20 Operational Deflection Shapes of a PWM-Fed Traction Motor 211 20.2.2 PWM Vibrations The interaction between a fundamental stator mmf and a harmonic mmf caused by a PWM time harmonic of the stator current, will result in so called PWM vibrations. Each harmonic of the stator current causes two groups of force harmonics, F pwm andFCpwm, with the properties listed in Table 20.2. fs is the fundamental stator frequency, f s n is the frequency of the PWM harmonic and s is ˙1 that indicates the propagation direction of the harmonic mmf. Since both the stator mmf wave and the PWM harmonic has spatial order p, the resulting force line will have spatial order 0or 2p, depending on the propagation directions of the interacting waves. That is, as indicated in Table 20.2, all pure PWM vibrations have a spatial order of 0 or 2p [9]. For PWM with triangular carrier, the harmonics f s n can be written as [4] f s n Dn1fs ˙n2fc; where n1 and n2have an opposite parity: (20.1) The amplitude of the harmonics of the stator current depends on both the carrier type of the PWM as well as the modulation index. For a triangular carrier, the amplitude of 2fc ˙fs is largest during the starting phase but at maximum speed the amplitude of fc ˙2fs is dominating, see Fig. 20.1 [4]. Substituting these frequencies in Table 20.2 gives the frequencies corresponding to the main peaks in the PWM related vibration response for a triangular carrier signal and their spatial orders as listed in Table 20.3 [9]. 20.2.3 Slotting PWM Vibrations The combination of a harmonic mmf caused by a PWM time harmonic of the stator current, a fundamental stator mmf, and slotting permeance harmonics will result in so called slotting PWM vibrations. Their characteristics are listed in Table 20.4 [10]. As can be seen, these vibrations have the same spatial order as the slotting vibrations but they occur at a higher frequency. Table 20.2 Frequency and spatial order of pure PWM force lines Name Frequency f Spatial order m F pwm fs sf s n p p D0 FCpwm fs C sf s n pCp D2p Fig. 20.1 Relationship between amplitude of harmonics and modulation index, fI is the fundamental stator frequency [4] fc ± 2fI 2fc ± f I 0 0 0.2 0.4 0.6 MODULATION INDEX D AMPLITUDE I(A) 0.8 1.0 3fc ± 2f I 4fc ± f I Table 20.3 Frequencies and spatial orders of main PWM-related vibration peaks for a triangular carrier (f s n D2fc ˙fs) (f s n D4fc ˙fs) (f s n Dfc ˙2fs) (f s n D3fc ˙2fs) Frequency f Frequency f Frequency f Frequency f Spatial order m FCpwm 2fc 2fs 4fc 2fs fc Cfs 3fc Cfs 2p F pwm 2fc 4fc fc C3fs 3fc C3fs 0 F pwm 2fc 4fc fc 3fs 3fc 3fs 0 FCpwm 2fc C2fs 4fc C2fs fc fs 3fc fs 2p
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