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Advanced AC Drives
Induction Motor Drives
Dr. M. Galea
Lecturer in Electrical Machines and Drives
Department of Electrical and Electronic Engineering
Part II
Concepts in Vector Control
2.1 Introducing the High Performance Drive
2.2 Concept of vectors
2.3 Vectors in a DC machine
2.4 Vector in an Induction machine
2.5 Stationary and Rotating Frames
Some definitions
A motor drive in which the motor torque obeys a torque demand within a few ms is called a torque controlled drive
An IM driven by a V-f PWM converter is not a torque controlled drive.
All the following are torque controlled drives Industry name Control Engineers name
Drive designed to go from one speed to another (or from one
position to another) as quickly as possible
Speed/position
Servo drive Servo drive
Drive designed to go from one torque to another as quickly
as possible Torque servo Servo drive
Drive designed to follow a speed or position trajectory Servo drive Tracking drive
Drive designed to follow a fixed speed Servo drive Regulator drive
Servo drives are often called high performance drives
Introduction the high performance drive
winHighlight
winHighlight
Examples of speed and position servo drives are
- machine tools speed and positioning (lathes, milling, laser cutting)
- Robots, object positioning systems
Examples of torque servo drives are
- drives for railways, ships, electric cars etc
- dynamometers (programmable torque loads for testing)
Examples of tracking and regulator drives are
- paper feeds, paper mills, coil winding, conveyors
- object position tracking (radar, telescopes, robots)
Examples of high performance drives
Traditionally, all torque controlled drives have been DC machines
- DC machine torque can be controlled directly and effectively
- AC machines traditionally steady-state AC machines driven from mains
AC vector controlled IM drives replacing DC drives for all high performance applications
- induction motor simpler, less expensive
- induction motor low maintenance (no brushes)
- capable of faster torque response
- capable of faster speed response due to lower rotor inertia designs
Cost of AC-AC PWM power converters more than AC-DC thyristor converters but silicon products reducing in price
- costs recovered through cheaper machine, maintenance, higher performance
High performance induction drives - 1
winHighlight
Vector drives need much higher P power, but now not a problem
- Now so cheap, that vector controlled drives even replacing V-f PWM drives
- Can run machine more efficiently (flux and torque control means lower losses), vector
drives common for large MW drives, even if not servo drives
- At lower powers, Manufacturers offer universal drives, for V-f, vector IM, vector PM etc
- Only extra cost is that of one or two extra current transducers (preferable to measure
all three phases)
What is main competition to the vector controlled IM?
- Permanent magnet machine has better torque density (torque:volume)
- e.g. PM drives in automotive and aerospace
- PM cost disadvantage (magnets!) reduces at lower power, so common at
Concept of a (space) Vector
Vector has magnitude and direction
The voltage vector V, the current vector I, flux vector is associated with a coil (actual coil or fictitious coil)
The instantaneous magnitude of the voltage/current/flux vector is the instantaneous size of the V, I, in the coil
