Werner Leonhard Control of Electrical Drives

180

10.

Symmetri

caI

Three-Phase

AC Machines

1

+

O'

_ 1 -

O'

1 -

~

~l

Is

=

Iso

[

2

O'

2

O'

1 +J

sp

1+0'

_1-0'

e-j2arctans

/

sp],

=

Iso

[

(10.73)

20' 20'

where

Il.

s

Iso

= j

Wl

Ls

(10.74)

is

the

ideal no load

current

valid for S =

O,

i.e. when

the

rotor

moves syn-

chronously

with

the

revolving

stator

field.

When

no speed difference exists,

no

rotor

current

will be induced

and

there

is no torque; hence

the

no-load

current

is a purely reactive magnetising current.

This

is in agreement with

the

equivalent circuits in Fig. 10.7 for S =

O.

Active current

(motoring)

!,Is

.I.

S

.

/"

Sp

/

- J

so

2

cr

,

JSD

2

cr

1

,

~

1-

cr

e -

j2

are

ta{

Sp

1

Js

cr

JSD

Reactive current

JSD

1+

cr

(ínductíve)

-1

Fig.

10.8.

Locus of

stator

current

with

stator

resistance neglected,

circIe

diagram

for

an

induction

motor

The

complex locus

Is

(S/

Sp) is depicted in Fig.

10

.8 with

the

voltage

phasor

assumed imaginary, U5 =

jUs,

in order to place

th

e circle in

the

customary

orientation.

The

curve covers

the

fllll

rall~

C

"00

<

S/Sp

< 00;

for S > O

the

motor

runs

below

synchrolli~IIJ

,

drawillg acl.ive power, while

for

S < O

iI,

op(

~

ral.(~s

as a gcnerat,ol' alld

Sl1ppli('

/ol

1\cl.iv(

~

pow(

~

r

to

t1w

lilln.

Reactiv(

~

pown

l'

ror lllil

(f,

lwtisatioll is

il.lwa

Y

~

1

11(.,..11'11 whi('h ,i!l n

dmr

a

('.I.(~ri!;tic

of

illdllt'l.ioll

1I

\f

I

Ii

Il

('1i

Jlil.vil

lj

(

110

illl."l'IlII.J v,,JI.

II

II.

'·

11

11111'("( '

whi('h

ir,

pn

·r;

(·IJI.

iII

HY

III·h,.IIl1

ll

il

Il

lhe

lllll(

'"

. til. S

II

1.1".

1'1

111"'11

1.

HIi!!

" " " '

!1

il.

rl

II

d

llillllllll

vIL

III"

10.2

Induction

Motor

in

Steady

State

181

representing

(for

Rs

=

O)

the

no-load

magnetising

current

as

mentioned

before.

Minimum

phase shift

cp

= arg(U5,1

5

),

i.e.

maximum

power factor is

reached when

the

current

phasor

forms a

tangent

to

the

circle.

The

optimal

power factor

then

follows from

the

geometrical relations of Fig. 10.8,

1-0'

COS(

() =

-_.

(10.75)

rn

1 +

O'

'

this is usually defined as

the

rated

or

nominal

operating

point.

The

pertinent

value of

Sn/

Sp

is

obtained

from

the

phase angle,

Eq

. (10.72),

7r 1 S S

cp

=

--

+

arctan

- - -

arctan-

(10.76)

2

O'

Sp Sp

by differentiation,

d(trSp)

=

O;

the

result is

~n

=

VU.

(10.77)

p

Rence,

not

only

the

geometry

of

the

current

locus,

but

also

the

maximum

power factor,

the

pull-out slip

and

the

normalised slip

at

the

nominal

oper-

ating

point

are all

determined

by

the

leakage fac

tor

0'.

The

magnitude

of

the

normalised

stator

current,

Is

V1+(~ff;r

(10.78)

Iso

VI

+

(ir

is

an

even function

of

slip, as seen in Fig. 10.9 a.

The

normal

operating

region

comprises only

the

central

portion

of

the

curve.

At

larger values of slip, for

instance

when

starting

the

motor

at

S = 1,

the

current

rises sharply so

that

the

simplifications employed become questionable; this

is

particlllarly

true

for

the

iron

saturation

that

is

not

negligible

at

high currents.

The

stator

current

at

nominal slip,

Sn/

Sp

=

y'ci,

is

IS

n

1

(10.

7!l)

Iso =

y'ci

,

while

the

asymptotic

value

of

the

current

is

Is

1

fim -

(lO.

BO)

5-400

Iso

O'

Tllis ull(krlillCS I.he

importance

of

lhe

feakage factor

O'

as a design

pararnd(

~

r.

'I'lw III<\ X

illllllll

;tJl(.l

ruilLimulU

val!i(':;

"f

til"

ac:t.i

ve

current

anel

acl.i

v,~

pO\V('r, ;1,S;;

lIl1lilll'.

Ii

..: ('OIlSt..,

;t

J'('

1'1'11..-111'11

H,

1.

S 1''''1''

i.".

in

tlw

v(

~

rl.i(

:

(·

n

(lI'

1,111'

..

ird,·.

I"

LIli'

III'

X

I.

r

ó(

·(·

I.j"lI

íl. 1:1 !

.h"wlI

I.h:d

.

LlI<'

:'w

a.re

al ~

()

1.11('

poilll.:.1

wl"'I"

1.111'

1."1

ii""

11

:.

/1

111111'11

iI

,

1I

P

('

I

t!

( vr

dll'·

:.

182

10.

Symmetrical Three-Phase

AC

Machines

~

ISO

~~--------------I1

-

---------------;~

--',

1

//;-

,---

/

RS=O

(J

=0,1

~

ISO

I I

-2

-1

2

S

-~

~

Sp

a

Sp Sp

,

mM

m

po

úJ,m

M

\

1-----

....

__

I

úJ

1

_

U

s

= 1

I

úJ

o

-

Uso

m

n

:

Motor

m

po

I

I I

-2

-1

2

S

Generator

Sp

------------'

-1

b

Fig.

10.9.

Stator current and torque

as

a function of slip

(Rs

=

O)

10.2.2

Steady

State

Torque,

Efliciency

An

expression for

the

steady

state

torque

is

obtained

by inserting

the

vec-

tors

for sinusoidal

symmetrical

currents,

Eqs. (10.62, 10.65),

into

the

general

formula for

the

torque,

Eq

. (10.48),

mM

= 3 Lo 1m (Is

IR)

,

(10.81)

yielding a

constant

value, as expected.

