AC Drives IMs Part2 3

Advanced AC Drives

Induction Motor Drives

Dr. M. Galea

Lecturer in Electrical Machines and Drives

[email protected]

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

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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

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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


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


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


-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


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


Stationary and rotating frames Dq rotating frame

Dq rotating frame

Vector components now dc values, and field orientated values


+

+

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



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:


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:


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:

Documents

AC Drives IMs Part2 3