1) RESULTADOS FEMM
bobina1
ferro1
bobina2
ar
ferro2
Caso 1: f=0
1.994e+000 : >2.099e+000
1.889e+000 : 1.994e+000
1.784e+000 : 1.889e+000
1.679e+000 : 1.784e+000
1.574e+000 : 1.679e+000
1.469e+000 : 1.574e+000
1.365e+000 : 1.469e+000
1.260e+000 : 1.365e+000
1.155e+000 : 1.260e+000
1.050e+000 : 1.155e+000
9.447e-001 : 1.050e+000
8.397e-001 : 9.447e-001
7.347e-001 : 8.397e-001
6.298e-001 : 7.347e-001
5.248e-001 : 6.298e-001
4.199e-001 : 5.248e-001
3.149e-001 : 4.199e-001
2.099e-001 : 3.149e-001
1.050e-001 : 2.099e-001
<0.000e+000 : 1.050e-001
Density Plot: |B|, Tesla
Caso 2: f=60Hz
1.973e+000 : >2.077e+000
1.869e+000 : 1.973e+000
1.765e+000 : 1.869e+000
1.662e+000 : 1.765e+000
1.558e+000 : 1.662e+000
1.454e+000 : 1.558e+000
1.350e+000 : 1.454e+000
1.246e+000 : 1.350e+000
1.142e+000 : 1.246e+000
1.038e+000 : 1.142e+000
9.346e-001 : 1.038e+000
8.308e-001 : 9.346e-001
7.269e-001 : 8.308e-001
6.231e-001 : 7.269e-001
5.192e-001 : 6.231e-001
4.154e-001 : 5.192e-001
3.115e-001 : 4.154e-001
2.077e-001 : 3.115e-001
1.038e-001 : 2.077e-001
<0.000e+000 : 1.038e-001
Density Plot: |B|, Tesla
Caso 3: f=180Hz
2.276e+000 : >2.396e+000
2.156e+000 : 2.276e+000
2.036e+000 : 2.156e+000
1.917e+000 : 2.036e+000
1.797e+000 : 1.917e+000
1.677e+000 : 1.797e+000
1.557e+000 : 1.677e+000
1.437e+000 : 1.557e+000
1.318e+000 : 1.437e+000
1.198e+000 : 1.318e+000
1.078e+000 : 1.198e+000
9.583e-001 : 1.078e+000
8.385e-001 : 9.583e-001
7.187e-001 : 8.385e-001
5.989e-001 : 7.187e-001
4.791e-001 : 5.989e-001
3.594e-001 : 4.791e-001
2.396e-001 : 3.594e-001
1.198e-001 : 2.396e-001
<0.000e+000 : 1.198e-001
Density Plot: |B|, Tesla
Caso 4: f=1kHz
2.593e+000 : >2.730e+000
2.457e+000 : 2.593e+000
2.320e+000 : 2.457e+000
2.184e+000 : 2.320e+000
2.047e+000 : 2.184e+000
1.911e+000 : 2.047e+000
1.774e+000 : 1.911e+000
1.638e+000 : 1.774e+000
1.501e+000 : 1.638e+000
1.365e+000 : 1.501e+000
1.228e+000 : 1.365e+000
1.092e+000 : 1.228e+000
9.554e-001 : 1.092e+000
8.189e-001 : 9.554e-001
6.824e-001 : 8.189e-001
5.459e-001 : 6.824e-001
4.095e-001 : 5.459e-001
2.730e-001 : 4.095e-001
1.365e-001 : 2.730e-001
<0.000e+000 : 1.365e-001
Density Plot: |B|, Tesla
2) RESULTADO MATLAB
Caso 1:f=0Hz
Caso 2: f=60Hz
Parte real:
Caso 3: f=180Hz
Caso 4 f=1kHz
Código Matlab:
N=703;
nel=1367;
freq=xx;
w=2*pi*freq;
nnel=3;
xx=[];
yy=[];
Bx=[];
By=[];
ind=[];
count=0;
count2=0;
load conec5.txt;
for i=1:nel
nodes(i,1)=conec5(i,1)+1;
nodes(i,2)=conec5(i,2)+1;
nodes(i,3)=conec5(i,3)+1;
end
load coord5.txt;
for i=1:N
gcoord(i,1)=(coord5(i,1))*1e-2;
gcoord(i,2)=(coord5(i,2))*1e-2;
end
bcdof=[];
bcval=[];
dirud=[];
for i=1:N
dirud_=(coord5(i,3));
dirud=[dirud dirud_];
if dirud(i)==0
bcdof_=i;
bcdof=[bcdof bcdof_];
bcval_=0;
bcval=[bcval bcval_];
end
end
for i=1:N
if dirud(i)==0
ind(i)=-1;
else
count=count+1;
ind(i)=count;
end
end
ff=zeros(count,1);
kk=zeros(count,count);
f=zeros(3,1);
