Werner Leonhard Control of Electrical Drives

200

10.

Symmetrical

Three-Phase

AC Machines

xs{t}

I

~

H

~.

I

II

t'

11111i

• I

IIIIIIIIII'~(\.~----------t-------

'Idlll"

~

I

II I o , I

!~~'Ys(t)

I

1

51

I

.!!L(t)

m

L

="2

m

po

Fi,

,,

.

10.20.

S

imulated

starting

transient

of

induction

motOr

wil.lt I<l W value of

inertia

iv

(t,)

=

li

e

j

wd

=

_1_

[1'IjJ

(t)

__

1_

1'IjJ

]

ejw1t

, . -s

(7

L s

-s

1 +

(7

R

-R

:

~

12

Uso

~

.-

-

--

xs

-

_1

XR)

+j

(YS

1

YR)]

.

__

ejw1

t

2

Wo

(7

Ls

[(

1 +

(7R

1 +

(7R

(10.140)

Wil,h

r

~

q.

(10.55) follows

'i:

>i

(l) = 1Re [is(t)]

(/

~

'II

f(

1 )

J

:~

'

:/

:,'; -

---

Xli

CO

:-)

W

I/'

1

lI

n

) S

ill

Ü)

lt

'l,

(A

i l )

ri

/,

::

1 I {r

ll

eI

S

(T

li

c

(10, 111)

I

r

./

úJo

.-.-1.

,_,_._,_

/

'./

.....

-.-

1'-

n

mM

I

'

~.-

I

/'-

-----

m

po

I

m

L

I

m

po

Sta

/1

at

no load

Load forque applíed

1

10.4

Indu

c

tion

Motor

with

Unsymmetrical

Line Voltages

in

Steady

State

201

w

~

"'o

a

W

"I

I;

\

W

o

" \

mM

/ ! 1m

/)

\

po

~

(.

r·

\ .

./

·

....

·-------

,/

"

b

Fig.

10.21.

(a)

Simplified block

diagram

of

induction

machine,

(b)

calculated

transients

This

is

also

plotted

in Fig. 10.20;

it

shows

the

c

haracteristic

large

inrush

current

which, as

the

motor

approaches synchronism, settles down

to

the

small sinusoidal no-load

current

until

the

load

torque

is applied,

The

currents

in

the

other

stator

phases are similar.

There

are

strong

alternating

components of

torque

which also affect

the

speed, even

though

with reduced

amplitude

because of

the

mechanical

inertia

.

If

the

inertia

is predominantly

on

the

load side,

the

shaft

and

the

mechanical

coupling

may

be severely stressed during

the

start.

The

speed, as a function

of time, follows a similar curve as

obtained

in

Sect. 3,2 when the electrical

transients

were

totally

ignored, Fig. 3.12; this explains also

the

absence of

the

overshoot in speed which is

pr

esent in Fig. 10.20,

As

an

intermediate

step

between

the

quasi-steady-state model

and

the

more complete model of the induction motor,

the

heuristic block

diagram

in Fig. 10.21 a may

be

helpful.

It

differs from

the

purely mechanical model

by

the

inclusion of a lag

T~,

following

the

steady-state

torque-slip-function.

For small values of slip, where

the

torque function

is

linear, this clearly re-

sembles

the

block diagram of a

De

machine. A comparison of

the

starting

transient

computed

with

this

model shows good qualitative agreement with

Lhe

transi

ellts o

bt

ained by integrating

the

Eqs. (10.127-10.131).

Of

course,

Lh

e os(:illa(,

iom;

Hl1p

erimposed on speed

and

torque

cannot

be

represented

but

I.I)(

~

lJl(

:

all

viI,

I'W

H

!

~

i'(

:

wdl

luo<!elkc!.

202

10.

Symmetrical

Three-Phase

AC Machines

10.4

Induction

Motor

with

Unsymmetrical

Line

Voltages

in

Steady

State

10.4.1

Symmetrical

Components

For

intermittent

duty

drives, such as cranes

or

hoists, where simplicity

of

hardware

is more

important

than

high efficiency, induction motors with un-

symmetrical

voltage supply

may

be used. By way

of

example, a phasor di-

agram

of

sinusoidal

but

unsymmetricaJ line-to-neutral volt ages is shown

in

Fig. 10.22

a.

Due to

the

isolated neutral

of

the

symmetrical

stator

winding,

the

sum

of

the

instantaneous phase voltages is still zero, Eq. (10.37); hence

the

phasor

diagram forms a elosed triangle,

fl

51

+ U

52

+ U

53

= O .

(10.142)

The

sarne holds

naturally

if

the

stator

windings are Ll-connected.

ejY!)s+

!)S1

~s+À

~S2

e -fYl}s-

!)S-

e -jy l}s+

a

b

c

Fig.

10.22.

Decomposition

of

unsymmetrical

balanced

three-phase

system

into

symmetrical

components

It

is known,

that

an

unsymmetrical three-phase system

may

be

de com-

posed into "symmetrical components" defined by

U

51

=

u

5

++

U

5-

+

fl.o

,

U

52

=

e-h

U5 + +

eh

U5 _ +

fl.o

,

211"

(10.143)

1=3'

U

53

=

ehfl5++e-hU5_+fl.o,

where U

5+

is

the

positive-, U

5-

the

negative-

and

fl.o

the

zero-sequence

volt age component

per

phase.

The

symmetrical components fl5+' U 5

.-

form

two symmetrical three-phase systems

oE

oppositc phase-sequencc. Solvilll( Eq.

(10.14.1) yi(dds

10.4

Induction

Motor

with

Unsymmetrical

Line Voltages

203

U

-

1

[U

-51

+ h

-52

U +

-h

U ]

,

-5

+

3"

e

e_53

U

-

3"

1

[u

+

e

-h

U +

e_53

h U ]

(10.144)

-5

-

-51 -52

,

fl.o

= 1

[U

51

+ U

52

+ u

53]

= O .

