Analytical Determination of Magnetic Fields Simple Cases of Conductors in Slots

In this paper La Place´s and Poisson´s equations are applied to cases of current carrying conductors in rectangular slots to show the flux distributions which obtain. The paper first treats with the general case which is a single conductor, completely surrounded by insulation, at the bottom of a slot. Next is taken up the more practical case of a slot containing two insulated conductors, one above the other, in which currents of equal magnitude are considered first to flow in the same direction, and then in opposite directions. In addition to these analyses, a few special cases are discussed. Methods are also discussed by which the flux distributions in slots containing an even number of coil sides may be obtained, or in slots the conductors of which carry currents not in time-phase. Equations are developed from which all of these fields can be calculated. A discussion is included to show the distribution of flux on the assumption that the lines go straight across the slot, and a comparison is made between the slot inductance determined mathematically from the developed equations and by the usual design formulas. In the case considered it is found that the design formulas give a value of slot inductance which is about 96 per cent of that obtained by the mathematical treatment which is within limits of engineering accuracy. An expression is developed which shows the error that may be expected in any particular case.