kel=zeros(3,3);
for iel=1:nel
nd(1)=nodes(iel,1);
nd(2)=nodes(iel,2);
nd(3)=nodes(iel,3);
x1=gcoord(nd(1),1);y1=gcoord(nd(1),2);
x2=gcoord(nd(2),1);y2=gcoord(nd(2),2);
x3=gcoord(nd(3),1);y3=gcoord(nd(3),2);
if conec5(iel,4)==0
[kel,f]=matriz_elementos(x1,y1,x2,y2,x3,y3,ind,nd,dirud);
end
if conec5(iel,4)==1
[kel,f]=matriz_elementos2(x1,y1,x2,y2,x3,y3,ind,nd,dirud);
end
if conec5(iel,4)==2
[kel,f]=matriz_elementos3(w,x1,y1,x2,y2,x3,y3,ind,nd,dirud);
end
if conec5(iel,4)==3
[f]=fontecorrente(x1,y1,x2,y2,x3,y3);
[kel,f]=matriz_elementos4(x1,y1,x2,y2,x3,y3,ind,nd,dirud,f);
end
if conec5(iel,4)==4
[f]=fontecorrente2(x1,y1,x2,y2,x3,y3);
[kel,f]=matriz_elementos4(x1,y1,x2,y2,x3,y3,ind,nd,dirud,f);
end
[kk,ff]=matriz_global2(kk,kel,nd,ind,ff,f);
end
fsol=kk\ff;
fsolk=[];
for i=1:N
if ind(i)==(-1)
fsolk(i)=0;
end
if ind(i)>0
count2=(count2)+1;
fsolk(i)=fsol(count2);
end
end
%
for iel=1:nel
nd(1)=nodes(iel,1);
nd(2)=nodes(iel,2);
nd(3)=nodes(iel,3);
x1=gcoord(nd(1),1);y1=gcoord(nd(1),2);
x2=gcoord(nd(2),1);y2=gcoord(nd(2),2);
x3=gcoord(nd(3),1);y3=gcoord(nd(3),2);
xx_=(x1+x2+x3)/3;
yy_=(y1+y2+y3)/3;
xx=[xx xx_];
yy=[yy yy_];
[Bnelx,Bnely]=inducaomagnetica(nd,fsolk,x1,y1,x2,y2,x3,y3);
Bx_=Bnelx
By_=Bnely
Bx=[Bx Bx_];
By=[By By_];
end
quiver(xx,yy,Bx,By);
hold all
Xlabel('x(m)');
Ylabel('y(m)');
Title('Distribuição de B para f=xx');
Axis([0 0.35 0 0.30])
grid
hold off
function [kel,f]=matriz_elementos(x1,y1,x2,y2,x3,y3,ind,nd,dirud)
f=zeros(3,1);
kel=zeros(3,3);
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
gradNi=[-1 1 0;-1 0 1];
K=[(y3-y1) (y1-y2);(x1-x3) (x2-x1)];
B=(1/det(J))*K*(gradNi);
kel=(-1)*(B')*B*(det(J))*(0.5);
if ind(nd(1))<0
f(2)=f(2)-(kel(2,1)*dirud(nd(1)));
f(3)=f(3)-(kel(3,1)*dirud(nd(1)));
kel(2,1)=0;
kel(3,1)=0;
end
if ind(nd(2))<0
f(1)=f(1)-(kel(1,2)*dirud(nd(2)));
f(3)=f(3)-(kel(3,2)*dirud(nd(2)));
kel(1,2)=0;
Kel(3,2)=0;
end
if ind(nd(3))<0
f(1)=f(1)-(kel(1,3)*dirud(nd(3)));
f(2)=f(2)-(kel(2,3)*dirud(nd(3)));
kel(1,3)=0;
kel(2,3)=0;
end
function [kel,f]=matriz_elementos2(x1,y1,x2,y2,x3,y3,ind,nd,dirud)
f=zeros(3,1);
kel=zeros(3,3);
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
gradNi=[-1 1 0;-1 0 1];
K=[(y3-y1) (y1-y2);(x1-x3) (x2-x1)];
B=(1/det(J))*K*(gradNi);
kel=(-1)*(1/1000)*(B')*B*(det(J))*(0.5);
if ind(nd(1))<0
f(2)=f(2)-(kel(2,1)*dirud(nd(1)));
f(3)=f(3)-(kel(3,1)*dirud(nd(1)));
kel(2,1)=0;
kel(3,1)=0;
end
if ind(nd(2))<0
f(1)=f(1)-(kel(1,2)*dirud(nd(2)));
f(3)=f(3)-(kel(3,2)*dirud(nd(2)));
kel(1,2)=0;
Kel(3,2)=0;
end
if ind(nd(3))<0
f(1)=f(1)-(kel(1,3)*dirud(nd(3)));
f(2)=f(2)-(kel(2,3)*dirud(nd(3)));
kel(1,3)=0;
kel(2,3)=0;
end
function [kel,f]=matriz_elementos3(w,x1,y1,x2,y2,x3,y3,ind,nd,dirud)
f=zeros(3,1);
c=1e7;
u0=4*pi*1e-7;
f=zeros(3,1);
kel=zeros(3,3);
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
gradNi=[-1 1 0;-1 0 1];
K=[(y3-y1) (y1-y2);(x1-x3) (x2-x1)];
