9 Harmonics in Induction Machines

Transcription

1 9 Harmonics in Induction Machines In attempting to understand e performance of an induction machine, we consider at e air-gap flux wave is purely sinusoidal. It is from at assumption e analysis of induced emf, sinusoidal currents, e expressions for generated torque etc. proceed. In practice, ere are deviations from is idealistic picture. 9.1 Time Harmonics The first non-ideality is e presence of harmonics in e input supply given to e ree phase machine. The source may contain 3 rd, 5, 7...harmonics. Note at due to e symmetry of e waveform (f(t) = f(t + T/2), where T is e period of e supply sine waveform, even ordered harmonics cannot exist. Let e R phase supply voltage be given by e expression v R = V 1m sin(ω 1 t + φ 1 ) + V 3m sin(3ω 1 t + φ 3 ) +V 5m sin(5ω 1 t + φ 5 ) + V 7m sin(7ω 1 t + φ 7 ) + (25) Being a balanced ree phase supply, we know at e waveforms of v Y and v B are 12 and 24 shifted from v R respectively. It is furer well known at if a waveform is shifted by φ degrees, its harmonics are shifted by nφ degrees, where n is e order of e harmonic. Thus e expressions for v Y and v B would be v Y = V 1m sin(ω 1 t + φ 1 2π 3 ) + V 3m sin(3ω 1 t + φ π 3 ) +V 5m sin(5ω 1 t + φ π 3 ) + V 7m sin(7ω 1 t + φ π 3 ) + (26) v B = V 1m sin(ω 1 t + φ 1 4π 3 ) + V 3m sin(3ω 1 t + φ π 3 ) V 5m sin(5ω 1 t + φ π 3 ) + V 7m sin(7ω 1 t + φ π 3 ) + (27) If we consider e ird harmonic components of e ree phase waveforms, and if v x3 (t) is e ird harmonic of phase x, we can see at v R3 = V 3m sin(3ω 1 t + φ 3 ) v Y 3 = V 3m sin(3ω 1 t + φ 3 ) v B3 = V 3m sin(3ω 1 t + φ 3 ) (28) 42

2 Therefore, all e ree ird harmonics are in phase. In a STAR connected system wi isolated neutral, ese voltages cannot cause any current flow since all ree terminals are equal in potential. If e neutral point is connected to some point, en en current can flow rough e neutral connection. Such a connection is however rare in induction machines. The machine is erefore an open circuit to ird harmonics. In fact, one can see at any harmonic whose order is a multiple of ree, i.e., e triplen harmonics, as ey are called, will face an identical situation. Since e machine is an open circuit to triplen harmonics in e excitation voltage, ese do not have effect on e machine. Let us now consider e fif harmonic. From e equations above, one can see at v RS = V 5m sin(5ω 1 t + φ 5 ) v Y S = V 5m sin(5ω 1 t + φ π 3 ) = V 5m sin(5ω 1 t + φ 5 4π 3 ) v BS = V 5m sin(5ω 1 t + φ π 3 ) = V 5m sin(5ω 1 t + φ 5 2π 3 ) (29) From eqns. 29 we see at e fif harmonic of e excitation forms a negative sequence system B phase lags R by 12 and Y phase lags R by 12. The MMF caused by a negative sequence excitation causes backward revolving flux pattern (compared to e direction of e fundamental). The torque which it generates will act as an opposing torque to at generated by e fundamental. Looking at e seven harmonic, we can see at v R7 = V 7m sin(7ω 1 t + φ 7 ) v Y 7 = V 7m sin(7ω 1 t + φ π 3 ) = V 7m sin(7ω 1 t + φ 7 2π 3 ) v B7 = V 7m sin(7ω 1 t + φ π 3 ) = V 7m sin(7ω 1 t + φ 7 4π 3 ) (3) 43

