Matlab PdetoolLezione 4

Induttanze con ferro

Ing. Flavio Calvano

Argomenti trattati

• Calcolo auto e mutua in presenza di ferro;Caso assialsimmetrico;Caso piano

Nucleo magnetico assialsimmetrico

J

x

y

Formulazione

( )

2

2

22 2 2 2

2

1 1cos ( )

10

2 2ˆ ˆ ˆ

1ˆ ˆ

z z

con t

con

A A A AA A A

AJ A A J

ρ ρ ϕ ρρ ρ ϕ

ϕϕ ϕ

µµ µ

µ

ρ ρ ϕ ρ ρ ϕ

µ ρ

Φ

Φ Φ Φ Φ

= →∇× ∇× = −∇ +∇ ∇ ⋅

∇ ⋅ = → ∇ = −

∂ ∂ ∇ = ∆ − − + ∆ − − + ∆ ∂ ∂

= ⇒ = ⇒ −∆ + =

A A A

A A J

A i i i

J i A i

Equazione risolvente

( )

( )

2

2

0 02 2

2

2 2 2

1 ( )

1 1( )

1 1

1 1 ( )

1 1( ) ( )

A A Az

A AA J A A J

z zAA A

A AA A A

A

Jz z

ϕ

ϕ ϕϕ ϕ ϕ

ϕ ϕ

ϕ

ρρ ρ ρ

µ ρ ρ µρ ρ ρ ρ ρ ρ

ρρρ ρ ρ ρ ρ ρ ρ ρ

ρρ ρ ρ ρ ρ ρ ρ ρ

ρ

ρ µρ ρ µρ

Φ Φ

Φ Φ

ΦΦ Φ

Φ ΦΦ

Φ

∂ ∂ ∂∆ = +

∂ ∂ ∂

∂ ∂ ∂ ∂−∆ + = ⇒ − + − =

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

− = − − =∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− − = − +

∂ ∂ ∂ ∂

⇓Ψ =∂ ∂ ∂ ∂

− Ψ − Ψ =∂ ∂ ∂ ∂

Geometria[pde_fig,ax]=pdeinit;pdetool('appl_cb',1);set(ax,'DataAspectRatio',[1 1 1]);set(ax,'PlotBoxAspectRatio',[1 1 1]);set(ax,'XLim',[-0.1 0.8]);set(ax,'YLim',[-0.4 0.4]);set(ax,'XTickMode','auto');set(ax,'YTickMode','auto');ymin=-1.e-2;ymax=1.e-2;xmin=0;xmax=0.018;xmax2=0.028;xmax3=0.032;dh=0.04;dh2=0.1;dr=0.002;dr2=0.01;l=0.001;pderect([xmax2-dr/2 xmax2+dr/2 -dh/2 dh/2],'R1');pderect([xmax3-dr/2 xmax3+dr/2 -dh/2 dh/2],'R2');pderect([xmin xmax -dh2/2 dh2/2],'R3');pderect([xmin xmax+dh2 dh2/2 dh2/2-xmax],'R4');pderect([xmin xmax+dh2 -dh2/2 -dh2/2+xmax],'R5');pderect([xmax+dh2-(xmax) xmax+dh2 -dh2/2 -l/2],'R6');pderect([xmax+dh2-(xmax) xmax+dh2 l/2 dh2/2],'R7');pderect([0 8*dh2 4*(-dh2) 4*dh2],'R8');

Esportazione geometria e meshset(findobj(get(pde_fig,'Children'),'Tag','PDEEval'),'String','R1+R2+R3+R4+R5

+R6+R7+R8');gd=get(findobj(get(pde_fig,'Children'),'flat','Tag','PDEMeshMenu'),'UserData');dl=decsg(gd);

[p,e,t]=initmesh(dl,'Hmax',5e-1,'init','off');[p,e,t]=refinemesh(dl,p,e,t,'regular');

box=find(t(4,:)==1);ferro1=find(t(4,:)==2);ferro2=find(t(4,:)==3);ferro3=find(t(4,:)==4);ferro4=find(t(4,:)==5);ferro5=find(t(4,:)==8 );ferro=[ferro1 ferro2 ferro3 ferro4 ferro5];spira1=find(t(4,:)==6);spira2=find(t(4,:)==7);

pdeplot(p,e,t(:,ferro))

