516 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Towards Accurate Measurement of the FrequencyDependence of Capacitance and Resistance
Standards up to 10 MHzShakil A. Awan and Bryan P. Kibble
Abstract—Progress toward an understanding of the frequencydependence of capacitance and resistance standards at frequenciesup to 10 MHz is presented. A qualitative comparison is also madefor capacitance and dissipation factor measurements betweenthe National Physical Laboratory (NPL) high-frequency fourterminal-pair (4TP) bridge and a commercial impedance analyzerfor the first time. A set of novel high-frequency calculable coaxialresistance standards, of nominal 100 and 1 k values, havebeen developed and their calculated frequency dependence up to1 MHz is given.
Index Terms—Calculable resistance standards, coaxial bridges,high-frequency impedance metrology, impedance analyzers, LCRmeters, quadrifilar resistors, quantum Hall effect.
I. INTRODUCTION
SEVERAL groups world-wide are currently engaged in ac-curate characterization of the frequency dependence of ca-
pacitance and resistance standards in order, ultimately, to pro-vide the LCR and impedance analyzer (IA) instrument manufac-turers with calibrations at frequencies other than 1 kHz [1]–[4].Most of the groups are investigating the Suzuki technique [1],which contrasts with the coaxial bridge approach we are cur-rently developing [4]. The two techniques differ in their im-plementation of the four terminal-pair (4TP) definition for animpedance standard [5], and interlaboratory comparisons be-tween the two techniques are planned. In this paper, we present acomparison between measurements from the National PhysicalLaboratory (NPL) high-frequency (HF) 4TP bridge and from acommercial 4TP IA. The comparison is only qualitative at thisstage because the instrument could not be operated under speci-fied conditions. This is due to the necessity of finite cable lengthsfor connections to the 4TP standards.
II. HF CAPACITANCE MEASUREMENTS
The frequency dependence of a fused-silica 100 pF capaci-tance standard measured on the NPL 1 MHz 4TP bridge againsta reference air-dielectric 1 nF standard (the frequency depen-dence of the reference standard was determined using a reso-nance technique detailed in [6]) is shown in Fig. 1(a). The solid
Manuscript received July 2, 2004; revised November 4, 2004. This workwas supported by the Electrical Program of the National Measurement SystemPolicy Unit of the Department of Trade and Industry, UK.
The authors are with the Division of Enabling Metrology, NationalPhysical Laboratory, Teddington, Middlesex TW11 0LW, U.K. (e-mail:[email protected]).
Digital Object Identifier 10.1109/TIM.2005.843582
line is a quadratic least-squares fit to the measured capacitancedata, and the dashed line is a linear fit to the measured dissi-pation factor results. The former shows a quadratic frequencydependence for 500 kHz (due to residual inductance), andthe latter shows a linear frequency dependence over the entire1 kHz–1 MHz range, as expected. The corresponding frequencydependence of capacitance and dissipation factor results for thesame 100 pF fused-silica standard using a commercial 4TP IAwith short 13 cm cables is shown in Fig. 1(b). For 500 kHz,the capacitance and dissipation factor results fluctuate betweentwo values, whereas above this frequency, the qualitative agree-ment with data in Fig. 1(a) is reasonably good. The basic accu-racy claimed by the manufacturer for the IA is 0.18%, whereasthe agreement between the two sets of measurements is betterthan 0.05%. A second set of measurements with twice the cablelengths is also shown in Fig. 1(b). The calculated correction forthe additional 13 cm cable lengths up to 1 MHz is approximately20 F/F, whereas the data in Fig. 1(b) shows a difference at 1MHz of 300 F/F. This difference increases as , so that at 10MHz, an approximate 2.8% error can be expected. This does notmean that the instrument is outside its basic accuracy, but ratherthe user must be aware of the limitations of the instrument athigh frequencies when finite length cables are used to connectthe instrument to a device under test.
It is also important to verify the scaling properties of both theNPL 1 MHz bridge and the IA instrument. Fig. 2 shows the re-sults from both bridge systems when measuring a 1 nF NPO-di-electric capacitance standard (in the case of NPL bridge mea-surements, the reference was an air-dielectric 100 pF standardwhose frequency dependence was also determined using the res-onance technique [6]). Although there are many interesting fea-tures to be observed in Fig. 2, the key result is that again (incomparison with Fig. 1 for 100 pF measurements) there is areasonably good qualitative agreement between the two bridgesystems to within 500 F/F over the entire frequency range. Thejumps in the IA measurements (which are not observed in resis-tance measurements as discussed in Section IV) may be due toits internal range switching as the frequency is swept higher. Inessence, the measurements of the 100 pF and 1 nF capacitancestandards verify the scaling properties of two completely dif-ferent bridge systems.
