Werner Leonhard Control of Electrical Drives

11111

!)

, (:olltrol of

Converter-supplied

DC

Drives

'Nb

y

!lN

ir-

i

ON

',

':!

'NV/

lN

k

~;

1'15

2

ii

t

)

-1

UN(t)

iit)

=

bN(t)

--

-

-.

...

-.-

.

5hort circuit Transfer

interval Interval

5

1

=5

2

=-1

5

1

=1,5

2

=-1

Jo'

i

~

.

9.1

7.

Principie

of

a single

phase

line-fed

four-quadrant

converter for

supplying

;)

1)( : lillk

'/'N

,

L N -

dt

+

RN

ZN

=

UN

-

UB

,

(9.10)

;1,lId

,Á

'I/,f)

, ,

'

=

~DN

-ZDL,

(9.11)

dI.

w

lwJ'\

>

'i/>/,

is

the

load-side

link

current;

furthermore

11./1

.:

(,<"

1 -

.5

2

)

uD

und

iDN

=

~(51

- 8

2

)

iN

(9.12)

I"

del

::,

II pprw::illlately

sinusoidalline

current

i

N

can

only

be

achieved

if

the

1

)(:

lildí,

11:'t

l<dly

operated

at

constant

volt

age

UD,

can

also

be

supplied

with

\'

II<'II"

,Y

w

ll'>1I

lhe

line

voltage

is

much

lower

than

UD,

i.e

.

near

the

zero

in-

I.I

'I'f

il

>('l.ioIlS

oe

'

U'N;

hence

it

is necessary,

as

shown

in

Fig. 9.17 b,

to

create

',

t'rll

iIll.crv;ds

of

the

volt

age

UB

for

storing

in

the

line-side

inductance

energy

I.Il

il.

1.

i:

,

slli>sequ(

~

ntly

transferred

in

the

form

of

a

current

pulse

to

the

link

I'II,p;u

'

iI.OI,

"'hus

it

is

necessary

to

employ

a

unipolar

or

cyclical

modulation,

w

I

w

J'\

~

I.hc COllverter

bridge

is

temporarily

short

circuitcd

at

a

DC

bus.

'1

'

11(>

switchillg

fr

c

quency

of

the

convert

er is best c

hosen

as

an

integer

1IIIdl.ipl(~

II

of

the

line

frequency

to

avoid

in

st

ea

dy

st.at(~

s

llhharmonics

of

the

l

i

lll

~

1'11/'1'<:111.;

with

a 50

Il7.

:m

pply

and

a clode

fn

:IlIIl'Jlcy ror

iu

sta.nce

of

2

kH

z,

n til)

swiLchillf,';

(:yd(~f>

would

fit

illto one

lillC

voll.iI,l'

:

(>

Jlcriod, rcsultillg in a

r

.d

rl.v

:n

llool.lt

w;l.vd<mll

()f

Lhe

('.111'1'<:111

:,

II

pll

:-ls

i

lll,'

co//Crol ;'

idlCIIll!

i

;{

:,

!t

U

WIl

iII

I"

i,,;

.

n,I

'

H,

whl'l'

tj

II. volt,ag., c

OJl

t

rol

1""11

I

~

II'

'

1/,11

iH rHlpI'I'illll)(IH,'d

OH

ali

iIIU

(

'l'

"

UI

T

I'

fll

'''

(

'I

II

1'\11'

'/:N

,-

'1'1

-

1\'

I'IIlTI

'

IJI.

,,'I

,'rl'

II/'I' ; N

11

,,1 i!

:1

d\,

ri

VI'd

1"

1'0111 t,h" li

"I'

v

ll

lt

HI'.~

'

ffll

~11,

I,nilli

l,,';

Il. H

il\

w,,,idal

Ii/\I'

"

11111

"

111.

11'(

" " '

111

"'" !-III'

IlI

ftl\

l

.!

I.I\/'" ,

II'

wld,

I,

ln dnl"I'

lll

hll'd

I,

y

LI

... v

..

II

';"F,"

9.2

DC

Drives

with

Force-co

rnrnutated

Converters

161

controller.

When

shifting

the

current

reference

by

to

with

respect

to

the

line

voltage

UN,

a

reactive

component

of

the

line

current

results.

Caused

by

the

single

phase

supply,

periodic

fiuctuations

of

the

link

voltage

UD

at

twice line

frequency

cannot

be

avoided

but

are

greatly

diminished

by

inserting

a

resonant

círcuit

tuned

to

double

line frequency,

as

is

indi

c

ated

in

Fig. 9.18;

this

is

the

usual

practice

on

traction

driv

es.

The

load-side

DC

link

current

iDL

acts

as

a

disturbance

to

the

volt

age

control,

its

effect

may

be

reduced

by

feed-forward

to

the

current

reference.

A

UN(t

+

to)!

U

N

i

N

i

NRef

i

OL

U

o

F

ig.

9.18.

Control

of a single

phase

line-fed

four-quadrant

converter

u / V

U

o

400

f----

U

N

11\

1\

A A A A

1\

A A

i

OL

200

~

l

k\IV

i

A\IV

i

A~Vl

k

~

i

-200

-400

"

';g.

tI.lU.

Si

lllllhL"d

loa<l

rcyC'r

sal

of

1\ lillc-fecl

fOllr

-lJ"adra

nt

convert

er

Ifi

;~

!I

. Control of Converter-supplied

DC

Drives

1\

~

iiIlIlIlated

load reversal is

plotted

in Fig. 9.19,

initiated

by inverting

the

I"IIITI'tll. i

,

lL

, which results

in

a phase reversal

of

the

line

current

in

steady

::Iil.l

c,

i. c. feeding power

back

to

the

line

[13].

The

resonant

circuit shown in

leig.

!l

.I8

is

included in

the

simulation model,

thus

reducing

the

harmonics

w ut.<:nt

of

the

link voltage.

10.

Symmetrical

Three-Phase

AC

Machines

The

asynchronous

or

induction

motor

is

the

most

widely used electrical drive

motor;

its

invention

at

the

end

of

the

last

century

has given a

strong

impetus

to

the

transition

from DC

to

AC

in

the

field

of

generation,

transmission

and

distribution

of

electrical energy.

Its

main

advantage

is

the

elimination

of

alI

sliding contacts, resulting

in

an

exceedingly

sim

pie

and

rugged construction.

Induction

machines are

built

in a variety

of

designs with

ratings

from a few

watts

to

many

megawatts.