The direction of the V, I, is that of the orientation of the coil; or the mmf direction set up by the current in the coil
A
A
A A
A
A
Voltage vector Va
(or current vector Ia or flux
vector a )
Voltage vector Vb
(or current vector Ib or flux
vector b )
Voltage vector Vc
(or current vector Ic or flux
vector c )
Concept of a Vector
A
A
A A
A
A
The size of the vectors V, I and will
vary with time
In steady state Va, Ia and a are AC
quantities
Eg. sinusoidal a shown below
A 4 or 6 etc pole machine:
The vector direction is along the axis of ONE OF THE POLES
The angle of all the vectors will be in ELECTRICAL radians, so we are always working with an equivalent 2-pole machine
The vector direction for e.g. Va, Ia and a in a 4-pole machine is shown
Difference between a vector and a phasor
A phasor relates only to steady state quantities
- phasor magnitude is the rms value of the SS sinusoid
- phasor direction is arbitrary, the angle between the phasors represents the phase
difference between sinusoids
A vector is in direction of mmf direction of coil
- This defines direction of voltage
across and current through coil
- Magnitude is size of voltage, current
etc
C:\ac vector drives\vector ill
Vector in DC machine and Torque production
If
B or If B
ia
ia
B or If or
f
ia
f = Lf If so that field flux and field current in the same direction
fafa
iii kkTm
famfamiiksiniikT
In terms of vectors, torque defined as:
Angle between ia and if kept at 90 by commutator action
Vector in DC machine and Torque production
DC Machine - Commutator action
f = Lf If so that field flux and field current in the same direction
fafa
iii kkTm
famfamiiksiniikT
In terms of vectors, torque defined as:
Angle between ia and if kept at 90 by commutator action
C:\acdrives\imd0044a.swf
Where is a flux in the Induction machine: consider
)(kTs
i
Induction machine equivalence to DC Machine
fafa
iii kkTm
For DC machine, we have: For Induction machine, we have: i
skT
Total B or
in machine Total flux in machine (no leakage)
Stator currents IS
Rotor currents IR
Flux S is the total field enveloping (linking) the stator windings
Flux R is the total field enveloping (linking) the rotor windings
Flux O is the total field crossing the airgap
If no leakage, then S = O = R =
In practice, S > O > R
Expression assumes no leakage
WARNING!
Total B or
in machine Total flux in machine (no leakage)
Stator currents IS
Rotor currents IR
Flux S is the total field enveloping (linking) the stator windings
It is NOT the flux due to the stator currents.
It is the sum of the fluxes (or fields) due to the stator AND rotor
currents: S = Lsis+Loir
Flux r is the total field enveloping (linking) the rotor windings
It is NOT the flux due to the rotor currents.
It is the sum of the fluxes (or fields) due to the stator AND rotor
currents: r = Lois+Lrir
If no leakage then Lo= L s = Lr and S = R =
Recall IM fields
Total field and field due to rotor and stator currents
C:\acdrives\elc0022.swf
Recall IM fields
Total field and field due to rotor and stator currents
C:\acdrives\elc0023.swf
Field due to stator
currents alone
Total Flux
= S = R
Field due to rotor
currents alone
Stator current
IS
Position of maximum mmf due
to Is - The is vector
Total
Position of
maximum
rotor current
Position of
maximum
stator current
Position of maximum mmf due to IR
The iR vector
Recall IM fields
Total field and field due to rotor and stator currents
sinik)(kTss
i dsqkiT
Fix d (direct) on total flux as shown
The dq axis exists as a concept in the control P