The

expression for

the

positive sequence component is similar

to

that

for

the

time

dependent volt age vector, Eq. (10.34). Since

the

volt ages between

the

motor

terminals

and

the

isolated

neutral

are

not

directly available

(the

motor

may

be

Ll-connected),

it

is

appropriate

to

introduce

the

line-to-line voltages

instead. By expanding Eq.

(10.144) a we find, analogous

to

Eq. (10,38),

U

5+

= 1[U

51

- U

52

-e-h

(fl52 - u

53)

+ u

52(1

+ e

h

+

e-h)]

v

!L

12

!L

23

'--v---'

=0

-

3"

1

[U

-

-hu]

-23

-

-12

e

(10.145)

and

correspondingly

fl5-

= t

[U

12

-

eh

U

23

]

(10.14G)

Two

of

the

line-to-line volt ages are sufficient

to

determine

the

positive ,Uld

negative sequence components, because

U12 + U

23

+ U

31

= O

(10.147)

is valid by definition. General1y

the

use

of

line-to-line volt ages does

not

ex-

elude

the

appearance

of

an

arbitrary

zero sequence component

fl.o

but,

as

seen before,

it

vanishes

with

a symmetrical load.

If

the

motor

is fed from a

line possessing noticeable internal impedance,

the

symmetrical components

of

the

terminal

volt ages depend also

on

the

load, i.e.

the

operating

state

of

the

motor.

Should

the

voltages U

12'

U

23'

~1

form a

symmetrical

three-phase system,

u =

e-j,,!

U

,

(10.148)

23

12

the

result is

e-

j7r

/

6

U- 1

-U

U-O

(10.H!))

-5+

U

-

J3

-12

-

-51'

-5

- - .

'l'hc case

of

a symmetricaJ induction

motor

fed by unsymmetrical snpply

vol

(.agci-i

is

fully containecl in the thcory developed in Scct. 10.1 bllt til<:

sllb

-·

~Cq'I('Ilt

dis(".I\~;:;i()Jl

oE

som(~

cX<tmpks

is

J'(~st.rid<:(l

t.o

steady-state.

WIlI'll l'''r1Ui

11/';

t.he

t.i"w-varyiHI

~

vol!.al(

(·

V(·('t.()f

H!'

.

(

~

()nlillf~

t.o

J

~;

qs.

(10

.;)'(

IO

.:,n,

IO

.

(iO)

lUtei

wit.it 1.

11('

ddillit.io!l (10.:\

.-

1),

WI'

lillel

204

10.

Symmetrical

Three-Phase

AC Machines

u (t) = 3

v2

[U

ei

Wl

t +

U'

e-i

Wl

tj

(10.150)

-s

2

-s+

-5-

,

a result,

that

can be

interpreted

as

the

superposition of two

constant

vectors

rotating

with equal

angular

velocity in opposite directioni hence

the

vectorial

sum

1!cs

(t) follows a periodic elliptical

path,

as seen

in

Fig. 10.23.

3{2

-j

w

1

t

-

Us_e

3..J2 U e

jW1

t

2 -

2- -s+

ldS(t)

///

/"

"-

----

Fig.

10.23.

Volt age vector of

unsymmetrical

balanced

supply

ln

steady

state,

assuming

constant

speed, similar definitions hold for

the

vector of

the

stator

currents,

i.(t)=3

v2

[I

eiw1t+I*

e-

iw1t

j

(10.151)

-s

2 -s+

-s-

,

w"cre

Is

t,

Is-

are

the

symmetrical components of

the

stator

currents which

;1.1"<:

ddined

in accordance

with

Eq. (10.143). Likewise

the

rotor

current vector

iII

sLal.or coordinates

may

be

written

as

í

(t)e

jo

=

3v2[I

eiw1t+I*

e-

jw1t

j

(10.152)

-lI

2

-R+

-R-

,

while

the

rotor

current vector in rotor coordinates is, with de:/dt = W

COllst.

,

,.

(t)

- 3

v2

[I

j

(WI-

W

) t +

1*

e-j

(wd-

w

)

tj

(10.153)

Ln

- 2

-R+

e

-R-

.

'I'hc spced w

i::;

counted in

the

direction of

th<:

positive sequClIce field. Rence

1.1)(',

frcqllcllcy 01' tlte positive and negativ(' ;'

('<I\lCI[('('

roLor

c

urrents

is W.l - W

;1.l1d C"'I 1 w

rl'~qJC(",(.jv(~I'y.

'1'111"

ph'y

~

:jnd

ill!.(:rprd,aÜol\

of

1':1)

11

.

(I

O.I~;

I,

1

(!.

IIJ

:.t

) iM

LIIILL

or

(.wo

li

cparal.c

II

11L1'

..

rl'i.ll

rll

li WII.V , ')i

"(J

I./tl.i

111

':

wi!."

I l

<i

l

iII

"PI,

olIl

!.(·

dlrC'd.i(H

Hl

ll

llld

l

( 'l

lIlH

illJ

~

cor

I'c'I

'(II

,

'lId

l

lll

\

1111

K

d('II:

d

!.y

WIL

VI'

1i

iII

1.1.

0

lI.lr

r'.II,p w

\"dl

i"

1.111

'

11

IlIclll""

i'.I

'l('

v"IL

l

lj

~('

10.4

Induction

Motor

with

Unsymmetrical

Line Voltages

205

components of Eq. (10.150).