Unfortunately,

the

speed

of

line- fed

induction

motors

cannot

be continu-

ously varied

without

additional

equipment

or

without

incurring heavy power

losses.

Even

though

the

problems

of

efficiently controlling

the

speed

of

induc-

tion

motors

have been investigated for decades, ali solutions realisable

until

some years ago were unsatisfactory with

regard

to

complexity, efliciency, dy-

namic

performance

or

cost.

It

is only

due

to

the

progress

of

semiconductor

technology in

the

last

30 years t

hat

suitable

static

frequency converters

can

now be

built

at

acceptable cost,

making

the

induction

machine a ve

rsatile

adjustable

speed drive for

many

applications.

The

substitution

of

the

mechanical

commutator,

by

a power electronic,

solid

state

converter reduces

the

AC

machine

solely

to

the

task

of

electrome-

chanical energy conversion, resulting

in

much higher power density

per

vol-

ume

or

mass.

This

is exemplified

by

two

main

line

traction

motors

,

the

out-

lines

of

which

are

shown in

Fig

. 10.1,

together

with some

characteristic

data.

The

motor

at

left is a proven design,

operated

for

many

years

in

Intercity

locomotives;

it

is a single

phase

series-wound

commutator

motor

as men-

tioned

in

Chapo 6, voltage controlIed

by

transformer

tap

changer.

At

the

right

is a

modem

AC

induction

motor,

fed

and

controlIed by a

GTO

volt-

age source converter. Clearly,

the

new

motor

is

superior

in every respect;

it

is also

suitable

for high

dynamic

performance

torque

control,

thus

alIowing

ful!

utilisation

of

the

adhesive frictional forces between wheel

and

rail. As a

result, a locomotive containing four

of

the

new

motors

can

produce

a similar

traction

force as

the

former

much

heavier locomotive with six single

phase

motors,

th

e torque of which is

pulsating

at

double

the

supply

frequency.

Th

c

tbcory

of

the

induction

motor

under

dynamic

conditions is

somewhat

iuvo

!V<'d

IlJ

:c

itlls(:

of

t.h

e

rot

ating

rn

af!,

lId.i

c

lidrl

s,

the

spatial relationships

of

whiclt dl'Jl<'lId

ou

::

1"

'

1:<1

nlld

loadj

i!.

..

11.11

Iw

In':Ü'

:

I[

h

c

l'(~

only

in

silllplifif~d

111

,1

1 II.

Symmetrical

Three-Phase

AC

Machines

1',

>I

'"

\,

()

II

tlIe

other

hand,

the

equivalent circuits, derived for

steady

state

"1)('r:ll.ioll with sinusoidal volt ages

and

currents,

are

inadequate

when dealing

wiUl

I.rallsients or when

the

motor

is supplied from a switching converter.

ln

I.h('

['ollowing,

the

steady

state

condition will

be

treated

as a special case

of

Uw

dyuamic

case.

Th

e

mathematical

model

to

be used is tailored

to

the

needs

of

controlled

d

ri

ves.

It

incorporates

most

of

the

qualitative

features of

an

actual

motor

lilll.

would not,

of

course,

be

accurate

enough for

the

purpose

of

designing

!.II(' Illaehine.

1

.-:

1 - -

--

,

.p.

11200

mm

§f,omm

b)

920mm

,I) , 870 mm

S"

r

i

e:

~

-wound

Three-phase

AC

-

!:1I11I1l11I

tator

motor

induction

motor

Lnll

kW,

7730

Nm

Conto

rating

1428

kW,

9155

Nm

,d,

1~):lIlJllin

-

l

at

1490

win-

1

H~dll

NIIl

Torque

(5

min)

11600

Nm

IlilUI

III;"

Max,

speed

4200

min-

1

:I

~

,

~

,II

h 1 "

Mass

2660

kg

1

~~

II

1'1

'. '

>I

;:

Inertia

22

kg

m

2

F

I,(

. 10. 1. ( ;lIlllparison

of

traction

motors

(Siemens).

(II ) S'

"I·

.I,·

I'I"\.~,,

o;eries-wound

AC

commutator

motor,

(I, )

'l'1>I

'

I"'-pll;c~e

AC

induction

motor

10. 1

Mathematical

Model

of

a

General

AC

Machine

I L i::

..

.

sslIllwd

that

the

stator

S

of

the

machine is a hollow iron cylinder with

,.;

rl'lda,

(TOSS

sectioIl, containing a concentric

rotor

R

$0

that

a llarrow

airgap

"I'

I·{)W;(.alll.

radial length h exists between

the

SIlI()oth

cyJillctrical surfaces

to

wlll<,h

sYlIlIlIl'I.ricaJ

(.!trp

c- plta:;c winctings

of

ll('I

'

:

lil~ihk

JI(~ight,

arp

assumed

to

III'

:d.J.;u

:lwd;

t.Iwir

sJliLl.ial

illllperetlll'llS

c1i

xtrillllli,,"::

IIlay

h(~

thOllght

o[

as

IlI'illl·.

prodll('('d

hy

slIiLallly

plal'cd

I.l,ill s

LI'1\!,,\:J

"I'

ax

ial

('Illldlldol's

or

COll-

""II'Liv,'

:d,,·..I,::. Illlth

1I(

·

nl.mls

"I'

UII'

:iI,i1,r

·

"()IIIl,

,,,\.,·d

wi

lldilll

(:-l

.U:(·

isola!.(·", l.\w

(.I

·,I·

I<I;Il

/

II

:, "

I'

1,

(" fOI."

..

willdÍlIj':

:U

',·

,·

,iI.ll<'l· "",,111" '

1.,,01

I.,!

:

:1

i

l>

l'

i"I''::

1)1

' :;b,,1'1.

('ir

-

t"Ilíl.,·d illl'

,"1

"

II

,v. N:

ó

•

Nu

1

1.1

'

1'

1.1 ...

11111111""'

;0

"I

1,,11

Iti

!.

..!1

1.111

'

1111

'1

11'

''

""

":

«'

10.1

Mathematical

Model

of

a

General

AC

Machine

165

winding.The

permeability

of

the

fully

laminated

stator

and

rotor

iron is as-

sumed

to

be

infinite;

saturation,

iron losses, end-windings

and

slot-effects

are

ignored.

The

following discussion applies to two-pole motors; with multi-pole

machines

the

synchronous speed is reduced correspondingly.

This

effect is

exploited with pole-changing windings, however

at

some loss in utilisation

of

the

motor.

a

0

s

(

ex,t)

, I

a.

