, the continual signal is measured, and the sample is held constant up the next measurement or sampling. The next sample is obtained from one of the next periods of the input signal, which is adjusted using the chosen simulation model parameters. Signals are multiplied and then integrated in time, thus obtaining the effective value (or active power). Since it has such an input block, Simulink enables the introduction of a deviation in the frequency of the processed signals. A separate program has been created in the Matlab. This program enables us (for a known spectral content of the processed voltage and current signals) to establish the desired sampling interval. It also enables us to determine the necessary number of samples to be processed in this manner, so that we can establish the power of the ac signal with a high precision. Table 1.1 shows the results obtained by the suggested procedure and the designed program for different cases of nonideal synchronization with a fundamental frequency of the processed signals. The results obtained by applying relation (1.107) were compared with those obtained using the practically realized instrument described in the following text. From the results given in Table 1.1, it can be concluded that the calculated relation for the uncertainties in the processing of ac signals in the case of nonideal synchronization provides satisfactory results. Thus, we can easily recalculate the uncertainties in the above described case.

26

Constant

Figure 1.3 Block diagram of simulation model for measuring effective value (or active power), based on the measuring concept suggested in this monograph

Table 1.1 Uncertainties in calculating RMS value of processing voltage signals obtained using relation (1.107) and fabricated instrument [1] (V = ..fi.220[V]; f=50Hz; W=40) Numberof measurements

Uncertainties in establishing sampling frequency

Uncertainties in establishing effectivevalue of voltagesignal using relation IE I [V]

t1ffHzl 1 2 3 4 5

0.05 0.04 0.035 0.1 0.2

0.1097 0.0878 0.0768 0.2193 0.4378

Uncertainties in establishing effectivevalue of voltage signal using fabricated instrument IE* I rVl 0.11 0.089 0.08 0.22 0.45

The obtained expression for the uncertainties (1.107) is in agreement with the results and uncertainties in the calculation of the basic ac values in [32-37].

1.5.2 Analysis of the Error Caused by Imprecision in Determining Sampling Interval For the proposed measurement concept, based on the usage of dual-slope ADC, we will analyze the error which introduces the sampling method and sampling interval generator, in order to completely recognize the possibility for practical realization of such a measurement method. Based on the realized analysis of the known harmonic content of the input signal (voltage or current), the necessary number of samples of the processed voltage and current signals is determined, which theoretically gives the correct result. All the remaining errors, which can appear in practice, practically come from the sampling interval generator. The error in the case of non-synchronized measurements is due to the fact that the measurement is not finished after the time period KT - because of the use of the "window' for measuring, in the most negative case, the measuring is performed subsequent to the time KT+ts, where K is the number representing the 27

number of periods in which we measured the time, T is the signal period, and ts is the interval between certain samples [15]. During the determination of the effective value we add the square value of the sampled data, while in case of determination of the active power we add the multiples of the sampled data regarding voltage and current. The analysis of the error (theoretically) in power measurement is performed in accordance with the following expression:

g =

1 1 VI

(

xr ,»,

)

J

[to+KT+t

S

(2)-Iu sin(2) ~ t - dt - (KT + t )VIcos(qJ)]

.[iv sin ~ t T

to

T

tp

s

(1.108)

The first member in the middle brackets is the work measured from the moment when sampling to , started and all through to the moment to+KT+ts. This further implies that the width of the 'window' in which the sampling occurs is different from the integer multiple of the number of signal samples. This is owing to the fact that sampling is not synchronized with the measured signal. The error is referred to the apparition power and not to the active power; this is necessary for error evaluation while we measure the active power in the presence of the reactive power. By solving the above integral for maximal error we obtain: (1.109)

The relative error in the calculation of the effective value of sine signals produced by the interval sampling generator can be represented by the equation:

E; = C1suml + C2sum2,

(1.110)

where sum1 and sum2 are the values which depend on the number W (samples) and relative error x in the realization of the sampling interval Ts : sumal = suma2 =

J.-. W

I cos(2JrT

2iTs J

i=O

(1.111)

J.-. ~sin(21! 2iTs ) W

i=O

T

t, =KT(l+x) Constants C t and C2 at synchronous sampling will depend in the simplest case only on the starting phase, so that the error is presented as: C1 =-cos2a,C2 =sin2a

(1.112)

E; = (- cos2a )suml + (sin 2a )sum2

To obtain a concrete result, we wrote the program in the C program language, which enables error analysis at synchronous sampling, in the case of calculating the effective value of sine signal, in relation to the relative error x in determination of the sampling interval, while W= I00 is fixed and phase u is given. Figure 1.4 presents the graphic dependency of the error Ep in calculation of the sine signal effective value from relative error x, in relation to the sampling interval in the range of -4% to 4%, for the two values of the starting phases: u=60° and u=90°. We can see that the error E; is almost a linear function of the error x in determination of the sampling interval, as soon as the x is in interval (-0,5%, 0,5%), which is important for synchronous sampling application. For the values of the relative error in the determination of the sampling period which is outside this range, the error in calculation of measured values with this method is unacceptably great. Another analysis which produces practical results, and can be significant in calculating the effective value of synchronous sampled sine signals is the analysis of the error dependent on the starting phase u, 28

for given Wand relative error x in determination of the sampling period. If synchronous sampling for a measuring system is realized by using a specified W, along with generating the sampling interval realized with the relative error x, then error in calculation is a function of the starting phase of the measured signal. By determining the derivation of the error function, we program-define the starting phases at which the error is maximal, together with their values. The results of this analysis for W=100 and the error in determination of sampling period x=±O,1%, are given in Figure 1.5 and graphic E; in function of the staring phaseo, We can conclude that the error Er, for a given Wand determined x, is a simple periodical function of the starting phase, and for some values of the starting phase it can be equal to O. The maximal value of this error is certainly dependent on x. Table 1.2 presents the concise program results for some characteristic values of x. The first column of the table shows the error value x, the second column is the starting phase at which the error in calculation of the effective value of sine signals E; is maximal, and in the third there is the maximal value of the error E: All the results are obtained for W= 100.

Synchronous sampling- Effectivevalues Dependency ofthe error Er*lE6from the relative error Ts EM=39267.20 ppm v(t)=V*sin(w*t+alpha), alpha=90, W=lOO (number ofsamples) alpha=60, W=100 (number ofsamples)

Figure 1.4 Dependency of the error in the calculation of the effective value from the error in determination of the synchronous sampling period

Sine function v(t) = V*sin(wt+alpha) E p=-cos(2 *alpha)*suml +sin (2 *alpha) *sum2,suml =f(W, x), sum2=f(W, x) 29

1009

E

-1009

1011

E

-1011

Figure 1.5 Error analyses in relation to the phase for certain errors in samplingperiod for synchronous sampling The third analysis is made for the effective value calculation of complex periodic signals (which, besides the fundamental harmonic contains the twentieth harmonic as well), while synchronous sampling is derived with W=200 samples, and with the error x=0,5% in the calculation of the sampling period. It infers that the error E; for certain values of x at synchronous sampling is not affected by the presence of the twentieth harmonic, which confirms the theoretical assumptionthat with a sufficiently large number of sampleswe eliminatethe systemic error which introducesthe measurementmethod. Table 1.2 The maximal value of error for certain errors in determinationof the samplingperiod for suggestedmethod of measurement

1

2 3 4

x(%) 0.001 0.005 0.5 4.0

nCO)

Er(ppm)

91.80 91.79 91.18 84.67

10.01 50.03 4977.61 38085.01

The programfor error analysisfor complexperiodicsignal in case of the suggestedmeasuring method 30

SYNHRONOUS SAMPLING - EFFECTIVE VALUES ERROR ANALYSIS IN THE CASE OF THE TWENTIETH HARMONIC FOR GIVEN W=200 (samples) AND RELATIVE ERROR FOR Ts=O.5% SINEFUNCTION v(t)= V*sin(wt+alpha) + V20*sin(20wt+alpha20) Sampling frequencyfs= 1OOHz+O.5% Fundamental harmonicf=50Hz, T=20ms, 20.harmonicj20=lKHz, T20=lms Number ofsamples W=200, the relative errorfor Ts errorTS=O.5% Er=El+E2+E3+E4 El =-1/2*cos(2*alpha)*sumll +1/2*sin(2*alpha)*sumI2 E2=-1 12 *cos(2*alpha20)*sum21+ 112 *sin(2*alpha20)*sum22 E3=cos(alpha20-alpha)*sum31-sin(alpha20+alpha)*sum32 E4=-cos(alpha20+alpha)*sum41 +sin(alpha20+alpha)*sum42 sumll=4975.l32521 ppm sumI2=-0.781492 ppm The maximal error for startingphase: alpha=O.OO, 90.00, 180.00, 270.00 Elmax(alpha)=-2487.566291 ppm sum21=4978.472857 ppm sum22=-15.640385 ppm The maximal error for startingphase: alpha20=0.09, 90.09, 180.09, 270.09 E2max(alpha20)=-2489.248712 ppm sum31=4975.863781 ppm sum32=-7.425271 ppm E3(alpha,alpha20)=4975.846969 ppm sum41=4976.028871 ppm sum42=-8.207152 ppm E4(alpha,alpha20)=-4976.035561 ppm The total error E=-4976.815186 ppm The results presented here for synchronous sampling represent concrete results for the error in the calculation of the measured value. This option was chosen due to the inability to effect an ideal realization of the sampling interval (tsampling which is defined earlier), i.e. an ideal synchronization of the sampling frequency with signal frequency which is the object of measuring, as it has the strongest influence on this sampling method. The remaining nonidealities are: due to the non-formed signal (to]), the delay SIR (t02) from the momentof issuing a command signalto the momentof holding a samplewill have the same influence as the starting phase. It is for this reason that we do not take this moment into consideration. Only the parameter t anmu, e.g. non-ideality of SIR (the uncertainty of sample catching) can give a different contribution to the error, because of its unexpected nature. The approximate analysis of the error, made in the MCAD-y program, for taptu in the range from 0 to 200ns and for W=300. The results of the analyses carriedout in this mannershow that the error ESHPPM(taptu);

n

ESHPPM(APT) = ESH(APT).10 6

APT value in range (0 - 2 * 10-7)s

ESHPPM(1) = -2.304

The error ESHPPM express in ppm

T

ESHPPM(1.5) = 12.999 ESHPPM(2) = 0.758 ESHPPM(0.5) = 2.695

The relative error E; is defined as a quotient of the absolute error and a theoretically accurate value of the measured value. By the application of the digital measuring methods, the calculation is taken as an important parameter which characterizes the performance of the measurement device. Synchronous sampling (the suggested measuring method in this book can be classified as a synchronous sampling method), in the case of a sine signal (which can appear in real electric utilities), when calculating the square of the effective value with W samples Veffusing the following equation:

