EEEB283 Electrical Machines & Drives
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Transcript EEEB283 Electrical Machines & Drives
Induction Motor – Vector Control or Field
Oriented Control
By
M.Kaliamoorthy
Department of Electrical Engineering
1
Outline
Introduction
Analogy to DC Drive
Principles of Field Orientation Control
Rotor Flux Orientation Control
Indirect Rotor Flux Orientation (IRFO)
Direct Rotor Flux Orientation (DRFO)
Stator Flux Orientation Control
Direct Stator Flux Orientation (DSFO)
References
2
Introduction
Induction Motor (IM) drives are replacing DC drives
because:
Induction motor is simpler, smaller in size, less maintenance
Less cost
Capability of faster torque response
Capability of faster speed response (due to lower inertia)
DC motor is superior to IM with respect to ease of control
High performance with simple control
Due to decoupling component of torque and flux
3
Introduction
Induction Motor Drive
Scalar Control
• Control of current/voltage/frequency
magnitude based on steady-state
equivalent circuit model
• ignores transient conditions
•
•
•
•
• for low performance drives
Simple implementation
Inherent coupling of torque and flux
• Both are functions of voltage and
frequency
Leads to sluggish response
Easily prone to instability
Vector Control or Field Orientation
Control
• control of magnitude and phase of
currents and voltages based on
dynamic model
• Capable of observing steady state
& transient motor behaviour
• for high performance drives
• Complex implementation
• Decoupling of torque and flux
• similar to the DC drive
• Suitable for all applications previously
covered by DC drives
4
Analogy to DC Drive
In the DC motor: Te = k f Ia
f controlled by controlling If
If same direction as field f
Ia same direction as field a
Ia and f always perpendicular
and decoupled
• Hence, Te = k f Ia
•
•
•
•
•
= k’ If Ia sin 90
= k’(If x Ia)
f
a
• Keeping f constant, Te
controlled by controlling Ia
• Ia, If , a and f are space vectors
5
Analogy to DC Motor
• In the Induction Motor:
s
c’
b
r
a
b’
c
Te = kr x s
• s produced by stator currents
• r produced by induced rotor
currents
• Both s and r rotates at
synchronous speed s
• Angle between s and r
varies with load, and motor
speed r
• Torque and flux are coupled.
6
Analogy to DC Motor
• Induction Motor torque equation :
3P
Te
ψ s is
22
3 P Lm
Te
ψr is
2 2 Lr
(1)
(2)
• Compared with DC Motor torque equation:
Te k I f I a k ψ f ia sin 90 k ψ f i a
'
(3)
• Hence, if the angle betweens orr andis is made to be
90, then the IM will behave like a DC motor.
7
Principles of Field Orientation Control
• Hence, if the angle betweens orr andis is made to be
90, then the IM will behave like a DC motor.
Achieved through orientation (alignment) of rotating dq frame
on r or s
Rotor-Flux
Orientation Control
Stator-Flux
Orientation Control
8
Principles of Field Orientation Control
Rotor-Flux
Orientation Control
Stator-Flux
Orientation Control
qs
qs
qr
qs
is
r
r
isq
i
s
dr
isqΨs
r
sd
ds
3 P Lm
Te
( rd isq rq isd )
2 2 Lr
is
ds
isdΨs
ds
3P
Te
( sd isq sq isd )
22
9
Principles of Field Orientation Control
• Summary of field orientation control on a selected flux vectorf
(i.e. either r , s or m):
1
2
3
• In revolving (rotating) dfqf - reference frame, obtain
• isqf* from given rotor speed reference r* (via speed controller)
• isdf* from given flux reference f*
• Determine the angular position f of f (i.e. reference frame
orientation angle)
• used in the dfqf dsqs conversion from vsdqf* (output of
isdqf* current controller) to vsdqs*.
• In the stationary dsqs - frame, obtain the reference stator voltages
vabcs*
• fed to the PWM inverter feeding the IM from vsdqs* using the
dsqs abc transformation.
