1958 Thompson: The Precise Measurement of Small Capacitances 245

and the generator revolutions given on the counter. By the Long Island Lighting Company at their Barrettsuitable calibration and selection of constants, the Power Station.meter and counter are calibrated in terms of volt-sec- Five tests were run with indicating wattmeters inonds, watt hours, etc. each of three phases and with one Inductronic Watt-

For long periods of time, such as 10 minutes or more, meter and integrator in phase A. The test time variedthe volt-seconds stored in the capacitor at the end of a from one to three hours. The integrator was read at thecycle do niot have to be read on the voltmeter. This is beginning and end of each test and the wattmeters werebecause the counter indication is large and the small read every minute.amount of volt-seconds indicated on the voltmeter can The largest difference between the two watt-hourbe neglected. For shorter periods of time, the sum of readings by each method was 0.12 per cent. The averagethe voltmeter and counter readings should be added to- difference for all five tests was 0.08 per cent. It is be-gether to give the total volt-seconds. lieved that using Inductronic Wattmeters in the otherThe basic accuracy for normal periods of time (10 two phases and feeding the outputs to three integrators

minutes or more) is completely dependent upon the would have provided total generator output in kilowattstability of the generator. This stability has been ob- hours to an accuracy of approximately 0.1 per cent. Thetained by good design practice and 0.1 per cent accuracy output of the three wattmeters can also be connected tois maintained. one integrator to provide the total three-phase energy.

For short periods of time, the accuracy of the instru- This may not be convenient where transformer connec-ment and readability of the counter are a factor and tion factors must be applied for each phase and where itmust be considered. A more complete description of the is desirable to know the load balance.precision integrator is given by Gilbert.6The combination of Inductronic Wattmeter and in- IX. CONCLUSIONS

tegrator has been tested under actual field conditions The product resolver can be used for the measurementto determine its suitability for measuring turbine effi- of volts, amperes, watts, and watt hours when used inciencies. This test was performed by the Engineers of conjunction with a precision integrator. It offers 0.1

per cent accuracy over a wide power frequency range

6 R. W. Gilbert, "The Weston Model 1473 Precision Integrator," and provides a dc output which can be measured orWeston Eng. Notes, vol. 10, pp. 6-9; December, 1955. used for control.

The Precise Measuremient of Small CapacitancesA. M. THOMPSONt

INTRODUCTION capacitor might provide values for the ohm and related

)HE practical electrical units are based on the units more accurate than those obtained by a direct2I electromagnetic system, and the starting point of electromagnetic determination.

most "absolute" determinations has been the con- An examination of a number of possible calculablestruction of an inductor and the calculation of its in- capacitors shows that they have one thing in common:ductance from its mechanical dimensions. If a capacitor the mechanical problem becomes easier as the capaci-is used as the starting point the capacitance is calculated tance is reduced; and optimum dimensions would givedirectly in electrostatic units, and a knowledge of the only a few picofarads. Considerable progress has al-velocity of light is necessary to convert to the practical ready been made with techniques for the measurementunits. Recent determinations of the velocity of light are of small capacitances, and it has been considered thatconsidered to be accurate to the order of 1 in 106 which these can be refined sufficiently to enable a 1-ohm re-is much better than has been claimed for absolute de- sistance standard to be related to a capacitance of theterminations of the electrical units. This fact, together order of 1 pF to an accuracy of 1 in 106. Accordingly, anwith the very much simpler geometry of some calculable electrostatic determination of the ohm based on a newcapacitors, suggests that a determination based on a type of calculable capacitor' is being made, and much of

the development work has been completed.* Manuscript received by the PGI, August 14, 1958. 1 A. M. Thompson and D. G. Lampard, "A new theorem in elec-t Electrotechnology Div., National Standards Laboratory, Chip- trostatics and its application to calculable standards of capacitance,"

pendale, Australia. NVature, vol. 177, p. 888; May, 1956.

