current transformer model

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007 187

Current Transformer ModelFrancisco das Chagas Fernandes Guerra and Wellington Santos Mota, Senior Member, IEEE

Abstract—This paper presents a current-transformer (CT)model that is useful for low-frequency applications. To describethe iron-core magnetic behavior, a hysteresis model is proposed,which is able to generate minor asymmetric loops and remanentflux. The effects of classic eddy current losses and anomalous lossesare represented by linear and nonlinear resistors, respectively.The obtained results are compared with those calculated by thePreisach’s model and measured in the laboratory. This modelmay be applied in power system protection studies, as it is the caseof numeric correction of distorted secondary currents in currenttransformers (CTs).

Index Terms—Current transformers (CTs), hysteresis, magneticlosses, power system protection.

I. INTRODUCTION

THE fault currents in electric power systems present a si-nusoidal component plus a dc component with exponen-

tial decline. The first produces a sinusoidal magnetic flux in thecurrent-transformer (CT) core. The second produces an initiallyincreasing flux, with a subsequent decline. The total flux, as-sociated with the remanent flux, can produce a high saturationlevel in the core. This fact usually causes severe distortions inthe secondary currents supplied to the protection relays. In con-sequence, the following problems can take place.

1) The relays can operate inadequately.2) The relays are not sensitized due to the distortions that

reduce the root-mean-square (rms) value of the secondarycurrent.

3) The relay operations can be delayed, for the reason men-tioned in the previous item.

4) The fault locators do not show correct indication.Those occurrences can cause thermal and electrodynamics

damages, loss of coordination in the protection relays, and diffi-culty of location of the faulted point, or loss of system stability.

In the literature, there are many works related to the in-fluences of the CT’s distorted secondary currents, as well asthe consequences of those distortions in the protection relaysperformance. However, in these works, the iron-core nonlinearproperties are not always modeled with sufficient accuracy.The magnetization characteristics are usually represented bysimple models, using the saturation curve.

The importance of the magnetic hysteresis representation isa point to be investigated, because it determines the values ofresidual flux in the CT magnetic core. Such phenomenon is moreimportant to be taken into account in the case of nongapped CTs,used in schemes with automatic reclosing.

Manuscript received June 14, 2005; revised October 28, 2005. Paper no.TPWRD-00353-2005.

The authors are with the Universidade Federal de Campina Grande, CampinaGrande PB 58109-095, Brazil (e-mail: [email protected]; [email protected]).

Color versions of Figs. 4–13 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRD.2006.887092

Some works that consider the hysteresis effects in CTs usemodels in which important nonlinear properties are not takeninto account. The model proposed in [1] is not able to reproducethe real behavior of the ferromagnetic materials.

In other works, studies of the CT’s transient behavior usingan Electromagnetic Transients Program (EMTP)/AlternativeTransients Program (ATP) model have been made [2]–[5]. Thismodel represents the core magnetization branch by a hystereticinductor (type 96). It was developed by [6], based on somemodifications made in the model proposed by [7].

A practical CT model was used in the EMTDC program [8](EMTP-type software developed by Manitoba HVDC ResearchCenter, Winnipeg, MB, Canada), in which the magnetizationbranch characteristic was represented by a nonlinear inductorwith a flux-current characteristic described by a nonintegerpower series. The eddy current loss and the hysteresis loss(associated with the Steimetz coefficient) are represented by anonlinear resistor in parallel to the magnetization branch. Thismodel has produced results close to field tests.

Mathematical methods of magnetic core modeling are re-vised by the IEEE Power System Relaying Committee [9]. Inthis paper, four models are presented, including those used inEMTP/ATP and EMTDC. In the Jiles–Atherton model, the an-hysteretic curve is represented by the Langevin function andother functions.

A novel method to improve the Jiles–Atherton model is pro-posed [10], where rational functions are used to represent theanhysteretic curve.

In order to use the Jiles–Atherton model, it is necessary todetermine five input parameters. There are several methods tosolve this problem, based on physical meaning [11] and trialand error [12]. However, these methods are very laborious. An-other way to determine the input data is to use the least squaresmethod [13] or the simulating annealing technique [14]. There-fore, if an efficient routine is not available, the difficulty to iden-tify the parameters is a disadvantage of this model.

