## 5443 Resultant mmf in a Balanced System

In the typical operation of an AC machine, stator windings are supplied with balanced set voltages, and because the windings are electrically symmetrical, a balanced set of currents flows through the windings. Let us assume that the rotor is open circuited, and all the current flowing through the stator winding is the magnetizing current required to establish the stator mmf. The three phase currents have the same magnitude and frequency, but are 120° shifted in time with respect to each other. The currents in the time domain can be expressed as

The space vector for the above balanced set of currents is as follows (using Equation 5.13): FIGURE 5.15 Resultant mmf space vector for t=-30°, 0°, and 90°.

The resultant stator mmf space vector is as follows (Figure 5.15):

The result shows that the stator mmf has a constant peak amplitude (because Ns and IM are constants) that rotates around the stator circumference at a constant speed equal to the applied angular speed of the applied stator voltages. This speed is known as the synchronous speed. Unlike the single-phase stator mmf (shown in Figure 5.10b), the peak of the stator mmf resulting in the three-phase AC machine is rotating synchronously along the stator circumference, with the peak always located at = t. The mmf peak position is time-varying for the three-phase winding, whereas the peak mmf position for the singlephase winding is not time varying. The mmf wave is a sinusoidal function of the space angle The wave has a constant amplitude and a space-angle t, which is a linear function of time. The angle t provides rotation of the entire wave around the air gap at a constant angular velocity . Thus, at a fixed time tx, the wave is a sinusoid in space, with its positive peak displaced tx from the reference =0. The polyphase windings excited by balanced polyphase currents produce the same general effect as that of spinning a permanent magnet about an axis perpendicular to the magnet, or as in the rotation of the DCexcited field poles.

The three-phase stator mmf is known as the rotating mmf, which can be equivalently viewed as a magnet rotating around the stator circumference at a constant speed. Note that with the vector sum of Fa( e), Fb( e), and Fc( e) as described in Equations 5.9 and 5.10, with ia(t), ib(t), and ic(t) replaced by the balanced set of Equation 5.20, we will arrive at the same result.

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