Veff 2(W)

V2

=-

2

-

V W-1 -L cos [2(mt 2

2W

i

+ a)]

(1.113)

i=O

The relative error is:

_ 1 E

=-

r

W

W -1

L

cos

2

(m

ti +

a )

(1.114)

i=O

In an ideal case the discrete sampling moment t. can be defined as: (1.115)

to- is the starting moment of the sampling process, 0 ---l1tM = nk"; 2- - - M = k"/\OJl1tM ~ 21C >- 1C; k" IS natural number Tharmonics Tharmonics If the last of the conditions (2.24) is satisfied, the step between two consecutive samples L1t must be taken as: flt -
bl b2 bM2 dl d, d, I

_

I

_

I

_

I

(2.28)

_

, (" ,) d' ,AI 'I' => d = gcd CI,C 2,.··,C M2 => d = A => 2 ml1tM = II" ¢::> A I1tM = IITharmonics

¢::>

,I kA

A'I1t ~ n; 2-M

= 11 /\ mI

I

I; is natural number

Apart from conditions (2.28) for equation (2.26), it is also necessary to analyze the second addend in (2.26). We can write for the minuend:

fCOS[(S-A~N8t+¢, -If/;]=cos(¢, -If/;)fcos[(s-A~NM]-sin(¢, -If/;)fsin[(s-{N8t] j=1

j=l

To equalize (2.29) to zero, it is necessary to: s - i r m/),.t -:;:. O/\sin-s - i r m/),.tM = O: Vr E { } sin-12 ... M => 2

2

"

, ,

,s-~

N - min(A: ) I

2

2

rm/),.t -< rc => /),.t -
sin~m/),.tM = sin dIS -Cr m/),.tM = 0; Vr E {1,2,...,MJ;Vs E {1,2,...,N I } d,

if e'= gcd{dls ~m/),.tM = l;rc,l; 2d,

E

N;

~ = Afl => Afl/),.tM = I;Tharmonics; M ~ Tha~;onics; k'Afl = I; ~

A

/),.t

The subtrahend in the second addend of (2.26) will also have to be equalized to zero. For this we can write:

fcos[(s + A~ ~Llt + ¢s + lj/~]= cos(¢s + lj/~ )fcos[(s + A~ ~Llt]- sin(¢s + lj/~ )fsin[(s + A~ ~Llt] j=l

j=l

The (2.31) is equal to zero if: . s + A'r m/),.t -:;:. O/\sm-. s + A'r (o/),.tM = 0; Vr E { } sm-1,2,...,M => 2

2

2

,

S

+
sm-mdtM = sm s r

=

d,

2

(2.31)

j=l

N + max(A' ) I rm/),.t -< 2

1C =>

,)

,

dt -
!tJ -< h

SIll

S

"

=> -1 -< Ar - JL s -< 1=>

Imax{A~ - JL: ~ 2

(2.42)

1 wtlt -< - wtlt -< N j wtlt -< 1C 2

Besides this condition,the followingmust be true as well (the subtrahendof(2.41»: . A~ + JL: . A~ + JL: { sIn--OJtlt;i::O/\SIll--OJtltM=O;'Vre 1,2,...} ,M ;'Vse {1,2,...,N }=> 2

2

2

max{ir + JL' }OJtlt -< 1C => tlt -
k' A = I => M ~ ~. I is natural number 6 AVI t:..t ' 6 VII VII tlt M = 1' => L' - JL 'I'Ar ;i:: Il ,; 1~ r S M /\1 ~ s ~ N }~ -OJt:..tM A if AVI = gcdlAr = 17, 1C => A - 2 2 s s 7 2 Tharmonics 2

AVI

(2.43)

=> k' AVII = I' => M > Tharmonics . I' is natural number 7 AVII t:..t ' 7

The seventh member that is the result of the applicationof the equation (2.6) is of the following form:

Mp;

M 2 N3

M

r=l s=l

j=l

VRIRLLk);L{cos[(P; -A~)jOJ~t+t/J~'

=

-Vl~]-cos[(u; +A~)jOJ~t+t/J~' +VI~»

(2.44)

It is necessaryto satisfy the following conditions(for the subtrahendin (2.44»: sin

A~ + P: OJM 0# 01\ sin A~ + P: OJMM = 0; Vr E {1,2,..., M 2 }; Vs E {1,2,..., N3}~ 2

2

max{ir + J-/} TharmOniCs . s mflt ~ 1r ::::::> flt ~ 2 max{A~ + Jl;}

AW ~ =gcd{A~ + Jl;ll ~ r ~ M 2 /\ 1~ s ~ N 3 }::::::> -OJfltM =1~1r::::::> AVlII _ _ t_ M =I~ ::::::>

if AVlII

2

::::::> k' AVIII =

l. ::::::> M > Tharmonics . 8 AHII flt '

max{Jl" -_r--:..mflt A' } ~ 1r ::::::> flt ~

_~_s

2

if AIX

t: is natural number TharmOniCs .

max{u; - A~}

IX I1~r~M2 /\l~s~N3 }::::::>-mfltM=191r::::::>A A ' I X ---M=19::::::> flt t

2

Tharmonics

Tho,

/x'

::::::> k' A

(2.45)

8

{" -A =gcdJls r I

Tharmonics

= I ::::::> M > ~. I is natural number 9

-

A/X flt '

9

If the condition shown in (2.45) is satisfied, it results in P7*=0. The condition for P8*is analogous to the one that has already been obtained (2.45). For the last member that is the result of multiplication of the voltage interharmonicand the current interharmonics (P9 *) we observe that: M3 N 3

M

r=1 s=l

j=l

Mp; = VRIRLLk~l;L {cos[(Jl; - A: )jOJ~t + t/J~' - VI:]- cos[~; + A: )jOJ~t + t/J~' + VI:» In order to guarantee that P9*=P9, it is necessary to have:

66

(2.46)

. A~ +- /1: mlit * O ' A~ +- /1: mlitM =O;V're {1 ,2,...,M };V'se {1,2,...,N }:::> sln/\sln3 3 2 2 max{l'r + r:II"}s mlit ~ J[ :::> lit ~ Tharmonics.

max{A~ + /1:}

2 .

X

{ "

if A =gcdAr

+/1s"I 1~r~M3

/\l~s~N3

}:::>-mfl.tM=/ AX ,

101C:::>A

2

X

, - -litM = /lO:::> Tharmonics

(2.47)

:::> k' AX = I' :::> M > Tharmonics . I' is natural number -

AXfl.t '

~

1C :::> lit ~

10

max{/1" - A"} mfl.t

_~~s----.;r~

2

10

TharmOniCs .

max{p: - A:}

• f -A"I" " if AXI =gcdVLs r /1s *Ar;1~r~M3 u

:::> k' AXI

= I'

/\l~s~N3

}:::>-mfl.tM=/ AAI , 2

I I1C:::>A

AI --M=/II:::> lit ' Tharmonics

:::> M > Tharmonics . I' is natural number

II

-

AAI fl.t '

II

2.1.1 Derived Conditions for Precisely Processing All the above conditions are summarized in the Table 2.1. The Tharmonics is the period of harmonic components of voltage and current signals. By accepting that a is the maximum value of all denominators in the second column of the Table 1, whereas b is the minimum of all the denominators in column 3, k' lowest common denominator (led) of 11k], llk3 , .•. , Uk» in column 4, then we come to a conclusion that it is necessary to satisfy the following condition:

~I -< Tharmonics

(2.48)

a

M~

Tharmonics

(2.49)

b~1

M = k'· Tharmonics ~t

~ k' a

(2.50)

By combining the conditions (2.48) and (2.49), we get the following: M ~~ (2.51) b from which, it becomes evident that the first natural number that satisfies the condition is defined by:

M

=[~] + 1;

[~ ] =m{~)

(2.52)

therefore we can conclude that with M chosen like this, we assume that L1 1 : 111 = Tharmonics • k' . M ' k'= lcd(ll A~, i = 1,...,M2;11 A~,j = 1,...,M3;11 /.1~,S = 1,...,N2;11 /.1;,r

(2.53)

= 1,...,N3 )

to fulfill the condition for the completely accurate recalculation of the active power, according to the suggested concept of synchronized measuring. The condition obtained for L1 1 is a complete equivalent of the one from the original concept of synchronized sampling (the number of samples M must be in accordance with the Shannon-Nyquist criterion that takes into consideration the highest frequency from the spectrum of the processed signal), because the member (2.53) for ~t, contained in the nominator of the expression, represents the period T of the complex signal which comprises both interharmonics and subharmonics: T = lcd(Tharmonics' Tinterharmonics' Tsubharmonics) (2.54) The conclusions (2.53) and (2.54) are completely new and unique. The viability of the method confirms the fact that we can apply this procedure on measurements described in [2, 3, and 4]. The

67

conclusions will be the same as before, with the exception that now the measuring time will be different, depending on the spectral contentof the processedsignals. Table2.1 Conditions to be fulfilled in order to satisfythe equation contained in relation(2.7) conditionto be satisfiedwhen selectingthe samplingintervalilt product of multiplying the voltage harmonicby current harmonic product of multiplying the current harmonicby the voltage subharmonic of product multiplying the current harmonicby the voltage interharmonic product of multiplying the voltage harmonicby current subharmonic product of multiplying the voltage harmonicby current interharmonic product of multiplyingthe voltage subharmonic by current subharmonic product of multiplying the voltage subharmonic by current interharmonic product of multiplying the current subharmonic by voltage interharmonic product of multiplying the voltage interharmonic by current interharmonic

J1f -
r ~) Sin(Wi~1r~) ] . W+!J. +2~ W+~ COS((W+I),r1r~)+~2 =c 2(,r ) . (A." M) . (x M) W +~ sm W+~ sin W+~ 2

p

p

P

1C

p1C

(2.83)

2.3.1.4 Zu-Liang Compensation Zu-Liang proposed a trapezoidal method with "end" correction as a possible processing method. He demonstrated that the ak and bi; a', and b '/, and a'~ and b"p terms were reduced considerably when compared to the other proposed methods. With this method the operator A was defined as: AVi(or

l ~ ] 1 [W-l 5vo+-LVi ii)= -1- [O.W +O.5vw +-(vw +vw+!J.) = - - LVi +o.s(vw

since vw+~ =

w+ ~

Vo

i=1

W +!1

2

i=1

for a measurement covering an integral number of cycles.

