34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1
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1 Electri c Machines and Control Vol. 22 No. 2 Feb DOI /j. emc TM A X Research of permanent magnet motors cogging force affected by interferential magnetic field ZHAO Wen-feng 1 2 ZHAO Bo 3 XU Yong-xiang 3 1. National Key Laboratory on Ship Vibration & Noise Wuxi China 2. China Ship Science Research Center Wuxi China 3. School of Electrical Engineering and Automation Harbin Institute of Technology Harbin China Abstract Aiming at the underwater acoustical directiity of permanent magnet PM motor theoretical and numerical simulation methods are used to study radial force characteristics of fractional slot motor when ideal sine wae power and containing harmonic current power supplied. It is found that radial force of the fractional slot PM motor with ideal sinusoidal phase current periodically appears after some teeth. When phase current contains noninterfering harmonic current the radial force is aperiodic in time domain but it is still periodic in frequency domain. When harmonic current in phase current is interferential the radial force is aperiodic in both time domain and frequency domain. The aperiodic law is erified by an experiment in which phase current containing interfering harmonic current caused by pulse width modulation PWM. The radial force is the driing source of electromagnetic ibration and noise. Different forces leads to different ibrations on tooth. The aperiodic law of radial force can explain the acoustic directiity. Keywords PM motors electromagnetic force radial force electromagnetic ibration and noise harmonic interference AA09A
2 34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1 Schematic diagram of stator tooth space position t + Δt B t A 2 3 p s - 0 ω 0 Δt t + Δt B t A 3 t A 3-7 t +Δt B f 0 t = F cos p 0 - ω 0 t - φ 1 f s t + Δt = F cos p s - ω 0 t - ω 0 Δt - φ 1 4 A B p s ω 0 Δt = k k 1 1 f w t f r t
3 z 0 p 0 z 0 = b + c 2mp 0 d = bd + c 6 d q = z = z 0 k p = p 0 k d 13 β = 360 /z 0 6 p 0 q = = 2 3k d 7 p s - 0 = ± 60 d 9 d = 6k d 10 p s - 0 = 6k ω 0 Δt d 11 6k ω 0 Δt = ± 60 k 2 k 1 = ± k 2 11 s ± 60 d ω I d f w t = F I cos p - ω I t - φ I I ω 0 Δt = ± 120 d ω 0 Δt 18 = ± 60 ω 0 Δt 60 d s t + Δt θ 0 t 12 f 0 t = f s t + Δt 12 s d 1 A B Λ p 0 t p s t + Δt 9 Λ 20 b t = f t + f t Λ θ 13 p t = b2 t 14 2 z 0 /6 Δt z ' r = ωδt /β = z 0 /6 = p 0 q = p 0 bd + c /d p s - 0 = 3k ω 0 Δt d 8 2 3k ω 0 Δt = ± 120 k 2 d z 0 2mp 0 k 1 k 1 = z 0 p 0 ± k 2 8 s 0 z 0 /m = m = 3 z 0 / 3 z 0 /6 d = p 0 15 d z r 16 z r = z ' r = bd + c 16 z 0 /6 d = 2p 0 d z r 17 ωδt = 120 z r = 2z ' r = bd + c 17 d bd + c F I I 5 Δt 19 k 3 ω I Δt = k ± 60 ω I Δt = k 3 } 360 ± ω 0 Δt = ± 60 d ω 0 Δt = ± 120 d ω I = 6k 3 ± 1 ω 0 ω I = 3k 3 ± 1 ω 0 } 20 ω 0 Δt 0 s ω 0 Δt 0 s 20 ω 0 Δt
