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© Copyright Ned Mohan 2001 1
Ned Mohan Oscar A. Schott Professor of Power Electronics and Systems Department of Electrical and Computer Engineering University of Minnesota Minneapolis, MN 55455 USA
Advanced Course onElectric Drives
© Copyright Ned Mohan 2001 2
Chapter 1
Introduction to Advanced Electric Drives
3
Graduate Course in Electric Drives
Objectives:Basics to Advanced Topics in 2 SemestersSeamless Continuation of the First CourseTopics: Dynamic Modeling and ControlApproach/Tools
dq-Windings based AnalysisDesign Examples Using SimulinkVerification in the Hardware Lab using dSPACE
© Copyright Ned Mohan 2001 4
Topics (Lectures)1. Introduction to Advanced Electric Drive Systems (2)2. Induction Machine Equations in Phase Quantities: Assisted by Space
Vectors (5)3. Dynamic Analysis of Ind. Mach. in terms of dq-Windings (7)4. Vector Control of IM Drives: A Qual. Examination (4)5. Mathematical Description of Vector Control (5)6. Detuning Effects in Induction Motor Vector Control (3)7. Space Vector PWM (SV-PWM) Inverters (3)8. Direct Torque Control (DTC) and Encoder-Less Operation of
Induction Motor Drives (5)9. Vector Control of Perm-Magnet Syn. Motor Drives (3)10. Switched-Reluctance Motor (SRM) Drives (3)
© Copyright Ned Mohan 2001 5
Continuation of Topics Discussed in the First Course
Switch-Mode Converters Average RepresentationMagnetics – TransformersFeedback Controller DesignSpace Vector Representation for AC MachinesBasic Calculations for Electromagnetic TorquePMAC Drives – Space Vector Based Steady State OperationInduction Motor Drives – Space Vector based Steady State Analysis
© Copyright Ned Mohan 2001 6
Design ExamplesInduction Machine Initially Operating in Steady state
Load Torque Disturbance at t=0.1 sControl Objective is to keep Speed Constant (design speed loop with a bandwidth of 25 rad/s and a phase margin of 60 degrees)
Permanent Magnet AC (PMAC) DrivesSwitched-Reluctance Motor (SRM) Drives
© Copyright Ned Mohan 2001 7
“TEST” INDUCTION MOTORNameplate Data:Power: 3 HP/2.4 kWVoltage: 460 V (L-L, rms)Frequency: 60 HzPhases: 3Full Load Current: 4 AFull-Load Speed: 1750 RPMFull-Load Efficiency: 88.5 %Power Factor: 80.0 %Number of Poles: 4
Per-Phase Motor Circuit Parameters:
2
1.77 , 1.34 , 5.25 (at 60 Hz)4.57 (at 60 Hz), 139.0 (at 60 Hz)
Full-Load Slip = 1.72 %, 0.025
s r s
r m
eq
R R XX X
J kg m
= Ω = Ω = Ω= Ω = Ω
= ⋅© Copyright Ned Mohan 2001 8
Chapter 2
Induction Machine Equations in Phase Quantities:
Assisted by Space Vectors
9
c axis
a axis
b axis
i c
i b
i a
θ
magneticaxis of phase a
ia
θ=0
′4′3
′2
′1
7
65 4 3
2
1
′7
′6
′5
θ π=
(a) (b)
Figure 2-1 Stator windings.
Sinusoidally-Distributed Stator Windings
(1) ( ) sin2s
sNn θ θ= 0 θ π≤ ≤
(2) ( ) i cossa a
g
NHp
θ θ= ( )(4) ( ) ( ) i coso sa o a a
g
NB Hp
µθ µ θ θ= =
( ) ( )(3) i cossa g a a
NF Hp
θ θ θ= =
© Copyright Ned Mohan 2001 10
Figure 2-2 Three-phase windings.
a'a
b
'b
c
'c
ai
bi
ci
axisa −
axisb−
axisc −/j4 3e π
/j2 3e π
j0e
θb
bi
ai
cic
a
(a) (b)
Three-Phase Sinusoidally-Distributed Windings
© Copyright Ned Mohan 2001 11
n
b
c
a
ia
a ax is−
ia
ia
r
leak age
m agn etiz in g
(a)
(b)Figure 2-3 Single-phase magnetizing inductance and leakage
inductance.
Single-Phase Magnetizing Inductance ,1m phaseL −
,,
,1
,(1) a leakagea as self
a a ai onlyaL Ls m phase
magnetizingLi i i
λλ λ
−
= = +
, ,1(2) s self s m phaseL L L −= +
2,1(3) ( )o sm phase
g
r NLp
πµ− =
© Copyright Ned Mohan 2001 12
n
b
c
a
ia
λ b
Figure 2-4 Mutual inductance .
Stator Mutual Inductance mutualL
0,,
(1) bmutual
a i i rotor openb c
Liλ
==
,11(3)2mutual m phaseL L −= −
, ,(2) cos(120 )ob duetoi a magnetizing dueto ia aλ λ=
© Copyright Ned Mohan 2001 13
a axis−
b axis−
c axis−
θm
ia
ib
ic
iA
iC
iB
A axis−
B axis−
C axis−
ωm
( )a
ia
ib
ic
−− +
+
+
va
vb
vc
Rs
−−
−
+
+
vA = 0
vB = 0
vC = 0
RriA
iB
iC
Stator circuit Rotor circuit
( )b
− +
Figure 2-5 Rotor circuit represented by three-phase windings.
(b)
Equivalent Windings in A Squirrel-Cage Rotor
,13(2)2m m phaseL L −=
23(3) ( )2
o sm
g
r NLp
πµ=
(4) s s mL L L= +
(1) ( ) ( ) ( ) 0A B Ci t i t i t+ + =
(1) ( ) ( ) ( ) 0a b ci t i t i t+ + =
,13(2)2m m phaseL L −=
(3) r r mL L L= +
,1(4) cosaA m phase mL L θ−= ⋅
Stator
Rotor
© Copyright Ned Mohan 2001 14
( )sF t( )
sF tθ
( )aF t( )bF t
( )cF t
axisc −
axisb −
axisa −
ci = −
bi =−
ai = +
( )aF t
( )bF t
( )cF t
axisc −
axisb −
axisa −
(a) (b)
Figure 2-6 Space vector representation of mmf.
Review of Space Vectors
( ) ( ) ( ) ( )a a a as a b cF t F t F t F t= + +
© Copyright Ned Mohan 2001 15
Figure 2-7 Physical interpretation of stator current space vector.
a'a
b'b
c
'cai
bi
ci
axisa −
axisb−
axisc −
/j4 3e π
/j2 3e π
j0e
sI
at time tmagnetic axis of hypothetical winding
ˆwith current sI
axisa −
(a) (b)
si
Physical Interpretation of Current Space Vector
( )0 2 / 3 4 / 3 ˆ(1) ( ) ( ) ( ) ( ) ( )j tia j j j ss a b c si t i t e i t e i t e I t eθπ π= + + =
(2) ( ) ( / ) ( )a as s sF t N p i t=
© Copyright Ned Mohan 2001 16
( )0 2 / 3 4 / 3 ˆ( ) ( ) ( ) ( ) ( )j tva j j j ss a b c sv t v t e v t e v t e V t eθπ π= + + =
( )0 2 / 3 4 / 3 ˆ( ) ( ) ( ) ( ) ( )j ta j j j ss a b c st t e t e t e t eθλπ πλ λ λ λ λ= + + =
Voltage and Flux-Linkage Space Vectors
© Copyright Ned Mohan 2001 17
Figure 2-8 Relationship between space vector and phasor in sinusoidal steady state.
