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398 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 49. NO.2. APRIL 2000

The Sensitivity of a Method to Predict a Capacitor'sFrequencyCharacteristic

~.;Svetlana Avramov-Zamurovic, Andrew D. Koffman, Member, IEEE, Nile M. Oldham, Fellow, IEEE, and

Bryan Christopher Waltrip, Member, IEEE

Abstract-A joint effort between the U.S. Naval Academy andthe National Institute of Standard and Technology (NIST) resultedin the development of a method to characterize the capacitanceand dissipation factor of a set of commercial standard four ter-minal-pair (4TP) capacitors. The method depends on network an-alyzer impedance measurements at high frequencies (40 MH-200MHz) and a regression of these measurements down to the fre-quency range of 10MHz-l kHz. This paper provides an analysis ofthe sensitivity of the regression parameters and the high-frequencyimpedance measurements.

Index Terms-Capacitance, dissipation factor, four ter-minal-pair capacitor, high frequency, impedance, precisionmeasurements, sensitivity analysis.

I. INTRODUCTION

SEVERAL groups have worked on the characterization offour terminal-pair (4TP) capacitance standards at high fre-

quencies [1]-[9]. The work originally started with CutkoskyandJones of the National Institute of Standards and Technology(NIST) and was followed by Suzuki, Aoki, Yokoi, Yonekura,and Wakasugi of HP Japan. This paper describes a variation ofthe technique described by Aoki et at. to predict a capacitor'sfrequency characteristic [1]-[3] The method is sensitive to re-gression parameter selection and the paper gives a detailed anal-ysis of the techniques used to calculate reasonable values forthese parameters.

The principles of a capacitor frequency characteristic predic-tion (CFCP) method is described in [1] and some practical as-pects are addressed in [3] and [9]. As a background introductionto our discussion, the practical realization of the CFCP methodis summarized.

II. PRACTICALREALIZATIONOF THECFCP METHOD

The following equation represents the calculation of a 4TPimpedance from driving-::point impedances (measurable with anetwork analyzer) [1], [3] (see Fig. 1):

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~

Z22 /

Z4tp = \/ Z Z (V (Zl1 - Zl1s2)(Z44 - Z44s3)11- l1s3

- V(Zl1 - Zl1s3)(Z44 - Z44s2)). (1)

Manuscript received May 26, 1999; revised January II, 2000.S. Avramov-Zamurovic is with the United States Naval Academy, Annapolis.

MD 21402 USA.

A. D. Koffman, N. M. Oldham, and B. C. Waltrip are with the ElectricityDivision. National Institute of Standards and Technology, Electronics and Elec-trical Engineering Laboratory, Technology Administration, U.S. Department ofCommerce, Gaithersburg, MD 20899-0001 USA.

Publisher Item Identifier S 00 18-9456(00)03433-1.

Four Terminal-Pair Capacitor

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Fig. I. Four terminal-pair capacitance standard.

Zii is the driving-point impedance at port i with all other portsleft open, Ziisj is the driving-point impedance at port i with portj shorted and all other ports left open.

Present measurement requirements at NIST have made it nec-essary to develop an updated approach for determining four ter-minal-pair capacitance and dissipation factor in the frequencyrange of 1 kHz-I0 MHz. In the proposed procedure, a networkanalyzer is used to measure the driving point impedance valuesat very high frequencies and a regression analysis is performedto predict the capacitor's frequency characteristic at frequenciesfrom 1 kHz to 10 MHz.

In order to measure the necessary impedances, namely, Ziiand Ziisj, the following procedure is performed.

Step 1) The network analyzer is verified and calibrated overthe frequency ranges used to conduct the driving-point impedance measurements.

Step 2) Impedances Zii and Ziisj are measured usingthe network analyzer over the specific frequencyranges. Equation (1) demonstrates the need to knowthe values of impedance differences: Zl1 - Zl1s2,Zl1 - Zl1s3, Z44 - Z44s2, and Z44 - Z44s3.

Throughout the rest of this paper Zii - Ziisj is usedto represent these difference measurements.