-TI

ex

b

~s(ex-4TI/3)

~s(ex)

~s(ex-2TI/3)

S zs: zs: 3

-TI

'S:6~SV

TI7KZ

c

....

'/;71

'

..........

"',\

.. a

o

-

TI

,

...

ti

Fil!;.

10.2.

Symmetrical

AC

machine

(Il)

CrIlSS

~cctioll

wilh

two-layer

airgap

windings,

(I.)

{lJ)wrapp"d slat.or

windings,

(t')

1\"'1)('1'1'1.

1"'(\1-1

di

:·l

l.rihIlLioll

or

xLa.l.nr

willdillgs,

(01)

'I'rllVl'II

;"I',

II.IIIJlI'II'I.III'IIH

W:\VI

'

ntll~(

,

d

h.v

t.Il1"

·

'·

'I''':''''

''

I'

.

II

..

r<:"L~

I{jG

10.

Symmetrical

Three-Phase

AC Machines

The

variables are defined in Fig. 10.2 a. a is

the

angular coordinate in

the

stator

wíth

reference

to

the

axis

of

stator

winding 1 which is indicated by

its

central

turn.

The

centres

of

the

identical windings 2

and

3

are

positioned

at

a =

-y

= 120

0

and

a = 2

-y

= 240

0

respectively. Corresponding definitions

hold for

the

rotor

windings, where

f3

is

the

angular

coordinate, again with

rderence

to

the

centre

of

winding

1.

é(t)

is

the

angle

of

rotation

of

the

rotor,

Il)(~asured

in

the

stator

frame of coordinates, w(t) =

dél

dt

is

the

instantaneous

;tIlgular velocity of

the

rotor.

The

magnetic field

in

the

airgap of the machine

has

radial

direction be-

LtllSC

of

the

smooth

parallel

stator

and

rotor surfaces and

the

postulated

irdillite permeability

of

the

iron; since end-effects are neglected,

we

are deal-

illg

with a two-dimensional magnetic field problem.

The

three

stator

currents

isl(t),

is

2

(t), iS3(t)

may exhibit any waveforms;

I'()r:

r(

~a

sons

of

symmetry

all currents are retained

throughout

the

calculations,

CV(~1l

though one of the currents is

redundant

because, due to

the

isolated

Ileutral,

'is1(t) +

is

2

(t)

+ iS3(t) =O

(10.1)

is

valícl

at

any

instant,

Le.

the

currents are balanced; this applies also

to

Ll _

COllllccted motors.

Wit,h these simplifications

and

definitions,

the

radial

ampereturns

wave

01'

t.11(

~

s

tator

windings consists

of

three spatially sinusoidal,

stationary

terms,

w li

ich

a]'(~

modulated

by

the

currents

2í'r

(-)"

(II

'.

I) ' N S [is

1

(t)

'!9(a)

+

is

2

(t)

'!9(a

-

-y)

+

iss(t)

'!9(a

- 2

-y)]

,-y=

3'.

(10.2)

!\\mrdillg

to Fig. 10.2,

8s(a,

t)

corresponds

to

the

stator

ampereturns

en-

<"1"

,,:

('<1

I,'y

ii.

ra.dial magnetic field line crossing

the

motor

at

angle

a;

because

"I

1.1)('

;1

,

SSIIllICd

high permeability

of

the

iron these ampere

turns

appear

as

III;q

~

"d()ltIotive

forces

at

the

two airgap crossings.

The

nondimensional

spatial

I'rrlll'l,i()1I

- 1

::;

'!9(a)

::;

1

in

Fig. 10.2 c characterises

the

ampereturns

distri-

1'IIt.io/l

()f

one

phase

of

the

stator

winding,

it

could be

written

as a Fourier

::

ni(!:

,;.

Wh

cn

lhe

windings are fed with

alternating

currents, each

term

in

Eq.

(

10.:J.)

o

~

cillat.

es

as a

standing

wave, fixed in space; their superposition results

iII

;t

,

travdling

wave

8s(a,

t), as seen in Fig. 10.2 d.

The

spatial harmonics

rllH.y

he

dÍluillished by reduced pitch and skew

of

the windings, resulting

in

/1

(

II

')

.,

cos cr.

I ty

illl

,roducillg rornplex llotation,

('OS

IY

I ( ,:in 'l .

:lU)

:

~

( ( ,

etc

,

(10.3)

IU

ld

J't·lI.r

,(,H

III·

.

ir'l

~

.

1';'1

.

(I().:~)

1>('

('

OJll(~S

()

'i (II .

I)

,I, N:

.'

1'

:1

(1

),' I " I /7,

(/

) "

I"

1

I f

~

" I 1

(

10.,1)

I

10.1

Mathematical

Model

oC

a General AC Machine

167

where

j2

is(t)

=

isl(t)

+

is

2

(t)

eh

+

iss(t)

e

"(

(10.5)

is a time-dependent current vector in

the

complex plane perpendicular

to

the

motor

axis

and

is(t)

=

is1(t)

+

is

2

(t)

e-h

+

iss(t)

e-

j2

"(

(10.6)

is

the

corresponding conjugate complex vector.

The

ampereturns

8s(a,

t)

are

of

course real, being a measurable physical

quantity

depending on a

and

t.

e

jy

2

'S2(t)

e

fY

is:ft)

!\/isi

t

)e

J2y

N 8

s

(ex,!)

s

Mt)e -ja

a

b

Fig.

10.3.

Complex

current

vector

The

construction

of

the

current vector

is(t)

is shown in Fig. 10.3 for

assumed values

of

iS

l

>

O,

is

2

, iss

<

O.

Magnitude

and

angle each vary

with

time

according

to

is(t)

=

is(t)

ej(t)

. (10.7)

The

current

vector determines

the

instantaneous

magnitude

and

angular

po-

sition

of

the

peak

of

the

sinusoidally

distributed

ampereturns

wave produced

by

the

three

spatially

displaced

stator

windings.

By

combining Eqs.

(1004,

10.7)

the

ampereturns

rnay be expressed as a travelling wave,

the

peak

of

which follows

the

angle a =

((t)

of

the

current vector,

8s(a

,

t)

=

Nsis(t)

cos(((t)

- a) .

(10.8)

Ir

t1w

st at.or currents are sinusoidal

and

form a syrnmetrical

three

phase

s'ysL(~1I1

I

LII(!

alllp

prct.lIfl1s wave revolves with cons

tant

magnitude

and

velocity

(v

I d( /

di,

i

II

I,)

I(

~

ai

q~ap

of

the

mo(;or

.