The measurements on the NPL 1 MHz bridge, shown inFigs. 1 and 2, were carried out using an improved bridgesystem compared to the original design [4]. Several modi-fications have been implemented; a new HF 10 1 voltageratio device has been built and calibrated using the permuting
AWAN AND KIBBLE: TOWARDS ACCURATE MEASUREMENT OF THE FREQUENCY DEPENDENCE 517
Fig. 1. Comparison of capacitance and dissipation factor measurements of a 100 pF standard between (a) the NPL 1 MHz bridge and (b) an IA instrument (1 V,� = 3 s time constant). All measurements have been normalized to 1 kHz values.
capacitors technique [7], an improved isolation transformerhas also been developed with significantly reduced capacitancebetween the primary and secondary windings at 1 kHz (about60 aF) and a more accurate realization of the defining condi-tions at the high potential ports of the main standards has beenachieved using new HF injection/detection transformers andbinary inductive voltage dividers for current source balances.
III. NEW 10 MHz 4TP CAPACITANCE BRIDGE
In addition to the improvements made to the 1 MHz bridge,a new 10 MHz 4TP 10 1 coaxial bridge system for accuratecharacterization of capacitance standards has also been devel-oped recently. A photograph of the bridge is shown in Fig. 3.The overall physical dimensions of the 10 MHz bridge systemare approximately a factor of five smaller than the 1 MHz bridge(which in turn is a factor of ten smaller than a typical 1 kHzcoaxial bridge system). The dimensional scaling of the high-fre-quency bridges is to be expected in order to keep the lengthof the bridge networks much smaller than the wavelength
of the bridge excitation signal (i.e., 4), making pos-sible a lumped (rather than distributed) parameter modeling ofthe bridge response at high frequencies. Table I shows the pre-liminary results measured for a commercial 1 pF capacitancestandard and the 10 1 voltage ratio device (VRD) from 0.5to 10 MHz. The table also shows the corresponding calibrationvalues of the 10 1 VRD (which was calibrated using a newdesign, near frequency independent, 0.1 and 1 pF air-dielec-tric capacitance standards with resonance frequencies estimatedto be approximately 700 MHz). The measured frequency de-pendence or change in capacitance between 1 kHz–10 MHz of
9.78 mF/F (the overall measurement uncertainty will eventu-ally be evaluated and further results published elsewhere) for theHP 1 pF standard compares reasonably well with 10 12 mF/Fmeasurement reported for a similar standard in [8].
IV. NOVEL HF CALCULABLE RESISTANCE STANDARDS
For accurate resistance measurements over a wide frequencyrange, a set of novel HF calculable coaxial resistance standards
518 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Fig. 2. Comparison of capacitance and dissipation factor measurements of a 1 nF NPO-dielectric standard between the NPL 1 MHz bridge and an IA instrument(1 V, � = 5 s, and 13 cm cables).
Fig. 3. Measurement of the commercial 1 pF capacitance standard (right)against an NPL 0.1 pF reference standard (center top) in the new 4TP 10MHz coaxial bridge. The center of the photograph shows a two-stage HF10 :�1 voltage ratio device and two injection/detection transformers. Theisolation transformer is shown on the left and a current source is shown in thebackground.
