IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-23, NO. 4, DF.CI:MBHLR 1974

in its defining connectors can be measured in a similarmanner provided that the admittances to be extrapolatedto zero are properly accounted for.Note also that "application E" provides a second method

for measuring precision transformer ratios. This methodshould be of special interest when precision transformersare available, but large numbers of nominally equaladmittances, as required in "application F," are notavailable.

It is hoped that four-pair admittanced will have a morepractical appeal and receive wider usage now that theycan be measured with two-pair bridges (including com-mercial bridges). At the other extreme, as pointed out byCutkosky, combining extrapolation techniques with four-pair bridge techniques provides exciting possibilities fordata acquisition and automation at the highest levels ofprecision.

REFERENCES

[1] R. D. Cutkosky, "Four-terminal pair networks as precisionadmittance and impedance standards," Commun. Electron., vol.70, pp. 19-22, Jan. 1964.

[2] , "Techniques for comparing four-terminal pair admittancestandards," J. Res. Nat. Bur. Stand., vol. 74C (Eng. an(d Instr.),No. 3 and 4, pp. 63-78, July-Dec. 1970.

[3] A. M. Thompson, "The precise measurement of small capaci-tances," IRE Trans. Instrum., vol. 1-7, pp. 245-253, Dec. 1958.

[4] M. C. McGregor et al., "New apparatus at the National Bureauof Standards for absolute capacitance measurement," IRE Trans.Instrum., vol. 1-7, pp. 253-261, Dec. 1958.

[5] R. D. Cutkosky, "Active and passive direct-reading ratio setsfor the comparison of audio-frequency admittances," J. Res. Nat.Bur. Stand., vol. 68C (Engr. and Instr.), no. 4, Oct.-Dec. 1964.

[6] D. N. Homan, "Applications of coaxial chokes to ac bridge eir-cuits," J. Res. Nat. Bur. Stand., vol. 72C (Eng. an(d Instr.),no. 2, pp. 161-165, Apr.-June 1968.

[7] R. D. Cutkosky and J. Q. Shields, "The )recisioni measurement oftransformer ratios," IRE Trans. Instrum., vol. 1-9, pp. 243--250,Sept. 1960.

[8] J. Q. Shields, "Voltage dependence of precision air capacitors,"J. Res. Nat. Bur. Stand., vol. 69C (Eng. and Instr.), no. 4, pp.265-274, Oct.-Dec. 1965.

Resistance Measurements at Frequencies Below 10 Hz

BRUNO FUHRMANN

Abstract-Several electronic circuits are described which increasethe input resistance of inductive voltage dividers up to 109 Q at fre-quencies below 10 Hz. One of the circuits has been used to buildan isolating inductive voltage divider consisting of 8 decades andhaving an input resistance of 109 U. The in-phase errors are less than2 parts in 10+8. This inductive voltage divider is the main part of anac potentiometer for the precise measurement of low value resistorswith comparatively high lead resistances. Lead resistances of 50 Qgenerate errors of only 1 part in 107.

Apart from the preceding application, the inductive voltage di-viders with high input impedance may also serve as a component of avoltage comparator. Such a comparator, for example, permits thecalibration of a resistive voltage divider which is used for main-taining the unit of voltage via the Josephson effect and which, forsuch a purpose, is kept at a temperature of 2 K. An uncertainty of5 parts in 10+10 is achieved for the measurement of a 1:1 ratio, evenif the ratios differ by a few parts in 103 from 1:1.

I. INTRODUCTION

INDUCTIVE voltage dividers are well suited for re-sistance measurements by ae methods due to their

smaller errors, higher input impedances, and lower outputimpedances compared with those of resistive voltagedividers. Further advantages of ae methods include a)no disturbing influences from thermal emfs and b) avail-

Manuscript received July 3, 1974; revised September 9, 1974.The author is with Institut fir Grundlagen der Elektrotechnik

und elektrische Mejltechnik, Technische Universitat Braunschweig,Postfach 33 29, 3300 Braunschweig, Germany.

ability of ac balance detectors with particularly highsensitivity.The disadvantages are the differences between the ae

and the de resistance of a resistor such as those caused,for example, by dielectric losses, eddy currents, skineffect, inductances and capacitances of the resistors, aswell as the leads.Such differences become very small at frequencies below

10 Hz. As the errors of the usual inductive voltage dividersincrease at low frequencies, Hill and Deacon [1] recom-mend that two-stage transformers be used as inductivevoltage dividers. The errors of two-stage inductive voltagedividers are sufficiently small, but their input impedances,which are mainly those of the exciting windings, are low.