From

Fig. 10.7 a

the

following

relation

between

stator-

and

rotor-current

is derived,

-j

w}

Lo I

(10.82)

I L

-s,

-R=

RR

I

S+jw}

R

which leads

to

' j

wl

I,o I.

mM

(10.8:3)

:\

r'lI

l

,~;

TIll

[

ali

IS

.i

W I [,1/

Wil.li

1.111'

111'11'

"I

1-:'1

,

(I

(J

,'

tH)

I,liil: ('

11 11

1"

,

)"',111

11

111'

,

""

L..

10.2

Induction Motor

in

Steady State 183

3 1 -

(J

U~

2 2

mM

=

-------

=m

p

---

(10.84)

2

(J

w?

Ls

SISp +

SpIS

SISp +

SpIS

'

where

mp

represents

the

peak

torque,

available

at

the

pull-out slip S = ±Sp.

The

function

of

torque

vs,

normalised

slip is

plotted

in

Fig. 10.9 b.

When

the

induction

motor

is

part

of

a variable

speed

drive,

the

stator

voltage U s

and

frequency

w}

may

differ from

their

nominal

values

Uso,

Wo

shown on

the

nameplate;

hence

the

following

extension

of Eq. (10.84) applies

Uslw}

2

mM

= mpo

(10.85)

(

Usolwo

)

SIS

p

+

SpIS

'

where

3 1 -

(J

U~o

mo=-------

(10.86)

p 2

w5

Ls

(J

is

the

maximum

torque

(for Rs =

O)

at

nominal

volt age

and

frequency.

This

indicates

that

the

torque

rises

with

the

square

of

the

flux;

the

reason is of

course

that

the

rotor

current,

being

caused

by

magnetic

induction,

is also

depending

on

fluxo

The

torque

vs. slip

curve

seen

in

Fig. 10.9 b

may

be

qualitatively

inter-

preted

as

follows:

When

the

rotor

speed

deviates

from

the

angular

velo city

w}

of

the

stator

field,

currents

are

induced

in

the

rotor

winding

producing

a

torque

with

the

tendency

of

reducing

the

speed

difference. Below synchronism

(S

>

O)

the

torque

acts

in

the

direction

of

the

rotating

field, i.e.

the

machine

operates

be-

tween

standstill

and

synchronous speed

as

motor;

above

synchronous speed,

the

torque

is opposed

to

the

direction of

rotation,

hence

the

machine

acts

as

generator,

feeding electrical power

to

the

line.

Steady

state

supersynchronous

speed

can

thus

only

occur

if

the

machine is mechanically driven. Because of

the

reactive power

problem,

induction

generators

are

rarely used

except

in

small

unattended

hydro

or

wind

power

stations,

but

temporary

regeneration

is a

normal

feature

of controlled AC drives.

The

occurrence

of

a

peak

torque

can

be

explained

by a simplified

model

of

the

rotor

circuit as shown

in

Fig. 10.10.

The

rotor

may

be

represented

by

a

R-L-circuit

that

is

supplied

from a voltage source,

the

amplitude

U

2

of

which is

proportional

to

frequency

W2.

The

current

phasor

U

~

20

Wo

U

20

1

----

(10,87)

I

2

= R

2

+ j

W2

L

2

jwOL2

1-j

~

W2

L2

as a

function

of

frequency

W2

is described by

the

semicircle shown in Fig.

10.10 b,

While

the

magnitude

h rises

asymptotically

with

W2,

the

active

compon

e

nl.

(}~

:

o

1

H,

('

/

'.1

(IO,RR)

II

~

ti/

:l

l

i

~

ii

.!

" I

J'

.~

101

.. 1, .

.1

/I

"

184

10.

Symmetrica!

Three-Phase

AC Machines

U

20

Re

'2

J

2

L

2

U

2

0~O

W

2

I

w

2

L

2

W

o

I

U

20

~

.R

2

j W

O

L2

a

b

C

Fi

g.

10.10.

Simplified mode!

of

rotor

circuito

(

a)

Circuitj

(h)

Locus of

current

phasorj

(

c)

Real

current

component

passes

through

a

peak

value;

the

curve,

plotted

in Fig. 10.

10

c,

has a similar

s

hape

as

the

torque characteristic in Fig. 10.9 b.

This points

to

the

reason for

the

decreasing

torque

of

the

induction mo-

tor

ato

higher slip frequency: Due

to

the

rotor

leakage,

the

rotor

current

is

increasingly lagging, causing

the

rotor

current wave

to

be shifted

and

eventu-

ally become orthogonal

to

the

stator

flux wave. Hence

the

torque

is

reduced

whereas

the

stator

and

rotor

currents continue

to

increase asymptotically.

TIl

view of

the

stability of

the

drive

and

the

rising losses,

steady

state

opnat.ion

of

the

motor

makes only sense below pull-out slip.

WhCll declaring

the

point, where

the

power factor

is

optimal, as

the

nom-

ill:!.1

op,'rat.ing point, Sn/Sp = fo,

the

ratio

of

the

peak

to

nominal torque

I"

'111

'

1.

0

_

1 +

(7

(10.89)

'

1I1

'

JI

- 2

fo

'

whidl

is a measure for

the

overload capacity

of

the

motor.

The

pull-out

I"rqllv

,

Eq

, (10.86), shows

that

the

rotor

resistance has no effect on

the

11I;\

,xiUIIIIll torque

but

only influences

the

pull-out slip. A wound

rotor

motor

wi

Ih

,

~xt:e

rnal

resistors in

the

rotor circuit

then

offers a possibility of modifying

lhe

torqllc

/speed

characteristics by changing

the

pull-out slip

Sp

through

the

rot.or

rc

sist.anc

e.

The

resulting family of curves is depicted in Fig. 10.

ll

in

the

;\,n :

ll

stoIllcd orientation.

ln

combination with

the

dashed curves for inverse

rol.aÜOll

of

th

e field by interchanging two

stator

terminais a fairly comple

te

wvnag<

~

of

the

four

quadrants

in

the

torque/

speed plane is achieved, albeit

1.\1('

s

liape

of S

OlllC

eurV($ is unfavourable.

Lowcrilll-!;

the

st.at.

or voltages, for

iIl

S

t,"'Il(,(

~

I.hwll

f.':

1t

a

tap

cltanl'-(illg

trans[oollcr,

l"(~dIlCCI

;

Lhe

peak

torqll

c by the

::

(\,,<1.1'"

0

1'

l,b

<'

voll.a!';

,'

\'(~dIlCl.i()lI, f\lrt.ll<~r

('olllpr('::: :

illl';

1.11('

(",IlJ'V(~S

itt

bori

7.

Olll.al

"íI"

'('

I.

