IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

[5] L. Crovini and C. G. M. Kirby, "Resistance comparisons at nano-volt level using an isolating current ratio generator," Rev. Sci.Instr., vol. 41, pp. 493-497, 1970.

[6] M. V. Chattle, R. L. Rusby, G. Bonnier, A. Moser, E. Renaot, P.

Marcarino, G. Bongiovanni, and G. Frassineti, "An intercompari-son of gallium flxed point cells," Temperature, Its Measurementand Control in Science and Industry, J. F. Schooley, Ed., vol. 5,part 1. New York: American Institute of Physics, 1982, pp. 311-316.

Impedance Measuring Device Based on an ACPotentiometer

GUNTHER RAMM

Abstract-This paper describes a novel ac potentiometer composed ofdigital sinewave generators and inductive voltage dividers (IVD's) forthe comparison of unlike impedances with different nominal valuesfrom a few 10 Q up to 1 kQ in the frequency range between 50 Hz and1 kHz. As an example, the resistance of a 100-SI standard resistor maybe compared with the capacitance of a 1-,gF standard capacitor at afrequency of 400 Hz with a relative uncertainty of about 1 X 106which is mainly due to the short-term stability of the digital sources.

I. INTRODUCTIONITH customary ac bridges, only like impedances (tworesistances or two capacitances) may be compared. In

the case of high-accuracy measurements the measured valuesof the components may not differ from their nominal valuesby a large amount. As an additional requirement, the timeconstants of the resistors or the loss factors of the capacitorshave to be of the same order of magnitude; otherwise thequadrature balance of the bridge will influence its main bal-ance. The measuring device described in this paper allowsunlike impedances with different nominal values to be com-pared. Time constants and loss factors which differ signifi-cantly do not contribute to the measuring uncertainty. Acomparison between a capacitance and a resistance is included.The resistance and the time constant of an ac resistor may,therefore, be derived from the capacitance and the loss factorof a standard capacitor.

IL. MEASURING METHOD

Fig. 1 shows a simplified diagram which explains the measur-ing method. A digital sine wave generator with the outputvoltage U3 produces the measuring current I which flowsthrough the two impedances Zx and ZN to be compared.With switch S in position "X" the voltage drop Ux across

Manuscript received August 20, 1984.The author is with the Physikalisch-Technische Bundenstalt, Bundes-

allee 100, Braunschweig, D-3300, Federal Republic of Germany.

ZN T 2 T"1 Va b=~~~~~~~~~~~~~~~~~~P 972

COMPUTeRd a tERimCe

Fig. 1. Simplified diagram of the impedance measuring device.

Zx is measured by means of a complex ac compensator com-posed of two sine wave generators with the output voltages U,and U2, two seven-decade inductive voltage dividers T1 andT2 with settings a and b and an isolating transformer T4 withratio k. In the same way the voltage drop UN across ZN ismeasured with switch in position "N." The three voltages U1,U2, and U3 are of the same frequency and may be adjustedin phase with respect to each other. They are formed bydigital synthesis in compliance with Fig. 2 [11, [2]. Thesource consists of three almost identical channels which arecontrolled by a common pulse generator. The output pulsesof this generator go to an address counter which producesconsecutive addresses for the subsequent ROM's 1, 2, and 3;the content of these memories is a sine wave in digital form.Thus for each pulse, a new value of the sine function reachesthe digital-to-analog converters (DAC's) and is transformedinto the corresponding voltage at the output. In this way thesource produces three independent output voltages as a stair-case approximation to a sine wave in analog form. The sinefunction is subdivided into 512 steps/period (9 bit) and 2048steps (11 bit) in the amplitude axis from peak to peak. Byadding preset values to the output signal of the address counter,voltages U1 and U2 may be digitally adjusted in phase (p, andp2) with respect to U3. The latches between the ROM's and

0018-9456/85/0600-0341$01.00 © 1985 IEEE

341

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

From settings ax,, bx1, aNl, and bNl of the dividers the im-pedance ratio ZX/ZN can be calculated only approximately,if the errors a and , of the source are not known

z (Zx\ ax, - lbxlZN IZN/1 aNl - ibNl

(6)

However, by digital permutation of the phase settings <p and'P2 it is possible to generate a second set of compensation volt-ages. With 'Pj = 900 and '02 = 0° and, in addition,

U1 = jU3 and U2 - (1 + a +j1)U3one obtains

U1 U2 U3

Fig. 2. Block diagram of the digital sine wave generators.

