IEEE TRANSACTIONS O N MAGNETICS,VOL. 2 5 , NO.5 , SEPTEMBER 1989
4153
FL%DAI”TAL MATIONS FOR EDDY CURRENT ANALYSIS BY USING THE A-# METI”@D AND 3-D ANALYSIS OF A a N D U C T I N G LIWID
Takatsugu Ueyama, Ihtsuhiko Shinkura, Ryuzo Ueda Kyushu Institute of Technology Nippon Steel Corporation Kitakyushu, Japan 804 Kit.akyushu,Japan 805 Abstract-The fundamental equation for the A-# method is solved simiiltaneously. However, an unknown variable p formulattd rising .?laxwell’sequations and t,he continuity is included eqn.(2). The number of the unknown variables equation for electric charge. Maxwell’s equations nema- is therefore more than the number of equations. Eqn.(S) sari ly hvolve t,he c.0ntinuit.y equation under the assump- and the continuity-eqn.(6) for electric charge can thus tiori that the displacement.current can be ignored. How- k generally itsel as R fundamental equation for the ever, in the FEM (FiniteElement Method), an acceptable analysis of the eddy current field. solution can not be given if the continuity equation is not i..reated cis an independent equation. This paper clardiv ( Jo+Je ) = 0 (6) ifies the situation theoretically and the physical sigriific!ance i s discitssed. A model of a r*onducting liquid Generally, div rot G = 0 ( G is a vector), and so the in a mo\ing magnetic field is analyzed and the validity divergence of both sides of eqn.I5 1 gives eqn. ( 6 ) . of the proposed formulation is verified by experiment. Therefore, eqns.(5) and ( 6 ) are not independent. INTRODUCTION The A-# method is most popular for analyzing magnetic fields. Various procedures for solving Maxwell’s quations composed of the variables A and # have been proposed [ I ] for 3-D (3-dimensional)ed4y current.analysis. Among these, a popular one is to take ?laxwell’s equations and t.he cont.inuity equation for electric charge irit.o account, but leaving out the Maxwell’s equation representing the relation between electric charge and electric flux der1sit.y [ 2 , 3 ] . However, the mathematical problems related to the fundimenta1 equations in the A-# met.hodand the physical meaning of the continuity equation have not been made clear. These problems are treattd in t h i s paper. In the problem treated here thr freqiiency is very low, so we can neglect the displacwnetit current term in Maxwell’s equations. The continuity equation is then included in Maxwell’s equations and is not additionally needed to solve them in the case of continuous equations, But when these equations are discretized, the continuity equation becomes independent of -ell’s equations. In addition, taking into arcourit.the boundary area in which the conductivity is changed, the equation relating electric charge and rlect.ric flux can not be neglected. 111 FEM, the continuity equation has its own role. This paper clarifies the physical meaning of this situation. A realistic solution regarding the eddy current is presented. FUNDAMENTAL EQUATIONS
Elaxwell’s equations are given by rot E+aB/at = 0, div B 0, rot H div D = p
J+aD/at.
y
.rot A
)
Jo+Je
The application of Galerkin’s method to eqns.(5) and ( 6 ) gives integral equations ( 7 ) and ( 8 ) -!;rad
Nix
Y
(
.rot A ).dV
+
.jwA+grad# ).dV
CT
(5)
where Ni,Q ,Sl,S2,nare the interpolation function, the space for analysis, the boundary surface of Q , the conductor surface and the unit.vector normal to S1 and 52, respectiwly. n Defining k = C N.j.Ax.i, Ay=C Nj*Ay.j etc., and writJ - I
J’1
ing t,heabove equations for one element in matrix form, r e have (10) K@x K @ y K ~ z K@#
where & = ( r l X 1 , . . . , A x n ) T and other A vectors are potentials at nodes: n is the number of the nodes; and, I(. are n x n matrices constituting a 4n x 4n stiffness matrix. K;! is defined as the (i,,j) component of K - a . The following equations are then obtained.
(2)
.
