The calculation of the magnetic field of rectangular conductors in a closed slot, and its application to the reactance of transformer windings

The magnetic vector potential of current-carrying conductors in a rectangular slot is the same as that due to an infinite number of images of these conductors from all the four walls of the slot. A solution of this problem can thus be obtained in the form of a Fourier series, periodic in both the x- and y-directions, the constants of the solution being determined by the boundary conditions, i.e. by the dimensions of the slot and of the conductors, and by the currents passing through them. Expressions for the magnetic vector potential due to any conductor arrangement can thus be developed, and these lend themselves to considerable simplification without undue loss of accuracy. Lines of magnetic flux density correspond to constant values of the vector potential, and this yields a ready method for plotting magnetic-flux maps. The results of this investigation can, with certain assumptions, be directly applied to transformer reactance problems and also to the equivalent problem of heat conduction of windings embedded in the slots of electrical machines. Formulae are developed for the reactance and forces in concentric-winding arrangements, showing the effects of slight asymmetries.