IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 2, APRIL 2007 453

Resistors With Calculable FrequencyDependencies Up to 1 MHz

Han Jun Kim, Rae Duk Lee, and Yu. P. Semenov

Abstract—Resistance standards from 1 Ω to 100 kΩ, withcalculable frequency dependencies of up to 1 MHz, havebeen developed for the calibration of commercial inductance–capacitance–resistance (LCR) meters and impedance analyzers.The resistors are designed on the basis of single bifilar loops.The typical resistance change from dc to 1 MHz is from 200 to800 µΩ/Ω. According to the measurement results, the frequencydependencies of the resistors are several times lower than themeasurement uncertainty of commercial LCR meters.

Index Terms—Calculable standards, calibration, capacitors,frequency, resistors.

I. INTRODUCTION

DURING the last few years, many metrological instituteshave been involved in works on the accurate determina-

tion of the values of resistance and capacitance standards, in thefrequency range of up to 10 MHz, with the purpose of provid-ing proper calibration of commercial inductance–capacitance–resistance (LCR) meters and impedance analyzers (IAs). Thiscalibration requires resistance standards from 1 Ω to 100 MΩwith a relative uncertainty of (100−500) × 10−6 depending onthe frequency and nominal values. Presently, LCR meters arecalibrated at 1 kHz over the whole resistance and capacitanceranges. At higher frequencies, the calibration is only performedfor some capacitance values in the range from 1 pF to 1 nF,for example, by the Suzuki or Callegaro methods [1], [2].Coaxial resistors of 100 Ω and 1 kΩ, whose estimated fre-quency responses do not exceed 0.01% at 1 MHz, are thereference standards for accurate resistance measurements overa wide frequency range [3], [4]. These resistors are high-frequency versions of low-frequency resistors with calculablefrequency dependencies (RCFDs) that have been studied forthe absolute reproduction of the ohm on the basis of theThompson–Lampard cross capacitor [5]–[7]. On the basis ofa bifilar model, we have developed RCFDs intended for thecalibration of commercial LCR meters and IAs at frequenciesup to 1 MHz.

II. DESIGN MODEL

The possibility of a theoretical analysis of the resistancebehavior in a wide frequency range is the main requirement

Manuscript received July 11, 2006; revised November 2, 2006.H. J. Kim and R. D. Lee are with the Korea Research Institute of Standards

and Science, Daejeon 305-600, Korea (e-mail: hanjun@kriss.re.kr).Y. P. Semenov is with the D. I. Medeleyev Institute for Metrology,

St. Petersburg 198005, Russia.Digital Object Identifier 10.1109/TIM.2007.891121

Fig. 1. Model of a single bifilar resistor with shield. (a) Element of length dxof a single three-wire line at distance x from its beginning. (b) Equivalent picircuit that is represented by a symmetrical quadrupole.

for the RCFD model. Several models meet this requirement;however, only two of them are applied in practice. One is ashort-circuited coaxial line, and the other is a uniform bifilartransmission line.

The basic model used in our investigations is a shieldedsingle bifilar (single loop), i.e., a two-wire transmission linewith a dielectric between the conductors and a shield aroundboth of them [7]. We use the model of a single loop, whose ratioof length to distance between the axes of conductors exceeds20–25. In this case, the error of applying the equations ofan infinitely long transmission line is only several percents,and the influence of the longitudinal edge effect is practicallynegligible. The model of a single bifilar resistor with shield isshown in Fig. 1.

The shield is part of a uniform line, so this model shouldbe considered as a uniform three-wire transmission line [8].In contrast to the well-known solutions [6], [9], this modelprovides the opportunity to change the shield potential. Theelement of length “dx” of a single three-wire line at distance xfrom its beginning is presented in Fig. 1(a). Currents i1 and i2pass through the bifilar conductors, while the lower conductorin Fig. 1(a) is the shield. The solution of the partial derivativedifferential equation system for this element is obtained using ashort circuit at the bifilar end. After some simple however cum-bersome transformations, it is possible to present the solutionin the form corresponding to a symmetrical quadrupole, as inFig. 1(b). The parameters ZΠ and YV of the pi equivalent circuit

0018-9456/$25.00 © 2007 IEEE

454 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 2, APRIL 2007

TABLE ISTRUCTURAL FEATURES OF THE RESISTORS

in Fig. 1(b) are described by the following equations, where wehave omitted the terms containing the conductance between thebifilar conductors g and between each conductor and the shieldgS due to their negligible value in the design of the consideredresistors