TABLE IFREQUENCY DEPENDENCE OF COMMERCIAL 1 pF STANDARD
AND VRD TO 10 MHz
(HF-CRS) have been developed in collaboration with NL Engi-neering. The schematic layout of the 4TP coaxial standards isshown in Fig. 4. Two 1 k and one 100 resistance standardshave been developed which will enable 10:1 and 1:1 ratio com-parisons. There are several differences between the HF-CRS
and the Gibbings calculable quadrifilars [9] and Haddad[10] resistance standards already in use in several nationalmetrology institutes. First, the “measurement plane” and the“calculation plane” are aligned precisely at the 4TP connectorsin the HF-CRS. This should permit accurate comparisons of thecalculated and measured frequency dependence of the HF-CRSstandards, eventually, up to about 100 MHz. Second, the 4TPconnectors of the HF-CRS are orthogonal to minimize mutualinductance between the current and potential paths—thus re-ducing the frequency dependence of the standards. Third, thereis no dielectric support of the inner resistance wire which alsominimizes the frequency dependence of the standards, and sim-plifies calculations. Finally, the end-effects are also minimizedin this design by displacing the potential connections awayfrom the current connections while maintaining the orthog-nality of the 4TP connectors (as illustrated by the side elevationdiagram in Fig. 4). These modifications to the Gibbings andHaddad designs are thought to result in a set of novel resis-tance standards which are near ideal (in respect of circular orcoaxial geometry, low frequency dependence, alignment of themeasurement and calculation planes, relative simplicity in theiranalytical resistance calculations, and high-frequency opera-tion). The near ideal design of the HF-CRS can, consequently,be represented by a relatively simple model, which contains themain sources of frequency dependence. The impedanceof the HF-CRS can be described in terms of a resistancewith a series inductance and an overall shunt capacitance
. The suitability of this model, to the exact geometricaldesign of the HF-CRS shown in Fig. 4, can be understood froma closer examination of the cross sectional layout (also shownin Fig. 4). Since the low-current and low-potential connectionsare made to the inner copper cylinder (or return conductor),it is part of the defined four terminal-pair resistance, whereasthe high-current and high-potential connections are made tothe Evanohm resistance wire. Therefore, the model described
AWAN AND KIBBLE: TOWARDS ACCURATE MEASUREMENT OF THE FREQUENCY DEPENDENCE 519
Fig. 5. Calculated resistance change from nominal values (frequencydependence) of a 100 and a 1 k HF-CRS based on (1).
above is the most appropriate electrical model to represent thegeometrical design of the HF-CRS. The outer copper cylinderforms a zero-impedance junction, and since the voltage acrossthe inner and outer copper cylinders is zero, then the capacitivecurrent through the supporting polytetrafluoroethylene (PTFE)dielectric is also zero. The impedance of the HF-CRS isthen given by
(1)
where 2 and the calculated values for the lumped param-eters are 0.22 H and 1.78 pF for the 100 resistance standard.The resistive part of is plotted in Fig. 5 and normalizedto the nominal value. Similarly, the lumped parameters for the1 k resistance standard, also plotted in Fig. 5, are 0.28 H and1.26 pF. Other contributions such as skin effect in the resistancewire, shield conductor, and the end plate are found to be neg-ligible (a more detailed treatment of all parameters giving riseto frequency dependence in the HF-CRS will be the subject ofanother article). The results in Fig. 5 show that the frequency de-pendence of the 100 standard varies as and is dominated
by the inductive terms in (1). Similarly, the frequency depen-dence of the 1 k standard also shows behavior, but the signof the variation is reversed indicating that the capacitive termsin (1) are now dominant. The time constant of the standards canalso be calculated from (1) and is given by .For the 100 and 1 k resistance standards it is found to be
2 ns and 1.03 ns, respectively.The frequency dependence measurements of the capacitance
standards using the NPL 1 and 10 MHz bridge systems are cur-rently being extended to enable accurate characterization of the100 and a 1 k HF-CRS. This will permit direct comparisonof the measured and calculated frequency dependence (and timeconstant) values to higher frequencies. It will also permit a sim-ilar comparison (to the qualitative capacitance comparison dis-cussed above) between the NPL HF bridge systems and the IAmeasurements for resistance standards. In addition, the accuratemeasurements and calculations of the HF-CRS to higher fre-quencies may lead to an improved understanding of the natureof the small but finite frequency dependence of the quantum Hallresistance (QHR) [11]–[13]. This would ultimately impact thepossibility of establishing a universal (both dc and ac) quantumstandard of impedance (QSI) in the near future which may alsorequire accurate closure of the metrological impedance triangle[6].
V. CONCLUSION
Progress toward accurate measurement of the frequency de-pendence of capacitance and resistance standards up to 10 MHzhas been presented. Comparison of capacitance and dissipationfactor measurements between NPL 1 MHz bridge and a com-mercial IA (when operated at or near specified conditions withregard to cable connections) showed agreement (to well withinthe stated uncertainty of the IA) over the entire 1 kHz–1 MHzfrequency range. A new 10 MHz 4TP coaxial bridge has alsobeen developed recently and preliminary measurements on aHP 1 pF capacitance standard have been discussed. A set ofnovel HF-CRS have been developed and calculation of their fre-quency dependence up to 1 MHz given. Possible application ofthe HF-CRS to accurate quantum Hall resistance measurementsat higher frequencies has also been discussed.
ACKNOWLEDGMENT
The authors would like to thank M. Lloyd and C. Lloyd,NL Engineering, Cambridgeshire, U.K. for the construction ofHF-CRS.
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520 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
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