EE595S: Class Lecture NotesChapter 14: Induction Motor Drives
S.D. Sudhoff
Fall 2005
Fall 2005 EE595S Electric Drive Systems 2
Overview of Strategies
• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control
Fall 2005 EE595S Electric Drive Systems 3
Volts-Per-Hertz Control
• History
• Advantages
• Disadvantages
Fall 2005 EE595S Electric Drive Systems 4
Volts-Per-Hertz Control
• Basic IdeaMachine will operate close to synchronous speed, thus speed controlled with frequencyWe must avoid saturation
• On Avoiding Saturation(14.2-1)(14.2-2)
asassas pirv λ+=
sesV Λ=ω
Fall 2005 EE595S Electric Drive Systems 5
Volts-Per-Hertz Control
• Elementary Volts Per Hertz Control
Fall 2005 EE595S Electric Drive Systems 6
Volts-Per-Hertz Control
• Example System50 Hp MachineRated for 1800 rpm, 460 V l-l rms, P=4, 60 Hzrs=72.5 mΩ; rr’=41.3 mΩLM=30.1 mH; Lls=Llr=1.32 mH
(14.2-3)⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+=
29.0)(1.0
bmrm
rmbL STTωωω
Fall 2005 EE595S Electric Drive Systems 7
Volts-Per-Hertz Control
• Performance Characteristics
Fall 2005 EE595S Electric Drive Systems 8
Volts-Per-Hertz Control
• Comments on Steady-State Performance
Fall 2005 EE595S Electric Drive Systems 9
Volts-Per-Hertz Control
• Problem 1: Resistance At Low Speed
• Solution: Set Voltage So Slope is Invarient
(4.9-19)
(14.2-4)2,
22,
2,
22,
estssbests
estsseestsbs
Lr
LrVV
ω
ω
+
+=
222
22
22
)()(
~2
3
rrsssrbe
rrssMbe
rs
asrbM
be
e
XsrXrXXXsrr
VsrXP
T
′+′⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡′−⎟⎟
⎠
⎞⎜⎜⎝
⎛+′
′⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
=
ωω
ωω
ωωω
Fall 2005 EE595S Electric Drive Systems 10
Volts-Per-Hertz Control
• Further Improvement – Current FeedbackLet’s approximate torque as
• (14.2-5)• (14.2-6)
Using (14.2-4) we have
• (14.2-7)
• This is not a function of synchronous speed
)( retve KT ωω −=
erre
tvTK
ωωω =∂∂
−=
)(2
3
2222
22
ssbsr
brMtv
Lrr
VrLP
Kω+′
′⎟⎠⎞
⎜⎝⎛
=
Fall 2005 EE595S Electric Drive Systems 11
Volts-Per-Hertz Control
Next, consider torque. We have• (14.2-8)
For steady-state conditions• (14.2-9)
• (14.2-10)
Thus• (14.2-11)
Where• (14.2-12)
)(22
3 eds
eqs
eqs
edse iiPT λλ −=
e
eqss
eqse
dsirv
ωλ
−=
e
edss
edse
qsirv
ωλ
−−=
)2(122
3 2*ss
eqs
eqs
ee IrivPT −=
ω
222
1 eds
eqss iiI +=
Fall 2005 EE595S Electric Drive Systems 12
Volts-Per-Hertz Control
Equating (14.2-7) and (14.2-11)
• (14.2-13)
Or• (14.2-14)
Where• (14.2-15)
• (14.2-16)
2/)2(3 2*2**
tvsseqs
eqsrr
eKIrivP −++
=ωω
ω
2),0max( 2**
corrrre
Χ++=
ωωω
corrcorr sH LPF χ)(=Χ
tvsseqs
eqscorr KIrivP /)2(3 2* −=χ
Fall 2005 EE595S Electric Drive Systems 13
Volts-Per-Hertz Control
Fall 2005 EE595S Electric Drive Systems 14
Volts-Per-Hertz Control
• Performance of Compensated Volts-Per-Hertz Control
Fall 2005 EE595S Electric Drive Systems 15
Volts-Per-Hertz Control
• Start-Up Performance of Compensated Volts Per Hertz Control
Fall 2005 EE595S Electric Drive Systems 16
Overview of Strategies
• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control
Fall 2005 EE595S Electric Drive Systems 17
Constant Slip Control
• Basic IdeaDefinition of Slip Frequency
(14.3-1)Our Strategy
res ωωω −=