Therefore, extremely low-lead resistances have to beused if the errors due to lead resistances are not to exceedthose of the divider itself.

This important limitation can be avoided by increasingthe input impedance of two-stage transformers by meansof electronic amplifiers. However, particular care mustbe taken to ensure that any changes of the parametersof the electronic components have little effect on theerrors of the divider.

Gibbings [2] describes an electronic circuit whichincreases the input impedance by a factor of about 100 at50 Hz. Still further increases, especially at frequenciesbelow 10 Hz, are difficult to achieve because of the stabilityrequirements of the whole system.

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FUHRMANN: RESISTANCE MEASUREMENTS

In contrast to the Gibbings's circuit, the followingcircuits are not insulated from the input of the divider.However, they require only one extra winding with theexception of the case where an extreme increase to theinput impedance is to be achieved. These circuits permita particularly large bandwidth. A floating input of thedivider is obtained by feeding the amplifiers from anisolated power supply.

II. INDUCTIVE VOLTAGE DIVIDERS WITHELECTRONICALLY INCREASED INPUT

IMPEDANCE

A very simple circuit [3] is shown in Fig. 1(a). Theexciting winding W2 as mounted on core 2 is fed by thevoltage follower A1. Neglecting the very high input imped-ance of A1, the input impedance of the inductive voltagedivider is that of the winding Wi. W1 and W2 have equalnumbers of turns. At frequencies of about 10 Hz, inputresistances up to 107 Q can be obtained if cores with 90 X60 X 20 mm3 of high permeability alloy are used with101 winding turns for W1 and W2.At low frequencies, it is not possible to increase the

input impedance any further. The reason is that theresistance of winding W2 together with the magnetizingimpedance of W2 form a voltage divider. Thus the fluxin core 2 originating from W2 is less than that of a windingwithout resistance. Also this flux has a different phaseas compared with that one generated by winding W1.The circuit shown in Fig. 1 (b) overcomes this problem

by electronically compensating for the resistance of W2. Acomplete compensation of the resistance of W2 is obtainedif the gain of A2 is set at 2 and the resistance of Rk is setequal to the resistance of W2. DC saturation of core 2is avoided if only a partial compensation is applied. Thisresults in input resistances up to 108 Q if cores such asdescribed previously are used [4].

Alternatively, the input irnpedance can be increasedas shown in Fig. 1 (c). A3 amplifies the difference betweenthe input voltage V1 and voltage V3, the latter beinginduced in the indicator winding W3 by the ac flux ofcore 2. This amplified voltage differenee is now fed to theexciting winding W2. Provided that the gain of A3 issufficiently high, most of the required flux is generatedby A3 and the input impedance of W1 will be high. How-ever, due to the de drift of the amplifier A3 and to thelow de resistance of winding W2, the gain of A3 must belimnited, even though high pass coupling can be used toprevent de saturation of core 2. Another limnitation ofthe gain of A3 is caused by the stability requirements of thewhole system. For the aforementioned reasons, the increaseof the input impedance does not exceed the magnitudeachievable by the circuits shown in Fig. 1(a) and (b).

Still further increase of input impedance can be obtainedby mneans of a combination of the circuits shown in Fig.1(b) and (c). This is shown in Fig. 2.Amplifier A4 serves to sum the output voltages of volt-

age follower A1 and anmplifier A3 and feeds the exciting

(a) (b) (c)Fig. 1. Increase of the input impedance of inductive voltage

dividers. (a) With a voltage follower. (b) As in (a), with compensa-tion of the resistance of W2. (c) With feedback around core 2Wl = W2 = W3 = N turns.

A

Fig. 2. Extreme increase of the input impedance of inductivevoltage dividers by a combination of a voltage follower and feed-back around core 2.

winding W2. Partial compensation of the resistance ofW2 is achieved by feeding the voltage drop across RKto the noninverting input of A4. The amplifiers shown inFigs. 1 and 2 are integrated circuits using appropriatefeedback to obtain the desired characteristics.

