Lectue Set 6 Bushless DC Machines S.D. Sudhoff Sping 2018
Reading Chapte 8, Electomechanical Motion Devices, 2 nd Edition 2
A Bushless DC Machine 3
Sample Applications Low Powe: Disk dive motos Medium Powe: Robot manipulatos Sevo systems Hybid/electic vehicles High Powe: Ship and submaine populsion Wind tubines 4
Disk Dive Moto 5
4 Hp BDC Machine 6
Chaacteistics The Good The Bad 7
Pemanent Magnet Synchonous Machines Radial Vesus Axial Suface Mounted Vesus Buied Magnet Sinusoidal Vesus Non-Sinusoidal 8
Radial Suface Mounted PMSM bs-axis Path of Integal fo Section 7.2.1 q-axis φ m θ m φ sm as-axis cs-axis d-axis 9
3-Phase PMSM 10
3-Phase PMSM Notation ( fabcs) = [ fas fbs fcs ] Voltage equations T dλ v i as as = s as + dt d v i bs bs = s bs + dt dλ v i cs cs = s cs + dt v = i + pλ abcs s abcs abcs 11
12 3-Phase PMSM Flux Linkage Equations whee m abcs s λ λ + = i L abcs + = = π θ π θ θ 3 2 sin( ) 3 2 sin( sin m csm bsm asm m λ λ λ λ λ + + + = ms ls ms ms ms ms ls ms ms ms ms ls s L L L L L L L L L L L L 2 1 2 1 2 1 2 1 2 1 2 1 L
PM Tems Intuitive Appoach 13
PM Tems Intuitive Appoach 14
PM Tems Intuitive Appoach 15
Inductances We will assume the following n n n as bs cs = N s sin( Pφsm / 2) = N sin( Pφsm / 2 2π/3) s = N sin( Pφsm / 2 + 2π/3) s It follows that w w as w bs cs 2N = s cos( Pφsm / 2) P 2N = s cos( Pφsm / 2 2π / 3) P 2Ns = cos( Pφ sm / 2 + 2π / 3) P 16
Inductances Recall λ α, m i β = L m, αβ = µ L Fom which we obtain 0 2π 0 w α ( φ) w β g( φ) ( φ) dφ L asbs 2 s 4πµ LN L asas = 0 = 2 P g 2πµ 0LN = 2 P g 2 s L ms = 1 2 L ms Doesn t include leakage 17
PM Tems Analytical Appoach 18
PM Tems Def. of Elec. Quantities θ ω φ φ s = P = P = P = P θ ω φ φ m m sm m / 2 / 2 / 2 / 2 19
PM Tems Suppose the B field due to the PM may be expessed B and suppose w due to PM as = 2N P Bpm 0 φ π = Bpm π φ 2π s cos( Pφ / 2) = sm 2N P s cos( φ ) s 20
PM Tems It can be shown that λ as due to PM = λ m sinθ whee λ = m 8LBpmNs P 21
PM Tems 22
PM Tems 23
PM Tems 24
PM Tems 25
PM Tems Comment: The sinusoidal tuns distibution gives ise to a sinusoidal flux linkage vesus electical oto position chaacteistic 26
Expession fo Toque It can be shown that T e P 2 [ i cos( θ ) + i cos( θ 2π / 3) + i cos( θ 2π / 3) ] = m as bs cs λ + 27
Expession fo Toque 28
Expession fo Toque 29
Expession fo Toque 30
Expession fo Toque 31
Expession fo Toque 32
Machine Equations in Roto Refeence Fame Conside the tansfomation Why? qd 0s f = K s f abcs Whee T ( fqd s) = [ fqs fds f0 K s = 0 s cosθ 2 sinθ 3 1 2 2 cos( θ π ) 3 2 sin( θ π ) 3 1 2 ] 2 cos( θ + π ) 3 2 sin( θ + π ) 3 1 2 33
Machine Equations in Roto Refeence Fame 34
Machine Equations in Roto Refeence Fame Voltage v v qs ds = si = i Flux Linkage λ qs Toque s = L qs ds ss i ds +ω λ + qs ω λ + qs λ = L i + λ T e ds ss ds m 3 P = λ 2 2 i m qs pλ qs pλ ds 35
36 Aside: Some Shothand 3) / 2 sin( 3) / 2 sin( ) sin( 3) / 2 cos( 3) / 2 cos( ) cos( π θ π θ θ π θ π θ θ + = = = + = = = + + s s s c c c
37 Aside: Some Tig IDs ) sin( 2 3 3) / 2 3) cos( / 2 sin( 3) / 2 / 3)cos( 2 sin( ) )cos( sin( ) cos( 2 3 3) / 2 3) sin( / 2 sin( / 3) 2 3) sin( / 2 sin( ) )sin( sin( ) cos( 2 3 3) / 2 3) cos( / 2 cos( 3) / 2 3) cos( / 2 cos( ) )cos( cos( 0 3) / 2 sin( 3) / 2 sin( ) sin( 0 3) / 2 cos( / 3) 2 cos( ) cos( y x y x y x y x y x y x y x y x y x y x y x y x x x x x x x = + + + + = + + + + = + + + + = + + + = + + + π π π π π π π π π π π π π π π π
Tansfomation of Voltage Equations 38
Tansfomation of Voltage Equations 39
Tansfomation of Voltage Equations 40
Tansfomation of Voltage Equations 41
