concerning the major development testsmade on this new breaker. To amplifysome of the application aspects of thebreaker, would the authors comment on:

1. No mention has been made of thebushings. Are they of an ASA oil-filledtype?

2. In Fig. 1, it appears that the bushingprotective gaps are furnished. What aretheir spacings and what is the expected im-pulse and switching-surge flashover of thegaps?

3. Are interphase tank barriers used?

4. Do the authors think that it will benecessary to clean the powders of the arcproduct from the bottom of the tank?

5. What is the form of the tank seal thatwill assure no leakage of SFe gas?

6. Can the breaker gas-handling systemalso transport the breaker gas to containersfor maintenance?

7. Are the overload capabilities of thisbreaker such that they can be applied to atransformer of equal current rating, andthen can carry the overloads that are allowedon the transformer? As an example, undercertain conditions a transformer is allowedto carry twice its current rating for 1 hour.

8. Will this breaker be derated for otherthan a standard duty cycle?

9. It is noted in Table VI that the open-ing time of the breaker is 3.3 cycles. Canthis be reduced?

10. The opening times in Table VIII arevariable. Would the authors further de-scribe the reason for this variance?

11. Would the authors indicate on Fig.5 the time at which the high-frequencytransient starts its journey?

I believe the answers to the above ques-tions will be a great aid in applying thisbreaker.

R. E. Kane and R. G. Colclaser, Jr.: Mr.Beehler has raised several questions whichcan best be answered individually:

1. The bushings are SF6 gas-filled andare an integral part of the breaker.

2. Bushing protective gaps are furnishedwith a gap spacing of 22 inches; the breakerwill meet standard ASA dielectric withstandtests with 60-cycle and impulse voltages of160 kv and 350 kv, respectively.

3. Interphase tank barriers are not re-quired, as demonstrated by the tests out-lined in Table VIII and discussed in thesubparagraph entitled "Full-Pole Bias-Voltage Tests."

4. It is recommended that whenever thebreaker is opened for routine maintenanceany powders be removed from the breaker.

5. The main tank seal is of the 0 ringdesign, which has proved itself throughmany years of trouble-free service.

6. The breaker gas-handling systemcan be used to pump gas from within thebreaker to storage bottles by means of thevalving which has been provided.

7. As there are no standards for over-load capabilities regarding circuit breakersat this time, this question would necessitatespecific data before a complete answer couldbe given.

8. Whenever this breaker is used forother than a standard duty cycle it shouldbe applied in accordance with ASA Stand-ard C-37.7-1960.

9. The opening time of the breaker listedin Table VI is 3.5 cycles. This breaker israted a 5-cycle breaker as listed in the pre-ferred ratings; however, it is possible to re-duce this time by means of a differentmechlanical operating system.

10. The opening times listed in TableVIII are variable. As shown in the tablethe tests covered an extended period in thetest program (tests 86421-JN through86421-QM). During this time modifica-tions were made to the operating mecha-nism which only affected the dead time of thebreaker and did not affect the contact speedor gas flow through the breaker.

11. In Fig. 5, the time at which the high-frequency transient starts its journey is notreadily indicated on the oscillogram, whichalso indicates the arc voltage across thebreaker. However, the exact point can bedetermined by mathematically developingthe envelope of the transient.

Sunmmary: A nonlinear differential equa-tion describing the steady-state operationof a current instrument transformer isformulated based upon a Rayleigh hysteresisloop approximation of the magnetic coreeffects. Using a harmonic-balance tech-nique, a suitable approximate solution ofthis equation provides expressions for theRCF (ratio correction factor) and phaseangle errors for the transformer directlyin terms of the magnetic properties of thecore material and the transformer instru-ment burden.

Paper 63-103, recommended by the AIEE Trans-formers Committee and approved by the AIEETechnical Operations Department for presentationat the IEEE Winter General Meeting, New York,N. Y., January 27-February 1, 1963. Manuscriptsubmitted August 3, 1962; made available forprinting December 5, 1962.

JBROME MEISEL is with the Case Institute ofTechnology, Cleveland, Ohio.

The author gratefully acknowledges the helpfuldiscussions with Dr. Paul L. Hoover and theencouragement of Mr. Frank Filo during thepreparation of this paper.

