Inductive Voltage Dividers with Calculable Relative Corrections

80 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT September

for this effect (14) is modified to the form

Vmn = kpa8w66s7T6I (17)

where k is a dimensionless constant somewhat less thanf. For the experimental unit f=0.83 and k=0.68.

The gage output is limited by the allowable powerdissipation. The input resistance is given by R, C=pCi,where C, is a dimensionless factor depending on thesample and contact dimensions. Expressing V7mn interms of input power Pi, we obtain

Vmn = Pos27r66s'T6k(Pin/Cl)'12. (18)

Thus a high Pos is desirable for a high output voltage.The equivalent circuit for the output terminals is a volt-age source Vmn, given by (17), in series with an output

resistance R2= C2p,,, where C2 is determined by thesample and contact dimensions. If the output terminalsare connected to a load resistance, the output power ismaximum when the load equals R2. The ratio of outputto input power for this condition is

Pout (k7r66' T6) 2

Pi, 4ClC2(19)

For the experimental gage, resistance measurementsgive C,C2,= 1.56. Then if T6= 5 X 101 dynes/cm2 andPin-=0.1 w, the calculated power output is 1.3X10-5watt. This power level is sufficient to give a full-scaledeflection on an inexpensive D'Arsonval meter. An ap-proximate analysis of the uniformly doped torque gageindicates that its available output power for the sameT6 and Pi, is an order of magnitude less.

Inductive VoItage Dividers with CalculableRelative Corrections*

THOMAS L. ZAPFt, SENIOR MEMBER, IEEE, CARL H. CHINBURGt, SENIOR MEMBER, IEEE,AND HARRY K. WOLF§, SENIOR MEMBER, IEEE

Summary-Inductive voltage dividers are now used in measure-ment laboratories for the production of audio-frequency voltageratios with errors (deviations from turns ratio) of a few parts in tenmillion of input. A major component of error arises from the interac-tion between distributed shunt impedances and leakage impedancesin the windings. Correction for this systematic error in dividers ofspecial design can result in an order-of-magnitude improvement inaccuracy.

A solution using network equations has been obtained for the cor-rections to the relative errors of inductive voltage dividers of specificdesign. Earlier theoretical considerations, confirmed by measure-

ments of limited accuracy, indicated an S-shaped curve of ratioerror vs nominal ratio, and quadrature component vs nominal ratio.The results from recent calculations are in agreement with the earliermeasurements and provide much better definition of the true shapeof the S-curve. An algebraic equation has been derived for the limit-ing form of this S-shaped characteristic curve. A resistance analogof an inductive voltage divider was constructed to represent thelumped circuit parameters equivalent to the distributed shunt ad-mittances and the winding impedances. Measurements of the grosserrors in the analog have yielded experimental results in excellentagreement with those calculated from the network equations.

* Received May 23, 1963.t National Bureau of Standards, Boulder, Colo.Colorado State University, Fort Collins, Colo.

§ California State Polytechnic College, Pomona, Calif.

INTRODUCTION

TvHE EXCELLENCE with which sinusoidal alter-nating-voltage ratios can be preserved by meansof inductive voltage dividers has stimulated con-

siderable interest in the establishment of more accuratevoltage ratios. Particularly successful results were ob-tained from an investigation of inherent systematicerrors in inductive voltage dividers of a specific design.This investigation was undertaken to account for cer-tain observed systematic errors' and to provide a basisfor correction or compensation.The study was confined to inductive voltage dividers

consisting of an even number of equal sections wound ona single core. A sinusoidal voltage applied to the ex-tremities of a divider having s equal sections is dividedinto s phasor components in series, each spanning a sec-tion of the divider. These phasor components may be re-solved further into two parts, a voltage induced in thewinding of the section by the resultant mutual flux, anda voltage drop produced by the current in the leakage

1 "Theoretical analysis of accuracy of ratio transformers," Engi-neering Bulletin No. 1 (Ratio Tran), Gertsch Products, Inc., LosAngles, Calif.