LO

II,

'1'11('

IlIn

i

ll

dI/

tiNI

,"

,

(,1,

01'

(·Olll.íIlIIOIl

!:

11

(

1("

'" '·"lI

l.

lt" I.y

rol.o!'

}"('SíH

I.O!'

K

(lI'

I',V l'q

l\I"III'i'

,

til"

H\ l tI,1i

1'

vnl

i'.H

I'."

t' b UI!'

<I"I

" "!,

'II

' 1

,1

d

ll<"i

"II"

'y

,

("!ly

/;,

1'

n

llll

r

l.

10.2

Induction

Motor

in

Steady

State

185

W

!:!L= U

s

,

RS=

O

Generator

S<O

Motor

0<S<1

mM

m

pO

Reverse braking

S>1

W1

2

/

-1

-0,5

~/

(:~-/S~=0,2

-1

--/-------

Operation

with reverse

-2

/

W

o

USO

/ /

----

/

phase sequence

Fig,

10.11.

Torquejspeed

curves

of

a

wound

rotor

induction

motor

time

duty, for instance when

starting

the

motor

or with crane hoist appli-

cations,

are

these los ses tolerablej sim pie control schemes of

this

type

are in

widespread use.

The

efficiency of

the

simplified symmetrically fed

induction

machine in

steady

state

operation

may

be

assessed as follows: Because of

the

neglected

iron

and

stator

copper losses

the

active power Ps supplied

to

the

stator

windings is passed on

to

the

rotor, where

it

is transformed into mechanical

power

and

heat

losses.

ln

the

steady

state

equivalent circuit (Fig. 10.7 a) alI

this

power is dissipated in

the

synthetic resistance

RR/

S,

P =

31

2

RR

(10.90)

R

RS'

ln

reality

the

resistance in each

rotor

phase

is

only

RR,

hence

the

difference in

power

must

correspond

to

mechanical power

that

cannot

be modelled directly

by a

static

circuit, Fig. 10.12,

[1

]2 2

l-S

(10.91)

Pm

=

31

R

RR

S

-1

=

31

R

RR

-S-

·

This

yields

an

estimate

for

the

efficiency in

the

motoring

region, O< S <

1,

1 '",

W

I - S =

-<

l ,

(10.92)

'II",

{'II

WI

H.

:;

w..I1

1

1f

1 /'",

"1"

"'

1

1.11111

: I

:

~ ;~

1

~

'

·

"(·l'

n

,

I

,

,,

I'

!

.'í

()

186

10.

Symmetrical

Three-Phase

AC Machines

PR

__

1_

=

Wl

< 1

rJg

=

Pm

- 1 - S w

(10.93)

which means

that

the

efficiency even of

an

idealised

motor

ü;

lowered in

proportion

to

speed, whether

the

reduction in speed

is

caused by additional

rotor

resistors or by reduced

stator

voltage. A similar situation exists in

the

generation region, where

the

losses rise also with

the

magnitude

of

the slip.

With

a real motor, having

primary

copper losses as well as iron

and

friction

losses,

the

efficiency will drop below this idealised case.

(JRLa

iR

Pv

Rotor lasses

~

RR

PR

RR

~

S

Inputpower

)

, ,

1

RR(S-l)

,

'r'

,

~

Pm

Mech. power

Fig.

10.12.

Power balance

in

the

rotor

The

explanation is, of course,

that

the induction

motor

functions like a

mechanical clutch, where

the

product

of

speed difference and torque causes

an

unavoidable power loss.

The

definition of efficiency in the braking region

at

reverse speed, S >

I,

does

not

make sense, because electrical as well as

mechanical power is supplied

to

the

rotor

and

converted

to

heat.

With

high slip control schemes for short

duty

drives the

main

problem is

the

dissipation of

the

rotor

heat

loss.

ln

this respect a wound

rotor

motor

with

external

rotor

resistors is preferable, however

at

the

cost of moving contacts.

These results give ample evidence

that

an

efficient adjustable speed AC

drive can only be realised when either

the

slip power is recovered by some

means

or

the

stator

frequency

Wl

is changed

to

track

the

mechanical speed,

thus restricting the

rotor

frequency to small values.

10.2.3

Comparison

with

Practical

Motor

Designs

The

simplified theory of a symmetrical induction motor in steady

state

points

to

the

totalleakage

factor

(j

as

the

most inflllcntial

panl.111

ctcr of

the

machine.

As

was mentioned before,

(j

depends on thc

:-;11,,1'(:

or

til!: Klots

and

increases

with

lhe

airgap.

Tlw

se

al,'

or

til,:

cír

c

nlar

c

urwlll

lonl~;

iH

d('I,

"l'Jllill""

lIy

Uw

i(!.:a,]

Ilo

..

load

('111'1'('111, l

.,.

"

wh

i('

h

lI

hw

"",)/,11,11

-1

1.0

a la

r/

:" ,' x

!.,'

..

t

"II

1,l

tI'

II

i

1'

1',

'a

." ,

1",,1'

11I

1'

c!uwi(';d

1"

'/li,I II "

',

1,111'

I

lI

rj',

!I

'

I'

LI 1

/1,

1

'/

',"1'

wil,11

IlJlrll.

l

l",)"

/I

lr

ll

"

Í,

W

II

1',,1

1'

1II;I('I,illl':

;

(Ir

"qllit.l

'1

1'

1,(

1

10.2

Induction

Motor

in

Steady

State

187

The

scale of

the

normalised

slip/torque

curve, Fig. 10.9 b, is fixed by

the

pull-out slip

Sp

and

the

peak torque m

p

;

in

the

interest of good efficiency

in

steady

state,

Sp

should

be

as small as possible.

The

peak torque is also

affected by

the

leakage factor.

ln

Table 10.2 some

of

the

characteristic

parameters

obtained with

the

simplified model are compared

to

those of

standard

motors, as taken from

manufacturers lists on medium size

(> 100 kW)

50

Hz motors, for a two pole-

and

an

eight pole-machine.

The

comparison indicates a fairly close agreement,

at

least from

the

standpoint

of

the

user.

Table

10.2.

Comparison

of

idealised

model

with

pararneters

of

standard

motors

Model

Real

Model Rea1

for a

= 0.05, for a = 0.1,

no = 3000

l/min

no = 750

l/min

Sn

= 0.22 0.20 0.32 0.30

--s;

fo

= 0.22 0.30 0.32 0.40

fs!

l-a

cos

'Pn

rra

= 0,90

0.90 0.82 0.84

mpo

1

~

= 2.35

2.30 1.82 2.0

m

n

2 a

10.2.4

Starting

of

the

Induction

Motor

When

selecting suitable values for efficiency, power factor and overload capac-

ity

at

the

nominal

operating

point,

the

starting

performance of the

motor

is

unsatisfactory; this is

apparent

from Fig. 10.13, where characteristic variables

of a

motor

with a leakage factor

(j

= 0.05 are

plotted

against normalised slip.

