A Microscopic Investigation of Force Generation in a Permanent Magnet Synchronous Machine S. Pekarek, Purdue University (W. Zhu UM-Rolla), (B. Fahimi University of Texas-Arlington) February 7, 25 1
Outline Torque and force characteristics Structure of permanent magnet (PM) synchronous machine Define macroscopic view of torque Define microscopic view of force Microscopic torque and force characteristics of PM synchronous machines Tangential and radial force ripple minimization Field reconstruction (FR) method Optimization using FR method 2
Permanent Magnet Synchronous Machine Has stator windings in stator slots Uses permanent magnets as magnetic source on the rotor 3
Macroscopic Machine Model v = r i d + λ dt λ = Li ' + λ abcs s abcs abcs abcs s abcs abcpm λ sin( ) λ ' 2π cpm λmsin( θr + ) 3 ' λapm λmsin( θr) ' ' 2π abcpm = λbpm = λm θr 3 Wc P λaspm λbspm λcspm Wpm Te = = ( ias + ibs + ics + ) θ 2 θ θ θ θ r r r r r 4
Machine Model in Rotor Frame of Reference v = ri + ωλ + pλ r r r r qs s qs r ds qs v = ri ωλ + pλ r r r r ds s ds r qs ds λ = λ r qs L i r ss qs = Li + λ r r ' ds ss ds m 3 P T i T 22 ' r e = λ mqs+ cog r ( θ ) Average torque not function of d-axis current 5
Field-Based Solution of Magnetic Forces Material 1 f B l Maxwell Stress Tensor (Neglecting z- component of flux density) Force Density: Material 2 f f = B B / µ t n t 2 2 n = n t f ( B B )/2µ Overall Force: t = t F f dl n = n F f dl Te = Ft Lstack Rcontour 6
Microscopic View of Electric Machines i1 i 2 i3 B n B t N S B 7
Cross-section of PM Machine Studied 3-phase, 4-pole, 12-slot, surface-mounted, 1 HP, 2 rpm 8
Flux and Force Density Distribution by PMs Distribution of B t and B n generated by permanent magnets Average Forces: F t =, F = 587 n N/m 9
Flux and Force Density Distribution r i qs = 4.6 A Distribution of B t and B n at single rotor r r position when A i ds Average Forces: =, i = 4.6 qs F t = 1219, F = 6239 n Distribution of f t and f n at single rotor position when r r i ds N/m =, i = 4.6 qs A 1
Flux and Force Density Distribution r i ds = 4. A Distribution of B t and B n at single rotor r r position when i ds Distribution of f t and f n at single rotor position r r when = 4., i = 4.6 i = 4., i = 4.6 qs ds qs Average Forces: F t = 1218, F = 9132 n N/m 11
r i ds Effect of on Average Components of Force d-axis current (A) Average F t (N/m) Average F n (N/m) -9. -4. 4. 9. 1148 1154 1157 116 116 1755 3852 6248 9278 13879 12
Effect of r i qs on Average Components of Force q-axis current (A) 1 3 4.6 6 8 Average of Ft (N/m) 249 748 1157 1495 1985 Average of Fn (N/m) 5837 5996 6249 6531 783 13
Detailed Force Density Expression Source of Magnetic Field: Phase currents Permanent Magnets If effect of saturation is neglected, then: B = B + B n npm ns B = B + B t tpm ts 1 f = B B B B µ + + t ts tpm npm ns 1 f B B B B µ 2 2 n = ( npm + ns) ( ts + tpm) 2 14
Components of Torque B tpm B B ns ts B B B ns B B npm ts tpm npm (zero average) (zero average) (non-zero average) (non-zero average) 15
Flux Densities Generated by Current in Single Stator Slot Magnet Bn [T] Φ s B n B t Airgap I slot Iron1 Bt [T] If set origin at center, then: f 1 is a even function. f 2 is a odd function. B ( φ ) = I f ( φ ) tsk s slot 1 s B ( φ ) = I f ( φ ) nsk s slot 2 s 16
Components of Tangential Force B = B cos( kφ ) npm npmk r k= 1 B = B sin( kφ ) B B B tpm tpmk r k = 1 ( φ ) = i F ( φ ) t _ as s as 1 s ( φ ) = i F ( φ 12) t_ bs s bs 1 s ( φ ) = i F ( φ + 12) t_ cs s cs 1 s F = F cos( kφs ) 1 1k k = 1 B B B ( φ ) = i F ( φ ) n _ as s as 2 s ( φ ) = i F ( φ 12) n_ bs s bs 2 s ( φ ) = i F ( φ + 12) n_ cs s cs 2 s F ( φ ) = F sin( kφ ) 2 s 2k s k = 1 17
Average Tangential Force Due to Fundamental Harmonics P 3 F π B F B F R I r t = ( npm1 11+ tpm1 21) 2π qs 2 µ 1 2π F21 = F2 ( φs ) sin( φs ) dφs π 1 2π F11 = F1 ( φs ) cos( φs ) dφs π d-axis current doesn t appear in average tangential force. 18
Components and Average Radial Force f n = 2 1 ( Bnpm + Bn as + Bn bs + Bn cs ) 2 µ 2 ( Bt as + Bt bs + Bt cs + Btpm) If consider only the fundamental components: Pπ R F F F i i B B 2 2 r 2 r 2 2 2 n = [9( 21 11 )( qs + ds ) + npm1 tpm1 4µ r F21Bnpm1 F11 Btpm1 ids + 6( ) ] 19