As

long as iron

saturation

is

ignored

and

the

speed

may

be assumed

constant

due

to

a sufficiently large inertia,

there

is

no

interaction

between

the

two symmetrical three-phase systems so

that

linear

relations exist between volt age

and

current

components,

(10.154)

Is+ = ys+ U

s+,

Is-

=

Ys-

U

5-

i

where

Ys+,

Ys-

are

the

positive

and

negative sequence

stator

admittances

per

phase. Because of

the

effective values

of

slip for positive

and

negative

sequence components, respectively,

Wl

- w = S

and

Wl

+W

= 2 _ S i

(10.155)

Wl

the

admittances,

derived from Eq. (10.72) with

Rs

=

O,

are

1

l+jl~

17

sp

Y

s+

= . . s

J

Wl

Ls

1 +

Js

p

1 + . 1

2-S

Y = 1

JuS;:

S-

j

Wl

Ls

1 + j 2

S

p

When

substituting

Eqs. (10.151, 10.152) into

the

torque

equation (10.48),

four

terms

arise, two of which are

alternating

torque

components having

the

frequency 2

Wl

i contributions

to

the

mean

torque

are only produced by

interactions between

stator

and

rotor currents

of

the

sarne sequencei hence

the

mean torque, counted in positive sequence direction,

is

(10.156)

mM

= 3

Lo

1m

[1

5

+

DH

+

Is-

IR-l =

mM+

+

mM-

.

These two components of

the

mean

torque

may

be derived by

the

sarne

reasoning as shown in Sect. 10.2, because each is produced by a symmetrical

three-phase voltage system.

ln

accordance

with

Eq. (10.85)

we

find

mM

mM+

mM-

-=--+--

mpú

mpo

mpü

2

~

+

sp

5

p

S

Us+

Wo

[

]

2

U50

Wl

2

2-S

+

~

5

p

2-S

[

Us-

wo] 2

Uso

Wl

,

(10.157)

where

mpo is again

the

peak

torque, Eq. (10.86),

at

nominal voltage

and

Sp

the

pull-out slip for

Rs

=

O,

Eq. (10.71).

Eqnation

(10.157)

may

be

interpreted

as

the

net

torque

produced by two

identical

mol.ors

coupled

to

the

same shaft

whidl

are supplied by syrnrnetrical

tltn'('-plt":;I'

voll.;!.J',"

sysl.('llls

"s

I,

U

S

(lf

(~Il'lid

fl't

~

ljlICm(

:

y

hllt

oppo:-;ite pltasc

:;('<lIII'II('C

' , j ,

C'

wil.iI

!.wo sl.nl.or

I.('flllill

l

tl

ll

clI"

til\' H

I'I"tllld

\1101.01"

illkrchnl1i~(:d,

\,

iI'.

I II '1\

206

10.

Symmetrical

Three-Phase

AC Machines

ls+

!

1s-

!

w

Us+!

L-+----J

I )

IUS-!

L-+----J

Fig.

10.24.

Induction

motor

with

unsymmetrical

supply

volt ages,

equivalent

scheme

Eq. (10.157) clearly demonstrates

the

basic disadvantage of alI control

schemes employing unsymmetrical supply; while

the

torque

is

reduced,

the

power losses are increased by

the

negative sequence voltage

and

current

com-

ponents.

By changing

the

rotor

resistance

(Sp)

and

with different circuit connec-

Lions,

resulting in speed-dependent positive

and

negative sequence voltages,

a variety

of

torque/speed-characteristics can be obtained.

This

is

discussed

with

the

help of three examples.

10.4.2

Single-phase

Induction

Motor

AI

------1

1

!..S1

=0

;'

----L-

--,--

Starting

1

capacitor

1

~

:l.C

'I

Ih3=

I}BC

Fi!!:.

10.25.

Symmetrical

induction

motor

with

single

phase

supply

Ily discolluec:ting one

stator

terminal of a Rylllluctrical induction motor,

1,11(,

siuglc phase circ:uit

in

Fig. 10.25 rct;ul

ts,

A,

II,

n

an

'

t:he

terminaIs

of

the

H.Y1l111I('!.ric:\.!

lill(~

voltages.

The

cowll.,rnint,

iUlpO

!-l

cd

!Jy

t.h(~

opcn

circlIit

is

1

";

1

)

'

~

"l

l

r

:-:

II

V"

j

lJ,

,

;

=

H;

(10.1

;)H)

r

Ulld,II

"! <'(

111<1

1

1,

;

<111

\;

dl<1

1N

11

rrnfll

1·:'1

J1

. (

1.11

. 1<

[1

;), ( IU. l·ll'i),

10.4

Induction

Motor

with

Unsymmetrical

Line Volt ages 207

-

1 ( h

-h)

_.

1 _

U U

3 e - e U

23

- J

v'3

U

23

- U

AM

,

(10.159)

-s+

-

-s-

-

where

U

AM

is

the

line-to-neutral volt age of

the

disconnected line-phase.

When

combining

the

two equations,

we

find

U -

Ys-

U - Zs+ U

-s+

- y + y

-AM

- Z + Z

-AM,

s+

s-

s+

s-

(10.160)

Ys+

zs-

U

s

_

=

Ys+

+

Ys-

U

AM

=

Zs+ +

Zs-

U

AM

;

Zs

=

l/Ys

is

the

impedance of a

stator

phase in symmetrical

three-phase

operation.

The

sarne volt age magnitudes would result if

the

stator

windings of

the

two coupled motors in Fig. 10.24 were connected in series

to

the

symmet-

rical three- phase volt ages. According

to

Fig. 10.26

the

positive

and

negative

sequence

admittances

can be

taken

from

the

normalised circular

diagram

(Fig. 10.8), where

Is/Iso

=

Ys/Y

so

was

plotted

for

Rs

= Oas a function of

slip.

This

indicates

that

a large difference exists between Ys+ = Ys(S)

and

Ys-

= Ys(2 -

S)

for small values

ofpull-out

slip, i.e. for a

motor

with

a cage

rotor.

jlm(~)

Y

o

Ys(S)

Y

s

+

y;;-=y;

R Y

s

1.

.(-)

a Y

o

Fig.

10.26.

Circular

diagram

showing positive

and

negative

sequence

admittance

If

the

motor

operates

near

the

positive synchronous speed

(S

~

O),

IYs+1

«

IYs-1

holds, resulting in U s+

~

U

AM,

U

s-

~

O.