246 IRE TRANSACTIONS ON INSTRUMENTATION December

In its present fornm the value of the calculable capaci- C12tor is only 0.25 pF. In this paper we propose to describe 11

some of the techniques which enable a standard of thissmall value to be used for the measurement of capaci- '

C

tance to an accuracy of 1 in 106 or better. These meas-urements are made at audio frequencies and are based 3on the use of 3-terminal capacitors and bridges withtransformer ratio arms. The principles involved are not Fig. 1-Eqtuivalent circuit of 3-terminal capacitor.new, but it is necessary to review these briefly if thelimitations of the system are to be appreciated. C12

2THREE-TERMINAL CAPACITORS

I ~~~~2-The capacitance between two conductors is only X

definite if one conductor completely surrounds the other.If there are exposed terminals the capacitance is de-pendent on the posi-tion of neighboring objects and vari- 3 3ations may be of the order of 1 pF. This lack of defini- GAS DIELECTRICtion may be eliminated by introducing a third conductorso that at least one of the conductors is completely sur- I Irounded. With three conductors the equivalent electri-cal circuit is indicated in Fig. 1, and the separate capac-itances are known as the direct capacitances between SMALL SMALLERthe pairs of conductors.2 The capacitor is called a 3Sterminal capacitor and may be so constructed that one Fig. 2-Construction of 3-terminal capacitors.of its direct capacitances is definite. The general princi-ple~~~~~ofsuc cntutoisilsrednFg. 2.Cnnc the hole. Both fixed and variable capacitors have beenple of such construction iS illustrated in Filg. 2. Connec-

dsge nti rnil.tions to the two ac-tive conductors (1 and 2) are made designed on this printciple.at points which are effectively shielded from each other For capacitances up to about 10,000 pF a gas di-by the third conductor. The direct capacitances are de- electric may be used, and, by a suitable construction,pendent on the positiOnS of all three of the conductors losses in the solid dielectric supports may be confinedand only one of them (C12) is definite, since the other to the ground capacitances. Under these conditions thetwo involve the capacitances of the leads. When for defined capacitance is practically loss free.4 The capaci-simplicity we refer to the capacitance of a 3-terminal tormustbesealed toeliminatethe effectsof atmosphericcapacitor, we mean the definite direct capacitance. The variations.other two are sometimes called the ground capacitances. MEASUREMENT OF DIRECT CAPACITANCEIt should be emphasized that the capacitances in theequivalent circuit are independent of the potential dis- Any method for the measurement of direct capaci-tribution and are not limited to any particular distribu- tance must make allowance for the presence of thetion which may simplify the computation of direct ground capacitances. Bridged-T and twin-T null cir-capacitance. cuits are not affected by the ground capacitances but

Three-terminal capacitors may be switched in such a the balance conditions are frequency sensitive, and suchway that no switch capacitances affect the defined direct circuits do not seem to be suitable for the measurementcapacitances, and since these are strictly additive this of very small capacitances. Most bridge methods can beoffers many advantages. For example, a switch capaci- adapted to 3-terminal measurements by the addition of

tance decade may be produced in which the units may components to balance the ground admittances, butalso be selected individually for calibration by substi- since the balance conditions for this and the main bridgetution. are interdependent, the balancing process can becomeA decade box of this type has a true zero. This follows very tedious.

from another important property of 3-terminal capaci- Campbell2 suggested a simple ratio device for a com-tors, in that the capacitance may be made as small as parison bridge, which would eliminate the effects of thewe wish by extending the third conductor as an electro-grudamtncswhotdiinlblnig.H

stti scee bewe the tw1ciecnutr.Ih suggested a T network as shown in Fig. 3. The operationscee is comlet th.aaiacszr.Asalhl of such a ratio device is most easily appreciated by

in the screen gives a correspondingly small value of considering the mesh equivalent.capacitance. The thicker the screen, the less effective

3F. K. Harris, "Electrical Measurements," John Wiley and Sons,Inc., New York, N. Y., p. 682; 1952.

2 G. A. Campbell, "Direct capacity measurement," Bell Sys. 4A. V. Astin, "Nature of energy losses in air capacitors at low fre-Tech. J., vol. 1, pp. 18-38; July, 1922. quencies," J. Res. NBS, vol. 22, pp. 673-695; June, 1939.