isq known as the torque current
isd known as the field producing current
All vectors, is , iR , rotate at e relative to the stator
Remember iR rotates at sl relative to rotor; rotor rotates at R relative to stator
Total = d
Position of iRq current iR vector = irq
iRd = 0
d axis
q axis
isq
isd
Position of isq current
Position of isd current
Torque and rotor and stator current vectors
Have leakage must choose a flux to fix dq axis on
For reasons to be shown, we select the ROTOR flux as the direction of the direct axis This is called Rotor Flux Orientation (RFO)
Can choose Stator Flux Orientation (SFO)
isq plays the same role as Ia in a DC machine
isd plays the same role as If in a DC machine
Induction machine flux definitions
N
S e
S
R
O
(a)
(a) No leakage: magnitudes equal S = O = R ; all point in same direction and
all rotate together
(b) Leakage: all rotate together, but magnitudes and directions slightly different;
situation shown is that of motoring
e
S
R
O
(b)
d axis (RFO)
q axis (RFO)
Remember: All vectors, is , iR , r rotate at e = d/dt relative to the stator
dq axis now fixed on rotor flux vector - will rotate at e = d/dt
Angle is the instantaneous angle between peak of rotor flux and the phase AA. It is called the rotor flux angle
isq is the torque current, and isd is the field producing current
Total r
d axis
q axis
isq isd
Summary of Rotor Flux Orientation
Stationary and rotating frames Mmf and vectors due to individual 3 phases
C:\acdrives\3phabc.avi
Red vector is voltage (or current or flux) due to phase A
Blue/yellow vector for phase B and C respectively
Stationary and rotating frames Mmf and vectors due to all 3-phases
Add blue and red together
Add in yellow vector
Note resultant is 1.5 times peak of phase vector
C:\acdrives\3phabc+.avi
Stationary and rotating frames Equivalent 2-phase system
Take resultant from the three phases
Same vector (mmf, flux etc) can be obtained from 2-phase motor
Can imagine IM with two phases , wound 90 apart and carrying currents, also 90 apart in time
C:\acdrives\2ph.avi
+A -A
+B
-B +C
-C
ia ib
t= 0 t= t1
ic
caii
cbaiii
At t=0: ia = 1A, ib = -0.5A, ic = -0.5A
Adding these together gives the current vector i
Resultant can be written
i= i+ ji where i = 1.5; i = +0
1a
i
5.0b
i
5.0c
i
5.0
5.0
1
c
b
a
i
i
i
+ -A
+
cbaiii
0
5.1
i
i
Stationary and rotating frames Going from 3-phase to 2-axis frame
-A
+B
-B +C
-C
ia ib
t= 0 t= t1
ic At t=t1 : ia = 0A, ib = 0.866A, ic = -0.866A
Adding these together gives the current vector i
Resultant can be written
i= i+ ji where i = 0; i = +1.5
cbaiii
866.0b
i 866.0c
i
87.0
87.0
0
c
b
a
i
i
i
+A + -A
+
5.1
0
i
i
Stationary and rotating frames Going from 3-phase to 2-axis frame
-A
+B
-B +C
-C
aii
i
+A +
+
aii
120
120cosb
i
bi
2
1
240
240cosc
i
ci
2
1
aii
2
3
cbaiii
-A
+B
-B +C
-C
+A
+
120
120sinb
i
bi
2
3
240
240sinc
i
ci
2
3
i
c
b
a
i
i
i
i
i
2
3
2
30
002
3
i
i
i
i
i
c
b
a
3
1
3
1
3
1
3
1
03
2
Stationary and rotating frames Going from 3-phase to 2-axis frame
In steady state motor operation, is , is are sinusoidal and 90 apart
For a given set of 3 phase currents, one can always find the equivalent 2-phase currents
(in a fictitious 2-phase winding) to give the same
mmf and flux conditions as the 3 phase currents
The flux linking the winding will be , . Similarly for voltages etc
Have stator winding and rotor winding
v , i,
C
B
A
v , i,
Have
240cos120cos
cbaiiii
240sin120sin
cbiii
3/43/2
)240sin240(cos)120sin120(cos
j
c
j
ba
cba
eieii
jijiijii
c