In [15], a correction technique of the distorted secondary cur-rents is presented based on a model that takes into account hys-teresis and eddy current effects. However, the hysteresis modelis not able to reproduce the properties of closing and elimina-tion presented by the minor loops [16]. The model applicationswere limited to steady-state operation.

A correction method is presented in [17], where the primarycurrent is estimated from measurements of secondary current,using the least squares technique. Despite the refined mathemat-ical methodology used in the work, the magnetic properties ofthe iron-core material was not considered.

A hysteresis model requests data that are not always avail-able. There are models that require a set of points of the dc hys-teresis loop as input data. However, it is not common to obtainthis information. The data that are more easily available are themagnetic induction and field intensity, characteristics of the core

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188 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007

Fig. 1. Proposed CT equivalent circuit.

material, usually supplied by the manufacturer. Normally, thesevalues are converted in magnetic flux and magnetizing current,using the geometric parameters of the core and the number ofturns of the windings. For power transformers, this practice pro-duces significant errors because, in the construction of the core,the sheets are cut, having an insertion of unknown gap lengths.So, the obtained magnetic characteristic is not the real charac-teristic [18]. Fortunately, this fact does not occur in the case ofCTs, which present simple geometry and a relative easiness ofdata obtainment. A CT core usually has a toroidal form (a ribbonmade of a silicon-iron alloy, coiled in a helicoidal way). Gapsthat are intentionally inserted have a known equivalent length.

The Preisach’s model [19] is selected for comparisons withthe proposed model. As input data, it requests only a set of pointsof the upper branch of the hysteresis loop. Moreover, this modelis one of the most used and effective tools to describe the mag-netic hysteresis phenomenon [20], [21].

II. CT MODEL

A. Equivalent Electric Circuit

The proposed CT equivalent electric circuit is shown in Fig. 1.and represent the resistance and the inductance of the sec-

ondary (winding and load). The nonlinear inductor is associ-ated with the saturation and hysteresis effects of the magneticcore. The effects of classic eddy current losses and anomalouslosses are represented by (linear resistor) and (nonlinear),respectively.

B. Hysteresis Modeling

The proposed model starts from the Talukdar and Bailey work[7]. A minor loop between the points and and a partof the descending branch of the major dc hysteresis loop areshown in Fig. 2.

is the vertical distance from a point , on the descendingtrajectory of the minor loop, to the point , on descending thetrajectory of the major loop.

It is assumed that varies as follows:

(1)

where and are the vertical distances from andto the descending branch of the major hysteresis loop;

, and are the values of flux linkage in the points, and is a parameter that depends on the core

material; and is the value of the flux linkages when the mag-netic core starts to saturate.

Fig. 2. Asymmetric hysteresis loops.

The descending branch of the major loop is fitted by the func-tion . For the points and , it follows that:

(2)

The parameter establishes different slopes for the magnetictrajectories. The best value of is obtained from comparisonsof magnetic paths generated by (1) and (2) with paths obtainedby laboratory measurements.

A similar way is applied to the generation of ascending pathsinside the major loop, considering the ascending branch.

The closing and elimination rules of the minor loops arereproduced by a stack that registers the magnetic history ofthe material. Initially, the stack is assumed to be empty. Whenthe magnetizing trajectory passes a reversal point, this point isplaced on the upper position of the stack. If the absolute valueof the last calculated exceeds the absolute value of the lastbut one value kept in the stack, a minor loop is closed, and thecoordinates of the last two reversal points are removed fromthe stack; so, the minor loop does not have any influence in thefuture magnetic state of the material.

In order to represent the major hysteresis loop, the followingfunction may be used:

(3)

The constants , and are determined from the major hys-teresis loop, using curve-fitting routines.

In order to approximate the initial magnetization curve in thefirst quadrant, the following equation is used:

(4)

The parameter reproduces the portion of the curve closestto the origin (Rayleigh region).