1958 Thompson: The Precise Measurement of Small Capacitances 247

Y' Y2 y'12 Z12 Zll 2 22 12 Z

- - -z,1 z22-z122Y23 22 Z22 + Z2a

IF '1 + Y2+Y3=O Y12 =z11Z22-z12Y13 3_ b Z11 +Z,2

2 3 2 = z

Fig. 3-Campbell's ideal ratio network.Fig. 4-Equivalent circuit of 2-winding transformer.

Since l

lv1 Y3 z12 KiiY2 + Y2± Z3 - \"l

( ~~~~= P'2

123 = V2 3 _ zl22t~ 13V1 + Y2 + 13

the effective bridge ratio is Y1/Y2 and this is independ- IDEAL TRANSFORMERent of the value of Y3. However, if Y3 is varied so that RATIOS 1 ' P2 P1 ~2-the sum Y1+ Y2+ Y3-*O the equivalent admittances z 2

Yn, Y23-x and the bridge ratio becomes independent >1 = 112 12 12of additional admittances shunting the ratio arms. 52 = Z2 - p2 Z12Campbell pointed out that a transformer with two Fi. 5-qivln cirui of2wndn. rnsomr

Fig. 4-Equivalent circuit of 2-winding tranisformer.

closely coupled windings was very nearly equivalent tohis ideal ratio network.

Tranlsformer ratios with more than two windings maybe used, and these also may have such a low effective _ E limpedance that the effect of admittances shunting theratio arms is in most cases negligible and no balancing z, z2 =of the ground admittances is necessary. In addition, j A1 P Ptransformers can give ratios of extraordinary precision, |iand with multiple windings there is considerable fiexi- Z11 Z22 3 i 92 l2bility in the design of measuring circuits. Some of thepossible configurations of transformer ratio arms and IDEAL TRANSFORMERtheir properties are outlined in the next section. RATIOS 1 Pp3 PtP2p3l

TRANSFORMER RATIO ARMS dj2 P2Z3 3Z1

The simplest case consists of two windings. If these z1, z13have a common connection the equivalent circuit shown = 2 Zin Fig. 4 may be used. If there is no common connection, z11z2panl equivalent circuit based on an ideal transformer is }2= 2sometimes useful. This is shown in Fig. 5, but there isno unique circuit of this type and one of the two leakage z Z" iimpedances or the ratio may be chosen to suit a particu-lar application. Fig. 6-Equivalent circuit of 3-winding transformer.

For three windings there is a unique equivalent circuitbased on an ideal transformerwhich is very useful. This nected in series. The simplest circuit of this type isis shown in Fig. 6. For more than three windings there shown in Fig. 7. If the windings have the same resistanceis no simple equivalent circuit. per turn and are wound to be symmetrical with respect

There are two types of bridge circuit in which a to a high permeability core, then multiratio windingstransformer may be used to give very precise ratios. The can be produced such that no ratio departs from thefirst type is obtained when the supply or detector is con- turns ratio by more than 1 in 1O5, and where close inter-nected across all the windings of the transformer con- leaving of the windings is possible this departure may

248 IRE TRANSACTIONS ON INSTRUMENTATION December

RATIO WINDINGS

Y,

Pl= p)2Y

Fig. 7-Bridge circuit with 2-winding transformer. PERMALLOY,=V STRIP-WOUNDC TOROIDAL CORE

l

P1 V

X 1lll

Il = p2'2YISUPPLY OR DETECTOR WINDING

P2 VY2Fig. 9-Cross section of electromagnetically

screened ratio transformer.

r 5 NWhichever type of circuit is used, transformer ratios

lp511 are best specified by the ratios obtained without any ex-

T ) | | L' 3 1 =ternal loading and by a set of effective impedances fromP1 Pa, = P2 PF 'Y which the change in the ratios due to external loading

|2 l g ;\// l l C Tmay be calculated. Such a specification only applies tothe particular method of connection, and it may varyslightly with frequency and with the level of excitation.