b
a
i
i
i
i
i
2
3
2
30
002
3
For a three phase system, this is the same as: 0 cba iii
Summary Going from 3-phase to 2-axis frame
Stationary and rotating frames Rotating dq axis frame
Look at the projection of the current vector onto two axis (at 90 degrees!) which are rotating at the same speed as all the vectors
Call these axis d and q. The components of the current vector on these two axes will have constant values in steady state. As shown below
But the dq axis is placed at an arbitrary angle to the rotor flux
Therefore the dq components of is dont mean anything
t= t2
i
t= t2 t= t1
i
isq= 3.6
is
r
d
isd= 3.8
d
q
isq= 3.6
is
r
t= t1
isd= 3.8
d
q
isd = 3.8A
isq = 3.6A
isd = 3.8A
isq = 3.6A
Stationary and rotating frames Rotating dq axis frame
Look at the projection of the current vector onto two axis (at 90 degrees!) which are rotating at the same speed as all the vectors
Call these axis d and q. The components of the current vector on these two axes will have constant values in steady state. As shown below
The dq axis is now placed with the d-axis on the rotor flux vector
The dq components of is now mean the field (producing) and torque currents
t= t2
i
t= t2 t= t1
i isq= 2.6
is
r
d
isd= 4.0
d
q
isq= 2.6
is
r
t= t1
isd= 3.8
d
q
isd = 4.0A
isq = 2.6A
isd = 4.0A
isq = 2.6A
isd= 4.0
Stationary and rotating frames 2-axis frame fixed to stator
Rotating current and flux in fixed reference frame
Vector components are sinusoidal
C:\acdrives\rotfield1.avi
Stationary and rotating frames Dq rotating frame
Dq rotating frame
Vector components now dc values, but not field orientated values
C:\acdrives\rotfield2.avi
Stationary and rotating frames Dq rotating frame
Dq rotating frame
Vector components now dc values, and field orientated values
C:\acdrives\rotfield3.avi
+
+
d
q
id
iq
i
i
cosi
sini
iq
i
i
+
+
d
q
id
d
i q
i
sini
cosi
i
i
i
i
q
d
cossin
sincos
jj
dq
qd
eejii
jjijijii
ii
)sin(cos)sin(cos
Stationary and rotating frames Transforming from to dq rotating frame
Magnitude of the stator current vector can be derived from the magnitudes of the
phase currents flowing in the a,b,c phases
is rotating and shown at two instances of time, the first instance has is is zero. The
vector is shown being resolved in to and components
acba
iiiii2
3
2
1
cbiii
2
3
2
3
)3/4cos(~
2
)3/2cos(~
2
cos~
2
tIi
tIi
tIi
Sc
Sb
Sa
ib
t= 0
ia ic
0
~2
2
3
i
IiS
At t=0,
2/~
2)3/4cos(~
2
2/~
2)3/2cos(~
2
~2
SSc
SSb
Sa
IIi
IIi
Ii
is= 0
Magnitude of current vector: the 3/2 times peak convention
Magnitude of the stator current vector
is rotating and shown at two instances of time, move to t = t1, (/2 radians later)
acba
iiiii2
3
2
1
cbiii
2
3
2
3
)3/4cos(~
2
)3/2cos(~
2
cos~
2
tIi
tIi
tIi
Sc
Sb
Saib
t= 0
ia
ic
t= t1
phphphIIIii
i
~2
2
3
2
3~2
2
3
2
3~2
2
3
0
At t=t1, /2 later
6
5cos
~2
3
4
2cos
~2
6cos
~2
3
2
2cos
~2
0
phphc
phphbb
a
IIi
IIii
i
is= 0
Magnitude of current vector: the 3/2 times peak convention
Magnitude of current vector: the 3/2 times peak convention
Evidently, the magnitude of
i.e. 3/2 times the peak of the phase current
SsIi~
22
3
ib ia ic
si
As sweep through time, current vector rotates in space
Can also project current vector onto d-axis
No matter what frame we define the co-ordinates, magnitude of vector is always
SdqsssIiii~
22
3
__
si
si
id
d
q
iq
si
Stationary and rotating frames phs X~
x 22
3
sdsq
r
iiL
LPT
2
0
23
2
The torque is:
The scaling of the transformation is arbitrary: FOUR conventions are in use in the world today.