Step 3) A circuit diagram [4], [5] of a 4TP capacitance stan-dard is shown in Fig. 1. It is assumed that the capac-itive component of this impedance does not changewith frequency. Capacitances Go, Ghg, and GIg (seeFig. 1) are measured at 1 kHz using a high-accuracycapacitance bridge. Capacitance Go is the nominalcapacitance measured between the high and low ter-minals, Ghg is the capacitance measured between thehigh and ground terminals, and GIg is the capaci-tance measured between the low and ground.

U. S. Government work not protected by U.S. copyright.

AVRAMOV-ZAMUROVIC et aJ.: SENSITIVITY OF METHOD TO PREDICT CAPACITOR'S FREQUENCY CHARACTERISTIC 399

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Fig. 2. Real and imaginary parts of the measured impedance for a 100 pF capacitor.

Step 4) It is assumed that the change in the capacitor's fre-quency characteristic is due only to the inductiveand resistive impedance components. For example,impedance Zii-iisj is represented in the form

solved using regression techniques. To reduce theinfluence of skin effect, the measured real compo-nent must first be normalized using the formula

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Step 5) This approach simplifies the regression process sothat resistive and inductive components may be an-alyzed separately, Le., the resistive components arecalculated from the real part of the network ana-lyzer measurements after being normalized using

In the above equation, circuit analysis techniquesare used to solve the equivalent capacitance at eachof the ports on a capacitor, Cii and Ciisj. Theremaining resistive and inductive components are

400 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO.2, APRIL 2

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(3). Likewise, the inductive components are calcu-lated from the imaginary part of the network ana-lyzer measurements according to (2). The form ofthe regression used is

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more detailed explanation on the selection of regressian parameters, as well as measurement frequencranges, is given below.

Step 6) The last step is to predict the value of each drivinpoint impedance at a desired frequency. The regresion (4) may be used to predict Rii-iisj and Lii-iis"for any frequency in the range 1kHz-l 0 MHz. Equ~'-tions (1) and (2) are then used to predict the resultinimpedance, Z 4TP .

III. REGRESSION PARAMETER SELECfION

(6)By using a network analyzer to measure the capacitor's i

pedances, several limitations in selecting the measurement frrt-quency ranges take place.

AVRAMOV-ZAMUROVIC et at.: SENSITIVITY OF METHOD TO PREDICT CAPACITOR'S FREQUENCY CHARACTERISTIC

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Network analyzers are calibrated using standard open, short,and load devices. The standard load is a calibrated 50 St re-

sistor. In order to maintain high accuracy, the network ana-lyzer should measure the test capacitor at frequencies whereits impedance is on the order of $0 St. To further reduce the

influence of any network analyzer offset errors, ZII and Z44are measured twice. For example, Z 11 is measured just priorto measuring ZI132 and also just before measuring ZI1,,3.

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4TP air dielectric capacitors exhibit extremely low dissipa-tionfactor behavior (on the order of 10-5 or lower). In gen-eral, network analyzers cannot resolve this behavior. To over-

come this limitation, the measurements may be made at ex-tremely high frequencies where the dissipation factor compo-nents dominate the behavior. However, these are the same fre-quencies where the real components of the measurements areseverely influenced by skin effect. A tradeoff must be made.

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402 IEEE TRANSACfIONS ON INSTRUMENTATIONAND MEASUREMENT.VOL. 49. NO.2. APRIL 2000

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;1'i-I

The third guideline in the selection of measurement frequen-cies is related to the frequencycharacteristic of each port mea-sured. Since the regression is used to estimate the capacitor'sfrequency characteristic, the measurements must be made faraway from any resonances (frequencies where the impedanceis nonreactive).The complexity of the problem lies in the fact that there are

seven different impedances to be measured in order to estimatethe capacitor's frequency characteristic. The real and imaginaryparts of the impedance for a number of air capacitors were mea-sured in the range 20 MHz-2oo MHz using a network analyzer.The example in Fig. 2 shows impedance measured for a 100pF air capacitor. The results sh<!wthe significant difference inmeasured values between the real and the imaginary parts ofthe impedances. This figure and subsequent figures are givento demonstrate the complexity of the optimization process, notnecessarily the details. To achieve optimal network analyzer

I'

measurements that are around 50 0, the frequency ranges hav,to be around 100MHz (note log(50) '" 1.7). Note also thnumber of resonant frequencies to be avoided. Air capacitorhave very small dissipation factor and the measured real component is very small. At lower frequencies the imaginary component of a capacitor (and thus, Q factor) is sufficiently higthat the network analyzer is unable to resolve the real component.