Fig. 8-Bridge circuits with 3-winding transformers. If a transformer has a large number of windings it is notpracticable to compute and apply corrections for the

be as low as 1 in 106. Close interleaving also reduces the effects of the separate loads on each winding, but theeffective impedances, and to keep these low a small num- effective impedances must be determined to confirmber of turns should be used. that the loading errors are within the desired tolerance.The other type of bridge circuit which gives precise This is best done by setting up the desired bridge circuit

ratios is derived frorn the first by the addition of a sepa- and then determining the change in the balance causedrate winding for the supply or detector. The simplest by shunting a winding with a known admittance.circuits of this type are shown in Fig. 8. This type ofcircuit has some advantages, but is best suited to the TRANSFORMER BRIDGES FOR THE MEASUREMENTcomparison of small admittances as the effective im- OF SMALL CAPACITANCESpedances of the ratio arms appear in series with the Sinice ratio transformers are made to have a low ef-components being compared. As before, the ratio wind- fective impedance to minimize loading errors, they doings should be svmmetrical with respect to a high per- not match the high impedances in the bridge when smallmeability core and also to the additional supply or de- capacitors are being compared. In this case it is foundtector winding, but in this case the resistances of the that the maximum sensitivity results from the use of awindings do not affect the open circuit ratios. By elec- voltage transformer only. A bridge of this type, for thetromagnetically screening the ratio windings from the comparison of two direct capacitances, is shown in Fig.core and the third winding it is possible to obtaini ratios 10. The third terminal of the capacitors is connected towhich are the same as the turns ratios to 1 in 107. The the ratio side of the detector. One of the ground capaci-cross section of such a transformer is shown in Fig. 9, tances is a shunt on the ratio arms which must be ofwhere the screening is provided by alternate copper and low effective impedance if this shunt is to have a neg-permalloy C screens. ligible effect on the ratio. The other ground capacitance

Transformer ratios which are connected to the supply is a shunt on the detector so that it does not alter theproduce knlown voltage ratios. They may operate with balance conditions but will have an effect on the sensi-quite high flux density, and are referred to as voltage tivity. The equivalent circuit of this bridge derived fromtransformers. Transformer ratios connected to the de- Fig. 6 is shown in Fig. 11. The balance conditions of suchtector define a current ratio and at balance operate at bridges may be analyzed in terms of the voltage distri-very low flux density, and are referred to as current bution referred to the ratio side of the detector. Thistransformers. gives for the voltage at the detector.

1958 Thompson: The Precise Measurement of Small Capacitances 249

C. ~~~~~~~~~~~~~Vi

Fig. 10-Transformer bridge for the measturement vxof small direct capacitances. (a)

V,Vi Vi

Cl

VlYl+V2+V2 cxYo + Y1 + Y2 Vx

(b)

YbY2 ~~~~~~~~~~~~~~V2vz ~~~~~~~~~~~~~~~~2

Fig. 11I-Equivalent circuit of bridge shown in Fig. 10. Kr 3

Vlyl + V2y2 CxV0 = VxYO + Vi + Y2

At balance V0 = 0 and (c)Fig. 12-Transformer bridges for small capacitances (a) VariableY, V2 capacitor. (b) Variable ratio. (c) Multiple ratio fixed capacitors.

Y2 V1There are a number of ways by which the separate

P2/P1 capacitance and conductance balances may be obtained,if zi and Z2 are both zero. Small corrections due to finite and some of these are described in the next section.values of zi and Z2 are given with sufficient accuracy by Capacitance Balancethe relations

V, = pjE[1 - zi(ya + Y1) There are three main practical forms of the generalbridge, and these are illustrated in Fig. 12 for the

V2 = -p2E[1 -Z2(Yb + Y2)] capacitance balance. In the first of these a subdivided

so that capacitor is used with a single fixed ratio. The capacitormay be of the switched decade type and there is no

-= - [V2 P2 limit to the subdivision which may be continued by- - - [I + Z1(ya + yl) - Z2(yb + Y2)I. adding further decades. However, this requires tenV2 VI Pi

separate capacitors per decade if they are to be indi-More complicated bridges may be analyzed in the same vidually intercompared by substitution, so that such away, since in all cases capacitor becomes bulky and expensive. In the second

ZvE Yn, form a single fixed capacitor is used and the ratio is ad-V/ -utbe Th sudvso avial on asinl trans-

>'1 former is limited by the number of turns, and additionaland the bridge is balanced when transformers are necessary if subdivision beyond 1 in

-O ~~~~~~~1000is required. In the third form there is a limited sub-n-jV *~=0 division of the ratio, and a number of fixed capacitors

In general, the corrections due to finite transformer im- are switched separately among these subdivisions. Thepedances are of the same form as for the case above, but equivalent of a decade capacitor is obtained by sub-more terms are necessary since all the ratios are affected dividing one of the ratio windings into ten equal partsby a load on one winding, and switching a single fixed capacitor for each decade.