ˆa js si I e α−=
@ t 0=
(b)
axisa −αα
ref
ˆa aI I α= ∠ −
(a)
Relationship between space vector and phasor in sinusoidal steady state
0
32
as ati I
== 3ˆ ˆ
2s aI I=
© Copyright Ned Mohan 2001 18
i a
ib
ic+−vc
+−vb
+− va
λ a t( )
λ b t( )
λ c t( )
a axis−
c axis−
b axis−
e j0
e j4 3π/
e j2 3π/ i s s,i
s s,i
B
Fs
s
,, λ
at time t' '
Figure 2-9 All stator space vectors are collinear (Rotor open-circuited).
All stator space vectors are collinear (Rotor open-circuited)
,due to leakage flux due to magnetizing flux
( ) ( ) ( ) ( )a a a as s m s s ss is
t L i t L i t L i tλ = + =
© Copyright Ned Mohan 2001 19
a axis−
b axis−
c axis−
θm
iA
iC
iB
A axis−
B axis−
C axis−
ωm
,
, ,
, Br r irFr r ir
i
λat time t' '
Figure 2-10 All rotor space vectors are collinear (Stator open-circuited).
a axis−
b axis−
c axis−
θm
iA
iC
iB
A axis−
B axis−
C axis−
ωm
,
, ,
, Br r irFr r ir
i
λat time t' '
Figure 2-10 All rotor space vectors are collinear (Stator open-circuited).
All rotor space vectors are collinear (Stator open-circuited)
,due to leakage flux due to magnetizing flux
( ) ( ) ( ) ( )A A A Ar r m r r rr ir
t L i t L i t L i tλ = + =
© Copyright Ned Mohan 2001 20
Making the case for dq-axis analysis
(3) ( ) ( ) ( ) ja a A ms s s m rt L i t L i t e θλ = +
(2) ( ) ( ) ja A mr ri t i t e θ=
a axis−
b axis−
c axis−
θm
ia
ib
ic
iA
iC
iB
A axis−
B axis−
C axis−
ωm
( )a
ia
ib
ic
−− +
+
+
va
vb
vc
Rs
−−
−
+
+
vA = 0
vB = 0
vC = 0
RriA
iB
iC
Stator circuit Rotor circuit
( )b
− +
Figure 2-5 Rotor circuit represented by three-phase windings.
(b)
(1) ( ) ( )a a as s s m rt L i t L iλ = +
(4) ( ) ( ) ( )a a as s s s
dv t R i t tdt
λ= +
© Copyright Ned Mohan 2001 21
Chapter 3
Dynamic Analysis of Induction Machines in Terms of dq-Windings
© Copyright Ned Mohan 2001 22
Representation of Stator MMF by Equivalent dq Windings
2 / 3 4 / 3(2) ( ) ( ) ( ) ( )a j js a b ci t i t i t e i t eπ π= + +
(3) ( ) ( )a ass s
NF t i tp
=
( )3/ 2(4) ds ssd sq s
N Ni ji ip p
+ =
( ) 2(5)3d
sd sq si ji i+ =
2,1
,1
(1) winding magnetizing inductance =( 3 / 2)
(3 / 2)
(using Eq. 2-12)
m phase
m phase
m
dq L
L
L
−
−=
=
axisa −
axisb −
axisc −
si
ai
bi
ci
at t
axisa −
axisb −
axisc −
si
at t
axisd −
axisq −
sdi
sqi
axisa −
axisb −
axisc −
si
at t
axisd −
axisq −
projection
projectionsq2i3
= ×projection
projectionsd2i3
= ×
(a)
(b) (c)
3
2 sN
Figure 3-1 Representation of stator mmf by equivalent dq winding currents.
daθ daθ
© Copyright Ned Mohan 2001 23
Representation of Rotor MMF by Equivalent dq Windings
2 / 3 4 / 3( ) ( ) ( ) ( )A j jr A B Ci t i t i t e i t eπ π= + +
( )( )/
AA rr
s
F ti tN p
=
Figure 3-2 Representation of rotor mmf by equivalent dq winding currents.
Ai
dAθrqi
a axis−
dωd axis−
q axis−
B axis− A axis−mω
daθ mθ
ri
••
rdi
projection2 projection3rqi = ×
projection2 projection3rdi = ×
© Copyright Ned Mohan 2001 24
a axis−stator
A axis−rotor
d axis−
ωm
isd
isq
q axis−at t
i rq
ird
θm
is
ir
32
isd
32
isq
32
irq 32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
a axis−a axis−stator
A axis−A axis−rotor
d axis−d axis−
ωmωm
isdisd
isqisq
q axis−q axis−at tat t
i rqi rq
irdird
θmθm
isis
irir
32
isd32
isd
32
isq32
isq
32
irq32
irq 32
ird32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
Figure 3-3 Stator and rotor representation by equivalent dq winding currents. The dq winding voltages are defined as positive at the dotted terminals. Note that the relative positions of the stator and the rotor current space vectors are not actual, rather only for definition purposes.
Mutual Inductance between dq Windings on the Stator and the Rotor
sd s sd m rdL i L iλ = +
sq s sq m rqL i L iλ = +
rd r rd m sdL i L iλ = +
rq r rq m sqL i L iλ = +
© Copyright Ned Mohan 2001 25
Figure 3-4 Transformation of phase quantities into dq winding quantities.
[ ]abc dqsT →
aibi
ci
sdi
sqi
(a) stator
[ ]ABC dqrT →
Ai
Bi
Ci
rdi
rqi
(b) rotordaθ dAθ
Figure 3-4 Transformation of phase quantities into dq winding quantities.
