{ MAGNET_COIL.PDE
**************************************
AXI-SYMMETRIC MAGNETIC FIELDS
This example considers the problem of determining the magnetic vector
potential A around a coil.
According to Maxwell's equations,
curl H = J
div B = 0
B = mu*H
where B is the manetic flux density
H is the magnetic field strength
J is the electric current density
and mu is the magnetic permeability of the material.
The magnetic vector potential A is related to B by
B = curl A
therefore
curl( (1/mu)*curl A ) = J
This equation is usually supplmented with the Coulomb Gauge condition
div A = 0.
In the axisymmetric case, the current is assumed to flow only in the
azimuthal direction, and only the azimuthal component of the vector
potential is present. Henceforth, we will simply refer to this component as A.
The Coulomb Gauge is identically satisfied, and the PDE to be solved in this
model takes the form
curl((1/mu)*curl (A)) = J(x,y) in the domain
A = g(x,y) on the boundary.
The magnetic induction B takes the simple form
B = (-dz(A), 0, dr(A)+A/r)
and the magnetic field is given by
H = (-dz(A)/mu, 0, (dr(A)+A/r)/mu)
Expanding the equation in cylindrical geometry results in the final equation,
dz(dz(A)/mu) + dr((dr(A)+A/r)/mu) = -J
The interpretation of the natural boundary condition becomes
Natural(A) = n X H
where n is the outward surface-normal unit vector.
Across boundaries between regions of different material properties, the
continuity of (n X H) assumed by the Galerkin solver implies that the
tangential component of H is continuous, as required by the physics.
In this simple test problem, we consider a circular coil whose axis of
rotation lies along the X-axis. We bound the coil by a distant spherical
surface at which we specify a boundary condition (n X H) = 0.
At the axis, we use a Dirichlet boundary condition A=0.
The source J is zero everywhere except in the coil, where it is defined
arbitrarily as "10". The user should verify that the prescribed values
of J are dimensionally consistent with the units of his own problem.
******************************************************
}
title 'AXI-SYMMETRIC MAGNETIC FIELD'
coordinates
xcylinder(Z,R) { -- Cylindrical coordinates, with cylinder
axis along Cartesian X direction }
Variables
A { -- the azimuthal component of the
vector potential }
Definitions
mu = 1 { -- the permeability }
rmu = 1/mu
J = 0 { -- the source defaults to zero }
current = 10 { -- the source value in the coil }
Bz = dr(r*A)/r
Initial values
A = 2 { -- unimportant unless mu varies with H }
Equations
curl(rmu*curl(A)) = J { -- FlexPDE expands CURL in proper coordinates }
Boundaries
Region 1
start(-10,0)
value(A) = 0 { -- specify A=0 along axis }
line to (10,0)
natural(A) = 0 { -- H<dot>n = 0 on distant sphere }
arc(center=0,0) angle 180 to close
Region 2
J = current { -- override source value in the coil }
start (-0.25,1)
line to (0.25,1) to (0.25,1.5) to (-0.25,1.5) to close
Monitors
contour(Bz) zoom(-2,0,4,4) as 'FLUX DENSITY B'
contour(A) as 'Potential'
Plots
grid(z,r)
contour(Bz) as 'FLUX DENSITY B'
contour(Bz) zoom(-2,0,4,4) as 'FLUX DENSITY B'
elevation(A,dr(A),A/r,dr(A)+A/r,A+r*dr(A)) from (0,0) to (0,1) as 'Bz'
vector(dr(A)+A/r,-dz(A)) as 'FLUX DENSITY B'
vector(dr(A)+A/r,-dz(A)) zoom(-2,0,4,4) as 'FLUX DENSITY B'
contour(A) as 'MAGNETIC POTENTIAL'
contour(A) zoom(-2,0,4,4) as 'MAGNETIC POTENTIAL'
surface(A) as 'MAGNETIC POTENTIAL' viewpoint (-1,1,30)
End