Residual impedance correction in impedance analyzers when using an asymmetrical test fixture needs three reference measurements, usually open circuit, short circuit, and load (meaning an impedance close to the impedance under test). This paper provides an uncertainty estimate for impedance measurements that apply a simple open/short correction in spite of using an asymmetrical test fixture. Experimental results show that the minimal uncertainty is obtained for impedance values close to the geometric mean of the short-circuit and open-circuit impedances, and that the theoretical prediction is indeed an upper limit for the actual uncertaintyPeer Reviewe

Pallàs Areny, Ramon

Torrents Dolz, Josep M.

English

Pallàs-Areny, Ramon

UPCommons. Portal del coneixement obert de la UPC

IEEE Instrumentation and MeasurementTechnology ConferenceBudapest, Hungary, May 21-23, 2001.Uncertainty Analysis in Two-terminal Impedance Measurements with ResidualCorrectionJ. M. Torrents and R. Pallàs-ArenyDept. Eng. Electrònica, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034 Barcelona, SPAINPhone +34-93-401-6758 Fax +34-3-401-6756 e-mail: elerpa@eel.upc.esAbstract – Residual impedance correction in impedance analyzerswhen using an asymmetrical test fixture needs three referencemeasurements, usually open circuit, short circuit, and load(meaning an impedance close to the impedance under test). Thispaper provides an uncertainty estimate for impedance measure-ments that apply a simple open/short correction in spite of usingan asymmetrical test fixture. Experimental results show that theminimal uncertainty is obtained for impedance values close to thegeometric mean of the short-circuit and open-circuit impedances,and that the theoretical prediction is indeed an upper limit for theactual uncertainty.Keywords – Impedance measurements, Residual correction, Un-certainty analysis.I. INTRODUCTIONAccurate impedance measurements require us to consider theeffects of residual impedance in test fixtures. Test fixtures canbe modeled as two-port networks described by their transmis-sion parameters (A B C D) [1]. Symmetrical (balanced) testfixtures they can be described by two ratios between trans-mission parameters because A = B. These ratios can be deter-mined from two reference measurements, for example open-circuit and short-circuit conditions (open/short correction),and the results allow us to correct for the unknown residualimpedance in electric contacts and cables connecting theimpedance under test to the impedance analyzer. When thetest fixture is not symmetrical, residual correction needs threereference measurements because we have to determine threeratios between transmission parameters [2]. Usually, a refer-ence impedance close to the impedance under test is meas-ured, hence the name open/short/load correction.However, it is sometimes difficult to obtain reference imped-ances close to the impedance under test, for example whenmeasuring electrolytes. Therefore, the question arises aboutthe uncertainty associated to impedance measurements thatuse an asymmetrical test fixture but implement a simpleopen/short correction procedure. This paper provides an esti-mate for the uncertainty in such impedance measurementsand substantiates the predictions by experimental resultsobtained in two-wire impedance measurements.II. UNCERTAINTY MODEL IN TWO-TERMINALIMPEDANCE MEASUREMENTSFig. 1 shows the impedance under test connected to an im-pedance analyzer through a test fixture modeled by its trans-fer parameters A, B, C, and D. The voltage and current at theinput of the impedance meter can be obtained from=2211IVDCBAIV (1)ImpedanceanalyzerA   BC   D Z xV1 V 2I1 I2Fig. 1. The test fixture connecting the impedance under test to the impedanceanalyzer is modeled by a two-port network described by its four transferparameters A, B, C, D.The impedance measured, disregarding the uncertainty of theimpedance meter, will be222211DICVBIAVIVZ xm++== (2)whereas the actual impedance under test is22IVZ x = (3)In order to estimate Zx from Zxm, we first rewrite (2) by re-placing V2/I2 = Zx to obtainDCZBAZZxxxm++= (4)Solving for Zx yields0-7803-6646-8/01/$10.00 ©2001 IEEE 1450xmxmx CZABDZZ−−= (5)Measuring at open-circuit condition (I2 = 0), (2) yields Zom =A/C. Measuring at short-circuit condition (V2 = 0), (2) yieldsZsm = B/D. Therefore, (5) can be rewritten asomsm1ZZZZADZxmxmx−−= (6)If we measure a known impedance Zl and obtain a result Zlm,we will haveomlmsmlml1ZZZZADZ−−= (7)Solving for D/A and replacing it in (6) yieldsommsmmsmlmomlmlu11ZZZZZZZZZZxxx−−−−= (8)where the subscript “u” has been added to indicate that Zx hasbeen measured with an unbalanced (asymmetrical) fixture andcorrected by three reference readings: Zom, Zsm, and Zlm. If thenetwork connecting Zx to the impedance analyzer were bal-anced (symmetrical), A = D and (6) would reduce toomsmb1ZZZZZxmxmx−−= (9)where the “b” in the subscript stands for balanced. Equation(9) differs from (8) by a gain factor. Therefore, using onlytwo reference measurements when the network connectingthe impedance under test to the impedance meter is asymmet-rical implies to consider a unity gain factor in (8), which isequivalent to use (9) to calculate the corrected impedancevalue from the reading Zxm and the two reference measure-ments Zom and Zsm. Hence, the relative uncertainty is( ) 11smlmomlmluub−−−=−=ZZZZZZZZZZuxxx (10)Because it is recommended to select Zl close to Zx, and wecan assume Zxm to be close to Zx, we can approximate Zl ≈ Zx≈ Zxm and Zlm ≈ Zxm. Under these assumptions, (10) leads to( )( )smom2smomZZZZZZZZuxmxm−−≅ (11)In summary, disregarding measurement uncertainties in theimpedance analyzer, if we use open/short impedance meas-urements to correct residual impedances for an unbalance testfixture, we can estimate the true impedance value corre-sponding to an instrument reading Zxm by applying (9) but thecalculated result has a maximal relative uncertainty given by(11). Furthermore, because impedance values are complexnumbers, the uncertainty because of an asymmetrical testfixture will affect both the amplitude and phase of the result.From (11), when the result is close to the geometrical mean ofthe open circuit and short circuit measurements,smomoptm ZZZ x = (12)the uncertainty because of the unbalanced connecting networkis minimal. Nevertheless, this calculated impedance value isnot necessarily close to any impedance of interest.III. EXPERIMENTAL RESULTS AND DISCUSSIONWe have applied the analysis above to predict the contribu-tion from incompletely corrected residual impedances to theuncertainty of impedance measurements performed with theHP4294A impedance analyzer. We assessed the influence ofan asymmetrical test fixture on the result by measuring sev-eral reference impedances, first with a symmetrical test fix-ture provided by the manufacturer (HP16047A) and then byconnecting each impedance under test to the instrument withRG214 cables 1 m long. Both measurement sets used thecorresponding open/short correction. The impedance valuescalculated from (9) when using a symmetrical test fixture andits open/short correction were considered the “true” value.The impedance analyzer averaged eight readings for eachimpedance value before transferring the results to a computerfor calculation and display. The impedance under test wasinside a grounded Faraday cage measuring 10 cm × 10 cm ×12.5 cm. Each cable shield was grounded too. Because theimpedance analyzer is a self-balanced bridge, the capacitancecontributed by the grounded shield did not affect the meas-urement result [3].Test impedance values were chosen according to the open andshort-circuit measurements when using the asymmetrical testfixture. We obtained Zom = 1/(jω × 2 pF) and Zsm = 60 mΩ +jω × 100 nH. From these values, (12) yields an “optimal”impedance that is difficult to implement. We have selectedtwo impedances close to that optimum: a 220 Ω resistor (Z1)and the series combination of that resistor and a 15 nF ca-pacitor (Z2). The remaining test impedances were off-the-1451shelf resistors: Z3 = 1 kΩ, Z4 = 51 Ω, Z5 = 5.1 kΩ, and Z6 =10 Ω.Fig. 2 shows the maximal absolute value of the relative un-certainty predicted by (11) for Z1 to Z6 in the range from100 Hz to 1 MHz. The theoretical uncertainty is obviouslyminimal for impedance values Z1 and Z2, which are close tothe optimum (12), and increases with frequency. The relativeuncertainty is similar for Z3 and Z4, whose values are (geo-metrically) symmetrical about that given by (12). The relativeuncertainty increases for impedance values that are very dif-ferent from (12): it is larger for Z5 and Z6 than for Z3 and Z4.Furthermore, at low frequencies, the relative uncertainty isproportionally larger for Z6 (small impedance) than for Z5(large impedance).Frequency (Hz)102 103 104 105 106Uncertainty02e-34e-36e-38e-3Z4Z1Z3Z5Z6Z2Fig. 2 Maximal absolute value of the predicted relative uncertainty for the sixtest impedance values according to (11).Figs. 3 and 4 respectively show the real and imaginary com-ponents of the predicted uncertainty