THE measurement of alternating cur-rents, particularly at power frequen-

cies, generally employs a single range am-meter in conjunction with a current in-strument transformer. The transformeraccuracy requirements for this system in-volve two factors:

1. Transformer primary to secondarycurrent magnitude ratio.

2. Phase angle difference betweenprimary and secondary currents.1

These two considerations are applied in-dividually to each of the harmonics of theprimary and secondary current wave-forms. The purpose of the transformer isto alter the magnitude of the primary cur-rent by a specified factor, without signifi-cantly changing the relative phase dis-placement between the harmonics of theprimary and secondary currents.

The major obstacle preventing anexact analytic approach to the currentinstrument transformer problem is thenonlinear double-valued magnetic coreeffects. Before the degree of success at-tained by an instrument transformer canbe examined, an adequate description ofthe familiar hysteresis phenomenon en-countered with all magnetic core materialsmust be developed. Previous methodsgenerally assume primary current com-ponents at fundamental frequency ad-justed in magnitude to account for theenergy dissipated by these hysteresiseffects.2 An analytic description of this

Fig. 1. Rayleigh hysteresis loop

Meisel-Current Instrument Transformer Error Calculations

Current Instrument Transrormer ErrorCalculationsJEROME MEISEL

MEMBER IEEE

] 082 DECEMBER 1963

Fig. 2. Current transformer constructed witha toroidal core of rectangular cross section

hysteresis phenomenon is possible be-cause of the relatively low core flux densitylevels found in normal current trans-former operation. An approximate solu-tion gives ratio and phase angle errorequations useful as an aid for the designand development of current instrumenttransformers.

Nomenlature

A = cross-sectional area of the core, meters2Bm = maximum core flux density, webers/

meter2Fc= mmf applied to the core, ampere-turnsFc=coercive mmf, ampere-turnsFm = maximum mmf applied to the core,

ampere-turnsH,=coercive magnetic field, ampere-turns/

meterHm =maximum magnetic field applied to

the core, ampere-turns/meterIp= rms value of primary current, amperesIs= rms value of secondary current, amperesip=primary current function, amperesis= secondary current function, amperesLs = total secondary circuit inductance,

henrys= core mean path length, metersNp=number of primary turnsNs=number of secondary turnsR =nominal or desired transformer ratioRCF = ratio correction factorRs=total secondary circuit resistance, ohmst= time, seconds13= phase angle error positive if secondary

leads the primary, radiansAs= power factor angle of the secondary

circuita =hysteresis function4) = normalized flux level functiono.=maximum value of flux level function,

webers=actual flux level function, webers

w=frequency of primary current, radians/second

Analytic Description of HysteresisCore Effects

The purpose of this section is the de-velopment of a suitable analytic ex-pression describing the magnetic flux in aferro-magnetic material upon the applica-tion of a given magnetomotive force(mmf). The forcing functions are re-

stricted to single-valued periodic func-tions. Upon application of such a func-tion, interest is centered only upon thesteady-state periodic portion of the mag-netic flux response, since instrument cur-rent transformers are primarily used todetermine steady-state values of desiredcurrent functions.

Rayleigh, in investigating hysteresisloops in the region of very low flux densitylevels, noted that the loops seemed to bevery nearly a parabolic deviation from astraight line, as shown in Fig. 1.3 Instru-ment current transformers are most gen-erally operated in such a fashion that theRayleigh loop is an excellent approxima-tion. A suitable analytic expression forthe loop shown in Fig. 1 is given by4

Fc= Fm4)+i Fc(1 -4)2) (1)where Fm is the maximum applied mmf,Fc is the coercive mmf (see Fig. 1), F,is the mmf applied to the magnetic cor-e,and 4) is the normalized flux level function;thus

4() (t)(2)

where 4) is the actual flux level func-tion, and 4)m is the maximum value ofthis function. The plus or minus sign inequation 1 can be eliminated by using

d4c-+1 if t >0

dt

=-1 if-<0dt (3)

Notice that equation 1 exactly matchesthe actual loop when I4) =|1 (i.e., b= 4),D)and also when )b=0. For values of 4) be-tween these two extremes the equationyields sufficient accuracy for a widerange of sampled transformer cores.5Substituting the function described byequation 3 into equation 1 gives

Fc= Fmnb+crFc(1 4-2)as a description of hysteresis core efor low flux density levels.