Directly related to the frequency range selection is the selection of the regression exponent parameter,p. A detailed modeof a capacitor is necessary in order to find the best regressiofit. Simulations were performed based on the model publisheby Yonekuraand Wakasugi [3].Using nominal values for resistive, inductive, and capacitive components of the model capacitor, a procedure was developed to simulatea wide range of network analyzer measurements.The model calculations were thecompared to the measurementsof a real capacitor.The measure

AVRAMOV-ZAMUROVIC ('I 01.: SENSITIVITY OF METHOD TO PREDICT CAPACITOR'S FREQUENCY CHARACTERISTIC

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parameters. (c) Dissipation factor results fo~ two different sets of regression parameters. (d) Power exponents for resistive and inductive regressions parameters

obtained by fittingeach driving-point impedance curve. (e) Power exponents for resistive and inductive regressions parameters obtained by fittingdissipation

factor curve and capacitance differences curve separately.

Q

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404 IEEE TRANSACfIONS ON INSTRUMENTATIONAND MEASUREMENT, VOL.49. NO.2. APRIL 2'

ment VS.simulation data show good agreement (see Fig. 3 forthe simulations and Fig. 4 for the measurements). Based on thisagreement, it was concluded that it is acceptable to use the givenmodel as a starting point in a search for an optimal set of regres-sion parameters.

The regression method used with a varied set of regression pa-rameters was applied to the simulated data to predict the capac-itor's frequency characteristic. Since the impedance ofthe modelcapacitor could be calculated at any frequency, the criterion foroptimal parameter selection was based on the best fit betweenthe predictions using the regression method and impedance cal-culations using the model.

A search program was developed to test a range of regres-sion parameter values using small incremental steps. Since thegoal of this method is to accurately predict a capacitor's dissipa-tion factor and capacitance change over a range of frequencies,these two properties were optimized independently and sets ofregression parameters for resistive and inductive componentswere found. One way to set a selection criterion is, for eachof the 4TP air capacitors evaluated in the study, to detenninea single exponent parameter for use in the inductance regres-sions and a single exponent parameter for use in the resistanceregressions. Another way to set a selection criterion is to searchfor the best fit for each driving-point impedance. These imped-ances are given in (1). In this way ten different values (five forthe resistive and five for the inductive components) for the fre-quency exponent parameter, p, were found for each 4TP capac-itor. Fig. 5(a)-(e) shows several plots comparing results fromtwo different sets of parameter values. Fig.5(a) compares totalimpedance results, showing better results when each drivingpoint impedance curve is addressed separately compared to thecumulative parameter for the dissipation and capacitance dif-ference result. Fig. 5(b) and (c) compare capacitance charac-teristics and dissipation factor results. These results are similarfor both selection procedures. Fig. 5(d) and (e) list power expo-nents for the resistive and inductive regressions obtained usingthe presented search procedure.

The search program was also tested using different networkanalyzer measurement frequency ranges. It was found thatthe values for the frequency exponent parameter p changesignificantly from range to range. This leads to the conclusionthat the CFCP method can not be implemented successfullywithout having a detailed model of a test capacitor. Futureresearch should address the results of simulations on the sensi-

tivity of the CFCP method due to variations in the regressionparameters.

IV. CONCLUSION

The results of the analysis of the CFCP method's sensitivityto exponent parameter variation have shown that this sensitivityis a significant contributor in the uncertainty analysis of NIST'scapacitance standard measurement system. A detailed descrip-tion and uncertainty analysis of this system are addressed in an-other paper.

ACKNOWLEDGMENT

The authors wish to thank K; Yokoi, Hewlett-Packard Japan,the original developer of the CFCP measurement system, for his

assistance during this project, and T. Aoki, who spent seve

rl

weeks at NIST providing engineering experience that saved eauthors many months in perfecting the measurement procedu e.

