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M.E. El-Hawary, Editor

PowerEngineering Letters

This section of the magazine offers a vehicle that speeds publica-tion of new results, discoveries, and developments. The sectionaffords authors the opportunity to publish contributions within a fewmonths of submission to ensure rapid disseminationof ideas and timelyarchiving of developments in our rapidly changing field. Original andsignificant contributions in applications, case studies, and research inall fields of power engineering are invited.

Submit contributions, criticism, and queries to the Power Engi-neering Letters editor: Dr. Mohamed E. El-Hawary, DalTech,Dalhousie University, P.O. Box 1000, Halifax, NS B3J 2X4 Canada,+1 902 494 6198 or +1 902 494 6199, FAX +1 902 429 3011, E-mailelhawary@dal.ca.

Editorial Board

The following editorial board members are responsible for the peer re-view of all Letters appearing in this section:

M.E. El-Hawary (editor), Dalhousie University, Canada A. Bose, Washington State University, U.S.A. M.T. Correia de Barros,Universidade Tecnicade Lisboa, Portugal A.M. DiCaprio, PJM Interconnection, U.S.A. A.R. El-Kieb, University of Alabama, U.S.A. R.K. Green, Jr., Central and Southwest Services, U.S.A. T.J. Hammons, University of Glasgow, UK S.I. Iwamoto, Waseda University, Japan

J.H. Jones, Southern Companies Services, U.S.A. P. Kundur, Power Tech Labs, Canada F.N. Lee, University of Oklahoma, U.S.A. J. Momoh, Howard University, U.S.A. M.M. Morcos, Kansas State University, U.S.A. A. Papalexopoulos, Pacific Gas and Electric, U.S.A. M. Poloujadoff, Universite Pierre et Marie Curie, France N. Rau, ISO New England Inc., U.S.A. A. Schwab, Universitat Karlsruhe, Germany M. Shahidehpour, Illinois Institute of Technology, U.S.A. W.L. Snyder, Jr., Siemens Energy & Automation, Inc., U.S.A. J.S. Thorp, Cornell University, U.S.A. S.S. Venkata, Iowa State University, U.S.A. B.F. Wollenberg, University of Minnesota, U.S.A.

In This Issue

This issue includes the following Power Engineering Letters: Fitting Saturation and Hysteresis via Arctangent Functions, by

Carlos Prez-Rojas Induction Motors Broken Bars Monitoring Using the H-H

Method, by M.S. Nat Sad, M.E.H. Benbouzid A Simplified Method for Outage Reduction Impact Estimation,

by D. Pedro Miquel, B. Jos R. Figueroa HIPER: Interactive Tool for Mid-Term Transmission Expansion

P la nn in g i n a D er eg ul at ed E nv ir on me nt , b y R .D .Cruz-Rodrguez, G. Latorre-Bayona

A Review of a Quasistatic and a Dynamic TCSC Model, by

Ricardo J. Dvalos, Juan. M. Ramrez

Fitting Saturation and Hysteresisvia Arctangent Functions

Carlos Prez-Rojas

Author Affiliation:Facultad de Ingeniera Elctrica, UniversidadMichoacana de San Nicols de Hidalgo, Michoacn, Mxico.

Abstract:In this letter an approach for fitting saturation character-istic and hysteresis using arctangent functions is presented. It has directapplication to modeling transformers and electric machines. The ap-proach is based on measurements of the of magnetizing curve of the

ferromagnetic material. The analytical form of the characteristic can becompletely described by only three parameters. These three parametersare enough to take into account: the slope of the linear region, the posi-tion of the knee, and the saturating slope. The appropriate selection ofthese parameters allows the fitting of a wide class of magnetizing char-acteristics in simulation programs.

Keywords:Magnetic saturation, hysteresis, arctangent function.Introduction: Saturation and hysteresis characteristics in ferro-

magnetic materials have been the subject of much research. The repre-sentations vary from empirical relationships to more sophisticatedanalytical expressions such as exponential, hyperbolae, polynomials,arctangent [1-3], and differential relay equations. The interest in a morerigorous modeling is justified in systems that require that the operationbe predicted accurately. This implies that their parameters should beknown with the minimum of uncertainty regardless of the operating

point and the size of the electric devices. A substantial amount of re-search has been done in this area with each method having its own ad-vantages and disadvantages for simulation purposes.

This work suggests an alternative based on the arctangent functionthat requires a minimum number of parameters and at the same time al-lows precise adjustment to any saturation characteristic. The simplicity

IEEE Pow er En gi n eer i n g Revi ew , Novem ber 2000 0272-1724/00/$10.002000 IEEE 55

Figure 1. Real and approximate by arctangent of the saturation curve

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of the analytic equation renders an algorithm that is notcomputationally time consuming.