[ ]abc dqsT →
aibi
ci
sdi
sqi
(a) stator
[ ]ABC dqrT →
Ai
Bi
Ci
rdi
rqi
(b) rotordaθ dAθ
[ ]abc dqsT →
aibi
ci
sdi
sqi
(a) stator
[ ]ABC dqrT →
Ai
Bi
Ci
rdi
rqi
(b) rotordaθ dAθ
[ ]
2 4 ( )cos( ) cos( ) cos( )( ) 2 3 3 ( )( ) 2 43 sin( ) sin( ) sin( ) ( )
3 3
ada da dasdb
sqda da da c
Ts abc dq
i ti t
i ti t
i t
π πθ θ θ
π πθ θ θ
→
− − = − − − − − [ ]
2 4 ( )cos( ) cos( ) cos( )( ) 2 3 3 ( )( ) 2 43 sin( ) sin( ) sin( ) ( )
3 3
AdA dA dArdB
rqdA dA dA C
Tr ABC dq
i ti t
i ti t
i t
π πθ θ θ
π πθ θ θ
→
− − = − − − − −
Mathematical Relationship between dq and phase Winding Variables
© Copyright Ned Mohan 2001 26
-axisa
-axisd
-axisα
-axisβ
daθ
i
ii
i
sλ
sdisqi
si α
si β- axisq
Figure 3-5 Stator and equivalent windings.dqαβ
dω
(1) s s s sdv R idtα α αλ= +
(2) s s s sdv R idtβ β βλ= +
_ _ _(4) s s s sdv R idt
α α ααβ αβ αβλ= +
_ _(6) ; etc.j das s dqv v e θααβ = ⋅
_(5) ; etc.s dq sd sqv v jv= +
Derivation of Stator Voltages in dq Windings
_(3) ; etc.s s sv v jvααβ α β= +
© Copyright Ned Mohan 2001 27
_ _ _ _(2) s dq s s dq s dq d s dqdv R i jdt
λ ω λ= + +
(3) sd s sd sd d sqdv R idt
λ ω λ= + −
(4) sq s sq sq d sddv R idt
λ ω λ= + +
[M ]rotate
0 1(5)
1 0sd sd sd sd
s dsq sq sq sq
v i dRv i dt
λ λωλ λ
− = + +
Derivation of Stator Voltages in dq Windings (continued)
_ _ _(1) s s s sdv R idt
α α ααβ αβ αβλ= +
© Copyright Ned Mohan 2001 28
-axisa
-axisd
-axisα
-axisβ
daθ
ii i
i
rλ
rdirqi
ri α
ri β
- axisq
Figure 3-6 Rotor and equivalent windings.dqαβ
dω
-axisA
dAθ
mθ
mω
Derivation of Rotor Voltages in dq Windings
(1) rd r rd rd dA rqdv R idt
λ ω λ= + −
(2) rq r rq rq dA rddv R idt
λ ω λ= + +
(3) dA d mω ω ω= −
(4) ( / 2)m mechpω ω=
[M ]rotate
0 1(5)
1 0rd rd rd rd
r dArq rq rq rq
v i dRv i dtλ λ
ωλ λ −
= + +
© Copyright Ned Mohan 2001 29
[M ]rotate
0 11 0
sd sd sd sds d
sq sq sq sq
v id Rv idt
λ λωλ λ
− = − −
[M ]rotate
0 11 0
rd rd rd rdr dA
rq rq rq rq
v id Rv idt
λ λωλ λ
− = − −
Obtaining Flux Linkages: Voltages as Inputs
[ ]_ _ _ rotate _[ ] [ ] [ ] M [ ]s dq s dq s s dq d s dqd v R idt
λ ω λ= − −
[ ]_ _ _ rotate _[ ] [ ] [ ] M [ ]r dq r dq r r dq dA r dqd v R idt
λ ω λ= − −
_s dqv
_r dqv
_s dqλ
_r dqλ
rotate[M ]
rotate[M ]
1s
1s
sR
rR
dAω
dω×
×
∑
∑
+
−
−
+
−
−
_s dqddt
λ
_r dqddt
λ
1[ ]M −
Eq. 3-62
dqi
_s dqi
_r dqi
Figure 3-7 Calculating winding flux linkages and currents.dq
© Copyright Ned Mohan 2001 30
a axis−
d axis−q axis−
rqi
rdi
sqi
d
d
add
subtract
due to and sq rqi i
due to rqileakage flux
i
i i
Figure 3-8 Torque on the rotor -axis.d
a axis−
d axis−q axis−
rqi
rdi
sqi
d
d
add
subtract
due to and sq rqi i
due to rqileakage flux
i
i i
a axis−
d axis−q axis−
rqi
rdi
sqi
d
d
add
subtract
due to and sq rqi i
due to rqileakage flux
i
i i
Figure 3-8 Torque on the rotor -axis.d
Electromagnetic Torque on the Rotor d-Axis
0 3/ 2ˆ(1) ( )s rrq sq rq
g m
mmf
N LB i ip L
µ = +
,3 / 2 ˆ(2)
2s
d rotor rq rdNpT r B ip
π
=
20
,3 / 2(3) ( )
2s r
d rotor sq rq rdg m
Np LT r i i ip L
µπ
= +
20
,3(4) ( )
2 2s r
d rotor sq rq rdg m
Lm
Np LT r i i ip L
µπ = +
,(5) ( )2 2d rotor m sq r rq rd rq rd
rq
p pT L i L i i i
λ
λ= + =
© Copyright Ned Mohan 2001 31
rqi
rdi
q
add
subtract
due to and sd rdi i
due to rdileakage flux
q axis−d axis−
a axis−
sdi
q
i i
i
Figure 3-9 Torque on the rotor -axis.q
rqi
rdi
q
add
subtract
due to and sd rdi i
due to rdileakage flux
q axis−d axis−
a axis−
sdi
q
i i
i
rqi
rdi
q
add
subtract
due to and sd rdi i
due to rdileakage flux
q axis−d axis−
a axis−
sdi
q
i i
i
Figure 3-9 Torque on the rotor -axis.q
Electromagnetic Torque on the Rotor q-Axis
, ( )2 2q rotor m sq r rq rq rd rq
rd
p pT L i L i i iλ
λ= − + = −
, ,(1) em d rotor q rotorT T T= +
(2) ( )2em rq rd rd rqpT i iλ λ= −
(3) ( )2em m sq rd sd rqpT L i i i i= −
(4) em Lmech
eq
T Tddt J
ω −=
Net Electromagnetic Torque
© Copyright Ned Mohan 2001 32
Equivalent d and q Axis Equivalent Circuits
( )sd s sd d sq s sd m sd rdd dv R i L i L i idt dt
ω λ= − + + +
( )sq s sq d sd s sq m sq rqd dv R i L i L i idt dt
ω λ= + + + +
0
( )rd r rd dA rq r rd m sd rdd dv R i L i L i idt dt
ω λ=
= − + + +
0
( )rq r rq dA rd r rq m sq rqd dv R i L i L i idt dt
ω λ=
= + + + +
sR
+
+
+
+
−−
− −
sR
sL
sL rL
rL
mL
mL
sdv
sqv
d sdω λ
d sqω λ
sdi
sqi
dA rqω λ
dA rdω λ
rdv
rqv
+
−
+
−
sdddt
λ
sqddt
λ
+
−
+
−
rdddt
λ
rqddt
λ
(a) -axisd
(b) -axisq
1 2
1 2
1 2
1 2
1
2
1
2
Figure 3-5 -winding equivalent circuits.dq
rdi
rqi
rR
rR
Figure 3-10 dq-winding equivalent circuits.
© Copyright Ned Mohan 2001 33
Figure 3-11 Per-phase equivalent circuit in steady state.