The square of the worst-case error coefficient is then given by:

77

~]

-vo)+-(vw +vo) 2

(2.84)

i Wk¢

Wk"" Wk¢ 2 sin ---'!.Wk¢ ;Wk¢ Wk"" e -iT sin Wk¢ .Wk; Wk; 2 +t:J/ 2 cos Wk; + je 2 sin-'" . 2 +!!:.e-;2 cos Wk; + je -;2 sin Wk¢J k¢ 2 k¢ 2 /2" sin e - i 2" sin

e (e har k

\2

1 J = (W + A)2 o

. 2

1

sm

Wk¢J

Y...

Y...

2

2

. Wk¢J sm

2 Y...

2 Y...

Wk¢

k¢

2

2

(2.85)

] . 2 Wk; 1 [. Wk¢ k¢ Wk¢J2 = - - sm-ctg-+!!:.cos2 (W + !!:.)2 2 2 2

2 k¢J

2

=

= - - --+2!!:.--cos-cos-+!!:. cos --sm

(W +!!:.f

[

sin 2

sin

2

(& r

2

2

g r= ( )2 [Sin (Wk 7r --'!!'-)ct W+A W+A (k7r --.!!.-) W+A 1

+A

cos (Wk 7r --.!!.-)]2 W+A = e2(k)

(c:!Ub) = _1 ()2 [Sin(WA,~7r--.!!.-)ctg(A,~7r~)+ ~COS(WA,~7r~)]2 W+~ W+A W+~ W+~ (&;tY =-(_1_ [Sin(WA~7r~)ctg(i~7r~)+ACOS(Wi~1r~)]2 W+A) W+A W+A W+A 2

Assuming Ik¢I-;(t {cos[(s - A, ~~t + ¢s -If/;]- cos[(s + A, }»jM + rPs + If/; n= 0 => r=1 s=1

j=1

p; =0

p;

p:

(2.96)

p;

Analogous to the above, we conclude that = 0; = 0; =0 For the following element of the equation (2.6) the conclusionis that:

~~ " I fk.JftCOSrAr ~(, - f..ls,)'iOJ~t+lf/r'-fA ,] ~( "n MP6* = 2VRI RLLk,/s'-cosr Ar + f..ls 'iOJ~t+lf/r +¢sJf= r=1 s=1 2 j=1 I

=

,)

VRIRr~>;d~:coS[(A, - Jl;)jmM+'I'; -¢;]= MVRIRI(( cos('I'; -¢;,)=> r=1 s==1

j=1

(2.97)

r=1

P: =VRIR}»;,cos('I';

-¢;,)=> P: = P6

r=1

p;

p;

Since already explained above that: = 0; = O. for the last member in the equation (2.6), we conclude that:

Mp; =2VRI

Rrl:((·.!.

2

r=1 s=1

~N3

M

r==1 s=1

j=1

f~os[~; -A;)jmM+¢; -'I':]-cos[~; +A;)jmM+¢; +'1':»= j=1

~

=VRIRLLk;I;LcOSL(f..l; -j,~)jm~t+¢;' -If/;

Q3

]

=

MVRIRL-k;l;r cos(¢;~ -If/;)=>

(2.98)

r=1

p; = VRIRIk);, cos(¢;, -'I':)=> p; = P9 r=1

This brings about the followingconclusion: p' = p

¢:>

~ Jv;(t)i;(t)dt = l..-

i>; (jM)i; (jM)

T o M j=1 T for ~t = M; T = lcdl;Tha,man;cs , Tin! erharmonics, Tsubharmonics ) 85

(2.99)

3. RECONSTRUCTION OF NONUNIFORMLY SAMPLED AC SIGNALS The reconstruction of trigonometric polynomials, a specific class of band limited signals, such as ac signals, from a number of integrated values of input signals is in the focus of this chapter of our monograph. It is widely applied in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. We are proposing a practical and new algorithm for signal reconstruction and discussing potential applications to recover band-limited signals in the form of Fourier series (with known frequency spectrum but unknown amplitudes and phases) from irregularly spaced sets of integrated values of processed signals. Based on the value of the integer of the original input (analogue) signal, a reconstruction of its basic parameters is performed by the means of derived analytical and summarized expressions. In this way, we create a possibility to conduct a subsequent calculation of all the relevant indicators related to the monitoring and processing of ac voltage and current signals. Computer simulation demonstrating the accuracy of these algorithms and potential hardware realization is also presented. The chapter investigates the errors related to the signal reconstruction, and provides an error bound around the reconstructed time domain waveform. In signal processing, reconstruction usually means determination of an original continuous signal from a sequence of equally spaced samples. The samples are measured by using a quantizer, i.e. ADC. It is a well-known fact that any real signal which is transmitted along a channel-like form will have a finite bandwidth. As the result, the spectrum of the received signal cannot contain any frequencies above a maximum value, fmax=Mf if is the fundamental frequency). Consequently, M frequencies provide a specification of everything we know about the signal spectrum in terms of a d.c. level plus the amplitudes and phases of just- i.e. all the information we have about the spectrum can be specified by 2M + 1 numbers. Any transmission of a signal will always presuppose the loss of information when it is sampled in the receiver. The same will occur with the processing of an analogue signal into digital circuits due to its sampling in an analogue to digital (AD) converter. We get jitter errors, sometimes we even lose data, which results in a larger sampling gap, and, of course, we get noise. Therefore it is a demanding task to reconstruct the original signal from its sampling values, and the applied uniform sampling and reconstruction methods will not ensure proper results. The available data are frequently insufficient to achieve a high resolution reconstruction. This is especially important for the problems occurring during the conversion of the observed signals, due to the non-ideal nature of the AD converter used here, as well as the transducer and other devices used in the process [1]. Therefore, many attempts have been made in respect of the application of sampling techniques supported by optimal methods of reconstruction of band-limited signals in the form of a Fourier series (trigonometric polynomials). Reference [2] describes a new technique for the reconstruction of a bandlimited signal from its periodic nonuniform samples. Using a model for the nonuniform sampling process, the reconstruction was viewed in terms of a perfect reconstruction multirate filter bank problem. The [3] considers sampling of continuous-time periodic bandlimited signals which contain additive shot noise. By modeling the shot noise as a stream of Dirac pulses, [3] shows that the sum of a bandlimited signal with a stream of Dirac pulses falls into the class of signals that contain a finite rate of innovation, that is, a finite number of degrees of freedom. Reference [4] introduces two new algorithms for perfect reconstruction of a periodic bandlimited signal from its nonuniform samples. The reconstruction can be obtained by using the Lagrange interpolation formula for trigonometric polynomials, and when the number of sampling points N is odd and it satisfies N 2 2M + 1, Lagrange interpolation for exponential polynomials results in a complex valued interpolation function. An iterative frame algorithm for the reconstruction of bandlimited signals from local averages with symmetric averaging function is described in [5]. This paper gives bounds estimates on the aliasing error when a direct frame reconstruction is used for the reconstruction of non-band-limited signals. In this chapter, the application of integrative sampling for periodic-signal reconstruction is analyzed. The reason for this approach is in the fact that sampling consists of a sample and hold (or track and hold) circuit followed by conversion to digital equivalent, and is implemented using an AD converter. In [6], attention is paid to the possibility of the elimination of a separate sample and hold circuit, which, in itself, 86

is an original approach. This paper provides the platform for the development of an entirely new algorithm for processing slow-changing ac signals. The described procedure (integration) is used to process the original input analogue signal, which can be (as it almost always occurs in practice) modified by the presence of noise, as this noise is also processed. The proposed method has one important advantage resulting from the integrative sampling - the input random and periodic disturbance are filtered. The algorithm in which the reconstruction of the signal is done through a system of linear equations is explained in detail in [7]. This kind of approach to the processing was also considered in [8] and [9], along with the problem of subsequent reconstruction of the processed signals. In [8] and [9], the standard matrix inversion is used as the method of reconstruction, which requires very intensive numerical calculation. In [10], it was noticed that the ac signal integration method produces a regular matrix form in the derived system of equations. Based on this, an assumption was made that a full analytical solution can be reached, which was pointed out in [10], and has been realized in this chapter. If the integrative samples were assumed to be measured without errors, the presented algorithm without any further modifications can be used for signal reconstruction of periodic band-limited signals, which occurs in simulation. In a real environment and when the measuring is performed in practice, the integrative samples are measured with error. This happens as a result of the error ,caused by the imprecision introduced by the voltage and current transducers used to adapt the measured signals to the measuring system - the error of conversion, generated by errors occurring during the processing of the analogue signal through various digital circuits, as well as during determining the digital equivalent, etc. The sampled values obtained in practice may not be the exact values of the signal at sampling points, but only averages of the signal near these points [5]. In this case, the suggested algorithm must be modified, in order to be able to determine the best signal estimate, according to the criterion assumed, as in [5], [9] or [11]. We are proposing an efficient implementation of the algorithm for signal reconstruction that yields a significant improvement in computational efficiency over the standard matrix implementation. The approach is based on the use of the values obtained as a result of the integral-processing of the continuous input signal, in precisely defined time periods; this kind of integral-processing was performed as many times as needed to enable the reconstruction of the multi-harmonic signal that is the subject of the processing operation. Its main advantage is the possibility to increase the time of measurement without using sample and hold circuits. The presented method is designed for very accurate RMS measurements of periodic signals, and can be applied for precise measuring of other important quantities such as power and energy.

Table 3.1 The nomenclature used in the following parts of the chapter Meaning

Symbol

f s(t) M ADC

fundamental frequency of processing signal bandlimited input analogue signal the number of signal spectrum ac (harmonic) analogue to digital converter

Qo

the average value of the input signal

Qk

the amplitude of the kth harmonic

k If/k

the number of the harmonic the phase angle of the kth harmonic

87

3. 2. Proposed Method of Processing Let us assume that the input signal of the fundamental frequency f is band limited to the first M harmonic components. This form of continuous signal with a complex harmonic content can be represented as a sum of the Fourier components as follows: M

(3.1)

s(t) = ao + La k sin(k2Jift+V!k) k=l

Integrating the signal (3.1) (similar to equations (7) [6]), and by forming a system of equations of the same form, in order to determine the 2M+ 1 unknowns (amplitudes and phases of the M harmonic as well as the average value of the signal), we get: M

x(t,}= ?[ao+ fak sin(k2JifT + r, )JdT = ao~t __ 1 ~[COS(k2if(tl + ~t) + 'Ilk) - COS(k2 if(tl At k=l 2if k=1 k 2

f

t'-2

=

~t) + 'lfk)] = 2

(3.2)

f

ao~t + ~ ~sin(2k1ift1 + 'l/k )sin(kif~t) if k=l k

where I

=

1,2,...,

2M+!. The time interval (tl+ ~t)_(tl_ ~t)=MiS an interval within which the

integration of the input analogue signal is done. The moment t, - tJ.t is defined as the initial moment,

2

from which the integration process begins. This moment must be shifted with any subsequent integration of the input signal. The x(tJ value represents the value of the integral and is measured according to any reference value used in the process of conversion, i.e. in the work of the circuit with which the integration is performed. The obtained relation can be represented in a short form as: M

AO,lGo + LAk,lGk(sina..

