The Pupin Theory of Asymmetrical Rotors in Unidirectional Fields: With Special Reference to the Goldschmidt Alternator

The case of a simple circuit having periodically varying inductance is first examined. The solution shows that the current has a constant component and an infinite series of convergently diminishing higher harmonics. Circuits having inductance, resistance, and variable mutual inductance are next considered. To solve the equations obtained, an infinite transformation is carried out, each variable being replaced by the sum of an infinite series of new variables, thus enabling an infinite number of arbitrary conditions to be imposed. As a result, an infinite series of equations is obtained, each of which can be solved if those preceding it have been solved. The solutions are worked out to the fourth harmonic in one circuit and the third in the other. In one circuit, only odd frequencies appear; in the other, only even. The general solutions are in the form of a Fourier's series, each amplitude of which is an infinite series. The convergence of the solutions is completely established. The solutions are then extended to the case where rotor and stator circuits contain capacities. It is shown that according to Pupin's theory, all currents in low resistance rotors and stators tend toward infinity if these circuits are appropriately tuned. This apparent discrepancy from practice is explained on the ground that the variable permeability of the iron in the Goldschmidt alternators automatically detunes the circuits and that the increasing losses of the iron tend further to limit all currents.