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

**Unit :**Induction machines

## Time Harmonics of Induction motors

**Time Harmonics of Induction motors:
**

To study the performance of an induction machine, we consider that the air-gap flux wave is purely sinusoidal. It is from that assumption the analysis of induced emf, sinusoidal currents, the expressions for generated torque etc. proceed. In practice, there are deviations from this idealistic picture.

The first non-ideality is the presence of harmonics in the input supply given to the three phase machine. The source may contain 3^{rd}, 5^{th}, 7^{th}. . . harmonics. Note that due to the symmetry of the waveform (f(t) = −f(t T/2), where T is the period of the supply sine waveform, even ordered harmonics cannot exist. Let the R phase supply voltage be given by the expression

Being a balanced three phase supply, we know that the waveforms of *v _{Y}* and

*v*are 120◦ and 240◦ shifted from

_{B}*v*respectively. It is further well known that if a waveform is shifted by ‑ degrees, its harmonics are shifted by

_{R}*n‑*

*Φ*degrees, where n is the order of the harmonic. Thus the expressions for v

_{Y}and v

_{B}would be

If we consider the third harmonic components of the three phase waveforms, and if v_{x3}(t) is the third harmonic of phase x, we can see that

Therefore, all the three third harmonics are in phase. In a STAR connected system with isolated neutral, these voltages cannot cause any current flow since all three terminals are equal in potential. If the neutral point is connected to some point, then then current can flow through the neutral connection. Such a connection is however rare in induction machines. The machine is therefore an open circuit to third harmonics. In fact, one can see that any harmonic whose order is a multiple of three, i.e., the triplen harmonics, as they are called, will face an identical situation. Since the machine is an open circuit to triplen harmonics in the excitation voltage, these do not have effect on the machine. Let us now consider the fifth harmonic. From the equations above, one can see that

From the above equation we see that the fifth harmonic of the excitation forms a negative sequence system — B phase lags R by 120◦ and Y phase lags R by 120◦. The MMF caused by a negative sequence excitation causes backward revolving flux pattern (compared to the direction of the fundamental). The torque which it generates will act as an opposing torque to that generated by the fundamental.

Looking at the seventh harmonic, we can see that

From the above eqns, it is evident that the seventh harmonic components of the excitation form a positive sequence system. The torque produced by these currents will therefore be additive with respect to the fundamental component’s torque. The actual effect of these harmonics on the induction machine would depend on the reactance of the machine since at high frequencies, it is the reactance component that dominates the inductance.

Excitation voltage waveforms with considerable harmonic content may result when induction machines are controlled through inverters. Apart from the effects on torque, these harmonics cause considerable heating in the machine and are hence a cause for concern. These harmonics are called time harmonics since they are generated by a source that varies non-sinusoidally in time.