Representation of magnetisation curves over extensiverange by rational-fraction approximationsG. F. T. Widger, B.Sc, Ph.D., D.I.C.

AbstractThe paper presents a simple mathematical expression which enables representation of all types ofmagnetisation curves over a very wide range. The equation given is ideally suited for problems whichrequire use of the B\H curve many times, since calculations using it require only a few operations of thebasic arithmetic adding, multiplying and dividing.

Although the equation is not linear in all the parameters, a method for obtaining the best parameters,in which the problem is first linearised and then restored to its original nonlinear form by a system ofiteration, is described.

A discussion of the meaning of the best parameters of the equation to fit the data curve, whether it isto be read from B to H, or H to B, or both ways, is given.

Comparison is given between several saturation curves and their mathematical counterparts overmagnetising m.m.f. from zero to, in one case, 300000AT/m.

List of symbolsa, b, c, d = arbitrary constants

B = flux density (any units)H = m.m.f. (any units)

Bh H) = co-ordinates of /th data point of B\H curve tobe represented

B], H] = Co-ordinates of/th point of B\H curve definedby equation

if = field current, A// = maximum power of variable (B or H) in

denominator of equationN = number of data points representing the Bjhf

curveP, = H,{\ + bx Bi •+• b2BJ + . .. + bnB",), i.e. de-

nominator of expression for error at pointBh H,

[M = permeabilityv0 . . . an = parameters of numerator of equation H = f(B)b{ . . . b,, = parameters of denominator of equation

H =f(B); parameters of equation B = f(H)have suffixes

e = error function to be minimised

as many as 40 different magnetisation characteristics mustbe stored within a single computer program (as in rotating-machine-design programs), the system of storing each asa series of points is very wasteful, and, if only a small com-puter is available, the curves must be stored externally and'read in' when required in the calculation.

There have been many attempts to find a satisfactory typeof equation to fit the magnetisation curve over the wholeuseful range, but all have proved inadequate. Recently, in abroader paper on saturation, Kavanagh1 made several sug-gestions, including exponential and hyperbolic functions.None of these, however, gives a satisfactory approximationover a very wide range. Trutt, Erdelyi and Hopkins,2 in apaper devoted entirely to finding a satisfactory equation forthe curve, come to the conclusion, after failing to find suitablepolynomial, exponential and hyperbolic functions, that theonly satisfactory way is by linear interpolation between aset of co-ordinate points. More recently still, Briansky3

suggests a single formula for the curves, but the complexityof the expression, which is a combination of polynomialsand exponentials raised to polynomials, must surely be areason why it is unsuitable for repetitive use. Moreover, noexamples of its use are given in Reference 3.

1 IntroductionIn the past, a need has been expressed for a simple

equation to represent magnetisation curves (e.g. References 4and 5). The requirements are twofold. First, the equationshould be usable in conjunction with other equations toobtain an analytical solution for a particular probleminclusive of saturation effects, such as eddy-current loss iniron. Secondly, for computer calculations, the magnetisationcurve should be capable of accurate representation over thewhole useful range by a single equation. The first requirementis a particularly stringent one, since it is then necessary thatthe equation is symmetrical in the positive and negativeregions, and furthermore, although not essential, that theequation suitably describes magnetisation to infinite fluxdensity. However, with the advent of the digital computer,with which problems may be solved speedily by numericalmethods, the first requirement is less important; the secondis the predominant one.

For many problems, it is adequate to represent the curveby a series of straight lines between fixed points on the curve(linear interpolation), but there have been instances5 wherethe 'kinks' caused by the junction of two straight lines havebeen magnified in the solution of a problem; and, in othercontexts involving iterative procedures, instability of flheiteration has resulted, for similar reasons. Moreover, where

Paper 5693 J, first received 5th June and in revised form 15th August1968Dr. Widger is with GEC-AEI Ltd., Rugby, War., England156

2 Criteria for best equation2.1 Choice of form of equation

To be more advantageous overall than the system oflinear interpolation, it was decided that the form of theequation must be extremely simple, so that, in repetitive oriterative calculations, the time consumed in using it wouldnot be significant. For this reason, hyperbolic and exponentialfunctions were not considered. Polynomials were also elimi-nated, since these had been found in the past to beunsatisfactory.

