induc machine
TRANSCRIPT
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