COSlf/k

+cosak,l Sinlf/k)=Bl

(3.3)

k=l

where:

I

sin [kifM] = Ak,i'

2k1iftl

(k = 1,2'00" M X(I = 1,2,...,2M + 1)

= ak,l; ;if~t = Ao,l;

(3.4)

1ifx(tl ) = B,

Variables Ao,l, Ak,l and ak,l are the result of integration of the signal in the real device as defined by the equation (3.1). When processing is based on a dual-slope ADC, as it was done in [6], the values of the parameters defined by the relation (3.3) adopt the form given in Appendix B. We can define the integration interval as:

(tl+ ~ )-(tl- ~) = M= const., for every I ~ (tl+ ~)+(tl- ~t)=2tl

(3.5)

Ao,I = iftJ.t = Ao and Ak ,I = ~ k sin (kiftJ.t ) = Ak

t, = tl _ 1 + tJ.t + t delay where tdelay is the delay in processing. The value of tJ.t=const. (constant) in the above relation (3.5) is arbitrary. The system determinant for the system of 2M+ 1 (unknown parameters) can be represented as:

88

~sin~,1

Azsin~,1

...

AMsinaM,1

~CO~,1

AzCO~,1

...

AMcosaM,1

~Sinal,2

Azsina2,2

... AMsinaM,2

~CO~,2

AzCO~,2

...

AMcosaM,2

~Sin~,2M+1

Az Sina2,2M+1

~CO~,2M+!

AzCOsa2,2M+i

...

AM COsaM,2M+!

X system =

(3.6)

=(~·Az

...

AM SinaM,2M+l

·... ·AM_1.AM)~M+!

where:

X ZM+1 =

sin ce.,

sinaz,1

sinaM,1

cosal,1

cosaz,1

cosaM,1

sinal,z

sinaz,z

sinaM,z

cosal,z

cosaz,z

cosaM,z

sinal,ZM+l

sinaZ,ZM+l

sinaM,ZM+I

cosaJ,ZM+1

cosaZ,ZM+1

cosaM,ZM+J

cosZzz, cosZzz,

cosMa z

sinal sin zz,

sin2a l

sin Mal

cos«,

sin Zcr,

sinMa z

cosrz,

cos Ma,

(3.7)

sin a ZM+J sin 2a ZM+I ... sin Ma ZM+J cos a ZM+J cos 2a ZM+J ... cos Ma ZM+I Here is:

a k,t

= kat = Zknft,

(3.8)

The form of the obtained determinant (3.7) was the object of study in [9], but this was only the first step for the estimation of Fourier coefficients which are estimated from integrative samples and subsequently, in the second step, the estimators were corrected using the method of least squares and precision measurement of rectified-signal average. In this chapter, we derived analytical and summarized expressions for solving system equations (3.3).

3.2.1 The Determinants ofthe Van der Monde Matrix Let us consider the Van der Monde determinant [12] of the M order:

11 M = w(XpXZ'''''X M) =

M-I XJZ xJ Z M-J XZ ... Xz

XI 1 Xz

2 XM x M

where xi,

X2, ... , XM Xz

M-I Xz

XM

Z XM

XM

SJ,M'

Xk E C

(3.9)

k=J

(X2X3"~M j:

Xz

M-Z Xz

XM

M-Z XM

=(x ZX 3, ,,X M )SZ,M =

=

-

= (X ZX3• .. X M)

IT ft~j -xk)=

j=k+1

are arbitrary variables. If the following symbol is introduced for the co-determinant:

Z XZ

~(M)_ II

M-\ XM

=

M-l

(3.10)

IT fi(x

j -Xk)

j=k+1k=Z

We can obtain the subsequent equations for the next co-determinants (see Appendix C for reference): (3.11) where: (3.12)

89

(3.13) where: 1 1 1 1 1 L(X2X3",1XM )2 =X2-1X3+ -X21+ " . + - - + - + - + ...+--+ ... + - X4 X2XM X3X4 X3XS X3XM XM-IXM

(3.14)

(thesumof all inverted double products of different indices). (3.15) (thesumof all inverted triple products of different indices).

s;(M) -_ (-1 )k+l S2,M (X2X3···XM )~ LJ (

1

(3.16)

)

X 2X3",XM k-l

Here k cantake anyvaluefrom 4 to M-l. (3.17) An analogous procedure can be conducted in order to finda solution to the determinants that have the form of ~\~) , where I takes thevalues from the set2 ~ I ~ 2M + 1.

3.2.2 Reconstruction ofBand Limited Signals in Form ofFourier Series In the case of the system of equations formed according to the suggested concept of processing of the input signals (equation (3.3)), instead of the x variables in the above expressions, it is necessary to take in the trigonometric values foral,az,a3, ...,azM+I' as it was defined in (3.8). The given determinant (equation (3.7)) canbe transformed in the following manner (by using Euler's formulas):

=

2~M e-Mfill e'" _ e- a 1i

= ~e-Mfill

eali _ e-ali

e

2a1i

_ e-

2a1i

e2ali _ e-2ali

•••

e

_

«":'

e

ali

+ e?"

•••

eMa\i _

«":'

eali

e2ali

Ma l i

22M

= (-I) e-Mfill

e-ali

e-2ali

•••

e-Mali

eali

e2a1i

•••

e

2a l i

.,.

+ e- 2a 1i

'"

eMa\i

+ e-Ma1il

e2ali +e-2a 1i

eMali

e2a2i +e- 2a 2i

eMa2i

+ e- Ma l i + e- Ma 2i

=

eMalil =

eMalil =

22M

(((M-I)

_ -I

2

-~

" e-M"2i\e-Mali

e-(M-I)a li

'"

e-a1i

e"

I

e2ali

'"

eMalil =

M(M-I)

= (-I)

2

e-Mfie-M(al +....+a2M+1 )ill

ea1i

e2ali

•••

e(M-I)ali

eMali

e(M+I)ali

•• ,

e2Malii =

22M M(M-I) -M'!.-i 2M+' 2M ( ) ( - 1)- 2 = - 2 - M- e 2 e-M(a'+ ....+a2M+tliIlIl\eaji _e a k i = 2

=

j=k+1 k=1

(_I(~+I)iM

e-M(al+ ...+a2M+tliTIf.r(eaj:aki2isin a j - a k J =

2

2

j=k+1 k=1

)M(~+I).M 2M(2M+I) 2,\1(2M+I) ( =_1_ _ ' e-M(al+ ....+ a2M+I)ieM(al+ ... +a2M+I~2 2 i 2 _

2M+1 2M

a. -

a k

JIlIlsin-

M

2

X

+ = (_I((~+I) 2 2M 2

2M 1

TITI sin j=k+lk=1

j=k+1 k=1

a,; -

~

(3.18)

2

a k

2

Theco-determinants that are necessary to reach the solution of thegiven system of equations (3.5) are: 90

XZM+I,I

BI Bz

sinal sinzr,

BZM +I

sina ZM +I

=

Bz

=

cosMal cosMa z

cos Zc,

cosal cosc,

cos2a z

(3.19)

sin 2a ZM +I ... sin Ma ZM +I cosa ZM +1 cos2a ZM +1 ... cosMa ZM +1

sinZe, sin2a z

BI XZM+I,Z

sin Mal sinMa z

sin Z«, sin Zc,

sin Mal

cos e,

cos2a l

cos Mal

sinMa z

cos«,

cos2a z

cosMa z

(3.20) BZM +I

sin2a ZM +I

...

cosa ZM +1 cos2a ZM +1 ... cos Ma zM +I

sin Ma ZM +I

and so on. The above given co-determinantsbased on the following development can be written as: (3.21) X~ ,X~ ,..., X~M+l are the cofactors, obtained from cofactor X2M+I,1 after the corresponding row as well as the first column has been eliminated. The second cofactor is derived from the expansion of X 2M+1 along such a column. For this purpose, we must determine x; as cofactors of X 2M+ I. After the intensive mathematical calculation (Appendix D) we obtain:

«

la

ZM ZrrM+lrr . -ak S 1 n2 --l + ...+a -I +a +1 + ...+aZ M+I ) M(M+I) p P M(M-I) Z x' = (-l)p+q (-l)-z- 2 j=k+1 k=1 . ~ cos 2 P ZM+I a p-a .t..J -k sin-(a l + ...+a p_1+a p+1+ ...+aZM+JM k=1 2

rr

(3.22)

ke p

(the summing is done by all of the M's of the set For 2~q~M +1: M(M+I) X q = (_1)p+q(_1)-z-2 ZM(M-I)+1 p

ZM . 1 n-ak ZrrM+lrr S ---

«

j=k+1 k=1

2

ZM+I

a - a sin - P__ k 2 l:5:k*p

rr

{al'az, ...,aZM+J).

