Question 3.AE.12: Nonideal induction machines may generate noninteger space ha......

with the angular velocities

\frac{d \theta_{h=1}}{d t}=\omega_1, \quad \frac{d \theta_{h=0.1}}{d t}=0.1 \omega_1, \quad \frac{d \theta_{H=13.5}}{d t}=\pm \omega_1 / 13.5, \quad \frac{d \theta_{H=13.5, h=0.1}}{d t}=\pm \omega_1 / 13.5        (E3.12-1)

Application Examples 3.10 to 3.12 (3.10, 3.11, 3.12) illustrate how sub- and noninteger harmonics are generated by induction machines. Such harmonics may cause the malfunctioning of protective (e.g., under-frequency) relays within a power system [42].

The calculation of the asynchronous torques due to harmonics and inter- and subharmonics is based on the equivalent circuit without taking into account the variation of the differential leakage as a function of the numbers of stator slots and rotor slots. It is well known that certain stator and rotor slot combinations lead to asynchronous and synchronous harmonic torques in induction machines. A more comprehensive treatment of such parasitic asynchronous and synchronous harmonic torques is presented in [30,43–69]. Speed variation of induction motors is obtained either by voltage variation with constant frequency or by variation of the voltage and the fundamental frequency using inverters as a power source. Inverters have somewhat nonsinusoidal output voltages or currents. Thus parasitic effects like additional losses, dips in the torque–speed curve, output power derating, magnetic noise, oscillating torques, and harmonic line currents are generated. It has already been shown [43–45] that the multiple armature reaction has to be taken into account if effects caused by delta-connection of the stator windings, parallel winding branches, stator current harmonics, or damping of the air-gap field are to be considered. Moreover, the influence of the slot openings [46] and interbar currents [30,47,48,68] in case of slot skewing is important. Compared with conventional methods [25,30,68] the consideration of multiple armature reactions requires additional work consisting of the solution of a system of equations of the order from 14 to 30 for a three-phase induction machine and the summation of the air-gap inductances. The presented theory [69] is based on Fourier analysis and does not employ any transformation. As an alternative to the analytical method described [69] two-dimensional numerical methods (e.g., finite-element method) with time stepping procedures are available [49–51]. These have the advantages of analyzing transients, variable permeability, and eddy currents in solid iron parts, but have the disadvantages of grid generation, discretization errors, and preparatory work and require large computing CPU time.