UX2 =ZxI= k(jaX2 - bX2(1 +a+jP))U3UN2 ZNI -k(laN2 - bN2(I + a-+MD))U3.

the DAC's are for synchronization purposes. They equalizetime delays in the three channels.

Setting U1 = U3 exp (i,pl) and U2 = U3 exp (hP2) and withepl.= 00 and P2 = 900, one obtains

U1= U3 and U2 =iU3.In this case, the two voltages necessary for compensation are

Ux =ZxI= k(axUi - bxU2) = k(ax jbx)U3 (1)

and

UN =ZNI=-k(aNU1 - bNU2)=-k(aN-/jbN)U3 (2)

and from these two equations the impedance ratio is calcu-lated to be

Zx ax - lbx =A +jB.

ZN aN - bN

Equation (3) is valid only for ideal, that means, faultlessDAC's. In practice amplitude and phase errors will appear, so

that U1 and U2 have to be written as

U1 = U3 exp (jip)and

U2 = U3(1 +a +j1)U3 exp(j'02)

where a = AU/U represents the relative amplitude error and,B= AipI<p the relative phase error. On the other hand, phaseangles 'Pl and 'P2 are almost free from errors, because they are

digitally adjusted.

III. MEASURING PROCEDUREThis section describes a method for eliminating the effect of

amplitude and phase errors by means of a permutation method[1], [2]. With pI =0° and 'P2 =900 and taking the source

errors into account, one obtains

U1 = U3 and U2 =j(1 +a +1j)U3 .

The voltages necessary to compensate for the voltage dropsacross Zx and ZN are given by

Uxy ZxI=k(axi -lbxl(1 + a+/f))U3 (4)

UN,1 ZNI = -k (aN1 - jbNl (I + Ci + M,B) U3 .(5)

From these two equations another approximate value for theimpedance ratio can be calculated:

_x Zx = bx2 -lax2

ZN \ZN /2 bN2 - laN2(9)

The influence of the source errors a and , on the impedanceratio can be reduced to second-order effects if the true imped-ance ratio is calculated as the arithmetic mean of the approxi-mate values given in (6) and (9), i.e.,

(10)

Measurements at a frequency of 400 Hz showed source er-

rors below 1 X 10-4 and a short-term stability below I X 106(relative value measured over a period of 8 min which corre-

sponds to the time necessary for one measuring cycle). Thesource errors, therefore, contribute to the impedance ratiocalculated from (10) only a few parts in 108 and may be ne-

glected, so that only the short-term stability of the sources

contributes to the measuring uncertainty.In order that the ratio errors of dividers TI, T2, and T4 do

not exceed the measuring uncertainty given by the short-termstability of the source, two-stage transformers with electroni-cally-increased input impedance [3], [4] were used. In thisway voltage sources U1 and U2 operate in a nonloaded condi-tion, and the ratio k of divider T4 is not influenced by theoutput impedances of T1 and T2 which vary with their settingsa and b. After having balanced the potentiometer, settingsa and b are read into a computer which calculates the desiredimpedance ratio according to (10), when a complete measuringcycle has been finished. In order to avoid operating errors, theposition of switch S and the settings for phase angles 'Pj and'P2 are also indicated by the computer. Fig. 3 shows a photo-graph of the impedance measuring device with the thermostatedcomponents, the compensation circuit and the inductive voltagedividers together with the digital sine wave generators (fromabove).