Siw('

1.111'

CC>lllp)('

X

v('d()r~;

ddillt'd

iu

)';(p

:. (10.;, In.S) rJ('scribe

t,h

e s

pat.i:tl

dl

ll

l.r

'

il'lll

,i

,,"

(Ir

II

1I

,

1I

11 ~

III'I

,

iI'

lidd

ill

II

,

1'""1<'

Iwrp"lIdi('lrlll

.r

I,,,

1.1".

11101,01'

Il.l

i i:l

,

169

1()8

lU.

SyJlllll(~Lrical

Thr

ee

-Phase

AC Machines

they are also called space-vectors or space-phasors [N7, 873, 31]. However,

they

are

not

to

be

confused

with

the

constant

complex phasors for describing

steady

state

sinusoidal

alternating

quantities.

The

sarne reasoning as before holds for

the

ampereturns

wave

produced

IJy

the

moving three-phase

rotor

windings,

0

R

(!3,

t) =NR [iRl(t) cosf3 +i

R2

(t)

cos(f3

-

,)

+iR3(t)

cos(f3

-

2,)]

.

(10.9)

WILcn

defining a

rotor

current

vector,

in(t)

=

iRl

(t) +iR2(t)

eh

+ i

R3

(t) e

i

2,

=iR(t) ei

ç(t)

,

(10.10)

fR(t) =

iRl

(t) +

iR2

(t)

e-h

+

iR3

(t)

e-i

2,

= iR(t) e-H(t) ,

(10.11)

(.]I(~

ampereturns-wave, excited by

the

rotor

currents

and

moving

with

the

ro(.or,

assumes

the

form

en(f3, t) =

~

N

R

[iR(t)

e-

i

{3

+

i~(t)

e

i

{3]

,

real;

(10.12)

i

Le;

drcct. ou

the

stator

is

obtained

by

substituting

li -- n - é )

(10.13)

()

II

(II',

(,

I.) - t

NR

[iR(t)

e-i

(a-c)

+

i~(t)

ei

(a-c)]

.

(10.14)

TI,,·

",:

:"ll.;ut!,

waw

is a

superposition

of

the

stator-

and

rotor-ampere

turns

H(<I',

"

1.)

=

0s(0,

t) + 0

R

(0,

é,

t) .

(10.15)

,';illC"

I.IL<~

pcrllleability

of

the

iron

is assumed infinite,

the

combined magne-

1.01l10!.i

vC

rOJ'(:e

becomes effective

at

the

two

airgap

crossings, causing a local

flIIX

dClIsit.y

at

the

stator

side of

the

airgap,

1

/I,

.

;(n,

s,

t) =

2h

110

[(1

+ 0"5)0

5

(0, t) +

0R(0,

é,

t)],

(10.16)

wli('I(~

(rs

il.CCOllnts

for magnetic leakage;

110

is

the

perrncability

constant.

A

""':til(~d

lIloddlillg of leakage effects is

not

possibk

with lhe simplifications

Jllad('.

WIL<

~

il'

calcll];.d

,

illg

llllx

lillkagcs,

the

sp;l.t.i;t!

di::I

,

ribIILioll

01'

thc conductors

1111

\1;

(.

IIC

(·oll

c.

id(·J'('d

;t:;

ill<li('a,(.(·d

i\l

Fig.

10

.

'1

1'''1'

1.1)('

(·x:l.Il1pk

01'

oue

slat.or

W1

Jld

i.

lll

i_

lIy

I

L"~

IJf

llill!,

:

li

(j1l;l.si-

(·,ol\t.iIlIIOIl:

i dl:i

t.

ri

lllll.i"ll

01'

I.lIrllH

wit.h

ali

"ill-

<'I'l·lW·L

II.

It

[

dv

II

I;: il

"y"

~

l

N

::

('OH

'\,

I.IL(

'

P"

:ll,lIll1.l.,·d

:il

llll

i!t

,j

,11I1

di::I.rlhlli.ioll

n

~~

I1III.

H

;

,I.

t.llI'

II

H!

I'"

t.1111<'

1.11<'

l,oUd 1IILIILlwr ,,!'

1.111

'

11

:1 III N;"

mi

p"" v",d hy

illl."I

',l'I

ül

"'1

10.1

Mathematical

Model

of

a General AC Machine

B(

0.)

Incremental

number

of

turns

N

.2.

cos

Â.

LLA..

2

Fig.

10.4.

Distributed

winding

in

the

interval - i < À < i.

Thus

the

ftux linkage

in

stator

winding 1 is

obtained

by a double

integration,

(10.17)

~s,

(t)

~

N

s

,L

'"'>.l2:

I r Bs(a, c, t)

da]

d>',

where I is

the

effective

axiallength

and

r

the

radius

of

the

rotor.

The

integra-

tion

over °is a consequence of

the

non-uniform field in

the

airgap while

the

integration

ove r À is required by

the

nonuniform

distribution

of

the

winding.

Inserting

Eqs. (10.4, 10.14, 10.16) results

in

1!.

.x+~

2 2 2

'l/J51(t)

=

~~

~

(1+0"5)110

/

[e

j

À+

e-j

À]

/ [i5(t)

e-ia

+

is(t)

e

i

a]

dodÀ

~~

À-~

~

À+~

j

+

N5.~~lr

110

/

[ejÀ

+e-

iÀ

] / [iR(t)

e-

(a-c)

+i~(t)ei(a-c)]

dodÀ.

_~

À-~

(10.18)

The

evaluation

of

the

integral is greatly simplified by

the

complex

notation

<lue

to

the

periodicity of

the

integrando

With

the

abbreviations

for self

and

lI111tual

inductances

.

N~

I r 1

(10.19)

(1

+

0"5)

~

71'

110

=

"3

L 5 ,

N

,r.;

NlIlr

1 M

(10.20)

H h -

71'

Ito =

:J

. ,

II. :.li

llli'I('

1'<'

11

1111.

'

lf

l

Olol.II

,ill('<l

171

170

10. Symmetrical

Three-Phase

AC Machines

'l/JSl

(t) = %

Ls

[is(t) +is(t)] + %M [iR(t) e

j

ê+iR(t)

e-j

ê]

(10.21)

'file

fiux linkage in

the

stator

winding 1 is a physical integral

quantity

and

Illust,

of

course,

be

real.

