Induction Machines – Asynchronous Machines

Induction Machines – Asynchronous Machines

Induction Machines – Asynchronous Machines

Induction Machines – Asynchronous Machines

rotor

stator

stator winding cage winding

terminal box stator frame

end shield

shaft

fan

Induction Machines – Asynchronous Machines

• the three-phase windings are displaced from each other by 120 degrees in space

• a phase coil produces a sinusoidally distributed mmf wave centered on the axis of the coil

• alternating current in each coil produces a pulsating mmf wave

• mmf waves are displaced by 120 degrees in space from each other

• resultant mmf wave is rotating along the air gap with constant peak

Rotating magnetic field

a m cosi I tω=

mb cos( 120 )i I tω= − °

c m cos( 120 )i I tω= + °

• balanced three-phase currents

Rotating magnetic field – Graphical method

a mi I=m

b 2Ii = −

mc 2

Ii = −

• at time t0

a maxF F=

maxb12

F F= −

c max12

F F= −

max32

F F=a 0,i =

mb3 ,2

i I=

c m3 ,2

i I= −

a 0F =

maxb32

F F=

c max32

F F= −

• at time t1

Rotating magnetic field – Graphical method

1202 60pfn f

p= =

• the resultant mmf vector retains its sinusoidal distribution in space

• it moves around the air gap – one revolution per period

• reversal of the phase sequence of the currents change the direction of rotation

max32

F F=

Rotating magnetic field – Analytical Method

a a( ) cosF Niθ θ=

b b( ) cos( 120 )F Niθ θ= − °

c c( ) cos( 120 )F Niθ θ= + °

a cb( ) cos cos( 120 ) cos( 120 )F Ni Ni Niθ θ θ θ= + − ° + + °

• analytical expressions for the mmf waves

• the resultant mmf wave

a m cosi I tω=

mb cos( 120 )i I tω= − °

c m cos( 120 )i I tω= + °

Rotating magnetic field – Analytical Method

m

m

m

( , ) cos coscos( 120 )cos( 120 )cos( 120 )cos( 120 )

F t NI tNI tNI t

θ ω θω θω θ

=+ − ° − °+ + ° + °

= − + +1 1cos cos cos( ) cos( )2 2

A B A B A B

m m

m m

m m

m

1 1( , ) cos( ) cos( )2 21 1cos( ) cos( 240 )2 21 1cos( ) cos( 240 )2 23 cos( )2

F t NI t NI t

NI t NI t

NI t NI t

NI t

θ ω θ ω θ

ω θ ω θ

ω θ ω θ

ω θ

= − + +

+ − + + − °

+ − + + + °

= −

Induced Voltages

/2p max/2

( ) 2B lr d B lrπ

πΦ θ θ

−= =∫

a p( ) cost N tλ ω Φ ω=

aa p maxsin sinde N t E t

dtλ ω Φ ω ω= − = =

• sinusoidal flux density distribution in space

• flux per pole

• flux linkage

• induced voltage

max( ) cosB Bθ θ=

cos( )cos cos sin sin

mB B tBm t t

ω θω θ ω θ

= −= +

( )2

2

cos cos sin sin

2 cos

p m

m

B t t lrd

lrB t

π

π

φ ω θ ω θ θ

ω−

= +

=

∫

Induced Voltages

maxb sin( 120 )e E tω= − °

c max sin( 120 )e E tω= + °

prms p p

2 4.442 2N fE N fN

ω Φ π Φ Φ= = =

Φ=rms ph p4.44 wE fN K

a p maxsin sine N t E tω Φ ω ω= =

• for distributed winding with winding factor Kw

0.85 0.95wK ≈ …

Standstill operation – phase shifter

•rotor open-circuited• rotating field in the air gap – speed ns

• field induces voltages in stator and rotor windings – same frequency

• stationary wound rotor induction machine can be used as a phase shifter

1 1 1 p W14.44E f N KΦ=

2 1 2 p W24.44E f N KΦ=W11 1 1

2 2 W2 2

KE N NE N K N

= ≈

Standstill operation - induction regulator

• variable polyphase voltage source• continuous variation of voltage• no sliding connection

1

0 1 2

in== +

E VV E E

+ Continuous variation of the output voltage+ No sliding electric connections- High leakage inductances- High magnetizing current- high cost

