Find the static magnetic field induced by the stator windings in a two-pole electric motor. The example uses the PDE 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, μ0 = 4π*10-7 H/m. In this example, use the magnetic permeability μ = μ0 for both the air gap and copper coil. For the stator and the rotor, μ is
where µmax = 5000, µmin = 200, and c = 0.05. The current density J is 0 everywhere except in the coil, where it is 10 A/m2.
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
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 app, follow these steps:
Set the x-axis limits to
1.5] and the y-axis limits to
1]. To do this, select Axes Limits from
the Options menu and set the corresponding
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,
(C1+C2+C3+C4+C5+R1+R2)*SQ1 in the Set
Remove unnecessary subdomain borders. To do this, switch to the boundary mode by selecting Boundary Mode from the Boundary menu. Using Shift+click, select borders, and then select Remove Subdomain Border from the Boundary menu 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 Show Subdomain Labels from the Boundary menu.
Specify the boundary conditions. To do this, select the boundaries along the x-axis. Select Specify Boundary Conditions from the Boundary menu. 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 app.
Specify the coefficients. To do this, select PDE Specification from the PDE menu or click the PDE button on the toolbar. Double-click each subdomain and specify the following coefficients:
Coil: µ =
= 10 A/m2.
Stator and rotor: µ =
ux.^2+uy.^2 equals to |∇A |2,
= 0 (no current).
Air gap: µ =
Initialize the mesh. To do this, select Initialize Mesh from the Mesh menu.
Choose the nonlinear solver. To do this, select Parameters from the Solve menu 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. To do this, select Solve PDE from the Solve menu or click 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 Parameters from the Plot menu and choose the contour and arrows plots in the resulting dialog box. Using the Axes Limits from the Options menu, 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.