+
−
(at )ωaV
+
−
sj Lω
maImaE
raI ′aIsR
rj Lω
rRs
( )m mj L jXω =
AI
Per-Phase Equivalent Circuit
_ _ _(1) s dq s s dq syn s dqv R i jω λ= +
_ _(4) 0 rdq syr n r dq
R i js
ω λ= +
_ _ _ _ _(2) ( )s dq s s dq syn s s dq syn m s dq r dqv R i j L i j L i iω ω= + + +
_ _ _ _(5) 0 ( )rr dq syn r r dq syn m s dq r dq
R i j L i j L i is
ω ω= + + +
(3) ( )a s a syn s a syn m a AV R I j L I j L I Iω ω= + + +
(6) 0 ( )ra syn r A syn m a A
R I j L I j L I Is
ω ω= + + +
© Copyright Ned Mohan 2001 34
Obtaining Currents from Flux Linkages
[ ]
sd s m sd
rd m r rdL
L L iL L i
λλ
=
[ ]
sq sqs m
m rrq rqL
iL LL L i
λλ
=
[ ]
0 00 0
0 00 0
sd sds msq sqs m
m rrd rd
m rrq rqM
iL LiL L
L L iL L i
λλλλ
=
1[ ]
sd sd
sq sq
rd rd
rq rq
ii
Mii
λλλλ
−
=
© Copyright Ned Mohan 2001 35
Induction Motor Model
_s dqv
_r dqv
_s dqλ
_r dqλ
rotate[M ]
rotate[M ]
1s
1s
sR
rR
dAω
dω×
×
∑
∑
+
−
−
+
−
−
_s dqddt
λ
_r dqddt
λ
1[ ]M −
Eq. 3-62
dqi
_s dqi
_r dqi
emT
LT
∑
Eq. 3-47
1
eqsJmechω
2p
∑
mω
+
−
+−
Figure 3-12 Induction motor model in terms of windings.dq
© Copyright Ned Mohan 2001 36
Calculation of Steady State Initial Conditions Using Phasors
ˆ
3ˆ ˆ(0)2
j ia a i s a
Is
I I i I e θθ= ∠ ⇒ =
ˆ
2 2 3 ˆ(0) (0) - = cos( )3 3 2sd s i
Is
i projection of i on d axis I θ = ×
ˆ
2 2 3 ˆ(0) (0) - = sin( )3 3 2sq s i
Is
i projection of i on q axis I θ = ×
© Copyright Ned Mohan 2001 37
(1) sd s sd syn sqv R i ω λ= −
(2) sq s sq syn sdv R i ω λ= +
(3) 0 r rd syn rqR i sω λ= −
(4) 0 r rq syn rdR i sω λ= + [ ]
0
0
000 0
s syn s syn m sdsd
syn s s syn m sqsq
syn m r syn r rd
rqsyn m syn r r
A
R L L ivL R L iv
s L R s L iis L s L R
ω ωω ω
ω ωω ω
− − = − −
[ ] 1
00
sd sd
sq sq
rd
rq
i vi v
Aii
−
=
Calculation of Steady State Initial Conditions Using Voltage Equations
© Copyright Ned Mohan 2001 38
Simlink-based dq-Axis Simulation of Induction Motor
© Copyright Ned Mohan 2001 39
Simulation Results
© Copyright Ned Mohan 2001 40
Chapter 4
Vector Control of Induction-Motor Drives: A Qualitative Examination
© Copyright Ned Mohan 2001 41
DC Motor Drive
aS
SN
aNemT
fφ
aφ
PPU
+
− aeav
aL
aR
ai
−
+
ai
0
0
t
em T aT k i=emT
t
Tem = kT ia
© Copyright Ned Mohan 2001 42
Brushless DC Motor Drive
( )rB t
( )si t
o90 NS axisa −
mθ
sI
0
0
tˆ
em T sT k I=emT
t
ˆ em T ST k I=
© Copyright Ned Mohan 2001 43
Vector-Controlled Induction Motor Drive
axisa −'( )ri t
'( )rF t( )rB t
( )rF t
( )msB t( )lrB t
rθ
rθ
( )sv t
speed feedback
currentfeedback
Tacho
IMPPU
( ) perpendicuar to ( ) and ( )r r rB t F t F t′'ˆ ˆ (keeping constant)em T r rT k I B=
© Copyright Ned Mohan 2001 44
Analogy to a Current-Excited Transformer With a Shorted Secondary
, 2m iφ , 1m iφ
1i l2φ
N:N
shown explicitly2R
short
net = 0
2v 0=2R
2i
0 t1i
+ -2 2(0 ) = (0 ) = 0λ λ
1 2
+ + +m,i m,i 2(0 ) = (0 ) + (0 )φ φ φ
+ +m2 1
2
Li (0 ) = i (0 )L
© Copyright Ned Mohan 2001 45
Analogy to a Current-Excited Transformer With a Shorted
Secondary
2 12
(0 ) (0 )mLi iL
+ +=
22 2( ) (0 ) ti t i e τ−+=
22
2
LR
τ =
φmi, 2 φmi, 1
i1 i1
i2
N1 N2
net =0
R2φ2
∴ = +φ φ φmi mi, ,1 2 2
0 t
t = +0
short
R shown licitly2 exp
( )a
i2
−
−+
+Lm
L2
+
−
e2
Nddt
L didt
φ2
2
2=
Nd
dtL di
dtmi
mφ , 2 2=
Noted
dtmi: ,φ
1 0=
t >0
( )b
0
0i2
i1
t
t
LL
im
21 0+d i
( )c
Figure 4-5 Analogy of A Current-Excited Transformer with a Short-Circuited Secondary;
0
1 2N N=
2 2mL L L= +
© Copyright Ned Mohan 2001 46
R1 R2i2
i2
00 tt im
i1i1
L1
L2
Lm
N N N1 2= =
Figure 4-6 Equivalent-Circuit Representation of the Current-Excited Transformer with a Short-Circuited Secondary.
Using the Transformer Equivalent Circuit
2 1 12 2
(0 ) (0 ) (0 )m m
m
L Li i iL L L
+ + += =+
2 21 1
2 2(0 ) (0 ) (0 )m
m
L Li i iL L L
+ + += =+
22 2( ) (0 ) ti t i e τ−+=
© Copyright Ned Mohan 2001 47
d- and q- Axis Winding Representation
sd s
2i = the projection of i (t) vector along the d-axis3
sq s
2i = the projection of i (t) vector along the q-axis3
a axis−stator
A axis−rotor
d axis−
ωm
isd
isq
q axis−at t
i rq
ird
θm
is
ir
32
isd
32
isq
32
irq 32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
a axis−a axis−stator
A axis−A axis−rotor
d axis−d axis−
ωmωm
isdisd
isqisq
q axis−q axis−at tat t
i rqi rq
irdird
θmθm
isis
irir
32
isd32
isd
32
isq32
isq
32
irq32
irq 32
ird32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
Figure 3-3 Stator and rotor representation by equivalent dq winding currents. The dq winding voltages are defined as positive at the dotted terminals. Note that the relative positions of the stator and the rotor current space vectors are not actual, rather only for definition purposes.
© Copyright Ned Mohan 2001 48
Initial Flux Buildup Prior to axisq −
axisd −
sqi 0=
sdi
t−∞ 0
, sdm iφ
si
rB
t 0−=
sdi
- - -a m,rated b c m,rated
1ˆ ˆi (0 ) = I and i (0 ) = i (0 ) = - I2
-sd ms,rated m,rated m,rated
2 2 3 3ˆ ˆ ˆi (0 ) = I = ( I ) = I3 3 2 2
sqi = 0
-t = 0
© Copyright Ned Mohan 2001 49
Step Change in Torque at m 0ω = +=t 0
rB sdi
net = 0
axisa −
slipωlrφ
, rm iφ, sqm iφ
slipω
sqi sqi
sdi
0−∞ t
ωm = 0isd unchangedStep-change in isq
Φq,net = 0sq r
+ + +m,i m,i r(0 ) = (0 ) + (0 )φ φ φ
mem r sq
r
LˆT α B , iL
-t = 0
© Copyright Ned Mohan 2001 50
Flux Densities at +t = 0
rB
t 0+=
rθ
lrBmsB
axisa −
© Copyright Ned Mohan 2001 51
Transformer Analogy –Voltage Needed to Prevent the Decay of Secondary Current
−
+
2i′
( ) ( )2 2 2v R i 0 u t+′ ′ ′=
2R′
mimL
l2L′l1L
( )1i u t⋅
1R t 0>
© Copyright Ned Mohan 2001 52
Currents and Fluxes at Sometime Later t > 0, Blocked Rotor
m 0ω =
t 0>
rBsdi
net = 0
axisa −
slipωlrφ
, rm iφ, sqm iφsqi
slipω
sqi
axisd −
axisq −
r m r sqslip
r
R , (L /L )i
Bω α
′
© Copyright Ned Mohan 2001 53
Vector-Controlled Condition With a Rotor Speed ωm
t 0>
rBsdi
net = 0
axisa −
synωlrφ
, rm iφ, sqm iφsqi
synω
sqi
axisd −
axisq −
mω
syn m slipω = ω + ω© Copyright Ned Mohan 2001 54
Torque, Speed, and Position Control
(measured)/d dt
encoder
Motor
PPUregulated current
to abcdq
to dqabc
*ai*bi*ci
aibici
sdisqi
emTˆrB
rBθcalculations
mθmω
(measured)mθ mω emT
(measured) (calculated)
P PI PI
∑
∑∑ ∑
−−+ +
−+
−
+ PImω(measured)
*ˆrB
ˆrB
mω
ˆrB (calculated)
*sdi
*sqi
*mθ *
mω *emT
(measured)mωmotor mathematical model
syn m slipω (t) = ω (t) + ω (t)
r
t
B syn0θ (t) = 0 + ω (τ) dτ∫ ⋅
© Copyright Ned Mohan 2001 55
Figure 4-14 (a) Block diagram representation of hysteresis current control; (b) current waveform.