B=(1/det(J))*K*(gradNi);
kel=(-1)*(1/1000)*(B')*B*(det(J))*(0.5);
kel(1,1)=kel(1,1)+(1i*w*c*u0*(det(J))*0.0833);
kel(1,2)=kel(1,2)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(1,3)=kel(1,3)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(2,1)=kel(2,1)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(2,2)=kel(2,2)+(1i*w*c*u0*(det(J))*0.0833);
kel(2,3)=kel(2,3)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(3,1)=kel(3,1)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(3,2)=kel(3,2)+(1i*w*c*u0*(det(J))*4.17e-2);
kel(3,3)=kel(3,3)+(1i*w*c*u0*(det(J))*0.0833);
if ind(nd(1))<0
f(2)=f(2)-(kel(2,1)*dirud(nd(1)));
f(3)=f(3)-(kel(3,1)*dirud(nd(1)));
kel(2,1)=0;
kel(3,1)=0;
end
if ind(nd(2))<0
f(1)=f(1)-(kel(1,2)*dirud(nd(2)));
f(3)=f(3)-(kel(3,2)*dirud(nd(2)));
kel(1,2)=0;
Kel(3,2)=0;
end
if ind(nd(3))<0
f(1)=f(1)-(kel(1,3)*dirud(nd(3)));
f(2)=f(2)-(kel(2,3)*dirud(nd(3)));
kel(1,3)=0;
kel(2,3)=0;
end
function [kel,f]=matriz_elementos4(x1,y1,x2,y2,x3,y3,ind,nd,dirud,f)
kel=zeros(3,3);
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
gradNi=[-1 1 0;-1 0 1];
K=[(y3-y1) (y1-y2);(x1-x3) (x2-x1)];
B=(1/det(J))*K*(gradNi);
kel=(-1)*(B')*B*(det(J))*(0.5);
if ind(nd(1))<0
f(2)=f(2)-(kel(2,1)*dirud(nd(1)));
f(3)=f(3)-(kel(3,1)*dirud(nd(1)));
kel(2,1)=0;
kel(3,1)=0;
end
if ind(nd(2))<0
f(1)=f(1)-(kel(1,2)*dirud(nd(2)));
f(3)=f(3)-(kel(3,2)*dirud(nd(2)));
kel(1,2)=0;
Kel(3,2)=0;
end
if ind(nd(3))<0
f(1)=f(1)-(kel(1,3)*dirud(nd(3)));
f(2)=f(2)-(kel(2,3)*dirud(nd(3)));
kel(1,3)=0;
kel(2,3)=0;
end
function [f]=fontecorrente(x1,y1,x2,y2,x3,y3)
u0=4*pi*1e-7;
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
f(1)=(det(J))*u0*6e6*(0.1667);
f(2)=(det(J))*u0*6e6*(0.1667);
f(3)=(det(J))*u0*6e6*(0.1667);
function [f]=fontecorrente2(x1,y1,x2,y2,x3,y3)
u0=4*pi*1e-7;
J=[(x2-x1) (x3-x1);(y2-y1) (y3-y1)];
f(1)=det(J)*u0*(-1)*6e6*(0.1667);
f(2)=det(J)*u0*(-1)*6e6*(0.1677);
f(3)=det(J)*u0*(-1)*6e6*(0.1667);
function [kk,ff]=matriz_global2(kk,kel,nd,ind,ff,f)
nnel=3;
for i=1:nnel
if ind(nd(i))>0
ii=ind(nd(i));
ff(ii)=ff(ii)+f(i);
for j=1:nnel
if ind(nd(j))>0
jj=ind(nd(j));
kk(ii,jj)=kk(ii,jj)+kel(i,j);
end
end
end
end
function [Bnelx,Bnely]=inducaomagnetica(nd,fsolk,x1,y1,x2,y2,x3,y3)
A=0.5*(x2*y3+x1*y2+x3*y1-x2*y1-x1*y3-x3*y2);
gradN1y=(x3-x2)/(2*A);
gradN2y=(x1-x3)/(2*A);
gradN3y=(x2-x1)/(2*A);
gradN1x=(y2-y3)/(2*A);
gradN2x=(y3-y1)/(2*A);
gradN3x=(y1-y2)/(2*A);
Bnely=real((-1)*fsolk(nd(1))*gradN1x+(-1)*fsolk(nd(2))*gradN2x+(1)*fsolk(nd(3))*gradN3x)
Bnelx=real(fsolk(nd(1))*gradN1y+fsolk(nd(2))*gradN2y+fsolk(nd(3))*gradN3y)
3) CONCLUSÃO
As perdas de Foucault no material com condutividade diferente de zero (ferro 2) são
proporcionais à freqüência. À medida que a freqüência aumenta, as perdas por Foucault neste
material aumentam e há uma redução da penetração das linhas do vetor B nesta região.