Modeling of Induction Motor

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(No Transcript)
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Modeling of Induction Motor
Construction - 3 phase stator - Identical
coils displaced 120o around
periphery -Magnetic field axes are displaced
120deg - Can have multiple poles
- Connected to three phase ac source
d
a
A B C
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Induction Motor
Construction Wound Rotor - 3 phase rotor -
Identical coils displaced 120o around
periphery -Magnetic field axes are displaced
120deg - Can have multiple poles
- Shorted through impedance or
connected to external source Squirrel cage -
embedded bars look like cage - shorted by a
ring - no external connection
d
a
q
q
A B C
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Induction Motor
Stator Magnetic Field
d
a
Current ia in phase a creates Sinusoidally
distributed flux density
Ba(a,t) Bmax cos( ? t)cos( a ) Bmax depends
on ia and design Schematically represented by
vertical arrow Maximum field at a0
d
Ba(a,0)
ia
A B C
0
p
p/2
a
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Induction Motor Stator Rotating Field
Net Stator Field Rotating Magnetic
Field- balanced positive sequence operation
d
a
d
a
Schematic of Net Stator Field Phases a,b,and c At
t0
q
q
Ba(a) (3/2)K Iamax cos(a)
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Induction Motor Stator Rotating Field
Net Stator Field Rotating Magnetic
Field- balanced positive sequence operation
d
a
d
a
Schematic of Net Stator Field Phases a,b,and c At
wt4p/3
q
q
Schematic of Net Stator Field Phases a,b,and c At
wt2p/3
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Induction Motor Stator Rotating Field
d
d
d
a
a
a
q
q
q
Bs(a,t) (3/2)K Iamax cos(?t- a) Observer at a
sees sinusoidal flux density in time Frequency
? lags ia by a At any time t flux density
is sinusoidally distributed in space Amplitude
is time independent (3/2)K Iamax Peak is at a
?/t
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Induction Motor Rotor Condition
d
a
Stationary Rotor phase a sees field Bs(a,t)
(3/2)K Iamax cos(?t- a) Because the field
ROTATES In a stationary rotor this induces
voltage Era(t) Eramax sin(?t- a) Phases b and
c see similar voltages displaced by 120deg Three
phase rotor has balanced three phase induced
Voltages at frequency ?
?
q
a-120deg
d
?
q
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Induction Motor Slip
d
a
Rotor turning at ?r, phase a, sees
field Bs(a,t) (3/2)K Iamax cos(?- ? r)t-
a) In a stationary rotor this induces
voltage Era(t) Eramax (?- ?r)/ ? sin(?-
?r) t- a Phases b and c see similar voltages
displaced by 120deg Three phase rotor has
balanced three phase induced Voltages at
frequency (?- ?r) The ,amplitude depends on
relative speed
?
?r
q
Define Slip s (?- ?r)/ ?
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Induction Motor Slip
d
a
Rotor turning at ?r, phase a, sees
field Bs(a,t) (3/2)K Iamax cos(?- ? r)t-
a) In a stationary rotor this induces
voltage Era(t) Eramax (?- ?r)/ ? sin(?-
?r) t- a Phases b and c see similar voltages
displaced by 120deg Three phase rotor has
balanced three phase induced Voltages at
frequency (?- ?r) The ,amplitude depends on
relative speed
?
?r
q
Define Slip s (?- ?r)/ ?
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Induction Motor
?
Three phase satator creates a rotating field
d
a
Rotor turning at ?r, phase a, sees
field Bs(a,t) (3/2)K Iamax cos(?- ? r)t-
a) In the rotor this induces voltage Era(t)
Eramax (?- ?r)/ ? sin(?- ?r) t- a Three
phase rotor has balanced three phase induced
Voltages at frequency (?- ?r) The ,amplitude
depends on relative speed Resulting current
produces a rotating field at speed (?- ?r) With
respect to rotr Rotor field rotates at speed ?
and is stationary with respect To stator field
?
?r
?
q
This creates The constant torque For motor action
Slip s (?- ?r)/ ?
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Induction Motor Transformer like Model
as Rs jXs
Rr jsXr ar
Vr -
Er -
Es -
Vs -
Is
Ir
Is
Frequency s?
jXm Rc
Frequency s?
Voltage ratio Es/ Er (a/ s) /a Current
ratio Is/ Ir 1/(a /a)
Es Is ? Er Ir Pmech 3 Re(Es Is - Er Ir)
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Induction Motor Transformer like Model
as Rs jXs
Rr jsXr ar
Vr -
Er -
Es -
Is
Ir
Is
Frequency ?
jXm Rc
Frequency s?
Vs -
as Rs jXs
Rr/s jXr ar
Vr -
Er -
Es -
Is
Ir
Is
Frequency ?
jXm Rc
Frequency ?