T
1k
*sI
*emT
*( )ai t*( )bi t*( )ci t
Σ ( )Aq t
phasea
( )ai t
dV+−
( )m tθ
+
−
(a)
(b)
reference current
actual current
t0
.From Eq 10 6−
using Eqs. 10-11 and 10-12
Power-Processing Unit (PPU)
© Copyright Ned Mohan 2001 56
Chapter 5
Mathematical Description of Vector Control
© Copyright Ned Mohan 2001 57
a axis−stator
A axis−rotor
d axis−
ωm
isd
isq
q axis−at t
i rq
ird
θm
is
ir
32
isd
32
isq
32
irq 32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
a axis−a axis−stator
A axis−A axis−rotor
d axis−d axis−
ωmωm
isdisd
isqisq
q axis−q axis−at tat t
i rqi rq
irdird
θmθm
isis
irir
32
isd32
isd
32
isq32
isq
32
irq32
irq 32
ird32
ird
dω
dω
daθ
dAθ
32 sN
32 sN
32 sN
Figure 3-3 Stator and rotor mmf representation by equivalent dq winding currents.
Figure 5-1 Stator and rotor mmf representation by equivalent dq winding currents. The d-axis is aligned with rλ .
Motor Model with the d-Axis Aligned with the Rotor Flux Linkage Axis
( ) 0rq tλ = mrq sq
r
Li iL
= −( ) 0rqd tdt
λ =
© Copyright Ned Mohan 2001 58
sR
+
+
+
+
−−
− −
sR
sL
sL rL
rL
mL
mL
sdv
sqv
d sdω λ
d sqω λ
sdi
sqidA rdω λ
0rdv =
0rqv =
+
−
+
−
sdddt
λ
sqddt
λ
+
+
−
rdddt
λ
0rqddt
λ =
(a) -axisd
(b) -axisq
rdi
rqi
rR
rR
−
+
−
rqi
Figure 5-2 Dynamic circuits with the d-axis aligned with rλ .
Dynamic Circuits with the d-Axis Aligned with the Rotor Flux Linkage Axis
Calculation of :dAωrq m
dA r sqrd r rd
i LR iωλ τ λ
= − =
Calculation of Torque
2 2m
em rd rq rd sqr
Lp pT i iL
λ λ = − =
:emT
© Copyright Ned Mohan 2001 59
rL
mLsdi
+
−
rdddt
λ rdi
rR
Figure 5-3 The d-axis circuit simplified with a current excitation.
D-Axis Rotor Flux Dynamics
(1) ( ) ( )mrd sd
r r
sLi s i sR sL
= −+
(2) rd r rd m sdL i L iλ = +
(3) ( ) ( )(1 )
mrd sd
r
Ls i ss
λτ
=+
(4) rd mrd sd
r r
Ld idt
λλτ τ
+ =
© Copyright Ned Mohan 2001 60
Motor Model
0( ) 0 ( )
tda dt dθ ω τ τ= + ∫
rFigure 5-4 Motor model with d-axis aligned with .λ
rdλrdλ1m
r
Lsτ+
rτ
DN
N/D1/ sdaθ
emTmω
mechω
dω dAωmL
sqi
sdi
2m
r
Lp
L⋅
2
p
rdλ
∑
×
++
© Copyright Ned Mohan 2001 61
Simulation with d-Axis Aligned with the Rotor Flux Linkage (line-fed machine; achieving vector control is not an objective)
© Copyright Ned Mohan 2001 62
Figure 5-6 Results of Example 5-1.
Simulation Results match that of Example 3-3
© Copyright Ned Mohan 2001 63
Vector-Controlled Induction Motor with a CR-PWM Inverter
Figure 5-7 Vector-controlled induction motor with a CR-PWM Inverter.
emT
isd*
isq*
ia*
ib*
ic*
Current −Regulated
PPU
Motor
( )mech measuredωmechθ
mod elMotor
of Fig.49
sdi
sqi
iaibic
Encoder
Transformations
Transformations
ddt
Eq. 107
Eq. 103
Motor Model(Fig. 5-4)
abctodq
dqtoabc
rdλ
daθ
daθ
© Copyright Ned Mohan 2001 64
Speed and Position Loops for Vector Control
Figure 5-8 Vector controlled induction motor drive with a current-regulated PPU.
(measured)/d dt
Motor
PPUregulated current
to abcdq
to dqabc
*ai*bi*ci
aibici
sdi
sqiemTrdλdaθ
Fig. 5-4
mechθmechω
(measured)
mechθ mechω emT
(calculated)
P PI PI
∑
∑∑ ∑
−−+ +
−+
−
+ PImechω *
rdλ*rdλ
mechω
rdλ (calculated)
*sdi
*sqi
*mechθ
*mechω *
emT
(measured)
daθ
mechωEstimated Motor Model
© Copyright Ned Mohan 2001 65
Design of Speed Loop
rd m sdL iλ =
2*
2m
em sd sqrk
LpT i iL
=
∑+
−
mechωk
( )sqi si
pkks
+ 1
eqsJ
*mechω emT
© Copyright Ned Mohan 2001 66
Simulation of CR-PWM Vector Controlled Drive using Simulink
© Copyright Ned Mohan 2001 67
Figure 5-11 Simulation results of Example 5-2.
Simulation Results of a Vector Controlled Induction Motor Drive
© Copyright Ned Mohan 2001 68
Calculation of Stator Voltages in Vector Control
2(1) 1 m
s r
LL L
σ = −
(2) msd s sd rd
r
LL iL
λ σ λ= +
(3) sq s sqL iλ σ=,
(5) msq s sq s sq d rd d s sd
rvsq vsq comp
Ldv R i L i L idt L
σ ω λ ω σ
′
= + + +
Figure 5-12 Vector control with applied voltages.
isd*
isq*
PI
PI
∑
∑ ∑
∑+
+
++
+
+
−
−
isd
isq
ia
ic
ib
va*
vb*
vc*
Motorvsd*
vsq*
Transformations
M o to rM o d e l
isd
isq
Encodermechθ( )mech measuredω
Transformations
Modulated
SpaceVector
PPU
rdλemT
daθ
daθ
,sd compv
,sq compv
sdv′
sqv′
,
(4) msd s sd s sd rd d s sq
rvsd vsd comp
Ld dv R i L i L idt L dt
σ λ ω σ
′
= + + −
© Copyright Ned Mohan 2001 69
Figure 5-13 Design of the current-loop controller.