4 36 22 ω I Δt I = ± 60 d ω I Δt I = ± 120 d ω I s 0 20 ω I 21 F I11 F I22 ω I ω 0 Δt ω I2 F I F ω I - ω 0 19 ω I - ω 0 = ω I1 + ω I2 ω I bd + c I Λ I I1 1 2 I I2 1 F I cos p - ω I t - φ I + F cos p - ω 0 t - φ 2 2 Λ 2 = F 2 Iν cos2 p - ω I t - φ I + F 2 cos2 p - ω 0 t - φ 2 + 2F I11 F I22 cos 1 p - ω I1 t - φ I1 cos 2 p - ω I2 t - φ I F 1 F 2 cos 1 p - 1 ω 0 t - φ 2 cos 2 p - 2 ω 0 t - φ F I F cos p - ω I t - φ I cos p - ω 0 t - φ 2 PMSM Δt = s ωδt = z = 12 2p = 8 q = 1 /2 d bd + d c = d r /min 2 ωδt = s 2. 2 bd + c = ~ z = 12 2p = Hz 330 Hz Fig. 2 2 Toothflux densities of PMSM with ideal sinusoidal phase current
5 Fig. 3 Air gap magnetic field of PMSM with ideal sinusoidal phase current 4 Fig. 4 Radial force of PMSM with ideal sinusoidal phase current 5 ωδt = 120 Fig. 5 5 Toothflux densities of PMSM with noninterference phase current = Hz = Fig. 6 Air gap magnetic field of PMSM with noninterference phase current phase current Hz 7 Fig. 7 Radial force of PMSM with noninterference
6 38 22 bd + c 2. 3 z = 12 2p = 8 80 Hz 160 Hz 8 2 s 0 ω 0 Δt = 120 Fig. 8 Radial force amplitude of PMSM with noninterference phase current ω 0 Δt Hz ω I 20 Fig. 9 9 Toothflux densities of PMSM with interference phase current 10 Fig. 10 Air gap magnetic field of PMSM with 11 interference phase current Fig. 11 Radial force of PMSM with interference phase current Hz 160 Hz 3 z = 12 2p = = ω I 20 14
7 2 39 PWM 15 A r /min A Fig. 15 Aphase current at speed r /min 12 Fig. 12 Radial force amplitude of PMSM with interference phase current r /min A Fig. 16 Spectrum map of A phase current at speed r /min Fig. 13 bd + c = Schematic diagram of sensor position ω I 20 PWM Fig Test loading installation diagram 16 A 800 Hz 80 Hz 160 Hz 320 Hz 160 Hz Hz 240 Hz Fig. 17 Amplitude of ibration acceleration on different 320 Hz 80 Hz teeth 240 Hz
8 bd + c d ω 0 Δt = ± 60 d ω 0 Δt = ± d ω I 6k 3 ± 1 ω 0 d ω I 3k 3 ± 1 ω 0 bd +c J C / / YANG Haodong CHEN Yangsheng. Electromagnetic ibration a nalysis and suppression of permanent magnet synchronous motor DRS J ZHU Z Q XIA Z P. Analytical modeling and finite-element computation of radial ibration force in fractional-slot permanent-magnet brushless machines J. IEEE Transactions on Industry Applications RAKIB I IQBAL H. Analytical model for predicting noise and ibrationin permanent-magnet synchronous motors J. IEEE Transactions on Industry Applications HUANG S AYDIN M LIPO T A. Electromagnetic ibration and noise assessment for surface mounted PM machines J. IEEE Power Engineering Society Summer Meeting J TANG Renyuan SONG Zhihuan. Study on source of ibration and acoustic noise of permanent magnet machines by inerter J. Electric Machines and Control J LI Xiaohua HUANG Surong. Calculation and analysis of ehicle ibration and noise of permanent magnet synchronous motor applied in electricehicle J. Electric Machines and Control FANG Yuan ZHANG Tong. Effect of tangential electromagnetic force on ibration and noise of electric powertrain J. Electric Machines and Control TAO Sun. Effect of pole and slot combination on noise and ibration in permanent magnet synchronous motor J. IEEE Transactions on Magnetics J with fractional slot combination J. Proceedings of the CSEE 11 D S. J. M D 2010.
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