10
Rotor Flux Orientation Control
qs
• d- axis of dq- rotating frame
is aligned with r . Hence,
qr
is
rd r
(4)
rq 0
(5)
r
r
r
isq
r
dr
r
• Therefore,
r
isd
ds
r
i
sq = torque producing current
r
isd
= field producing current
3 P Lm
Te
( rd isq )
2 2 Lr
Similar to
ia & if in
DC motor
(6)
Decoupled
torque and
flux control
11
Rotor Flux Orientation Control
From the dynamic model of IM, if dq- frame rotates at general
speed g (in terms of vsd, vsq, isd, isq, ird, irq) :
vsd Rs SLs
vsq g Ls
vrd
SLm
vrq ( g r ) Lm
g Ls
Rs SLs
( g r ) Lm
SLm
SLm
g Lm
Rr ' SLr
( g r ) Lr
g Lm isd
SLm
isq (7)
( g r ) Lr ird
Rr ' SLr irq
r rotates at synchronous speed s
Hence, drqr- frame rotates at s
Therefore, g = s
These voltage equations are in terms of isd, isq, ird, irq
Better to have equations in terms of isd, isq, rd, rq
(8)
12
Rotor Flux Orientation Control
• Rotor flux linkage is given by: rdq Lmisdq L'r irdq
• From (9):
rdq Lm
irdq
'
r
L
'
r
L
isdq
(9)
(10)
• Substituting (8) and (10) into (7) gives the IM voltage
equations rotating at s in terms of vsd, vsq, isd, isq, rd, rq:
vsdψr Rs SLs
sLs
ψr
Rs SLs
vsq sLs
vrdψr Rr ' Lm Lr '
0
ψr
0
Rr ' Lm Lr '
vrq
S Lm Lr '
s Lm Lr '
Rr ' Lr ' S
sl
s Lm Lr ' isdψr
ψr
SLm Lr ' isq
ψr
rd
sl
ψr
Rr ' Lr ' S rq
(11)
13
Rotor Flux Orientation Control
• Since rqr 0 , hence the equations in rotor flux
orientation are:
Lm d ψr (12)
d ψr
ψr
ψr
ψr
vsd Rs isd Ls isd sLsisq s
rd
dt
Lr ' dt
L
d
(13)
vsqψr Rs isqψr Ls isqψr sLs isdψr s m rdψr
dt
Lr '
Rr ψr d ψr Lm
v 0 rd rd
Rr isdψr
Lr '
dt
Lr '
ψr
rq
Lm
v 0 sl
Rr isqψr
Lr '
Important equations for
Rotor Flux Orientation Control!
ψr
rq
ψr
rd
(14)
(15)
Note:
Total leakage factor =
2
L
1 m '
Ls Lr
sl = slip speed (elec.)
14
Rotor Flux Orientation Control
ψr
rdψr Lmimrd
• Let
• Using (16), equation (14) can be rearranged to give:
Lr ' d ψr
ψr
ψr
isd imrd
imrd
Rr dt
(16)
(17)
ψr is called the “equivalent magnetising current” or “field
• imrd
current”
Lr '
ψr
ψr
(18)
• Hence, from (17): isd 1 S r imrd where r R
r
• Under steady-state conditions (i.e. constant flux):
ψr
isdψr imrd
(19)
15
Rotor Flux Orientation Control
qs
• r rotates at synchronous speed s
• drqr- frame also rotates at s
• Hence,
qr
is
r
r
isq
r
dr
r
isd
ds
dq- reference frame
orientation angle
s dt
r
• For precise control, r must be
(20)
obtained at every instant in time
• Leads to two types of control:
– Indirect Rotor Flux
Orientation
– Direct Rotor Flux
Orientation
16
Indirect Rotor Flux Orientation (IRFO)
• Orientation angle: r s dt
• Synchronous speed obtained by adding slip speed and
electrical rotor speed
r s dt sl r dt
• Slip speed can be obtained from equation (15):
Lmisqψr
(21)
isqψr
Lm Rr ψr
sl
i
ψr
ψr sq
ψr
Lr ' rd
r rd r imrd
ψr
ψr
• Under steady-state conditions (imrd isd ):
(22)
sl
isqψr
i
ψr
r sd
(23)
17
Indirect Rotor Flux Orientation (IRFO)
- implementation
Closed-loop implementation under constant flux condition:
1. Obtain isdr* from r* using (16):
ψr *
ψr *
isdψr * imrd
rd
(24)
Lm
Obtain isqr* from outer speed control loop since isqr*
Tm* based on (6):
*
2
Te
3 P Lm
ψr*
(25)
isq ψr* where kt
kt isd
2 2 Lr
Obtain vsdqr* from isdqr* via inner current control
loop.
