THE BEHAVIOUR OF INDUCTION COILS IN THE HIGH FREQUENCY RANGE

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THE BEHAVIOUR OF INDUCTION COILS IN THE HIGH FREQUENCY RANGE

P. Marty, D. Delage

To cite this version:

P. Marty, D. Delage. THE BEHAVIOUR OF INDUCTION COILS IN THE HIGH FREQUENCY RANGE. Journal de Physique Colloques, 1984, 45 (C1), pp.C1-933-C1-936.

�10.1051/jphyscol:19841190�. �jpa-00223667�

T H E BEHAVIOUR OF INDUCTION COILS IN THE HIGH FREQUENCY RANGE

P. Marty and D. Delage

GIs MADYLAM, B. P. N '68, 38402 S a i n t .Martin .d 'Hzres Cedex, France

Resum6 - Une m6thode numerique permettant de calculer la distribution de cou- rant electrique dans une bobine fonctionnant en haute frgquence est propos6e.

Abstract - We present a numerical technique for computing the distribution of theelectrical density of current in a high frequency coil.

I - INTRODUCTION

The problem of predicting power losses in the numerous turns of high current coils and the current distribution in circular billets of arbitrary cross-sections beco- mes increasingly interesting as the number of industrial plants using the induc- tion heating syste,,~ grows.

As is well known,high frequency alternating currents are not uniformly distribu- ted along the conductors, but tend to be concentrated near the conductor edges.

Consequently, the interior power loss of the conductors rises above the dc value.

In magnetohydrodynamics for example, in order to obtain shaping or regulation of the rate of flow of a liquid metal, a high frequency magnetic field is needed

(10 to 500 Khz). With the increasing frequency, ac-dc resistance ratios quickly become prohibitive (20 and more). For this reason,further research would be desi- rable in order to develop an optimized coil design. Creating a higher intensity of the magnetic field could be obtained by the addition of a magnetic circuit to the system. Firstly, the electric streamlines distribution in the copper conductor has to be calculated.

Previous works /I/ have discussed methods of calculating distribution in circular geometries. BIRTNGER et al,/2/ in an earlier paper have also improved a numerical method of solving such a problem in the case of straight conductors. Unfortunately, these methods become inapplicable when the electromagnetic penetration depth is much smaller than the typical dimension of the conductor.

On the other hand, quite recent works describe new techniques for analyzing linear skin effect phenomena in multi-conductor systems by a finite elements method / 3 , 4 / .

These methods are hamperedbythefact that the typical dimension of the elements must be of similar magnitude as the electromagnetic skin depth.

This paper presents a numerical procedure to calculate the current distribution for an axi-symmetric coil of any cross-section and for high frequencies. A particu- lar case will be discussed in details : the current distribution in a single turn coil of rectangular cross-section. The numerical results will be compared to those obtained with a finite elements method (FLUX 2D - Laboratoire dlElectrotechnique de Grenoble).

11 - HYPOTHESIS -EQUATIONS

In the following analysis, it is assumed that the conductors are of non-magnetic material and that the temperature is held constant. The return leads should be far enough removed from the conductors of the studied coil so that proximity effects may be neglected. For the sake of convenience, the electric current will be consi- dered to be sinusoidal. Since the studied systems are linear, non-sinusoidal pro- blems may be solved by superposition.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841190

Cl-934 JOURNAL DE PHYSIQUE

The induction equation, when combined with the Ohm's law, leads to :

(1)

-

-t

a3/,

where A is the vector potential and where b ,, ~respectively denote the electri- cal conductivity of the conductors, the angular frequency and the complex number such as : i2= -I.

According to equation (I), it follows, in polar coordinates (n , e ) : (2) J/, = - ^{i w A + av/n2,}

Where V is the electricalpotential which is supposed to decrease linearly along each turn so that : ~ V / $ J Q =

where d is a coiletailt.

In the co-axial system presently studied, the electrical density and vector poten- tial only have an azimuthal component.loreover, it is convenient to express equa- tion (2) in a curvilinear frame /5/ with the boundaries C1, C2 ... taken as geodesic lines (see figure 1) : Let x and y be a system of local coordinates respectively tangential and perpendicular to the boundaries C1, C2, ...