1965 Zapf, Chinburg and Wolf: Inductive Voltage Dividers with Calculable Corrections

inductance and effective resistance of the section. Thedivision of the voltage is ordinarily considered to be ac-curate if the voltage ratio is equal to the ratio of turnsin the windings. Accuracy is adversely affected by in-complete and disproportionate coupling among the sec-tions with consequent disproportionality among theleakage inductances, and by the concomitance of seriesimpedances in the sections with internal loading fromshunt admittances between sections or with externalloading.2A winding dlesign was used that practically eliminated

the systematic disproportionality in coupling arisingfrom core inhomogeneity; the remaining coupling errorsresult only from small residual irregularities in coupling.Methods of using inductive voltage dividers are avail-able in which errors from external loading can be madeinsignificant or correctable. This leaves the effect of in-ternal loading to be explored. Internal loading is usuallyof no concern at the very low audio frequencies, but cancause significant errors at 1000 cycles per second, and athigher frequencies can severely limit the accuracy ofvoltage division. The internal loading errors result fromvoltage drops in the distributed series impedances pro-duced by currents from the distributed shunt admit-tances. It is this source of error that was examined. Theanalysis was accomplished by developing a lumped-parameter equivalent circuit comprising representativefactors contributing to internal loading effects, writingthe independent node equations, and mathematicallysolving the equations for the voltage-ratio corrections.

DESIGN CRITERIA

The component of input voltage across each section ofan inductive voltage divider is impressed on the effec-tive resistance and leakage inductance of the section inseries with the electromotive force induced in the sec-tion by the resultant mutual flux. Between the sectionwindings there exists distributed capacitance and con-ductance which allow current to by-pass elements of thewinding. The exact distribution and magnitude ofthese series and shunt parameters depend on the spe-cific design, which mav be selected to permit represen-tation by tractable forms of equivalent circuits.The winding design chosen for analysis may be de-

scribed concisely as a single-layered, toroidal helix ofbundled wires of equal length and diameter in whichthe wires are intermingled randomly along their length.The bundle is usually twisted to improve the mutualproximity of the wires and to facilitate handling andwinding. After the bundle is wound on an insulated,high-permeabilitv core the ends of the wires in thebundle are separated as close to the core as possible, ap-propriately paired, and joined in series to form an induc-tive voltage divider having nominally-equal sections.

2 T. L. Zapf, "The calibration of inductive voltage dividers," In-strument Society of America Transactions, vol. 2, pp. 195-201; July,1963.

Before the sections are connected together each sec-tion winding is electrically indistinguishable from anyother section winding, and the equivalent leakage fluxof each section is practically equal to that of every othersection. Pursuing this analysis to a greater depth, it canbe reasoned, for example, that the leakage flux cuttingtwo section windings at any one cross section of thebundle has equal counterparts coupling each of thesetwo windings to every one of the other windings at othercross sections. This network of flux components con-tributes to the equivalent leakage flux of each sectionas well as the net mutual flux, but there are no forcestending to create systematic inequalities among theleakage inductances of the sections. Furthermore, thewinding of every section makes a single complete helicalexcursion around the core, thus reducing to a minimumthe net effect on leakage inductance of winding positionaround the core and magnetic inhomogeneity of thecore material.The random intermingling of wires in the bundle not

only achieves uniformity of magnetic coupling amongthe sections and consequent uniformity in the leakageinductances but also insures uniformity in the dis-tributed shunt capacitances and conductances amongthe sections. These admittances may be lumped and,for convenience, considered as terminating at the junc-tions of the sections rather than at ill-defined pointswithin each section. This convenience is not withoutdisadvantage, but the disadvantage can be compen-sated in practice as explained later. In practice, com-plete uniformity in the series and shunt parameterscannot be achieved and the lumped-parameter conceptis not without limitations; nevertheless, this approachto a satisfactory, equivalent circuit has been notablysuccessful.

Because of the considerable length of the wires andtheir proximity within the bundle it is reasonable toexpect that the shunt admittances are quite large rela-tive to the admittances from these wires to the core orto the case surrounding the divider. The effect of thelatter adnmittances are of secondary significance to thisinvestigation and will not be treated here.

EQUIVALENT CIRCUIT

The distributed magnetic and electrical propertiesdiscussed in the preceding section can be represented bylumped parameters. These lumped parameters may beused to develop an equivalent circuit from which may bederived the in-phase and quadrature components of thevoltage-ratio phasor. The equivalent circuit and deriva-tion of corrections are discussed in general terms byZapf.2 The in-phase voltage-ratio correction is a func-tion of the product of the leakage inductance and shuntcapacitance as well as the product of winding resistanceand shunt conductance. The quadrature component is afunction of the product of the leakage inductance andthe shunt conductance as well as the product of the

