Approximate Parametric Fault Diagnosis

Zden k KINCL, Zden k KOLKA

Dept. of Radio Electronics, Brno University of Technology, Purky ova 118, 612 00 Brno, Czech Republic

xkincl01@stud.feec.vutbr.cz, kolka@feec.vutbr.cz

Abstract. The paper shows a procedure for testing large analog circuits or circuits with parasitic parameters via multi-frequency parametric fault diagnosis. To simplify and accelerate calculations the approximate symbolic analysis is used and unknown tested parameters are analyzed in separate frequency bands. Classification of unknown parameters into frequency bands is based on sensitivities. The proposed procedure is shown on parametric diagnosis of an EMI filter.

Keywords Fault diagnosis, frequency set selection, symbolic approximation, testability analysis, EMI filter.

1. Introduction The paper is focused on parametric fault diagnosis of

linear analog circuits in the frequency domain using approximate symbolic analysis. The goal of the fault diagnosis is to locate the fault components in circuit or to estimate actual values of some network parameters from measurements of network characteristics [1].

An arbitrary network function of a lumped linear time-invariant circuit can be written in symbolic form as

)(...)()(...)(),(

0

0

ppppp

bsbasa

sH mm

nn (1)

where s is the complex frequency and the polynomial coefficients ai and bi are nonlinear functions of network parameters p = [p1,...,pR]T. It is advantageous to formulate (1) for normalized network parameters because their usual values not commensurable among different types of components (resistors, capacitors, etc.).

The actual values of some network parameters pi can be estimated upon the measurement of one or more different network functions on several frequencies [2]. The other parameters are considered to have (chosen) nominal values. The estimation of unknown parameters represents solving a set of nonlinear fault equations based on network functions Hk, k = 1,..., K and measurements Mk,i evaluated on frequencies k,i, i = 1,.., Fk, [3]

ikikk MjH ,, ),( p . (2)

The procedure can be divided into two separated phases. The first one determines a group of testable circuit parameters related to the chosen test point [4]. The total number of testable parameters is referred to the testability degree T. In the case where only one test point is chosen, it is the rank of the Jacobi matrix of (1)

j

i

pjH

rankT),( p . (3)

The testability degree T is theoretically independent of the nominal values of components and test frequencies [4]. But in the real world the boundary between testable and untestable parameter is not sharp. The effective testability (determined numerically) depends on nominal values of components and test frequencies [3]. In the case when the testability degree T is lower than the total number of potentially faulty components, the system of nonlinear equations does not have a unique solution and additional test point must be chosen.

The set of linearly dependent columns of Jacobian matrix determines ambiguity groups [4]. Inside those groups the effects of individual parameters on the network function are indistinguishable from one another. Only one parameter of each group can be tested.

The second step of parametric fault diagnosis consists in solving the system of nonlinear equations (2) to obtain unknown circuit parameters. An appropriately chosen set of test frequencies reduces measurement errors and errors caused by the uncertainty of fixed network parameters [1].

A robust method for the selection of test frequencies has not been determined yet. A procedure based on maximizing the sensitivity in the frequency domain was proposed [2]. Other authors proposed the use of genetic optimization [5] or a simple heuristic solution [1], [6]. In this paper the test frequency selection is based on minimizing the Test Index measure (T.I.) [5]

MIN

MINMAXIT 2.. (4)

using the Particle Swarm Optimization (PSO) [7]. Symbols MAX and MIN represent the maximum and minimum

978-1-61284-324-7/11/$26.00 ©2011 IEEE

singular values of the Jacobi matrix. Due to the large range of values of T.I., it is usually given in the logarithmic scale.

The system of equations (2) with a small T.I. is well-conditioned, while a large T.I. leads to an ill-conditioned system. The T.I. should be minimized by choosing an ap-propriate set of test frequencies.