This

means

that

the

negative sequence field is almost completely suppressed by

the

rotor

wind-

ing due

to

the

large relative speed 2 - S

~

2;

hence

the

motor

operates

with

llearly

symmetrical

volt ages despite

the

fact

that

one

terminal

is

not

con-

nected.

ln

the

vicinity

of

the

negative synchronous speed

the

conditions

are

illverted.

Th('!

st.('ady-st.a.te

torque

of

the

single phase

motor

is

obtained

by substi-

LilLillg

Eqs.

(10.

HiO)

illlo Eq. (10.157)

208

10.

Symmetrical

Three-Phase

AC Machines

I

Ys.

I

Ys++Y

s

.

Sp

I

Ys+

I

Y

s

++

Y

s

.

2·

Sp

2 S

Fig.

10.27.

Symmetrical

components

of

single-phase

motor

.Q!.

I

mM

I

11

úJ,

I

" m

M

+

, m

pO

I

mM· I

I

mpO

I

m

po

I

I _

m

----,---+

m

pO

'

.....

I

.....

-

I

-1

Fig.

10.28.

Torquejspeed

characteristic of single-phase

motor

mM

=

(10.161)

mpo

2

UAM

Wo

1

2

1

--s

YS-

[

~+s 1

YS+

+Y

s

-

2-S

+~

IYs+Y:+yJ]

[

Uso

wd

.

sp

s;-

2-S

Since

the

torque rises with

the

square

of

the

voltage,

the

negative sequence

torque

is

even

further

reduced

than

the

voltage. Some

of

the

f1.Jnctions

con-

tained

in Eq. (10.161)

are

sketched in Fig.

10

.

27;

by combining these curv

es

the

torque/speed

characteristic of

the

singlcvlt;1.se

motor

is obtained, drllwll

in

the

usual oricntation, Fig.

10

.28.

1'11('

ClIJ

V(~

is s'yl1llllctrica.l with

rm;ped

to

the

orif~ill

hc("all

s

(:

aI.

S =

1.,

ys, = yC; holdli a

lld

I,\l(~

o[l[los

iJl[!;

t.orqU(·

("()[lIpOIl(·IIf.

~ 1

(

'1

1.11<",

,1. 1I.'lIcl;

lhe

IlItlI.Ol" t!.

·V

••

I"

pH

IjO

.1

...

1,

I.ol"tliw

;ü

!;

I.

;lIId'!il.iU a

li.!

(

'11

1111<>1

i.I,

)

1I

I, wlt.lulllf,

('

xI,"l'lInl

ll

Nl

ii

tl

l,

IIIU

'O.

Ir

1\

/1

11 11

111

ItIllt.OI :d, 1;l.:l.Ild n

t.i\ll

1M

•

_.#'

10.4

Induction

Motor

with

Unsymmetrical

Line Voltages

209

turned

manually in

either

direction,

the

corresponding sequence prevails

and

the

motor

continues

to

accelerate

at

light load

up

to

the

operating

speed.

To make

the

motor

self-starting, either some

asymmetry

has

to

be

built

into it, as is done with shaded-pole motors, where a

short

circuited

turn

is

built into

the

stator

winding,

or

a

starting

capacitor

must

be

connected

to

the

third

or

an

auxiliary winding, as is shown

in

Fig. 10.25; in

both

cases

the

direction

of

rotation

is fixed.

The

value

of

the

starting

torque depends

on

the

design

of

the

auxiliary winding

and

the

capacitor.

Besides

the

starting

problems,

the

drawbacks

of

single-phase

operation

lie

in

the

residual

pulsating

torque

as well as reduced utilisation

and

efficiency

of

the

motor

which

are

always below

that

of

a symmetrically fed motor.

It

is for these reasons

that

single-phase induction motors

are

only used

at

low

power ratings, for example in household appliances.

10.4.3

Single-phase

Electric

Brake

for

AC

Crane-Drives

Q

3

!l23

= O

Fig.

10.29.

Single

phase

supply

of

AC

motor

for

braking

duty

A simpIe

intermittent-duty

AC crane-drive required

to

operate

as elec

t.ri

...

brake

in

the

lowering direction may have

the

asymmetrical circuit

COllllec

~

tion shown in Fig.

10

.29, where two

stator

terminals

of

a symmetrical

wOIIJld

rotor

motor

with

large

rotor

resistor are short-circuited while single

-pha

H('

voltage is applied between

the

short-circuit connection

and

the

remaillillf

~

st

a

tor

terminal

[G23].

This

is one of several possible connections having silJli

lar properties.

The

analysis of

the

circuit is easy with

the

help

of

symm

etric

il.1

U

lInponcnts

.

J.llt,rodll<'ill{~

t.1l!:

c

Oll

st

raint

s

()f

t.

hc

circlIle

(/

I ::

/I

.·

111'

/I

'

.1

:1

.-

{)

(I().I(;~)

illl,

..

I'~q

t!

(III

H !,) ,

(10

I

Iii)

[,

":

11111.:

1

iII

4

10

10.

Symmetrical

Three-Phase

AC

Machines

1 _ 1

Us+ =

Us-

=

"3

UAB

-

-v'3

UAM

,

(10.163)

illdicating

that

the

symmetrical

voltage

components

are

of

equal

magnitude,

illdependent

of speed. Hence,

the

two machines

in

the

equivalent

circuit,

Fig

.

10.24,

would

have

to

be

supplied

by

constant

and

equal

three-phase

voltages

with

two

terminaIs

of one

motor

interchangedj

this

renders

the

superposition

(,f

the

two

opposing

torques

particularly

simple.

úJ

,

úJo

,

1

, -

" m

M

+

',m

po

I

\

m

M

. \

"

,

(b)

Braking

with

fn

po

\

De

exeitation

\

------"-\---

m

_

1.

\

1.