1

.~sin

z:

a

l

)

+ ...+a p_1+a p+1+ ...+a ZM+I

2

(

)

(3.23)

- a l + ...+a p-I +a p+1 + ...+a ZM+I M+q-I

q = M +r+ll\l::;; r::;; M-l ZM+I ZM a.-a J sin - 2_k M(M-I) . Z M(M-I)+1 X" = (_1)-z-2 J=k+1 k=1 ~ cos P ZM+I a -a.t..J sin - P__ k 2 l:5:bp

ITIT

rr

q=2M+l

rrrr

ZM+I ZM

X" = (_I)M(~-I) 2 ZM(M-I)+1 j=k+1 p

k=1

ja -

l

+ ...+a p-I +a p+1 + ...+aZM+I

)

2 ( ) a l + ...+a p -I +a P +1 + ...+a ZM+ 1 M-r

(3.24)

a-a

sin _j_k 2 cos a l + ...+a p_ 1+a p+1+ ...+aZM+I

ZM+I a -a rrsin-P__ k l:5:k*p 2

2

Based on (analytical) relation derived in this way the unknown parameters of the signal (amplitude, phase), can be determined through a simple division of the expression that represents a solution of the adequate co-determinants with the expression that represents an analytical solution to the system determinant: X2M+I,M+k+l

If/k = arctg - - - X 2M +1,k+ l

k

(3.25)

~ X 22M +1,k + l + X 22M +1,M +k+l

1 «; = A X 2M+l

91

The analyzed determinants and co-determinants of the system (3.3) were used only as the starting function (of the polynomial form), to the purpose of obtaining explicit and summarized analytical expressions which can be used further on to perform the calculation of the unknown signal parameters. It is a fact that the obtained system of equations (3.3) can be described, after the processing, by a special form of the determinant (which is summarized as the Van der Monde's determinant). This fact enables factoring and application of transformations that can be applied only on determinants. Any other procedure would lead to a much more complex calculation and to relations that are mathematically much more demanding. Owing to this, any subsequent calculation conduct towards reconstructing a periodic signal will not be related either to determinants themselves or to the procedures that are typically used in their solving. The obtained result clearly suggests that it is not necessary to use the standard procedure for solving the system of equations, as suggested in [8] and [9]. This procedure in the case of an extremely complex spectral content of a signal would require a powerful processor and enough time for processing. The derived expression, which is practically on-line for the known frequency spectrum of input signals, ensures determining all unknown parameters (amplitudes and phases). The proposed algorithm for signal reconstruction is presented in the form of a flow-chart in the text below. For the proposed algorithm, the order of the highest M harmonic component in the processed signal spectrum is required to be either known or adopted in advance, accepting that an M determined in this manner is higher than the expected (real) value. For the estimation of the frequency spectrum one of wellknown methods can be used. In [13], two accurate frequency estimation algorithms for multiple real sinusoids in white noise based on the linear prediction approach have been developed. The first algorithm minimizes the weighted least squares (WLS) cost function subject to a generalized unit-norm constraint. Similarly, the second method is a WLS estimator with a monic constraint. Both algorithms give very close frequency estimates whose accuracies attain Cramer-Rao lower bound for white Gaussian noise. A modified parameter estimator based on a magnitude phase-locked loop principle was proposed in [14]. It showed that the modified algorithm provided tracking improvements for situations in which the fundamental component of the signal became small, or disappeared for certain periods of time. In order to recalculate unknown parameters (amplitude and phase) of the processed periodic signals, it is necessary to have the results of the integration of the input analogue signals Bi; (equation (3.3) and (3.4)) obtained by means of integration in a precisely defined time interval of the signal that is the object of the reconstruction. The moment II -

~I , from 2

which the integration process begins, is referred to in

relation to the detected zero-crossing of the input analogue signal. It is possible here to equal t1 -

~I

with 2 o (the moment from which the integration process begins in the first passage). Since any subsequent integration begins immediately after the previous one, the total integration time in every passing is practically determined witht int .,! = f1t + t de/ay ' According to the suggested algorithm, the frequency of the carrier signal is recalculated in every passage, in a manner that takes into consideration a possible change, introducing it in the process of recalculating the new integration interval for the input analogue signal, the new variables Ao, Ak , az and the value of the input analogue signals integral Bi. Thus we reduce the possibility of error in the reconstruction process that occurred as a result of the variation in the frequency of the processed signal. The value f:J.t = canst. =

(1

f2M+l

) is to be taken as the constant value in order to

make the proposed method possible. By choosing its value in this manner the integration process can comprise the whole period of the complex input signal. Thus the processing will comprise all the possible changes to the processed signal at its period. Similarly, the error in calculation of the variable Ak becomes smaller. The procedure described in [7] can be used in order to have precise measuring of the period of the carrier signal. The proposed procedure depends on the harmonic frequency f since all of the relevant parameters calculated in the proposed algorithm depend onf However, the problem of synchronization is much less expressed than with the standard techniques of synchronized sampling, since this is a case of a real reconstruction of a signal, i.e. determining its basic parameters (phase and amplitude). 92

As an error occurs when determining the integrative samples BI, and variables Ao, Ak and ai; which is caused by their dependence on the carrier frequency f of the processed signal (Fig. 1-5), in the practical applications of the proposed algorithm we need to provide the best estimate of the given values, according to the criterion assumed. This can be done by the means of recalculation of the values BI, Ao, Ak and ai, through N passages (N is arbitrary). In this process, we form series Bnl, Ano, Ank, and ani (n=I, ...,N), as is given in the proposed algorithm. Random errors tJ,.n of measurements are unbiased, E(tJ,.n)=O have the same variance var(tJ,.n)=d, and are not correlated. Under these assumptions, the least squares (LS) estimator minimizes the residual sum of squares [15]: N

N

2

S(SI)= L>n(SI-Bnlf;S(~)= L>n(~-Ano) ; n=1

S( A

k )

n=1

N

= ~>n (Ak n=1

2

-

(3.26)

N

A ;S(Ill) = L>n (Ill -anl nk )

)2

n=1

where Pn = k / (J'2 ,k >- 0 (k is arbitrary). By minimizing the function S, we obtain the LS estimators

B,,~,Ak,al of the values BI, Ao, Ak and al as:

(3.27) n=1

n=1

n=1

n=1

The value of N will depend on the required speed of processing - the higher the N, the more precise the estimation of the value. Specifically, the LS method has been applied to a variety of problems in the real engineering field due to its low computational complexity. In this particular case, the estimation procedure does not require the matrix inversion and is considerably less demanding from the processor aspect than the methods described in [16] and [8]. In addition, when the proposed algorithm is used in simulations, the estimation of the given variables (3.27) is not necessary, which significantly reduces the processing time needed for its realization. It is important to notice that the values of the determinants and co-determinants (equations (3.7) and (3.21)) are governed by the measured frequency of the fundamental harmonic of the processed signal, as well as by the adopted constant and the value of the initial moment t1 - tJ,.t from which the integration 2 process begins. This is due to the fact that the calculation of the values of the determinant elements is based on coefficient ai, according to equation (3.8). The computational load of the iterative step involving FFTs does not change with the number of sampling values for some of the non-matrix implementation, but the speed of convergence is improved if a greater number of points are available [17]. A larger number of sampling points will result in the appearance of large matrices; the same occurs in the case of a large-spectrum input signal. Therefore the computational load for standard matrix methods (either iterative or those using pseudo-inverse matrices) increases quickly. Thus they may be extremely effective in situations with few sampling points, but fairly slow if there are many sampling points. Quite contrary effects are observed with the matrix methods proposed here. In the suggested algorithm, the determination of the inversion matrix is not required - a fact that makes it much faster and have a much better convergence than other matrix-based methods. Inverting the Van der Monde matrix requires calculating very high powers of the coefficients, which is always a problem with single precision or even double precision calculations. Apart from this, the suggested algorithm is non-iterative and therefore much faster. This is the key reason for our view that the analytical solution that we derived is more computationally attractive for moderately sized problems. Moreover, this feature makes it feasible for large reconstruction problems. 93

(

START

Frequency spectrum estimation of specified input signal (volatge/ current) derived by one of known technique s [12] and [13] 'f Defined value of M-the number of signal spectrum ac (harmonic ) component s

, ,

for n=! to N A special circuit detects the zero-crossing of the processing signal using comparator described in [6].

,

,

Integration time (const.) is defined as: M =

canst. =

rt:

I

I)

2M +!

AnO= :r -f -til

,

,

~

I

for k=1 to M Ank = .!.sin(k;iftil) k

¢> ~

for/ =! t02M+1 'J

if I = 1then I I II

~ = O;a. 1 = Lnft, else

= 11_ 1 + LlI

.,

ani = 2:rft l

As a result of integration of input analogue signal in a defined time interval we obtain the value Bnl

~

,---.j

for I=! to 2M+I and for k=! to M LPnBnl

Lp.A.o

LPnAnk

LPna nl

~>n n.

~>n

LPn n.

LPn

B =~; Ao = ~; A* = ~ ; al = ~ J

n=

¢>

.

Determinanti on of all necessary determin ants and co-determinant s based on derived analitical solutions with known coefficients I he unknown parameter s ot the SIgnal (amplitude , phase) cctermmco through a simple dividi on of the adequate co-determinats with the svstem determinat similarv to the well-known Cramer's rule 'f Calculation ofRMS in input signals, active power, energy

yo Continue=? No

(

94

STOP

)

This procedure can also be used for the spectral analysis where it is possible to find out the amplitude and the phase values of the signal harmonic, based on the set (predicted) system of equations. By taking the step-by-step approach in conducting the described procedure, it is possible to establish the exact spectral content and subsequently perform the optimization of the proposed algorithm. With this, the algorithm will be adapted to the real form of the signal.

3.3. Simulation Result and Error Analysis Additional testing of the realized calculations was carried out by simulation in the Matlab program package (version 7.0). In Appendix E, to provide better understanding of the record of derived results the form of solutions of obtained determinants of fifth order (M=2) is given. Table 3.2 presents a comparison of the results obtained through the application of the derived relations for solving the observed system of equations (3.3) {the relations (3.18) and (3.22)-(3.24)), and GEPP algorithm (Gaussian elimination with partial pivoting), offered in the Matlab program package itself (all of the calculations are done in IEEE standard double floating point arithmetic with unit round off u ~ 1.1x 10-16 ) . This represents a practical verification of the proposed algorithm for a case of ideal sampling (without an error in taking the value of the integral sample and determining the frequency of the processed signal). The values in columns separated by commas correspond to the solution for derived relations with different orders (taking that M=7, .f=50 Hz, tFO.OOl s). This means that in the column for x: values 77.3751539; 65.8192372; -96.0730198; 82.9370773; etc. correspond to x: = 77.3751539; x;=65.8192372; X~=-96.0730198; x;=82.9370773, respectively. Table 3.2. Verification of the derived expression for solving a system of equations with which the . 0 fh reconstruction teob served si signal is IS done X~; I

X 2M +1

proposed algorithm

96.2746562

GEPP algorithm

96.2746562

= 1,...,15

77.3751539; 240.7369977; 585.7387744; 1048.6023899; 1601.6571515; 2122.6636989; 2515.8984526; 2655.5007221; 557.8121810; 2190.4815359; 1686.1064501; 1129.5098525; 644.9058063; 279.7307319; 94.0066154 77.3751539; 240.7369977; 585.7387744; 1048.6023899; 1601.6571515; 2122.6636989; 2515.8984526; 2655.5007221; 557.8121810; 2190.4815359; 1686.1064501; 1129.5098525; 644.9058063; 279.7307319; 94.0066154

As it can be seen from Table 3.2, the derived relations produce solutions that are practically identical to the procedure that is most commonly used in solving systems of linear equations. The difference in the obtained values was equal to ixro" . Since the measuring becomes corrupted by noise, the reconstruction is an estimation task, i.e. the reconstructed signal may vary, depending on the actual noise record. The estimated input signal of the measurement system consists of two types of errors: systematic and stochastic ones. In the proposed system for the signal reconstruction, we eliminated a sample-and-hold circuit as a possible source of systematic errors. We investigate the issue of noise and jitter on the measurement method. The occurrence of the noise and jitter causes false detection of signal zero-crossing moments, giving an incorrect calculation of determinants and co-determinants. The error analysis was performed both in the program and as a SIMULINK (Figure 3.7 shows a SIMULINK model of a possible hardware realization used for simulation and uncertainty analysis), closely matching the flow-chart of the proposed algorithm. Figure 3.1 shows the influence of the error in determining the frequency of the carrier signal on the relative error in determining the integration interval and the A o and A k coefficients (equation (3.4)), for various harmonic content of the input periodic signal. 95