18
Indirect Rotor Flux Orientation (IRFO)
- implementation
Closed-loop implementation under constant flux condition:
2. Determine the angular position r using (21) and
(23):
ψr*
i
P
sq
*
sl r dt ψr* m dt
2
r isd
s dt
r
(26)
where m is the measured mechanical speed of the
motor obtained from a tachogenerator or digital
encoder.
r to be used in the drqr dsqs conversion of
stator voltage (i.e. vsdqr* to vsdqs* concersion).
19
Indirect Rotor Flux Orientation (IRFO)
- implementation
drqr dsqs transformation
Rotating frame (drqr)
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
Staionary frame (dsqs)
vsdr*
vsq
ejr
PI
-
-
isdr* isqr*
Eq. (23)
NO field
weakening
(constant flux)
slip
+
vsd
s*
r
r
+
isdr
isqr
vas*
vsqs*
PI
r*
2-phase (dsqs )
to 3-phase (abc)
transformation
2/3
vcs*
isqs
PWM
VSI
IRFO
Scheme
P/2
m
ias
isds
e-jr
vbs*
3/2
ibs
ics
20
Indirect Rotor Flux Orientation (IRFO)
- implementation
drqr dsqs transformation
vsqs*
vsdr*
vsqr*
ejr
vsds*
xsds cos r
s
xsq sin r
sin r xsdr
cos r xsqr
dsqs drqr transformation
isdr
isqr
isds
e-jr
isqs
xsdr cos r
r
xsq sin r
sin r xsds
cos r xsqs
21
Indirect Rotor Flux Orientation
(IRFO) - implementation
• 2-phase (dsqs ) to 3-phase (abc) transformation:
vas*
vsqs*
vsd
s*
2/3
1 s
xabc Tabc
xdq
vbs*
vcs*
• 3-phase (abc) to 2-phase (dsqs ) transform is given by:
ias
isds
isq
s
3/2
s
xdq
Tabc xabc
ibs
ics
where:
Tabc
1 0 0
1 1
0 3 3
and
1
Tabc
1 0
12 23
12 23
22
Example – IRFO Control of IM
• An induction motor has the following
parameters:
Parameter
Symbol
Value
Rated power
Prat
30 hp (22.4 kW)
Stator connection
Delta ()
No. of poles
P
6
Rated stator phase
voltage (rms)
Vs,rat
230 V
Rated stator phase
current (rms)
Is,rat
39.5 A
Rated frequency
frat
60 Hz
Rated speed
nrat
1168 rpm
23
Example – IRFO Control of IM ctd.
Parameter
Symbol
Value
Rated torque
Te,rat
183 Nm
Stator resistance
Rs
0.294
Stator self
inductance
Referred rotor
resistance
Ls
0.0424 H
Rr’
0.156
Referred rotor self
inductance
Lr’
0.0417 H
Mutual inductance
Lm
0.041 H
24
Example – IRFO Control of IM ctd.
The motor above operates in the indirect rotor field orientation (IRFO)
scheme, with the flux and torque commands equal to the respective
rated values, that is r* = 0.7865 Wb and Te* = 183 Nm. At the
instant t = 1 s since starting the motor, the rotor has made 8
revolutions. Determine at time t = 1s:
1.
2.
3.
4.
the stator reference currents isd* and isq* in the dq-rotating frame
the slip speed sl of the motor
the orientation angle r of the dq-rotating frame
the stator reference currents isds* and isqs* in the stationary dsqs
frame
5. the three-phase stator reference currents ias*, ibs* and ics*
25
Example – IRFO Control of IM ctd.