When the core reaches high saturation levels, (3) and (4) maybe inaccurate. A way to improve the fitting is to use sequences of

GUERRA AND MOTA: CURRENT TRANSFORMER MODEL 189

points of the initial magnetization curve and the major hysteresisloop. A value of corresponding to an intermediate value ofis obtained using a search routine in an ordinate table [22] andlinear interpolation. In this case, it is not necessary to determinea value for .

C. Core Losses Modeling

For a flux density , the total specific mag-netic losses (W/kg), per cycle, is [23]

(5)

where is the frequency and is the specific hysteresis losses(W/kg). and are, respectively, the constants of classiceddy current losses and anomalous losses.

, and the specific hysteresis losses for cycle aredetermined by a fitting curve routine, from a set of points of thecurve versus , obtained by the use of a variable frequencysource and a low distortion amplifier, with the peak inductionvalue sustained in the nominal value of operation.

The induction in the core can be written as follows:

(6)

where is the number of turns of the secondary winding andis the core cross-section area.The instantaneous eddy current losses and anomalous losses

in the magnetic core in Fig. 1 are given, respectively, for theexpressions [24]

(7)

(8)

The parameter is the density of the magnetic material andis the mean length of the magnetic path. Replacing (6) in (7),

results in

(9)

Considering that

(10)

the resistance of the classic parasitic losses is

(11)

Replacing (6) in (8), gives

(12)

The resistance of anomalous losses is

(13)

From (12) and (13), the results is

(14)

(15)

D. Circuit Equations

For the circuit in Fig. 1, the following equations can bewritten:

(16)

(17)

(18)

(19)

(20)

Equation (20) describes the proposed hysteresis model. Inte-grating (16) and (17), it is obtained

(21)

(22)

Using the trapezoidal integration rule and considering a timestep , the discretized equations of the circuit are written asfollows:

(23)

(24)

(25)

(26)

(27)

where and are given by the following equations:

(28)

(29)

The nonlinear system is solved by use of the Newton–Raphsonmethod, using a time step s.

III. RESULTS

A toroidal core was used in the laboratory measurements. Itis made of a grain-oriented silicon-iron alloy, with the followingdata: mean diameter, 0.125 m; cross-section area, 0.001 m ;lamination thickness, 0.3 mm; stacking factor, 0.96; weight, 3kg; number of turns in primary and secondary windings, 60;


Fig. 3. Magnetic losses p=f as a function of frequency; B = 1:5 T.

windings resistances, 0.2 ; windings leakage inductances,negligible.

The measured curve of the magnetic losses as a function offrequency, in the range 0–130 Hz, is shown in Fig. 3, for a peakvalue induction T.

The parameters of (5) were determined using a curve-fittingroutine based on Levenberg–Marquardt’s method [22]. Theresults are

In 60 Hz, with a peak induction of 1.5 T, the total calculatedlosses are 1.289 W/kg. According to the manufacturer (ACE-SITA), these losses are 1.280 W/kg.

The measured and calculated hysteresis loops are shown inFig. 4 (flux linkages, , versus excitation current

), for T and . In the measured dc loop,the flux linkages were obtained by numerical integration of theinduced voltage in the secondary winding; in the ac loop, it usedan RC circuit.

Comparisons between simulated and experimental primaryand secondary currents waveforms are presented in Figs. 5 and6, for different conditions of excitation. It was considered a pri-mary time constant of 23.7 ms and burdens ofand .

The excitation was increased to saturate the core and after,it gradually decreased in such a way that the residual flux wasmade equal to zero.

The fault inception angle was set to 0 by a synchronousswitch. This device is composed by a triac controlled by an elec-tronic circuit.

The proposed CT model was compared with another modelin which the hysteresis is represented by using the Preisach’s

Fig. 4. Hysteresis loops B = 1:72 T, � = 0:3.

Fig. 5. Currents. Burden 2:40 + j0:10 .

theory. For this purpose, a new routine was carried out in sucha way that (27) was replaced by , accordingto the formulation presented by [19]. Simulations were per-formed considering a fault at the receiving end of a transmissionline with lumped RL series parameters. The transmission-lineimpedance was . At the sending end, it was con-sidered a 115-V, 60-Hz voltage source. In all cases, .