(
Di2cTgtizgziion of the equations with the FIN
(1)
where E,B,H,J,D,p are electric field intensity, magnetic flux density, magnetic field intensity, current density, electric flux density, electric charge density, respectively. Let f i , 6 ,E. be denoted as, respectively, magnetic permeability, conductivity, dielectric constant: then the following equations are defined: (3) B = ,uH, J = a E , D = E E The A-q5 method introduces a magnetic vector potential A and an electric scalar potential q5 , and represents B and E with equations as described below: B = rot A, E = -aA/at-grad# (4) Eqns.(l), ( 3 ) and (4)can be transformed into eqn. ( 5 ) by ignoring the displacement current, and replacing a /at by j w rot
EQUATIONS
INDEPFMENCE IN F ” D A . . A L
(5)
where Je -6 ( .jwAtgrad qj ) ; Jo.Je,y are forced current density, ed4v current density and magnetic resistivity, respectively. Eqns. ( 5 ) and ( 2 1 must be
K j=
k(
v
aN’
aNj
aN‘
aNj
az
az
ax
ax
(11)
,
X -+ v 1 2 I -+ .I o
U
yNi
Nj ) d v
An element of a rectangular solid with 8 nodes is considered and a function Ni at node i for interpolation is represented as follows: 112) Ni ( 1t.E 6 i ) ( 1 + q v , ) ( 1+c t , ) / 8 where E = ( x-xc )/a, 7 = ( y-yc )/b, { = ( z-zc )/c and the cent.erof the element is defined as (sc,yc,zc), with sides length 2a, 2b and 2c. The integrands in eqn.(ll) art: denoted as k : ; , . . . , and eqn.(12) is substituted into them. Eqn.(l3) is then obtained.
kl:(g X
,?)
(1+0
,c
)
-Y
7 j)ll+{
Z
E
jq
,(l+E 5
I )
t, ,)(l+f5 j)/64ab
etc.
(13)
Independence Integrands in eqn.I 11) obey the linear relations as f o l lows
0018-9464/89/0900-4153$01 .OO01989 IEEE
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4154
has beer, chosen here only for the purposes of explanation, but the gat ge condition 1.s not required for FE1 analysis in most analytic, models. The i.oLe (-If the continuity equation of electric charge in F F 1 is n o w discussed. By putting C2 in eqn.(9) t :eix,, I tic. sunmmtion of current, n-Je which flows out of t h e boimdar~surface S2 becomes zero. Eqn. (8)does !lot Iiold .in the region c<fa =O. Therefore the surface 52 I eprcsents t l i + boundary between the conductor and iiis;tlator,X I I ~ the relation C2=0 implies making the current f l w i r g out of the cnrlduct.or equal to zero. This Lwimdar> c o i d i t i o n is inevitable for the solution of @ 1-rpresentiiig the electric charge distribution. We can prove this by (.omputer simulation. Fig.1 shows a model f i t for. intlicatin; the role of a scalar potential @ [SI. 'I'liis ns.dt.1 w s anaLyz4 iriththe FIPl formulated from etps. ( 5 ) am1 ( 5 ) . F i s . Z ( a ) shotis the result of the analysis 1 1 1 :.he copper. plate and suggests that the airdlysis ik reasonable. The value of @ is not equal to .:q<ri> s lid i epi.esents the esistence of electric charge on thc2 cond~ictorh u n r h r y on which t.he conductivity d is < + i a i ~ ~ rt l. ( ~ ~ ~ r o (hi . the other hand, tie solve eqns.( 5 ) :itid e q r i . ( 2 2 1 simultxneously instead of solving eqns.( 5 ) :+:id t,hP (.on1 I t n i i t y equatj.on for electric charge (eqn. ,.I
where A=& ,/a(l+& E I ),B=q ,/bc1 + 7~ I ) ,C={ ,/c(1+5 5 , ) Numerical integrals are carried out in eqns.(ll). They are represented by polynomials of 1 3 terms, where I is the order of each integral. ( 4 n example shown below has eight terms.)
,cm
%here w l i , w I , w are weight coefficients and 6 k , q 1 are coordinates of an int.egra1 point.. Each term of eqn.( 15) has a linear relation represented as w
I * W i*{AI*k::IE +C1.kzy(E1 1 7 I , <
1.w
i,q 1 ) )
1 . 5 i)+Bl.k::(E 1 , q = kg y ( g 1 , q 1 . 5 I )
I , <
I )
etc.(l61
(6)).
