Laboratory of Power Electronics of Lille, France
IEEE-IAS’2004 Seattle
Right Harmonic Spectrum for the Back-Electromotive Force of a n-phase PM Synchronous Machine …in order to use efficient algorithms developed for sinusoidal 3-phase machines 1
E. Semail, X. Kestelyn and A. Bouscayrol
Outline
IEEE-IAS’2004 Seattle
Introduction 1>Multi-machine Modeling of n-phase machine (equivalence to a set of fictitious 2-phase machines) 2>Deduced ideal harmonic spectrum 3>Experimental control for a 5-phase machine Conclusion 2
Introduction Generalization of 3-phase machine concepts: IEEE-IAS’2004 Seattle
Efficient vector control of wye coupled 3-phase machines with SINUSOIDAL back electromotive force
i*a ib* = * ic
3
cos (θ r ) 2 2π cos θ r − 3 3 cos θ r + 2π 3
− sin (θ r ) 2π − sin θ r − 3 2π − sin θ r + 3
1 2 ids 1 iqs 2 1 0 2
Equivalent to a 2-phase machine Concordia-Park transformation
(from « AC Motor Speed Control », T.A. Lipo, Karel Jezernik)
Introduction Generalization by using Multimachine Modeling. IEEE-IAS’2004 Seattle
n-phase PM machine equivalent to a set of 2-phase machines Each 2-phase machine with a sinusoidal back EMF ! Particular trapezoidal back EMF required 3-phase machine control used For the machine designer: trapezoidal EMF easier to achieve than sinusoidal EMF (Windings and spatial repartition of Permanent Magnet ) For the drive designer: 4
to achieve several times the same control of 3-phase machines
Introduction • Support: a 5-phase PM motor ν IEEE-IAS’2004 Seattle
i1
1
Back-electromotive force 50 40 30 20 10 0 -10
Vbus
-20 -30 -40 -50
8cm
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
D-Space1103 board
5
0.04
1>Multi-machine modeling of the machine
IEEE-IAS’2004 Seattle
Multi-Machine modeling of n-phase machines
im is 6
vs
Tm
im
vm em
Tem =Tm +Ts
MM Ts
is
es
SM
Ω
Ω
1>Multi-machine modeling of the machine
IEEE-IAS’2004 Seattle
Assumptions for the machine: • regular manufacturing ; • no reluctance effect ; • no damper winding effects ; • no saturation effect. 7
1>Multi-machine modeling of the machine Tools to get the results ? IEEE-IAS’2004 Seattle
* Vectorial formalism ; * Energetic Macroscopic Representation EMR; For a wye coupled 3-phase machine3-phase machine For a perfect electromechanical wye coupled
Actions on the basic electromechanical converter
im
Tm Puts MM No resistance and no inductance…only energy conversion Emphasis on em Ω energetic laws 8
im . e m = Tm Ω
Mechanical load
Reactions of the electromechanical converter
1>Multi-machine modeling of the machine
IEEE-IAS’2004 Seattle
Tools to get the results ? * Energetic Macroscopic Representation For a wye coupled 3-phase PM machine
Electrical source
im
vm im Voltage Source Inverter
em
Tm MM Ω
2 1 d 2 Lm im 2 + Rm im = vm .im − em .im dt
9
Mechanical load im . em = Tm Ω
Power = Dot product of vectors (action and reaction)
1>Multi-machine modeling of the machine EMR for a 5-phase motor:
IEEE-IAS’2004 Seattle
Equivalence with two 2-phase fictitious drives FOR a 5-phase motor
Two fictitious machines
vm im is
vs
Tm MM em Ts is SM Ω es im
Fictitious machines 10
Ω
Mechanical coupling is . es = Ts Ω
di Ls d iss + Rs is = vs − es Ls dt + Rs is = vs − es i . e = T Ω s s s
dt
Tem=Tm+Ts
Load
IEEE-IAS’2004 Seattle
1>Multi-machine modeling of the machine
For a 5-phase machine: Origin of equivalence with two 2-phase fictitious machines
11
1>Multi-machine modeling of the machine Equivalent set of two 2-phase fictitious drives Deduced from properties of stator inductance matrix: IEEE-IAS’2004 Seattle