3

\

m

po

\

,

3

,

(a)

Single phase braking

,

-----~

-1

':'~--\;)S;m~etrieal

voltages

reversed sequenee

"'l

IJ:.

JO.30.

Torquejspeed

curves

of

AC

motor

in

unsymmetrical

connection

'l'he

torque

curves

are

depicted

in Fig. 10.30

assuming

Sp

= 2.0

caused

by

;\

. large

external

rotor

resistor.

Due

to

the

reduced

voltages

the

peak

torques

i

UT

only

kof

the

nominal

peak

torque,

hence

the

motor

is

again

only

partially

IIl.iliscd. Since

the

resulting

torquejspeed

curves lie

in

quadrants

2

and

4

the

drive

can

only

operate

as

an

electric brake,

regeneration

is

not

possible

with

I.I.is

scheme.

An

advantage

of

this

connection, when used as a lowering

brake

in

crane

applications, is

the

fact

that

the

resulting

torquejspeed

curve (a) passes

I.I.

rough

the

origin;

unintentional

speed

reversal (raising)

at

light

load

does

1101.

occur

as would

be

possible

with

symmetrical

supply

and

the

rotor

re

sis-

I.allcc

sLill

furl'.hcr illcreased.

SillJilar

hrakill

~

:

dmfacleris(,ics

aJ"(

~

liI/I.il.ill,'d with H

sllIaller

ro(,or

J"(~SiH-

1.1~1I1·"

11

'y

r,'('djllf

~

I,II!

' til,nl.ol" windiJlI{ wi

l.It

,11,

'

("<'1,

(' lI.rT

'·

.fJl.

(h).

TIl('

lIlol.or

1'''''

11

"1'

('1'11

1."

' 1 r

l'

i 1

111

\'

dd

,v

1'

111

('('

111.

I,rl

l.h

'.

10.4

Induction

Motor

with

Unsymmetrical

Line

Voltages 211

Finally,

Fig

. 10.30 also

contains

the

torquejspeed

curve

with

symmetrical

voltages

but

inverted

sequence

and

short-circuited

rotor

resistance (c).

This

operation

allows

regeneration,

however

at

higher

than

synchronous

speed

in

lowering direction.

It

could

be

used as

the

"Full speed, Lowering"

set-point

of

a

contactor-controlled

AC

crane

drive.

10.4.4

Unsymmetrical

Starting

Circuit

for

Induction

Motor

Smooth

acceleration

during

start-up

is

important

in

some

applications,

such

as

textile

machines, where

there

is a

danger

of

breaking

fibre

threadsj

an

asymmetrical

circuit

like

the

one shown

in

Fig. 10.31

can

provide a

sim

pIe

solution

for

this

problem.

An

external

impedance

ZI

is

inserted

between a line

phase

and

one of

the

stator

terminaIs

of a

symmetrical

cage- or

wound-rotor

induction

motor;

the

circuit is one

of

several

similar

connections.

Assuming

the

impedance

ZI

to

be

constant,

all volt ages

and

currents

remain

sinusoidal

in

steady

statej

the

following

equations

hold:

ZI

I

Sl

+ U

12

=U

AB

=

(1

-

e-h)

U

AM

,

(10.164)

U

23

= U

BC

=

(e-h

-

eh)

U

AM

.

(10.165)

A

!J

23

=

!Jae

Fig.

10.31.

Unsymmetrical

starting

circuit

The

s

tat.or

current

can

be

expressed

with

symmetrical

components,

Eq.

( I

O.

1

~H),

;;,

I ' :i I

;;,

I

1\',

.

,'

I

1/

,'

.

,'

I I Y·; (l

·.'

.I ;

(10. I (iii)

~!

I

~~

10.

Symmetrical Three-Phase

AC

Machines

likcwise

the

line-to-line voltages

obtained

from Eqs. (10.143)

are

U

12

= U

51

- U

52

=

(1

-

e-h)

U

5+

+

(1

-

eh)

U

5-

,

U

- U - U - (

-h

-

h)

U + ( h -

-h)

U

-23

-

-52

-53

- e e

-5+

e e

-5-

.

811bstituting these equations into Eqs. (10.164, 10.165)

and

solving for U5+'

lT.

s

_

results

in

U - 3 +

Zl

Y

s

-

U

(10.167)

-5+

- 3 +

ZI

(Ys+ +

Ys-)

-AM

,

u =

-Zl

Ys+

U.

(10.168)

-5-

3 +

Zl

(Y

s

+

Ys-)

-AM

'l'ltis contains

the

limiting cases

of

the

symmetrically fed

motor

(Zl

=

O)

aud

the

single phase

induction

motor

(Zl

-+

00)

discussed

in

the

ln

general,

the

positive

and

negative sequence voltages

are

com-

plicated functions of

the

motor

admittances

which in

turn

depend

on

speed

and

pull-out slip.

The

volt age across

the

series impedance

Zl

I

S1

=

Zl

[Ys+

U

5+

+ Y

s

- us-l =

-3

u

5-

(10.169)

is

a measure

of

the

asymmetry

of

the

stator

voltages.

With

these results

the

phasor

diagram

of

the

motor

can

be

drawn for

any

chosen

operating

point.

Tlte

torquejspeed

curves

may

be

computed

from Eq. (10.157);

their

shape

dq)ends

on

the

impedance

Zl

and

the

motor

parameters,

but

they

lie inside

i.Ill~

:IIea

which

is

bounded

by

the

curves for

Zl

= O

and

Zl

-+

00.

This

is

qll:dil.atively indicated in Fig.

10.32 for a low-power induction

motor

having

I'ni

rly large pull-out slip

and

assuming a resistive impedance,

Zl

=

RI.

Of

('1111

rsp,

this

and

all similar

arrangements

are associated

with

additionallosses

iII t.hc

external

components as well as

the

motor

itself

and

are only

suitable

ror

short

duty

operation.