Fall 2005 EE595S Electric Drive Systems 18
Derivation of Constant Slip Control
• Consider Our Expression for Electromagnetic Torque
• (14.3-2)
• Solving for the Current Yields
• (14.3-3)
• The Plan
22
22
)()(2
3
rrsr
rsMse
Lr
rILP
T′+′
′⎟⎠⎞
⎜⎝⎛
=ω
ω
estrrestMs
estrrsestres
rLP
LrTI
,2
,
2,
2,
*
||3
))((||2
′
′+′=
ω
ω
Fall 2005 EE595S Electric Drive Systems 19
Avoiding Saturation
• Before Proceeding …
• Consider the Steady-State Equivalent Circuit
(14.3-4)(14.3-5)
Thus(14.3-7)
)~~(~~arasMarlrar IILIL ′++′=λ
srLjLjII
rrreMe
asar /~~
′+′−=′
ωω
222'
rrrs
rMsr
rL
rLI′+′
′=
ωλ
Fall 2005 EE595S Electric Drive Systems 20
Avoiding Saturation
Taking the Magnitude(14.3-7)
Combing (14.3-7) and (14.3-2)
(14.3-8)Thus
(14.3-9)So For Large Torque Commands
(14.3-10)
222'
rrrs
rMsr
rL
rLI′+′
′=
ωλ
rrs
rePT ′=
2'
23 λω
estr
rsetsthreshe r
PT,
2max,,
, 23
′=
λω
2max,
,*
3
2
r
estres
P
rT
λω
′=
Fall 2005 EE595S Electric Drive Systems 21
Constant Slip Control
Fall 2005 EE595S Electric Drive Systems 22
Selection of Slip Frequency
• Control for Maximum Torque Per AmpFrom (14.3-1)
(14.3-11)
Optimizing with Respect to Slip Frequency
(14.3-12)
2,
2
2,
2 )()(2
3
rsetsr
rMsets
s
eLr
rLP
IT
′+′
′⎟⎠⎞
⎜⎝⎛
=ω
ω
estrr
estrsets L
r
,
,, ′
′=ω
Fall 2005 EE595S Electric Drive Systems 23
Maximum Torque Per Amp Control
Fall 2005 EE595S Electric Drive Systems 24
Maximum Efficiency Control
• To Optimize Efficiency Note That(14.3-13)
So(14.3-14)
Comparison to (14.3-14) to (14.3-2)(14.3-15)
Now (14.3-16)
)~Re(3 assin VIP =
22
222
)(33
rrsr
rsMesssin
LrrLIIrP
′+′
′+=
ω
ωω
eessin TP
IrP ω23 2 +=
erout TP
P ω2=
Fall 2005 EE595S Electric Drive Systems 25
Maximum Efficiency Control
So(14.3-17)
Finally
(14.3-18)
Minimizing Yields(14.3-19)
sessoutin TP
IrPP ω23 2 +=−
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
′+
′= s
mr
rrss
ms
sreloss
LrLr
LrrT
PP ωω
ω 2
2
22
1
1
,
,2
,
2,,
,,
+′′
′
′=
estr
ests
estrr
estmestrr
estrsets
rr
L
LLr
ω
Fall 2005 EE595S Electric Drive Systems 26
Maximum Efficiency Control
Fall 2005 EE595S Electric Drive Systems 27
Comparison of Constant Slip Controls
• Maximum Efficiency
• Maximum Torque Per Amp
Fall 2005 EE595S Electric Drive Systems 28
Speed Control with Constant Slip Control
Fall 2005 EE595S Electric Drive Systems 29
Transient Performance of Constant Slip Controlled Drive
Fall 2005 EE595S Electric Drive Systems 30
Overview of Strategies
• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control
Fall 2005 EE595S Electric Drive Systems 31
14.4 Field-Oriented Control
• Objective
• Challenges
Fall 2005 EE595S Electric Drive Systems 32
The Basic Idea
θsin2BiNLrTe −=
Fall 2005 EE595S Electric Drive Systems 33
Torque Production in Induction Motor
• Now(14.4-2)
(14.4-3)
)(22
3qrdrdrqre iiPT ′′−′′= λλ
θλ sin||||22
3qdrqdre iPT ′′−=
Fall 2005 EE595S Electric Drive Systems 34
The Plan
• Objective
• ObservationIn Steady-State, This is Always True
• (14.4-4)
• (14.4-5)
• (14.4-6)
Thus
edrre
reqr ri λωω )(1
−′
−=
eqrre
redr ri λωω )(1
−′
=
edr
edr
eqr
eqr
eqdr
eqdr iii ′′+′′=′⋅′ λλλ
Fall 2005 EE595S Electric Drive Systems 35
The Plan (Continued)
• But.. We need flux and current to be perpendicular all the time. To do this ..
