• The fundamental component of resultant airgap flux density distribution in interaction with the fundamental of stator (or rotor) mmf produces the fundamental (working) electromagnetic torque in the IM.
• The placement of windings in slots (even in infinitely thin slots) leads to a stepped-like waveform of stator (rotor) mmfs which exhibit space harmonics besides the fundamental wave (with p1 pole pairs). These are called mmf harmonics (or step harmonics); their lower orders are called phase-belt harmonics ν = 5, 7, 11, 13, … to the first “slot” harmonic order νsmin = Np1s ±1. In general, ν = (6c1 ± 1).
• A wound rotor mmf has its own harmonic content µ = (6c2 ± 1).
• A cage rotor, however, adapts its mmf harmonic content (µ) to that of the stator such that
ν −µ = c N2 r p1
• ν and µ are electric harmonics, so ν = µ = 1 means the fundamental (working) wave.
• The mechanical harmonics νm, µm are obtained if we multiply ν and µ by the number of pole pairs,
νm = p1ν; µm = p1µ;
• In the study of parasitic torques, it seems better to use electric harmonics, while, for radial forces, mechanical harmonics are favored. As the literature uses both concepts, we thought it appropriate to use them both in a single chapter.
• The second source of airgap field distribution is the magnetic specific airgap conductance (inversed airgap function) λ1,2(θm,t) variation with stator or/and rotor position. In fact, quite a few phenomena are accounted for by decomposing λ1,2(θm,t) into a continuous component and various harmonics. They are outlined below.
• The slot openings, both on the stator and rotor, introduce harmonics in λ1,2(θm,t) of the orders c1Ns/p1, c2Nr/p1 and (c1Ns ± c2Nr)p1 with c1 = c2 = 1 for the first harmonics.
At high currents, the teeth heads saturate due to large leakage fluxes through slot necks. This effect is similar to a fictitious increase of slot openings, variable with slot position with respect to stator (or rotor) mmf
• with respect to stator (or rotor) mmf maximum. A λ1,2(θm,t) second order harmonic (4p1 poles) traveling at the frequency of the fundamental occurs mainly due to this aspect.
• The main flux path saturation produces a similar effect, but its second order harmonic (4p1 poles) is in phase with the magnetization mmf and not with stator or rotor mmfs separately!
• The second permeance harmonic also leads to a third harmonic in the airgap flux density.
• The rotor eccentricity, static and dynamic, produces mainly a two-pole harmonic in the airgap conductance. For a two-pole machine, in interaction with the fundamental mmf, a homopolar flux density is produced. This flux is closing axially through the frame, bearings, and shaft, producing an a.c. shaft voltage and bearing current of frequency f1 (for static eccentricity) and Sf1 (for the dynamic eccentricity).
• The interaction between mmf and airgap magnetic conductance harmonics is producing a multitude of harmonics in the airgap flux distribution.
• Stator and rotor mmf harmonics of same order and same stator origin produce asynchronous parasitic torques. Practically all stator mmf harmonics produce asynchronous torques in cage rotors as the latter adapts to the stator mmf harmonics. The no-load speed of asynchronous torque is ωr = ω1/ν.
• As ν ≥ 5 in symmetric (integer q) stator windings, the slip S for all asynchronous torques is close to unity. So they all occur around standstill.
• The rotor cage, as a short-circuited multiphase winding, may attenuate asynchronous torques.
• Chording is used to reduce the first asynchronous parasitic torques (for ν = −5 or ν = +7); skewing is used to cancel the first slot mmf harmonic ν = 6q1 ± 1. (q1–slots/pole/phase).
• Synchronous torques are produced at some constant specific speeds where two harmonics of same order ν1 = ±ν′ but of different origin interact.
• Synchronous torques may occur at standstill if ν1 = +ν′ and at
S = +1 c N22p1r ; -1≥ C2 ≥1
for ν1 = −ν′.
• Various airgap magnetic conductance harmonics and stator and rotor mmf harmonics may interact in a cage rotor IM many ways to produce synchronous torques. Many stator/rotor slot number Ns/Nr combinations are to be avoided to eliminate most important synchronous torque. The main benefit is that the machine will not lock into such speed and will accelerate quickly to the pertinent load speed.
• The harmonic current induced in the rotor cage by various sources may induce certain current harmonics in the stator, especially for ∆ connection or for parallel stator windings (a >1).
This phenomenon, called secondary armature reaction, reduces the differential leakage coefficient τd of the first slot harmonic νsmin = Ns/p1 ± 1 and thus, in fact, increases its corresponding (originating) rotor current.
• Such an augmentation may lead to the amplification of some synchronous torques.