Un

3E

EID

Fig. 3. Required coerciveand peak magnetic fields for aspecified peak flux density for

supermalloy

(4)

ffects

Transformer MMF Balance Equation

The circuital equilibrium equation forthe seconldary of the current instrumenttransformer is expressed by

Ls d+Rsis=Nsbmdtdt dt (5)

where Ls and Rs are the burden plusleakage inductance and resistance respec-tively of the secondary circuit, is is thesecondary current, and N. is the numberof secondary turns. * The steady-statesolution of equation 5 gives the secondarycurrent as

N5m e-RSt/Ls db dtis(t)= ,JeeRs*/Lsd (6)

Applying Ampere's circuital law inintegral form given by

JfH.dl=f,f J-d. (7)

to the current transformer cornfigurationshown in Fig. 2 gives

Fc = Npip- Nsis (8)

where Np and ip are the primary numberof turns and current respectively. Sub-stituting equations 4 and 6 into equation8 after some manipulation yields

i =Kie-Rat/Ls f eRst/LS -dt+4P dt

X4)+,E(1 -42) (9)

upon defining

K1= NI2L.mNpIpL,s

FmNplp

Fc=-NpIp

(10)

(11)

(12)

where Ip is the rms value of the primarycurrent function ip(t).

* For a precision current transformer a toroidalcore is generally employed to minimize leakageinductance.

1~~~~C.0ll_w==_:, _I111

IT --COERCIVE MAXIMU

I'

- - - FIELD(Hc FIELD (Hm)

.0--- I I

.03--- -.02 01S T

.01 .02.0C .05 .10 2 .3 .5 1.0 2 -3 5 10

H IN AMPERE TURNSMETERS

Meisel-Current Instrument Transformer Error CalculationsDECEMBE-R 19g63 1083

,1 IN MINUTES

Fig. 4. RCF and phase angle error limits forASA 0.5 accuracy class

Equation 9 is the desired equilibriumequation of the current instrument trans-former. Normally the primary currentip(t) is specified and interest is centeredupon the resulting secondary current is(t).Equation 9 can be used to determine thenormalized flux level function 4>(t) andthen equation 6 can be employed to findis(t). Equation 9 has (I as the dependentvariable, since this is the only con-venient way in which the hysteresis coreeffects can be formulated.The effect of eddy currents induced in

the core material is being neglected inthis analysis on the premise that propercare has been exercized in the laminationdesign. Hysteresis effects are a propertyof the magnetic material, and, for a givencore, with a specified flux density, cannotbe altered. However, in the case of eddycurrents the designer has the ability toreduce these effects until they are insigni-ficant. If the eddy current effects arenot negligible, then they can be includedas a component of the external trans-former burden.

Harmonic Balance Solution

An exact analytic solution of equation9 is most difficult. The nonlinear terminvolves the dependent variable to thesecond power in addition to a coefficientwhose value is a function of the time rateof change of this same dependent variable.The primary current forcing function isquite reasonably taken to be sinusoidal,thus

ip(t) =V2Ip sin (cut-,8) (13)

where cw is the radian frequency of thesystem and /3is an arbitrary phase anglein radians.

With the left-hand side of equation 9specified by equation 13 a steady-statesolution for 4(t) is required. If the de-vice is to operate successfully as a currentinstrument transformer, the secondarycurrent must be of the same form as theprimary current with a magnitude differ-ing only by a constant ratio. Since anarbitrary phase angle is inserted in equa-tion 13, the secondary current is taken asreference and is given by

is(t) = x/2IS sin cot (14)

Equation 14 is in effect proposing a solu-tion to equation 9 on the basis that a suit-able transformer should result. Furtherconsiderations will show the conditionsunder which this assumption is valid.