The

function of

the

circuit shown in Fig. 10.31 is

further

illustrated

by

I IiS!:\lssing

the

phasor

diagram

of

the

stator

volt ages in locked

rotor

condition.

AI,

8 = 1

we

have Y

s

+ =

Ys-

=

Ys(l).

Hence

it

follows from Eqs.

(10.167-

10.169)

with

Zl

=

RI

R I = 3

RI

Ys(l)

U

O

:S

RI

<

00

;

(10.170)

.

.L

-SI

3 + 2

RI

Ys(l)

-AM

,

as

a limit.ing case (RI

-+

00)

this

includes again

the

single phase

motor

hm

.'

(l{IL

";

I) =

'l

3

U

AM

,

(10.171)

U

1

H'XI

wil.11

1.11<'

1l11

1l

.'

:

oJ'

d i a

l'

/H

III

(Ir

thc

f

;l.a

l.or

vO!l.n

W·

!4

ai,

,<.,',

I d

q(

' 'II ('I't

ll.

i

111':

1

111,(1

:.

11

1.

1'

11

,1

1'

,

111.1

1

11

1

'

(

10'11

'

\'

IO

,:I:\)

1':'1

, (

10

, I'/(l)

1'

1'/11'1'1

11

'111,::

li

,

lilll'

llI'

1'

1I11.-1.i'lll

wil

ieh

10.4 Induction Motor with Unsymmetrical Line Volt

ages

213

maps

the

real

Rl-axis

into

a circular

arc

in

the

complex plane,

extending

from

RI

= O

to

RI

-+

00.

When

the

rotor

is running, O < 8 < 1,

the

phasor

diagram

changes

with

the

tendency

of

reducing

the

volt age

R1I

s1

,

thus

achieving improved symmetry.

This

corresponds

to

the

situation

found

with

the

single-phase motor.

úJ

o

mM

m

pQ

Fig.

10.32.

Torquejspeed curves of unsymmetrical starting circuit

for

large pull-out slip

A

{\--Rj=O

ZI=R

I

S=1

\AB

AR

3,C

R

I

=oo

2,8

!J

Be

Fig.

10.33.

Phasor diagram of asymmetrical circuit at standstill

The

shaded

area

AR

of

the

triangle (Fig. 10.33), formed by

the

terminal

voltages fl

12

, U

23'

~l

at

standstill

(8 =

1),

can

be

expressed as

AR

=

~

1m

[fl

23

U;ll

;

(10.172)

whjeh

hy

illsI'l'l.illl';

Eqs.

(10.145, 10.14ô) ma,y

be

written

as

ii

'"

I,;

III} 1

(!?,

I,

(10,'173)

"

;, I '1 I

()

. SymmetricaI

Three-Phase

AC Machines

A co

mparison

with Eq. (10.157) shows

that

AR

is a measure

of

the

torque

at

standstill

[28J;

it

vanishes for

Rl

-t

00

and

assumes its

maximum

value

at

/lI

=

O,

i.e.

with

symmetrical supply.

The

circuit shown in Fig. 10.31 can

be

extended by

additional

controllable

i

III

pedances

to

allow continuous speed control, even four-quadrant

operation

hlll.

llot regeneration. This has been realised

in

the

past

with

saturable

reac-

lo!s, the impedance

of

which could be varied by direct current

in

specially

c, >llllccted control windings. Because

of

the

high losses these drives were re-

~;I.!ictcd

to

short

duty

operation; also the control was

rather

slow due

to

the

si

11J';gi

sh response

of

the

large reactors, making this

type

of

equipment obsolete

I.<>day

[L29J.

A similar control effect, however

with

greatly reduced size

and

("os

I.

of

the

components as well as much more

rapid

response

can

be achieved

hy

ill

sc

rting

antiparallel thyristors into the

stator

supply leads

of

the

motor;

l.

!t

is

causes considerable nonlinear distortion

of

the

motor

voltages, resulting

i

II

yet increased losses

but

it

may

be

acceptable

in

short

duty

applications.

11.

Power

Supplies

for

Adjustable

Speed

AC

Drives

An

important

result

of

the

preceding

chapter

is

that

the

application

of

AC

motors in continuous

duty

adjustable speed drives calls for

static

converters

of

adequate

power, generating three-phase volt ages

of

variable

amplitude

and

frequency.

This

is necessary

to

maintain

at

all speeds a low

rotor

frequency,

which is a precondition for acceptable overall efficiency

of

the

drive. Con-

verters

of

this

type

are available

today

employing thyristors,

gate

turn-off

thyristors

(GTO),

or

switched power transistors

(MOSFET

and

IGBT),

but

the

complexity

and

cost

of

the

converter equipment

and

control still exceeds

that

of line-commutated converters

of

a similar rating. While

part

of

the

cost

for

the

converter can be recovered by

the

reduced cost

of

the

AC

motor

as

compared

to

a

DC

machine

of

comparable

rating,

the

overall cost

of

the

AC

drive

may

still

be

somewhat higher

than

that

of

a

standard

DC drive.

Of

course, AC drives

exhibit

a number

of

important

advantages,

most

promi-

nent

in

the

case

of

an induction

motor

with cage rotor,

that

will inftuence

the

decision

in

their

favour:

• Power

rating

and

speed

of

an

AC

motor

is

not

limited by a mechanical

commutator,

• Reduced

axiallength,

volume

and

inertia

of

an

AC

motor,

as seen in Fig.

10.1,

• Rugged design, reduced maintenance

and

service requirements,

• Drive is applicable in explosive

and

contaminated

environment.

The

last features are

of

particular

interest

with

traction

and servo drives

where

many

driven axes are incorporated into

the

machine tool or

robot.

Of

course, there are also applications

at

high power

and

speed, where

DC

drives

could

not

be

built

and

only AC drives can provide a solution. Since

the

cost

proLlems will diminish as

the

converter-

and

AC drive- technology

matures,

it is likely

that

DC

drives will eventually

be

completely superseded by

the

m

or

e

adv

anced AC drives.

