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Magnets, electric motors, and transformers are areas where problems involving magnetostatics can be found. The "statics" implies that the time rate of change is slow, so we start with Maxwell's equations for steady cases,
$$\nabla \times H=J$$
$$\nabla \cdot B=0$$
and the relationship
$$B=\mu H$$
where B is the magnetic flux density, H is the magnetic field intensity, J is the current density, and µ is the material's magnetic permeability.
Since $$\nabla \cdot B=0$$, there exists a magnetic vector potential A such that
$$B=\nabla \times A$$
and
$$\nabla \times \left(\frac{1}{\mu}\nabla \times A\right)=J$$
The plane case assumes that the current flows are parallel to the z-axis, so only the z component of A is present,
$$A=(0,0,A),\text{\hspace{1em}}J=(0,0,J)$$
You can impose the common gauge assumption (Lorenz gauge or Coulomb gauge, see Wikipedia [2])
$$\nabla \xb7A=0,$$
and then the equation for A in terms of J can be simplified to the scalar elliptic PDE
$$-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla A\right)=J,$$
where J = J(x,y).
For the 2-D case, we can compute the magnetic flux density B as
$$B=\left(\frac{\partial A}{\partial y},-\frac{\partial A}{\partial x},0\right)$$
and the magnetic field H, in turn, is given by
$$H=\frac{1}{\mu}B$$
The interface condition across subdomain borders between regions of different material properties is that H x n be continuous. This implies the continuity of
$$\frac{1}{\mu}\frac{\partial A}{\partial n}$$
and does not require special treatment since we are using the variational formulation of the PDE problem.
In ferromagnetic materials, µ is usually dependent on the field strength |B| = |∇A|, so the nonlinear solver is needed.
The Dirichlet boundary condition specifies the value of the magnetostatic potential A on the boundary. The Neumann condition specifies the value of the normal component of
$$n\cdot \left(\frac{1}{\mu}\nabla A\right)$$
on the boundary. This is equivalent to specifying the tangential value of the magnetic field H on the boundary.
Visualization of the magnetostatic potential A, the magnetic field H, and the magnetic flux density B is available. B and H can be plotted as vector fields.
[1] Popovic, Branko D., Introductory Engineering Electromagnetics, Addison-Wesley, Reading, MA, 1971.
[2] Wikipedia entries on Gauge fixing.
As an example of a problem in magnetostatics, consider determining the static magnetic field due to the stator windings in a two-pole electric motor. The motor is considered to be long, and when end effects are neglected, a 2-D computational model suffices.
The domain consists of three regions:
Two ferromagnetic pieces, the stator and the rotor
The air gap between the stator and the rotor
The armature coil carrying the DC current
The magnetic permeability µ is 1 in the air and in the coil. In the stator and the rotor, µ is defined by
$$\mu =\frac{{\mu}_{\mathrm{max}}}{1+c{\Vert \nabla A\Vert}^{2}}+{\mu}_{\mathrm{min}}.$$
µ_{max} = 5000, µ_{min} = 200, and c = 0.05 are values that could represent transformer steel.
The current density J is 0 everywhere except in the coil, where it is 1.
The geometry of the problem makes the magnetic vector potential A symmetric with respect to y and antisymmetric with respect to x, so 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. The field outside the motor is neglected leading to the Dirichlet boundary condition A = 0 on the exterior boundary.
The geometry is complex, involving five circular arcs and two rectangles. Using the PDE app, set the x-axis limits to [-1.5 1.5] and the y-axis limits to [-1 1]. Set the application mode to Magnetostatics, and use a grid spacing of 0.1. The model is a union of circles and rectangles; the reduction to the first quadrant is achieved by intersection with a square. Using the "snap-to-grid" feature, you can draw the geometry using the mouse, or you can draw it by entering the following commands:
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')
You should get a CSG model similar to the one in the following plot.
Enter the following set formula to reduce the model to the first quadrant:
(C1+C2+C3+C4+C5+R1+R2)*SQ1
In boundary mode you need to remove a number of subdomain borders. Using Shift+click, select borders and remove them using the Remove Subdomain Border option from the Boundary menu until the geometry consists of four subdomains: the stator, the rotor, the coil, and the air gap. In the following plot, the stator is subdomain 1, the rotor is subdomain 2, the coil is subdomain 3, and the air gap is subdomain 4. The numbering of your subdomains may be different.
Before moving to the PDE mode, select the boundaries along the x-axis and set the boundary condition to a Neumann condition with g = 0 and q = 0. In the PDE mode, turn on the labels by selecting the Show Subdomain Labels option from the PDE menu. Double-click each subdomain to define the PDE parameters:
In the coil both µ and J are 1, so the default values do not need to be changed.
In the stator and the rotor µ is nonlinear and defined by the preceding equation. Enter µ as
5000./(1+0.05*(ux.^2+uy.^2))+200
ux.^2+uy.^2 is equal to |∇A |^{2}. J is 0 (no current).
In the air gap µ is 1, and J is 0.
Initialize the mesh, and continue by opening the Solve Parameters dialog box by selecting Parameters from the Solve menu. Since this is a nonlinear problem, the nonlinear solver must be invoked by checking the Use nonlinear solver. If you want, you can adjust the tolerance parameter. The adaptive solver can be used together with the nonlinear solver. Solve the PDE and plot the magnetic flux density B using arrows and the equipotential lines of the magnetostatic potential A using a contour plot. The plot clearly shows, as expected, that the magnetic flux is parallel to the equipotential lines of the magnetostatic potential.
Equipotential Lines and Magnetic Flux in a Two-Pole Motor