Hysteresis minor loops are shown in Fig. 7, for a burden of.

Three intervals of time were considered: first closing, deadtime, and reclosing, each of them corresponding to two cycles,with a fault inception angle .


Fig. 6. Currents. Burden 1:52 + j1:70 .

Fig. 7. Hysteresis minor loops. Burden 0:25 + j0:15 .

The flux linkages and the currents waveforms are shown inFigs. 8 and 9, for a burden of . It was consid-ered closing (3.5 cycles), dead time (5 cycles), and reclosing (5cycles), with fault inception angles (first closing) and

(reclosing).In order to evaluate the influence of the residual flux in the

magnetic core, the fault inception angle in the reclosing periodwas changed to 180 . The results are shown in Figs. 10 and

11.At the beginning of the reclosing period, the flux imposed by

the excitation increases with the same polarity than the residual

Fig. 8. Flux linkages � = 0 ; � = 0 ; burden 1:00 + j0:50 .

Fig. 9. Currents � = 0 ; � = 0 ; burden 1:00 + j0:50 .

flux in the core. Therefore, a higher level of saturation is reachedand the distortion in the secondary current is more pronouncedthan those indicated in Fig. 9 (in this case, the flux imposed bythe excitation increases with opposed polarity than the residualflux).

Figs. 12 and 13 show the comparisons with the re-sults obtained by measurements, considering the burden of

. It is observed that the deviations are moresignificant in the reclosing period.

These results show that the model is able to reproduce the hys-teresis and magnetic losses phenomena, including remanenceand minor loops.


Fig. 10. Flux linkages � = 0 ; � = 180 ; burden 1:00 + j0:50 .

Fig. 11. Currents � = 0 ; � = 180 ; burden 1:00+ j0:50 .

The deviation between a secondary current value given by theproposed model and a correspondent current value obtainedby another way is calculated as follows:

(30)

where is the rated secondary current (1 A); and is the over-current factor (20, according to international standards). The de-viation is taken as a percentage of because the CT per-formance must be evaluated in fault conditions.

The maximum deviations occurring in the analyzedcases are summarized in Table I. The greatest deviations occur

Fig. 12. Currents � = 0 ; � = 0 ; burden 1:00+ j0:50 .

Fig. 13. Currents � = 0 ; � = 180 ; burden 1:00 + j0:50 .

for the wavefoms of Fig. 12. Fig. 14 shows the values of forthis case.

The international standards specify the CT’s accuracy interms of ratio and phase errors. This practice is based on thephasor quantities, which is unsuitable for nonsinusoidal wave-forms. In this case, it is preferable to define the ratio error foreach calculated current value, as follows [5]:

(31)

In order to verify the correlation between the CT accuracy andthe degree of saturation, the reactance of the lumped parameter


TABLE IMAXIMUM SECONDARY CURRENT DEVIATIONS

Fig. 14. Deviations between experimental and calculated current values.

transmission line was fixed in 14.4 and the primary time con-stant was varied from 5 to 200 ms. It was considered sec-ondary burdens of , , and .Reclosing was not considered. The maximum errors , as afunction of , are shown in Fig. 15.

For the last case of Table I, the variation of the anomalousloss resistance with the time is shown in Fig. 16.

IV. CONCLUSION

A new CT model was presented, which includes the represen-tation of the magnetic hysteresis and the dynamic losses in thecore. The results are close to those obtained by the Preisach’smodel, and acceptable if compared with laboratory measure-ments. The model has a simple formulation, with easy under-standing and implementation. Thus, the number of calculations

Fig. 15. CT maximum error as a function of the primary time constant.

Fig. 16. Resistance R ; � = 0 ; � = 180 ; burden 1:00 + j0:50 .

is less than the Preisach and Jiles–Atherton models, becauseonly simple algebraic equations are involved.

This method is valid for any toroidal nongapped iron-coreCT, regardless of size, rating, and power system topology.Moreover, it is suitable for multiple CTs connected to the sameburden. For applications in CTs with complex magnetic cores,it is necessary for additional developments.

