计算电流线圈产生的磁场1

1、（二）求截面为矩形的圆线圈周围产生的磁场一、数值方法（一）数学模型：所研究的电流圆线圈产生磁场的问题在柱坐标系下研 究，根据磁场强度跟矢势之间的关系，得到磁场；磁场为B，矢势为A（由A具有轴对称得到）所以 B = V x A = V x A。匕在柱坐标系中，由公式f =顷(Vx f)r+ f e + f e9 9z z1 dfdf=一-上r d9 dz(VX f)9(Vx f)z1 6frr 69B = V x A = -9 e +(rf )-6z- r 6r 9 z即 B =一普，B = - 6- (rA9)（1）先求矢势A一日 IdlA = J圳4兀 rL一个电流为I，半径为.的线圆环周围

2、空间产生的磁场，其矢势表示为cos 9d 9r2 + z2 + a2 - 2 ar cos 9推广到截面为矩形的圆环线圈中A （r, z）=9尚之兀4兀s0R1 z.d 9 dz drr2 + (z 一 z)2 + r2 一 2rr cos 9其中S为矩形截面的面积，R , R为矩形截面的两边距圆环中心的距离，z , z为1212矩形截面的上下面的z轴坐标。（二）数值模型离散化（均匀网格有限差分）（1）高斯方法计算三重积分（参考书:徐士良常用算法程序集第二版）（2）根据一阶三点公式来求解磁场的分布:dA(A ) 一(A )B = -0 =0 i, j -10 i, j + 11 a(A )-

3、(A )(A )B =(rA ) = 010i hj + 0，, ji，j对于r 0点处1 aaa aa a( a )- ( a )lim B = lim(rA ) = lim ( + ) = 2 = 0 i + 1，j0 】，j0 z 0 r a r 0 0 r a r a r rB = ：B 2 + B 2二数值计算程序SUBROUTINE FGAUS(N,JS,X,FS,F,S, Z,R)DIMENSION JS(N),X(N)DIMENSION T(5),C(5),D(2,11),CC(11),IS(2,11)DATA T/-0.90617,-0.538469,0.0,0.538469

4、,0.90617/DATA C/0.2369,0.47862,0.568889,0.4786,0.2369/M=1D(1,N+1)=1.0D(2,N+1)=1.010 DO 20 J=M,NCALL FS (J,N,X,DN,UP)D(1,J)=0.5*(UP-DN)/JS(J)CC(J)=D(1,J)+DNX(J)=D(1,J)*T(1)+CC(J)D(2,J)=0.0IS(1,J)=1IS(2,J)=120 CONTINUEJ=N30K=IS(1,J)IF(J.EQ.N)THENP=F(N,X, Z,R)ELSEP=1.0ENDIFD(2,J)=D(2,J+1)*D(1,J+1)*P*C(

5、K)+D(2,J)IS(1,J) = IS(1,J)+1IF(IS(1,J).GT.5)THENIF(IS(2,J).GE.JS(J) THENJ=J-1IF(J.EQ.0) THENS=D(2,1)*D(1,1)RETURNENDIFGOTO 30ENDIFIS(2,J) = IS(2,J)+1CC(J)=CC(J)+D(1,J)*2.0IS(1,J)=1ENDIFK=IS(1,J)X(J)=D(1,J)*T(K)+CC(J)IF(J.EQ.N) GOTO 30M=J+1GOTO 10ENDEXTERNAL FS,FDIMENSIONJS(3),X(3),Z(30),R(30),a0(30,

6、30),BB(30,30),BR(30,30),BZ(30,30)DATA JS/4,4,4/N=3 c=1H=0.5Z(1)=1.5DO I=1,29Z(I+1)=Z(I)+HENDDOR(1)=1.5do j=1,29R(J+1)=R(J)+HENDDODO I=1,30DO J=1,30CALL FGAUS(N,JS,X,FS,F,S,Z(I),R(J)a0(i,j)二c*sWRITE(1,*)Z(I),R(J),SOPEN(1,FILE=DUHAI.DAT)ENDDOENDDO!求解磁场的Brdo i=1,30Br(i,1) = (-3*A0(i,1)+4*A0(i,2)-A0(i,3

7、)/2*hBr(i,30) = (A0(i,28)-4*A0(i,29)+3*A0(i,30)/2*hdo j=2,29Br(i,j) = (A0(i,j-1)-A0(i,j+1)/2*hend doend do!求解磁场的Bzdo j=1,30Bz(1,j) = (-3*A0(1,j)+4*A0(2,j)-A0(3,j)/2*h+A0(1,j)/r(1)Bz(30,j) = (A0(28,j)-4*A0(29,j)+3*A0(30,j)/2*h+A0(30,j)/r(30)do i=2,29Bz(i,j) = (A0(i+1,j)-A0(i-1,j)/2*h+A0(i,j)/r(i)end

8、doend dodo i=1,30do j=1,30BB(i,j)=sqrt(Br(i,j)*2+Bz(i,j)*2)end doend dodo i=1,30do j=1,30write(*,*)z(i),r(j), bb(i,j)write(2,*)z(i),r(j), bb(i,j)open (2,file=cichang.dat)enddoEnddoEndSUBROUTINE FS(J,N,X,DN,UP)DIMENSION X(N)IF(J.EQ.1) THENDN=0.5UP=1.0ELSEIF(J.EQ.2)THENDN=0.5UP=1.0ELSEIF(J.EQ.3) THENDN=0.0UP=2*3.1415926ENDIFRETURNENDFUNCTION F(N,X,Z,R)DIMENSION X(N)f=x(2)*cos(x(1)/sqrt(z-x(3)*2+R*2+x(2)*2-2*R*x(2)*cos(x(1)RETURNEND三计算结果矢势A的分布如下：磁场B的分布如下68101214