The

conditions for acceptable performance

at

the

nominal operating

POill(.

are:

a)

The

normalised slip

at

the

nominal operating point should be

about

Snl

Sp

~

..j(i

to

achieve a good power factor

and

adequate overload

ca

-

pacity

_

b)The

need for good efficiency

rJn

< 1 -

Sn

calls for a minimal value

of

nominal and, hence, pull-out slip.

With

motors above 100 kW, a nomill;d

slip

Sn

~

0.

02

is

realisable; the pull-out slip would

then

be

about

SI'

~

O,lO.

This llw:tns

that

an

indllctioll

motor

with a short circuited, consta.lli

f':

"

:

ii

st,

nIlC<'

rotor

wOllld llitvo to

Ht.ari

fnllll

fi

lalldstill

(8

=

1)

at

SI

Sp

~

10

whirll

c'o['n 'R!JOlldN !.o n

ll

op

e

ra.l.ill

~

:

POilll, "

,I,

1.111'

fil

J'

rll':hl,

01'

!.h,

~

circula,!'

dia

,

/

~

nulI

,

I"i/',.

I O

.H.

'1'

1(('

ill

,lIr

l.ill/

\ 11I'1'f'mfll ll llc'l'

;11

, S I ,

"!a

nrll l'I,(', j

!1

c·t1

"y

188

10.

Symmetrical Three-Phase

AC

Machines

IS

mM

COS

<{>

'm

n I

Sn

IS

4

COS

<{>n

R==O

0=

0,05

3

Snc

-==-'0=

0224

S '

p

m

n

~

l-O

COS

<(>n

=~

==

0,9

2

",

I

Sr

,

Nominal

operating

point

,

'........

mM

........

---_'!!.n_

S

, 0,5

2 3 4

5

Yci

Sp

Fig.

10.13.

Characteristic curves of induction motor with constant rotor resistance

Is 1

I

~

-lU

~

4.5 , cos

cp

~

0.095 ,

~

0.47,

(10.94)

Sn

m

n

would

be

very poor, since low

torque

is combined with a high

predominantly

reactive current.

When

the

motor

is fed from a grid

with

a relatively large

internal

impedance,

there

could

be

an

appreciable volt age drop, reducing

the

current

linearly

but

the

torque

with

the

square

of

the

voltage. This

may

even

lead

to

a

situation

where

the

motor

does

not

start

under

loadj

the

protective

devices would

then

have

to

disconnect

it

from

the

line

to

prevent

damage

due

to

overheating.

The

conditions

are

particularly

severe in small grids such as

on

board

ships, where

the

generators are

not

by orders

of

magnitude

larger

than

the

largest motorsj this also calls for fast volt age control

of

the

generators.

Two

options are available

to

improve

the

starting

performance:

a) Use

of

a

motor

with

a wound

instead

of a cage

rotor

which

permits

the

rotor

resistance and, hence,

the

pull-out slip

Bp

to

be

temporarily

increased

at

will,

creating

optimal

conditions for

starting

as well as

aI;

operational

speed.

The

rotor

resistance

may

be

shorted in steps

by

COllt.a.ctors

controlled

in

a

fixed

time

sequence

or

depending on

stator

CHr)'(!Il(.

01'

spced

(sec also Sect.

3.1.4).

The

larger

part

ofthe

power losses dmil'l'.

st.il.l't.

o<:(',llrs

ill

Lhe

startiug

resistor, i.e. outside

the

motor,

so

UlilJ

1.11<'1'1'

;1,1\'

110

addit.iollal coolillg

problC:lll~;,

Wll(~u

t.\l(~

operat.iIlI~

s(

)('('

d

11

11:1

("

'V

II

1'

(·

n.

cll('d,

i,ll(' slip-rin

l(S

call

1>

c

s!lol'l,,'d

illt.l'I'l

lII.

lly

wit.h

ii

~

(I('('i

ll.

l

1111

'1' 1

11

11 11

11

1

11

wl

li

ll'

t.l1(,'IH'II

:d

ll': i

11.1'

11

lil'l,('"

1.(,

!-tv

" i

,1

LI,, ·,

1I1111<

1I'

('

WII

II'y fl'lt- Ll

oll

(

lI

,,1 W(' 1'.I .

fi.

1""1.

, ,,·

wl

l.

"

/I

o,

()

r" II I.It l'

l.

I

"I

"

10.2

Induction Motor in Steady State

189

with

Bp

= 1.0 would show

the

following

starting

performance

at

B = 1

Is

-I

~

3.2,

coscp

~

0.66,

~

2.35,

Sn

m

n

signifying a decisive improvement over

the

case

with

a

short

circuited ro-

tor

winding.

With

larger

motors

the

starting

resistor

may

have

the

form

of

a

tank

of

water

with

enhanced

conductivity

into

which

the

electrodes

connected

to

the

sliprings

are

progressively submerged.

b) A

more

elegant

method

requiring no moving

contacts

is

the

utilisation

of

eddy

currents

in

the

rotor

bars

leading

to

a

concentration

of

the

current

in

the

outer

portions

of

the

conductors

at

higher

rotor

frequency (skin

effect),

thus

increasing

the

effective

rotor

resistance

at

larger slip. As a

consequence,

the

starting

torque

is improved

with

an

accompanying reduc-

tion

of

current,

while

in

normal

operation

at

low slip

the

desirable features

of

the

motor

with

short

circuited

rotor

are

maintained.

Double cage

R

2

-1

L

2

b

R

I

"R

2

a

L

I

"L

2

Fig.

10.14.

Cross section of eddy current rotors and equivalent circuit

There

are

various designs which

tend

to

emphasise

the

effect

of

eddy

currents

in

the

rotor

bars.

Qne

solution employing two cage windings is shown

in

Fig. 10.14 a.

The

starting

cage (1), being

of

higher resistivity, is placed

in

open

slots

immediately

below

the

rotor

surface

and

exhibits low

magnetic

leakage; as a consequence

this

winding produces a high pull-out slip

BpI.

Because of

the

low leakage

this

winding

dominates

at

high

rotor

frequency,

i.e.

at

low speed.

The

operational

rotor

winding (2),

in

contrast,

is characterised by a larger

cross section

of

the

conductors, possibly employing also

material

of

higher

conductivitYj

the

leakage reactance is increased because

the

bars

are

now

embedded

in

the

rotor

iron. Clearly, this winding becomes

mainly

effective

at

low slip

fre(]IlPIlcy,

Figure

10.14 b shows

the

equivalent circuit

with

the

two

rotor

Willdilli\S COllIH',c[,cd in parallcl.