The

flux in

the

other

stator

windings is

computed

lilcewise,

with

the

exception

that

the

integration over À is shifted

to

, ±

~

and

2,

±

~

respectively, reflecting

the

position

of

the

windings. This results

111

'l/JS2(t)

= %

Ls

[is(t)

e-h

+

is(t)

eh]

+%M

[iR(t)ej(ê-')') +

iR(t)e-

j

(ê-')')]

,

(10.22)

'

tPS3(t)

= %

Ls

[is(t)

e-

j2

')'

+

is(t)

e

j2

')']

+%

111

(iR(t) e

j

(ê-2')')

+iR(t)

e-j

(ê-2')')]

(10.23)

Tlw

symmetry

of these equations gives rise to

the

definition

of

a complex

v(

'd

or

of fiux linkages

!fs(t)

=

'l/JSl(t)

+

'l/JS2(t)

eh

+

'l/JS3(t)

e

j2

')'

,

(10.24)

allowing

Eqs. (10.21)-(10.23) to

be

combined

and

disposing

of

the

conjugate

I.cJ'lIlS

'~

:.'

Jt)

=

Ls

is(t)

+ M iR(t) e

j

ê(t)

.

(10.25)

','his

IIIIX: vector describes

the

magnitude

and

angular

position

ofthe

peak

of

1.1(('

:-;

illllsoidal

flllX

density in

the

airgap

ofthe

machine.

The

exponential

term

aIJ;wllcd

L()

in

(t) indicates

that

the

rotor

current vector

must

be

rotated

by

(.11"

11.1'1,;1('

()r

IIlcchanical

rotation

before its effect

can

be

superimposed

upon

I.hal.

01'

L1w

stator

current vector

is(t).

'1'11('

IIllx

linkage of

the

moving

rotor

windings is

computed

in exactly

the

~:;UII('

w;ty. COllversion of

the

stator

current vector into rotor coordinates by

IIl!'illI~;

or

Eq

. (10.13) yields

(

"

~:;

((J,

c, t) = tN s [is(t)

e-j

(.B+ê) +

is(t)

e

j

(.B+

ê

)]

(10.26)

wilich

produ

ccs a fiux density on the rotor surface, corresponding

to

Eq.

(IO,IIi)

/I}{(!1, c, t) = 2

1

h

/.Lo

[(1 +

O'R)8R(!3,

t) + 8

s

(fJ

, c,

t)]

(10.27)

r'

li

ilf'.ajll

si.a.llds

ror

(",he

tmavoidahle

nmglldic

1r·;i1q

~

(

' .

"d

'

('I~raLioll

aWlIlld

t

,

h(~

(

~

iI'Clllllf(~rclI(,('

(lI'

ti",

rul,ol',

II.

g

ll.

iu

prcsnpposing

a

lil

,

l"ly

d

i

~

tri"IlI.('(ll'()tor

WilldiJlI!',

kad

s

to

1111

(

'

~

pl

'

('

f1:il

o

ll

fo

r til(' JlIlX

lillkitl~(

~

iII

ro

t"r

will"il"'. I,

10.1

Mathematical

Model

of

a General AC Machine

(10.28)

'l/JRl

(t) = tLR [iR(t) +iR(t)] + tM [is(t)

e-j

ê+

is(t)

e

j

ê]

,

which has a similar form as Eq. (10.21). However, as this

fillX

is defined

in

rotor

coordinates,

the

stator

current

vector is now

turned

backwards relative

to

the

position of

the

rotor.

N

2

lr

(1

+

O'R)

; h 7r

/.Lo

= tLR (10.29)

defines

the

self inductance of a

rotor

winding.

Likewise

the

flllX

linkages of

the

other

rotor

windings

are

obtained,

'l/JR2(t)

= tLR [iR(t)

e-h

+iR(t)

eh]

+tM[i

s (t)e-j(ê+')')+is(t)ei(ê+')')] , (10.30)

j2

'l/JR3(t)

= %LR [iR(t)

e-

')'

+iR(t) e

')']

+t

M

[is(t)e-

j

(ê+2')')

+

is(t)e

j

k+

2

')')]

(10.31 )

These expressions

are

again simplified by forming a complex vector for

the

rotor

flux .

!fR (t) =

'l/J

Rl

(t) +

'l/J

R2

(t)

eh

+

'l/J

R3

(t) e

j

2

')'

(10.32)

= L

R

iR(t) + M

is(t)

e-j

ê •

The

magnetic linkages described by Eqs. (10.21)-(10.23) are now used

to

derive

the

voltage equations for

the

stator

and

rotor

circuits, depicted

in

Fig.

10.5.

i

S1

i

R1

i

S2

'l'SI

'l'R)

i

R2

iS3

'l'S2

'l'R2

i

R3

'l'S3

I

'l'R3

RR

Rs

RR

Rs

I

uR)

usd

RR

["R'

u~

l

R,

U

R3

U

S3

_________________

L

________________

_

~

Fig.

10.5.

Magnetic

linkages

and

voltages

The

tine-to-neutral voltages

in

the

stator

circuit

are

.

d'l/JSl

RSZ81 + ----;[t = USl(t)

d'ljJ

S2 _

'tL'i2(t)

,

(10.33)

U:.;

'i.:;

',~

(li.

dl

f'

:;

;1

U ;

'

;"

. 1

1I

;

::

I(t.) ,

ti,

ln

lO.

Symmetrical

Three-Phase

AC Machines

wlwre

Rs

is

the

stator

resistance

per

phase

and

USl,

US2,

US3

are volt ages of

a.,iJitrary waveforms. These equations may again be combined by introducing

("/II\ljJlex

vectors. By formal definition of a voltage vector, eventhough there

is

110

immediate physical

interpretation

as with

the

currents,

"Ils

(t) = USl(t) +

US2(t)

eh

+

US3(t)

e

j2

'Y,

(10.34)

;tllcI

with Eqs.

(10.5,10.10,

10.25), this results in

.

d'IjJs

. dis d

..

"S'/..

5 +

dt

=RSIs+Ls

dt

+Mdt(1ReJé)=:J!s(t).

(10.35)

1'

;lIlploying

the

rules of differentiation, this

equation

assumes

the

form

dis

di

R

· .

ll

s

is

+ L S dt + M dt e

J

é + j w M iR e

J

=

:J!s

(t)

(10.36)

é ,

w!tere w =

dê

/

dt

is

the

angular

velo city of

the

rotor.

The

two

terms

containing

t!t(,

rotor

current

may

be

interpreted

as voltages due

to

mutual

induction

and

r/

lI.ation, respectively.

l1y

inserting Eqs. (10.21-10.23) into Eq. (10.33)

and

considering Eq. (10.1)

whic:h

is

due

to

the

isolated neutral,

it

is

found

that

the line-to-neutral volt-

;

tf~('s

aI'

the symmetrical machine are also balanced

at

any

instant,

'/ISI

(t) +

US2(t)

+

US3(t)

=O .