81


winding resistance and the shunt capacitance. The leak-age inductance and the winding resistance are similarlydistributed, and the shunt capacitance and the shuntconductance are similarly distributed. Hence, the char-acteristic curves for both in-phase corrections andquadrature components of the voltage-ratio phasor havethe same shape. For purposes of analysis it is sufficientto consider only the product of the winding resistanceand the shunt conductance, which is the basis of theequivalent circuit shown in Fig. 1. The equal voltagesinduced in each section are designated by the letter "e".The resistors in series with the induced voltages repre-sent the winding resistances. The array of resistorsdrawn between the taps represents the distributed shuntadmittances between the sections. These are drawn asequal resistors between the taps. There are 21 taps, butonly 20 sections between the end taps to contribute tothe shunt admittance. The additional necessary admit-tances required by the equivalent circuit are easily ob-tained in practical construction. This is done by includ-ing in the bundle an extra wire, the ends of which areconnected to the 0 and 1 taps. The midpoint of thisextra wire is cut within the bundle and the two endsthus formed are separated and left unconnected. Theextra shunt admittances provided by these "compensa-tion tails" at the end of the divider, above 1 and below0, approximate the admittances required by the equiva-lent circuit of Fig. 1. With end effects reduced by thiscontrivance, the results of analysis of a 20-sectionequivalent circuit may also be considered as closelyrepresentative of dividers having 10 sections.The errors introduced by omitting the compensation

tails are very small and approach zero as a limit whenthe series impedances or shunt admittances approachzero. In practical construction of inductive voltagedividers, in which the series impedances are small, thecompensation tails can be omitted, and the limitingform of the characteristic curve described in this papercan be applied with negligible error.The terminals, or taps, are identified by decimal frac-

tions corresponding to the nominal output-to-inputvoltage ratio, the zero end of the divider being the com-mon reference. The voltage ratio at each tap is shownas a nominal voltage ratio plus a correction. Because ofthe symmetry of the circuit the numerical corrections tocomplementary voltage ratios (e.g., 0.4 and 0.6, 0.15and 0.85, etc.) are equal in magnitude, but of oppositesign. The corrections at 0, 0.5, and 1 taps are zero inthis equivalent circuit. Thus a plot of corrections vsnominal ratio falls on an S-shaped curve. The exactform of the S-curve is characteristic of the design. TheS-shaped characteristic curve can be normalized todepict the magnitude of the various corrections relativeto a reference correction which may be chosen arbi-trarily. The correction at the 0.2 tap was selected as thereference for the relative corrections described in thispaper because the correction at the 0.2 tap is near themaximum point of the characteristic curve. The abso-

SERIESRESISTORS, a SHUNT RESISTORS, b

VOLTAGERATIO1.0

0 0.95 - C 05'0.9 - C0.85-C.15O.8 -C 20.75 - c 25'0.7 -C'30.65 - C 35

D 0.6 -C40.55 - C 4

0.50.45 + C 45'04 + c04D 0.35 + C 350.3 + c 3O 25 + C 25'02 + C 20 15 + C ,5

'.O + CD 0.05 + C 05D0

Fig. 1-An equivalent circuit for a 20-sectioninductive voltage divider.

lute corrections for different dividers of the same specificdesign may be somewhat different, but the relative cor-rections for all of these dividers will be similar.

CIRCUIT EQUATIONS

A mathematical calculation of the relative correctionsfor a 20-section divider has been made bv solving theindependent node equations written for each nonzero-correction tap of the divider. The corrections sought areC.Oa5, C,C15, * C.45 The input voltage, applied to theextremities of the divider, was considered as unity forsimplicity. The common, or zero, tap was chosen as thereference node and nine equations were written for thetaps 0.05 through 0.45. The voltage and correction atthe reference node are both zero.

Fig. 2 shows the equivalent circuit for writing thenode equation at the 0.2 tap. The equivalent voltagesources shown in Fig. 1 have been converted to equiva-lent current sources in Fig. 2. In the node method anegative sign is prefixed to the current (voltage-conduct-ance product) terms involving mutual conductancesbetween the node under consideration and other nodes,and a positive sign is prefixed to the current (voltage-conductance product) terms involving the self con-ductance of the node under consideration. For example,when the applied voltage is unity, the node equation is

-(0.05 + Cs0)g - (0.1 + c.i)g- (0.15 + c.15)(G + g) + (0.2 + c.2)(2G + 20g)

