I. DETERMINATION OF THE DYNAMIC INDUCTANCE The total magnetic energy W can be determined as

The Computation of the Dynamic Inductance of Magnet Systems and Force Distribution in Ferromagnetic Region on the Basis of3–D Numerical Simulation of

Magnetic Field

N.Doinikov,V.Kukhtin,https://www.wendangku.net/doc/f22642051.html,mzin,B.Mingalev,Y u.Severgin,S.Sytchevsky,

The D.V.Efremov Scienti?c Research Institute of Electrophysical Apparatus,

189631,St.-Petersburg,Russia

Static and dynamic inductances are ones of the main technical parameters of magnet systems at the designing stage.Pondero-motive force distribution is required for mechanical stress calcu-lations.

The static inductance is used for evaluations of the stored en-ergy,magnetic?ux linkage in coils at instantaneous currents. The dynamic inductance allows to de?ne the interrelation be-tween the instantaneous?ux linkage and currents determining a transient process in coils.

In the given paper a technique for determination of the dy-namic inductance for magnet systems on the assumption of no eddy currents in ferromagnetic elements of a construction is pro-posed.This technique necessitates the evaluation of magnetic energy at two rather close values of current in a coil on the mag-netization curve,i.e.static parameters of a magnet system are applied for the determination of the dynamic inductance.As, at present,magnet systems are more frequently designed on the basis of a magnetic?eld distribution analysis,obtained as a re-sult of numerical simulation,the calculation of magnet energy involves no dif?culties.

Algorithmic aspects of numerical simulation of speci?c and surface ponderomotiveforce loads for practical needs for design-ing electrophysical devices are given.

I.DETERMINA TION OF THE DYNAMIC

INDUCTANCE

The total magnetic energy W can be determined as follows[1], [2]

where,-are the magnetic induction and strength vectors;

-is the vector potential();

,,-are the density vector,total current and?ux linkage of the K-th coil;

-is the transverse cross-section of the K-th coil;

-is the loop of an elementary current?lament with cross-section.

In the general case the dynamic inductance of a current coil is known[4]to determine the velocity of the magnetic?ux linkage with this coil

Let us de?ne:

where-is the static inductance.

Without limitation of the commonness let us consider the case of one coil().Then

The?nal expression for is the following:

(1) where

The determination of according to(1)necessitates the two-fold computation of the problem for a magnet system to de-?ne the energy increment.However,due to a small cur-rent increment in a coil the results of the previous numerical simulations are rather well initial approximation for subsequent computations.The ef?ciency of a similar procedure is substan-tially increased,if determination of dependency is needed.

In the case,when and are known,can be de?ned using the central difference of the form

(2) where

For the case of coils,matrix of the dynamic inductances is cal-culated in the similar way to(2).

II.PONDEROMOTIVE FORCE SIMULA TION The problem of ponderomotive force determination has been discussed in[1],[6],[3],[7].

In using the?nite element method for spatial magnetic?eld simulation it is assumed that magnetic permeability to be con-stant in each?nite element.Such an approach permits a required accuracy of calculations of magnetic induction components and magnetic intensity ones,as well as?eld,energy,inductance and so on.In this case it is naturally to use a linear dependence be-tween magnetic inductance B and magnetic intensity H for cal-culations of the ponderomotive force.Thus,detailed distribu-tion of”the equivalent density”of ponderomotive force in ferro-magnetic[1]can be constructed by using the”Maxwell Stress”, .In the given model all of the?nite element sides are ”strong break surfaces”[6]in electromagnetic?eld.For real ge-ometry of magnet systems such surfaces are interfaces,on witch the surface density of ponderomotiveforce has physical sense[6] De?ning the outward normal from media”1”to media”2”one can obtained the expression for the ponderomotive force density, acting upon the interface

(3) Thus,an algorithm of ponderomotive force calculations per-mits to?nd the speci?c ponderomotive force density,as well as the surface one,acting upon the interfaces of electromagnetic value break.Also it is possible to?nd the resultant force applied to the whole body,as well as to the part of the body taking into ac-count small construction gaps.Such an approach has been used and appropriate software FERROPON(Finite Element,FERRo-magnet continua,PONderomotive force distribution)has been developed.Though this software is a part of the KOMPOT pro-gram package[5],it can be easily used separately for pondero-motive force calculations for available distribution of magnetic ?eld.

References

[1]I.E.Tamm.The Foundations of The Electricity Theory.M.:

Nauka,1976,p.616.

[2]H.Buchholz.Elektrishe und magnetische Potentialfelder

Springer-verlag,1957

[3]J.Simkin.Recent developments in?eld and force compu-

tation.Jorn.de Physique,c1,v45,No1,1984

[4]L.R.Neiman,K.S.Demirchyan.The Theoretical Founda-

tions of Electrical Engineering.V.2.L.:Energia,1975, p.407.

[5]Doinikov N.I,Lamzin E.A.,Sytchevsky S.E.On Computa-

tion of3-D Magnetostatic Fields of Electrophysical Appa-ratus Magnet Systems.IEEE Trans.on Magnetics,V.28, No.1,Jan.1992,pp.908-911.

[6]L.I.Sedov.Mechanics of continua.v.I,M.:Nauka,1983,

p.528.

[7] A.V.Ivanov–Smolensky.Electromagnetic forces and en-

ergy transformation in electrical machines.M.:Vishaia shkola,1989,p.312.