Figures 3.2, 3.3, 3.4 and 3.5 show another dependence: the manner in which 1) the relative error in determining the value of the input signal integral and 2) the value of system determinant both depend on the error in determining the frequency of the carrier signal and on the error in determining the initial moment from which the integration process begins, for various harmonic contents of the input periodic signal. The error in determining integration interval is not governed by the harmonic content; similarly, the value of the integral will depend on the chosen moment for the start of the integration and the harmonic content of the input periodic signal. The analyses showed that the starting moment of the integration is best determined in accordance with the zero-crossing of the input signal (t 1 - !J.t = 0). The 2 values of the A oand Ak coefficients prove to be less dependent on the fundamental signal frequencies.

l

0.5 ,----.---r---.----.,--~r_-;:==;====::;:'===.===.=1l

I

inl egrenon interval • Ak , M=5 --- Ak , M=7

-0 .1

-0 .2

·0 .3

L .0.t.2-

-

,-L- -

-0. 15

--,.L.--0 . 1

--:c':-::- -0 .05

----'-0

-

-,--l-::0 .05

-

,-L- 0 .1

----::--'-:- 0.15

--:c'

02

frequ ency deviatio n [Hz}

Figure 3.1 The relative error in determination of integration time and variables Ak as function of error in synchronization with frequency of fundamental harmonic of the input signal

1.5

..... ~

~ i

0.5

~

.~

-0.5

., · 1.5 1

0 .2

96

Figure 3.2 Relative error in integral calculation of inputsignal as function of error in synchronization with frequency of fundamental harmonic of the inputsignal and error in definition of the startingmoment (for M=5)

' 5

~

0 .5

~

t

0

~

~ t

-0.5

., 0.2

Figure 3.3 Relative error in integral calculation of the inputsignalas function of error in synchronization with frequency of fundamental harmonic ofthe inputsignaland error in definition of the startingmoment (for M=7)

97

0.2 0 .15 0. 1

-~

0 .05

.~

-0.05

~

~

-0 .1 -0 . 15

-0.2 1 0 .2

-0.2

Figure 3.4 Relative error in calculation of the system determinant as function of error in synchronization with frequency of fundamental harmonic ofthe input signal and error in definition ofthe starting moment (for M=5)

0 .6

--"!. ,

0 .4

02

S

.~

~

-0.2 -0 .4 1

0.2

Figure 3.5 Relative error in calculation of the system determinant as function of error in synchronization with frequency of fundamental harmonic of the input signal and error in definition of the starting moment (for M=7) The immunity of algorithm could be improved by applying more complex algorithm for the detection of signal zero-crossing moments [7]. Special attention is given to uncertainty analysis for the calibration 98

of high-speed calibration systems in [18]. The effect of the uncertainty created by the time base generator (jitter) can be modeled as non-stationary additive noise. Reference [18] also develops a method to calculate an uncertainty bound around the reconstructed waveform, based on the required confidence level. The error that appears as a result of the supposed non-idealities occurring in the suggested reconstruction model is within the boundaries specified by the [18] and [19]. A sensitivity function is commonly formulated assuming noise-free data. This function provides point-wise information about the reliability of the reconstructed signal before the actual samples of the signal are taken. In [19], the minimum error bound of signal reconstruction is derived assuming noise data. The quantization error is a very important problem because the reconstruction algorithm proposed here is of quite sophisticated form and some operations, like determinant calculation for example, are badly conditioned task and may considerably amplify the quantization errors. However, the error of quantization that appears here is much smaller, due to the described concept of measuring (integration) of the input signal and proposed hardware realization (Figure 3.6).

3.4. Possible Hardware Realization of the Proposed Method of Processing The block diagram of the digital circuit for the realization of the proposed method of processing is shown in Figure 3.6. As shown in Figure 3.6, there is no special circuit for sample-and-hold - the analogue signal is directly taken to the integrating circuit by the means of an analogue multiplexer which practically determines the type of processing. Signals x(t) and xlt) are the input periodic signals that are the object of reconstruction according to the suggested algorithm (e.g. voltage and current signal), after being adapted to the measuring range within which the processing components operate.

SIGN

Counter and control logic

"'"

, --

Figure 3.6 Block scheme ofthe digital circuit for the realization ofthe proposed method of processing Within this precisely defined interval, the counter will count up to the value of a certain content remembered inside the accompanying logic (the DSP that controls the operation of the complete system in figure 6, and within which the proposed processing algorithm is implemented). After this, the counter is reset. The digital equivalent to the calculated value is established by the means of the counter and the reference voltage signal, following the same principle by which this is done inside a dual-slope ADC or sigma-delta ADC. Namely, dependening on the speed of the counter (counter's tact signal) being used, after the integrator performs the re-calculation according to the equation (2), in the second time interval, a numerical equivalent of the calculated integral of the analogue input signal is determined, based on a known reference value. The value determined in this manner is transferred to the controlling processor by the "DATA" lines (Figure 3.6). At the end of the second interval, the comparator placed at the integrator 99

output will identify the moment of completion of the conversion process as the moment in which the total voltage at the integrator output reached the 0 value. In this manner a signal is generated, by which the counter is reset, to form a digital equivalent of the input analogue signal. In the same process, the condenser in the integrating circuit is reset, and the suggested system is ready for a new integration process. Special selection lines control the operation of the analogue multiplexer at the input, which determines the type of the signal (voltage or current) being processed. The proposed solution can be modified for the purpose of reducing the error in determining the integral of the input signal. Owing to this, the proposed algorithm (even without a special addition by which to perform the best estimation of the integrative samples B, of observed signal) can perform a nonideal but feasible signal reconstruction. In [20] it was shown that implementation of sampling and reconstruction with internal antialiasing filtering radically improves performances of digital receivers, enabling reconstruction with much lower error. The B[ value (equation (3) and (4)), represents the value of the integral of the input analogue signal. We should note that the sign of VREF is dependent on the sign of B/, i.e., if B/ is positive, then VREF must be negative, and vice versa. The line "SIGN" (Figure 6) represents the detected polarity of the input signal which enables the polarity of the reference voltage to be chosen via a separate switch. The sign of the expression B[ can be verified by a program check, thereby adjusting the sign of VREF, which can be avoided if the operation is conducted using the absolute values of the operation. In order to verify the effectiveness of the proposed hardware solution, a simulation was performed in the SIMULINK program package (Figure 3.7).

Figure 3.7 SIMULINK model of the proposed hardware realization Figure 3.7 shows the addition of the white Gauss noise, thermal and IIJnoise, and jitter to the complex periodic signal. In the original toolbox, all the possible noise sources (mainly the contributions of the operational amplifiers and of the voltage references) were supposed to be white. The noise power in the block "Band-Limited White Noise" is of the height of the PSD (Power Spectral Density) expressed in V2/Hz. The power of the noise in the digital circuit suggested for the realization of the proposed method of processing (Figure 3.6) was well-described and defined in [21]. Considering that the noise power is additive, the PSD can be considered as the sum of a term due to flicker (1/j) noise and one due to thermal noise, associated to the sampling switches and the intrinsic noise of the operation amplifier. However, in order to have the most precise analysis of the sampling noise and the op-amp' s thermal and flicker noise at low frequencies, these were modeled by separate blocks in Figure 3.7. Here, k, T, and Care Boltzmann's constant, the temperature in Kelvin, and the sampling capacitor, respectively [22]. The Vn denotes the input-referred thermal noise of the op-amp. Flicker (1/j) noise, wide-band thermal noise and 100

2

dc offset contribute to this value. The total noise power V n can be evaluated, through transistor level simulation, and during the simulation it is assumed that Vn=30 JlVRMS, while the value of the sampling capacitance was C=2 pF. The effect of jitter was simulated by using a model separately shown in Figure 3.7, under the assumption that the time jitter is an uncorrelated Gaussian random process having a standard deviation f!,. t. This model of the jitter was described and analyzed in detail in [23] and [24], where it was shown that the noise floor is dependent on the input sinusoidal frequency. During the simulation, the value of jitter was Ins (standard deviation f!,. i) [22], [25]. The parameters of the input signal correspond to the values given in Table 3.3. In the course of the simulation conducted in this manner, the signal-to-noise distortion ratio (SNDR) ranged between 56 dB and 96 dB. By using the simple Relay block shown in Figure 3.7, the frequency of the fundamental harmonic was measured in the signal formed in this manner, which produced a series of samples to be used as input data for "S-Function Builder". Inside it, and based on the proposed algorithm the integration interval is defined, which practically generated the signal action (enable signal) in the block "If Action Subsystem". This (enable signal) action determines time interval in which integration of the input signal was done. After the values of the input signals integrals were determined through the above procedure, these were introduced in the "S-Function" block, so that the values of the unknown amplitudes and phases of the input periodic signal can be established, named on the derived relations. The simulation results for signal reconstruction are shown in Table 3.3. In the simulation, a signal containing the first 7 harmonics was used (with the fundamental frequency of 50Hz). Table 3.3 defines the amplitude and phase values of the signal. In order to achieve a realistic simulation, it is necessary to scale the given values of the amplitudes of the harmonic components, by adjusting then to the working range of the proposed digital circuit (figure 6), in the way it was done in [7]. The superposed white noise and jitter will, in simulation performed in this manner, cause an error in detection on fundamental frequency of 0.022%. We can observe that the accuracy of the proposed algorithm is very promising and results are better then presented in [26]. The error in the amplitude and phase detection are mainly due to the error in measuring of the integrative samples and the error in determination of the necessary determinants and co-determinants needed for the solution of the obtained set of equations. Table 3.3 Simulation results of signal reconstruction by the proposed algorithm Harmonic number

Amplitude

1 2 3 4 5 6 7

311 280 248 217 186 155 155

Phase [rad] 1t 'Tt/3

0 1t/6 1t/4 1t/12 0

Proposed reconstruction algorithm Amp.error [%] 0.016 0.026 0.023 0.025 0.023 0.025 0.017