1\';<:;1.11:';1'

01'

II"

n~,

DI

~

L~

1.111'

I.wo

willdiJlgs

ar<~

cOlllplemclLtary.

'1'11('

('tlnl·1I1.

1",'11::

,,(' !.II('

1IIi1

,dlilll: iH,

"r

,'

111

II

'

:!

" , II<'

1011/':('1'

;1.

('irdc,

'I'ypic<t.\

---------

1'111

III.

,"iY"lIlldrical

Three-Phase

AC Machines

I,.

111

M

/

11111

,'.

11

..

, /

~

'Sn

?

mM

m

n

Sr

1 S

Jo'i(.;.

10.15.

Characteristic

curves

or

induction

motor

with

eddy

current

rotor

('llrves

of

an

induction

motor

with

double cage

rotor

are

seen in Fig. 10.15;

;\

wmparison

with Fig. 10.13 indicates

the

improvements achieved.

Another

d(~si~1I

of

a

rotor

winding with

strong

eddy-current effect is a cage

with

narrow

I)ar,~

extending deep into

the

rotor

iron; this is also seen in Fig. 10.14 a.

WIt(~n

line-starting larger motors

it

is sometimes necessary

to

redu

ce

the

ílllllSIt

ClllTCnt

by lowering

the

stator

voltages

or

by using only

part

of

the

,·;L:!i.(I/, willdillg, thus increasing

the

leakage factor.

ln

both

cases

the

starting

(,(lrqll('

is

fllrLllCr

reduced which limits this

method

to

starts

with light load.

'I 'II\' ,;i"'lll('sl. possibility for reducing the voltage

during

the

start

is,

of

course,

" y

1/

',

1'I

'(

'(lIIIJ{:ction

of

the

stator

windings.

It

entails a reduction

of

voltage

1\11<1

"111'1('11(

,

hy

1/../3

and

of

input

power

and

torque

by

1/3.

'(I.;,

IHd

Ilction

Motor

SlIpplied

by

Impressed

Voltages

of

Arbitrary

Waveforms

Arl.,'!'

I.his

discussion

of

the

steady

state

with sinusoidal symmetrical three-

phasc

voltages let us now

return

to

the

more general model of

the

symmetrical

i

11<1

Ilcl.iol1

motor

in dynamic condition, when torque

and

speed are varying.

Wc

assume

that

the

stator

is

supplied by impressed volt ages having, in

prill('iplc:, variable frequency

and

arbitrary

waveforlll, as long as Eq. (10.37)

is

rlllfillNI.

lu

fact, due to

the

isolated

neutral

point

o[

the motor, even a

viobl.ioll

of

thi::;

eondition would

be

of

110

COllS(:qll('l1n:;

iI.

would simply mean

I.ha!'

I.he

llel1tral

of

t:he

WilldillgH aSSlIm(

',';

a

dilf(:n:lIt

pol.(

'lllial

aoS

the

neutral

"r

1.11('

slIpply

vol(,ap;<'s

'

/1',

'

:1,

'/},:;2, '1/.,'::1' A

lHO,

til('

H'

H

IlI!,

!!

IlIlly

III'

g'(:n('raliscd to

rI.

lill/,/,Iy

h;wjl'l

~

('()II~I.;l.Id

,

:tlld

(:'111111

i

lll.('!'ll

ld illlj"'t!HIII'\'H iII (';LeI. pha.

;.;

(';

1.I"

:y

('

",ii.!

I

...

ill<'llI,[(',1

iII

U

..

,

,,

1.11.1."1'

illlJII'dl

u""'

1I

(1,'

,'1

,

,,

::

1,(

,)

"r

1.111'

111"1.,,1',

10.3

Induction

Motor

with

Impressed Voltages

191

Should

the

motor

be

fed

with

sinusoidal

symmetrical

three-phase voltages

of

constant

frequency

and

loaded with

constant

torque,

the

transients

will

eventually decay

and

the

steady

state

condition discussed in Seet. 10.2 will

emerge, where alI quantities in

the

stator

are periodic funetions

with

the

angular

frequency

Wl'

A

better

insight into

the

dynamic behaviour of

the

machine is

obtained

by choosing a moving system

of

coordinates, where these

steady

state

oseillations disappear.

A suitable choice of coordinate system, independent

of

the

waveform

of

the

voltages, is

that

defined by

the

vector of

stator

voltages, Eqs. (10.34),

(10.38),

j2

'Y,S(t)

= USl(t) + US2(t)

eh

+

US3(t)

e

--y

(10.95)

= UI2(t) -

U23(t)

e-h

= Us(t) e

j

À(t) ;

its

instantaneous

angular velocity

d>'

Wl(t) = dt

(10.96)

corresponds

to

the

angular frequency

of

the

stator

voltages. Should

the

mo-

tor

be

supplied with sinusoidal symmetrical three-phase voltages of

constant

frequency

and

amplitude, Eq. (10.60) again becomes yalid

'Y,S

(t) = 3 y'2 U S e

j

(Wl

t+Tl)

(10.97)

2 '

resulting in

3y'2

us(t)

=

-2-

Us,

>.(t)

=

Wl

t +

TI

,

(10.98)

where

U

s

is

the

RMS line-to-neutral voltage.

All electrical variables are now defined in this coordinate system; i.e.

they

are transformed as

they

appear

to

a moving observer positioned

at

angle

>.(t);

by

making

use

of

Eqs. (10,7, 10.10)

we

define

is(t)

e-j

À =

is(t)

e

j

(-À)

=

lis(t)

(10.99)

;l.Ild

iR(t) e

j

ê

e-

jÀ

= iR(t) e

j

(e+ê-À)

=

li

R

(t)

(10.100)

as

the

current vectors in moving coordinates.

With

sinusoidal symmetrical

I.hree-phase

stator

voltages

of

fixed frequency

and

constant

load torque, these

qll<1l1ti

(,ies are constant in

steady

state,

When

multiplying Eq. (10.50) by

e-

j À

w(~

JiJld

, \

Ih

-;

"À

d(

",

'

,\

/(,

"

1"

. 1'

.1

I {

",.

.

t:

J

1/'(1

1./

1

1'.1

)1

' J

'Il.

s

r-·

i

>--

(10,101 )

",

'

.II.

III

192

10. Symmetrical

Three-Phase

AC Machines

which is transformed by

substituting

l

d

is

d(.

_j>..)

di

s

_

jÀ

.d>'.