d i v A = 0 (C:oulombgauge) (22) By applying Gal.erkin's method t.o eqn. (22), we obtain where Ah = e 1 /a(l+E k t i ),Bk = q i/b( 1+q k q i ) I eqn.(23) and solve eqri.(S) together with eqn.(23) inCk = i/C(l+C k < I ) However, w e n if there exists a linear relation in each stead of eqn.(8). JQA*grad Ni dV =where C3 c3 = S2i.n-A~ S = O(231 term, the sane linear relation is not always valid in their sinmation. For example, consider a linear relation, given in eqn.(17), between two sets of Xl,Yl,Zl and X2,Y2,Z2. In this case we can not find A aid H shown in eqn. (18). AI-Xl+Bl.Yl= 21 A2*X2+B2*Y2= 22 (17) (18) A.(Xl+X2)+B.(Yl+Y2)= Zl+ZZ That is, any linear relations among .Y1+?12,Yl+Y2 and %1+ 22 do not hold. The same situation is observed in the set of ~K::,K;:,..-) In more concrete terms, even if there exists a linear relation among the integrands {ki; ,k;i, * * ) , we cannot find the linear relation in the sumnation of the integrands. Therefore, each equation in eqn. (101,which is derived after numerical integration, is independent of the others. This is true even when the order of integral is infinite o r even when another Power Source:60 Hz element. such as a pyramid is adopted. Eqn. (10) thus .?lametomotive force of coil:7200 AT gives a unique solution in any case. This independence Conductivity of copper plate:5.814x107S/m is caused by making an approximation for the potentials Fig.1 The model(from Ref.161) A wid ai 11sin.gthe internolation function. This indenendence breaks down if the degree of the approximation is increased or if the element division is made sufficiently fine.
.
F'HYSICAL
MEAJING
OF THE LDN'l'INC'ITY
EQUATION OF ELECTRIC
BA!EE In this section, it is shown that the continuity equation of electric charge (eqn.(6)) has an important significance in FEM. Modifying eqn.(6) and substituting eqns.(2) and ( 3 ) into it, then we obtain eqn.(19), diva E = E.grada +a .divE = B.grada + ( 6 / E ) p = 0 ( e /a )-E.gradu -p (19) Eqn.(19) implies the presence of electric charge &here a gradient.o r discontinuity in 6 occllrs. Since the equation representing the electric charge is eqn.(2), we can not. obtain the solut,ion f o r the electric charge by using eqn.(5) derived only from eqn.(l). From eqn.(2) we obtain the following 8 -div E = - e {(a/at).div A+div.grad @ } = p (20) Assuming the @ulomb gauge relation div A = 0, eqn. ( 2 0 ) is :educed t.o E .div.grad g = - p ( 21 ) Eqn.(21) shows that the electric scalar potential @ is t.hn i-ariablerepresenting the electric charge. That is, in determining @ , eqn.(2) determining the distribution of elect-ric charge has t.obe irivolved. The (:.xhxnb gauge
(a) eqns.(5)and(61 (b) eqns.(5)and(l9) Fig.:! Eddy current in a copper plate (Thick lines indicate location of coil, arrows show the imaginary part of eddy current) The result is indicated in Fig.2(b) but does not give ?tie i*orrect solution. The eddy current is not prevented
by the slit. The value of @ is reduced to zero in the wholr spce. The electric charge p h*ich should exist 0 1 1 the boundary of the conductor and prevent the current â&#x201A;Ź 1 owing c+ut of the conductor is not taken into account. This can lx exiplained as follows: the boundary condition (y3 ,given h> r q r 1 . ( 2 3 ) does riot include the boundary Condition C2 given by eqn.(8), meaning that the current. flowing out. <,f conductor is zero. In addition, when
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4155
shohn in Fig.tJ. I n type 1 , induced current in the fluid f m m s paths .going ro\cnd on the sides of the flow to satisfy the continuity condition, whereas in type 2 , p 1 t . 1 ~of~ c w r e r t t are formed on the electric poles on the sides nf the floh, and the secondary current has only a :' coinlmnerlt (the electromagnetic force has only an S compnent). In this case, the effect from both sides tlc.les not appear auvl the result of 3-D analysis is almost equcil t o t,hst2 rtf 2-n analysis 1x2 plane), whereas in t y p I , the l-ffect~ive elec;tl.omagnetic force is decreased by 1.his effe-t. In this example, the results of 3-1) :malysisgive values abollt, 75% of that of 2-D analysis, agreeing k-ell h'ith the result of the experiment.