First observation of a magnetisation curve shows that thematerial appears to exhibit a very high permeability at lowmagnetising force, saturating to a low constant permeabilityat very high magnetising force. The permeability of thematerial changes between the two limiting values inverselywith the magnetising force. Assume that the permeabilityJX changes in an inverse polynomial manner:

a + bH + cH2 + dH3 + . . .• (I)

The permeability \x given by eqn. I exhibits a constantvalue at low magnetising forces, but, at high magnetisingforces, tends to zero, instead of to a constant low value. Theaddition of a constant term C is therefore necessary:

L r (2)a + bH + cH2 + dH3 + . . . " ' K '

PROC. IEE, Vol. 116, No. /, JANUARY 1969

Replacing fj, = B\H, and replacing the combined constantsby single ones, the equation for the magnetisation curve isobtained:

a[H a2H2 a'nH

\ +b[H b'2H2 b'nH"

(3)

The equation is seen to be of suitable form, since, forsmall H, the function reduces approximately to the straight

line B = — H, the initial slope of the magnetisation curve,

and, at very high H, to the line B — (a'Jb'n)H, the final slopeof the curve. Indeed, it is possible to consider the function

as acting, first, as the straight line B = -j H, then as the line

B = (a'Jb\)H, then as the line B = (a'2lb'2)H etc., with asmooth transition from one line to the next. However, inpractice, a rational fraction of this type is capable of manygraphical forms, and it may be that the best approximationto a magnetisation curve over a given range is obtained wherethere is no such direct relationship between the parametersof the equation and the permeability of the material.

2.2 Direction of reading

Now that the form of equation is established, it is importantto consider the direction of reading. Almost without excep-tion, it is true to say that, once a problem is formulated,requiring use of the magnetisation curve, the reading of thecurve is unidirectional; i.e. if H is required from B, thereverse will not be required. The direction, however, maydiffer in different problems; e.g., in machine design, H isrequired from B, but, when using an open-circuit saturationcurve, the saturating quantity, voltage, is required fromexcitation current. It is important, therefore, to state thealternative function:

a0 axB + a2B2 anB"

I + + + • • • + bnB"(4)

2.3 Definition of best fit

It is necessary, when required to find the values of theparameters a0, ax . . . ; b{, b2 . . . , which give the bestapproximation to a given curve, to define what is meant by thebest approximation. There are three predominant definitions:

(a) The sum of the magnitudes of the errors, between pointsof the data curve and corresponding points calculated bythe equation, is a minimum:

N

2 \(H'; — Hj)\ is a minimum/ = i

(b) The sum of the squares of the magnitudes of the sameerrors is a minimum:

;v2 (H'j — Hj)1 is a minimum

;= i

(c) The maximum error is a minimum:

MAX(A/- — Hj) is a minimumN

This definition corresponds to 2 (//• — Z/,)00 is aminimum. / = 1

Of the three definitions, b is chosen, because it is both agood compromise between a and c, and also because it isthe easiest to handle mathematically. The errors are estimatedeither in the H sense or in the B sense, whichever is appro-priate to the manner in which the curve is to be used. If thecurve is required to be read in both directions, B to H andH to B, with equal accuracy, a weighting factor may beintroduced, by which each error is multiplied. One example

of weighting factor for this purpose is M +-777) for theerror at the /th point. >'

The most practical mathematical equation is obtained ifrelative or percentage errors, rather than absolute errors,are minimised.

PROC. 1EE, Vol. 116, No. I, JANUARY 1969

2.4 Accuracy of approximation

Fig. 1 shows a typical magnetisation curve and amathematical approximation. At point P, the relative error

Fig. 1Relative errors in B & H sense

true curvemathematical approximation

in the B sense, (B — B')/B, is very nearly equal to therelative error measured in the //sense, (H — H')\H. However,at point Q, the relative error (H — H')\H is seen to be manytimes larger than the relative error in the B sense. Thus, thesum of the relative errors, or the mean error, in one sensemay be very much larger than the sum of the errors, or meanerror, in the other.

The parameters of the equation for all the examples givenhere are derived by minimising the relative errors in the sensein which the errors appear largest:

N / / / • - H\2

2 ( — L T T — ' ) ' s minimisedi= |V Hj J

However, extra weighting, i.e. more points, is introducedat higher values of H to obtain a fit to the data curve, whichhas higher percentage errors, though small absolute errors, atlow // , and the reverse at high H.

The overall accuracy of the mathematical approximationis considered to be satisfactory when the greatest error iseither:

(a) beyond the accuracy of the metering with which themagnetisation characteristic is measured

or(b) less than the variations that occur between samples of

the same material

In practice, it is not unusual to find that the magnetisingforce required to produce a given flux density varies by asmuch as 15% from sample to sample of the same magneticmaterial. Thus, b is the predominant criterion for fitting theequation to the characteristic of magnetic materials, and athe criterion for finding the parameters to fit a particularelectrical-machine characteristic.

The criteria a and b define how many terms of the rationalfraction (eqns. 3 and 4) are required to find the mathematicalapproximation.

3 Method for determining best parametersof equationThe method is similar to that used by Sanathanan and

Koerner6 for finding the parameters of a rational fractionfor fitting a frequency locus.