Characteristic of Saturation:The actual saturation characteristicof a magnetic component can be approximated using an arctangentfunction of the form

m m n m mi m i i( ) ( )= +atan . (1)

The constants m , m, and, which are calculated, form the actualcurve shown in Figure 1, as follows:

n x

= 2

(2)

mi i s

=

( )

( )

2 1

2 1 2 (3)

= [ ( )]s n s

s

m i

i

atan

(4)

where

x is the value of at the intersection of slopes, n is the value ofx normalized to the maximum of the arctangentfunction,/2,

s is the saturated value ofin the actual magnetizing characteris-tic,

m is the curves initial slope, normalized to the maximumarctangent value/2 and to the saturation value s ,

isis the value ofi corresponding to s ,

is the linear increment of m, m is the magnetizing flux linkages, andi

m is the magnetizing current.

Note that evaluating the parameters of (2)-(4) uses a simple proceduredirectly from the actual characteristic of Figure 1. The effect of eachparameter is demonstrated in Figures 2-4.

In modeling electromagnetic and electromechanical systems the

magnetization inductance Lmis defined as

L d i

dim

m m

m

= ( )

.(5)

From (1) we have

L d i

di

m

m im

m m

m

n= =+

+

( )

1 2 2 .

(6)

These equations are applicable when the formulation of the systemsis developed on the currents taken as state variables. The result can beextended to the case of flux linkages taken as state variables. For thiscase the nonlinear relationship is i fm m= ( ) , as shown in Figure 5.

From (1) we obtain

im

im m

m

n n

m( ) tan

=

1

(7)

and for this case the inverse of the magnetization inductance is

1 11 2

L

di

d mi

m

m m

m n

m

n n

m= = +

( )tan

.

(8)

Hysteresis:A simple hysteresis model is used consisting of a resis-tance, representing the iron losses, in parallel with a nonlinear induc-tance, as shown in Figure 6. It has been proven that a reactors

nonlinearity is easily represented with the arctangent. It will be shown

56 IEEE Pow er En gi n eer i n g Revi ew , Novem ber 2000

Figure 4. Relative movement varying

Figure 3. Relative movement varyingn

Figure 2. Relative movement varying m

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that the hysteresis phenomenon is also efficiently represented with thisformulation.

Application to a Reactor: Let us consider the reactor of Figure 7.Figure 1 illustrates the nonlinear characteristic of the reactor. Once theindicated constants are obtained (see Appendix A) the arctangent func-tion is calculated as

i im mm

m( ) . tan.

.

=

115710 3215

0 0115 .

(9)

using theflux linkages formulation because of its simplicity. As a resultwe obtain a model for the circuit of Figure 7.

Solving the model of the reactor, the results are illustrated in Fig-ures 8-9, where we plot the current and the hysteresis cycle. The resultsshow how hysteresis is modeled in the reactor circuit.

Conclusions: An alternative for representing saturation and hyster-esis characteristics has been presented. Only three parameters are

enough to characterize a wide class of saturation curves. The representa-tion is based on the arctangent function. The usefulness of the approachhere presented has been illustrated with a simple but practical case.

References:[1] M.M. Hassani, G. Lachiver, and G. Jasmin, Numerical simula-

tion of the magnetic core of a transformer in transient operations, Ca-nadian Conf. Elect. Comput. Eng., pp. 289-291, 1989.

[2] K.A. Corzine, B.T. Kuhn, S.D. Sudhoff, and H.J. Hegner, An

improved method for incorporating magnetic saturation in the q-d syn-chronous machine model, IEEE Trans. Energy Conversion, vol. 13,pp. 270-275, 1998.

[3] J.R. Lucas and P.G. Mclaren, B-H loop representation for tran-sient studies,Int. J. Eng. Educ., vol. 28, pp. 261-270, 1991.

Appendix A: Constants for the arctangent method, s =0 3215.m= 08642. =0 0037.Copyright Statement:ISSN 0282-1724/00/$10.002000 IEEE.

Manuscript received 4 April 2000. This paper is published herein in itsentirety.

IEEE Pow er En gi n eer i n g Revi ew , Novem ber 2000 57

Figure 5. Characteristic i fm m

= ( )

Figure 7. Reactor equivalent circuit

Figure 6. Hysteresis equivalent circuit

Figure 8. Reactor magnetizing current

Figure 9. Reactor hysteresis cycle