• Answers:
26
Indirect Rotor Flux Orientation (IRFO)
– field weakening
• Closed-loop implementation under field weakening
condition:
– Employed for operations above base speed
– DC motor: flux weakened by reducing field current if
vf
Lf d
if
if
Rf
R f dt
imrd*
– Compared with eq. (17) for IM:
imrd (rated)
L
'
d
ψr
ψr
isdψr imrd
r
imrd
Rr dt
– IM: flux weakened by reducing imrd
(i.e. “equivalent magnetising current”
or “field current)
r (base)
r
27
Indirect Rotor Flux Orientation (IRFO)
– field weakening implementation
With field
weakening
r*
Rotating frame (drqr) Staionary frame (dsqs)
imrd r *
+
imrd
r
1
1 S r
isd
PI
r* +
r*
PI
-
+
isqr* +
imrdr*r
vsdr*
PI
vsqr*
ejr
PI
-
isq *
Eq. (22)
vsqs*
slip
+
r
r
Same as
in slide 20
+
isdr
isqr
vsds*
isds
e-jr
isqs
28
Indirect Rotor Flux Orientation
(IRFO) – Parameter sensitivity
Mismatch between IRFO Controller and IM may occur
due to parameter changes with operating conditions (eg.
increase in temperature, saturation)
Mismatch causes coupling between T and producing
components
Consequences:
r deviates from reference value (i.e. r*)
Te deviates in a non-linear relationship from command
value (i.e. Te*)
Oscillations occurs in r and Te response during torque
transients (settling time of oscillations = r)
29
Direct Rotor Flux Orientation (DRFO)
• Orientation angle:
tan
1
r
rq
s
rd
s
(27)
obtained from:
1. Direct measurements of airgap fluxes mds and mqs
2. Estimated from motor’s stator voltages vsdqs
and stator currents isdqs
Note that: ψ s 2 s 2
r
rd
rq
(28)
30
Direct Rotor Flux Orientation (DRFO) –
Direct measurements mds & mqs
1. Direct measurements of airgap fluxes mds and mqs
mds and mqs measured using:
Hall sensors – fragile
flux sensing coils on the stator windings – voltages induced in
coils are integrated to obtain mds and mqs
The rotor flux r is then obtained from:
s
s
L'r
'
(29)
rdq
mdq Llrisdq
Lm
Disadvantages: sensors are inconvenient and spoil the
ruggedness of IM.
s
31
Direct Rotor Flux Orientation (DRFO) –
Direct measurements mds & mqs
Rotating frame
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
-
r
NO field
weakening
(constant flux)
(drqr)
Stationary frame
vsdr*
vsq
r*
ejr
PI
DRFO r
Scheme tan-1
isdr
isqr
r
e-jr
vsd
s*
2/3
vbs*
vcs*
mds
rds
rqs Eq. (29) mqs
P/2
PWM
VSI
m
ias
isds
isqs
Flux sensing coils
arranged in quadrature
vas*
vsqs*
PI
(dsqs)
3/2
ibs
ics
32
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
2. Estimated from motor’s stator voltages and currents
sds and sqs obtained from stator voltage equations:
sdq vsdq Rs isdq sdq 0
The rotor flux r is then obtained from:
s
s
s
s
(30)
s
s
L'r
rdq
sdq Ls isdq
(31)
Lm
Disadvantages: dc-drift due to noise in electronic
circuits employed, incorrect initial values of flux vector
components sdq(0)
s
33
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
2. Estimated from motor’s stator voltages and currents
This scheme is part of sensorless drive scheme