-jÀ

--=-zse

=-e

-J-zse

dt

- dt

dt

-

(10.102)

Developing

the

expression

d

·

dI'

is

-jÀ

iS

+.

1.

-e

=--

JW1

Zs

dt dt -

(10.103)

correspondingly,

. 'ê) d

1

i.

1.

d(lR

e)

-jÀ

=

---=!},

+ J

W1

iR,

--e

dt

(10.104)

and

inserting Eqs. (10.103), (10.104) into Eq. (10.101),

the

stator

voltage

equation

in

the

moving reference frame results,

d

Rs

1is +

dt

(Ls

1is +

Lo

1i

R

)

+ i

W1

(Ls 1is +

Lo

1i

R

)

=

us(t)

.

(10.105)

The

rotor

equation Eq.(10.51) is transformed accordingly, with

Y.R(t)

==

O,

RR

1i

R

+

~

(L

R

1i

R

+

Lo

1is) + j

(W1

-

w)

(L

R

1

iR

+

Lo

1is) =

O.

(10.106)

These equations may be simplified by

substituting

the

flux vectors, defined

in Eqs. (10.25), (10.32) which are also transformed into moving coordinates,

1.

/.

./.

-j

À

LI'

'f..s = 'f..s e = s

is

+ o

iR,

(10.107)

1.

/. =

./.

e

j

(ê-À)

= LR

li

+ L

li

.

'f..R

!:f.R

-R

o_s

(10.108)

Hence

the

current vectors are

1·

_

(1./.

1

1./.)

1

zs---

'I-'

---

'I-'

-

(7

Ls

-s

1 +

(7R

-R

(10.109)

1·

_

(1./.

1

1./

.)1

ZR-

--

'I-'

---

'I-'

-

(7

L

R

-R

1 +

(7s

-s

(10.110)

Inserting Eqs. (10.107-10.110) into Eqs. (10.105, 10.106) results in

d

1./.

T

'

1:...s

[1

+ J

.

W1

T

'J

1./.

- -

1

o

1

4)

I

1"

..

.<;

Us

t s

-d

- + S

'l-'s

- =

()

,

t -

l+(7R

- I.

(10.111)

111.4)

J

T/~

lU

-H

-I-

rI

I· j

(WI

-

w)

TRJ

''1/'/1

0.,

(10.112)

I .1

(f

;

,'

'1/'

"

W

1",\"

,

10.3

Induction

Motor

with

Impressed Voltages 193

(7

Ls

(7

LR

(10.113)

T~

=

Rs

'

T~=

RR

are

transient

time

constants.

The

expression for

the

electrical torque, Eq. 10.48, can be

written

in

the

form

mM

(t) =

~

Lo

1m

[is (iR e

j

ê)*]

=

~

Lo

1m

[lis

ei

R

)*]

(10.114)

the

torque

is independent of

the

coordinate system because

the

inverse

trans-

formations

with

e

jÀ

are cancelling. Hence, using Eqs. (10.109, 10.110)

we

have

mM

(

t)

= -

2

Lo

1m

[(1

7jJ

-

--

1 1 )

7jJ

(1

7jJ

-

--

1 1 )

7jJ

* J

.

3

(72

Ls L R

-s

1 +

(7

R

-R

1 +

(7

S

-s

(10.115)

Since only

the

mixed

terms

of

the

product

contribute

to

the

imaginary

part,

this

reduces

to

mM(t) =

~

Lo

1m

[17jJ

17jJ*

+ 1

17jJ

17jJ*

J

3

(72

Ls

L

R

-S

-R

(1

+ (7s)(l +

(7R)

-R

-S

=

~

1 -

(7

~

1m

[17jJ

17jJ*]

. (10.116)

3

(7

Lo

-s

-R

This

is inserted into Eqs. (10.52, 10.53) for

the

mechanical motion.

To arrive

at

dimensionless real equations,

the

transformed flux vectors

are

nOw

normalised and splít in real

and

imaginary

parts,

representing

the

components

that

are parallel (x)

and

orthogonal

(y)

to

the

voltage vector.

The

flux linkages in steady

state

operation

with

sinusoidal voltages

and

nominal

frequency, as indicated on

the

nameplate,

and

neglected

stator

resistance

serve as flux references,

(t)

-2-

3

.12

Uso

[xs(t) + j ys(t)]

(10.117)

17jJ

-s

=

Wo

,

3

.12

Uso

[XR(t)

+ j YR(t)] .

17jJ

-R

(t) = 2

Wo

(10.111'\)

Tlle

magnitude

of

the

voltage vector in Eq. (10.98) is also referred to

I,IH

'

Illagnitude

at

nominal

operating

conditions,

3.12

Us(t)

(10.] I!))

'Ils(t)

=

-2-

Uso Uso

.

II.

is llutccl

that

no restrictions with regard

to

waveform or frequency

of

t.hc

:i/,;II,()I' voH'lgcs lmve been imposed so far.

WiI

.h

1.11('1:('

ddillitiollS

thr

V()Jl.al~('

(~q1\atiolls

(10.111, 10.112)

and

thc e.' ·

pr":i

r:

i"ll

rOl'

LIli'

I,

")'qll('

1-;'1'

(IO.llli)

:1.':: :11111<'

1.11(~

r()llowillf~

fonn

I

!11

10.

Symmetrical

Three-Phase

AC Machines

1'8

~

(xs

+

jys)+

[1

+

jWl

T;J

(xs

+

jys)

1 (

')

I Us

-

-1--

XR

+J

YR

=

Wo

T

s

-U

'

(10.120)

+

(TR

50

T

R

~

(XR

+

jYR)+

[1

+j

(Wl

- w)TRJ

(XR

+

jYR)

1 .

-

-1--

(xs

+J

YS)

=O ,

(10.121)

+

(TS

mM

=2

(1

+ (Ts)mpo

(Ys

XR

- XSYR) ,

(10.122)

wltcre

mpo

is

the

nominal

peak

torque for

Rs

=

O,

as defined in Eq. (10.86).

With

the

following abbreviations for

Pull-out

slip

at

nominal

frequency

and

Rs

= O

RR

SpR

=

Wo

(T

L R =

1

Wo

T R '

(10.123)

Pull-out

slip

of

rotor-fed

motor, nominal frequency

Rs

1

and

RR = O

(10.124)

Sps =

Wo(T

Ls

=

woT;

,

M

(~chanical

time

constant

T _ Jwo

m - -

(10.125)

mpo

:

1.11<1

lhe

instantaneous normalised frequencies

"t.at.or frequency

Wl

Wo

lIlecltanical

rotational

r['(~qll

e

Ilcy

(speed)

W

Wo

I'U

lor

frcq

U(~llcy

(,);\ W I W

(IO.1

2

Ci)

(,)

0 (.II

I!