ZΠ∼=R

[1+

23ω2

(τLτC− 1

5τ2C

)− 1

3ω2τS

(τL− 3

10τC +

140

τS

)]

+ jωR

τL − 1

3τC +

13

(ω

ω0

)2 (τL − 6

5τC

)

+16τS

[1 − ω2τL

(τL − 9

5τC +

320

τS

)]

YV∼= 1

24ω3τSCS

(1 +

15ω2τLτS

)

+12jωCS

[1 +

112

ω2τS

(τL − 1

10τS

)]. (1)

Referring to Fig. 1(a), τL = L/C, τC = RC, and τS = RCS

are the time constants; ω0 is (LC)−1/2; and L, R, C, and CS

are the inductance, resistance, capacitance between conductors,and capacitance between conductors and shield, respectively,i.e., the values of the distributed parameters of the bifilarresistor.

The complex impedance of the bifilar resistor with an arbi-trary value of the shield potential VS is determined by

ZAB =ZΠ

1 + KYV ZΠ(2)

where K = VS/(V1 − V2), 0 ≤ K ≤ 1, and (V1 − V2) is thepotential difference at the terminals of the bifilar resistor.

The frequencies, at which the RCFD is used, are much lowerthan the resonance frequency ω0/2π, so all the equation termscontaining (ω/ω0)2 can be neglected.

Due to current density variation in the conductor cross sec-tion caused by different eddy current effects, the values of L, R,C, and CS do not remain constant when the frequency changes.To estimate the variation of the active resistance of a bifilarresistor with circular cross-sectional conductors as a functionof frequency, the following equation is used [10]:

Rac =R0

1+

(ωσµ)2

192

[1 + 3 ·

(d

a

)2

+18·(d

a

)4

+ · · ·](3)

where d is the conductor diameter, a is the distance betweenthe conductor axes, σ is the electric conductivity of the resistivewire, µ is the magnetic permeability of the resistive wire, R0

is the dc resistance, and ω = 2πf . At a → ∞, this equationdescribes the ac resistance of a single isolated conductor. Thecalculation according to (3), carried out with the data fromTable I, has demonstrated that both eddy current effects—skineffect and proximity effect—cause negligible resistance vari-ations for the 100-Ω, 1-kΩ, and 10-kΩ resistors when thefrequency changes from 0 to 1 MHz.

No limits have been introduced in the calculations for thecross-sectional shape of the conductors, so (1) and (2) are alsoapplicable to ribbon-type bifilars, which are used for buildinglow-resistance 1- and 10-Ω RCFDs. The distance t betweenthe conductors in a ribbon-type bifilar resistor is chosen sosmall that one can assume that the magnetic field is locatedbetween the bifilar surfaces facing each other. The magneticfield is practically absent outside the resistor except in a nar-row zone at the edges. At a low frequency, the current isdistributed uniformly over the whole conductor cross section,and the magnetic field strength is the same at any point betweenthe conductors. With the frequency increase, the uniformity ofthe current distribution over the conductor cross section is bro-ken due to skin effect and proximity effect. The current densityδ is increased therewith in two directions, namely 1) to the fac-ing surfaces of the conductors caused by proximity effect and2) to the conductor edges caused by transverse edge effect. The

KIM et al.: RESISTORS WITH CALCULABLE FREQUENCY DEPENDENCIES UP TO 1 MHz 455

Fig. 2. Distribution of current density δ and magnetic field intensity H in aribbon-type bifilar resistor.

current density δ(y) and δ(x) distribution diagrams are shownin Fig. 2.

Due to edge effect, the current density in the middle part ofthe conductor cross section is decreased, which leads to thedifferent dc and ac resistances as well as to the phase shiftincrease with the frequency [11]. Unfortunately, we do notknow the body of mathematics that could be used to estimatethe contribution of the transverse edge effect to the frequencycharacteristic of the bifilar resistor. Obviously, the edge effectdecreases with the decrease of the thickness of the conductorsh and the distance t between them. In case of edge effect,the impedance per unit length for a ribbon-type bifilar resistor,when the conductor width b is much larger than the conductorthickness h, is determined by [11]

ZRb =2p(1 + j)

bσcth [(1 + j)ph]

p =√

(ωσµ)/2 (4)

where p is the damping coefficient of the flat electromagneticwave in the conductor.