250 IRE TRANSACTIONS ON INSTRUMENTATION December

Combiniations of the above techniques may be used G2 yl2to obtain the best solution to a particular measurement G 2problem. I AAA-2 =

Conductance Balance cn C2A conductanice balance may be provided by tech- 3

niques similar to those used for capacitance, but since IF G = G2the phase angles of the capacitors and of the transformer C, C2ratios that we are considering are very small, a four-dial _G C2voltage divider and a fixed conductance standard are Ci+C2usually adequate. It is difficult to obtain a very small Fig. 13-T network for a small direct conductanceconiductance of good phase angle as a single component. of good phase angle.If a relatively large conductance is used then the sensi-tivity may be reduced considerably. The equivalent of La small direct conductance of good phase angle may be X I- 2obtained by the use of a T network such as the oneshown in Fig. 13. In practice G1 is a large conductance TCof good phase angle and Ci such that GK<<coCj. Under 3these conditions the magnitude of the direct conduct- IF G,L (c + 2) = = G C2ance 1 2

tG C2VICl+ C22 ,,(G1 ~~~~~~1+~~~~2) ~~2G, 2G, C2

is to a first order independent of the very small con-ductance G2 whose function is to provide a small phase Clangle correction. For operation at a fixed frequency G2 3 ,Tmay be dispensed with and the small phase angle correc-tion may be obtained in a number of ways, two of which 2 3are illustrated in Fig. 14. In this case the voltage on the IF C= 4 = G1C2capacitor C2 must be in quadrature with that of the (c1C 2) 2 +C2 C3main ratios. A source of quadrature voltage obtained Fig. 14-Direct conductance networks for fixedfrom an amplifier with appropriate feedback may be frequency operationi.used to replace the rest of the network, and this has ad-vantages in some cases.5 cies it is usually better to evaluate the loading errors of

the latter. A good ratio transformer has an effective im-ACCURACY pedance of less than 100 mQ, so a variation in the ground

In considering the accuracy with which two capaci- capacitances of 100 pF would produce an error of onlytors may be compared, the two cases of most interest 1 in 107 for a supply frequency of 1592 cps.are when the capacitances are equal and when they have With 3-terminal capacitors it is necessary to considera ratio of 10 to 1. The most accurate method for the the effect of the leads which are used to connect thecomparison of two nominally equal capacitances is by capacitor to the terminals of the bridge. A simplifiedsubstitution, and in this case very little is required of case is illustrated in Fig. 15 where a single impedancethe bridge other than adequate sensitivity and stability has been added to represent the impedance of the con-for the short time required to substitute one capacitor nection between the ground terminal of the capacitorfor the other. This substitution should be accomplished and the bridge. If the T network formed by Ca, Cb, andby switching the detector ends either to ground or to z is replaced by its mesh equivalent we obtain an addi-the detector so that the load on the transformer remains tional admittance of approximately -W2Ca,Cbz at theconstant. When measuring incremental values this may bridge terminals. If C1= C2 = 100 pF and z= 1 AiH thennot be possible. The effect of load variatio-is may be for w = 104 the capacitance error would amount to 1 ApF.eliminated by choosing a bridge circuit in which the This type of error may be avoided by providing thesupply is connected across the same ratio arm as the capacitor with two sets of coaxial connlections and usingcapacitors being substituted.fi However, such ratios are the bridge arrangement shown in Fig. 16. However, thisnot as stable as those obtained by using a separate ex- bridge does not solve the problem of the direct additioncitation winding, and for precise work at low frequen- of what are now the transfer admittances of two-

