End-Winding Leakage of High-Speed Alternators by Three-Dimensional Field Determination

The rapid increase in electric and magnetic loadings of electrical machines demands improved methods of predicting the end zone field distribution. This paper presents a numerical method for the determination of end zone fields and for the calculation of end-leakage reactance of high-speed alternators. The method takes account of all boundaries, simplified in regard to their geometries and magnetic nature. As a first step, the partial differential equations of the electromagnetic field in the different regions of the end-winding region of homopolar alternators are developed, based on the concept of the magnetic vector potential. These are transformed into finite difference equations, which are retained in a general form allowing for the nonlinearities of iron and the effect of current carrying regions. The theory of the energy method to evaluate the end-leakage reactance is explained. The difficulties of a numerical solution in three dimensions are presented. Then a mathematical model is set up taking the discrete nature of the windings into account. This model makes possible adequate consideration of the effects of the surrounding boundaries and also the effects of the air gap, slots, and of the short-pitched armature winding. An orthogonal lattice system is then fitted to the model. The boundary conditions in differential and difference forms are developed in terms of the magnetic vector potential. An iterative procedure consisting of successive point relaxation of the vector potentials is described.