Problems in PM Synchronous Machine Applications Acoustic noise and vibration caused by: Torque ripple Harmonics in radial force Solutions for torque ripple mitigation: Improve machine design to obtain better back-emf waveform Employ excitation control methods to eliminate torque harmonics 2
Force Optimization using FEA 21
Field Reconstruction Rotor Magnet Air-gap Stator Slot# k k+1 B = B + B n npm ns B = B + B t tpm ts B = B ( B, B ) n n nsk npm B = B ( B, B ) t t tsk tpm B B ns ts L = B k = 1 L = B k = 1 nsk tsk B ( φ ) = I f ( φ ) tsk s slot 1 s B ( φ ) = I f ( φ ) nsk s slot 2 s 22
Field Reconstruction Method 23
Field Reconstruction Results Normal Component 24
Field Reconstruction Results Tangential Component 25
Force Optimization Using Field Reconstruction 26
Operation with Fixed Ft and Minimal Copper Loss Min i i i F t = 1 ias + ibs + ics = max 2 2 2 ( as + bs + cs) Subject to: N/m i i, i, i i as bs cs max ias, ibs, ics [A] 5 4 3 2 1 1 2 3 4 5 5 1 15 2 25 3 35 4 Angle [Electrical degree] Phase current waveform ias ibs ics 27
Waveform of Ft, Fn and Comparison with Sinusoidal Excitation Ft [N/m] Ft [N/m] 14 135 13 125 12 115 11 15 1 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] 1 8 6 4 2 optimized sinusoidal currents optimized sinusoidal currents 2 4 6 8 1 12 14 16 18 2 Harmonic order Fn [N/m] Fn [N/m] 7 65 6 55 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] 5 4 3 2 1 optimized sinusoidal currents optimized sinusoidal currents 2 4 6 8 1 12 14 16 18 2 Harmonic order Peak-to-peak Optimal Ft (N/m) 7.3/111 Fn (N/m) 991/6198 Sinusoidal 145/1157 24/6248 28
Operation with Fixed Fn and Ft Min[( F F ) + F ias + ibs + ics = max t F 2 ( n n ) ] Subject to: i i, i, i i t as bs cs 2 max ias, ibs, ics [A] 5 4 3 2 1 1 2 3 4 5 5 1 15 2 25 3 35 4 Angle [Electrical degree] Phase current waveform ias ibs ics 29
Waveform of Ft, Fn and Comparison with Sinusoidal Excitation Ft [N/m] 14 13 12 11 optimized sinusoidal currents Fn [N/m] 65 64 63 62 optimized sinusoidal currents Ft [N/m] 1 2 4 6 8 1 12 14 16 18 1 5 Angle [Electrical degree] optimized sinusoidal currents Fn [N/m] 61 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] 1 5 optimized sinusoidal currents 5 1 15 2 Harmonic order 5 1 15 2 Harmonic order Peak-to-peak Optimal Ft (N/m) 7.3/112 Fn (N/m) 16/6297 Sinusoidal 145/1157 24/6248 3
Operation with Fixed Fn, Ft and Minimal Copper Loss Min i i i Subject to: F t = 1 N/m F n = 62 i i, i, i i max 2 2 2 ( as + bs + cs) N/m as bs cs max ias, ibs, ics [A] 5 4 3 2 1 1 2 3 4 ias ibs ics 5 5 1 15 2 25 3 35 4 Angle [Electrical degree] Phase current waveform 31
Waveform of Ft, Fn and Comparison with Sinusoidal Excitation Ft [N/m] Ft [N/m] 14 135 13 125 12 115 11 15 1 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] 1 8 6 4 2 optimized sinusoidal currents optimized sinusoidal currents 2 4 6 8 1 12 14 16 18 2 Harmonic order Fn [N/m] 65 645 64 635 63 625 62 615 61 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] Fn [N/m] 1 8 6 4 2 optimized sinusoidal currents optimized sinusoidal currents 2 4 6 8 1 12 14 16 18 2 Harmonic order Peak-to-peak Optimal Ft (N/m) 6.6/11 Fn (N/m) 23/6293 Sinusoidal 145/1157 24/6248 32
Trapezoidal back-emf case: Fixed Fn, Ft and Minimal Copper Loss Min i i i Subject to: F t = 1 F n = 62 i i, i, i i max 2 2 2 ( as + bs + cs) as bs cs max ias, ibs, ics [A] 5 4 3 2 1 1 2 3 4 5 5 1 15 2 25 3 35 4 Angle [Electrical degree] Phase current waveform ias ibs ics 33
Waveform of Ft, Fn and Comparison with Trapezoidal Excitation Ft [N/m] 13 12 11 optimized trapzoidal currents 1 2 4 6 8 1 12 14 16 18 Angle [Electrical degree] Ft [N/m] 5 4 3 2 1 optimized trapzoidal currents 5 1 15 2 Harmonic order Fn [N/m] Fn [N/m] 8 75 7 65 6 55 5 2 4 6 8 1 12 Angle [Electrical degree] 14 16 18 12 1 8 optimized trapzoidal currents optimized trapzoidal currents 6 4 2 2 4 6 8 1 12 14 16 18 2 Harmonic order Peak-to-peak Optimal Ft (N/m) 11/11 Fn (N/m) 31/6289 Trapezoidal 97/1185 2773/6427 34
Conclusions Microscopic investigation of forces leads to result that both d- and q-axis currents influence radial force (quadratic) Area of tangential force density relatively small leading to larger radial than tangential force Opens the question - are alternative designs/excitation strategies possible to provide a more effective force profile? Field reconstruction is a time-efficient tool to consider alternative methods of excitation for control force profile 35