6A. M. Thompson, "<A bridge for the measurement of permit- terminal pair networks. The analysis given in the Ap-tivity,"' Proc. lEE, vol. 103, pt. B, pp. 704-707; November, 1956. pendix shows that two such networks may be connected

hA. C. Lynch, "A br-idge network for the precise measurement of in parallel so that the transfer admittances add directly,direct capacitance," Proc. IEE, vol. 104, pt. B, pp. 363-366; July, prvddteipdneotegonopiag1957. s 'poie h meac ftegon opi ag

1958 Thompson: The Precise Measurement of Small Capacitances 251C Y

02

CI2D

on the separate two-e Fig. 16-Comeparison of 3-terminal capacitors withCa 0b ~~~~~~~~separationi of the grou-nd leads.

Fig. 15-Three-terminal capacitor with one finite lead i fpedanceanid approxim-Fate equivalen-it circuit.

enyough. The ground loop impedance may be increasedconsiderably by threading the coaxial leads through a Ghigh permeability core, and this should have no effecton the separate two-terminal pair parameters. The addi- -____________tional admittance that results from the completion of a Fig t circuitfor a sml unane

to 109 pF nd isnot dtectale i a lo voltge brdgev ent crren1Eqivalen cicifor1amp Thsmlshuiaale mae

ground loop is given in the Appendix as Ya Ybrrrb/Zo. If of a capacitance bridge.reasonable values are substituted in this expression,namely, The noise produced by G at room temperature may bema = Yib = 100 pF ra = rb = 10Mn represented by an equivalent current generator whose

root mean square value is rmsi= 1.3 X10i10(Guf)t2. Ifand Zo = 1 mH, this additional admittance is equivalent G = 10bimho and the bandwidth Af= 1 cps this equiva-to 10r9pF and is not detectable in a low voltage bridge, lent current is 4 X -15 amp. This should be comparedThe impedance of each separate lead gives rise to the with a current of 10-14 amp produced by 1 volt across

same type of error as the impedance of the ratio arms. 1 ppF at a frequency of 1592 cps.This is a relative error which is only significant when A thermionic tube amplifier connected directly tomeasuring large capacitances and does not set a lower the bridge is the best detector for the measurement oflimit to capacitance measurements. small capacitances. If a grid resistor is used to control

For the comparison of unequal capacitances, the error the grid bias it should be of the order of t0 oohms. Thein the ratio of the transformer must be determined. If amplifier should be tuned and the usual precautionsthe nominal-ratio is n toi the error can be determined taken against microphony and interference. A very nar-by intercomparing n +1 equal capacitors by substitu- row bandwidth may be obtained by the use of a phasetion and then using n in parallel against the remaining sensitive detector and low-pass filter or integrating cir-one to form a known capacitance ratio of n to 1. The cuit. The measured noise characteristic of an amplifieraccuracy of this process requires the capacitances to be with selected tubes is shown in Fig. 18 as a function ofstrictly additive, and is limited by the stability of the the total capacitance at the input. It should be notedrelative capacitances. With similar capacitors and reason- that there are noise limitations to both the current andable temperature stability, an accuracy of1i in 106 is the voltage sensitivity. When large capacitances arereadily obtained. Small capacitors 10-100 pF are most being compared the voltage sensitivity is usually ade-convenient and with these the loading errors are small. quate, but greater sensitivity may be obtained by usingA shielded ratio transformer of the type shown in Fig. 9 a suitable coupling network between the bridge and themay be used as a standard to calibrate other ratios bY detector amplifier. The most direct way of increasingusing the ratio of a pair of capacitors as a transfer. the sensitivity is to increase the voltage supplied to the

SENSITIVITYc e tht rh l c obridge. Voltages up to about 100 are easily obtained, butSENoITIVITY ne t c is c u cabove this level the transformers become rather cumber-

If the type of bridge shown in Fig. 12 is initiallY some or their effective impedances are increased, andbalanced and then unbalanced by a small change in C, their loading errors become significant. The improve-

ofynisel inathis ciruit iste conductmhoance G=This mapiaFlmttin.hsensitivitylimitasuinteh isue aboutbe ofpFw

252 IRE TRANSACTIONS ON INSTRUMENTATION December

-13 4mm10 _o ~~10

E V E X E REFERENCE A 0. Imm

oO8 1IC4 Fig. 19-Three-terminal capacitanice probe for the nmeasurenmelntI-:, of mechanical displacements.