REFERENCES

[I] K. Suzuki, "New calibration method for four-terminal pair admitta?cestandards," IEEE Trans. Instrum. Meas., vol. 40, pp. 420--422, ~pr.1991.

[2] K. Suzuki, T. Aoki, and K. Yokoi, "New Calibration Method for Fo~r-Terminal Pair Capacitance Standards," Yokogawa-Hewlett-Pack~rd,Ltd., Apr. 1991.

[3] -, "Calibration method for four-terminal-pair capacitance standarps:Progression report," presented at the Proc. CPEM'98, Washington, QC,1998.

[4] T. Yonekura and T. Wakasugi, "Frequency characteristics of fqmrterminal pair air-dielectric capacitors," in Proc. NCSL Workshop dJldSymp., 1990, pp. 469-483.

[5] R. N. Jones, "Evaluation of three-terminal and four-terminal pair capac-itors at high frequencies,", NBS Tech. Note 1024, Sept. 1980.

[6] -, "A technique for extrapolating the I kc values of secondary ~a-pacitance standards to higher frequencies,", NBS Tech. Note 201, Nov.1962.

[7] R. D. Cutkosky, "Four-terminal-pair networks as precision admitta~ceand impedance standards," IEEE Trans. Commun. Electron., vol. 70, qp.19-22, 1964. I

[8] -, "Techniques for comparing four-terminal-pair admittance stan-dards," J. Res. Nat. Bur. Stand., vol. 74C, pp. 63-77, 1970.

[9] A. D. Koffman, S. Avramov-Zamurovic, N. M. Oldham, and B. C. W:

ll-

trip, "Capacitance and dissipation factor measurements from I kHz to 0MHz," presented at the Proc. 1998 NCSL Workshop and Symp., 199 .

Svetlana Avramov-Zamurovic received the B.S. and M.S. degrees in electri

~

al

engineering from the University of Novi Sad, Yugoslavia, in 1986 and 19 0,respectively, and the Ph.D. degree in electrical engineering from the Univers tyof Maryland, College Park, in 1994.

From 1990 to 1994, she was involved in developing a voltage ratio brid

~

e

for the NASA Zeno experiment. She was a Guest Researcher at the Natio alInstitute of Standards and Technology (NIST), Gaithersburg, MD, from 19 0to 1994. Currently, she is an Associate Professor at the United States Na alAcademy, Annapolis, MD. Her recent work as a Guest Researcher at NIST i -volves the development of impedance bridges and measuring techniques.

Andrew D. Koffman (S'86-M'90) received the B.S. degree from the Univqr-sity of Maryland, College Park, in 1988, and the M.S. degree from Vanderbl1tUniversity, Nashville, TN, in 1990, both in electrical engineering.

He joined the Electricity Division at the National Institute of Standards a

id

Technology (NIST), Gaithersburg, MD, in 1990. He has worked to develop a dapply model-based strategies for testing complex electronic systems and c -rently works in the area of developing ac impedance measurement systems a dimpedance calibration services.

Nile M. Oldham (M'73-SM'92-F'98) received the B.S. degree from VirginkaPolytechnic Institute and State University, Blacksburg, in 1966.

Since then, he has been a Physicist (and later an Electronics Engineer) in

~

. e

Electricity Division, National Institute of Standards and Technology (NIS ),Gaithersburg, MD. He is recently retired and working part-time. His recent wo kat NIST includes the development of electrical standards of voltage, curre t,phase angle, power and energy using digital synthesis techniques, and the i -vestigation of Internet-based metrology techniques.

Bryan Cristopher Waltrip (M'87) received the B.S. degree in electrical

~-

gineering and computer science from the University of Colorado, Boulder, dthe M.S. degree in electrical engineering from The Johns Hopkins Universi ,Baltimore, MD.

Since 1984, he has been an Electronics Engineer at the National InStitu

~

eof Standards and Technology (NIST) Electricity Division, Gaithersburg, M ,where he is currently working on precision waveform synthesis and measur-ment standards in the dc to 1 GHz frequency range.