using machine parameters, voltages and currents to
estimate flux and speed
sdqs calculations (eq. 30) depends on Rs
Poor field orientation at low speeds ( < 2 Hz), above 2 Hz,
DRFO scheme as good as IRFO
Solution: add boost voltage to vsdqs at low speeds
Disadvantages: Parameter sensitive, dc-drift due to
noise in electronic circuits employed, incorrect initial
values of flux vector components sdq(0)
34
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
Rotating frame (drqr) Stationary frame (dsqs)
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
-
r
NO field
weakening
(constant flux)
vsdr*
PI
vsq
r*
ejr
PI
DRFO r
Scheme tan-1
isdr
isqr
vas*
vsqs*
r
e-jr
vsd
s*
2/3
vbs*
vcs*
PWM
VSI
sds
rds
vsdqs
rqs Eq. (31) sqs Eq. (30) isdqs
m
P/2
ias
isds
isqs
3/2
ibs
ics
35
Direct Rotor Flux Orientation (DRFO) –
field weakening implementation
With field
weakening
r*
Rotating frame (drqr) Stationary frame (dsqs)
imrd r *
+
imrd
r
1
1 S r
isd
PI
r* +
r*
PI
-
+
isqr* +
vsdr*
vsqs*
PI
vsqr*
ejr
PI
-
r
tan-1
isdr
isqr
r
e-jr
vsds*
rds
rqs
r
Same as
in
slide
26 or 29
isds
isqs
36
Stator Flux Orientation Control
• d- axis of dq- rotating frame
is aligned with s. Hence,
qs
qs
is
s
isqΨs
i
ds
ψsdψs ψs
(32)
ψ 0
(33)
ψs
sq
• Therefore,
Ψs
sd
ds
Ψs
i
sq = torque producing current
isdΨs = field producing current
3P
Te
( sd isq )
22
Similar to
ia & if in
DC motor
(34)
Decoupled
torque and
flux control
37
Stator Flux Orientation Control
From the dynamic model of IM, if dq- frame rotates at general
speed g (in terms of vsd, vsq, isd, isq, ird, irq):
vsd Rs SLs
vsq g Ls
vrd
SLm
vrq ( g r ) Lm
g Ls
Rs SLs
( g r ) Lm
SLm
SLm
g Lm
Rr ' SLr
( g r ) Lr
g Lm isd
SLm
isq (7)
( g r ) Lr ird
Rr ' SLr irq
s rotates at synchronous speed s
Hence, dsqs- frame rotates at s
Therefore, g = s
These voltage equations are in terms of isd, isq, ird, irq
Better to have equations in terms of isd, isq, sd, sq
(8)
38
Stator Flux Orientation Control
• Stator flux linkage is given by: Ψsdq Lsisdq Lmirdq
• From (9):
Ψsdq L
irdq
Lm
s
Lm
isdq
(35)
(36)
• Substituting (8) and (36) into (7) gives the IM voltage
equations rotating at s in terms of vsd, vsq, isd, isq, sd, sq:
vsdψs
Rs
0
ψs
0
Rs
vsq
vrdψs Ls 1 S r
sl rLs
ψs
Ls 1 S r
vrq sl rLs
S
s
1 S r
sl r
s isdψs
S isqψs
ψs
sl r sd
ψs
1 S r sq
(37)
39
Stator Flux Orientation Control
• Since ψ 0 , hence the equations in stator flux
orientation are:
d ψs
ψs
ψs
vsd Rs isd sd
(38)
ψs
sq
dt
vsqψs Rsisqψs s sdψs
vrdψs 0 sdψs r
(39)
d ψs
d
sd Ls isdψs r isdψs sl r Ls isqψs (40)
dt
dt
d ψs
ψs
v 0 Ls isq r isq sl r sdψs Lsisdψs
dt
Important equations for
Stator Flux Orientation Control!
ψs
rq
(41)
40
Stator Flux Orientation Control
• Equation (40) can be rearranged to give:
1 S r sdψs 1 S r Lsisdψs sl r Lsisqψs
• ψ
(42)
ψs
sd
ψs
i
should be independent of torque producing current sq
ψs
ψs
ψ
ψ
• From (42), sd is proportional to isd and isqs .
ψs
ψs
ψ
• Coupling exists between sd and isq .
