• In absence of rotor currents, the stator winding currents produce an airgap flux density which contains a strong fundamental and spatial harmonics due to the placement of coils in slots (mmf harmonics), magnetic saturation, and slot opening presence. Essentially, FEM could produce a fully realistic solution to this problem.
• To simplify the study, a simplified analytical approach is conventionally used; only the mmf fundamental is considered.
• The effect of slotting is “removed” by increasing the actual airgap g to gKc (Kc > 1); Kc–the Carter coefficient.
• The presence of eventual ventilating radial channels (ducts) is considered through a correction coefficient applied to the geometrical stack axial length.
• The dependence of peak airgap flux density on stator mmf amplitude (or stator current) for zero rotor currents, is called the magnetization curve.
• When the teeth are designed as the heavily saturated part, the airgap flux “flows” partially through the slots, thus “defluxing” to some the teeth extent.
• To account for heavily saturated teeth designs, the standard practice is to calculate the mmf ( )F1 300 required to produce the maximum (flat) airgap flux density and then increase its value to F1m = ( )F1 300 /cos300 .
• A more elaborate two dimensional analytical iterative model (AIM), valid both for zero and nonzero rotor currents, is introduced to refine the results of the standard method. The slotting is considered indirectly through given but different radial and axial permeabilities in the slot zones. The results are validated on numerous low power motors, and sinusoidal airgap and core flux densities.
• Further on, AIM is used to calculate the actual airgap flux density distribution accounting for heavy magnetic saturation. For more saturated teeth, flat airgap flux density is obtained while, for heavier back core saturation, a peaked airgap and teeth flux density distribution is obtained.
• The presence of saturation harmonics is bound to influence the total core losses (as investigated in Chapter 11). It seems clear that if teeth and back iron cores are heavily but equally saturated, the flux density is still sinusoidal all along stator bore. The stator connection is also to be considered as for star connection, the stator no-load current is sinusoidal, and flux third harmonics may occur, while for the delta connection, the opposite is true.
• The expression of emf in a.c. windings exhibits the distribution, chording, and skewing factors already derived for mmfs in Chapter 4.
• The emf harmonics include mmf space (step) harmonics, saturation-caused space harmonics, and slot opening (airgap permeance variation) harmonics νc.
• Slot flux (emf) harmonics νc show a distribution factor Kqν equal to that of the fundamental (Kqν = Kq1) so they cannot be destroyed. They may be attenuated by increasing the order of first slot harmonics νcmin = 2qm – 1 with larger q slots per pole per phase or/and by increased airgap, or by fractional q.
• The magnetization inductance L1m valid for the fundamental is the ratio of phase emf, to angular stator frequency to stator current. L1m is decreasing with airgap, number of pole pairs, and saturation level (factor), but it is proportional to pole pitch and stack length and equivalent number of turns per phase squared.
• In relative values, l1m increases with motor power and decreases with larger number of poles, in general.
REFERENCES of The Magnetization Curve and Inductance
1. F.W. Carter, Airgap Induction, El. World and Engineering, 1901, p.884.
2. T.A. Lipo, Introduction to AC Machine Design, vol.1, pp.84, WEMPEC – University of Wisconsin, 1996.
3. B. Heller, V. Hamata, Harmonic Field Effects in Induction Machines, Elsevier Scientific, Amsterdam, 1977.
4. D.M. Ionel, M.V. Cistelecan, T.J.E. Miller, M.I. McGilp, A New Analytical Method for the Computation of Airgap Reactances in 3 Phase Induction Machines, Record of IEEE – IAS, Annual Meeting 1998, vol. 1, pp.65 – 72.
5. S.L. Ho, W.N. Fu, Review and Future Application of Finite Element Methods in Induction Motors, EMPS vol.26, 1998, pp.111 – 125.
6. V. Ostovic, Dynamics of Saturated Electric Machines, Springer Verlag, New York, 1989.
7. G. Madescu, I. Boldea, T.J.E Miller, An Analytical Iterative Model (AIM) for Induction Motor Design, Rec. of IEEE – IAS, Annual Meeting, 1996, vol.1., pp.566 – 573.
8. G. Madescu, “Contributions to The Modelling and Design of Induction Motors”, Ph.D. Thesis, University Politehnica, Timisoara, Romania, 1995.