Assume that the curve to be fitted is represented by/V points, whose co-ordinates are Bh Hj (/ = 1, N). The errorfunction e, to be minimised, is defined as:

- HA2

(5)

157

Substituting H\ from eqn. 4 gives:

bxB, + b2B] + . . . + bnB? "'2 + a2B) + . . . + anBi+i) -

///(I + 6,5/ + M? + • • • +

The minimum value of e may be found by equating the

differentials -̂—, r—, .oa0 oo, =rr, =rr" e t c- t 0 ze ro> a n d solvingoo | ob2

the (2/7 — I) equations for the (In — 1) parameters a0, a{, ...a,,; bx . . . bn. However, the equations obtained are nonlinear,and, therefore, are not easily solved.

The system of equations may be linearised if the deno-minator of eqn. 6 is replaced by a constant P,:

b2Bj bnB'l) (7)

and, initially, P, is set as unity. This corresponds to multiply-ing the error at each point by the value of the denominatorof the error term for that point. The effect of this is to con-sider each single point Bh H, to be replaced by H,{\ + bxB-, +b2Bj + . . . + bnBf) identical points. If the values of b{ . . . bn

were known, it would be possible to see immediately by howmany points each single point is replaced. Fig. 2b shows by

Fig. 2

Iteration to find best fit

best fit at each stage

a Data points defining curve to be fittedb First cycle of iteration Pi = Ic A subsequent cycle of iteration(I Final fit

clusters of points what the effect is. Of course, each pointin a cluster is really on the same spot, but they are separatedto show the principle.

158

• (6)

The error function for this artificial situation is defined by

.2

- (//,_+ bxB,H, + b2BjHi (8)

With Pj constant, eqn. 8 may be easily differentiated withrespect to the (2/7 — 1) parameters, and the differentialsequated to zero give {In — 1) linear equations, from whichthe parameters are easily solved. These parameters give thebest fit for the situation of Fig. 2b. Clearly, the curve passesclosest to the place where the 'density' of the points is thegreatest.

Now that an initial assessment of the parameters has beenmade, the values of the function P-t may be calculated forevery point, and the (2/; — 1) linear equations resolved withnew constant values of P,. The problem is now one of findingthe best parameters for the situation with fewer points ineach cluster (Fig. 2c). The new values of 6, . . . bn give newvalues of />,, and the process is repeated until the (2// — I)parameters converge onto consistent values. The requiredapproximation is then obtained (Fig. 2d). The computer isused to apply the procedure.

4 Examples of rational fractionapproximationFig. 3 shows a synchronous-machine open-circuit

magnetisation curve adequately approximated by a rational

700

600

500

400

300

200

100

0 20 40 60 SO 100 120 140field current if,A

Fig. 3

Open-circuit magnetisation curve of500hp synchronous motor

.. test points26-56 0-00575// .

I H-0-0315//

fraction, in a manner which enables simple evaluation ofopen-circuit voltage from field current.

The magnetisation curves of materials used for examplesare typical of those employed by the author in the design ofrotating machines. Part of the curves have been derived bymeasurement, but the extreme upper parts have been linearly

PROC. IEE, Vol. 116, No. /, JANUARY 1969

extrapolated to give greater range. The equation has beenfitted to the curves over the whole range.

Fig. 4 shows a magnetisation curve of a typical laminated

tion curves using the function in the form H as a functionof B. They are the most extreme ones found available. Theexample of Fig. 5 was chosen because of its low initial

2 8 r

30 000H.AT/m

50 000 60 000 2000 0 -2000error,H, AT/m

15 0error, °/o

b

-15

-0-110 000 20000 30000

H,AT/m40000 50000

-1060000

Fig. 4

Magnetisation curve of typical electrical sheet steel426-1615 - 759-5979B + 438-51I5fl2

a H =

B r=

1 - 0-802)4675 + 0- 169835552

8-969025 x IO"5W + 9-165906 x I

B

I + 7078978 x 10"3« -!- 5-539592 xb absolute error

percentage erc absolute error

•°r \H:error/

percentage error B

-f{B)

sheet steel. Approximations are shown, with their associatederror curves, for the best rational fraction with five para-meters, both H as a function of B, and B as a function of H.The initial part of the curves is expanded in scale 20 times.Clearly, Was a function of B gives the best approximation.

Figs. 5 and 6 show two more approximations to magnetisa-

permeability over a large range of //; the example of Fig. 6because of its very steep initial slope, its sharp knee, and itsvery shallow final slope. The latter proved the most difficult,and seven parameters had to be employed to give a satis-factory approximation. The initial part of the curves in Fig. 6is expanded in scale 100 times.