3 From eqns. 3, it is evident at e seven harmonic components of e excitation form a positive sequence system. The torque produced by ese currents will erefore be additive wi respect to e fundamental component s torque. The actual effect of ese harmonics on e induction machine would depend on e reactance of e machine since at high frequencies, it is e reactance component at dominates e inductance. Excitation voltage waveforms wi considerable harmonic content may result when induction machines are controlled rough inverters. Apart from e effects on torque, ese harmonics cause considerable heating in e machine and are hence a cause for concern. These harmonics are called time harmonics since ey are generated by a source at varies non-sinusoidally in time. 9.2 Space Harmonics Apart from is, ere is anoer kind of harmonic generated in machines called space harmonic. To understand at is behaviour, it is necessary to consider MMF/flux production in e machine. It may not be out of place to recall once again at in all our foregoing analysis we have assumed at bo air-gap mmf and flux are sinusoidally distributed in space. Let us consider a single full-pitched coil excited by a sinusoidal voltage. The current flowing rough it is sinusoidal and hence e time variation of e mmf produced by it is sinusoidal. But if we travel around e span of e coil, e MMF variation at we would encounter is square. It is e amplitude of is square wave at varies sinusoidally in time. The behaviour is depicted diagrammatically in fig. 36. Let one more coil be connected in series to is, which is spatially displaced by e slot angle. This is shown in fig. 37. The same current passes rough bo and hence e mmf pattern generated by bo would vary in tandem. However, ey will be displaced by a slot angle as far as spatial distribution is concerned. The resulting situation is also shown in fig. 37 at a particular time instant. It can be seen at e resultant mmf waveform a non-sinusoidal function of e space angle θ. The harmonics are functions of e space angle. These are called space harmonics. Let us consider e effects of ese. Let e net flux waveform as a function of angle at an instant of time when unit current flows in e coils be described by f(θ). Clearly f(θ) is a periodic function of θ wi a period equal to 2π. Therefore one may express is as a fourier series. If f A (θ) is e distribution function for phase A, f A (θ) = A 1 sin(θ + φ 1 ) + A 3 sin(3θ + φ 3 ) + A 5 sin(5θ + φ 5 ) + (31) 44

4 current, A MMF 9 o 18 o 27 o 36 o θ time, s (a) θ=9 ο (b) ο θ= (c) Figure 36: Coil MMF wi sinusoidal excitation The distribution functions for phases B & C will be displaced from at of A phase by 12 and 24 respectively and hence are given by f B (θ) = A 1 sin(θ + φ 1 2π 3 ) + A 3 sin(3θ + φ 3 ) + A 5 sin(5θ + φ 5 4π 3 ) + (32) f C (θ) = A 1 sin(θ + φ 1 4π 3 ) + A 3 sin(3θ + φ 3 ) + A 5 sin(5θ + φ 5 2π 3 ) + (33) Note at we have written ese at a given instant of time when unit current flows. We know at is current variation is sinusoidal in time. Considering e fif harmonic, let e resultant fif harmonic variation is given by, f 5 (t, θ) = A 5m sin ωt sin(5θ + φ 5 ) + A 5m sin(ωt 2π 3 ) sin(5θ + φ 5 4π 3 + A 5m sin(ωt 4π 3 ) sin(5θ + φ 5 2π 3 ) + (34) 45

5 Upon simplification, we get Figure 37: MMF in a distributed winding in two slots f 5 (t, θ) = 3 2 A 5m cos(ωt + 5θ) (35) Consider e behaviour of is function. At t=, e function a value of 3A 2 5m at θ =. Now let ωt = π. At is instant, we find at e function reaches a value 3 3A 2 5m at θ = π. In oer words e function f 3 5 5(t, θ) has shifted by an angle which is a fif of e value of ωt, in e negative direction. The fif harmonic erefore rotates opposite to e direction of e fundamental at a speed which is one-fif of e fundamental. Similarly, if we consider e seven harmonic, it can be shown at e resultant distribution is f 7 (t, θ) = 3 2 A 7m cos(ωt 7θ) (36) By similar arguments as above we conclude at e seven space harmonic rotates in e same direction as at of e fundamental at one seven e speed. In general, we may have harmonics of e order h = 6n±1, where n is an integer greater an or equal to 1. Harmonics orders generated by e addition operation move in e same direction as e fundamental and ose generated by e subtraction operation move in e opposite direction. The speed of rotation is ω 1 /h, where ω 1, is e synchronous speed of e fundamental. 46

6 The space harmonics, it may be noticed are a result of non-sinusoidal distribution of e coils in e machine and slotting. These have eir effects on e speed torque current of e machine. An example speed-torque characteristic of an induction machine is compared wi its ideal characteristic in fig. 38. The effect of 5, 7, 11 and 13 harmonics have been considered. It can be seen at ese harmonics result in kinks in e speed-torque characteristic near starting region wiout harmonic wi harmonic 2 Torque, Nm speed, rpm Figure 38: Ideal and actual speed-torque curves To understand e effect of ese kinks, consider fig. 39, which shows e same speed torque characteristic in e motoring region. A load characteristic is also shown, which intersects e machine characteristic at various points. Note at point 1 is stable and hence e machine would have a tendency to operate ere, ough e intended operating point might be point 5. This tendency is referred to as crawling. A momentary reduction in load torque conditions might accelerate e machine to print 2, which is unstable. The operating point would en settle down at 3. The intended operating point may be reached if favourable torque variations are ere. 47

7 8 7 line Torque, Nm speed, rpm Figure 39: Effect of harmonics on loaded machine 48