-(0.25 + c.25)(G + g)- (0.3 + c.3)g - (0.35 + C.35)g - (0.4 + G.4)g

- (0.45 + C.45)g - 0.5g - (0.55 - C.45)g- (0.6 - C4)g - (0.65 - C,35

- (0.7 - C.3)g - (0.75 - C.25)g - (0.8 - C.2)g- (0.85 - c.i5)g

- (0.9 - c.)g - (0.95 - c.05)g- g = eG- eG = 0. (1)


I

(JV G = I/a

(F G= I/C

+

III F I - A;=: ? LL LLIL LLIH LL

- 1.0- 0.95 -

0 0.9 -

0 0.85-- 0.8 -

° 0.75 -o 0.7 -

- 0.65 -o 0.6 -

- 0.55 -

- 0.5- 0.45 +- 0.4 +- 0.35 +- 0.3 +- 0.25 +

°0.2 +. rN c; -u , _.1_I I I . 0 U.lb

= e/a , oO.I

o 0.0

Fig. 2-Equivalent circuit for writing thenode equation at the 0.2 tap.

+

By expanding and collecting terms the equationduces to

-Gc.15 + (2G + 21g)c.2 -Gc 25 = 6.3g

or

-C.15 + (2 + 21k)c.2 C.25= 6.3k

AGE where K=(2+21k) in this 20-section divider. The

C*5 equations can be solved through the use of Cramer'sC., rule in which A appears as the denominator in all of theC.'c25 solutions. The numerator of each solution is formed byC 35 replacing the appropriate column in A by the column ofC.35 terms on the right side of equations (4). The relativeC 45 corrections were obtained directly from the numeratorsC45 by dividing each of them by the numerator for c.2. ThisC4 procedure was followed with each set of equations ob-C.3C 35 tained for values of k= 10-4710 5, 10-6, - 10-6, -10-5,C 25 -10-4, and for the limit as k- 0. The results are tabu-

C2

C15 lated in Table I.C.05 As the value of k varies, the relative corrections

change by small amounts. These changes are made evi-dent bv subtracting the relative corrections for k-*0from the relative corrections for k of other magnitudeslisted in Table I. The results are shown in Table II. The

re- data for k= 160- are plotted as a function of nominalvoltage ratio in Fig. 3. One should not use the differ-ences shown in Table II without warning that the

(2) equivalent circuit may not be adequate to justify highlycritical use. However, the trends shown in Table II, andthe double-S-curve in Fig. 3, are of qualitative interest.

(3)

where k = g/G is the ratio of the series to the shunt im-pedances. Similar equations can be written for each ofthe other taps. The complete array of equations is

(2 + 21k)c.os- C.1 = 9.45k

-c.05 + (2 + 21k)c.i -c.15 = 8.40k

-c.1 + (2 + 21k)c.1s-C.2 = 7.35k

-C.15 + (2 + 21k)c. 2 - C.25 = 6.30k

-C.2 + (2 + 21k)c.2s- C.3 = i.25k

-6.25 + (2 + 21k)c.3 - C.35 = 4.20k

-C.3 + (2 + 21k)c.35- C.4 = 3.15k

-C635 + (2 + 21k)c4 - C.45 = 2.10k

-C,4 + (2 + 21k)c.45 = 1.05k.

For any specified value of k the above equations can

be solved by determinants. The determinant of theabove set of equations is

K -1 0 0 0 0 0 0 0

-1 K -1 0 0 0 0 0 0

0-1 K -1 0 0 0 0 0

0 0-1 K -1 0 0 0 0

0 0 0-1 K -1 0 0 0

0 0 0 0-1 K -1 0 0

0 0 0 0 0 -1 K -1 0

0 0 0 0 0 0 -1 K -10 0 0 0 0 0 0 -1 K

RESISTANCE ANALOGAn analog voltage divider was constructed in con-

formance with the circuit shown in Fig. 1 to obtain, bymeasurement, the relative errors at the various taps.Initially a 10-section analog was constructed in whichthe balanced shunt resistances were omitted, as it wasexpected that their contribution to the error measure-

ments would be negligible. The immediate success ofthis analog stimulated the construction of a 20-sectionanalog voltage divider of similar designi, but with bal-anced, as well as unbalanced, shunt resistances includedfor completeness.

