Calculation of three-dimensional electromagnetic fields involving laminar eddy currents

The formulation of a method to solve steady-state electromagnetic field problems involving laminar eddy currents is described. Iron nonlinearity is included. The formulation is implemented using finite elements and the problems associated with the solution of the resulting nonlinear equations discussed. Experiments with a Newton-Raphson iteration and modifications to it are described and a best alternative is defined. The matrix equation which requires solution is asymmetric, consequently a matrix equation solver involving the use of a preconditioned incomplete LU decomposition followed by biconjugate gradients has been developed. The application for which the method has been developed is to calculate the stator end-region fields in turbinegenerators and examples of the results from this work are presented. List of principal symbols B = magnetic flux density H = magnetic field strength E = electric field strength J = current density T = electric vector potential <p = magnetic scalar potential \x — permeability a = conductivity co = circular frequency N = finite element shape functions L = upper half decomposition of a matrix U = lower half decomposition of a matrix t = time x, y, z = cartesian coordinates r, z, 0 = cylindrical polar coordinates n = unit normal vector Subscripts x, y, z = components in cartesian form r, z, 0 = components in cylindrical polar form R = real component Paper 5449A(S8), first received 1st September 1986 and in revised form 10th April 1987 Dr. Jack is with the Electrical and Electronic Engineering Department, Merz Laboratory, The University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, United Kingdom Dr. Mecrow is with the Electrical Design Department, NEI Parsons Ltd, Heaton Works, Newcastle upon Tyne NE6 2YL, United Kingdom / = imaginary component i, j , k = associated with node i, j or k c = conducting region nc = nonconducting region