(14.4-7)(14.4-8)
• Achieving (14.4-7)By choice of θe
0=′eqrλ
0=′edri
Fall 2005 EE595S Electric Drive Systems 36
The Plan (Continued)
• Achieving (14.4-8)All we need to do is to keep d-axis stator current constant. Proof:
• Consider(14.4-9)
• The q-axis rotor flux linkage is zero (by choice of reference frame)
(14.4-10)• Substitution of d-axis rotor flux linkage equation
into (14.4-10)(14.4-11)
• Thus the d-axis rotor current goes to zero
edr
eqrre
edrr pir λλωω ′+′−+′′= )(0
edr
edrr pir λ′+′′=0
eds
rrMe
drrrre
dr piLLi
Lrip −′′
−=′
Fall 2005 EE595S Electric Drive Systems 37
Some Observations
• Since the d-axis rotor current is zero
(14.4-12)
(14.4-13)
• Combining these results with our flux linkage equation
(14.4-14)
edsss
eds iL=λ
edsM
edr iL=′λ
eqr
edre iPT ′′−= λ
223
Fall 2005 EE595S Electric Drive Systems 38
Some Observations
• Since the q-axis rotor flux linkage is zero
(14.4-15)
• Thus
(14.4-16)
eqs
rrMe
qr iLLi −=′
eqs
edrrr
Me i
LLPT λ′=
223
Fall 2005 EE595S Electric Drive Systems 39
Generic Rotor Flux-Oriented Control
Note: θe must be selected so as to keep q-axis rotor flux equal to zero.
Fall 2005 EE595S Electric Drive Systems 40
Some “Minor” Details
• Determination of position of synchronous reference frame
• Determination of the rotor flux
Fall 2005 EE595S Electric Drive Systems 41
Aside Types of Field Oriented Control
• OrientationsStatorMagnetizing (Air-Gap)Rotor
• MethodsDirectIndirect
• ModificationsRobust
Fall 2005 EE595S Electric Drive Systems 42
Direct Field Oriented Control
• Basic IdeaMeasure the flux directly (more or less)
• Consider
(14.5-1)
• We need to set q-axis rotor flux to zero. Thus
(14.5-2)
⎥⎥⎦
⎤
⎢⎢⎣
⎡
′
′⎥⎦
⎤⎢⎣
⎡ −=
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
′
′sdr
sqr
ee
eee
dr
eqr
λ
λθθθθ
λ
λcossinsincos
2)( πλλθ +′−′= s
drsqre jangle
Fall 2005 EE595S Electric Drive Systems 43
Direct Field-Oriented Control
• As a Side Effect
(14.5-3)
• The problem: We can measure the rotor flux linkages. We can measure the stator flux linkages.
22 )()( sdr
sqr
edr λλλ ′+′=′
Fall 2005 EE595S Electric Drive Systems 44
Estimation of Rotor Flux Linkages
• Consider(14.5-4)
• Thus(14.5-5)
• Recall(14.5-6)
)( sqr
sqsM
sqm iiL ′+=′λ
M
sqsM
sqms
qr LiL
i−′
=′λ
)( sqr
sqsM
sqrlr
sqr iiLiL ′++′=′λ
Fall 2005 EE595S Electric Drive Systems 45
Estimation of Rotor Flux Linkages
• So(14.5-7)
(14.5-8)
sqslr
sqm
Mrrs
qr iLLL ′−′
=′ λλ
sdslr
sdmM
rrsdr iL
LL ′−′
=′ λλ
Fall 2005 EE595S Electric Drive Systems 46
Rotor Flux Estimator
Fall 2005 EE595S Electric Drive Systems 47
Direct Rotor Field-Oriented Control
Fall 2005 EE595S Electric Drive Systems 48
Robust Direct Field-Oriented Control
• Sensitivity of Rotor Flux Estimator(14.6-1)
• Sensitivity of Q-Axis Current Calculation
(14.6-2)
sqdsestlr
smqdestM
estrreqdr iL
LL
,,,
,ˆ ′−′
=′ λλ
edrestrr
estMee
qs
LLP
Tiλ′
=ˆ
223
,
,
**
Fall 2005 EE595S Electric Drive Systems 49
Robust Direct Field-Oriented Control
• Sensitivity of the D-Axis Current Injection
(14.6-3)estM
edre
ds Li
,
** λ=
Fall 2005 EE595S Electric Drive Systems 50
Flux Control Loop
• Since we have a flux estimator, consider
• It can be shown that
(14.6-5)1
1,*
+
+=
′
′
sL
Ls
M
estMedr
edr
λ
λ
τ
τ
λ
λ
Fall 2005 EE595S Electric Drive Systems 51
Torque Control Loop
• Torque EstimationRecall