0- , 1 *,*.-EIo1 Y 2B

10 100 1000A B

INPUT CAPACITANCE pF 3

Fig. 18-Noise charaLcteristics of a thermionic tube amplifier; EO= Lofrequency 1592 cps, bandwidth 1 cps. I

applicatioiis become practicable. Apart from measure- Fig. 20-Network representing a 3-terminal admittanceapplications ~~~~~~~~~~~~~~withseparate coaxial leads and ground loop.ments where the capacitance itself is of primary interest,there are numerous possibilities for the application of3-terminal capacitance transducers to measurement circuited) the voltage drop in the input lead is (Y+ Ya)zaproblems. As an example, consider the sensitivity that and in the output lead YZb. The resulting current con-could be obtained with a capacitance probe for the tribution to i2 would bemeasurement of mechanical displacement. The probe -(Y + Ya)Zay - YZb(Y + Yb).could take the form of a small disk with guard as shownin Fig. 19. A disk diameter of 4 mm spaced 0.1 mm from Hencethe reference surface would give a direct capacitance of i21 pF. The gap would easily support 100 volts giving a YAB = - = Y[1 - Y(Za + Zb) - YaZa - YbZb] (1)sensitivity of 10-7 pF. This corresponds to a change of E

only 10-1 mm in the separation of the surfaces. Capaci- and this is the equivalent transfer admittance as an iso-tance probes of this type and others with spherical ends lated two-terminal pair network. The potential appear-have been particularly useful in the assembly of a pre- ing between terminals A' and B' is approximatelycise calculable capacitor where measuring sensitivities jira +t2rb where ra and rb are the resistances of the outerof a few microinches were required. They have the great conductors, since there is very little inductive couplingadvantage that they do not contact the reference to an external circuit including the outers, providedsurface. these are not too thick. Hence VA'B'W- (Y+ Ya)ra+ Yrb. If

the external circuit is now closed and Eo=0, the currentAPPENDIX in the ground loop is io = VA'B'jZO where Z0 is the im-

A 3-terminal admittaince defined at terminals 1, 2, pedance of the ground loop. The voltage drops in theand 3 is connected by coaxial leads to separate coaxial leads due to this current in the outers only are iora andterminations AA' and BB'. It is required to find the iorband the resulting addition to i2 is ioraY+iorb( Y+ Yb).transfer admittance YAB and the effect on this admit- Hence, the additional admittance due to connectingtance of an additional circuit joining A' and B'. This A'B' isnetwork, shown in Fig. 20, may be solved in terms of 1the loop currents indicated and associated loop im- - [Y(r(, + rb) + Yara][Y(r. + rb) + Ybrb]pedances, but there is some effort involved in reducing Zothe result. Because of the very small effect of the lead /ra + rb\ Yimpedances (usually <106),e a first approximation to = - [Y( +r)+Yr+Yr l-YrYr.the correction is all that is necessary. This can be ob-°°tamned by approximating the current distribution and By threading the coaxial leads through a high perme-hence the potential drops in the leads, and super- ability core, Zo can be made »>ra+rb and the first termimposing the current distribution that such potential is much smaller than the leads corrections given bydrops would give in the rest of the network. For 1 volt (1). The term YaYbrarb/Zo is independent of Y and mustapplied to AA' andl for io =0 the lead currents are be considered when very small capacitances are beingii~ Y+ Ya and i2~ Y. If the lead impedances are za and compared.Zb (as measured from one end with the other end short When two admittances are joined in parallel so that

1958 McGregor, et al.: New Apparatus at the NBS for Absolute Capacitance Measurement 255

the ground leads form a loop, the current in this grounid iniduced in the ground loop from stray fields, the result-loop is due to the difference between the two separate ing current in the ground loop would contribute thevalues of VA'B' that would be obtained before making equivalent of an additioilal admittancethe parallel connection. In this case, the additional ad-mittance independent of Y and Y' is z- [ Y(r, + rb) + Ybrb].

