Analysis of Synchronous Machines under Unbalanced Operation

Considering only the fundamental frequency components of currents and voltages, an idealized synchronous machine under any unbalanced load conditions can be represented in phase coordinates, by the sum of a constant impedance matrix and an impedance matrix whose elements are functions of current phase angles. A general method of solving for the unknown current phase angles and hence the total impedance matrix is given when the external impedances are known. The results are shown to be consistent with 1) two-reaction theory for the case of balanced steady- state loads, and 2) symmetrical-component theory using sequence- impedance representation under the assumption of zero resistances in all circuits. Numerical comparison shows that although the current magnitudes can be calculated by the approximate sequence- impedance representation quite accurately, current phase angles could differ, in certain cases, up to about 10°.