For the particular case of small-thickness conductors (ph 1), this equation can be simplified by expanding the hyperbolicterms and neglecting the terms of a higher order, i.e.,

ZRb ≈ 2pbσ

(sh2ph + sin 2phch2ph− cos 2ph

+ jsh2ph− sin 2phch2ph− cos 2ph

). (5)

The inductance per unit length of the bifilar resistor can befound by

LRb ≈ (µ/b) · (h/3) + t (6)

where the first term comes from the internal inductance ofthe conductor, and the second one is from the inductance ofthe single loop.

One of the characteristics of the developed resistors is thesmall value of the ratio of the distance between conductor axesto the shield diameter. Keeping the minimum value of this ratiomakes it possible to decrease the edge effects and to excludethe eddy current losses in the shield from consideration. Withthe fixed value of the shield diameter D, this condition is metby decreasing the distance between the conductor axes a. Theresistance due to eddy currents in the shield is proportionalto (2a/D)n, where n = 2, 4, and 8 for bifilar, quadrifilar,

Fig. 3. Multisection resistive elements of two 1-kΩ RCFD differing by thecurrent direction in four series-connected wire bifilars.

and octofilar, respectively. That is why in “narrow” bifilars,values of 0.001–0.06 for (2a/D) provide for a negligibly smallresistance component due to eddy currents in comparison toconventional RCFDs, where this ratio is from 0.09 to 0.4 [6],[12], [13]. The decrease of the distance between the conduc-tors also reduces the end effect and the influence of lumpedcapacitance load in the end plane of the short-circuited trans-mission line.

The design of any resistor on the basis of a folded wire hasone or several joints where the wire changes its direction. Thecapacitance between each of those wire parts and the shieldcontributes to the frequency dependence of the resistor. Incase of a bifilar, this contribution can be estimated using thesimplified equation [6]

R4TP ≈ R[1 + k2(1 − k)2R2ω2C2

1

](7)

where R is the wire resistance, C1 is the lumped capacitance,and k is the fraction of the wire length (not the transmissionline!) from the beginning to the point of the wire’s fold.

This equation can be transformed for the case of a trans-mission line containing n wires connected in series, such as aquadrifilar (n = 4) or an octofilar (n = 8). One can transform(7) by using the superposition of all n− 1 load effects andtaking into account that ki = i/n, where i = 1, 2, . . . , (n− 1)

R4TP = R ·[1 +

R2ω2C2l

n2

i=n−1∑i=1

i2(

1 − i

n

)2]. (8)

In this equation, the value of the coefficient is equal to1/16, 17/128, and 337/1024 for bifilar, quadrifilar, and octofilar,respectively. So the lumped capacitance effect increases withthe number of wire in the transmission line.

Each developed resistor, except 1 kΩ, presents a single bifilarfixed to the dielectric support that separates its conductors. Theresistive element of the developed 1-kΩ RCFD is built as amultisection unit in the form of four similar bifilars located ona single tubular support and connected in series. The layout isshown in Fig. 3. This model does not present an octofilar inthe sense that the distance between the conductors of one pairt = (D0 −Din)/2 is dozens of times smaller than the distancebetween the axes of different pairs. As a result, the mutualinfluence of electric and magnetic fields of different pairs ofconductors can be neglected, and each pair of conductors canbe calculated as a single bifilar. The error caused by this

456 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 2, APRIL 2007

TABLE IIFREQUENCY DEPENDENCE AT 1 MHz AND TIME CONSTANT, WITH CORRESPONDING k = 2 UNCERTAINTIES, AND THEORETICAL

ESTIMATION OF COMPONENTS OF THE RELATIVE FREQUENCY DEPENDENCE AT 1 MHz

simplification of the model was estimated experimentally bymeans of a comparison of the frequency characteristics oftwo similar resistors differing from each other only by thecurrent direction in bifilars, in accordance with Fig. 3. Themeasurements taken at frequencies of 1 kHz and 1 MHz do notdisplay any differences within 0.005%, which is the accuracylimit of the experiment.