Substituting equation 14 into equation5 specifies the normalized flux level func-tion to be

4?(t)=-cosQt+tan-± Ls (15)

with the maximum value of the flux levelfunction given bv

'

(Rs2 +co2Ls2)"12coNs V2.I8 (16)

The time rate of change of the normalizedflux levelfunction obtained by differentiat-ing equation 15 is thus

d =W sin (wt+tan-'LsR) (17)dt /s

The hysteresis function, defined by equa-tion 3, is positive unity if

2irm< cot +tan -1 R-) <7r(2m+ 1) (18)

and negative unity if

2rm> ot+tan-l R-) >wr(2m-1) (19)

where m equals any integer includingzero. A Fourier series representation ofthe hysteresis function o- is given by

4 sin F(2a-1)(cot+tan'-R-)L]a=1 (2a-1)

(20)

Substituting equations 13, 14, 15, 17, and20 into equation 9 yields

sin (wt- 3)=NsI sin cot+

7A1X2+(3, ) sin cot+tan-l'R_37r ~ 3 Rs\

tan 8 J (21)8E

upon equating or balancing fundamentalfrequency components.

Equation 21 can now be solved for thephase displacement 3 between the primaryand secondary currents and also for therms magnitude of the secondary currentI.. After some trigonometric manipula-tion, assuming ,B to be a small angle, thesetwo coefficients are given by

Fm\2NpIp

-8 Fc sinl083w V2 NpIp

(22)

p- 11+8

c_CosOs +Is Np 3wr v/2 NpIpFm

sin Os) (23)

where 68=tan-' u;L8/R, and equations 11and 12 have been substituted for X and erespectively. Equation 22 expresses thephase angle in radians by which thesecondary current leads the primary cur-rent.The RCF for instrument current trans-

formers is defined to be6

RCF Ip/IsR

(24)

where R is the desired or marked ratio ofthe transformer. If the transformer isconstructed such that the turns ratio isequal to the desired ratio, meaning

R= Ns (25)

then from equation 23, the ratio correctionfactor is given by

/ 8 FcRCF= 1+3

- N 'cos s±+

mN sin Os) (26)

Equations 22 and 26 constitute the per-formance standards upon which the ac-curacy class of the transformer depends.

Approximate Error Equations

The core material data commonly avail-able often do not present hysteresisloops taken for the very low magnetizationlevels at which a current transformerusually operates. Thus, the coercive fieldH, as a function of the maximum coreflux density B,a as illustrated in Fig. 3,might not be readily available. A usefulapproximation to the error equations 22and 26 can be made by realizing that thebounds on coercive mmf are specified by

Fc>O

and

Fc< Fm

referring to Fig. 1.

(27)

(28)

Meisel-Current Instrument Transformer Error Calculations1084 DF-CF-MBF-R 1963

A conservative estimate for the phaseangle error of the transformer can bemade by using the equal sign of expres-sion 27, therefore,

X< - Cos0V\2 Np.Ip

(29)

This approximation is quite reasonablesince the second term on the right-handside of equation 22 is usually less than halfthe value of the first term, for most prac-tical core materials and secondary circuitpower factor angles 0,. A conservativeestimate of the RCF can be formulatedby using the equal sign of expression 28,thus giving

RCF<Ll+V NI K3 cos Os +sin Osj[ N.plp 3r )

(30)

Using equations 29 and 30 an estimateof the transformer errors can be made withonly a knowledge of the magnetizationcurve for the core material. This curvegives the maximum flux density Bmplotted as a function of the maximumvalue of the applied field Hm, and is gen-erally available. A practical example willshow how the complete equations 22 and26 are used and how the results of theapproximate equations 29 and 30 comparewith the complete results.

EXAMPLE

A current instrument transformer op-erating on a 60-cps (cycles per second)line is desired with a one-turn primary(Np= 1) and a 100: 5-ampere ratio.The total secondary burden, includingthe secondary winding, is 5 va (volt-amperes) at 0.9 power factor lagging.The core material is to be supermalloyin the form of a tape-wound toroid withan outside diameter= 15.2 cm (centi-meters), ID (inside diameter) = 10.2 cm.,and height= 5.08 cm. Determine thephase angle error and ratio correctionfactor for this design and compare theseresults with the ASA (American Stand-ards Association) limits for accuracyclass 0.5, as shown in Fig. 4.7

Cross-sectional area of the core is

A H(OD*-ID) 5.08(15.2-10.2)2 2

= 12.7 cm2

Mean length of the core is

7r(OD +ID) r(15.2 +10.2)2 2

=39.9 cm

From equationl 16,

(D=2 ZsIswNs

where

Zs =(RS2+ct2Ls2)1/2

V-\2 5 va )

, 5Mamperes27rX60X20 1.88X10webers

with the secondary turns, N2 =20 andZj, = secondary va/I8

.'m 1.88X10-4 weberBm = - = 0.148

A 12.7X 1°-4 meter2

Fig. 3 shows an approximate plot ofthe maximum magnetic field Hm and thecoercive field He as a function of the maxi-mum flux density for supermalloy.