(10.37)

()IJ

IIIOst

motors the

neutral

point

of the

stator

winding

is

not

accessible,

;:

0 J.ltaJ.

oltly

the

terminal voltages

US12

=

USl

-

US2

etc. are available for

/'XI,CTII:d

IIS(\

while

the

line-

to

neutral

volt ages are dependent quantities.

'1'11

i::

COIT(

:s

ponds

to

an

actual

situation, where

the

motor

may

in fact be

L1

/·/)/III('CI.(·d

.

With

the

terminal volt ages

U12

=

USl

-

US2

etc.

the

voltage

VI'/'LIlI'

iiI

Eq. (10,34) can also be

written

as

(/

::

(/) =

1t12(t)

-

U23(t)

e-h,

(10,38)

'l'1c('

S<t.IJW

arguments are valid for the rotor winding, where

the

currents are

:tlso

lJ<t.lall

ced, due

to

the

assumed fictitious

neutral

point

i

lll

(I)

+ iR2(t) + iR3(t)

==

O ;

(10,39)

/.Iw

clashc:d

lines connecting

the

neutraIs in Fig, 10,5 are

redundant

in reality.

11/

:

111:('

t!te voltage equations are

"

(l1/Jra

Ic./I'/./li

+ - ,- =

uw(t)

,

dt

({c/I

11.'1.

fluiu/.

·

UII~(i.)

,

ri/.

(10.110)

11

1/'

/,

' .1

11'1l11l

~ 1

'1/

/n

(I) .

,(/

10.1

Mathematical

Model

of

a

General

AC

Machine

173

With

the

help of Eqs. (10.10, 10.32) they are

again

combined

to

form a

vectorial equation,

.

d'IjJ

R . di

R

d.

_ .

RR

IR

+

di

= RR

IR

+ L R

di'

+ M

dt

(1s

e J

é)

=

:J!R

(t)

,

(10.41)

where alI variables are defined in

rotor

coordinates.

The

vector of

the

rotor

voltages,

:J!R(t)

= URl(t) +

UR2(t)

eh

+uR3(t)e

j2

'Y

(10.42)

may

be impressed by

an

external

voltage source.

ln

case of a cage

rotor,

the

voltages are zero

and

the

rotor

circuit

is

internalIy

short

circuited.

The

vector differential equations (10.35, 10.41) describing

the

electromag-

netic interactions of

the

symmetrical

AC

motor

in

steady

state

and

transient

condition are now

to

be supplemented by equations for

the

electrical

torque

and

the

mechanical transients.

On

a real

motor

the

tangential forces are act-

ing

at

the

sides of

the

slots, where

the

magnetic field enters

the

iron. Since

there are no slots

in

our

simplified model with

airgap

windings,

the

torque

is

computed

with

the

help of

the

tangential Lorentz- forces exerted

on

the

ax-

ial

current

carrying conductors orthogonally crossed by

the

radial

magnetic

field.

The

component of

the

flux density

on

the

rotor

surface which is caused

by

the

stator

currents folIows from Eq. (10.27)

B

RS

((3,

é,

t) =

N:~o

[i

s

(t)e-

j

(!3+é)

+

is(t)

e

jUHé

)]

. (10.43)

The

magnetic field produced by

the

rotor

currents themselves does

not

gen-

erate

any

net

tangential

forces with

the

rotor

currents

since there

is

no reluc-

Lance

torque

with

the

uniform air-gap; this could be shown by carrying

out

t!te caIculations with B

R

instead

of

BRS.

The

current

distribution

aR((3,

t)

along

the

circumference

of

the

rotor

is

;t!so,

with

the

assumptions made, sinusoidal. According to Fig. 10.6,

it

is

cldined as

the

spatial

derivative

of

the

rotor

ampereturns,

-

~

8GR((3, t)

__

. NR [. -j{3 _

.•

j{3]

((3

t) (10.44)

aR

, - 2

8(r

(3)

- J 4 r

lR

e

lR

e .

T!te tangential force df acting

on

an

axial

strip

of width r

d(3

of

the

rotor

sur-

,

'

aC(~

is

the

product

of radial flux density

and

axial

rotor

current

distribution

d! =

-B

RS

((3,

é,

t)

aR(!3,

t) 1r

d(3

,

(10.45)

w!tidl Ilpon

int

eg

ration

along

the

circumference yields

the

electric

torque

in

!.II(' clin:ctioll of rotation,

271'

'III'M{I.)

l'/IU

,/:lZ

( Uus

(f'i

,

It

,

/.)

11.11((1,

/.)

d(1

.

( 10.46)

/,'

o

J74 10.

Symmetrical

Three-Phase

AC Machines

1nserting Eqs. (10.43, 10.44) results in

2".

mM(t)

= -

M.

/

[is

e-

j

(t3+

ê

) +

i*

e

j

(t3+

ê

)]

[iR

e-j

13

- iR e

j

13]

df3

,

6 'Ir J

-s

o

(10.4

7)

where M

is

the

coefficient

of

mutual

inductivity defined in Eq. (10.20).

When

integrating

over

the

full circumference of

the

rotor,

the

exponential

terms

containing

f3

canceI. Hence

we

find

j

M

/2".

is

iR

e-j

ê -

fSiR

e ó

df3

=

~

MIm

[is(t)

(iR e

j

Ó)*]

mM(t)

= - 2 . 3

3 'Ir J

o

(10.48)

the

imaginary

part

of

the

bracket

is

equivalent

to

a vector-product, being

proportional

to

the

product

of

the

two

current

vectors

and

the

sine of

the

angular

displacement

X,

as seen in Fig. 10.6 b.

dr

a

R

(f3,t)

B

Rs

(j3,t)

is

ic

~

e

a

b

Fig.

10.6.

Rotor

current

distribution

and

torque

When

assuming

that

the

stator

and

rotor windings have

an

equal

number

of turns,

N~

= N

s,

and

introducing

the

usual leakage factors,

Ls

=

(1

+

O"s)

Lo, L

R

=(1+O"R)L

o

,

M;;-:Lo

(10.49)

the

compld<' matlwntatical model of

Lhe

sYUIJllI'Lrjeal

I~ellera.l

AC machine

with

til(' llllllJwd illNI.ia J

t.JWIl

aSSlllll(~K

1.

11('

followilll':

fOl'lll,

wil.h

mI,

being

t.lw

1Id.