The frequency limits of this model are the same ones estab-lished by standards such as ANSI and IEC (25 up to 400 Hz).For high-frequency applications, it is necessary to include thedistributed capacitances in an appropriate way.

Due to the simplicity, the authors believe that this methodmay be used in other available tools for current transformersmodeling, such as EMTP and EMTDC.


The penalty for using this model is in the need to know a setof points of the descending segment of the major dc hysteresisloop. The necessity of determining and constants constitutean additional problem. However, those constants vary inside anarrow range of values for a given material.

The magnetic core model presented here is also useful forother purposes, such as studies of ferroresonance and inrush cur-rents in power transformers.

ACKNOWLEDGMENT

The authors would like to thank Prof. D. Fernandes Júniorof Federal University of Campina Grande for his stimulatinginterest in this work and his suggestions.

REFERENCES

[1] M. Poljak and M. Kolibas, “Computation of current transformertransient performance,” IEEE Trans. Power Del., vol. 3, no. 4, pp.1816–1822, Oct. 1988.

[2] T. Conrad and D. Oeding, “A method to correct the distorted secondarycurrents of current transformers,” in Proc. PSCC, 1987, pp. 311–315.

[3] M. Kezunovic, C. W. Fromen, and F. Philips, “Experimental evalua-tion of EMTP-based current transformer models for protective relaytransient study,” IEEE Trans. Power Del., vol. 9, no. 1, pp. 405–412,Jan. 1994.

[4] A. K. S. Chaudhary, K. Tam, and A. G. Phadke, “Protection system rep-resentation in the Electromagnetic Transients Program,” IEEE Trans.Power Del., vol. 9, no. 2, pp. 700–708, Apr. 1994.

[5] Y. C. Kang, J. K. Park, S. H. Kang, A. T. Johns, and R. K. Aggarwal,“An algorithm for compensating secondary currents of current trans-formers,” IEEE Trans. Power Del., vol. 12, no. 1, pp. 116–124, Jan.1997.

[6] J. R. Frame, N. Mohan, and T. Liu, “Hysteresis modeling in an electro-magnetic transients program,” IEEE Trans. Power App. Syst., vol. 101,no. 9, pp. 3403–3412, Sep. 1997.

[7] S. R. Talukdar and J. R. Bailey, “Hysteresis model for system studies,”IEEE Trans. Power App. Syst., vol. PAS-95, no. 4, pp. 1429–1434, Jul./Aug. 1976.

[8] J. R. Lucas, P. G. McLaren, W. W. Keethipala, and R. P. Jayasinghe,“Improved simulation models for current and voltage transformers inrelay studies,” IEEE Trans. Power Del., vol. 7, no. 1, pp. 152–159, Jan.1992.

[9] D. A. Tziouvaras, P. McLaren, G. Alexander, D. Dawson, J. Ezster-galyos, C. Fromen, M. Glinkowski, I. Hasenwinkle, M. Kezunovic, Lj.Kojovic, B. Kotheimer, R. Kuffel, J. Nordstrom, and S. Zocholl, “Math-ematical models for current, voltage and coupling capacitor voltagetransformers,” IEEE Trans. Power Del., vol. 15, no. 1, pp. 62–72, Jan.2000.

[10] U. D. Annakkage, P. G. McLaren, E. Dirks, R. P. Jayasinghe, and A.D. Parker, “A current transformer model based on the Jiles-Athertontheory of ferromagnetic hysteresis,” IEEE Trans. Power Del., vol. 15,no. 1, pp. 57–61, Jan. 2000.

[11] D. C. Jiles, J. B. Thoelke, and M. K. Devine, “Numerical determinationof hysteresis parameters for the modeling of magnetic properties usingthe theory of ferromagnetic hysteresis,” IEEE Trans. Magn., vol. 28,no. 1, pp. 27–35, Jan. 1992.

[12] S. Prigozy, “PSPICE Computer modeling of hysteresis effects,” IEEETrans. Educ., vol. 36, no. 1, pp. 2–5, Sep. 1993.