The magnitude of the current vector is 3/2 x peak of the phase stator current
Called the 3/2 times peak convention
The voltage vector will also be 3/2 x peak of the phase stator voltage
phsX~
x 22
3 For any vector:
For alternative rms convention see Worked Example 2
To calculate the rated values of is , isd, isq etc see Worked Example 2
Transforming from 3 phase to 2 phase
Transforming from 2-phase to dq
3/2
is
is
isa
isb
isc
isd
isq
is
is
je
)(2
3)( titi sas
)(2
3)(
2
3)( tititi scsbs
3
4
3
2
0
j
c
j
b
j
aeieieijii i
First transform measured 3-phase currents into 2-phase currents
Then transform 2-phase currents into dq currents
sin)(cos)()( tititisssd
cos)(sin)()( tititisssq
Numbers are for 3/2 times peak convention. For rms x all by 2/3
j
e
is
is
isd
isq
2/3
is
is
isa
isb
isc
)(3
2)( titi ssa
)(
3
1)(
3
1)( tititi sssb
)(
3
1)(
3
1)( tititi sssc
sin)(cos)()( tititi sqsds
cos)(sin)()( tititisqsds
Can transform from dq currents into 2-phase currents
Inverse transformations
Can transform 2-phase currents into 3-phase currents
Numbers are for 3/2 times peak convention. For rms x all by 3/2
j
e
vs
vs
vsd
vsq
2/3
vs
vs
vsa
vsb
vsc
)t(v)t(vssa
3
2
)t(v)t(v)t(vsssb
3
1
3
1
)t(v)t(v)t(vsssc
3
1
3
1
sin)t(vcos)t(v)t(vsqsds
cos)t(vsin)t(v)t(vsqsds
ALL transformations can be applied to voltages and fluxes etc
e.g. from dq voltages into 2-phase voltages:
Inverse transformations
And from 2-phase voltages into 3-phase voltages:
Fundamental structure of vector control
All vector controllers first transform measured currents to dq domain
- Measured voltages can be transformed to dq if necessary (not shown below)
Vector controller controls the currents in the dq domain and outputs dq voltage demands Voltage demands are inversed transformed into 3-phase demand voltages for PWM The transformations need the angle at every point in time
vs*
vs*
isd
isq
3/2
is
is
isa
isb
isc
j
e
vsd*
vsq*
j
e
2/3
v*sabc
PWM
IM
Vector
Controller
Microprocessor Vector controller needs to calculate
DIRECT VECTOR CONTROL - in which the rotor flux angle is derived from measured stator
voltages and currents INDIRECT VECTOR CONTROL - in which is derived from the vector controlled constraint equation
Finding the Rotor Flux angle
r
d
q
)()( tdt
dt
e
dttt e )()(
dq axis frame rotates at instantaneous speed e:
Part III
Dynamic Equation of Cage IM
3.1 Dynamic Equations for Cage IMs
3.2 Vector Control Equations
3.3 Equivalence to a DC machine
3.4 Slip gain and Motor Torque
The 3 stator coils A,B,C can be represented by TWO stationary coils
Each stationary coil has resistance and inductance and:
ssss
dt
dRiv ssss
dt
dRiv
ssssdt
dRiv
The 3 rotor coils A,B,C can be represented by TWO moving coils
Each moving coil has resistance and inductance and:
rsrr
dt
dRiv
rsrrdt
dRiv
rsrrdt
dRiv
These are the equations of coils moving round at rotor speed
Dynamic Equation of Cage IM - Introduction
S
S
R
R
R
Dynamic Equation of Cage IM Stator
ssss
dt
dRiv
s
q
e
xd xq
x
d
s
sqesdssdsddt
dRiv
sdesqssqsqdt
dRiv
j
sdqe
jsdq
s
j
sdq
j
sdq
j
sdqs
j
sdq
j
sdq
ejedt
dReiev
edt
dReiev
sj
ssdqevv
sj
ssdqeii
sj
ssdq e
sj
sdqsevv
sj
sdqseii
sj
sdqse
sdqe
sdq
ssdqsdqj
dt
dRiv
The equations of the TWO coils rotating at r can be written in a coordinate frame rotating with e