The system of nonlinear equations is solved by the Newton-Raphson iterative method [8] to obtain actual values of network parameters. For the sake of simplicity, we will consider only one network function (one test point) and only magnitude measurements in subsequent analysis. The Newton-Raphson iteration scheme will be

)()~,()~,(~~ 11 inininnn jjj MpHpJpp (5)

where p~ is a chosen subset of network parameters to be found, |H(j i, pn)| and M(j i) are column vectors of actual and measured magnitudes, respectively, and n is an adap-tively chosen damping parameter.

The entry of Jacobian matrix J represents a sensitivity of the magnitude in decibels to normalized parameter pj on frequency i [5]. It can be expressed as

)~,(

1)~,(Relog20p

pij

i,j jHp

jHeJ . (6)

The success of iterating procedure (5) strongly depends on the initial guess and the number of parameters being tested. The basic idea of method proposed in the paper is to exploit frequency-selective sensitivity of the network function to tested parameters. For example, some parameters may be important on low frequencies, while they do not practically influence the function on high frequencies, i.e. system (2) can be divided into a few band-limited problems of lower dimension.

Section 2 of the paper gives overview of the proposed method and Section 3 deals with an example analysis of an EMI filter.

2. Frequency-selective Identification The main idea of the proposed procedure is to divide

the large system of fault equations (2) into a few band-limited subsystems of lower dimension. Generally, the solvability of low-order systems of equations is better. The solution is found faster and the results should be less affected by numerical errors. The final solution of original system (2) consists of partial solutions of subsystems that are solved independently.

The all unknown tested parameters are classified into individual frequency intervals using frequency-selective sensitivity of original network function (1) to tested pa-rameters. For each group of tested parameters and selected frequency interval an appropriate set of test frequencies is determined by PSO. This stochastic optimization mini-

mizes the T.I. criterion of Jacobian matrix based on original transfer function (1).

A reduction method of Simplification Before Genera-tion (SBG) [9], originally developed for the approximate symbolic analysis, can be used for division of original system of fault equations (2) into a few subsystems. The original network function (1) is simplified in limited fre-quency band by replacing insignificant components in circuit by an open or short circuit. Approximated network function is valid only in selected frequency interval; in other intervals it may not be valid.

For any circuit parameter pi, the network function (1) can be expressed in form [3]

dcpbap

Hi

i (7)

where a, b, c, d are complex numbers for a fixed frequency (identical with an appropriate set of test frequencies for each group of tested components).

The large-change variations of H can be determined from the bilinear form (7)

db

HHiplim

00 ,

ca

HHiplim . (8)

In case of a two-pole element, formula (8) represents its replacing by an open or short circuit. In the case of controlled sources, H corresponds to replacing the source with an ideal operational amplifier.

If the circuit parameter pi has a significant influence in more frequency bands, the approximated network functions must also contain this parameter, but the actual value of this parameter is calculated only in one selected frequency band, in others is considered fixed.

The SBG method often returns a large number of different approximated functions. The optimal choice will need to undergo further examination.

Using a set of approximated functions and set of test frequencies the all unknown circuit parameters can be estimated by parametric fault diagnosis.

3. Application Example The above mentioned approach will be demonstrated

on a simple example of an EMI filter, which is used for the reduction of electromagnetic interference in electric power transmission lines. The actual values of network parame-ters may strongly affect its frequency characteristics, especially the maximum attenuation in stop-band and the attenuation on high frequencies [11].

The basic circuitry of the EMI filter is shown in Fig. 1. A reduced circuit model based on asymmetrical measurement, shown in Fig. 2, is valid up to approximately 100 MHz [11].

The circuit in Fig. 2 can be considered as two resonant circuits. The first one, consisting of inductor L1 and capacitor C2, provides low-pass filtering of interferences, while resistors R1, R2, inductor L2, and capacitor C1 are the parasitic parameters of the real EMI filter. Resistor R3 is the output impedance of the generator, resistor R4 is the input impedance of the network analyzer. Both of them have a nominal value of 50 and they are considered to be known.

Fig. 1. Basic model of the EMI filter (Schurter 5110.1033.1).