(14.6-6)Now
(14.6-7)So
(14.6-8)Which Suggests
(14.6-9)
)(22
3 sds
sqs
sqs
sdse iiPT λλ −=
sqdm
sqdsls
sqds iL λλ +=
)(22
3 sds
sqm
sqs
sdme iiPT λλ −=
)(22
3ˆ sds
sqm
sqs
sdme iiPT λλ −=
Fall 2005 EE595S Electric Drive Systems 52
Torque Control Loop
• Now that we can estimate torque, how about
Fall 2005 EE595S Electric Drive Systems 53
Torque Control Loop
• Analysis of Torque Control LoopDefine
(14.6-10)(14.6-12)
Now, assuming the control works(14.6-11)
This, assumption, coupled with our torque control loop yields
(14.6-13)
*,
,, 22
3 edrestrr
estMestt L
LPK λ′′
=
edrrr
Mt L
LPK λ′′
=22
3
*eqste iKT =
1
1,*
+
+=
sK
Ks
TT
t
esttt
t
e
e
τ
τ
Fall 2005 EE595S Electric Drive Systems 54
Robust Direct Field-Oriented Control
Fall 2005 EE595S Electric Drive Systems 55
Performance of Robust Direct Field Oriented Control
Fall 2005 EE595S Electric Drive Systems 56
Performance of Constant Slip Drive
Fall 2005 EE595S Electric Drive Systems 57
Performance of Volts-Per-Hertz Drive
Fall 2005 EE595S Electric Drive Systems 58
Indirect Field Oriented Control
• Consider (14.7-1)
Since the q-axis rotor flux is zero
(14.7-2)
Which may be expressed
(14.7-3)
eqr
edrre
eqrrr pir λλωω ′+′−+′′= )(0
edr
eqr
rrei
rλ
ωω′
′′−=
ede
eqs
rrr
rei
iLr′′
+= ωω
Fall 2005 EE595S Electric Drive Systems 59
Indirect Field Oriented Control
• ThoughtWe know what the electrical frequency will be if we have field orientation (i.e. field orientation causes (14.7-1))Does it work backward ? Will using (14.7-1) cause field-orientation ?The answer: Yes !
Fall 2005 EE595S Electric Drive Systems 60
Indirect Field Orientation
• ProofConsider the rotor voltage equations
(14.7-5)(14.7-6)
Now substitute in our expression for electrical frequency
(14.7-7)
(14.7-8)
eqr
edrre
eqrr pir λλωω ′+′−+′′= )(0
edr
eqrre
edrr pir λλωω ′+′−−′′= )(0
eqr
edre
de
eqs
rrre
qrr pi
iLrir λλ ′+′′′
+′′= *
*0
edr
eqre
de
eqs
rrre
drr pi
iLrir λλ ′+′′′
−′′= *
*0
Fall 2005 EE595S Electric Drive Systems 61
Indirect Field Orientation
• Continuing On …Now put in terms of q-axis rotor flux and d-axis rotor current (and stator currents)This yields
(14.7-9)
(14.7-10)
[ ] eqr
edsM
edrrre
ds
eqs
rr
r
rr
eqsM
eqr
r piLiLi
iLr
LiL
r λλ
′++′′′′
+⎥⎥⎦
⎤
⎢⎢⎣
⎡
′
−′′= *
*
**0
[ ]**
*0 e
dsMe
drrreqre
ds
eqs
rr
redrr iLiLp
i
iLrir +′′+′′′
−′′= λ
Fall 2005 EE595S Electric Drive Systems 62
Indirect Field Orientation
• Now, keeping the d-axis stator current fixed we have
(14.7-11)
(14.7-12)
• Comments
edre
ds
eqs
reqr
rrre
qr ii
ir
Lrp ′′−′′′
−=′*
*λλ
eqre
ds
eqs
rr
redrrr
redr i
i
Lri
Lrip λ′
′
′+′
′′
−=′*
*
2)(
Fall 2005 EE595S Electric Drive Systems 63
Indirect Field Oriented Control
• Thus, our control becomes
Fall 2005 EE595S Electric Drive Systems 64
Indirect Field Oriented Control
• Advantages of Indirect Field Oriented Control over Direct Field Oriented Control
• Disadvantages of Indirect Field Oriented Control over Direct Field Oriented Control
Fall 2005 EE595S Electric Drive Systems 65
Performance of DetunedIndirect Field Oriented Control
• Overestimate magnetizing inductance by 25% (we started to saturate machine)
• Underestimate rotor resistance by 25% (rotor resistance increased with temperature)
Fall 2005 EE595S Electric Drive Systems 66
Performance of Detuned Field Oriented Control