IV. MEASUREMENT RESULTS

The characteristics of the measuring device were investigatedexperimentally by means of two thermostated standard capaci-tors (Zl and Z2) and two standard resistors (13 and Z4) withnominal values of I ,uF and 100 Q, respectively. These com-

(7)(8)

342

ZX1ZN;Z:. ' [(ZXIZN)l + (ZXIZN)21 -

RAMM: IMPEDANCE MEASURING DEVICE

Fig. 3. Photograph of the impedance measuring device with the thermo-stated components, the compensation circuit, and the inductivevoltage dividers together with the digital sine wave generator (fromabove).

Zi Z2 Z3 A

'122 02\4 5 _76IL 1> 13

Fig. 4. Procedure for testing the measuring device.

ponents were compared at a frequency of 400 Hz in six possi-ble combinations as indicated below and shown in Fig. 4:

(1) Z,/Z2 (2) Z, /Z3 (3) Z, /Z4(4) Z2/Z3 (5) Z2l/Z4 (6) Z3/Z4.

All these measurements were self-checked by permuting thecomponents. In the ideal case, the multiplication of the mea-

surement results obtained with the normal (ZXIZN)fl and thereversed (ZXIZN)r measurement should yield unity. In prac-

tice, the deviations in the real and imaginary part were typicallybelow 2 X 10-6. From the six measurements, the impedance

TABLE IRESULTS OF A COMPARISON BETWEEN Two I-jiF CAPACITORS AND Two

100-fS RESISTORS AT A FREQUENCY OF 400 Hz

(Z2/Z)a = 1 + 20.6 106 -j.1.2.106(z21Z )b = I + 21.3 10-6 _ j.1.5.10-6

(Z2/Zd)c = 1 + 20.0 10-6 -j, 91o-6

(Z3/Z4) = 1 + 210.5-10-6 + j71.110-62~~3!Aa ~ -6 -

(Z3/Z4)b= 1+ 210.7 10 + j.71.I-106(Z3/Z4)c= 1 + 209.4.10-6 +j-70.7-10-

ratios Z2/Zl and Z3/Z4 can be calculated in three differentways, the mean values of the normal and reversed measure-ments being taken (see Fig. 4, lower part):

a) directly from measurements (1) and (6):

(Z2/Zi)a and (Z3IZ4)a,b) indirectly, by combining the results of measurements (2)

and (4) and measurements (2) and (3):

(Z2/Z3)/(Z1Z3) = (Z2IZl)b(Z1/Z4)f(Z11Z3) = (Z3/Z4)b, and

c) indirectly, by combining the results of measurements (3)and (5) and measurements (4) and (5):

(Z2 /Z4)/(Z /Z4) = (Z2 IZI )c(Z2/Z4)/(Z2/Z3) = (ZA3Z4)c.

The results are given in Table I. The real parts of the imped-ance ratios Z2/lj and Z3/Z4 represent the capacitance ratioCl/C2 and the resistance ratio R31R4, respectively. Theimginary parts give an indication of the differences in the lossfactors of the capacitors and the time constants of the resis-tors. As may be seen from Table I, the real and imaginaryparts of the impedance ratios differ by less than 1 X 10-6from their mean values, irrespective of the fact that they weredetermined in different ways. Therefore, this measuring deviceis suitable for the measurement of both like and unlike imped-ances. For higher frequencies the deviation increases nearlylinearly with frequency due to the dominating influence ofthe stability of the DAC's phase errors. Since the stability ofthe amplitude errors is nearly frequency-independent, thedeviations do not decrease for frequencies below 400 Hz.

V. CONCLUSIONIn the preceding sections a novel impedance measuring device

based on an ac potentiometer for the comparison of unlikeimpedances with differing nominal values has been described.With this method a relative measurement uncertainty (1 U) ofabout 1 X 10-6 can be obtained, a value which is surpassedin comparison of unlike impedances only if much more com-plicated bridges operating at a fixed frequency and measuringimpedances of selected nominal values are used. On the otherhand, with simpler methods, arbitrary impedances can only bemeasured with a relative uncertainty which is some two orthree orders of magnitude higher than the value mentionedabove. Therefore, this measuring method suitably fills a gap inmodern impedance measuring techniques.