10.3

Induction

Motor

with

Impressed

Voltages

195

six real differential equations emerge for

the

direct

and

quadrature

flux com-

ponents as well as

motor

speed

and

positionj

eo

is

an

arbitrary

angle

of

reference.

,

dxs

1

Wl

1 1 Us(t)

T

s

--

=

-Xs

+ -

-Ys

+

---XR

+

---

(10.127)

dt Sps

Wo

1 +

(TR

Sps

Uso

, dys 1

Wl

1

T

s

- =

--

-

Xs

- Ys +

---

YR,

(10.128)

dt Sps

Wo

1 +

(TR

,

dXR

1 1

W2

T

R

--

=

---Xs

-

XR

+

--

-YR,

(10.129)

dt 1 +

(T

5

SpR

Wo

,

dYR

1 1

W2

T

R

--

=

---Ys

-

--

-XR

-YR,

(10.130)

dt 1 +

(TS

SpR

Wo

d

C~o)

T

m

= 2

(1

+(Ts)(Ys

XR

-

Xs

YR)

- -

mL

(w,

c, t) ,

(10.131)

.

mpo

eo

d

(eco)

W

(10.132)

Wo

dt

Wo

These equations

may

be represented in graphical form by

the

block dia-

gram

in Fig. 10.16, where

the

integrators

with feedback are redrawn

to

form

unity

gain lag elements. Due

to

the

rotational

symmetry

of

the

machine

the

angles

e(t),

À(t) do

not

enter

the

right

hand

side

of

the

model equations Eqs.

(10.127-10.131) with

the

exception

of

load

torque

mL

that

may

depend

on

the

mechanical angle.

The

block

diagram

describes

the

dynamic behaviour

of

the

symmetrical

motor,

fed by impressed

stator

volt ages independent

of

waveformj

the

mag-

nitude

us(t)

of

the

volt age vector is given by Eq. (10.119),

its

instantaneous

angular

frequency

wdt)

by Eq. (10.96).

As

special case with Us

=const.,

Wl

= dÀ/dt = const.,

the

transient

condition

of

the

motor

operating

on

the

symmetrical

three-phase supply results, where U5 is

the

RMS value of

the

line- to-

neutral

volt ages.

Because of

the

transformatíon

into

the

voltage based coordinate system

moving

with

stator

frequency,

the

steady

state

values

of

the

flux components

and

speed, assuming

constant

load torque, are

constant

.

The

steady

state

values are derived by

setting

the

derivatives in Eqs. (10.127-10.131) equal

to

zero;

the

resulting system

of

algebraic equations is equivalent

to

the

results

obtained

in Sect. 10.2.

Of

course, under dynamic conditíons

the

Hux

components are

no

longer

CO!l:;tant beeause

there

may

be

transient

flux waves which are linked

to

the

Hlator

01'

rotor

windings and, hence, move with

the

angular velo city

Wl

or

L,)~

«.I

I

é.)

rdat.ive

to

lhe

c()ordjnat.(~

s'ys

t.<!IJl,

whcre

the.Y

appea.r as

AC

11

1I

II).

:

~

yllllll(

,

Lrical

Three-Phase

AC Machines

l1.

, {I)

OJdt)

USO

XSYR

I--=,

I+-t---ó+ - - - - I

YSXR

OJ2

Wõ

mM

m

po

.!!!.b...

mpO

.!:!!....

OJo

"'

i

,

~.

I O.

LO

. lllock diagram

of

symmetrical

induction

motor

r.."

h

.v

illq .r<'

.'i.'i

ed

volt ages

of

arbitrary

wave form

t.

111111<

.J1CIII.

S

OL

'Lppropriate frequencies; as a result there

are

torque

and

speed

Vi

l.!

iill.ioll

S.

WII( '

JI

ill

.'i

pc

cting

the

general

structure

of

Fig. 10.16

it

is recognised

that

i

I.

(

':

UI

I>(

!

I"(

!

<iuc

ed to

the

form shown in Fig. 10.17 which reminds one

of

a DC

"':i.("!t

i

ll("

I"ig

. 5.9.

The

generation

of

the

electric

torque

mM

is now much more

illvolved, which is

due

to

the

fact

that

the

currents

in

the

rotor winding

are

"I:lglld.ically induced

rather

than

impressed

through

a galvanic contact; also,

1.11('

all[.';Jc between

the

stator-

and

rotor-ampereturn

waves depends on

the

op(!lal.illg condition, whereas

it

is essentially fixed by

the

field-

and

armature-

i i

,

(

~S

iII

the case of a

compensated

DC machine.

The

presence

of

AC

quantities

;

t./

:;

o

1.(

~

11(1s

to blur

the

view, even though

this

has been alle

viated

by

the

( ' (

I(

H'(

I

ill

a

Le

transformation.

Wh

iII!

110

<Lllal

ytical solution

of

the

nonlinear

clifr

crential equations exists,

I

;OIlI(~

01'

t/w

illteraetions

in

Fig. 10.16 can

1)('

<]lIalil.al

.iveIy illt.crpreted. For

nalll"I('

, I.W()

r(~r:dllad(

loops

ar

e

rcco

!',

lIis;dlJe,

011('

IlI'!ollgiug

to the s

tator

i!.lld

I.lI e

ol.lll'J"

1.0

I.JI(~

rol

.

or,

Lha.!.

cOlIl.ail.l

IIl1dl,il'li

cnl.i

ulIs

11'y

WI

J/ WO

alld

w

~/w

()

J"(

·

Y;

I'

('('l.ivd

)'

,

1,-\1111

'

11

1U'1':I,'diIlI

C(

li

o!"

n

HIOII

IV

III.

1.11

1"

<'1'

1111

1;

(

'

ollpliIlI

~S

drawlI

iII

dllf

dll'(!lill,

oll

,

II

h<lllllll

'·

,<'

II4"HI

!I

/'l

1'

·"('lI.d u r

dI.'!

\'q

Ill!I U '11!

111

1'

)1

I

I('

l

',nin(',1

d(

'ri(' 1 il.iIlI'.

10.3

lnduction

Motor

with

Impressed Voltages

197

.!:!.§...

I

mL

Uso

mpo

Oh

úJo

úJ

úJo

mpo

Fig.

10.17.