Fig. 2. Reduced mathematical model of the EMI filter in

asymmetrical measured system.

The effective testability degree T of this filter with respect to selected test point is seven. Since some columns of the Jacobian matrix are linearly dependent, there is one ambiguity group (R3, R4).

Since the number of unknown parameters is lower than the testability degree T, the system of nonlinear equa-tions has a unique solution and all unknown circuit pa-rameters can be estimated. In this case, six test frequencies must be chosen.

The normalized magnitude of partial derivative of the original network function is shown in Fig. 3. Two fre-quency subintervals <104 Hz, 106 Hz >, <106 Hz, 108 Hz> were selected. In the lower band, three parameters L1, C2, R1 are tested. An appropriate set of test frequencies was obtained by PSO based on the original transfer function

f1 = 69.2 kHz, f1 = 0.275 MHz, f3 = 1 MHz. (9)

These frequencies were also used for the SBG approximation of network function (1). The approximated

function should contain only unknown parameters of each frequency subinterval. If there are other parameters, they are considered constant. The maximum error between the original network function and its approximation was less than 0.5 dB on measured frequencies. The polynomial coefficients of the approximated transfer function can be obtained, for example, using the SNAP program [12]

.,

,,,

214321412

131124311

41310

141

410

CLRRCLRRbLRLRCRRRb

RRRRbLRaRRa

(10)

103

104

105

106

107

108

109

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

|Jn|

[-]

R1R2R3R4L1L2C1C2

Fig. 3. Normalized magnitude of partial derivative of original

network function.

In the next step, the unknown parameters L1, C2, R1 were estimated by the parametric fault diagnosis. The sys-tem of nonlinear equations is based on test frequencies (9) and polynomial coefficients (10). Fig. 4 shows a compari-son between the estimated first-band approximation and measurements.

104

105

106

107

108

109

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

|Ku dB

| [dB

]

MeasuredSolvedTest frequencies

Fig. 4. Estimated results for L1, C2, R1 parameters compared

with measurements (Red curve represents the approximated transfer function for the first band).

In the second frequency interval, remaining unknown parameters L2, C1, R2 are tested. The whole procedure is identical to the procedure for the first band.

The set of test frequencies and the polynomial coefficients of the approximated transfer function for the second group are

f1 = 13.2 MHz, f1 = 21.9 MHz, f1 = 36.3 MHz, (11)

.,

,,,

,

2122

2132121

20

2122

2121

10

CCLbCCRCCRb

CbCCLaCCRa

Ca

. (12)

The approximated transfer function contains also parameter C2, which has a significant influence in the second frequency subinterval. But its value was estimated in the previous step and now is considered constant.

103

104

105

106

107

108

109

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

|Ku dB

| [dB

]

MeasuredSolvedTest frequencies

Fig. 5. Estimated results for L2, C1, R2 parameters compared

with measurements. (Red curve represents the approximation for the second band.)

103

104

105

106

107

108

109

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

|Ku dB

| [dB

]

MeasuredSolvedTest frequencies

Fig. 6. Comparison of results for full model of the EMI filter.

Estimated parameters of this filter are R1 = 678 , R2 = 127 m , L1 = 300 uH, L2 = 4 nH, C1 = 54 pF, C2 = 14 nF. The maximum error of solution in 100 MHz frequency band was 1.2 dB and could be caused by inaccuracy of the model, approximations of transfer function or measurement errors.

4. Conclusion The classical least-square identification of the same

model required approximately 300 measurements to achieve similar results. The proposed method decreases computational cost of the identification and decreases the cost of measurement. For the particular EMI filter only six test frequencies must be chosen to determine six unknown network parameters.

Acknowledgements The research was financially supported by the Czech

Science Foundation under grant no. 102/08/H027 and P102/10/1665. The research is a part of the COST Action IC 0803, which are financially supported by the Czech Ministry of Education under grant no. OC09016 and MSM0021630513.

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