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The symmetry of the induction machine can be exploited to obtain general closed form expressions for the small signal transfer functions describing speed, voltage, frequency, or load perturbations by utilizing the complex time variables introduced by Ku and Lyon in the 1950's. After a brief introduction to complex variables, the linearlzed complex variable equations describing small signal dynamic performance are presented. These equations are used to obtain transfer functions in which the effects of excitation level are isolated in the gain factors. The speed and frequency dependence of the poles and zeros is expressed in closed form employing a useful non-dimensional parameter system. To illustrate the application of these results, the dynamic behavior of the induction machine without feed-back control is analyzed. It is shown that the general dynamic response can be characterized by the non-dimensional loop gain and stator frequency. A set of general non-dimensional root loci are presented which permit rapid estimation of the relative stability (dominant eigenvalues) and the frequency of minimum damping of any specific machine. The application of the transfer functions to cases involving feedback control of the machine is also discussed.