For

weberBm= 0.148 -

meter2

Hm 0.82 ampere-turnmeter

and

He = 0.31 ampere-turnmeter

therefore

Fm =Hml = 0.82 XO.399 = 0.327 ampere-turn

Fc = Hcl = 0.31 XO.399 = 0.124 ampere-turn

From equation 22,

0.327 8 0.1241 X= 00X0.9--X X0.435V\-2Xioo0 3w VO~Xioo

3 = (2.08-0.33)X 10 -3 = 1.75X 10-3 radians

( = 6.01 minutes at 100% of rated current

From equation 26,

8 0.124RCF = 1+--X XO09+

0.327\/2xiooX 435

RCF = 1 +(0.67+ 1.01) X 10-3

RCF= 1.00168 at 100% of rated current

Table I summarizes the calculations for10% and 150% of rated current.

Plotting the values obtained for thephase angle error (3and for RCF, on theplot of Fig. 4 shows all values to be withinthe parallelogram from 10% to 150% ofrated current, thus the design is accept-able for accuracy class 0.5. Notice thatthe most critical point is at 10% ofrated current, as is typical with trans-formers designed with a nominal ratioequal to their turns ratio.

Table I

10% Rated 150% RatedCurrent Current

Bm.weber 0.0148 ....... 0.222meter2

HM ampere-turns. 0.138. 1.12meter

HC ampere-turns. 0.0317.... 0.428meter

Fm ampere-turns 0.0551............. 0.447Fe ampere-turns. 0.0126 ....... 0.171f3 minutes............ 10.9 ....... 5.48RCF ................. 1.00438. 1.00153

The approximate equation 29 for phaseangle error yields

0.327\/3< xX X0.9 radian

(3<7.16 minutes at 100% of rated current

Equation 30, for the RCF, gives an upperbound as

F 0.327 (8RCF<L1+\[ sX100,3-X 0.435)]RCF<1.00277 at 100% of rated current

These results for 3 and RCF provide anextremely simple method for checking aproposed transformer design

Conclusions

Equations for current instrument trans-former phase angle error and ratio cor-rection factor are developed. These twoperformance criteria, ( and RCF, can becomputed from a knowledge of the Ray-leigh hysteresis loops of the core material,the physical dimensions of the trans-former, and the electrical properties ofthe instrument burden. The simplicityof the calculation makes these equationsmost useful in the design of precisioncurrent instrument transformers.

References

1. INSTRUMENT TRANSFORMERS (book), B. Hague.Sir Isaac Pitman and Sons, Ltd., London, England,1936, pp. 7-10.

2. Ibid., pp. 40-41.

3. FERROMAGNETISM (book), R. M. Bozorth.D. Van Nostrand Company, Inc., Princeton, N. J.,1951, pp. 489-91.

4. ANALYSIS OF THE MAGNITUDE AND PHASERELATIONSHIPS IN MULTI WINDING CURRENTTRANSFORMERS, J. Meisel. Ph. D. Thesis, CaseInstitute of Technology, Cleveland, Ohio, June1961, pp. 7-22.

5. Ibid., pp. 22-26.

6. BASIC ELECTRICAL MEASUREMENTS (book),M. B. Stout. Prentice-Hall, Inc., EnglewoodCliffs, N. J., second edition, 1960, pp. 396-99.

7. AMERICAN STANDARD FOR INSTRUMENT TRANS-FORMERS. Standard C-57,13-1948, AmericanStandards Association, New York, N. Y., p. 12.

Meisel-Current Instrument Transformer Error Calculations

* Outside diameter.

DECEMBER 1963 1085