∑+−
( )sdi s( )sdv s′* ( )sdi sPI
1
s sR s Lσ+
Design of the Current-Loop Controller
(1) sd s sd s sddv R i L idt
σ′ = +
(2) sq s sq s sqdv R i L idt
σ′ = +
1(3) ( ) ( )sd sds s
i s v sR s Lσ
′=+
1(4) ( ) ( )sq sqs s
i s v sR s Lσ
′=+
© Copyright Ned Mohan 2001 70
Simulation of Vector Controlled Drive with supplied Voltages
© Copyright Ned Mohan 2001 71
Figure 5-15 Simulation results of Example 5-3.
Simulation Results of Vector Controlled Drive with supplied Voltages
© Copyright Ned Mohan 2001 72
Chapter 6
Detuning Effects in Induction Motor Vector Control
© Copyright Ned Mohan 2001 73
sdi
0sqi =
rλ-axisa-axisd
-axisq
0t −=
-Figure 6-1 windings at 0 .dq t =
Initial conditions
0t −=*
sd sdi i=
, 0da da estθ θ= =
© Copyright Ned Mohan 2001 74
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
Figure 6-2 windings at 0; drawn for 1.dq t kτ> <
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
Figure 6-2 windings at 0; drawn for 1.dq t kτ> <
Effect of Detuning due to Incorrect Rotor Time Constant
,(1) r
r estkτ
ττ
=
( ) ( ), ,,_ _ _(3) j jd d est d estda da est errs dq s dq s dqi i e i eθ θ θ− − −= =
,(4) err da da estθ θ θ= −, * *_(2) d est
sd sqs dqi i ji= +
© Copyright Ned Mohan 2001 75
* *(3) cos sinsd sd err sq erri i iθ θ= +
* *(4) cos sinsq sq err sd erri i iθ θ= −
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
Figure 6-2 windings at 0; drawn for 1.dq t kτ> <
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
-axisd
-axisestd
-axisa
-axisq
-axisestq
_s dqi
sdi
*sdi
*sqi
sqi
sdi
*sdi
*sqi
sqi
,da estθdaθ
0t >
Figure 6-2 windings at 0; drawn for 1.dq t kτ> <
Effect of Detuning due to Incorrect Rotor Time Constant (continued)
( ) ( ), ,,_ _ _(1) j jd d est d estda da est errs dq s dq s dqi i e i eθ θ θ− − −= =
,err da da estθ θ θ= −
, * *_(2) d est
sd sqs dqi i ji= +
© Copyright Ned Mohan 2001 76
Estimated Motor Model (Rotor Blocked)
*sdi
*sqi
dqtoabc
CR-PWMInv.
*ai
*bi
*ci
ai
bi
ci
abctodq
sdi
sqi
1m
r
Lsτ+
mL
rτ
DN ND
dAω 1s
dAθ
rdλ
,rd estλ,1
m
r est
Lsτ+
,r estτ
,dA estθ
,dA estθ
abctodq
mLN
D,dA estω
ND
1s
*,sd est sdi i=
*,sq est sqi i=
(ideal)
Estimated Motor Model(Rotor Flux and Slip Estimator)
Actual Motor Model
Figure 6-3 Actual and the estimated motor models (blocked-rotor).
© Copyright Ned Mohan 2001 77
Simulation of Vector Control with Estimated Motor Parameters
© Copyright Ned Mohan 2001 78
Figure 6-5 Simulation results of Example 6-1.
Effect of Detuning in Dynamic and Steady States
© Copyright Ned Mohan 2001 79
-axisd
-axisestd
-axisa
-axisq-axisestq _s dqi
sdi
*sdi
sqi
*sqi
daθ ,da estθ
0t >
Figure 6-6 windings at 0; drawn for 1.dq t kτ> <
,i estsθ
isθ
Calculations of Steady State Errors
1 sqdA
r sd
ii
ωτ
=
*
, *1
,sq
dA estsd
i
ir estω
τ=
*
*,
,
1 1sq sq
r est r sdsddAdA est
i iii
ωω
τ τ=
2 22 2 * *_ˆsd sq sd sq s dqi i i i I+ = + =
© Copyright Ned Mohan 2001 80
-axisd
-axisestd
-axisa
-axisq-axisestq _s dqi
sdi
*sdi
sqi
*sqi
daθ ,da estθ
0t >
Figure 6-6 windings at 0; drawn for 1.dq t kτ> <
,i estsθ
isθ
Calculations of Steady State Errors (continued) *
*sq
sd
im
i=
2
* 2 21
1sd
sd
i mi k mτ
+=+ ⋅
* *sq sd
sq sd
i iki i
τ=
1 1tan ( ) tan ( )err m k mτθ − −= − ⋅
,err i est is sθ θ θ= −
2
* 21
1 ( )em
em
T mkT k m
ττ
+=+ ⋅
© Copyright Ned Mohan 2001 81
Chapter 7
Space-Vector Pulse-Width-Modulated (SV-PWM) Inverters
Advantages
• Full Utilization of the DC Bus Voltage
• Same simplicity as the Carrier-Modulated PWM
• Applicable in Vector Control, DTC and V/f Control
© Copyright Ned Mohan 2001 82
Synthesis of Stator Voltage Space Vector
Figure 7-1 Switch-mode inverter.
aq bq cq
+
−
dV
ab
c
N
+−av
bv+
cv +
ai
bi
ci
0 2 / 3 4 / 3(1) ( ) ( ) ( ) ( )a j j js a b cv t v t e v t e v t eπ π= + +
(2) ; ;a aN N b bN N c cN Nv v v v v v v v v= + = + = +
0 2 / 3 4 / 3(3) 0j j je e eπ π+ + =
0 2 / 3 4 / 3(4) ( )a j j js aN bN cNv t v e v e v eπ π= + +
0 2 / 3 4 / 3(5) ( ) ( )a j j js d a b cv t V q e q e q eπ π= + +
© Copyright Ned Mohan 2001 83
0 7Figure 7-2 Basic voltage vectors ( and not shown).v v
-axisa
sv
1(001)v
3(011)v2(010)v
6(110)v
4(100)v5(101)v
sector 1
sector 2
sector 3
sector 4
sector 5
sector 6
Basic Voltage Vectors
00
12 / 3
2/ 3
34 / 3
45 / 3
5
6
7
(000) 0
(001)
(010)
(011)
(100)
(101)
(110)
(111) 0
asa js da js da js da js da js da js das
v v
v v V e
v v V e
v v V e
v v V e
v v V e
v v V e
v v
π
π
π
π
π
= =
= =
= =
= =
= =
= =
= =
= =
© Copyright Ned Mohan 2001 84
Synthesis of Voltage Vector in Sector 1
01
jdv V e=
/ 33
jdv V e π=
ˆ j ss sv V e θ=
sθ
1xv
3yv
Figure 7-3 Voltage vector in sector 1.