Running operation

s

s

n nsn−=

2 s s 1( )120 120p pf n n sn sf= − = =

2s 2 2 p W2 1 2 p W2 24.44 4.44E f N K sf N K sEΦ Φ= = =

2 12 s

120 120f sfn snp p

= = =

( )2 1 s s sn n s n sn n+ = − + =

• rotor circuit is closed• induced voltages produce rotor currents that interacts with air gap field

to produce torque• rotor starts to rotate• relative speed decreases - induced voltage decreases – torque balance

• slip

• frequency of induced rotor currents

• induced rotor voltage

• speed of induced rotor field with respect to the rotor is

• and with respect to the stator is

Running operation

• motoring

• generating

• plugging

Equivalent Circuit Model

• a three-phase wound-rotor induction machine• the cage winding can be represented by an equivalent three-phase winding

• in steady-state the produced magnetic fields rotate at the synchronous speed• resultant air gap field will induce voltages in both stator and rotor windings• supply frequency f1 in stator; slip frequency f2 in rotor

• form of the equivalent circuit appears to be identical to that of a transformer

Equivalent Circuit Model

• stator winding per-phase quantities

V1 terminal voltageR1 winding resistanceL1 leakage inductanceE1 induced voltageLm magnetizing inductanceRc core loss resistance

• Equivalent circuit similar to transformer primary• Magnetizing current can be 20 – 50 % of stator current (1 – 5 % in transformer)• X1 larger than in transformer due to airgap and distributed windings

Equivalent Circuit Model

• rotor winding per-phase quantities

E2 induced voltage at standstill (f1)R2 winding resistanceL2 leakage inductancef2 = s f1

• at given slip s

22

2 2

sEIR jsX

=+

22 2 2P I R=

• rotor circuit (f2 = s f1)

• rotor current can be expressed as

Equivalent Circuit Model

22

2 2( / )EI

R s jX=

+

2 2 22R PP Is s

= =

• although the amplitude and phase are the same the frequency is different !

• power associated with the equivalent circuit (air-gap power)

Equivalent Circuit Model

• complete equivalent circuit

- same frequency in stator and in rotor

- turns ratio has to be taken into account1

2

NaN

=• to stator referred quantities

2 1 2

22

E E aEIIa

′ = =

′ =

22 2

22 2

R a R

X a X

′ =

′ =

Equivalent Circuit Model

2 22 2ag 2 2 2 2 mech(1 )R RP P I I R s P P

s s⎡ ⎤= = = + − = +⎢ ⎥⎣ ⎦

22 2 2 agP I R sP= =

ag 2 mech: : 1 : : 1P P P s s= −

= − = −2 2mech 2 ag(1 ) (1 )RP I s s P

s

• air gap power crosses the air gap• it includes rotor copper loss P2 as well as mechanical power developed Pmech

• a fraction s is dissipated in rotor resistance P2• the fraction (1-s) is converted into mechanical power Pmech

Equivalent Circuit Model• approximate equivalent circuit

• IEEE-recommended equivalent circuit

Equivalent Circuit Model

• Thevenin equivalent circuit

m m1 1 1th th1/22 2 1 m1 1 m( )

X XV V V K VX XR X X

= ≈ =+⎡ ⎤+ +⎣ ⎦

m 1 1th th th

1 1 m

( )( )

jX R jXZ R jXR j X X

+= = +

+ +

22m

1 1th th1 m

XR R K RX X

⎛ ⎞=⎜ ⎟+⎝ ⎠

1thX X

due to small resistance

due to large magnetizing inductance

Equivalent Circuit Parameters

• no-load test at nominal voltage and frequency, no load• blocked rotor test at nominal current, reduced voltage and frequency, rotor blocked

• DC-resistance measurement

• no-load power core losses + windage and friction losses• blocked rotor reactances

• the equivalent circuit is used to predict performancescharacteristics at steady state:

• efficiency• power factor• current• starting torque• maximum (pull-out) torque, etc…

Performance characteristics – Torque

2 22mech mech mech (1 )RP T I ss

ω= = −

mech260nπω =

synsynmech (1 ) 2 (1 )

60n

s sω ω π= − = −

1syn

120 4260f f

p pπω π= × =

2 2syn 2 agmech

RT I Ps

ω = =

2 22 2ag 2 2mech

syn syn syn

1 1 1R RT P I Is sω ω ω

′′= = =

• torque per phase Tmech

Performance Characteristics – Torque profile2th 2

mech 2 2syn 2 2th th

1( / ) ( )

