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Find the static magnetic field induced by the stator windings in a two-pole electric motor. The example uses the PDE Modeler app. Assuming that the motor is long and end effects are negligible, you can use a 2-D model. The geometry consists of three regions:

Two ferromagnetic pieces: the stator and the rotor (transformer steel)

The air gap between the stator and the rotor

The armature copper coil carrying the DC current

Magnetic permeability of the air and copper is close to the
magnetic permeability of a vacuum, *μ*_{0} =
4*π**10^{-7} H/m.
In this example, use the magnetic permeability *μ* = *μ*_{0} for
both the air gap and copper coil. For the stator and the rotor, *μ* is

$$\mu ={\mu}_{0}\left(\frac{{\mu}_{\mathrm{max}}}{1+c{\Vert \nabla A\Vert}^{2}}+{\mu}_{\mathrm{min}}\right)$$

where *µ*_{max} =
5000, *µ*_{min} = 200, and *c* =
0.05. The current density *J* is 0 everywhere except
in the coil, where it is 10 A/m^{2}.

The geometry of the problem makes the magnetic vector potential *A* symmetric
with respect to *y* and antisymmetric with respect
to *x*. Therefore, you can limit the domain to *x* ≥
0, *y* ≥ 0 with the Neumann boundary condition

$$n\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla A\right)=0$$

on the *x*-axis and the Dirichlet boundary
condition *A* = 0 on the *y*-axis.
Because the field outside the motor is negligible, you can use the
Dirichlet boundary condition *A* = 0 on the exterior
boundary.

To solve this problem in the PDE Modeler app, follow these steps:

Set the

*x*-axis limits to`[-1.5 1.5]`

and the*y*-axis limits to`[-1 1]`

. To do this, select**Options**>**Axes Limits**and set the corresponding ranges.Set the application mode to

**Magnetostatics**.Create the geometry. The geometry of this electric motor is complex. The model is a union of five circles and two rectangles. The reduction to the first quadrant is achieved by intersection with a square. To draw the geometry, enter the following commands in the MATLAB

^{®}Command Window:pdecirc(0,0,1,'C1') pdecirc(0,0,0.8,'C2') pdecirc(0,0,0.6,'C3') pdecirc(0,0,0.5,'C4') pdecirc(0,0,0.4,'C5') pderect([-0.2 0.2 0.2 0.9],'R1') pderect([-0.1 0.1 0.2 0.9],'R2') pderect([0 1 0 1],'SQ1')

Reduce the model to the first quadrant. To do this, enter

`(C1+C2+C3+C4+C5+R1+R2)*SQ1`

in the**Set formula**field.Remove unnecessary subdomain borders. To do this, switch to the boundary mode by selecting

**Boundary**>**Boundary Mode**. Using**Shift**+click, select borders, and then select**Boundary**>**Remove Subdomain Border**until the geometry consists of four subdomains: the rotor (subdomain 1), the stator (subdomain 2), the air gap (subdomain 3), and the coil (subdomain 4). The numbering of your subdomains can differ. If you do not see the numbers, select**Boundary**>**Show Subdomain Labels**.Specify the boundary conditions. To do this, select the boundaries along the

*x*-axis. Select**Boundary**>**Specify Boundary Conditions**. In the resulting dialog box, specify a Neumann boundary condition with*g*= 0 and*q*= 0.All other boundaries have a Dirichlet boundary condition with

*h*= 1 and*r*= 0, which is the default boundary condition in the PDE Modeler app.Specify the coefficients by selecting

**PDE**>**PDE Specification**or clicking the**PDE**button on the toolbar. Double-click each subdomain and specify the following coefficients:Coil:

*µ*=`4*pi*10^(-7)`

H/m,`J = 10`

A/m^{2}.Stator and rotor:

*µ*=`4*pi*10^(-7)*(5000./(1+0.05*(ux.^2+uy.^2))+200)`

H/m, where`ux.^2+uy.^2`

equals to |∇*A*|^{2},`J = 0`

(no current).Air gap:

*µ*=`4*pi*10^(-7)`

H/m,`J = 0`

.

Initialize the mesh by selecting

**Mesh**>**Initialize Mesh**.Choose the nonlinear solver. To do this, select

**Solve**>**Parameters**and check**Use nonlinear solver**. Here, you also can adjust the tolerance parameter and choose to use the adaptive solver together with the nonlinear solver.Solve the PDE by selecting

**Solve**>**Solve PDE**or clicking the**=**button on the toolbar.Plot the magnetic flux density

*B*using arrows and the equipotential lines of the magnetostatic potential*A*using a contour plot. To do this, select**Plot**>**Parameters**and choose the contour and arrows plots in the resulting dialog box. Using**Options**>**Axes Limits**, adjust the axes limits as needed. For example, use the**Auto**check box.The plot shows that the magnetic flux is parallel to the equipotential lines of the magnetostatic potential.

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