The resistive elements are not longer than 110 mm andmounted on printed circuit boards (PCBs), so the four-terminal-pair (4TP) “definition plane” of the resistors is located ex-actly at the ends of the inner conductors of the four coaxialBayonet Neill–Concelman connectors. The geometry of theconductors of the PCB provides for extremely small mutualinductance—less than 3 nH—between the conductors. Theresistors could be connected to commercial IA and LCRmeters without additional connecting cables. The dielectric ofthe PCB is made of microwave composite material based onpolyphenilene oxide filled with fiberglass. This material hasa relative permittivity value of 2.8 and a dissipation factorvalue lower than 20 × 10−4 over a wide frequency range upto 10 GHz. The resistive elements are located in rectangular(120 × 80 × 40 mm) or cylindrical 50-mm diameter boxes.The materials and main parameters of the resistors are givenin Table I.

In Table I, ρ is the resistivity of the conductor material, d isthe diameter of the wire, h is the thickness of the conductor ina ribbon-type resistor, t is the thickness of the support, and ε isthe relative permittivity of the support material.

III. FREQUENCY DEPENDENCE

The resistance of any resistor at frequency f can be repre-sented by Rf = R0[1 + δfΣ(f)], where R0 is the dc resistance,and δfΣ is the total frequency correction at frequency f .

In the frequency range not exceeding 10% of the resonantfrequency, the individual components of δfΣ can be consideredmutually noncorrelated, and the value of δfΣ can be givenby an algebraic sum of components, each of them relating toone of the influencing factors. Table II presents the measuredfrequency dependence at 1 MHz and time constant, with corre-sponding k = 2 uncertainties, and the theoretical estimation of

TABLE IIIMEASUREMENT RESULTS OF 1-Ω RESISTOR BY FOUR COMMERCIAL

INSTRUMENTS A–D. RESULTS ARE PRESENTED AS THE

RELATIVE CHANGE, IN PERCENT, OF THE READINGS

AT FREQUENCY f FROM THE READINGS AT 1 kHz

the components of the relative frequency dependence at 1 MHzfor all considered standard resistors.

In Table II, δf0 is the effect due to the inherent equivalentparameters of the uniform line L, R, C, or CS ; δec is due to skineffect and proximity effect in the resistive element; δesc is dueto eddy currents in the shield; δsh is due to shunting caused bythe presence of a dielectric with losses; δC is due to the lumpedcapacitance between the resistive element and the shield; δend

is due to the edge effect of the uniform line; and δM is dueto the mutual inductance between the current and the potentialconductors in the resistive element.

IV. RESULTS OF EXPERIMENTAL INVESTIGATIONS

Our investigations are restricted, by the absence of appro-priate metrological equipment, to measurements up to 1 MHz.Information about the characteristics of the resistors can beobtained from the comparison of the measurement resultsat frequencies from 0.1 kHz to 1 MHz using four differentcommercial 4TP instruments—three LCR meters and one IAof two manufacturers. The basic uncertainties of these LCRmeters are 0.05%, and the basic uncertainty of IA is 0.07%.Real uncertainty is a composite function of the measurand, fre-quency, and energy condition. As an example, Table III showsthe results of 1-Ω resistor measurements by the aforementionedcommercial instruments (A–D). Results are presented as therelative change, in percent, of the readings at frequency f fromthe readings at 1 kHz.

KIM et al.: RESISTORS WITH CALCULABLE FREQUENCY DEPENDENCIES UP TO 1 MHz 457

TABLE IVCOMPARISON OF THE MEASURED RESISTANCE CHANGE, IN PERCENT,WITH RESPECT TO THE VALUE AT 1 kHz, AND THE CORRESPONDING

MEASUREMENT UNCERTAINTY ACCORDING TO

SPECIFICATION OF THE INSTRUMENT

Table IV compares the relative differences in readings atfrequency f and 1 kHz, i.e., (1 −Rf/R1kHz) of one of thesemeters and the uncertainty calculated according to the manu-facturer’s instruction. At 1 kHz, the measurement results for the1- to 100-Ω resistors did not agree by less than 0.02%; for the1- and 10-kΩ resistors, the difference did not exceed 0.005%,and for the 100-kΩ resistor, it is 0.01%. The measurementresults demonstrate that the manufacturers use a rather conser-vative estimation that exceeds the probable uncertainty value.The meters, to which the resistor to be measured is connectedby means of cables, showed an error proportional to the ca-pacitance between the low-current terminal and the shield. Oneof the instruments demonstrated an error of 0.0064% per 1 pFirrespective of the measured impedance. This error was prob-ably caused by the interaction between the output impedanceZLC of the resistor and the input admittance YLC of the meter.

