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This example shows how to solve a 3-D magnetostatic problem for a solenoid with a finite length iron core. Using a ferromagnetic core with high permeability, such as an iron core, inside a solenoid increases magnetic field and flux density. In this example, you find the magnetic flux density for a geometry consisting of a coil with a finite length core in a cylindrical air domain.

The first part of the example solves the magnetostatic problem using a 3-D model. The second part solves the same problem using an axisymmetric 2-D model to speed up computations.

Create geometries consisting of three cylinders: a solid circular cylinder models the core, an annular circular cylinder models the coil, and a larger circular cylinder models the air around the coil.

```
coreGm = multicylinder(0.03,0.1);
coilGm = multicylinder([0.05 0.07],0.2,'Void',[1 0]);
airGm = multicylinder(1,2);
```

Position the core and coil so that the finite length core is located near the top of coil.

coreGm = translate(coreGm,[0 0 1.025]); coilGm = translate(coilGm,[0 0 0.9]);

Combine the geometries and plot the result.

gm = addCell(airGm,coreGm); gm = addCell(gm,coilGm); pdegplot(gm,'FaceAlpha',0.2,'CellLabels','on')

Zoom in to see the cell labels on the core and coil.

figure pdegplot(gm,'FaceAlpha',0.2,'CellLabels','on') axis([-0.1 0.1 -0.1 0.1 0.8 1.2])

Create an electromagnetic model and assign air geometry to the model.

model3D = createpde('electromagnetic','magnetostatic'); model3D.Geometry = gm;

Specify the vacuum permeability value in the SI system of units.

model3D.VacuumPermeability = 1.2566370614E-6;

Specify a relative permeability of 1 for all domains.

`electromagneticProperties(model3D,'RelativePermeability',1);`

Now specify the large relative permeability of the core.

electromagneticProperties(model3D,'RelativePermeability',10000, ... 'Cell',2);

Assign an excitation current using a function that defines counterclockwise current density in the coil.

electromagneticSource(model3D,'CurrentDensity',@windingCurrent3D, ... 'Cell',3);

Specify that the magnetic potential on the outer surface of the air domain is 0.

electromagneticBC(model3D,'MagneticPotential',[0;0;0],'Face',1:3);

Generate a mesh where only the core and coil regions are well refined and the air domain is relatively coarse to limit the size of the problem.

```
internalFaces = cellFaces(model3D.Geometry,2:3);
generateMesh(model3D,'Hface',{internalFaces,0.007});
```

Solve the model.

R = solve(model3D)

R = MagnetostaticResults with properties: MagneticPotential: [1×1 FEStruct] MagneticField: [1×1 FEStruct] MagneticFluxDensity: [1×1 FEStruct] Mesh: [1×1 FEMesh]

Find the magnitude of the flux density.

Bmag = sqrt(R.MagneticFluxDensity.Bx.^2 + ... R.MagneticFluxDensity.By.^2 + ... R.MagneticFluxDensity.Bz.^2);

Find the mesh elements belonging to the core and the coil.

coreAndCoilElem = findElements(model3D.Mesh,'region','Cell',[2 3]);

Plot the magnitude of the flux density on the core and coil.

pdeplot3D(model3D.Mesh.Nodes, ... model3D.Mesh.Elements(:,coreAndCoilElem), ... 'ColorMapData',Bmag) axis([-0.1 0.1 -0.1 0.1 0.8 1.2])

Interpolate the flux to a grid covering the portion of the geometry near the core.

x = -0.05:0.01:0.05; z = 1.02:0.01:1.14; y = x; [X,Y,Z] = meshgrid(x,y,z); intrpBcore = R.interpolateMagneticFlux(X,Y,Z);

Reshape `intrpBcore.Bx`

, `intrpBcore.By`

, and `intrpBcore.Bz`

and plot the magnetic flux density as a vector plot.

Bx = reshape(intrpBcore.Bx,size(X)); By = reshape(intrpBcore.By,size(Y)); Bz = reshape(intrpBcore.Bz,size(Z)); quiver3(X,Y,Z,Bx,By,Bz,'Color','r') hold on pdegplot(coreGm,'FaceAlpha',0.2);

Now, simplify this 3-D problem to 2-D using the symmetry around the axis of rotation.

First, create the geometry. The axisymmetric section consists of two small rectangular regions (the core and coil) located within a large rectangular region (air).

R1 = [3,4,0.0,1,1,0.0,0,0,2,2]'; R2 = [3,4,0,0.03,0.03,0,1.025,1.025,1.125,1.125]'; R3 = [3,4,0.05,0.07,0.07,0.05,0.90,0.90,1.10,1.10]'; ns = char('R1','R2','R3'); sf = 'R1+R2+R3'; gdm = [R1, R2, R3]; g = decsg(gdm,sf,ns');

Plot the geometry with the face labels.

pdegplot(g,'FaceLabels','on')

Zoom in to see the face labels on the core and coil.

figure pdegplot(g,'FaceLabels','on') axis([0 0.1 0.8 1.2])

Create an electromagnetic model for axisymmetric magnetostatic analysis and assign the geometry.

model2D = createpde('electromagnetic','magnetostatic-axisymmetric'); geometryFromEdges(model2D,g);

Specify the vacuum permeability value in the SI system of units.

model2D.VacuumPermeability = 1.2566370614E-6;

Specify a relative permeability of 1 for all domains.

`electromagneticProperties(model2D,'RelativePermeability',1);`

Now specify the large relative permeability of the core.

electromagneticProperties(model2D,'RelativePermeability',10000, ... 'Face',3);

Specify the current density in the coil. For an axisymmetric model, use the constant current value.

electromagneticSource(model2D,'CurrentDensity',5E6,'Face',2);

Assign zero magnetic potential on the outer edges of the air domain as the boundary condition.

electromagneticBC(model2D,'MagneticPotential',0,'Edge',[2 8]);

Generate a mesh.

generateMesh(model2D,'Hmin',0.0004,'Hgrad',2,'Hmax',0.008);

Solve the model.

R = solve(model2D);

Find the magnitude of the flux density.

```
Bmag = sqrt(R.MagneticFluxDensity.Bx.^2 + ...
R.MagneticFluxDensity.By.^2);
```

Plot the magnitude of the flux density on the core and coil.

```
pdeplot(model2D,'XYData',Bmag)
xlim([0,0.05]);
ylim([1.0,1.14])
```

Interpolate the flux to a grid covering the portion of the geometry near the core.

x = 0:0.01:0.05; y = 1.02:0.01:1.14; [X,Y] = meshgrid(x,y); intrpBcore = R.interpolateMagneticFlux(X,Y);

Reshape `intrpBcore.Bx`

and `intrpBcore.By`

and plot the magnetic flux density as a vector plot.

Bx = reshape(intrpBcore.Bx,size(X)); By = reshape(intrpBcore.By,size(Y)); quiver(X,Y,Bx,By,'Color','r') hold on pdegplot(model2D); xlim([0,0.07]); ylim([1.0,1.14])

function f3D = windingCurrent3D(region,~) [TH,~,~] = cart2pol(region.x,region.y,region.z); f3D = -5E6*[sin(TH); -cos(TH); zeros(size(TH))]; end

[1] Thierry Lubin, Kévin Berger, Abderrezak Rezzoug. "Inductance and Force Calculation for Axisymmetric Coil Systems Including an Iron Core of Finite Length." *Progress In Electromagnetics Research B, EMW Publishing* 41 (2012): 377-396. https://hal.archives-ouvertes.fr/hal-00711310.