MMF: Magnetomotive force provided by PM, A * turn [R.sub.l]: Magnetic reluctance of the rod, H [R.sub.g]: Magnetic reluctance of the air gap from GMM bar to wall, H [R.sub.w]: Magnetic reluctance of wall, H [phi]: Magnetic fluxes, wb A: Magnetic vector potential, wb/m [phi]: Magnetic scalar potential
, wb/m [H.sub.i]: Magnetic field in each element, A/m [H.sub.avg]: Average magnetic field, A/m [L.sub.GMM]: Total length of GMM patches, m [L.sub.PM]: Total length of PM patches, m [lambda]: Magnetostriction of GMM [[lambda].sub.s]: Saturated magnetostriction of GMM [M.sub.s]: Saturated magnetization of GMM, A/m M: Magnetization of GMM, A/m [A.sub.GMM]: Section area of GMM rod, [m.sup.2].
where [H.sub.0] is the field due to source currents and can be easily computed by Biot-Savart-Laplace relation, and [phi] is reduced magnetic scalar potential
Using the conformal mapping method and considering the boundary conditions of magnetic scalar potential
[U.sub.m] for the magnet-pole and stator surfaces in z-coordinates, at first the magnetic complex potential W(w) for the conformal complex w-coordinates may be determined by Schwarz' integral [5, 6].
where g(r, r') = [1/4[pi]][1/[absolute value of r - r']] is the Green's function for the Laplace equation, and [[phi].sup.pr] is the magnetic scalar potential of the primary magnetic field.
where N is the linear nodal function and [[phi].sup.k] is the magnetic scalar potential on the kth vertex of [[GAMMA].sub.e].
To satisfy the boundary condition, a radially directed magnetic line dipole distribution [P.sup.[rho].sub.m]([phi]) = [??][P.sup.[rho].sub.m]([phi]) is introduced along the centre-line of the wire, giving rise to the magnetic scalar potential
With these solutions, the magnetic scalar potential expression in each region can be expressed in a Fourier series.
According to (39) and (40), the magnetic scalar potential distributions at both the inner and outer air-gaps can be calculated as shown in Figure 5 in which the effects due to PMs on the high-speed and low-speed movers are determined separately.
where [OMEGA] is a magnetic scalar potential
. And, since
A recurring theme is the formulation of and algorithms for the problem of making branch cuts for computing magnetic scalar potentials
and eddy currents.
Similarly, the integral expressions for the electric vector and magnetic scalar potentials
, related with the magnetic source currents, become