343

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985

ACKNOWLEDGMENTThe author wishes to thank H. Bachmair for many stimulat-

ing discussions and R. Vollmert and R. Harke for their assis-tance in the experimental work.

REFERENCES

[1] H. Bachmair, and R. Volimert, "Comparison of admittances by

means of a digital double-sinewave generator," IEEE Trans. In-strum. Meas., vol. IM-29, pp. 370-372, 1980.

[21 , "Digitaler Generator speist Impedanzme1briicke," Elektronik,no. 20,pp.95-99, 1983.

[3] B. Fuhrmann, "Induktiver Spannugsteiler mit hoher Eingangsim-pedanz," PTB-Mitteilungen, no. 3, pp. 196-201, 1975.

[4] G. Ramm, "Hochauflosende Wechselstrom-Mei3brucke fur Wider-stande," Elektronik, no. 13, pp. 101-104, 1983.

LC Comparison System Based on a Two-PhaseGenerator

FRANCO CABIATI AND GIAN CARLO BOSCO

Abstract-A solution to the problem of measuring a standard inductorin terms of the unit of capacitance using a stable ac two-phase generatoris presented. Accurate separation of inductive and conductive compo-nents is obtained by means of a self-adjustment feature of the system,whose quadrature phase, in principle not essential for LC comparison,allows exclusion of any reference resistor. Standard deviations s < 1 X10-6 and overall uncertainties near to 1 X 10-6 (la) are obtained witha test implementation of the system.

I. INTRODUCTION

THE development of two-phase generators, favored by theuse of digital techniques for sinewave synthesis [1], [2],

gives new suggestions for designing ac measurement systems,and particularly for admittance comparisons [3], [4]. Thepresent paper concerns itself with an application of a stabletwo-phase generator to the problem of measuring a standardinductor in terms of the unit of capacitance. Bridges using aquadrature phase, obtained by passive phase-shifting networks,are well known [5] -[8]. More recently, as in the above citedsystems, the quadrature phase is supplied by a low-impedancesource belonging to the generator. In a broad sense the systemdescribed below can be included among the quadrature bridgesfor unlike admittance intercomparisons, and, with regard tothe generator, to the latter group.The measurement circuit is based on inductive ratios, variable

voltages and fixed reference standards deflned as two-terminal,or four-terminal N-pair devices. In fact the extension ofcoaxial techniques to inductance measurements has becomemore convenient since the introduction of shielded standardinductors with toroidal winding. The choice of a parallel res-

Manuscript received August 20, 1984.The authors are with the Istituto Elettrotecnico Nazionale "Galileo

Ferraris," 1-10135 Torino, Italy.

onant circuit, with a current summing point on the detectorside, seemed to be the more suitable to the use of admittance-defined standards.Of course, measuring a standard inductor as an admittance

(that is referring to its parallel equivalent circuit) requires acurrent balance at the low-voltage terminal by means of twoorthogonal current components, one of which in quadraturewith the voltage. These current components can be obtainedby means of either two unlike reference admittances with asingle-phase generator, or two like admittances with a two-phase generator. In the first case the measurement systemmay have the basic structure of a transformer bridge for likeadmittances, except that an accurate separation of the maincomponent of the admittance is more critical than for capaci-tors or resistors, due to large impurity component of precisionstandard inductors. In particular a standard resistor is neces-

sary whose time constant must be determined by other means

and also its ac conductance must be known if the parallel to

series conversion of the equivalent inductance is required. Tothis class of bridges recently presented solutions belong [9],[10] . In the second case, two reference capacitors can be used,at least one of which should be a gas dielectric capacitor withnegligible loss angle. It will be shown that the apparent redun-dancy of a two-phase generator can be justified by some self-adjustment and self-calibration features of the system.

II. PRINCIPLE OF THE METHOD

The schematic diagram of Fig. 1 shows the structure and themain components of the system. The circuit appears in itsnormal configuration; a second configuration is obtained bychanging both the switches from the normal (N) to the reverseposition (R).

0018-9456/85/0600-0344$01.00 © 1985 IEEE

344