ψs
ψs
ψ
i
Varying sq to control torque causes change in sd
ψs
Torque will not react immediately to isq
41
Stator Flux Orientation Control
– Dynamic Decoupling
• De-coupler is required to
–
–
ψs
ψ
overcome the coupling between sd
ψs
ψ
no effect on sd )
ψs*
Provide the reference value foris d
ψs
sq
and i
(so
ψs
that sq
i
has
• Rearranging eq. (42) gives:
isdψs*
ψs*
1 sdψs*
*
S
sl isqψs*
r Ls
1
S
r
(43)
• is q can be obtained from outer speed control loop
• However, eq. (43) requiressl*
42
Stator Flux Orientation Control
– Dynamic Decoupling
•
can be obtained from (41):
*
sl
ψs*
sd
• ψ
1
S
r ψs*
*
sl ψs*
isq
sd
isdψs*
Ls
(44)
in (43) and (44) is the reference stator flux vector ψ
*
s
• Hence, equations (43) and (44) provide dynamic decoupling
ψs
of the flux-producing isψs*
and
torque-producing
isq currents.
d
43
Stator Flux Orientation Control
– Dynamic Decoupling
• Dynamic decoupling system implementation:
s*
1
Ls
S
1
r
isqs*
from speed
controller
1
+
+
S
1
r
isds*
isqs*
S
1
r
x
x
sl*
1
ψ
ψs*
isd
Ls
*
s
44
Stator Flux Orientation Control
dsqs- frame also rotates at s
For precise control, s must be
obtained at every instant in time
Leads to two types of control:
qs
qs
is
s
isqΨs
s
isdΨs
dq- reference frame
orientation angle
ds
Indirect Stator Flux
Orientation
Direct Stator Flux
Orientation
s easily estimated from motor’s
stator voltages vsdqs and stator
ds
currents isdqs
Hence, Indirect Stator Flux
Orientation scheme
unessential.
45
Direct Stator Flux Orientation (DSFO) implementation
Closed-loop implementation:
1. Obtain isds* from s control loop and dynamic
decoupling system shown in slide 38.
Obtain isqs* from outer speed control loop since
isqr* Te* based on (34):
*
Te
3P
i ψs* where kt
(45)
kt isd
22
Obtain vsdqs* from isdqs* via inner current control
loop.
ψs*
sq
46
Direct Stator Flux Orientation (DSFO) implementation
Closed-loop implementation:
2. Determine the angular position s using:
ψ tan
s
s
sq
1
(46)
sd s
sds and sqs obtained from stator voltage equations:
sdq vsdq Rs isdq sdq 0
s
Note that:
s
s
s2
ψs sd sq
s
s2
(47)
(48)
Eq. (48) will be used as feedback for the s control loop
47
Direct Stator Flux Orientation (DSFO) implementation
Closed-loop implementation:
3. s to be used in the dsqs dsqs conversion of
stator voltage (i.e. vsdqs* to vsdqs* concersion).
s estimated from pure integration of motor’s stator voltages
equations eq. (47) which has disadvantages of:
dc-drift due to noise in electronic circuits employed
incorrect initial values of flux vector components sdqs(0)
Solution: A low-pass filter can be used to replace the pure
integrator and avoid the problems above.
48
Direct Stator Flux Orientation (DSFO) implementation
r
r* +
s*
isqs*
+
-
PI
-
Decoupling
system
+ i s*
sd
1
+
PI
- | |
S
vsqs*
vsds*
1
r
PI
s
tan-1
isqs
s
Eq. (48)
sds sqs
ejs
isds
s
e-js
vsds*
sds
2/3
PWM
VSI
vbs*
vcs*
vsdqs
Eq. (47) isdqs
sqs
ias
isqs
isds
m
vas*
vsqs*
PI
+
-
+
P/2
3/2
ibs
ics
Rotating frame (dsqs ) Stationary frame (dsqs )
49
References
• Trzynadlowski, A. M., Control of Induction Motors, Academic
Press, San Diego, 2001.
• Krishnan, R., Electric Motor Drives: Modeling, Analysis and
Control, Prentice-Hall, New Jersey, 2001.
• Bose, B. K., Modern Power Electronics and AC drives, PrenticeHall, New Jersey, 2002.
• Asher, G.M, Vector Control of Induction Motor Course Notes,
University of Nottingham, UK, 2002.
50