calculated from (11). Thereal component slightly depends on frequency for Z5 and Z6(the largest and smallest impedance values). The imaginarycomponent also increases with frequency for Z3, Z4, Z5, andZ6, but starts to increase earlier than the real component. Also,the farther the impedance value is from the optimum given by(12), the sooner the frequency dependence starts. By com-paring the relative uncertainties in Figs. 3 and 4, we infer thatthe imaginary component is responsible for most of the de-pendence of the uncertainty with the frequency.Frequency (Hz)102 103 104 105 106Uncertainty-2e-302e-34e-36e-38e-3Z4Z1Z3Z5Z6Z2Fig. 3 Real component of the theoretical relative uncertainty predicted from(11) for six test impedance values.Frequency (Hz)102 103 104 105 106Uncertainty-4e-3-2e-302e-34e-3Z4Z1Z3Z5Z6Z2Fig. 4 Imaginary component of the theoretical relative uncertainty predictedfrom (11) for six test impedance values.Fig. 5 shows the absolute value of the relative uncertainty ofthe impedance for these six test impedance values measuredin the range from 100 Hz to 1 MHz. The relative uncertaintyis minimal for Z1 and Z2, as predicted, and is smaller than thepredicted uncertainty in Fig. 2. The relative uncertainty in-creases from 100 kHz up, particularly for the largest andsmallest impedance values (Z5 and Z6), whereas the predictedrelative uncertainty increases above 10 kHz. The uncertaintyis similar for Z3 and Z4, the same as in Fig. 2. Also, at lowfrequencies the relative uncertainty is smaller for Z5 than forZ6, which is consistent with the theoretical prediction.1452Frequency (Hz)102 103 104 105 106Uncertainty02e-34e-3Z4Z1Z3Z5Z6Z2Fig. 5 Absolute value of the relative uncertainty for the six test impedancevalues measured with the HP4294A.Figs. 6 and 7 respectively show the real and imaginary com-ponents of the actual relative uncertainty. They reveal thatboth the real and the imaginary components of the actualrelative uncertainty are smaller than the (maximal) predicteduncertainty (Fig. 3 and Fig. 4) and that their frequency de-pendence is also smaller than predicted. Furthermore, theincrease in relative uncertainty at high frequencies in Fig. 5 isattributable to the frequency dependence of the imaginarycomponent of the relative uncertainty, the same as for thetheoretical prediction.Frequency (Hz)102 103 104 105 106Uncertainty-2e-302e-3Z4Z1Z3Z5Z6Z2Fig. 6 Real component of the relative uncertainty for the six test impedancevalues in Fig 5.Frequency (Hz)102 103 104 105 106Uncertainty-4e-3-2e-30Z4Z1Z3Z5Z6Z2Fig. 7 Imaginary component of the relative uncertainty for the six test imped-ance values in Fig. 5.IV. CONCLUSIONAccurate measurements using impedance analyzers require usto compensate for impedance residuals in the test fixture.When the test fixture is symmetrical, this correction is usuallyperformed by measuring short-circuit and open-circuit im-pedance, and using the results to compute the actual imped-ance according to (9). When the text fixture is asymmetrical,we need a third reference measurement in order to compen-sate for impedance residuals according to (8). Using simpleopen/short correction instead adds gain uncertainty to that ofthe impedance analyzer. Equation (11) estimates the maximalrelative uncertainty added and (12) determines the impedancevalue having the minimal relative uncertainty. Experimentalresults confirm that: (a) actual relative uncertainties whenperforming simple open/short correction are smaller than thetheoretical limit calculated from (11); (b) relative uncertain-ties for impedance values close to the optimum calculatedfrom (12) are very small; and (c) at low frequencies, relativeuncertainties are larger for small impedance values than forhigh impedance values.ACKNOWLEDGMENTThis work has been funded by the Spanish CICYT, ProjectTAP99-0742.REFERENCES[1] M. Honda. The Impedance Measurement Handbook. A Guide to Meas-urement Technology and Techniques. Tokyo: Yokogawa-Hewlett-Packard LTD, 1989.[2] Anonymous. Effective impedance measurement using open/short/loadcorrection. Application Note 346–3. Hewlett Packard, 1998.[3] Y. Narimatsu. Impedance measuring instruments. Chapter 7 in: C. F.Coombs (ed.): Electronic Instrument Handbook, 2nd Ed. New York:McGraw-Hill, 1995.1453

Uncertainty analysis in two-terminal impedance measurements with residual correction

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Uncertainty analysis in two-terminal impedance measurements with residual correction

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