Asynchronous machines 1

Asynchronous machines

Rotor: magnetic circuit and electriccircuit...

Stator: magnetic circuit and polyphasedwinding (usually 3-phase), with p pairs ofpoles, with polyphased currents

... either made of a polyphased winding,(with star connection in order to avoidcurrents between rotor phases)connected to short-circuit rings on therotor axis; brushes connect these ringsto external resistances® wound rotor

... or made of copper or aluminum bars,short-circuited at each extremity of therotor® squirrel-cage rotor

rings

09/02/2020

2

General principle

p)g1( w

-=q!

psw

=q!

1g0 ££

pg

pw

=q-w !

When 3-phase currents of pulsation w flowin the stator windings they create a rotatingmagnetic flux density with p pairs of poles,rotating with angular speed w/p.

This field induces 3-phased e.m.f.s ofpulsation w in the (stationary) rotorwindings, and thus 3-phase currents ofpulsation w.

The interaction betweem the rotating statorflux density and the induced currents in therotor creates a torque, which puts the rotorinto movement

Wound rotor machine initially at rest ... 1g =

0g =

glissementgdef=

Synchronous speed

Rotation speed

Angular speed of the stator rotating field with respect to the rotor

e.m.f.s and induced currents withpulsation gw, and torque

no e.m.f., no current, no torque(asynchronous machine, or induction machine)

Asynchronous machines

slip

3

General equations

11111 I)XjR(EU ++= 2222 I)XgjR(E +=

Stator equation Rotor equation

Link equation

÷÷ø

öççè

æ-= µ

tr

211 n

IIXjE

Rotating magnetic fluxdensity in the airgapleads to e.m.f. E1 in statorwinding, with pulsation w

Rotating magnetic fluxdensity in the airgap leadsto e.m.f. E2 in rotorwinding, with pulsation gw.

X1 = Leakage reactance of one stator phaseR1 = Resistance of a stator phase

For one phase

X2 = Leakage reactance of a rotor phase at pulsation wR2 = Resistance of a rotor phase

r11 kjE Fw-= r22 kgjE Fw-=

gn

EE tr

2

1 =

221121r I'kI'kFFF +-=+=resulting m.m.f.

ep

rr R

F=Fwith

resulting fluxXµ = Magnetizing reactance

tr2121 n'k/'kk/k == ...

Asynchronous machines

4

Equivalent circuit

( )211 'IIXjE -= µ

(a)

(b)

(c)

(a) (b)(c)

gn

EE tr

2

1 =

11111 I)XjR(EU ++=

222

1 'I'Xjg'RE ÷

ø

öçè

æ+=

µ<< X'X,X,'R,R 2121(ratios ~ 40 X, 600 R)

÷÷ø

öççè

æ-= µ

tr

211 n

IIXjE21 'III -=µ

magnetizing current

2222 I)XjgR(

gE

+=

€

E 1 = ntr2 R2

g+ j ntr

2 X2

⎛

⎝ ⎜

⎞

⎠ ⎟

I 2ntr

Simplified equivalent circuit

Asynchronous machines

/ g

5

Equivalent circuit parameters

Experimental determination of the parameters

No load test

Short-circuit test

magv PP =µ

=XU3Q21

v

( ) 2121cc I'RR3P += ( ) 2121cc I'XX3Q +=

<<Þ<< primaireJoule1 PI

<<Þ<< µ mag1 PetIU

0g =

1g =

Under nominal voltage and at synchronous speed

Under reduced voltage at at zero speed (stationnary rotor)

in the resistances of the stator and rotor windings

in the leakage reactances of the stator and the rotor

magnetic losses in the stator laminations

Asynchronous machines

6

Power, torque and efficiency

÷ø

öçè

æ+=j= 2

222

1111 'Ig'RIR3cosIU3P

22

2211rotst 'I

g'R3IR3PP =-=®

211 IR3

222

222rotstelm 'I'R

gg13'I'R3PP -

=-= ®

222 'I'R3

22

2elmelm 'I

g'Rp3PC

w=

q= ! g1)g1( mecstat -<hh-=h

g1'I'R

g13

'I'Rgg13

PP

222

222

rotst

elmrot -=

-

==h®

Rotor efficiency (2)

Asynchronous motor efficiency(1)x(2)x(3)

Electromagnetic torque

Electric power

PP rotst

stat®=h

elm

mecmec P

P=h

Stator efficiency (1)

Mechanical efficiency (3)

Asynchronous machines

Stator

Airgap

Rotor

Absorbed power

Stator Joule losses

Rotor Joule losses

Power transmitted from stator to rotor

Electromagnetic power

7

Mechanical characteristic

Evolution of the electromagnetic torque C as a function of theslip g (or the rotation speed), for a given stator voltage U1and pulsation w