V. CONCLUSION

The developed resistance standards can become useful toolsfor the investigation and calibration of commercial LCR me-ters up to 1 MHz. According to the measurements, one canassume that the frequency dependences of the resistors areseveral times lower than the measurement uncertainty of thementioned meters. Measurements by more precise equipmentare necessary to check that the frequency characteristics of theresistors correspond to the theoretical estimations.

REFERENCES

[1] K. Suzuki, “A new universal calibration method for four-terminal-pair-admittance standards,” IEEE Trans. Instrum. Meas., vol. 40, no. 2,pp. 420–422, Apr. 1991.

[2] L. Callegaro and F. Durbiano, “Four-terminal-pair impedances and scat-tering parameters,” Meas. Sci. Technol., vol. 14, no. 4, pp. 523–529,Apr. 2003.

[3] S. Awan, B. Kibble, I. Robinson, and P. Giblin, “A new four-terminal-pair bridge for traceable impedance measurements at frequencies upto 1 MHz,” IEEE Trans. Instrum. Meas., vol. 50, no. 2, pp. 282–285,Apr. 2001.

[4] S. Awan and B. Kibble, “Towards accurate measurement of the frequencydependence of capacitance and resistance standards up to 10 MHz,” IEEETrans. Instrum. Meas., vol. 54, no. 2, pp. 516–520, Apr. 2005.

[5] R. J. Haddad, “A resistor calculable from dc to ω = 105 rad/sec,” M.S.thesis, George Washington Univ., Washington, DC, 1969, p. 57.

[6] D. L. H. Gibbings, “A design for resistors of calculable ac/dc resistanceratio,” Proc. Inst. Electr. Eng., vol. 110, no. 2, pp. 335–347, Feb. 1963.

[7] Y. P. Semenov, I. Klebanov, R. D. Lee, and H. J. Kim, “Bifilar ac–dcresistor using a microwire,” IEEE Trans. Instrum. Meas., vol. 46, no. 2,pp. 333–336, Apr. 1997.

[8] G. Alekseyev and Y. Scripnic, “Shield effect on frequency depen-dence of microwire resistors,” in Microwire and Resistance Instruments.Kishinev, Moldova: Shtiinca, 1962, pp. 188–203. (in Russian).

[9] B. Hague and T. R. Ford, Alternating Current Bridge Methods, 6th ed.New York: Pitman Publication, 1971, p. 602.

[10] P. A. Ionkin, Theory of Electrical Engineering. Moscow, Russia: HigherSchool Publication, 1976, p. 384. Band 2, (in Russian).

[11] M. S. Veksler and A. M. Teplinsky, AC Shunts. Moscow, Russia:Energoatomizdat Publication, 1987, p. 120. (in Russian).

[12] J. Bohacek and B. M. Wood, “Octofilar resistors with calculable frequencydependence,” Metrologia, vol. 38, no. 3, pp. 241–247, Jun. 2001.

[13] J. Melcher, J. Bohacek, J. Ríha, A. von Campenhausen, and E. Pesel,“Intercomparison of resistance standards with calculable frequency de-pendence for the characterization of quantum Hall devices,” in Proc.CPEM Conf. Dig., 2000, pp. 176–177.

Han Jun Kim was born in Korea in 1957. He received the B.S. and M.S.degrees in electronics from Chung-Ang University, Seoul, Korea, in 1982 and1984, respectively.

In 1987, he joined the Electricity Laboratory, Korea Research Institute ofStandards and Science (KRISS), Daejeon, Korea, where he has been workingon impedance standards at low frequency.

Rae Duk Lee was born in Korea in 1945. He received the B.S. and M.S. degreesfrom Soong-Jun University, Taejon, Korea, in 1968 and 1980, respectively, andthe Ph.D. degree from Han-Nam University, Taejon, in 1991, all in physics.

In 1978, he joined the Electricity Laboratory, Korea Research Institute ofStandards and Science (KRISS), Daejeon, Korea. His present research activitiesare associated with impedance standards at low frequency.

Yu. P. Semenov was born in Leningrad, Russia, in 1940. He received theacademic degree from the Electrotechnical University, Leningrad, in 1963 andthe Ph.D. degree from the D. I. Mendeleyev Institute for Metrology (VNIIM),St. Petersburg, Russia.

Since 1962, he has been with the VNIIM. His present research activities arefocused on the improvement of the methods and facilities used to realize theSI farad and ohm on the basis of the cross theory.