So . eauation (2) is written :

( 3 )

J"ryc

~ L u S ~ U . . .

Av is the voltage drop along the turn which contains the point (x, y ) . ^{The func-}

tion F can be expressed in terms of LEGENDRE elliptical integrals 11 and I2 of the first and second kind, respectively.

111 - SOLUTION TO THE GENERAL EQUATION

The electromagnetic skin depth, $ , is supposed small enough to set the local density J (x, y ) with the form :

( 4 ) JC-,+ ^{= J(=).} ~ ~ F [ - ( ~ + ~ ) " a / l J

which represents the classical solution of' the diffusion of an alternating magnetic field in an infinitely plane boundary. From equation ( a ) , and after integration of equation (3) through the skin depth, it yields :

( 5 )

J ( x y f = - i w

The boundaries C1, C2, ...

density ~ ( x ) is assumed to be constant. Considering the subsection numbered i , the density J; is now given by :

In the case where the coil is composed of a set of several (say Nt) turns, all the subsections of the same turn will be interrelated, this given the fact that total current of all these elements must be equal to the total conductor current. The set of the N unknown densities and the Nt voltage drops can now be obtained by finding the solution of a N + Nt) x (N + Nt) complex linear system. The resulting system is of the form : [A]. [J] = [B]. This matrix equation has been solved by inverting the A matrix.

The phase angle ( ? ) between the complex quantities AV (total voltage drop) and I (total current) leads to the values of the electrical resistance and inductance :

( 7 ) R = AV. MY/, - z, A v . A ~ ~ ' P / ~ . ~

On the other hand, the value of the resistance R can also be calculated by the mean

: our solution

apc=43p ; % = A ~ ~ H

-+*-*-: ~ c l u t ~ o n o b t e l n e d - w t t b

a f l u t e eiemen,tts method

%bL=35pa ; kf= 4 3 p H

FIGURE 2 : The electrical density at the surface of the conductor is plotted against the linear abscissa.

Cl-936 JOURNAL DE PHYSIQUE

THE SINGLE TURN OF RECTANGULAR CROSS-SECTION

Let us consider a copper single turn carrying a 5 Khz sinusoidal current. Figure 2 shows the surface density of current along the half perimeter of the conductor,firs- tly computed with the method presented here and secondly with a finite elements technique /6/. The agreement between the two methods is quite good although the ac - dc resistance ratio is 19 _% higher with our method. This point can be ex- plained by the slight discrepancies appearing near the inner edges of the conductor, where very high gradients of electric current and magnetic field occur.

IV CONCLUSION

A method of predicting current distribution in circular geometries h&s been deve- lopped. The particular case of a rectangular single turn has been studied : the current distribution have been found with a good accuracy except near the corners where some discrepancies have been observed. This last point could be later improved by choosing a smaller mesh in these regions. A method considering corner effects as a separate problem could be eventually used /2/.

On the other hand, more efficient formulas could be tested for calculating 'he mu- tual and self inductance values of the sub-segments (see /1/ for example).

Anyway, the technique described above is interesting for the study of multi-turns coils for whose a great deal of computing time can be saved.

ACKNOWLEDGMENT

The authors wish to thank Mr G. MEUNIER and Mr B. MOREL for many useful discussions and helpful comments in the use of their interactive computer system : FLUX 2D

R E F E R E N C E S

/1/ BURKE P.E., RYFF P.F., BIRINGER P.P., SOLGER E., Proceedings of the IEEE PICA Conf. (1969) pp 464-482.

/2/ BIRINGER P.P., RYFF P.F., SEGSWORTH R.S., Proceedings of the IEEE IGA Group Ann. Meeting (1968) pp 105-116.

/3/ KONRAD A., CHAR1 M.V.K., CSENDES Z.J., Proceedings of the IEEE Conf. on Magne- tics, Vol. MAG-18, n02 (1982), pp 450-455.

/4/ CSENDES Z.J., KONRAD A., The Mathematical Challenge, Eds. Philadelphia, SIAM 1980, pp 460-467.

/5/ FAUTRELLE Y.R., J. Fluid Mech., vol. 102 (1981), pp 405-430.

/6/ GIRARD P., MOREL B., SABONNADIERE J.C., 8th Int. Conf. on Magnet Techn., (19831, Grenoble, France.