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A comprehensive approach to the analysis of the fundamental behavior of the three-phase, cylindrical-rotor synchronous machine in the steady state is presented. Beginning with a very basic and understandable time-domain model of the machine, a straightforward matrix analysis leading to the phasor representation is given. This, in turn, is carried ultimately to the symmetrical-component model which is obtained by means of a mathematically rigorous similarity transformation. This collection of features makes the approach significantly different from those found in the standard undergraduate texts. This concise explanation emphasizes the role of the synchronous machine as both an individual device and as a system element, and has been found by the author to be a useful supplement to the material presented in the customary undergraduate and beginning graduate treatments of power and machinery.