Plot della mesh relativa alla regione ferro

Condizioni al contorno

h=findobj(get(pde_fig,'Children'),'flat','Tag','PDEBoundMenu');bl=get(findobj(get(h,'Children'),'flat','Tag','PDEBoundMode'),'UserData');

pdetool('changemode',0)pdesetbd(4,...'dir',...1,...'1',...'0')pdesetbd(5,...'dir',...1,...'1',...'0')pdesetbd(6,...'dir',...1,...'1',...'0')pdesetbd(16,...'dir',...1,...'1',...'0')pdesetbd(17,...'dir',...1,...'1',...'0')pdesetbd(15,...'dir',...1,...'1',...'0')pdesetbd(13,...'dir',...1,...'1',...'0')pdesetbd(14,...'dir',...1,...'1',...'0‘)

Nspire=1;I1=1;I2=I1;J1=zeros(size(t,2),1);J2=zeros(size(t,2),1);mu0=4.e-7*pi;mur=4000;mu(box)=mu0;mu(ferro)=mur*mu0;mu(spira1)=mu0;mu(spira2)=mu0;J01=Nspire*I1/(dh*dr);J02=Nspire*I2/(dh*dr);J1(spira1)=J01;J2(spira2)=J02;

%coordinata ρ del baricentro dei triangolirc=pdeintrp(p,t,p(1,:)');

area=abs(pdetrg(p,t));%area di ogni triangolo

Phi1 = assempde(bl,p,e,t,1./(rc.*mu),'0',J1','0');PhiT1=pdeintrp(p,t,Phi1);

figure;pdegplot(dl)hold on,pdecont(p,t,Phi1,20),axis equal

Energia12=2*pi*sum(J2(spira2)’.*PhiT1(spira2).*area(spira2));

M12energia=Energia12/(I1*I2)

S=pi*((0.118)^2)-pi*((0.1)^2);R1=l/(mu0*S);Manalitica=1/(R1)

2 2

21 2 1 2 2 1 2 2 2 2 21 2 1 2

1 2 ( , )V S

M dV J A z d dzi i i i

π ρ ρ ρ= ⋅ =∫ ∫J A

Calcolo Mutua

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

RisultatiIn presenza di ferro

M-analitica = 1.549e-005 H con mur->inf nel ferro

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

M = 1.517e-07 H-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

-0.15

-0.1

-0.05

0

0.05

0.1

M = 1.103e-06 H

M = 1.630e-05 HM = 7.097e-06 H

µr=10 µr=100

µr=1000 µr=10000

Nucleo magnetico 2D

Formulazione

2

2

1cos ( )

ˆ ˆJ A10 A J

z z

z z

con t

con

µµ

µ

= →∇× ∇× = −∇ +∇ ∇⋅

= ⇒ =

∇⋅ = → ∇ = −

z z

A A A

J i A i

A

XZ

Y

Geometria[pde_fig,ax]=pdeinit;pdetool('appl_cb',1);set(ax,'DataAspectRatio',[1 1 1]);set(ax,'PlotBoxAspectRatio',[1 1 1]);set(ax,'XLim',[-0.8 0.8]);set(ax,'YLim',[-0.4 0.4]);set(ax,'XTickMode','auto');set(ax,'YTickMode','auto');ymin=-1.e-2;ymax=1.e-2;xmin=0;dh=0.02;dh2=0.1;dr=0.002;dr2=0.01;l=0.001;xmax=0.008;xmax2=dh2-0.002;xmax3=xmax+dh2+0.002;pderect([xmax2-dr/2 xmax2+dr/2 -dh/2 dh/2],'R1');pderect([xmax3-dr/2 xmax3+dr/2 -dh/2 dh/2],'R2');pderect([xmin xmax/2 -dh2/2 dh2/2],'R3');pderect([xmax/2 xmax+dh2 dh2/2 dh2/2-xmax],'R4');pderect([xmax/2 xmax+dh2 -dh2/2 -dh2/2+xmax],'R5');pderect([dh2 xmax+dh2 -dh2/2+xmax+l dh2/2-xmax-l],'R6');pderect([-xmax2+dr/2 -xmax2-dr/2 -dh/2 dh/2],'R7');pderect([-xmax3+dr/2 -xmax3-dr/2 -dh/2 dh/2],'R8');pderect([-xmin -xmax/2 -dh2/2 dh2/2],'R9');pderect([-xmax/2 -xmax-dh2 dh2/2 dh2/2-xmax],'R10');pderect([-xmax/2 -xmax-dh2 -dh2/2 -dh2/2+xmax],'R11');pderect([-dh2 -xmax-dh2 -dh2/2+xmax+l dh2/2-xmax-l],'R12');pderect([-8*dh2 8*dh2 4*(-dh2) 4*dh2],'R13');