Simplified block

diagram

of

symmetrical

induction

motor

the

fiux components, for

instance

in

the

statori

by

eliminating

Ys

we find

, 2 d

Xs

'

dxs

1

Wl

2

[(

)

2]

(10.133)

T

s

di2

+ 2 T

s

dt

+ 1 +

S;s

Wo x S =O ,

with

the

eigenvalues

.Wl)

81,2

= Wo

(

-Sps

± J Wo

(10.134)

hence

the

pertinent

transient

is a

damped

oscillation with

the

angular

fre-

quency

Wl,

belonging

to

a fiux wave

that

is linked

to

the

stator

winding

and

thus

remains

fixed

in

space.

The

damping

is achieved by

the

dynamic

short

circuit

through

the

volt age source

at

the

stator

terminals. Likewise

the

lower feedback circuit in Fig. 10.16 is found to

be

described by a differential

e

quation

with

the

eigenvalues

.

W2)

(10.135)

83,4

= Wo

(

-SpR

± J Wo

The

associated physical effect is a flux wave linked to

the

rotor

winding which

moves with slip frequency relative to

the

coordinate

axis.

Considering now

the

special case

Wl

=

W2

=

O,

when

the

mechanically

locked

motor

is fed

with

direct voltage,

and

taking

the

dashed cross-couplings

iI/to account, two new

non-interacting

feedback loops

are

formed, where each

(~lHbrace

s

stator

and

rotor.

The

differential

equations

for

transients

in

the

:,:-axis

may

be

combined to

,

,d?-xs

'

,dxs

TsT

R

-2-

+ (T

s

+T

R

) -

+axs

= O.

(10.136)

dt

!10th

(

·

~

i

g

envalues

are

negative real, belonging

to

aperiodic transientsi a cor-

IT

:';

POlldiug

equation

is valid for

the

quadrature

axis of

the

machine. Clearly

I.II1'S(: ('qllal.iolls eo

rrespond

to

those of two-winding transformers, describing

I.I/\'

d

l'

(,()l/l'bll"

ilW

\(

~

l.ic

illl.(~rad

:

ions

in

the

two axes

ofthe

machinei

under

the

,dn

l/lIlI't!

1,

('

1:

1,

('

llIlIlil.i"IIH

;1.11.1

ill

vil:w

or I.lw

voIt.a.g

e-based coordinate system,

19i;

10.

Symmetrical

Three-Phase

AC

Machines

'm

(!o)

j1

1

(01

~

jO,5

(02

'=

0,05

'=

const

(00

'

(00

0,15

-0,1

Re

(!o)

Fig.

10.18.

Root

locus

of

induction

machine for

W2

= 0.05 = consto

at

different

stator

frequencies

and

speeds

the

:/lux

in

the

quadrature

axis

must

be zero,

Ys

=

YR

=

O,

hence

there

is

no

torque.

This

is in agreement with

the

practical experience

of

a symmetrical

induction

motor

with

DC excitation

at

standstill.

When

analysing

the

block diagrarn in Fig. 10.16 more closely, i.e. for

Wl,

W2

f:.

O,

the

mutual

interactions

of

the

two

damped

oscillators, coupled

dircctly

and

through

the

speed,

must

be

taken

into account. This leads to a

complicated resonant system

of

6

th

order,

the

response

of

which is diflicult

1.0 assess. However, assuming

Wl

=const.

and

W2

~

const., i.e. large inertia,

l.hcre are considerable simplifications because

the

multiplications with

Wl,

W2

<tr<' rcduced

to

constant

factors; assuming complete

symmetry

between

stator

<tud

rotor,

5

ps

= 5

pR

=5

p

,

(J"S

=

(J"R

=

(J"o,

T;

=

T~

=

To

,

(10.137)

tbe

characteristic equation pertaining to Eqs. (10.127-10.130) is

[(To

H

1)'

-

(1-

u) +

;;

(:;)']

[(To

H

1)'

-

(1-

u)

+

;;

(::)']

1 - (J"

(Wl

+

W2)2

_ O

+ 52 2

-.

p W

o

(10.138)

Tlle

solnl.iOI\S

or

I.hi~

(~q1l'ati()n,

i.e.

th~~

('igcllvalllf's

or

(.h(~

systf:tn,

havc

heen

cOlllplll.

ed

1l1l111

\

'rir:dly

1(lr

di/fcJ"(~nl.

valtll's

or

:JI.III.()r

rn~qll(

~

l1C:y

WI,

a.:

;S

lllllill~

!.II(

' fiol[\l

wl

l'

IJ'.

1,III'

hu

lC'!..'r:·:

10.3

Induction

Motor

with

Impressed Voltages

199

5

p

=

0.10,

(J" = 0.08,

W2

= 0.05 = consto

(10.139)

Wo

The

result is

plotted

in

Fig. 10.18

in

the

form

of

a root locus

in

the

complex

s - plane,

omitting

the

conjugate branch

of

the

locus.

The

shape

of

the

curve

indicates

that

there

is

little

interaction between

the

two oscillators in Fig.

10.16 for

Wl

»

W2,

approximately leading

to

the

"decoupled" eigenvalues

mentioned before; on

the

other

hand,

the

interactions are

strong

at

low val-

ues

of

the

stator

frequency,

Wl

~

W2,

resulting

in

slowly decaying

transients.

Figure

10.19 depicts

the

decay of a

:/lux

component

at

different

stator

fre-

quencies, assuming

the

sarne

initial

conditions;

the

transient

torque

is also

shown.

There

is a remarkable effect

of

stator

frequency

and

speed

on

the

dynamic behaviour

of

the nonlinear planto Therefore

the

induction machine,

when

operated

in

a wide speed range, represents a very difficult control

plant,

which calls for special control procedures; some will

be

discussed in Chapo

12.

W

2

=0,05 =

const

W

o

tlxs

tlmM

- w

1

m~i

Fig.

10.19.

Flux

and

torque

transients

for

W2

= 0.05 = consto

at

different

stator

frequencies

and

speeds

As a

qualitative

test

for

the

validity of

the

mathematical

model

of

the

induction machine, which was derived with considerable simplifications, a

line

start

at

nominal volt age

and

frequency has been computed by numerical

intcgration

of

Eqs. (10.127-10.131);

the

load

torque

is initially zero

but,

aftcr

stcady

state

speed has been reached, half

the

peak

torque is applied as load.

Tlte results are

plotted

in Fig. 10.20, assuming a low value

of

inertia

so

that

I.!te

start

is completed in a few periods

of

line voltage;

the

other

parameter:;

are

the

sarne as before.

Besides

the

flux components in direct

and

quadrature

direction,

the

stator

Cllrrents are of

main

interest.

The

associated vector is produced by inverne

1.1':tIlSrolllliltion with

the

help

of

Eq. (10.109)

Werner Leonhard Control of Electrical Drives

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