10

;

1.<1

1.01"qlll'

:d.

I.h(~

I'ollplillf':

01'

!.II('

1I1C/!.IH,

10.2

Induction

Motor

with

Sinusoidal Volt ages 175

· L di

s L d ( . j

ê)

( )

(10.50)

R

Sls

+ s

dt

+ O

dt

1R

e =

'Y<s

t ,

· L

di

R d (' j

ê)

( )

R

(10.51)

R1R

+ R

dt

+

Lo

dt

ls

e- =

'Y<R

t ,

J

~~

=

mM(t)

-mdt)

= í

Lo

1m

[is (iR e

j

ê)*] - mL (ê,

w,

t),

(10.52)

dê

(10.53)

dt

=

w.

Since

the

first two equations

can

be split into real

and

imaginary

parts,

this

represents a

set

of 6 scalar nonlinear differential equations.

They

are valid for

any waveforms

of

voltages,

currents

and

at

varying load torque

and

speed.

The

physical currents, for example in

the

stator

windings, are easily de-

rived from

the

vectorial representation.

With

Eq. (10.1),

the

current

vector,

defined in Eq.

(10.5),

is

is(t)

=

~

is1(t)

+j

V;

[is

2

(t) -

is

3

(t)]

(10.54)

hence

is1(t)

=

íRe[is(t)]

,

(10.55)

and

correspondingly,

is

2

(t) = í Re [is e-h] , etc.

(10.56)

Likewise

the

currents in

the

rotor

winding are obtainedi of course, in case of

a squirrel cage rotor,

the

notion of phase currents loses its physical meaning,

lwen

though

the

simplified theory remains applicable.

The

Eqs. (10.50-10.53) are

the

starting

point

for ali subsequent discus-

sions of

transient

and

steady

state

phenomena.

They

describe

the

electrome-

chanical interactions of a symmetrical

AC

machine

and

can

be

adapted

to

various constraints, imposed by

the

type of machine used, as long as

the

basic

assllmptions on which

the

model was based, remain

valido

This

mainly refers

1.0

ra.dial

symmetry

and

constant

airgap, excluding saliency of

the

stator

or

rotor.

Table

10.1 contains

the

rotor circuits of different nonsalient AC machines

IIs('d

in controlled drives

that

may be represented by

the

mo dei equations.

'l'hc

details of

the

constraints are discussed in

later

chapters.

'fil

e model equations contain, of course, also

the

special case of

steady

::

!.al.

e opcration, when

the

AC

motor

is

fed with sinusoidal impressed volt ages

1J)' a s'yulluetrica.l

three-phase

supply

and

is loaded with

constant

torque.

To

f01"l1l a

link

to tbis commonly used presentation

and

to

gain a

better

IIlId('l"st.alldilll

~

of

\.III:

1llOdel <'quntiolls,

t,his

cas!! will

bc

discussed first.

With

1111

0,

H

(

'I

~

I

\

"

IIIO!.Ol"

or

a wC/ulld rot.ol'

1'1

'S II 1

1.

:-1

,

w!tI'J"('

('xLerna)

~ym

mdrical

IO!.OI

' 1'

.'

:Ii: I

!."I'

::

III!!

}

Itn

,v,'

1""'11

,1.11<11

'

01

.

,'

(1;

'II

.

~-;

y

"'lfIetrical

Three-Phase

AC Machines

'1'

11

1

01.

·

10.1.

Mathematical

models for nonsalient

rotors

,,'

di'I'"I" '

''1.

AC

machines

used

in

controlled electrical drives

.

3/

.,

iR

=

'2

F

+lR

IJI

R1 i

R1

i

RI

i'RI

RR

IJI

R2

3 / .

lR='2

F

IJI

R3

i

R3

=const.

a Synchronous motor

b Synchronous motor with

with permanent magnets

permanent magnets and

on rotor surta ce

damper winding

U

RI

i

RI

-

~

~IUF

i

R2

~

!dR

= O

!dR=

UF

i

R3

c Induction motor with

d Synchronous motor

wound

ar

cage rotor

wíth fie/d and

damper windings

e Doub/y-fed inductíon motor

with impressed rotor vo/tages

10

.2 l.

lHluction

Motor

with

Sinusoidal

Symmetrical

V olta/,!;es

in

Steady

State

IO.:l

.l

Stator

Current,

Current

Locus

II.

:

i'y

lIlllletrical three-phase system of sinusoidal voltages having

the

angular

I'r(:<jIH:JlC:y

Wl

may be defined as follows, using complex

notation,

'll.S J (t) =

-/2

uS

COS(Wl

t +

Td

= Re

[-/2

Us e

j

TI

e

j

Wl

t]

=

V2

[u

ejwlt+U*e-jWlt]

2

-8

- s ,

(10.57)

w

IIC:r

e

rr

o.;

=

(Js

(;j

T I is

t.hc

constant

volLagc I'IJ:tJ;or; correspomlinG exprcssions

ii ..

!"

for

I'lIt

uws

~

1IJl<!

:~,

10.2

Induction

Motor

in

Steady

State

177

US2(t)

=

USl

(t

-

:J

-/2

[U

e

j

(Wl

t-,)

+

U·

e-j

(WI

t-,)]

21l'

2

-s

,=

""3

(10.58)

US3(t)

=

USl

(t

-

~~)

-/2

[U

e

j

(WI

t-2,)

+

U·

e-j

(Wl

t-2

,)]

(10.59)

2

-s

The

voltage

phasor

US is a complex

constant,

having a

magnitude

equal

to

the

RMS

line-to-neutral

voltage

and

an

arbitrary

phase angle

TI

which

may

be

set

at

will because

it

is defined

by

the

chosen

instant

for

time

zero.

Wh

en

the

complex voltage vector according

to

Eq. (10.34) is formed, a

very simple

result

is

obtained,

. .

3V2

.

Jfs

()

t =

USl

+

US2

e

,

+

US3

e

J

2,

=

--

U e

J

Wl

t .

(10.60)

J

2

-s

,

the

conjugate

complex

part

vanishes, leaving a vector

with

constant

mag-

nitude

and

constant

angular

velocity describing a circular

path

around

the

origino

The

stator

and

rotor

currents

form also

symmetrical

three-phase systems

in

steady

state.

