**Subject :**Electrical Machines II (AC Machines)

**Unit :**Induction machines

## Alternate Mathematical Analysis for Rotating Magnetic Field

**Alternate Mathematical Analysis for Rotating Magnetic Field:**

*Fig: 1*

The three-phase sinusoidal currents produce fluxes Φ_{1}, Φ_{2} and Φ_{3} which vary sinusoidally. The resultant flux at any instant will be the vector sum of all the three at that instant. The fluxes are represented by three variable magnitude vectors [See Fig. 1]. In Fig. (1), the individual flux directions arc fixed but their magnitudes vary sinusoidally as does the current that produces them. To find the magnitude of the resultant flux, resolve each flux into horizontal and vertical components and then find their vector sum.

Thus the resultant flux has constant magnitude (= 1.5 Φ_{m}) and does not change with time. The angular displacement of Φ_{r}relative to the OX axis is

Thus the resultant magnetic field rotates at constant angular velocity w( = 2 pf) rad/sec. For a P-pole machine, the rotation speed (ω_{m}) is

Thus the resultant flux due to three-phase currents is of constant value (= 1.5 Φ_{m}where Φ_{m}is the maximum flux in any phase) and this flux rotates around the stator winding at a synchronous speed of 120 f/P r.p.m. For example, for a 6-pole, 50 Hz, 3-phase induction motor, N, = 120 ´ 50/6 =1000 r.p.m. It means that flux rotates around the stator at a speed of 1000 r.p.m.