III. ISOLATING INDUCTIVE VOLTAGEDIVIDER

As shown previously, the input impedance of inductivevoltage dividers can be increased electronically by severalpowers of ten, which means a significant reduction of theexciting current flowing through the input winding. Thusinductive voltage dividers with very small errors at lowfrequencies can now be built in the form of autotrans-foriners as well as isolating transformers. By using thecircuit shown in Fig. 2, an isolating inductive voltagedivider which consists of 8 decades (Fig. 3) has beenconstructed. The first three decade windings are mountedon a two-stage core which also carries the prinmary windingand one turn to drive the exciting winding of the secondtwo-stage core. The primary winding consists of 10 strandsof 0.5-mm diameter enamelled wire surrounded by a thinpolyester foil and another 10 strands of 0.5-mm diamieterenamelled wire forming the first decade. The whole cablehas been twisted about 30 times per meter and woundaround the 90 X 60 X 20 mm3 cores with 100 turnsequally distributed around the circumference. 10 turns

353

-1

v,I

354IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, DECEMBER 1974

slit rE12 -=-,bL

A]

Fig. 3. Isolating inductive voltage divider with electronicallyinereased input impedance.

of a 10-strand cable form the second decade, another turnof a 10-strand cable being the third. The exciting windingand the indicator winding are wound on core 2, bothwindings consisting of 103 turns of wire with diametersof 0.3 mmn and 0.1 mm, respectively, and both windingsbeing screened with a thin copper foil. The following threedecade windings are mounted on a two-stage core whilethe seventh and the eighth decade windings are wound onsingle cores.With the gain of A1 at 1 and that of A3 at 33, the input

resistance of the divider is 104-Q shunt by a capacitanceof 1.8 X 109 F. The output resistance is a function of thedial setting with a maximum of 30 Q. All in-phase errorsare smaller than 2 parts in 10§ of input and are, to a largeextent, caused by the voltage drop across the resistanceof the primary winding. The in-phase linearity errors areless than 10-8 of input. A bootstrap mnethod has beenused to calibrate the divider. It is similar to Hills andDeacons [5] method, but the quadrature balance isachieved by feeding an appropriate current through asmall mutual inductance and by adding the voltage of itssecondary winding to the reference voltage.

IV. AC POTENTIOMETER FOR RESISTANCEMEASUREMENTS

A new measuring circuit for low-value resistors is shownin Fig. 4. Its main part is the isolating voltage dividerdescribed previously. The staindard resistor R8, the un-known resistor R. and the ac current source I at a fre-quency of 81 Hz form a closed loop. The isolating dividerD divides the voltage drop across R8 by n and this iscompared with the voltage drop across Rx With theinput resistance R, of the divider D (if we neglect theproducts of very smnall terms) the equationi for the in-phase balance reads

RX -nR8 I--RS-)RS1R)Rx is a linear function of RS,. R81Rp will be very small andcan easily be eliminated.

Errors due to lead resistances R,, can also be kept smallbecause Rp is 109 Q. The lead resistances R1i only reducethe performance of the balance detector. Lead resistancesof 50 Q produce errors of 10Q. Thus the influence of leadresistances is of little importance for applications de-manding long or thin leads as, for example, in resistancethermometry.The balance detector suppresses quadratuire signals

which are 60 dB above the in-phase signal and is of ex-

Rs

Fig. 4. AC potentiometer for resistance measurements.

cellent linearity. Together with the linear behavior ofthe potentiometer, this permits the recording of resistancechanges by a potentiometer type recorder. A quadraturebalance is not necessary.

V. CALIBRATION OF A RESISTIVE VOLTAGEDIVIDER AT 2 K

For the comparison of the Josephson de voltage withthe ENIF of standard cells, a resistive voltage dividerhaving a ratio of several hundred to one is needed. Toavoid errors due to thermal EMF's, this divider should beat the temperature of the Josephson junction which isat 2 K [6]. DC measurements of the resistance ratiowill be more difficult if a measurement system at roomtemperature is used because of thermal EMFs, and thusan ac voltage comparator is a good solution to the prob-lem. If the electronic circuits shown in Figs. 1(b) and 2are used, the input resistance of the voltage comparatorwill be about 101 Q. Thus the influence of the lead resist-ances will be sufficiently small. The use of a fixed dividerand an isolating divider with a ratio of 103 will assurelow values for both the output resistance as well as thenoise. The isolating divider is fed by a variable 8 decadedivider. Therefore, balance can be achieved even if theresistors deviate by several parts in 10+3 fromn their nominalvalues.