Th

e basic lay-

out

of

an

adjustable speed AC drive is shown in Fig.

11

.

1;

iI.

collsists

of

an

AC

motor,

a

stati

c converter, which generates a variable

voU

.age, variable frequency AC system and

the

associated control equipment.

'I

'II<

: i

11(".)"(

':

1:-:

«:<1

col1Ipk

x:

it.y

of

th

e

c:onv

erter is callsed by

the

fact

that

inver-

I."I'

H

wil.h

fOI'( ·,'d

("()

li

11

II

Latio

l1

ma

y

n:q\lÍn

' :ul<lil.iollal component.s

and

a more

"o

l

lll'l

i" l

d,

,·d

IIIO.!"

01'

,,))("r

lt.

l,ioll.

Si

lllllll.l· I\1 1':

11111<'111.:

4 iI.l'l'l.v

al~o

1.0

1.1«:

SI,I"II<:

I.III'(

:

______

}I(i

11. Power Supplies for

Adjustable

Speed

AC Drives

'J'1I1,lc

11.1.

Adjustable

speed

AC drives, synopsis

UJ

1'1

tJ

'"'

W';3

°

e-

~

m

ç1

;.::::

~

8..<:1

13

2'~

13

u ". °

;,.,. 0

u

•

I H S

~

t) J;

13

Q)

....

..<:10

t-'

.~

Q)

-+J

<.J

~

H

;>

'>

I

e-

~§"'~

§

~

o

..<:1

U o.

~

..<:1

°

....,

UI,

. - S

ç1::;~~S

~t;O~o

.....

-

t-<

....,

U

::Jt::,CJ

~O)

UI;...·~;>

U

'"

tt1

I 8

o

ç1

tE

....

" o

,

1

"

-,

~

,.)

u

....

'

, n

'

-,

'

Q)

~

U

;

~

o

....

.

• I

~/

~

.

~I

I-'

;:

I

"

I ,

,,

'

I'

~'

Ü

~

U

r1

:~

íl

, ,

'--,

r

O

i).1

I"

:1

110

""'"

~

(I)

'

''

~

....

~

.

~

§

'

~

e-<

°

,...

....

--u

'"

J,

'1

I:

'u

I-

(,,1

"

VI

V

O)

S s

....

~-ro~

~-:.

~~

~ro~

~iSal~

~~]~~~~~

0"",0.0.

0.,<:;

'"

• 0.2:.-::;

....

0.2: d

(fJ

..<:11.()~ç1

..<:1013].<::

~>,

bOo;'::::

Q)

.bOl.()

O"d

.~~

0.8

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II

)

-

11. Power Supplies for

Adjustable

Speed

AC Drives

217

ContraI

Fig.

11.1.

General scheme of AC

motor

control

of

the

inner

torque

controlloop

which is much more complex

than

for a DC

motor.

The

reason

is

that

an

AC

motor

with its simple mechanical design

represents

an

involved nonlinear multivariable control

plant.The

motor

must

be fed with

alternating

currents of variable

amplitude,

frequency

and

phase;

in addition,

the

rotor

currents

cannot

be measured with ordinary cage motors.

This

is in

contrast

to

a DC machine, which has a very simple decoupled

control

structure

but

a more complex mechanical designo

The

mechanical

part

of

the

control scheme is,

apart

from

the

smaller

inertia

of

AC

motors,

the

sarne for DC-

and

AC-drives.

Fig. 11.2 indicates

that

the

mechanical

commutator

of a DC machine

acts like

an

electromechanical converter which feeds

the

armature

conduc-

tors

with

alternating

current

of a frequency

proportional

to

speed, Fig. 5.2 .

On

an

AC drive,

the

functions of current conversion

and

torque

generation

are mechanically separate,

thus

creating new options for electromechanical

interfaces .

Supply

DC - machine

/

~.--_.-.-._-----_.----

U

o

'

úJo

constant

Commutator

Line-sde

converter

1-

DC

- link

-----+-1

/

AC

-

motor

converter >

0

I

machine-side I W

....

____

---I.

UI'

úJ

I

L-

___

~

variable

I

~--------------

I

•

mM'

úJ

1

I

Fig.

11.2.

Controlled

AC-drive

with

DC-link

converter

Table 11.1 presents

an

overview of controlled AC-drives, showing five ba-

:;ic fonns of power electronic converters for changing from

the

fixed line-

to

v;l.rial'](',

1I10Lor"

frcqm:ucy.

FOlll'

of til(: circuits contain a DC-link for decou-

1'1

i "I',

1.11('

I.W() :;id4'1 :

411'

I.!t\:

CO

\lVerLCT , w

114'1'(

';\:;

I

,

h(

~

cydo-collverter performs

this

"114'1

ii. Li

4111

iII

;

1.

:1

;111' .

.100

C'O\lVC'rnillll

:il,

;lI'.4'

f\

II

01'

1.

114'

4'nllvc'rl.4'l'

:-1

are

;t.dapl

,

;~hk

to

219

218 11. Power Supplies for

Adjustable

Speed AC Drives

four-quadrant

operation,

they

could also

be

used

with

variable speed genera-

t.ors, for

instance

in wind energy plants.

ln

the

vertical column

of

Table 11.1

li

ve basic

types

of

AC-machines

are

listed

that

may

be

used for controlled

dr

ives;

with

the

exception

of

the

reluctance

motor,

which is characterised by

salient poles,

they

can alI

be

treated

with

the

mathematical

model derived

iii

Sect. 10.1.

ln

principIe,

most

of

the

25

possible converter-machine combinations could

1)('

r

ea

lised

but

only those, where comments are entered, are presently applied

i

II

practice; since each

type

exhibits

particular

advantages,

it

is unlikely

that

I.h(' lield will eventually narrow to

just

one

or

two

major

circuits as has

been

t

.