Tansfomation of Voltage Equations 42
This yields v Whee Tansfomation of Voltage In expanded fom Equations qd 0s = si qd 0s + ωλ dqs + pλ qd 0s ( λ v v v qs ds dqs = = ) T s s ds = [ λ λ i i qs ds + ωλ qs 0s = si0 s + pλ 0s ωλ ds qs + + pλ pλ 0] qs ds 43
Tansfomation of Flux-Linkage Equations 44
Tansfomation of Flux-Linkage Equations 45
Tansfomation of Flux-Linkage Equations 46
Tansfomation of Flux-Linkage Equations 47
Tansfomation of Flux-Linkage Equations 48
Tansfomation of Flux-Linkage Equations 49
Tansfomation of Flux-Linkage Equations This yields 3 L + L 0 0 ls ms 2 iqs 0 3 qd 0 s = 0 Lls + Lms 0 ids + λ m 1 2 i0s 0 0 0 L ls λ O in expanded fom λ qs = L Whee ss i qs 3 Lss = Lls + 2 L ms λds = ss ds + λ m λ 0 s = Llsi 0 s L i 50
Tansfomation of Toque Equation Stat with T e P 2 [ i cos( θ ) + i cos( θ 2π / 3) + i cos( θ 2π / 3) ] = m as bs cs λ + 51
Tansfomation of Toque Equation Finally, we aive at 3 P Te = λ 2 2 m i qs 52
Zeo Sequence 53
Zeo Sequence 54
Relationship of RMS Value and Phase to QD Components 55
Relationship of RMS Value and Phase to QD Components 56
Relationship of RMS Value and Phase to QD Components 57
Relationship of RMS Value and Phase to QD Components 58
Voltage Souce Opeation In this mode, idealized voltage applied is vas = 2v s cosθesv 2 v bs = 2v s cos( θesv π ) 3 2 v cs = 2v s cos( θ esv + π ) 3 Whee θ = θ + φ esv v 59
Applied Voltage in QD Vaiables We can show that vqs = 2vs vds = 2vs cos sin φ φ v v 60
Applied Voltage in ABC Vaiables 61
Applied Voltage in QD Vaiables 62
Analysis of Steady State Opeation Pediction of Q- and D-Axis Cuents 63
Analysis of Steady State Opeation 64
Analysis of Steady State Opeation 65
Analysis of Steady State Opeation 66
Example 1 Conside a machine with the following paametes s = 3.1 Ω P = 4 L ss = 12.1 mh λ m = 0.156 Vs N =3 Futhe suppose V s = 100 φ v = 0 ω m = 1800 RPM Find the toque and efficiency 67
Example 1 68
Example 2 Conside the machine with paametes of example 1. Plot the toque speed and ms cuent speed cuves 69
Example 2 21.35 25 T e ( 0, ω i ) T e T e π, 4 ω i π, 2 ω i ( ( ), ω i ) T e φ vmt ω i 20 15 10 5 0 0.418 5 0 500 1000 1500 2000 0 ω i 2 10 3 70
Example 2 40 32.258 i s ( 0, ω i ) i s i s π, 4 ω i π, 2 ω i ( ( ), ω i ) i s φ vmt ω i 30 20 10 0.025 0 0 500 1000 1500 2000 0 ω i 2 10 3 71
Optimization of Phase Advance 72
Optimization of Phase Advance 73
Cuent Souce Opeation Intepetation 1 (ABC Vaiable) 74
Cuent Souce Opeation Intepetation 2 (Toque Tansduce) 75
Desied D-Axis Cuent 76
Example 3 Conside a machine with the following paametes s = 3.1 Ω P = 4 L ss = 12.1 mh λ m = 0.156 Vs N =3 Plot the voltage equied and efficiency fo the following conditions Toque command: 2 Nm, d-axis cuent 0 A Toque command: 6 Nm, d-axis cuent 0 A Toque command: 6 Nm, d-axis cuent -6 A 77
Example 3 78
Example 3 150 ( ) (, ( ), 0.0 ) (, ( ), 6 ) v s ω i, i q ( 2), 0.0 v s ω i i q 6 v s ω i i q 6 100 50 0 0 200 400 600 800 1 10 3 ω i 79
Example 3 1 0.8 ( ) (,, 0) ( ) η ω i, 2, 0 η ω i 6 0.6 η ω i, 6, 6 0.4 0.2 0 0 200 400 600 800 1 10 3 ω i 80
Effect of D-Axis Cuent on Voltage 81
D-Axis Injection 82
D-Axis Injection 83
D-Axis Injection 84
D-Axis Injection 85
D-Axis Injection 86
Example 4 At 2000 pm, the zeo-to-peak line-to-line voltage has a 100 V amplitude and a fequency of 100 Hz. Compute λ m and P. At standstill and at 60 Hz, the impedance looking into the a- to b-phase is 0.2+2j. Find s and L ss. 87
Example 4 Pat 1 88
Example 4 Pat 1 89
Example 4 Pat 1 90
Example 4 Pat 1 91
Example 4 Pat 2 92
Example 4 Pat 2 93
Example 4 Pat 2 94