-[Yara - Ya'ra'F[Ybrb -Yb rb'] If the error aYbrbJZo is not to be detected, then theZ6 voltage induced in the ground loop should be less than

aind in some cases it may be worthwhile to balailce the about Zo/rbtimes the voltage senisitivity of the detector.leads and the ground admittances. If there is a voltage With a few turns oli a high permeability core Z0orb> 100.

New Apparatus at the National Bureau of Standardsfor Absolute Capacitance Measurement*

M. C. McGREGORt, J. F. HERSH$, R. D. CUTKOSKY§, F. K. HARRIS§, AND F. R. KOTTER§

INTRODUCTION work of Thompson and his group at the NationalT HE use of tightly coupled inductive ratio arms Standards Laboratory of Australia. By combining the

rather than resistive ratio arms in a 4-arm bridge best techniques for constructing ratio transformers,for the comparison of impedances was suggested completely shielded 3-terminal capacitors and detec-

by Blumlein' in 1928, and the use of a 3-winding trans- tors of high sensitivity, together with a cylindrical crossformer in such a bridge circuit was described by Starr2 capacitor as a calculable standard, there is now promisein 1932. Other bridges using Blumlein's principle have of being able to assign values to capacitance standardsbeen described by several workers3 in the past 30 years. comparable with, or perhaps even better than, the ac-Historically, it is of interest to note that conjugate curacy assigned to our present standards of electro-bridges making use of 3-winding transformers were de- motive foree and resistanee.scribed by Elsas4 in 1888 for resistance comparison and The present paper deseribes a transformer bridge eon-by Trowbridge5 in 1905 for capacitance and inductance structed at the National Bureau of Standards for meas-comparisons. uring the direct capacitance of 3-terminal capacitorsThus the basic principle of operation and the general ranging in values up to 1 yf and having a least count of

arrangement of transformer bridges have been known 1 lspf. Although the transformers and network com-for many years. However, the possibilities of such ponents described below were designed specifically forbridges for the precise comparison of very low value operation at 1 kc, the operation is by no means limitedcapacitors had never been fully exploited before the to this frequency. Voltage output of the ratio trans-

formers constitutes the most serious limitation at lowerfrequencies, but it is reasonable to suppose that, with

* Manuscript received by the PGJt,AUlUNsn,Auustr14,i1958 relatively minor modificationls, satisfactory operationt National Standards Lab., Chippendale, N.S.W., Australia. reailymnr odfctos,aifcoyopain$ General Radio Co., Concord, Mass. should be possible over the audio-frequency range to at§ National Bureau of Standards, Washington, D. C. least 10 kc.IBritish Patent No. 323037.2 A. T. Starr, "A note on impedance measurement," W. Eng. and While some of the present bridge components differ

Exp. W., vol. 9, pp. 615-617; November, 1932. sbtnilyfromtercutrat tNL tsol3 C H. Young, "Measuring inter-electrode capacitances," Bell substantially om their counterparts at NSL it shouldLabs. Rec., vol. 24, pp. 433-438; December, 1946. be understood that no more is involved generally than

H. A. MI Clark and P. B. Vanderlyn, "A. C. bridges with induc- modifications and in some cases improvements of de-tively coupled ratio arms," Proc. lEE, vol. 96, pp. 365-378; May,1949. signs already proven by Thompson and his group

C. XV1. Gatley and J. G. Yates, "Bridges with coupled inductive in Sydney.ratio arms for the comparison of standards of resistance or capaci- Thr a enltl eaie nomtoulsetance," Proc. IEE, vol. 101, pp. 91-100; March, 1954. Teehsbe itedtie nomto ulsetVA. ,M.OThompson, "A ,bridgBe for 7the measuroembent of pemt- up to now concerning these components, and the present

4A. Elsas, "Ueber Widerstandsmessungen mit demn Differential- paper must be considered primarily as a discussion of

ind5uctor,"Ann.riPhys., vnol. 3e5,dipfp. 82n8-a83t3r;ar18f8o8.mr e the constructional details and performance of the NBSvol. 20, pp. 65-76; 1905.trnfme-aibid.