1 31(1) [ 0]a
s s s ss
v xT v yT v zTT
= + + ⋅
1 3(2) asv xv yv= +
(3) 1x y z+ + =
0 / 3ˆ(4) j j jss d dV e xV e yV eθ π= +
© Copyright Ned Mohan 2001 85
Synthesis using Carrier-Modulated PWM
0 7Figure 7-4 Waveforms in sector 1; .z z z= +
sT
/ 2sT7z
0 / 2z 0 / 2z
/ 2y/ 2y
/ 2x / 2xaNv
bNv
cNv
0
0
0
dV
dV
dV
0
,control avtriv
,control bv
,control cv
,
,
,
ˆ / 2
ˆ / 2
ˆ / 2
control a a k
dtri
control b b k
dtri
control c c k
dtri
v v vVV
v v vVV
v v vVV
−=
−=
−=
max( , , ) min( , , )2
a b c a b ck
v v v v v vv +=
( ) ( ) ( ) 0a b cv t v t v t+ + =
© Copyright Ned Mohan 2001 86
Synthesis of Space Vector using Carrier-Modulated PWM in Simulink
© Copyright Ned Mohan 2001 87
Figure 7-6 Simulation results of Example 7-1.
Control Waveforms for Carrier Pulse-Width-Modulation
© Copyright Ned Mohan 2001 88
Limit on the Amplitude of the Stator Voltage Space Vector
,max,max
ˆ(4) ( ) 3 0.707
2 2phase d
LL dV VV rms V= = =
,max3(5) ( ) 0.612
2 2LL d dV rms V V= = (sinusoidal PWM)
030,maxsV
dV
dV
ˆFigure 7-7 Limit on amplitude .sV
,max ,maxˆ(1) ( ) j ta syns sv t V e ω=
0,max
60 3ˆ(2) cos( )2 2s d dV V V= =
,max ,max2ˆ ˆ(3)3 3
dphase s
VV V= =
© Copyright Ned Mohan 2001 89
Chapter 8
Direct Torque Control (DTC) and Encoder-less Operation of Induction
Motors
© Copyright Ned Mohan 2001 90
DTC System Overview
Measured Inputs: Stator Voltages and Currents
Estimated Outputs: 1) Torque, 2) Mechanical Speed, 3) Stator Flux Amplitude and 4) its angle
+−dV aq
bq
cq
aibici
IM
∑ ∑PI+ +− −
+∑
−
*mechω
mechω
*emT
*ˆsλ
ˆsλ
emT
sθ∠
Estimator
Selectionof
sv
Figure 8-1 Block diagram of DTC.
© Copyright Ned Mohan 2001 91
Principle of DTC Operation
Figure 8-2 Changing the position of stator flux-linkage vector.
sθ
rθ
mθ
( )s tλ
( )s t Tλ − ∆sv T∆i
( ) ( )r rt t Tλ λ≈ − ∆
-axisa
Rotor -axisAArθ
2ˆ ˆ sin
2m
em s r srLpTLσ
λ λ θ=
sr s rθ θ θ= −
© Copyright Ned Mohan 2001 92
22 3
ˆ2em
slip rr
TRp
ωλ
=
m r slipω ω ω= −
(2 / )mech mpω ω=
ˆ( ) ( ) ( )t j ss s s s s s
t Tt t T v R i d e θλ λ τ λ
−∆= − ∆ + − ⋅ =∫
ˆ( ) jr rr s s s rm
L L i eL
θλ λ σ λ= − =( ) ( )r r
r rt t Td
dt Tω
ω
θ θω θ − − ∆= =∆
Im( )2
conjem s s
pT iλ=
21 m
s r
LL L
σ = −
Calculation of Stator Flux:
Calculation of Rotor Flux:
where
Estimating Torque:
Estimating Mechanical Speed:
s s s sdv R idt
λ= + ⇒
© Copyright Ned Mohan 2001 93
Inverter Basic Vectors and Sectors
a-axis1(001)v
3(011)v2(010)v
6(110)v
4(100)v 5(101)v
b-axis
c-axis
1
23
4
5
6
Figure 8-3 Inverter basic vectors and sectors.
© Copyright Ned Mohan 2001 94
Stator Voltage Vector Selection in Sector 1
sector 1sλ
1v
3v2v
4v 5v6v
Figure 8-4 Stator voltage vector selection in sector 1.
© Copyright Ned Mohan 2001 95
Selection of the Stator Voltage Space Vector
sector 1sλ
1v
3v2v
4v 5v6v
Figure 8-4 Stator voltage vector selection in sector 1.
sector 1sλ
1v
3v2v
4v 5v6vsector 1sλ
1v
3v2v
4v 5v6v
Figure 8-4 Stator voltage vector selection in sector 1.
Effect of Voltage Vector on the Stator Flux-Linkage Vector in Sector 1.
increasedecrease
decreasedecrease
decreaseincrease
increaseincreasesv
3v
2v
4v
5v
emT ˆsλ
a-axis1(001)v
3(011)v2(010)v
6(110)v
4(100)v 5(101)v
b-axis
c-axis
1
23
4
5
6
Figure 8-3 Inverter basic vectors and sectors.
© Copyright Ned Mohan 2001 96
Effect of Zero Stator Voltage Space Vector
sθrθ
mθ
( ) ( )s st t Tλ λ≈ − ∆
-axisa
Rotor -axis at t- tA ∆
Arθ Rotor -axis at tA
( )r t Tλ − ∆( )r tλ
mθ∆
sin ( )sr s rθ θ θ≈ −
( )em s rT k θ θ−
0sθ∆
0Arθ∆
Ar m r mθ θ θ θ∆ = ∆ + ∆ ∆ ( )em mT k θ∆ − ∆
© Copyright Ned Mohan 2001 97
DTC in Simulink
© Copyright Ned Mohan 2001 98
Fig. 2 Torque Waveforms.
© Copyright Ned Mohan 2001 99
Fig. 3 Speed Waveforms.
© Copyright Ned Mohan 2001 100
Fig. 4 Stator Flux.
© Copyright Ned Mohan 2001 101
Fig. 5 Stator and Rotor Fluxes.
© Copyright Ned Mohan 2001 102
Chapter 9
Vector Control of Permanent-Magnet
Synchronous-Motor Drives
© Copyright Ned Mohan 2001 103
Non-Salient Permanent-Magnet Synchronous Motor
sd s sd fdL iλ λ= + sq s sqL iλ =
axisc−
axisa−
axisb−
N
S
bi
ai
ci
mθ
( )a
axisc −
axisa −
axisb −
N
S
bi
ai
ci
mθ
rB
'a
a
-axisq
( )b
-axisd
Figure 9-1 Permanent-magnet synchronous machine (shown with =2).p
© Copyright Ned Mohan 2001 104
sd s sd sd m sqdv R idt
λ ω λ= + −
sq s sq sq m sddv R idt
λ ω λ= + +
2m mechpω ω=
( )2em sd sq sq sdpT i iλ λ= − [( ) ]
2 2em s sd fd sq s sq sd fd sqp pT L i i L i i iλ λ= + − =
em Lmech
eq
T Tddt J
ω −=
Non-Salient Permanent-Magnet Synchronous Motor (Continued)
© Copyright Ned Mohan 2001 105
sd s sd m s sqv R i L iω= −
Per-Phase Steady Stead Equivalent Circuit
sq s sq m s sd m fdv R i L iω ω λ= + +
3/ 2s s s m s s m fd
e fs
v R i j L i jω ω λ= + +
23a s a m s a m fd
E fa
V R I j L I jω ω λ= + +
2ˆ3fa fd m E m
kE
E kλ ω ω= =
23E fdk λ=
sR
aV
m mLωm sLω
m lsLω
ˆ( )fa fa E mE E k ω=
−
+
aI
−
+
Figure 9-2 Per-phase equivalent circuit in steady state ( in elect. rad/s)mω
© Copyright Ned Mohan 2001 106
Controller in the dq Reference Frame
compd
( )sd s sd s sd m s sqdv R i L i L idt
ω= + + −compq
( )sq s sq s sq m s sd fddv R i L i L idt
ω λ= + + +
22, ,
3ˆ ˆ( )2sd sq dq rated a ratedi i I I+ ≤ =
Figure 9-3 Controller in the dq reference frame.
ai
bi
ci
Motor
+ Σ
+−
Σ
+−
Σ*sqi
*sdi
( )2m mechpθ θ=
sensorposition
abctodq
+ Σ+
+
( )m s sd fdL iω λ+
m s sqL iω−
*sdv
*sqv Inverter
PWMi
pkks
+
mωsqi
sdi
decouplingterms
1s
toabc
dq
*av
ipkks
+
*bv
*cv
© Copyright Ned Mohan 2001 107
Vector Control of a Permanent-Magnet Synchronous-Motor Drive
© Copyright Ned Mohan 2001 108
Figure 9-5 Simulation results of Example 9-1.