[13] H. G. Brachtendorf, C. Eck, and R. Laur, “Macromodelling of magneticphenomena with SPICE,” IEEE Trans. Circuits Syst. II—Analog Digit.Signal Process., vol. 44, no. 5, pp. 378–388, May 1997.

[14] E. D. M. Hernandez, C. S. Muranaka, and J. R. Cardoso, “Identifica-tion of the Jiles-Atherton parameters using random and deterministicsearches,” Phys. B: Condensed Matter, vol. 275, no. 1–3, pp. 212–215,Jan. 2000.

[15] N. L. Locci and C. Muscas, “Hysteresis and eddy currents compensa-tion in current transformers,” IEEE Trans. Power Del., vol. 16, no. 2,pp. 154–159, Apr. 2001.

[16] J. Tellinen, “The simple scalar model for magnetic hysteresis,” IEEETrans. Magn., vol. 34, no. 4, pp. 2200–2206, Jul. 1998.

[17] S. Bittanti, F. A. Cuzzola, F. Lorito, and G. Poncia, “Compensation ofnonlinearities in the current transformer for the reconstruction of theprimary current,” IEEE Trans. Control Syst. Technol., vol. 9, no. 4, pp.565–573, Jul. 2001.

[18] M. Elleuch and M. Poloujadoff, “New transformer model includingjoint air gaps and lamination anisotropy,” IEEE Trans. Magn., vol. 34,no. 5, pp. 3701–3711, Sep. 1998.

[19] S. R. Naidu, “Simulation of the hysteresis phenomenon usingPreisach’s theory,” Proc. Inst. Elect. Eng., vol. 137, pp. 73–79, Jul.1990.

[20] F. Liorzu, B. Phelps, and D. L. Atherton, “Macroscopic models of mag-netization,” IEEE Trans. Magn., vol. 36, no. 2, pp. 418–428, Mar. 2000.

[21] D. A. Philips, L. R. Dupré, and J. A. Melkebeek, “Comparison of Jilesand Preisach hysteresis models in magnetodynamics,” IEEE Trans.Magn., vol. 31, no. 6, pp. 3551–3553, Nov. 1995.

[22] W. H. Press, B. P. Flannery, S. A. Tewkolsky, and W. T. Wetterling, Nu-merical Recipes—The Art of Scientific Computing. Cambridge, MA:Cambridge Univ. Press, 1986.

[23] F. Fiorillo and A. Novikov, “An improved approach to power lossesin magnetic laminations under non-sinusoidal induction waveform,”IEEE Trans. Magn., vol. 26, no. 5, pp. 2904–2910, Sep. 1990.

[24] J. G. Zhu, S. Y. R. Huy, and V. S. Ramsden, “Discrete modelingof magnetic cores including hysteresis, eddy current and anomalouslosses,” Proc. Inst. Elect. Eng. A, vol. 140, no. 4, pp. 317–322, Jul.1993.

Francisco das Chagas Fernandes Guerra wasborn in Antenor Navarro, Brazil, in 1954. He re-ceived the B.Sc. and M.Sc. degrees in electricalengineering from the Federal University of Paraiba(UFPB), Campina Grande, Brazil, in 1977 and 1982,respectively.

Currently, he is teaching in the Departmentof Electrical Engineering, Federal University ofCampina Grande (UFCG), Campina Grande, Brazil,where he is currently pursuing the D.Sc. degree. Hisresearch interests include magnetic materials and

power system protection.

Wellington Santos Mota (M’76–SM’02) was bornin João Pessoa, Brazil, in 1946. He received the B.Sc.and M.Sc. degrees in electrical engineering fromthe Federal University of Paraiba (UFPB), CampinaGrande, Brazil, in 1970 and 1972, respectively, andthe Ph.D. degree in electrical engineering from theUniversity of Waterloo, Waterloo, ON, Canada, in1981.

Currently, he is a Senior Professor with the De-partment of Electrical Engineering, Federal Univer-sity of Campina Grande (UFCG), Campina Grande.

From 1973 to 1977, he was with Sao Francisco River Hydro (CHESF), Recife,Brazil, in power system planning. His research interests include power systemcontrol and stability, including wind farms.

current transformer model

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