With

jw1t

+ r

jw1t

is

(t)

=

-/2

[I

e

e-

] etc. (10.61)

1

2-S

-s

,

we

find

i

(t)

= 3

-/2

I e

j

WI

t

(10.62)

-s

2

-s

,

wh

ere

the

current

phasor

Is

is still

to

be

determined

. A similar definition

('xists for

the

rotor

currents

oscillating

with

slip frequency

W2

=

Wl

- W

i

(t)

=

-/2

[I

e

j

(WI

-w)

t+ r

e-j

(WI-W)

t]

etc. (10.63)

RI

2

-R -R

,

n"

. . _ 3

-/2

I

ej(WI-W)t.

L/I(t) - 2 - R

(10.64)

TI\(:

rotor

Cllrrcnt

in s

tator

coordinates, i.e. as seen by a

stationary

observer,

III with

~

- wl. lWC;lUSC of

constant

sp(:ed,

.,

J/.

i

(",I

i /{(I) ,

,,

I

tI

{l}

.. I/{ t

..

(IO.(;[i)

'0

I

'/

H

lO ''-;'y"11I<'l.rical

Three-Phase

AC Machines

II(~II(,(

'

Ilw

rotor

ampereturns

wave, moving with slip frequency across

the

rotor

slIfface,

rotates

in synchronism with

the

stator

current wave; this

is

a

prccondition for

constant

electrical torque. By introducing these expressions

into

Eqs. (10.50, 10.51)

and

omitting

the

redundant

terms,

the

differential

equations are converted with

UR

= O

to

algebraic equations for

the

phasors,

(Rs

+j

Wl

as

LoH

s

+j

W}

Lo

(Is

+

IR)

=Us

(10.66)

(RR

+ j

w2

aR

LoH

R

+ j

W2

Lo

(Is

+

IR)

= O.

Norlllalising

the

second equation with

the

slip frequency

s =

W2

=

W}

-w

(10.67)

Wl

n'snlts in

(

~R

+j

Wl

aR LO)

IR

+ j

Wl

Lo

(Is

+

IR)

= O.

(10.68)

I~:qu

a

tions

(10.66, 10.68) lead immediately

to

the

familiar single phase equiv-

alcllL

circuit seen

in

Fig. 10.7 a;

its

validity is restricted

to

steady

state,

asslllning sinusoidal symmetrical currents

and

constant

load torque.

/Is

as

Lo

aR

Lo

lR

RS

aL

s

(1+a

R

J1R

~.

J.s

is

RR

1/.,'

, 1

RR

LO

Y.

s

(1-aJ

Ls

--2-

S

(1+a

R

J S

Is

1.1/1

=.1

m

lmR

;/

N

s

=N'R

b

1

..!!....L

--lR

1-/

S

1-a s

1+a

s

I

II

i

is

y.

s

Lsl

1ms

O

(1+a

s

J

2

S

R

1+a

s

Fi

H·

10.7.

Silll\k

pI

IlL~

('

1.''1"ivaklll.

c.ircllit,j

"I' i,,<lII"(,;

II

"

111111.111'

iII

"t

..

;\

.

dy

,'

jl.nl.c

(II) w;l.h

1"

lt!t;'1',n 1Il<lIl

("

1.

1

11"

·

('1I

'"I

Il

llLl,,,' ,

,,"1

1'

0(.,,,

111

.1,

',

(h)

w

il.h

1"111

(1

11

1

'"

;,,<111<'1.

111

,

11'

"

'"I

::

I,at"

..

Il

i

<l

,',

(.-) wi

!.11

1

(l1I

10

I/

J.

4' I

lIdllc

(,

11

1" '

1\

jl!l

1'0(,01

' ri

idc'

10.2

Induction

Motor

in

Steady

State

179

By

inspecting

the

equivalent circuit in Fig. 10.7 a,

the

stator

impedance

per

phase

is

derived.

It

may

be

written

in

the

form

·SWII7L

R

1 +J

RR

(10.69)

Zs=Rs+jw}Ls

l+js'1:

R

'

where

1

(10.70)

a = 1 -

(1

+

as)(l

+

aR)

is

the

totalleakage

factor of

the

motor, which has

an

important

influence

on

the

characteristics of

the

machine;

it

is

chosen by

suitable

design

parameters

such as

the

shape

of

the

slots

and

the

length of

the

airgap. Normal values of

a range between 0.05 for low leakage machines to

about

0.20.

The

expression in Eq. (10.69) gives rise to

the

definition of yet

another

important

parameter

RR

1

LR

(10.71)

Sp

=

W}

a

LR

wlaTR'

with

TR =

RR

'

which is called pull-out slip because a

motor

with zero

stator

resistance pro-

duces

maximum

torque

at

S =

Sp,

as will be shown later.

The

pull-out slip

of

normal

medium

size

motors

is

Sp

< 0.25.

The

steady

state

equivalent circuit in Fig. 10.7 a contains three

synthetic

inductances which, because of

1m

=

Is+IR'

are

not

representing

independent

energy storages; hence

it

must

be

possible

to

redraw

the

circuit

diagram

with

only two inductances. One possibility

is

shown in Fig. 10.7 b, where all

the

lea.kage is concentrated

at

the

stator

side.

There

is

a different fictitious

rotor

resistance leading

to

a different

current

in

the

right

hand

side

branch

of

the

circuit

but

the

stator

impedances are identical.

The

new synthetic

rnagnetising

current

ImR represents

the

rotor flux, as will be shown

in

a

later

chapter

. A

third

equivalent circuit is seen in Fig. 10.7 c, where all

the

leakage

is referred

to

the

rotor

side

and

where

Ims

stands

for

the

stator

flux.

This

arrangement

can

also be used for designing controls.

The

stator

current

is now

computed

with

the

added simplification

Rs

= O

which is

of

little

significance with larger machines

operating

with low slip

at

line frequency.

Thus

the

normalised expression for

the

stator

current

phasor

becomes

. 1 S

U 1 +J Ü s.

I =

-s

, s

(10.72)

-s

j

Wl

Ls

1 +J

sp

whj

r

Jt

is n liucar cowplex function umpping the real S/Sp-axis onto

the

Is-

pIHII(

~.

A('('ordilll

~

,

to

t.Iw

mie

s

or

confoJ'lllal

Illapping,

the

result is a circular

lorwl,

1.111'

('irei" di:"',1 il,

tIl

, ..

.l

S

(l

('1I.lI('d

( h

rm

llll

ll.'H

(lI'

I!('ylaud's

circl

e.

This

is

""lIlirll"'.!

11

1'y

LI",

1,')It>\

v

ll,,~

I.r;

~

II:lr(lrlll

l l

,

I

,

i

,,,,

c

Werner Leonhard Control of Electrical Drives

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