Phase error r%1 0.017 0.021 0.025 0.024 0.026 0.023 0.021

The suggested algorithm can be applied in operation with sigma-delta ADC, thus enabling high resolution and speed in processing of input signals. This is an important difference to be taken into consideration when comparing its implementation in this approach to the processing, to the results presented in [6] and [7]. This practically enables the reconstruction to be performed on-line, thus enabling high resolution and speed in processing of input analogue signals. To realize the circuit with which the integration is done, it is possible to use a separate circuit of the integrator (with the addition of a resetting option, so as to annihilate the impact that the time constant has upon the precision in the process of integrating the value of the input analogue signal; this in fact is performed internally in a dual-slope and sigma-delta ADC). Subsequent to this, the counter will form a digital equivalent to an integral obtained in this manner. Based on this, it can be assumed that the circuit can be realized in a very simple manner, as anIC. 101

The speed of the proposed algorithm makes it nearly as fast as the recently proposed algorithms [27], [28]. A computer with CPU 2.4 G, 256Mb memory, and Windows XP 2002 operating system was used for the verification of the real-time characteristic of the proposed algorithm. The time required for the performance of the necessary number of the integrations of the input signal that is the object of reconstruction is defined as (2M + 1Xlit + t delay). It is directly dependent on the working tact of the system, and can be practically limited to the period of the processed signal, which represents the value approximate to the time needed for the reconstruction (in simulation). In practical applications of the proposed algorithm, the determined time for the reconstruction of the processing signal ought to be increased by the time testimation; necessary to estimate the variables EI, A o, Al and al. This time is directly governed by the value N, although it can be limited to the interval of 1 second. The time interval tdelay is directly governed by the speed at which the condenser in the integrator circuit in Fig. 6 reaches the value of O. It also depends on the speed of the switches and the analog multiplexer. This time can be reduced, in accordance with the processing principles used in sigma-delta ADC. All calculations were done by floating point arithmetic, thus eliminating any software uncertainty contributions to the measuring method. Reference [29] gives a measurement of the required processor time, in the realization of the matrix method in the reconstruction of signals, in the form in which it is implemented in many program packages. The method suggested by the authors neither requires any special memorization of the transformation matrix, the way it happens in [29], nor does it require recalculation of the inversion matrix. Thus, it is much more efficient in implementation, and it is not limited only to sparse matrices. In addition to this, the proposed solution becomes easier for hardware realization, while the proposed algorithm can be practically implemented on any platform. The accuracy of signal reconstruction can be guaranteed in practical applications in noisy environments with the use of a powerful processor with adequate filtering. The realized algorithm is a promising approach for determining the signal reconstruction and for highly accurate measuring of periodic signals [30-31]. The implementations we propose make this algorithm significantly more computationally attractive for moderately sized problems, and even make it feasible for large-reconstruction problems. It is based on integrative sampling of input analogue signals. The derived analytical expression opens a possibility to perform on-line calculations of the basic parameters of signals (the phase and the amplitude), while all the necessary hardware resources can be satisfied by a DSP of standard features. What has been avoided here is the use of a separate sample-andhold circuit which, as such, can be a source of a system error. The suggested concept can also be used as separate algorithm for the spectral analysis of the processed signals. Based on the identified parameters of the ac signals, we can establish all the relevant values in the electric utilities (energy, power, the RMS values). The uncertainty bound is a function of the error in synchronization with fundamental frequency of processing signal, owing to the nonstationary nature of the jitter-related noise and white Gauss noise. The analysis shows that the proposed algorithm retains high accuracy in reconstructing periodic signals in a real environment.

References [1]. 1. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, Applications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [2]. R. S. Prendergast, B. C. Levy, and P. 1. Hurst, "Reconstruction of Band-Limited Periodic Nonuniformly Sampled Signals Through Multirate Filter Banks", /EEE Trans. Circ. Syst.-/, vol. 51, no. 8, pp.1612-1622, 2004. [3]. P. Marziliano, M. Vetterli, and T. Blu, "Sampling and Exact Reconstruction of Bandlimited Signals With Additive Shot Noise", IEEE Trans. Inform. Theory, vol. 52, no. 5, pp.2230-2233, 2006. [4]. E. Margolis, Y. C. Eldar, "Reconstruction of nonuniformly sampled periodic signals:algorithms and stability analysis", Electronics, Circuits and Systems, 2004. /CECS 2004. Proceedings of the 2004 11th IEEE International Conference, pp. 555-558, 13-15 Dec. 2004. [5]. W. Sun and X. Zhou, "Reconstruction of Band-Limited Signals From Local Averages", IEEE Trans. Inf. Theory, vol. 48, no. 11, pp. 2955-2963, 2002. 102

[6]. P. Petrovic, S. Marjanovic, and M. Stevanovic, "Measuring of slowly changing AC signals without sample and hold circuit", IEEE Trans. Instrum. Meas., vol. 49, no. 6, pp.1245-1248, 2000. [7]. P. Petrovic, "New Digital Multimeter for Accurate Measurement of Synchronously Sampled AC Signals", IEEE Trans. Instrum. Meas., vol. 53, no. 3, pp.716-725, 2004. [8]. P.Pejovic, L.Saranovac, and M. Popovic, "Comments on "New Algorithm for Measuring 50/60 Hz AC Values Based on the Usage of Slow AID Converters" and "Measuring of Slowly Changing AC Signals Without Sample-and-Hold Circuit"", IEEE Trans. Instrum. Meas., vol. 52, no. 5, pp.l688-1692, 2003. [9]. A.K. Muciek, "A Method for Precise RMS Measurements of Periodic Signals by Reconstruction Technique With Correction", IEEE Trans. Instrum. Meas., vol. 56, no. 2, pp.513-516, 2007. [10]. P. Petrovic, M. Stevanovic, "A Reply to Comments on "New Algorithm for Measuring 50/60 Hz AC Values Based on the Usage of Slow AID Converters" and "Measuring of Slowly Changing AC Signals Without Sample-and-Hold Circuit"", IEEE Trans. Instrum. Meas., vol. 55, no. 5, pp.1859-1862, 2006. [11]. A.V.D.Bos, "Estimation of Fourier Coefficients", IEEE Trans. Instrum. Meas., vol. 38, no. 5, pp.1005-1007, 1989. [12]. V.E. Neagoe, "Inversion of the Van der Monde matrix", IEEE Signal Processing Letters, vol. 3, no. 4, pp.119-120, 1996. [13]. H. C. So, K. W. Chan, Y. T. Chan, and K. C. Ho, "Linear Prediction Approach for Efficient Frequency Estimation of Multiple Real Sinusoids: Algorithms and Analyses", IEEE Trans. Signal Proc., vol. 53, no. 7, pp.2290-2305, 2005. [14]. B. Wu and M. Bodson, "Frequency estimation using multiple source and multiple harmonic components", American Control Conference, 2002. Proceedings of the 2002, vol.1, pp. 21-22, 8-10 May 2002. [15]. G. Seber, Linear Regression Analysis, New York; Wiley, 1977. [16]. SJ.Reeves and L. P. Heck, "Selection of Observations in Signal Reconstruction", IEEE Trans. Signal Proc., vol. 43, no. 3, pp.788-791, 1995. [17]. H. G. Feichtinger, "Reconstruction of band-limited signals from irregular samples, a short summary", 2nd International Workshop on Digital Image Processing and Computer Graphics with Applications, pp. 52-60, 1991. [18]. T. Daboczi, "Uncertainty of Signal Reconstruction in the Case of Jitter and Noisy Measurements", IEEE Trans.on Instrum.Meas., vol. 47, no. 5, pp.1062-1066, 1998. [19]. G. Wang, W. Han, "Minimum Error Bound of Signal Reconstruction", IEEE Signal Proc. Lett., vol. 6, no. 12, pp. 309-311,1999. [20]. Y. S. Poberezhskiy and G. Y. Poberezhskiy, "Sampling and Signal Reconstruction Circuits Performing Internal Antialiasing Filtering and Their Influence on the Design of Digital Receivers and Transmitters", IEEE Trans. Circ. Sys.-I, vol. 51, no. 1, pp. 118-129,2004. [21].E. Alon, V. Stojanovic and M. A. Horowitz, "Circuits and Techniques for High-Resolution Measurement of On-Chip Power Supply Noise", IEEE Journal of Solid-State Circuits, vol. 40, no. 4, pp.820-828, 2005. [22]. H. Z. Hoseini, I. Kale and O. Shoaei, "Modeling of Switched-Capacitor Delta-Sigma Modulators in SIMULINK", IEEE Trans. Instrum. Meas., vol. 54, no. 4, pp.1646-1654, 2005. [23]. G. Vendersteen and R. Pintelon, "Maximum likelihood estimator for jitter noise models", IEEE Trans. Instrum. Meas., vol. 49, no. 6, pp.1282-1284, 2000. [24]. K.J.Coakley, C.M.Wang, PD. Hale and T.S. Clement, "Adaptive characterization of jitter noise in sampled high-speed signals", IEEE Trans. Instrum. Meas., vol. 52, no. 5, pp.1537-1547, 2003. [25]. G.N. Stenbakken, and J.P. Deyst, "Timebase Distortion Measurements Using Multiphase Sinewaves", IEEE Instrum. Meas. Techn. Conf., Ottawa, Canada, pp.1003-1008, May 1997. [26]. N.C.F. Tse and L.L. Lai, "Wavelet-Based Algorithm for Signal Analysis", EURASIP Journal on Advances in Signal Processing, vol. 2007, Article ID 38916, 10 pages, 2007. [27]. T. Cooklev, "An Efficient Architecture for Orthogonal Wavlet Transforms", IEEE Signal Proc. Lett., vol. 13, no. 2, pp. 77-79,2006. 103

[28]. Y. C. You; L. 1. Fei and Y. Q. Xun; "A Real-Time Data Compression & Reconstruction Method Based on Lifting Scheme" Proc. Trans. Dist. Conf. Exh., 200512006 IEEE PES, pp. 863-867, 21-24 May 2006. [29]. S. 1. Reeves, "An Efficient Implementation of the Backward Greedy' Algorithm for Sparse Signal Reconstruction", IEEE Signal Proc. Lett., vol. 6, no. 10, pp. 266-268, 1999. [30]. P. Petrovic, "A new matrix method for reconstruction on band-limited periodic signals from the sets of integrated values", IEICE Transactions on Fundamentals of Electronics, Communications and ComputerSciences, vol.E91-A, no.6, pp.1446-1454, Jun 2008. [31]. P. Petrovic, "New Approach to Reconstruction of Non uniformly Sampled AC Signals", Proceedings of 2007 IEEE International Symposium on Industrial Electronics (ISlE 2007), 1-4244-0755-9/07/$20, Vigo, Spain, pp. 1693-1698, 4-7 June 2007. [32]. P. Petrovic, M. Stevanovic, "New algorithm for reconstruction of complex ac voltage and current signals", author art with number 2478, Serbian and Montenegro Patent, Belgrade, 30. August 2006. [33]. P. Petrovic, M. Stevanovic, "Digital analizer of complex and nonharmonic ac signals", Serbian and MontenegroPatent No.2006/0558, Belgrade, 9.0ctober 2006.