Simulation Results
© Copyright Ned Mohan 2001 109
Salient-Pole Synchronous Machine
sd md sL L L= +
sq mq sL L L= +
fd md fdL L L= +
rd md rdL L L= +
rq mq rqL L L= +
sd sd sd md rd md fdL i L i L iλ = + +
sq sq sq mq rqL i L iλ = +rd rd rd md sd md fdL i L i L iλ = + +
rq rq sq mq sqL i L iλ = +
fd fd d md sd md rdL i L i L iλ = + +
-axisa
-axisd
-axisb-axisq
-axisc
•
× -axisa
-axisd-axisq
sqi
rqi
rdisdi
fdi
( )a
( )b
Figure 9-6 Salient-pole machine.
© Copyright Ned Mohan 2001 110
sd s sd sd m sqdv R idt
λ ω λ= + −
sq s sq sq m sddv R idt
λ ω λ= + + ( 0)rd rd rd rd
dv R idt
λ=
= +
( 0)rq rq rq rq
dv R idt
λ=
= +fd fd fd fd
dv R idt
λ= +
Winding Voltages
+
−
+
−
+
−
+
−
+
−
( ) -axisa d
( ) -axisb q
sdv
sqv
sqi
sdi
rqi
rdi
fdi
rdv
rqv
fdvsR m sqω λ sL rdL rdR
mdL
sR m sdω λ sL rqL rqR
mqL
fL fdR
Figure 9-7 Equivalent circuits for a salient-pole machine.
© Copyright Ned Mohan 2001 111
field+damper in d-axis saliency
[ ( ) ( ) ]2em md fd rd sq sd sq sd sq rq sdmqpT L i i i L L i i L i i= + + − −
Electromagnetic Torque
( )2em sd sq sq sdpT i iλ λ= −
sd sd sd md rd md fdL i L i L iλ = + +
sq sq sq mq rqL i L iλ = +
© Copyright Ned Mohan 2001 112
Space Vector Diagram in Steady State
sd sd sd md fdL i L iλ = + sq sq sqL iλ =
sd s sd m sq sqv R i L iω= − sq s sq m sd sd m md fdv R i L i L iω ω= + +
sd sq s sd s sq m sd sd m md fd m sq sqv jv R i jR i j L i j L i L iω ω ω+ = + + + −
23sd sq sv jv v+ =
23sd sq si ji i+ =Figure 9-8 Space vector and phasor diagrams.
-axisd
-axisq
sdv 2 / 3 sv
2 / 3 si
sqv
aV
aI
Re
Im
( )a ( )b
© Copyright Ned Mohan 2001 113
Chapter 10
Switched-Reluctance Motor (SRM) Drives
© Copyright Ned Mohan 2001 114
mechθo0mechθ =
da
ad
c c
b
b
Figure 10-1 Cross-section of a four-phase 8/6 switched reluctance machine.
Cross-Section of a Switched-Reluctance Machine
© Copyright Ned Mohan 2001 115
mechθo0mech
al
θθ
==
b
a
a
b
c c
d
d
mechθo0mechθ =
b
a
a
b
c c
d
d
unθ
(a) (b)
Figure 10-2 Aligned position for phase a; (b) Unaligned position for phase a.
Aligned and Unaligned Positions for Phase-a
© Copyright Ned Mohan 2001 116
unaligned position
aligned position
(V-sec)aλ
(A)aiFigure 10-3 Typical flux-linkage characteristics of an SRM.
Typical Flux-Linkage Characteristics
© Copyright Ned Mohan 2001 117
Figure 10-4 Calculation of torque.
aλ
ai1I
1λ2λ
0
12
1θ1 mechθ θ+ ∆
Calculation of Torque
mech em mechW T θ∆ = ∆
1 2(1 2 1)elecW area λ λ∆ = − − − −
2 1(0 2 0) (0 1 0)storageW area areaλ λ∆ = − − − − − − −
mech elec storageW W W∆ = ∆ − ∆
1 2 2 1
1 2 1 2(0 1 2 0)2
(1 2 1) (0 2 0) (0 1 0) (1 2 1) (0 1 0) (0 2 0)
(0 1 2 0)
em
area
T area area areaarea area area
areaλ
θ λ λ λ λλ λ λ λ
− − − −
∆ = − − − − − − − − − − − −= − − − − + − − − − − − −
= − − −
(0 1 2 0)em
mech
areaTθ− − −=
∆ constantem
mech ia
WTθ =
′∂=∂
© Copyright Ned Mohan 2001 118Figure 10-5 Performance assuming idealized current waveform.
( )ai A
( sec)a
Vλ−
,
( )em aTNm
( )ae V
(deg)mechθ
unθ
unθalθ
alθ
Waveforms Assuming Ideal Current Waveforms
a a av Ri e= +
( , )a a a mechde idt
λ θ=
a aa a mech
a mech imech a
d de ii dt dtθ
λ λ θθ
∂ ∂= +∂ ∂
a aa mech mech
mech mechi ia amech
dedtω
λ λθ ωθ θ∂ ∂= =
∂ ∂
© Copyright Ned Mohan 2001 119
(A)ai
(V-sec)aλ
,
(Nm)em aT
onθ unθ
onθ unθ
onθ unθ offθ alθ
offθ alθ
offθ alθ
offθ alθonθ unθ
emT
,em aT,em bT ,em cT ,em dT(Nm)
Figure 10-6 Performance with a power-processing unit.
Performance with a Power Processing Unit
© Copyright Ned Mohan 2001 120
Figure 10-7 Flux-linkage trajectory during motoring.
0
emW
ai
aλ
fW
unalignedposition
alignedposition
offθ
Role of Magnetic Saturation
em
em f
WEnergyConversion FactorW W
=+
© Copyright Ned Mohan 2001 121
Dc1
Sc1
Pha
se a
Dd1
Dc2
Sa1
Sc2
Pha
se d
Sd1
Pha
se b
Sa2
Pha
se c
Db2
Da1 Db1
Da2
Sd2Sb2
Dd2
Sb1
dcV
+
−
Figure 10-8 Power converter for a four-phase switched reluctance drive.
Power Processing Unit
© Copyright Ned Mohan 2001 122
∫v
−+ Σ
λ
i( , )iθ λ
mechθ
iR
Figure 10-9 Estimation of rotor position.
Determining Rotor Position for Encoder-less Operation
© Copyright Ned Mohan 2001 123
positions
*mechω
+−
refIPI
currentcontroller
powerconverter
gate
signals
aλ
Σ
mechω
Figure 10-10 Control block diagram for motoring.
Control in Motoring Mode