In both analogs the voltage sources for the sectionswere obtained from electrically insulated windings on a

common toroidal core. The series elements were 10-ohm,5 per cent, carbon resistors which were chosen forequality within a range of 2 per cent. The shunt ele-ments for the 10-section analog were 10,000-ohm, 5 per

cent, carbon resistors; and for the 20-section analog theshunt elements were 100,000-ohm, 5 per cent, carbonresistors. No attempt was made to match the shunt ele-ments, the great number of them tending to reduce theeffect of individual deviations to a large extent.The analog dividers were calibrated with a standard

inductive voltage divider by a comparison method. Thevoltage-ratio errors in the standard divider were neg-

ligible in comparison with the gross voltage-ratio errors

exhibited by the analog. These large errors were meas-

ured, the errors at complementary ratios averaged with-out regard to sign, and then normalized by dividing bythe mean error at the 0.8 and 0.2 taps. The results are

shown in Table III. It is interesting to compare the ana-

log results shown in Table III with the mathematicalresults in the k = 10-4 column of Table I.

I - e/a

p

T

t, t

I

I- II

?

ti~ +_ t t t t ' Xllb ' .........il I I I I + + 'I

I- 'q= I/b


TABLE IRELATIVE CORRECTIONS FOR VARIOUS VALUES OF k

CreI

kk=10-4 k=10-6 k=l10-6 k->0 k=-10-6 k=-10-6 k=-10-4

0.05 0.447 409 0.445 523 0.445 334 0.445 312 5 0.445 291 0.445 102 0.443 2010.1 0.752 195 0.750 220 0.750 022 0.750 000 0 0.749 978 0.749 779 0.747 7850.15 0.930 950 0.929 814 0.929 700 0.929 687 5 0.929 675 0.929 561 0.928 4120.2 1.000 000 1.000 000 1.000 000 1.000 000 0 1.000 000 1.000 000 1.000 0000.25 0.975 442 0.976 450 0.976 551 0.976 562 5 0.976 574 0.976 675 0.977 6980.3 0.873 175 0.874 816 0.874 982 0.875 000 0 0.875 018 0.875 184 0.876 8500.35 0.708 936 0.710 736 0.710 917 0.710 937 5 0.710 958 0.711 139 0.712 9680.4 0.498 332 0.499 832 0.499 983 0.500 000 0 0.500 017 0.500 168 0.501 6930.45 0.256 872 0.257 718 0.257 803 0.257 812 5 0.257 822 0.257 907 0.258 767

TABLE II

DIFFERENCE BETWEEN THE RELATIVE CORRECTIONS FOR k OF VARIOUS MAGNITUDES AND THE RELATIVE CORRECTION FOR k-0O

k=l10- k=10- k=10-6

0.05 0.002 0970.1 0.002 1950.15 0.001 2620.2 0.000 0000.25 -0.001 1210.3 -0.001 8250.35 -0.002 0010.4 -0.001 6680.45 -0.000 940

q}

z0

LULcr0

LLI>

-LJ

0.000 2100.000 2200.000 1270.000 000

-0.000 113-0.000 184-0.000 201-0.000 168-0.000 095

0.000 0210.000 0220.000 0130.000 000

-0.000 011

-0.000 018-0.000 020-0.000 017-0.000 009

k=-10Of

-0.000 021-0.000 022-0.000 0130.000 0000.000 011

0.000 0180.000 0200.000 0170.000 009

+1.2+1.0 _ LIMITING FORM

+0.8 / > s/ AS k-O_+0.6 _

+0.4 /

+0.2 -

-0.2 ---0.4 - DIFFERENCES FROM t

-0.6 _ LIMITING FORM FOR _

-0.8 - k- IO-6 MULTIPLY

-1.0 SCALE AT LEFTX 10- ._-12_sI,I, I, I, ~~~~~~~~~~~~~~~I I I I _

-.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1NOMINAL RATIO, AN

k=-10-5

-0.000 210-0.000 221-0.000 1270.000 0000.000 1130.000 1840.000 2020.000 1680.000 095

k=-10-4

-0.002 112-0.002 215-0.001 2760.000 0000.001 1350.001 8500.002 0300.001 6930.000 955

Fig. 3 Relative corrections for a 20-sectioninductive voltage divider.