Dynamic Equation of Cage IM Rotor
0 rrr
dt
dRi
rqslrdsrddt
dRi 0
rdslrqsrqdt
dRi 0
The equations of the TWO coils rotating at r can be written in a coordinate frame rotating with e
rdqslrdqsrdqj
dt
dRi 0
r
r
q
-r
e
xd xq
x
d
r
r
)r(j
rrdqevv
)r(j
rrdqeii
)r(j
rrdq e
)r(j
rdqrevv
)r(j
rdqreii
)r(j
rdqr e
The stator and rotor fluxes of the two coils in the dq rotating frame are:
rdosdssdiLiL
rqosqssqiLiL
rdrsdordiLiL
rqrsqorqiLiL
These can be used to eliminate ird, irq,sd ,sq
The stator equations
Dynamic Equation of Cage IM contd
rq
r
o
erd
r
o
sqsesdssdssdL
L
dt
d
L
LiLi
dt
dLiRv
rd
r
o
erq
r
o
sdsesqssqssqL
L
dt
d
L
LiLi
dt
dLiRv
rdq
r
o
erdq
r
o
sdqsesdssdqssdq
L
Lj
dt
d
L
LiLji
dt
dLiRv In terms of is and r :
sdqe
sdq
ssdqsdqj
dt
dRiv
Had:
Comparing real and
imaginary terms :
These are the stator equations in a rotating dq axis frame
The stator and rotor fluxes of the two coils in the dq rotating frame are:
rdosdssdiLiL
rqosqssqiLiL
rdrsdordiLiL
rqrsqorqiLiL
These can be used to eliminate ird, irq,sd ,sq
rqslrdsdr
r
o
rd
r
r
dt
diR
L
L
L
R0
The rotor equations
Dynamic Equation of Cage IM contd
rdqslrdqsrdqj
dt
dRi 0
rdslrqsqr
r
o
rq
r
r
dt
diR
L
L
L
R0
rdqslrdqsdqr
r
o
rdq
r
rj
dt
diR
L
L
L
R0
Had:
In terms of is and r :
Comparing real and
imaginary terms :
These are the rotor equations in a rotating dq axis frame
Consider equations in dq rotating frame orientated on rotor flux:
Dynamic Equation of Cage IM
rd
r
o
sqsesdssdssddt
d
L
LiLi
dt
dLiRv
rd
r
o
esdsesqssqssqL
LiLi
dt
dLiRv
The stator equations under ROTOR FLUX ORIENTATION (RFO)
is
r
d
q
rq= ?
rq= 0!!
Hence all rq terms go zero
Consider equations in dq rotating frame orientated on rotor flux:
Dynamic Equation of Cage IM
rdsdr
r
o
rd
r
r
dt
diR
L
L
L
R0
The rotor equations under ROTOR FLUX ORIENTATION (RFO):
rdslsqr
r
oiR
L
L0
is
r
d
q
rq= 0!!
Hence all rq terms go zero
The vector control equations (RFO)
rdsdr
r
o
rd
r
r
dt
diR
L
L
L
R0
Consider rotor equations under ROTOR FLUX ORIENTATION (RFO)
rdslsqr
r
oiR
L
L0
mrdosdr
r
o
mrdo
r
ri
dt
dLiR
L
LiL
L
R0 sdmrdmrd
r
riii
dt
d
R
L
mrdoslsqr
r
oiLiR
L
L0 sq
mrdr
r
sli
iL
R
mrdordiLDefine the magnetising current imrd as:
These are often called the vector control equations
Equivalence to DC machine
Consider the d-axis vector control equation: sdmrdmrd
r
riii
dt
d
R
L
fffffi
dt
dLiRV
f
f
ff
f
f
R
Vii
dt
d
R
L
Comparing terms we see that: fmrd ii ffsd R/Vi
Lf
ia
if
Vf
Rf
In steady state i.e. under constant field conditions sdmrd
ii
isd is really the field forcing current, but is commonly called the field current
Be aware of difference between isd and imrd under TRANSIENT conditions
Slip gain and motor torque
Consider the q-axis vector control equation:
The value of k depends on the convention used for the 3-2 (3-phase to transformation) scaling:
sq
mrdr
r
sli
iL
R
sq
mrdr
r
sli
iL
R
sqslsq
mrdr
sliki
i
1
This is true IF and ONLY IF we are orientated on the rotor flux axis
Slip is directly proportional to the torque component of stator current
The slip gain ksl is depends on field level and the rotor time constant r
The motor torque was:
rdsqkiT
mrdordiLAnd if mrdsqo iikLT
r
o
L
LPk
23For the rms
convention: r
o
L
LPk
23
2
For the 3/2 x peak convention: