THE MAGNETIZATION CURVE AND INDUCTANCE
As shown in Chapters 2 and 4, the induction machine configuration is quite complex. So far we elucidated the subject of windings and their mmfs. With windings in slots, the mmf has (in three-phase or two-phase symmetric windings) a dominant wave and harmonics. The presence of slot openings on both sides of the airgap is bound to amplify (influence at least) the mmf step harmonics. Many of them will be attenuated by rotor-cage-induced currents. To further complicate the picture, the magnetic saturation of the stator (rotor) teeth and back irons (cores or yokes) also influence the airgap flux distribution producing new harmonics. Finally, the rotor eccentricity (static, and/or dynamic) introduces new harmonics in the airgap field distribution. In general, both stator and rotor currents produce a resultant field in the machine airgap and iron parts. However, with respect to fundamental torque-producing airgap flux density, the situation does not change notably from zero rotor currents to rated rotor currents (rated torque) in most induction machines, as experience shows. Thus it is only natural and practical to investigate, first, the airgap field fundamental with uniform equivalent airgap (slotting accounted through correction factors) as influenced by the magnetic saturation of stator and rotor teeth and back cores, for zero rotor currents. This situation occurs in practice with the wound rotor winding kept open at standstill or with the squirrel cage rotor machine fed with symmetrical a.c. voltages in the stator and driven at mmf wave fundamental speed (n1=f1/p1). As in this case the pure traveling mmf wave runs at rotor speed, no induced voltages occur in the rotor bars. The mmf space harmonics (step harmonics due to the slot placement of coils, and slot opening harmonics etc.) produce some losses in the rotor core and windings. They do not influence notably the fundamental airgap flux density and, thus, for this investigation, they may be neglected, only to be revisited in Chapter 11. To calculate the airgap flux density distribution in the airgap, for zero rotor currents, a rather precise approach is the FEM. With FEM, the slot openings could be easily accounted for; however, the computation time is prohibitive for routine calculations or optimization design algorithms. In what follows, we first introduce the Carter coefficient Kc to account for the slotting (slot openings) and the equivalent stack length in presence of radial ventilation channels. Then, based on magnetic circuit and flux laws, we calculate the dependence of stator mmf per pole F1m on airgap flux density, accounting for magnetic saturation in the stator and rotor teeth and back cores, while accepting a pure sinusoidal distribution of both stator mmf F1m and of airgap flux density, B1g. The obtained dependence of B1g(F1m) is called the magnetization curve. Industrial experience shows that such standard methods, in modern, rather heavily saturated magnetic cores, produce notable errors in the magnetizing curves, at 100 to 130% rated voltage at ideal no load (zero rotor currents). The presence of heavy magnetic saturation effects such as teeth or back core flux density flattening (or peaking), and the rough approximation of mmf calculations in the back irons are the main causes for such discrepancies.