TABLE IIIRELATIVE CORRECTIONS FOR A 20-SECTION INDUCTIVEVOLTAGE DIVIDER DERIVED FROM MEASUREMENTS ONAN ANALOG CIRCUIT. THE ESTIMATED UNCERTAINTY

IS ± 1 PER CENT OF THE CORRECTION

Nominal Ratio Relative Correction

0.05 0.4480.1 0.7540.15 0.9320.2 1.0000.25 0.9760.3 0.8730.35 0.7070.4 0.4980.45 0.255

AN-1 1.. 1..--

CreI-Cre k-

.0


LIMIT EQUATIONThe internal loading effect in practical inductive volt-

age dividers is very small indeed, and it is interestingand informative to obtain the solution to the limitingform of the relative corrections as k approaches zero. Ifthe series impedances could be reduced to zero, or theshunt impedances could be made infinitely large, theerrors from internal loading of the type described wouldbe zero. In practical cases neither of these conditionshas been attained. For simplicity, a 10-section dividerwas chosen to illustrate the procedure for solviiig thesimultaneous equations to yield the limit as k ap-proaches zero. The four independent node equations ofa 10-section divider are

(2 + llk)c. -C.2 = 4.4k

-c., + (2 + llk)c.2 - C.3 = 3.3k

-C.2 + (2 + llk)c.3 - C4 = 2.2k

-C.3 + (2 + llk)C.4 = 1.1k.

The relative corrections are

C.i 4K3+ 3K2 6K--2

C.2 3K3+ 6K 2- 2K--4

C.2-=1

C.2

C.3 2K3+ 4K2+ 2K-1

C.2 3K3+ 6K 2- 2K--4

C.4 K3 + 2K2+ K + 2=_ ~~~~~~~~~(7)C.2 3K3+ 6K 2-2K-4

where K= (2+ I 1 k) in this 10-section divider. The limit-ing ratios as k approaches zero (K approaches 2) are

C.1/C.2= 30/40 = 0.75

C.2/C.2 = 40/40 = 1.

C.31C.2= 35/40 = 0.875

C.4/C 2 = 20/40 = 0.5. (8)

A complete set of limiting ratios for a 20-section dividerare shown in Table I in the column headed k-*0, andthe relative corrections are plotted as a function of nom-inal ratio in Fig. 3.The arravs of independent node equations (4) and (6)

are shown for 20-section and 10-section dividers. In theprocess of developing these equations guidelines wereformed for writing directly the equations for an s-sectiondivider (s even). It may be helpful to describe the proc-ess. On the left side of equations (4) and (6) the sub-

scripts on the c terms are for identification purposesonly. Note the orderly progression of the c subscriptsand that the coefficient of the first and third terms is- 1. The corrections co and C.5 are zero and are omittedfrom the array. The coefficient of the middle term is2-+ (s-+-1)k. Notice the orderly progression of the multi-plier of k on the right side of the equation. Each succes-sive multiplier, progressing upward from the bottom, isan integral multiple of the bottommost multiplier,which is (s+ 1)/s. Using these simple guidelines thearray of equations for any divider having an even num-ber of equal sections can be written without difficulty.The plotted points of the k->0 curve in Fig. 3 indicate

that the curve is prima-facie cubic. A generalized cubicequation was written in which the relative correction,Crel, was a cubic function of nominal ratio, AN. Coeffi-cients of the A v terms were obtained by the simulta-neous solution of a set of three independent equations ofpoints on the S-curve of Fig. 3. It is unnecessary todetail the method because the resulting cubic equationcan be checked quite easily by substitution of data. Theequation is

Crel = 20.833 ... AN(i - AN)(0.5 - AN) (9)

and describes the limiting form of relative corrections ask approaches zero. In practical inductive voltage di-viders, in which k is very small, (9) provides a veryclose approximation to the actual relative corrections.

CONCLUSIONThe relative-error characteristic of intrinsic loading in

a specific design of inductive voltage divider has beenderived from both analog measurements and a mathe-matical analysis of equivalent circuits. The limitingform of the S-shaped characteristic has been identifiedas a cubic equation. The mathematical analysis is par-ticularly valuable because it provides the excellent reso-lution necessary to determine the nature and magnitudeof higher-order effects which otherwise could not havebeen discerned.

For a particular inductive voltage divider of thespecified design the relative-error characteristic com-bined with the results of appropriate measurements bycomplementary methods2 yields an approximation tothe absolute errors. Compensation or correction of thesystematic portion of these errors reduces the limits ofuncertainty to a level commensurate with the uncer-tainty from unpredictable irregularities in the magneticand electrical parameters. Thus, the results from thisinvestigation account for a major source of intrinsic sys-tematic error in both the in-phase and quadrature com-ponents of the voltage-ratio phasor.

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Inductive Voltage Dividers with Calculable Relative Corrections

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