AppendixB When processing is based on the usage of dual slope ADC, we have the following:

i"sin(k;if2ntJ= At, (k=1,2,...,M) k1if(2 + tc + 2to )=a k ,/ ; 1if2 tc n

n 1

(3.28)

= Ao; 1ifVREP (t2,/ - t1,/)=B/

where T2 = tz: - tu = it., the time in which the counter registered i clock impulses, VREF is reference voltage, B, is result of input signal integration, tc is the sampling rate with which the counter inside the AD convertor operates, 2n is the total number of the state that can be occupied by the counter (n-bit), while to is the initial moment of the conversion process. After developing a system of equations formed in this manner, we obtain a determinant equivalent in form to the one described in relation (7), where the qJz value is defined as: (3.29) a k ,/ = kip, = kif(2 n+1 t c + 2to )

Appendix C Based on relations (3.9-3.10), we can concluded that: ••• •••

M-l

X2

M-l

X3

(3.30)

M-l

•••

XM - 1

•••

XM

M-l

where a coefficient is still unknown, and will be determined through the iterative procedure, while: M-l

•••

X2

•••

X

•••

XM - 1

M-l

3

= (-1) (X2X3,,,XM_IYW(X2,,,,,XM_J

(3.31)

M-l

which yields the following written expression: (M)A(M-l) a 12 U12

)82,M-l

( - - X 2 .. ·X M _ 1

(3.32)

From this equation, it follows that: 104

A(M-l) _ ( -

Ll 12

X

\ (X

X M - l -X2 XM - l -X3 }

··

_ (- I)M - 2 ( X2",XM - 2 )a(M-l)A(M-2) 12 Ll l2 -

\I

M - 1 -XM - 2J\.X M - l

(

-

\A(M-2) -a12(M-l) P12

I)M-l( )2 X2·"XM - 2 S2,M-2

(3.33) (3.34)

where: (3.35) By continuingthe same procedure (by reducing the order of the determinant), we obtain the following:

11\~ = (x4 -X3XX4 -X2XX4 -ag)Xx3 -X2XX3 +X2 )

(3.36)

(3.37) where: 11(4) -

12 - -

S T· 2,4 2,4'

(3.38)

In order to determinethe unknown coefficients,we will have the following: A(5) - (

Ll 12

-

X. X

\I

X5 - X2 Xs - X3 Xs - X4J\.XS -

a l2(5)\A(4) P12

(3.39) (3.40) (3.41)

After a sufficientnumber of steps, we get: A(M-l)

Ll 12

S

= 2,M-l

T

(3.42)

M-3,M-l

(3.43) Finally, we can write that: (3.44) where: (3.45) If we assume that: 1

x2

1

x3

A(M) _ Ll l3

(3.46)

-. M-l

1

X M-1

1

xM

X M-1 M-l

xM

where:

105

1 (

-

M- l ( )a(M)A(M-l) _ ( I)M( I) 2···XM _ 1 2···XM - 1 13 t i 13

X

X

1

(3.47) X~

X~-2

a(M)i1(M-l) __ 1 13

13

(3.48)

-

X~_1

M-2 XM - 1

(3.49) (3.50)

(3.51)

Apart from this, the following form is applied:

- (X2X3 XX3 - X2 )ag) = (X2X3 XX3 - X2 XX3 + x2 ) ~ ag) = -(X2X3 )

(3.52)

From now on, we can write that:

i1\~ = S2,4 (x 2 + X3 + x4 ) = S2,4~,4

(3.53) (3.54) X4X2

+ X4X3 + X3X2

(3.55)

X2 +X3 +X4

What we conclude fromthis is:

i1~1 = S2,sT;,s

(3.56)

After a sufficientnumberof iterations, we conclude that: A(M-l) S T t i l3

=

2, .•1-1

M-4,M-l

(3.57) (3.58)

Finally,we have: (3.59)

(the sum of all the inverteddoubleproductsof differentindices). For the next co-determinant in this succession, we can write that:

106

-~~)=

1

X2

x 22

1

X3

X3

x 24

M-l

X2

M-l

4

2

X3

X3

-( (M)~(M-I) - X M -X2 X X M -X3 }.. (X M -X M - 1X X M -a14 14

• XM - 1

X~_1

XM

XM

X~_1

2

(3.60)

M-l XM - 1 M-l XM

4

XM

(3.61)

X~_1

Fromthis, it can be concluded that: (M)A(M-l) S T a 14

L.l 14

(3.62)

2,M-l M-4,M-l

- -

A(M-l) _ ( - X M-1 - x2

L.l 14

(M-l)A(M-2) a 14 Ll 14 - -

X

\(

\I

(M-t) lA(M-2)

X M-1 - x3},. X M-1 -XM-2!\XM-1 - a14

S

P14

T

(3.63) (3.64)

2,M-2 M-S,M-2

After conducting an iterative procedure, with which we reduce the order of the obtained expressions, we get the following form:

~~1 = (x5 - x2 XX5 - x3 XX5 - al(~) Xx s - x4 XX4 - x2 XX4 - x3 XX3 - x2 ) (X4X3X2)al~)(X3 -

X 2XX4 - X 3XX4 - x 2)

= -(X4X3X2

1

X2

11

x3

1x 4

x~ x~

(3.65)

x;

where:

(3.66)

What is also appliedis that - (X2X3

~(5)

14

=

)p(x3 -

ST.' 2,5 1,5'

x 2)

t

= (X2X3 XX3 - x2 XX3 + x 2) => p = -(x2 + x3 a}~) = -(x2 + x3 + x4 )

~(6) 14

=

S2,6T2,6

(3.67) (3.68)

An analogous conclusion is that A(M-l) _ -

L.l 14

ST'

(M) 2,M-l M-S,M-l' a 14 -

TM - 4,M - l

. A(M) - -

--T--' L.l 14

S

T

2,M M-4,M

(3.69)

M-5,M-l

If we performa substitution here, we get (3.70)

(the sum of all invertedtriple productsof differentindices). Repeating the procedure described above,we obtainethe following equations

107

(3.71)

(3.72)

AppendixD We must determine

x~

for l~q~M+l. Therefore:

x~ = (-lr+qE~

(3.73)

where E~ is obtainedfrom XZM+l overtumingp row and q column. We know that: X ZM+1=EzM+l. -M!!..i 2

E I = _e_ _ leati P 22M

_e-ati

•••

e'"

eMati _e-Mati

+e-ali

e2ali +e-2ati

•••

(3.74)

eMati +e-Mati\p

The index p shows that p row is eliminated fromthis determinant. -M!!..i 2

I

- e -2M - eati -e -at i EI p 2

...

e

Mati

I

+ e-Mati = p

(3.75)

(3.76) (3.77) (3.78) A

L.l

If. 2M+I 2M a - a· ap" (l)M 2M(2M+I) M-;t M(at+a2+...+ U2M+dirrrr· j k( e = e e· Sln--- Xj = ) 2M+I ( j=k+1 k=1 2

i)" 2-2M e -Mapi e 2'at+.·.+ ap-l+ap+I+···+U2M+I, • 1/

\.

(3.79)

rr

2M+l r2M r

EI

M(M+I) = (_1)-2-22M(M-I)

p

sin a j

-ak

2

j=k+1 k=1 2M+I

a

I(

).

(

\.

.e2at+...+ap-l+ap+I+...+a2M+11 • ~ e-at+...+ap-l+ap+I+...+a2M+lJiI

L..J

- a

M

rrsin-P--k

2

k=1 kv p

I

or: E I = (_l)M(~+I) 22M(M-I) P

nIT

sin a

j

-ak

2 . ~ cos

j=k+1 k=1

rr

2M+I k=1 ke p

a

sin-p -

- a -

k

£..J

ja l

+ ••• +a p_1 +a p+I + ••• +a 2M+I

2 (

-\a l

) + ••• +a p_1 +a p+1 + ••• +a 2M+1 M

2

It follows that: 108

(3.80)

(- I )P+1 ______ 1 . ZM+I a -a k

ITsin-P-

Lcos{a + ...+a l

p- I

+a P+1 + ...+a ZM+1 2

(3.81 )

-

2

ISk..p

Now, we can determine E~ for 2~q~M+l:

(3.82)

Now is:

(~P,M-q+Z - ~ p,M+q XXj = e

aji

)

nIT

E" = (_I)M(~+I) 2ZM(M-I)+1 j=k+1 P

hI

la

j

sin a -ak 2 . "" sin L..J

ZM+I a -a IT sin~2 k

(3.83) + ...+ap_1 +ap+1+ ...+aZM+1

l

( -

)

2 )

a l + ...+a p_ l +a p+ 1 + ...+a ZM+1 M+q-I

ISk.-p

It follows that:

X~ __ (-I)p+q _ ZM-1 X ZM+I

2

1:

1 ------.

IT ZM+I

a -a sin-P- - -k 2 ISk..p

.la

SIn

l

+ ...+a p_1+a p+1+ ...+a ZM+ 1 2

)

( ) - a l + ...+ap_1+a p+1+ ...+a ZM+1 M+q-I

;2 s q s M

(3.84)

while: x~

X ZM+I

_ (-l)p+q

-

2

ZM-1 ZM+I

IT'

«. -ak SIn---

ISk..p

. a l + ...+a _I +ap+1 + ...+a ZM+1 . SIn p ; for q = M +1 2

2

Forq=M+r+l:

109

(3.85)

(I) = __ -_ _ M- l

2M

1!

e

()

-M 2i(_I)M

~-I

e

M(Q]+...+ ap-l+ap+l+...+a2M+l~(~ p,M-r+l

+

~

) p,M+r+1

=>

q=M+r+ll\l~r~M-l 2M+12M

a-a

TITI sin- 2 L {a cos ----p-=----=----k

j

-

E q = ( -I

M (M - l ) 2 )

2

2M(M-l\'1 j=k+l k=1

r

1 + ••• + a

1 +ap+l + ••• + a 2M+1

a -a TIsin-P - - k

2

2M+I

P

2

l$.b'p

(3.86)

q=2M+I 2M+12M

a-a

TITI sin-

k

j

E" = (_I)~22M(M-l)i.l

-

j=k+1 k=1

cos a

2

a -a TIsin-P - - k

1 + ••• + a p_1 +ap+, + ••• + a2M+ 1

2

2M+I

p

2

l$ko