îíì

=w=

=constanteconstanteU

avec)g(fC 1

Mechanical characteristic

22

2 'Ig'Rp3C

w=

22

222

21

2'Xg'R

Ug'Rp3C+w

=

222

22

1

22

22

12

'Xg'R

U

'Xg'R

E'I

+

»

+÷÷ø

öççè

æ=with

Nonzero startup torque (g = 1)

Zero torque at synchronous speed (g = 0)

Maximum torque ¹ f(R'2)

2

21

max 'X2Up3

Cw

=

2

2'X'Rg ±=at

2

21

g 'RUgp3C

w=<<

generator notor

above synchronous speed (g < 0)

Asynchronous machines

with

8

Stability zone

0dgdC

ddC

<-=q!

0dgdC

ddC

>-=q!

A decrease (resp. increase) in speed, or an increase(resp. decrease) in the slip g, leads to an increase(resp. decrease) of the motor torque, from pointM®M' (resp. M®M''). The machine will thusaccelerate (resp. decelerate) to reach back point M,where the motor and resistant torques matchÞ stable.

A decrease (resp. increase) in speed, or an increase(resp. decrease) in the slip g, leads to a decrease(resp. increase) of the motor torque, from point N®N'(resp. N®N''). The machine will thus decelerate(resp. accelerate), further increasing the mismatchbetween the resistant and motor torques (resp. leadingto an evolution towards the stable stable N''®M)Þ unstable.

Stable Unstable

Asynchronous machines

9

Speed controlModification of the rotor resistance R'2

Reduction of speed at the cost of efficiency (h < 1–g)

Modification of the voltage U1

Small variation range

Fixed limit

Modification of the frequency

w»

w=

w=l=F µµ

µµ11

rUEIX

I

Interest in keeping the magnetic flux(hence U1/w) constant...

Cmax constant

Asynchronous machines

10

Asynchronous motor startup

22

22

21

2'X'R

Ug'Rp3C+w

=

Startup torque (g = 1)

usually quite small, since R'2 <<

( ) ( )2212

21

1d

'XX'RR

UI+++

=

Sartup current

very large, should be limited…

Squirrel cage motor... thanks to reduced voltage(transformer, electronic converter, transient star coupling)

Startup torque divided by 3

Wound rotor ... resistances in series with the rotor winding (progressively removed)

Startup torqueincreases and

startup currentdecreases

Startup possible only for small resistant torque at low speed (e.g. pumps, ventilators, air conditioners) Asynchronous machines

11

Special asynchronous motorsDouble cage rotor

Exterior (e) andinterior (i)

R'e >> R'i

X'e << X'i

Startup (g=1):R'i,e << gX'e << gX'iÞ Ce >> Ci

Nominal speed (g <<):R'e >> R'i >> g X'i,eÞ Ci >> Ce

2i,e

22i,e

21

i,ei,e'Xg'R

Ug'Rp3C+w

=

Skin effect rotor

Elongated rectangular rotorconductors

Þ same behavior as double cage

Startup (g=1):high apparent resistanceÞ large C

Nominal speed (g <<):low apparent resistance

Asynchronous machines

12

Circle diagram

( ) 22121 'Ig/'RR'IXjU ++=

µµ = Xj

UI 1

(0)

(1)

X = X1 + X'22

212

1 'IXj

g/'RR'IXjU +

+= Þ Locus of I'2 is a circle of diameter U1/X

constanteU1 =

(2)-(3)

(2)

(3)

21 'III += µ

(4)AMU3cosIU3P 111 =j=

1U3PAM = (5) (5)

A'OU3sinIU3Q 111 =j=

1U3QA'O =

(7) = (7a) + (7b)(7)

Active power

Reactive power

(7b)(7a)

OAO'OU3

QQA'O

1

XX+=

+= µ

Reactive power in Xµ and X

...'IX/U

OA'I

2

12÷÷ø

öççè

æ=

(0)

XUOAR3'IR3P 1

1221Js ==

OAXR

U3PAB 1

1

Js ==

(5a)(5a)

(5b)

(5c)

Stator Joule losses

XUOA'R3'I'R3P 1

2222Jr ==

OAX'R

U3PBC 2

1

Jr ==

(5b)

Rotor Joule losses

Electromagnetic powerCM (5c)

BMBCg = (6)

Asynchronous machines

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Operating modesThe machine• absorbs electric power (A1M1 ³ 0) ;• produces mechanical power (C1M1 ³ 0).

Asynchronous motor

The machine• produces electric power (A2M2 £ 0) ;• absorbs mechanical power (C2M2 £ 0).

Asynchronous generator

The machine• absorbs electric power (A3M3 ³ 0) ;• absorbs mechanical power (C3M3 £ 0).

Radiator

Asynchronous machines