eqn.( 8 ) is applied to at least two elements whose conducti\-itiesare 6 1 and 6 2 , eqn.(24)[41 is derived as the continuity condition of current on each boundary br putt irig t~het.ermrorrespnding to C2 equal to zero. d l e ( .joA+srad # 1 1-n d 2 - ( j oA+grad # Z )-n (24) E q n . 1 6 ) is indispensable in the F'RI because the term C2 in eqn.(8) is not represented in eqn.(7). In slnronary, at such places where d changes, the continuity condition of current represented by C2=0 in eqn.( 9 ) enables the xalues of A and qi to be obtained reflecting the existence of the electric charge on the boundary. Hence in the FEN, it can be concluded that eqn,(6)plays the same role as eqn.(2), which is physirally independent of eqn.( 1 ) . After A and # are obtained in the above way, eqn.(2) is reduced to only a definit.ionof p ANALYSIS AND EXF'ERIMENT FOR A CONDUCTING FLUID
Apmratus and conditions of experiment As an application the flow control of alloy of low i a ) 1 r l s l ~ l : i i - d chanrlel (b) Copper-po melt.ing point was carried out by using a LIM (Linear In( T . w 1) (T;vpe 2) Linear motor:pole pitch 69 mm,power 220V.150A duction %tor) with an apparatus shown in Fig.3 as a fundamental study before proceeding to the practical .Alloy of low melting point:composite 50Bi-30Pb-20Si flow control of molten steel. The result of experiment nleltirg point 95째C ,conductivity 8.OxlO5S/m was compared with that.of analysis. Molten alloy flows specific gravity l.O~104kg/m~,weight 160kg from an upper container, through a flat flow channel, Fig 4 Analytic model int.0a lower container. A IJM, installed at the center 8M of the channel along the flow, generates electromagnetic I force which acts on the alloy flowing through this = 0 Measured Measured .4z:.; s MO region. Alternating magnetic field penetrates verti20,1 9D cally through the flow and moves parallel to it. Two 5 t-ypes (of .:harmel shown in Fig.4 were used in the 4w "m experiment. The experiment was carried out by pouring 5 I 5 the alloy into the upper container and 1et.ting it flow , mo int.n the lower container. The electromagnetic force * h ( m n head) with and without the effect of the LIM was 1 calculated by measuring the flow time and using eqn. ( 2 5 ) On 50 120 LBO uo 3w '0 M iao 180 uo am h i 1-( toff/ton ) 1 . H (25) FrequencdHz) Frequency (Hz) where H is the difference in meniscus level between the (a)TS-pe 1 ( b ) Type 2 ripper arid the lower container; toff: the flow time F i g 5 Hlectmmagnetic force on the fluid without the effect of the LIM; and, ton: the flow time with the effect of the LIM. The analysis employs 3-D FEM with eqns.(5) and 16) as the fundamental equations. Conventional analyses of 'IIII' ' a LIM include a quasi 3-D walysis[7] which combines FEN for 2-D flow of secondary current with an anal.vtica1 1111' * solution. However, when a LIM has a conductor of general form where distributed force is important, a st.rictly3-D electromagnetic analysis is necessary. Examples of this typ? of analysis are few. The analysis I .82;lO7 A h ' space is half of a plane cutting the fluid vertically in (a)Type 1 (b) Type 2 Fig.5. It is divided into 1680 (21~8x10)elements with Fig.6 Distribution of eddy current in the fluid 2178 nodes.
= 9
k
/+'
d
im/i
coNcLus1oN
Upper c o n t a i n e r
This paper deals with the importance of the continuity equation of the electric charge for FEM. An analytical example is given for a conducting fluid. In mcist analytic models where the FEM is applied, A is we1.I-defined by boundary conditions, and this, we ctinsider, makes the gauge condition unnecessary. Further st,udieswould elucidate this point. REFERENCE
Fig.3 Outline of experimental apparatus Results of analysis and experiment Fig.5 shows the result of experiment and analyses of electxomagnetic force in types 1 and 2. In 3-D analysis, the result,of calculation agrees quite well r;ith measurements f o r both types. The distribution of eddy current in the fluid obtained by the analysis is
[ 11 Hashizme, " A study on the gauge",The electromagnetics symposium preprint in Japanese,(1987) [2] S.J.Salon,J.P.Peng,IEEE Trans.Magnetics,MAG20,NO.5 ppl992,(1984) r 3 ] T.Yakata, " Magnetic fields ",The Journal of The Electrical engineers of Japan,Vol.108 N0.3 pp213,(1988) 141 C,W.Trowbridge,F'roceeding of BIS3"88,Beijing,(1988) [5] T.Onuki,IEEJ Trans.Vol.108D N0.11 pp1049,(1088) [6] T.E'oshimoto,S.Yanada,Ii.Bessyo,IEEJ Trans.,Vol.107D NO.6 pp796,(1987) [7] S.Nonaka,IEEJ Traris. Vol.lO5B N0.3 pp211,(1985)
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