Esportazione geometria e meshset(findobj(get(pde_fig,'Children'),'Tag','PDEEval'),'String','R1+R2+R3+R4+R5+R6+R7+R8+R9+R10+R11+R12+R13');gd=get(findobj(get(pde_fig,'Children'),'flat',...

'Tag','PDEMeshMenu'),'UserData');dl=decsg(gd);[p,e,t]=initmesh(dl,'Hmax',5e-1,'init','off');[p,e,t]=refinemesh(dl,p,e,t,'regular');box=find(t(4,:)==1);ferro1=find(t(4,:)==2);ferro2=find(t(4,:)==5);ferro3=find(t(4,:)==4);ferro4=find(t(4,:)==3);ferro5=find(t(4,:)==6);ferro6=find(t(4,:)==13);ferro7=find(t(4,:)==16 );ferro8=find(t(4,:)==11);ferro9=find(t(4,:)==12 );ferro=[ferro1 ferro2 ferro3 ferro4 ferro5 ferro6 ferro7 ferro8 ferro9];spira21=find(t(4,:)==8);spira22=find(t(4,:)==7);spira11=find(t(4,:)==9);spira12=find(t(4,:)==10);spira2=[spira21 spira22];spira1=[spira11 spira12];pdeplot(p,e,t(:,ferro)); axis equal

Plot della mesh relativa alla regione ferro

Condizioni al contorno% Boundary conditions:pdetool('changemode',0)pdesetbd(2,...'dir',...1,...'1',...'0')pdesetbd(3,...'dir',...1,...'1',...'0')pdesetbd(15,...'dir',...1,...'1',...'0')pdesetbd(16,...'dir',...1,...'1',...'0')% PDE coefficients:h=findobj(get(pde_fig,'Children'),'flat','Tag','PDEBoundMenu');bl=get(findobj(get(h,'Children'),'flat','Tag','PDEBoundMode'),'UserData');

Calcolo Autoinduttanza

Nspire1=1;Nspire2=1;I1=1;I2=1;J1=zeros(size(t,2),1);J2=zeros(size(t,2),1);J01=Nspire1*I1/(dh*dr);J02=Nspire2*I1/(dh*dr);

spira11=find(t(4,:)==9);spira12=find(t(4,:)==10);spira21=find(t(4,:)==8);spira22=find(t(4,:)==7);spira2=[spira21 spira22];spira1=[spira11 spira12];J1(spira11)=J01;J1(spira12)=-J01;J2(spira21)=J02;J2(spira22)=-J02;mu0=4.E-7*pi;mur=1000;

mu(box)=mu0;mu(ferro)=mur*mu0;mu(spira1)=mu0;mu(spira2)=mu0;area=abs(pdetrg(p,t));%area di ogni triangoloA1 = assempde(bl,p,e,t,1./mu,'0',J1','0');Figure,pdegplot(dl),hold on, pdecont(p,t,A1,20), axis equalA1T=pdeintrp(p,t,A1);

Energia11=sum(J1(spira1)’.*AT1(spira1).*area(spira1));Energia12=sum(J2(spira2)'.*AT1(spira2).*area(spira2));

L1energia=Energia11/(I1*I1)M12energia=Energia12/(I1*I2)

lz=1; % spessore unitarioS=xmax*lz;R1=2*l/(mu0*S); Lanalitica=1/(R1)

111121

11121

11

11

1 dydxJAil

dVJAi

LS

z

V∫∫ ==

Risultati

L1-numerica = 7.22e-06 H, M=2.77e-07 H

L1-analitica = 5.03e-006 H con mur->inf nel ferro

mur=1000