SYMMETRY and CIRCULARITY Linear operator L M1 M 2 M 2 M1 L M1 M 2 Ls = φ =Λ i M 2 M1 L M 1 M M 2 M1 L 2 Flux Stator current • ONLY three eigenvalues Lm, Ls and L0 of Λ
( )
[ ]
M1 M2 M2 M1
• three orthogonal eigenspaces Gm, Gs and G0 of Λ
12
For every vector:
y = y m + y s + y0
1>Multi-machine modeling of the machine Equivalent set of two 2-phase fictitious drives Three independent flows of energy? IEEE-IAS’2004 Seattle
φ = φm + φs + φ0 = Lm jm + Ls js + L0 j0
r ddφi m + e v =imR+i L+ vm = R + em m dt Voltage vector
eigenvalues (II) Back electromotive vector
Current vector
Electrical stator power:
(I)
ps =
k =5
∑v
k ik
= v .i
k =1
Orthogonality 13
p s = v . i = vm .im + v s .is + v0 .i0
(III)
1>Multi-machine modeling of the machine Equivalent set of two 2-phase fictitious drives Three independent flows of energy IEEE-IAS’2004 Seattle
From (III)
i0 = 0
Wye coupling
2 1 d Lm im 2 2 vm .im = + R im + em . im dt 2
14
1 d Ls is 2 2 v s .i s = + R i s + es . i s dt
Tm Ω = em . im Tem=Tm+Ts
Ts Ω = es . is
IEEE-IAS’2004 Seattle
2>Deduced ideal harmonic spectrum
Deduced ideal harmonic spectrum of back electromotive force
15
2>Deduced ideal harmonic spectrum
IEEE-IAS’2004 Seattle
Harmonic analysis of fictitious machines Analysis of projections of
For a 5-phase
spatial periodic vector onto Gmand Gs
motor
• the Main Machine (MM) is associated with odd harmonics of order: 1, 9, 11, 19,…, 5 h ± 1 • the Secondary Machine (SM) is associated with odd harmonics of order: 3, 7, 13, 17,… 5 h ± 3 Other fictitious machines if n=7, 9, 11,... 16
(same families by Klingshirn: time harmonic analysis)
2>Deduced ideal harmonic spectrum Harmonic analysis of fictitious machines
IEEE-IAS’2004 Seattle
Fictitious machine First 2-phase machine
Sinusoidal EMF
Eigenspace G2
Families of odd harmonics 1, 5, 7,…, 3h ± 1
3-phase machineImply Torque Ripples
Generalization: only one harmonic per fictitious machine If each 2-phase fictitious machine has a sinusoidal back electromotive force … 17
no torque ripples, usual 3-phase algorithms
2>Deduced ideal harmonic spectrum Harmonic analysis of fictitious machines
IEEE-IAS’2004 Seattle
Fictitious machine Eigenspace Families of odd harmonics First 2-phase machine G2 1, 9, 11,…, 5 h ± 1 Second 2-phase machine G4 3, 7, 13…, 5h ± 2
Possible trapezoidal EMF
5-phase machine
Fictitious machine Eigenspace Families of odd harmonics First 2-phase machine G2 1, 13, 15,…, 7h ± 1 Second 2-phase machine G4 5, 9, 19…, 7h ± 2 Third 2-phase machine G6 3, 11, 17…, 7h ± 3
18
Possible trapezoidal EMF
7-phase machine
IEEE-IAS’2004 Seattle
Deduced Control of experimental set-up for a 5-phase PM motor Multimachine Control Structure
8cm
19
Deduced Control of experimental set-up Multimachine Control Structure Electrical coupling
IEEE-IAS’2004 Seattle
DC bus
vem
vbus
iem ibus
Mechanical Fictitious machines coupling
Fictitious VSI
ies m
ves
vm im is
vs
Tm MM em Ts is SM Ω es im
Load
Tem =Tm +Ts Ω
Ts_ref Tem_ref T 20
m_ref
??
Blue control blocks obtained by systematic inversion
Deduced Control of experimental set-up Multimachine Control Structure Electrical coupling
IEEE-IAS’2004 Seattle
DC bus
vem
vbus
iem ibus
Mechanical Fictitious machines coupling
Fictitious VSI
ies m
ves
vm im is
vs
Tm MM em Ts is SM Ω es im
Load
Tem =Tm +Ts Ω
T s_ref i s_ref im_ref 21
Tem_ref Tm_ref
??
Blue control blocks obtained by systematic inversion
Deduced Control of experimental set-up Multimachine Control Structure DC bus
IEEE-IAS’2004 Seattle
m
Electrical coupling
vem
vbus
iem ibus
Mechanical Fictitious machines coupling
Fictitious VSI
ies m
ves
vm im is
vs
Tm MM em Ts is SM Ω es im
Load
Tem =Tm +Ts Ω
Ts_ref vs_ref vm_ref 22
is_ref im_ref
Tem_ref Tm_ref
??