3 0 r

2 0

10

10000 20000 30 000H.AT/m

3-0r

40 000 600 O -600error, H.AT/m

15 0 -15error. °/o

Fig. 5

Magnetisation curve of black steel plate across graina data curve

H =2593-260 - 3793-9715 + 1614-688B2

B- O-8O36855fl + 0-I950866S2

b absolute errorpercentage error

PROC. 1EE, Vol. 116, No. I, JANUARY 1969

100000 1500001000 H,AT/m

200000 250000 300000 20000 -2000error. H.AT/m

Fig. 6

Magnetisation curve of grade 190 electrical sheet steel

data curve= 251 1622 - 512 2454B + 344-8631B* - 56-3281 \B>~ I - 12-22686B + 0-4998543fi2 - 0 06820061fl*

absolute errorpercentage error

159

5 ConclusionsA rational fraction has been shown to be capable of

fitting saturation curves over an extremely wide range. Theequation is very fast to use, because only a few operations ofthe basic arithmetic are required. A method has been givenfor the derivation of the best parameters of the equation fora particular data curve.

In general, magnetisation curves of electric machinery maybe approximated by the equation with three parameters. Five,occasionally seven, parameters give a satisfactory representa-tion of the magnetisation curves of iron.

Although the equations of curves given here are notsymmetrical with respect to the positive and negative regions,if this is a requirement, the equation may be derived usingterms of even powers only.

The hysteresis loop may be expressed by two equations ofthe rational form.

6 AcknowledgmentsThe author wishes to express his thanks to the directors

of AEI Large Electrical Machines Ltd. for permission topublish the paper.

Particular thanks are due to a former employee of thecompany, M. G. Cox, for many helpful discussions on thegeneral principles of curve fitting.

References1 KAVANAGH, R. J.: 'An approximation to the harmonic response of

saturating devices', Proc. IEE, 1960, 107 C, pp. 127-1332 TRUTT, F. c , ERDELYI, E. A., and HOPKINS, R. E.: 'Representation of

the magnetisation characteristic of d.c. machines for computer use',Proceedings of the IEEE winter power meeting, New York, 1967,Paper 31, pp. 67-77

3 BRIANSKY, D.: 'Single analytical formula for the entire B/H curve',Internal. J. Elect. Engng. Educ, 1967, 5, (2), pp. 199-201

4 GIBBS, w. j . : 'Induction and synchronous motors with unlaminatedrotors', J. IEE, 1948, 95, Pt. II, pp. 411-420

5 MASTERO, s.: 'Contribution a la predetermination des reptes dans lestoles ferromagnetiques', Ph.D. thesis, Faculte Polytechnique deMons, Belgium, 1968

6 SANATHANAN, c. K., and KOERNER, i.: Transfer-function synthesisas a ratio of two complex polynomials', IEEE Trans., 1963, AC-9,pp. 56-58

South-East Scotland Sub-Centre: Chairman's AddressA. J. Bode, C.Eng., M.I.E.E.

This is my second term on the South-East ScotlandSub-Centre Committee, and I look back with awe at my pre-decessors. I am one of the less distinguished members of theSub Centre and make no apologies for the common-or-gardenviews which, I feel, should sometimes be aired in committees.

The Scottish Centre has done well in the production ofrecent papers in the sphere of supply-industry activities, andI feel it is my function to carry these advancements into thefield. I really have nothing to offer in a formal sense whichcould be regarded as unique, and I have no intention to addto the mass of literature in present production.

The failure of our times is communication. There are the• learned societies, universities and seats of education on theone hand, and public and private industries and services onthe other, talking different languages. We have failed to getthe full fruits of research onto the shop floor to present.

The Scottish Centre has been the vanguard of IEE reforms.Activities were decentralised a number of years ago by theformation of sub-centres to deal with the extensive geo-graphical area. Within the terms of the IEE's recommenda-tion to arrange district meetings, we have for some time been

Abstract 5745 of Address delivered at Edinburgh, 1st October 1968Mr. Bode is with the South of Scotland Electricity Board, Edinburgh 2,Scotland

arranging, with electrical societies, meetings to meet localrequirements.

The graduates and students are being encouraged to takea part in senior activities, and in this way senior membersmay be able to keep up to date while the juniors gainexperience in selling the idea.

Programmes are sometimes unjustly criticised by members.We have been more adventurous this year, and I hope toreceive support from your attendance and observations inthe discussions.

\n line with the Council's recommendations for lectures ona wider horizon we have already heard from Mr. A. M. Palmer,MP, who has described some of the activities in Parliamentthrough the eyes of a professional (electrical) engineer.

Clearly it is emerging that successive governments andparliaments will seek professional advice from the seniorinstitutions. Our Institution must prepare to play a moreexacting role in influencing government policy. I refer to therecent report of the joint committee on professional manpower.

I believe we are an important medium not only in theadvancement of electrical engineering but also in conveyingthe products of research and original thought into theindustrial field. We must do these things not only in theinterests of our industry, but also in the interests of our fellowmen.

160 PROC. IEE, Vol. 116, No. I, JANUARY 1969