It(~

case

with

DC-drives.

ln

fact, new combinations

may

be

added

as new

~;lViLcbing

devices

and

converter circuits

appear

on

the

scene. An example

of

Ih('

oIlgoing development

are

resonant-línk converters, presently

restricted

to

Imv

power applications such as switched mode power supplies; cyclo convert-

crs with forced

commutation

(matrix

converters)

are

also a possible

future

opLioll.

Only a

few

typical converter topologies

that

show promise

to

stay

\ViII

be discussed in this chapter.

11.1

Pulse

width

modulated

(PWM)

Volt

age

Source

Tr

ansistor

Converter

(IGBT)

TI\(~

rOIlf-quadrant

DC/DC

converter shown

in

Fig. 9.13 a

can

be

extended

by

aclclillf~

\,wo

more

arms,

as seen in Fig. 11.3,

to

form a three-phase converter

fo

r 1."(' slIpply

of

AC-motors

in

the

lower

and

medium

power range, from

::11111.11

hiV;h

d'yl1amic performance servo drives with speed

and

position control

(. I

()

I,

W)

Lo

lIloSt auxiliary drives in industry, ranging up

to

several

hundred

li

'v

V,

II\.

Itil;hcr power

this

two-Ievel circuit is also extended to multi-leveI

('

oll/il',1I

r:l.t.iOllS.

The

converter is

suitable

for supplying induction as well as

Il

ylI('h

\(

)1l01lS

motors.

W

iLh

Cite

voltage source converter shown,

the

terminal

voItages

of

the

mo-

\'()f' (

';

\.11

assume

the

three possible values

UD,

O

and

-UD.When

the

motor

is

r('I'.(

·

II(

'l'

at.illg,

the

mean

link

current

I

D

is reversed, calling for a

two-quadrant

:;lIl'pl'y of

the

DC link. As long

as

the

power levei is low

or

when regenera-

Lioll

OCCIll'S

only occasionally, for instance

during

dynamic

braking

of

a servo

drive',

Lh

e DC link is normalIy supplied by diode rectifiers

and

the

reverse

I"

,wcr

is

clis

s

ipated

in a balIast resistor,

Fig

. 9.14, as indicated also

in

Fig.

I 1.:\ w

iLh

dashed lines.

The

pulse-width

modulated

ballast

circuit

then

be-

I'.IIIS

L()

di

schargc the link capacitor as soon as

the

link volt age rises above a

pr('s('\.

value.

Of

course, any

other

source

of

direct voltagc could also serve

1;)1

I'(

'(

'dillg t.hp converter, such as a

battery

011

an

electrically driven vehick,

wl\('re

r(~I

~:

('II(

~

rntion

is il1lJ)ortant

t.o

sav(

~

(m('rr~.Y.

Wi\.h

Il

ipolnr

t.mllsi s

t.!ll'K,

Lhe

:;wit.dlÍlI

~~

fr('qlwlI('.y

i:-;

lilllil.(~dto

n

r{

~

W

1\1

-

/(

"

IlIll.

\\'il.ll

low

1"'111'('1'

11I:;III:t.I,(

,

·cI

C a

L{'

1111'01

1

11

'

1'1'

1I.

1I1

1

íl~I

.

"

r

H

(leIIT),

H

:-i

:;

ho

w

lI

iII

11.1

Pulse

width

modulated

Voltage Source

Transistor

Converter

U

o

io

-',

: : R

B

" I

,

I

I'

,

I~TB

.....

I

i

SI

i

S2

iS3

Fig.

11.3.

Switched

IGBT-

converter for

three-phase

AC

motor

(simplified circuit)

i

SI

Fig. 11,3,

or

Field Effect

Transistors

(MOSFET),

having a switching

time

in

the

order

of

a microsecond,

the

mean

switching frequency

may

be

above

16 kHz, i.

e.

beyond

the

audible threshold, so

that

no objectionable acoustic

noise is

emitted

from

the

drive.

An

additional

advantage

is

that

the

converter

then

exhibits a large

bandwidth

for controI.

If

the

link volt age UD is

of

sufficient

magnitude,

fast

current

control loops can

be

designed which keep

the

stator

currents

in line

with

the

alternating

reference values;

this

effectively

produces

near

ideal

current

sources for

the

stator

windings

of

the

motor,

thus

eliminating

the

effects

or

the

stator

volt age

equation

Eq. (10.50) on

the

dynamics

of

the

drive. As a consequence, considerable simplification

of

the

drive control is achieved because

the

stator

voltage interactions

are

then

dealt

with

by

the

current

controlIers. These could

either

employ pulse-width

modulators

operating

at

constant

frequency

or

simple

On/Off

comparators

having a

narrow

hysteresis

band,

also

indicated

in

Fig. 11.3.

The

resulting

current

waveform

with

a single-phase load

and

On/Off-control (symmetrical

modulation)

was shown

in

Fig. 9.16;

it

has

the

advantage

of fast response

but

exhibits variable switching frequency which

may

be

undesirable in view

of interference problems.

When

extending

the

principIe

to

three-phase

loads, as

in

Fig. 11.3,

the

hys-

t.eresis

band

is helpful

to

avoid interactions between

the

controUers because,

in

vicw of

the

isolated

neutral

of

the

machine winding,

the

stator

currents

ar

e balanced

and

one of

the

controUers

is

redundant,

it

is only

retained

for

sylllmetry.

To

produce a prcfcrred switching freqllency, a

triangular

clock signal

may

Iw

added

aI.

LIt

~

\l\lJl\lillg

poillt

or

t.Jj('

cmrclIL cnntrollers [829].

Another

pos-

::ihilit

.y j::

1.1)(

,

>1

YII('hlOllj'l,aLwJI

(lI' UI('

(,

()IILIOII('\"

~:,

i.t:.

:dl()wiJli~

a

t,rallsj(.joll

to

Werner Leonhard Control of Electrical Drives

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