For the first experiments, a divider with a ratio of 0.5was chosen because this divider can be calibrated by itself.With this choice, the resistors of the resistive dividerhave to be weighted in a binary fashion. A simple sketchof this circuit is shown in Fig. 5. The voltage dividerconsisting of R1 to Rn is fed by the voltage source V witha frequency of 8' Hz. Switch &1 selects that part of theresistive divider the ratio of which is to be m-ieasured.Divider A2 with a ratio of n2 = 0.5 and divider 1)3 with8 decades are connected in parallel. The voltage at theceiltre tap of that part of the resistive divider connectedby SI with the mieasuring circuit is compared with thevoltage at the center tap of D2. Quadrature differenceswill be eliminated by adding a correction voltage whichis generated hy the current through the resistive dividerand the mutual inductance Ml. In-phase differences willbe caneelled by a correction voltage generated by thedivider D3 and the isolating divider DI wliiehih has a ratioof ni = 10-3.

3.534

FUHRMANN: RESISTANCE MEASUREMENTS

Fig. 5. Clalibration of a resistance ratio by ani ac voltage com-

paratcor; ni = 10-3, n2 = 0.5, 0 < n3 < 1 with Dl, D2, D3 havinginiput resistaiices > 108 R.

The equation for the in-phase balance of the first partof the resistive divider reads

R,a = n2 n1n31 = 0.5 i-10-3n31,

R, + R2

0<n31<1. (2)

Trhe equation for the ill-phase balance of the second part is

a2 R ,+ R+ - ±n2i n1n32 = 0.5 i 10-3n31,

0 < n32 < 1. (3)

The required ratio a of the resistive divider results fromthe following equation:

R1aR +R.2+ =.+R =aa2 * an-l- (4)

a is a product of the mieasured ratios a,- ban- -. Highaccuracy of ratio a is ensured if the uncertainties of themeasured ratios a,,.. a,,-1 are kept very sm-all as theirgeometrical sum is. by approximation, the probableuncertainty of ratio a. The relative uncertainty bnl/n1of divider DI is smaller thani I part in 10- 7. The errors ofdivider DA are smialler than 5 parts in 10+9 of the iinput. Toeliminate the error of divider D2, the average of twomeasuremi.ents has to be taken, one measuremeint withD2 in normal l)osition anid the other rneasurement withD2 reversed by switch S2. Thus an uncertainty of 5 partsin 10+10 of input for the individual ratios can be achievedwvith iniput voltages larger thani 50 rnV. At lower inputvoltages, the uncertainty wvill increase due to noise fromthe balance detector as well as from the output resistancesof the dividers and from the lead resistances. The noisevoltage is about 25 IV at a banidwidth of 9.1 Hz

ACKNOWLEDGMIENTThe author wishes to thaink H1 H ELmscherrnarn for

his helpful suggestions and encouragement.

REFERENCES[1] T. A. Deacon and J. J. Hill, "Two-stage inductive voltage

dividers," Proc. vol. 115, no. 6, pp. 888-892, June 1968.[2] D. L. H. Gibbings, "A circuit for reducing the exciting current

of inductive devices," Proc. vol. 108(b), pp. 339-343, 1961.[3] R. D. Cutkosky, "An ac resistance thermometer bridge," J. Res.,

vol. 74C, nos. 1 and 2, pp. 15-18, 1970.[4] B. Fuhrmann, "Widerstandsmessunig mittels Wechselstromver-

fahren unter besonderer Beriicksichtigung der Widerstands-thermotrie," dissertation, Technische Universitat, Braunschweig,Germany, 1974.

[5] J. J. Hill and T. A. Deacon, "Theory, design, and measurementof inductive voltage dividers," Proc. vol. 115, nio. 5, pp. 727-735, 1968.

[6] V. Kose, "Maintaining the volt at PTB via Josephson-effect,"presented at CPEM, 1974.

3.5.5