Most rotating electrical machines have cylindrical air gaps. A traveling magneto motive force (MMF) can be generated in the air gap when multi-phase sinusoidal currents flow through the multi-phase coils. Since the coils are confined in the slots, the MMFs have quantized levels. The MMF harmonics are analyzed by Fourier series. The voltage, the current, and the flux of the three phase system can be identified with some rotating vectors in the complex plane. Interestingly, the direction of the flux vector coincides with that of real flux. Coordinate transformation maps are defined, and illustrated with examples.