Blue control blocks obtained by systematic inversion
Deduced Control of experimental set-up Multimachine Control Structure Electrical coupling
IEEE-IAS’2004 Seattle
DC bus
vem
vbus
iem ibus
Mechanical Fictitious machines coupling
Fictitious VSI
ies m
ves
Tm MM em Ts is SM Ω es im
vm im
vs
is
Load
Tem =Tm +Ts Ω
Ts_ref vs_ref
is_ref
vm_ref im_ref 23
Tem_ref Tm_ref
??
Blue control blocks obtained by systematic inversion
Deduced Control of experimental set-up Multimachine Control Structure Electrical coupling
IEEE-IAS’2004 Seattle
DC bus
vem
vbus
iem ibus
SM branch 3rd harmonic MM branch 1st harmonic
24
Fictitious machines
Fictitious VSI
ies
ves
Tm MM em Ts is SM Ω es im
vm im
Mechanical coupling
vs
is
Load
Tem=Tm+Ts Ω
Ts_ref vs_ref
is_ref
vm_ref im_ref
Tem_ref Tm_ref
??
Vector control for 2-phase machine with sine-wave EMF
Deduced Control of experimental set-up Multimachine Control Structure 0
IEEE-IAS’2004 Seattle
Tem-ref
Ωréf
Ω
vsα-ref
PIs
sd-ref
R(3θr)v + -
sq-ref
vpβ-ref
+ -
0
− sin pθ r i cos pθ r
PIm
θ i
cos pθ r R( θ ) = sin pθ r
R(θr) +
imq-ref
v1ref
vpα-ref
PIm
-
imd-ref
Repartition of reference torques
IP
+ -
+
PIs
sβ-ref
Trans forma tion
v 2ref
v3ref
PWM VSI
v 4ref
C v5ref
2 C= 5 r r
1
0 4π sin 5 6π sin 5 12π sin 5 16π sin 5
1 0 1 2 (θr4)π cos(θ1 r ) cos 2π sin−2πsincos 5 5 5 2 1 4π 4π 6π 2 π 2 π cosθ −2 cos 5 −sinsin5θ cos 5 − 1 3cos6π sin 6π rcos 123π 5 5 5 2 2π π 1 2π 8π 8π 16 cosθ + cos −sin cos θ sin + r 5 5 3 2 3 5
1 2 1 2 1 2
Concordia ’s transformation 25
machine θr Resolver
R -1(θr) θr
R -1 (3θr)
Trans forma tion
If 3-phase machine
C-1
i1, i2, i3, i4
Deduced Control of experimental set-up Using the two fictitious machines 4 1
isq
IEEE-IAS’2004 Seattle
3
isq-ref
0 -1
2
-2
SM
SM
1
imd-ref
imd
-3
MM MM
-4
-5
0
imq-ref
imq
-6
-1
-2 0
isd-ref
isd 0.005
0.01
0.015
0.02
0.025
0.03
0.035
-7
0.04
0.045
0.05
-8 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Currents in the two fictitious 2-phase machines ‘MM’ and ‘SM’. 26
Ts-ref =10% Tm-ref
0.05
Deduced Control of experimental set-up Using the two fictitious machines 25
Ωmes
IEEE-IAS’2004 Seattle
20
15
Ωref
imq
10
5
0
-5
isq -10
-15
-20
-25 0
27
1
2
3
4
5
6
7
8
9
Experimental active currents in MM and SM
10
Deduced Control of experimental set-up Using the two fictitious machines 56
4.5
IEEE-IAS’2004 Seattle
Tem-ref
4 4
Fundamental of i1: i1-1 i1
3.5 2
=
Ts-ref
3
100%
2.50
Tm-ref
-2
+ Ts-ref
=
2
3rd harmonic of i1: i1-3
1.5
29%
0.5 -6 00 0
10% Tm-ref
SM
1 -4
0.005 50
0.01 100
0.015 150
0.02 200
7th harmonic 0.025 250
0.03 300
0.035 350
0.04 400
0.045 450
0.05 500
Experimental i1 current in phase n°1 at 1250 rpm 28
repartition between MM and SM: isq-ref = (29%) imq-ref.
Conclusion For n-phase (n odd number) PM machines, IEEE-IAS’2004 Seattle
generalization of 3-phase sinusoidal machine high performance controls
if back EMF of 5-phase machines has only 1st, 3rd. if back EMF of 7-phase machines has only 1st, 3rd, 5th. then Independent implementations of 3-phase machine algorithms. (1 time for 3-phase; 2 times for 5-phase ; 3 times for 7-phase) 29