V RTsR R s X Xω′

=′ ′+ + +

• torque is proportional tothe square of the voltage

• total torque is obtained bymultiplying the per phasetorque by the number of phases

• 5 % difference in torque predictiondepending on the kind of equivalent circuit

2th

mechsyn 2

1 VT sRω

≈′

2th 2

mech 2syn 2th

1( )

V RTsX Xω′

≈′+

Performance Characteristics – Maximum Torque

2 2th th

max 1/22 2syn syn 2th2th th th

1 12 2( )

V VTX XR R X Xω ω

= ≈′+⎡ ⎤′+ + +⎣ ⎦

effect of rotor resistance on torque characteristics

′=⎡ ⎤′+ +⎣ ⎦

′≈

′+

max2

1/22 2th th 2

2

th 2

( )T

RsR X X

RX X

• maximum torque corresponds to: 0dTds

=

• maximum torque independent from rotor resistance

• corresponding speed depends on rotor resistance

Performance characteristics – Torque ratio

maxmax

2 22 2max th th

2 22 2th th

( / ) ( )( / ) ( ) TT

R R s X XT sT sR R s X X

′ ′+ + +=′ ′+ + +

maxmax

2 22 2max th

2 22 2th

( / ) ( )( / ) ( ) TT

R s X XT sT sR s X X

′ ′+ +′ ′+ +

max max

max maxmax

max

max

2 2 2 22 2max

22

( / ) ( / )22( / )

12

T T

T TT

T

T

R s R s s sT sT s s sR s

s ss s

′ ′+ += × =

′

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠

R1 small

Performance characteristics – Current and Power factor

• stator current• typical startingcurrent 5 – 8 timesrated current

• power factor

21 1 1 2

11

´// ´mRZ R jX X jXs

Z θ

⎛ ⎞= + + +⎜ ⎟⎝ ⎠

=

11 2

1

´VI I IZ φ= = +

1PF cosθ=

Performance characteristics – Efficiency

in 1 1 13 cosP V I θ=

out

in

Eff PP

=

• ideal efficiency = only rotor resistive losses

( )out

idealin

Eff 1P sP

= = −

in ag

2 ag

out mech ag (1 )

P P

P sP

P P P s

=

=

= = −

Power flow

• motoring

• generating

• plugging mode• rotor losses !

Effects of rotor resistance

• small rotor resistance+ high efficiency+ small nominal slip- small starting torque- large starting current

• large rotor resistance- poor efficiency- large nominal slip+ large starting torque+ small starting current

• in wound-rotor externalresistance can be connected tothe rotor windings through theslip rings

Effects of rotor resistance – Deep-Bar Squirrel-cage

• rotor frequency changes with speed• effective rotor resistance changes withfrequency if the shape of rotor bars isadequate (skin effect)

Rotor Resistance - Double-Cage Rotors

• two rotor cages each with its own end ring• outer cage with small cross section and high resistivity material• inner cage with larger cross section and low resistivity material

• at standstill most of rotor current flows in outer cage large resistance• at small slip (full-load) current flows in both cages smaller resistance

• equivalent circuit formed by additional branches in the rotor

Speed Control

• speed is determined by supply frequency, number of pole-pairs and slip

• pole changing• synchronous speed can be changed in discrete steps• motor is more expensive• normally speed changed in ratio 2:1• cage induction machine only

• line voltage control• torque is proportional tothe square of the terminal voltage

• increased slip inefficient operation• method is used with fans and pumps

( )12fn sp

= −

Speed Control – Line Voltage Control

• auto transformer

• solid-state controller

• closed-loop operation• precise speed control

Speed Control – Line Frequency Control

2fnp

=

Speed Control

pEf

Φ ∝ pVf

Φ ∝

• below the nominal speed• voltage-frequency proportion is kept constant to avoid saturation• constant flux / torque

• above the nominal speed• voltage is kept constant to avoid electric breakdown• constant voltage / power

Speed Control – Rotor Resistance Control

• efficiency ?

• open loop

• closed loop

Speed Control – Rotor Slip Energy Recovery

• open loop

• closed loop

Starting of Induction Motors

• usually started by direct connection• starting current 5…8 IN• line voltage drop• long starting time – overheating• reduced-voltage starting

• step-down autotransformer• star-delta method

• reduced torque

• solid state voltage controller