Calculation of the magnetic field in dynamic-electric machines by Southwell's relaxation method

A method of computing magnetic and electric fields is explained with reference to the problem of the magnetic flux distribution between stator and rotor of dynamo-electric machines. Mathematically, the problem is that of solving Laplace's or Poisson's equation. It is shown how approximate solutions are obtained with the help of a simple arithmetical technique. The differential equations are first replaced by a system of algebraic equations which are then solved by means of a computation method due to R. V. Southwell. The computation process is outlined and illustrated by a simple example which shows all the essential steps involved. A solution is given for two idealized examples of pole shoe and armature for which rigorous solutions are known. The agreement between the rigorous and approximate solutions is found satisfactory. Sharp corners of pole shoe and armature present a problem for which various solutions are possible; some of these are indicated. The flux distribution of a complete machine is computed and presented as an example of a complex problem not amenable to a rigorous solution.