• The stator harmonics currents circulating between phases (in ∆ connection) or in between current paths (in parallel windings), whose order is multiples of three may be avoided by forbidding some Ns, Nr combinations (Ns−Nr) ≠ 2p1, 4p1, ….
• Also, if the stator current paths are in the same position with respect to rotor slotting, no such circulating currents occur.
• Notable differences, between the linear theory and tests have been encountered with large torque amplifications in the braking region (S > 1). The main cause of this phenomenon seems to be magnetic saturation.
• Slot opening presence also tends to amplify synchronous torques. This tendency is smaller if Nr < Ns; even straight rotor slots (no skewing) may be adopted. Attention to noise for no skewing!
• As a result of numerous investigations, theoretical and experimental, clear recommendations of safe stator/rotor slot combinations are now given in some design books. Attention is to be paid to the fact that, when noise is concerned, low power machines and large power machines behave differently.
• Radial forces are somehow easier to calculate directly by Maxwell’s stress method from various airgap flux harmonics.
• Radial stress (force per unit stator area) pr is a wave with a certain order r and a certain electrical frequency Ωr. Only r = 0, 1, 2, 3, 4 cases are important.
• Investigating again the numerous combination contributions to the mechanical stress components coming from the mmfs and airgap magnetic conductance harmonics, new stator/rotor slot number Ns/Nr restrictions are developed.
• Slip ring IMs behave differently as they have clear cut stator and rotor mmf harmonics. Also damping of some radial stress component through induced rotor current is absent in wound rotors.
• Especially radial forces due to rotor eccentricity are much larger in slip ring rotor than in cage rotor with identical stators, during steady state. The eccentricity radial stress during starting transients are however about the same.
• The circulating current of parallel winding might, in this case, reduce some radial stresses. By increasing the rotor current, they reduce the resultant flux density in the airgap which, squared, produces the radial stress.
• The effect of radial stress in terms of stator vibration amplitude manifests itself predominantly with zero, first, and second order stress harmonics (r = 0, 1, 2).
• For large power machines, the mechanical resonance frequency is smaller and thus the higher order radial stress r = 3, 4 are to be avoided if low noise machines are to be built.
• For precision calculation of parasitic torques and radial forces, after preliminary design rules have been applied, FEM is to be used, though at the price of still very large computation time.
• Vibration and noise is a field in itself with a rich literature and standardization. [11,12]
• standardization. [11,12]
10.12. REFERENCES
1. B. Heller, V. Hamata, Harmonics Effects in Induction Motors, Chapter 6, Elsevier, 1977.
2. R. Richter, Electric Machines, Second edition, Vol.1, pp.173, Verlag Birkhäuser, Basel, 1951 (in German).
3. K. Vogt, Electric Machines–Design of Rotary Electric Motors, Chapter 10,
VEB Verlag Technik, Berlin, 1988. (in German)
4. K. Oberretl, New Knowledge on Parasitic Torques in Cage Rotor Induction Motors, Buletin Oerlikon 348, 1962, pp.130 – 155.
5. K. Oberretl, The Theory of Harmonic Fields of Induction Motors Considering the Influence of Rotor Currents on Additional Stator Harmonic Currents in Windings with Parallel Paths, Archiv für Electrotechnik, Vol.49, 1965, pp.343 – 364 (in German).
6. K. Oberretl, Field Harmonics Theory of Slip Ring Motor Taking Multiple Reaction into Account, Proc of IEE, No.8, 1970, pp.1667 – 1674.
7. K. Oberretl, Parasitic Synchronous and Pendulation Torques in Induction Machines; the Influence of Transients and Saturation, Part II – III, Archiv für Elek. Vol.77, 1994, pp.1 – 11, pp.277 – 288 (in German).
8. D.G. Dorrel, Experimental Behaviour of Unbalanced Magnetic Pull in 3phase Induction Motors with Eccentric Rotors and the Relationship with Teeth Saturation, IEEE Trans Vol. EC – 14, No.3, 1999, pp.304 – 309.
9. J.F. Bangura, N.A. Dermerdash, Simulation of Inverter-fed Induction Motor Drives with Pulse-width-modulation by a Time-stepping FEM Model–Flux Linkage-based Space Model, IEEE Trans, Vol. EC – 14, No.3, 1999, pp.518 – 525.
10. A. Arkkio, O. Lingrea, Unbalanced Magnetic Pull in a High Speed Induction Motor with an Eccentric Rotor, Record of ICEM – 1994, Paris, France, Vol.1, pp.53 – 58.
11. S.J. Yang, A.J. Ellison, Machinery Noise Measurement, Clarendon Press, Oxford, 1985.
1. P.L. Timar, Noise and Vibration of Electrical Machines, Technical Publishers, Budapest, Hungary, 1986.