Voltage ratio Es/ Er (a) Current ratio
Is/ Ir 1/a
Es Is Er Ir Pmech 3 Re( Er Ir) 3
Ir2 Rr
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Induction Motor Transformer like Model
as Rs jXs
Rr/s jXr ar
Vr -
Er -
Es -
Is
Ir
Is
jXm Rc
as Rs jXs
Rr jXr Rr(1-s)/s ar
Vr -
Er -
Vs -
Es -
Is
Ir
Is
jXm Rc
Voltage ratio Es/ Er (a) Current ratio
Is/ Ir 1/a
Pmech 3 Re( Er Ir) 3 Ir2 Rr Pmech 3
Ir2 Rr (1-s)/s
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Induction Motor Transformer like Model
?r0 Standstill s 1 Rotor Shorted
as Rs jXs
Rr jXr ar
Vs -
Vr -
Er -
Es -
Is
Ir
Is
jXm Rc
?r ? Synchronous speed s 0 Rotor
Shorted/Looks Open
as Rs jXs
Rr jXr ar
Vs -
Vr -
Er -
Es -
Is
Ir
Is
jXm Rc
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Induction Motor Balanced Steady State Performance
Reflect to stator
as Rs jXs
Rr jXr ar
Vs -
Vr -
Er -
Es -
Is
Ir
Is
jXm
Rsth jXsth Rr
jXr
Rr(1-s)/s
Vth -
Ir
Is
Vth Vs (jXm)/(RsjXsjXm) RthjXth (RsjXs)jXm
Rr, Xr are rotor resistance and leakage reactance
referred to Stator
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Induction Motor Balanced Steady State Performance
Reflect to stator
Rsth jXsth Rr
jXr
Vth -
Ir
Is
Rr(1-s)/s
Ir Vth /(RsthjXsthjXrRr/s) IsVth/ (RsjXs
(jXm)(Rr/s j Xr) Pmech 3 Ir2 Rr(1-s)/s
W T Pmech/?r 3 P Rr(1-s)/2s?r Vth2 /
(RsRr/s) 2 (XrXs) 2 T Pmech/?r 3 P
/2 ? Vth2 (Rr/s)/ (RsRr/s) 2 (XrXs) 2
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Induction Motor Balanced Steady State Performance
4160 V, 60 Hz, 670 HP 3 phase 2 pole motor
Rated slip 3.5
Synchronous speed Rated speed
T 3 P /2 ? Vth2 (Rr/s)/ (RsRr/s) 2
(XrXs) 2
Max Torque
Stable
G
0
?
2 ?
Speed
M
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Induction Motor Balanced Steady State
Performance-Starting
Starting
Large starting current ( 6xrated) and low power
factor causes a large Voltage drop (Dip, Sag)
when motor starts May Require starter -
autotransformer - wye-delta - impedance -
electronic
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Induction Motor balanced Steady State
Running at rated load
HP
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Induction Motor Balanced Steady State
Half Load Running Condition
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Induction Motor Unbalanced Steady State
Phase domain admittance matrix can be developed
from Sequence Admittance Matrix YM A Y012-1
A
From sequence networks ( next Slide)
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Induction Motor Sequence Model
Rs jXs Rr/s jXr
Vs1 -
Ir1
Is1
s (?- ?r)/ ?
jXm
Normal operation,low slip Rr/s is large
Rs jXs Rr/s- jXr
Vs2 -
Ir2
Is2
S- (??r)/ ? 2- s
jXm
Normal operation,low slip Rr/s- is small Motor is
very sensitive to Negative sequence
voltage Because the zero sequence Impedance is
just leakage Impedance motors are delta or
wye-ungrounded
Rs jXs
Vs0 -
Is
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Induction Motor Unbalanced Steady State-
Feeder analysis
287 KVA 0.0.87 pf lag
13.8 4.16 kv
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C3 b
1 mi
Sub
C1 5000 C1 5000
2
3
1
C3 a
618 KVA 0.874 pf lag
13800 kV
732 KVA 0.82 pf lag
1 mi
Source impedance j0.005 pu
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221 KVA 0.9 pf lag
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System Admittance Matrix with Motor
YOH12 YS
-YOH12
YOH12 YOH23 Yugb25 Yuga24
-YOH12
-Yuga24
-YOH12
-Yugb25
Y
YOH23 YM/ a2
-YOH12
Yuga24
-Yuga24
Yugb25
-Yugb25
a13.8/4.16 turns ratio of motor transformer
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Induction Motor Unbalanced Steady State-
Feeder analysis
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Running No Motor Starting
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Induction Motor Unbalanced Steady State-
Feeder analysis
A small voltage unbalance Can create A large
current unbalance