Formula Sheets: Three-Phase Induction Motor | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Three-Phase Induction Motors
(Electrical Machines) – GATE
1. Basic Concepts
• Three-Phase Induction Motor: AC motor with rotating magnetic ?eld created
by three-phase stator currents, self-starting.
• Types: Squirrel-cage and wound-rotor.
• Operation Principle: Rotating magnetic ?eld induces currents in rotor, produc-
ing torque.
2. Synchronous Speed
• Synchronous Speed:
N
s
=
120f
P
where f: Supply frequency (Hz), P: Number of poles.
• Synchronous Speed in rad/s:
?
s
=
2pN
s
60
3. Slip
• Slip:
s =
N
s
-N
r
N
s
where N
r
: Rotor speed (RPM).
• Rotor Frequency:
f
r
=sf
4. Equivalent Circuit (Per Phase)
• Parameters:
– Stator: R
1
: Resistance, X
1
: Leakage reactance.
– Rotor (referred to stator): R
'
2
: Resistance, X
'
2
: Leakage reactance.
– Magnetizing branch: X
m
: Magnetizing reactance, R
c
: Core loss resistance.
• Rotor Impedance:
Z
2
=
R
'
2
s
+jX
'
2
1
Page 2


Formula Sheet for Three-Phase Induction Motors
(Electrical Machines) – GATE
1. Basic Concepts
• Three-Phase Induction Motor: AC motor with rotating magnetic ?eld created
by three-phase stator currents, self-starting.
• Types: Squirrel-cage and wound-rotor.
• Operation Principle: Rotating magnetic ?eld induces currents in rotor, produc-
ing torque.
2. Synchronous Speed
• Synchronous Speed:
N
s
=
120f
P
where f: Supply frequency (Hz), P: Number of poles.
• Synchronous Speed in rad/s:
?
s
=
2pN
s
60
3. Slip
• Slip:
s =
N
s
-N
r
N
s
where N
r
: Rotor speed (RPM).
• Rotor Frequency:
f
r
=sf
4. Equivalent Circuit (Per Phase)
• Parameters:
– Stator: R
1
: Resistance, X
1
: Leakage reactance.
– Rotor (referred to stator): R
'
2
: Resistance, X
'
2
: Leakage reactance.
– Magnetizing branch: X
m
: Magnetizing reactance, R
c
: Core loss resistance.
• Rotor Impedance:
Z
2
=
R
'
2
s
+jX
'
2
1
• Total Input Impedance:
Z
in
=R
1
+jX
1
+
?
?
?
?
1
1
Rc
+
1
jXm
+
1
R
'
2
s
+jX
'
2
?
?
?
?
-1
5. Power Flow
• Input Power:
P
in
= 3V
1
I
1
cos? where V
1
: Phase voltage, I
1
: Stator current, cos? : Power factor.
• Air-Gap Power:
P
ag
= 3I
2
2
R
'
2
s
where I
2
: Rotor current (referred to stator).
• Rotor Copper Loss:
P
cu,r
=sP
ag
= 3I
2
2
R
'
2
• Developed Mechanical Power:
P
mech
= (1-s)P
ag
• Output Power:
P
out
=P
mech
-P
mech,loss
where P
mech,loss
: Friction and windage losses.
6. Torque
• Torque:
T =
P
ag
?
s
=
3I
2
2
R
'
2
/s
?
s
where ?
s
=
2pNs
60
.
• Torque Expression:
T =
3
?
s
V
2
1
sR
'
2
(R
1
+R
'
2
/s)
2
+(X
1
+X
'
2
)
2
• Maximum Torque:
s
max
=
R
'
2
v
R
2
1
+(X
1
+X
'
2
)
2
T
max
=
3V
2
1
2?
s
(
R
1
+
v
R
2
1
+(X
1
+X
'
2
)
2
)
• Starting Torque (s = 1):
T
start
=
3V
2
1
R
'
2
?
s
[(R
1
+R
'
2
)
2
+(X
1
+X
'
2
)
2
]
2
Page 3


Formula Sheet for Three-Phase Induction Motors
(Electrical Machines) – GATE
1. Basic Concepts
• Three-Phase Induction Motor: AC motor with rotating magnetic ?eld created
by three-phase stator currents, self-starting.
• Types: Squirrel-cage and wound-rotor.
• Operation Principle: Rotating magnetic ?eld induces currents in rotor, produc-
ing torque.
2. Synchronous Speed
• Synchronous Speed:
N
s
=
120f
P
where f: Supply frequency (Hz), P: Number of poles.
• Synchronous Speed in rad/s:
?
s
=
2pN
s
60
3. Slip
• Slip:
s =
N
s
-N
r
N
s
where N
r
: Rotor speed (RPM).
• Rotor Frequency:
f
r
=sf
4. Equivalent Circuit (Per Phase)
• Parameters:
– Stator: R
1
: Resistance, X
1
: Leakage reactance.
– Rotor (referred to stator): R
'
2
: Resistance, X
'
2
: Leakage reactance.
– Magnetizing branch: X
m
: Magnetizing reactance, R
c
: Core loss resistance.
• Rotor Impedance:
Z
2
=
R
'
2
s
+jX
'
2
1
• Total Input Impedance:
Z
in
=R
1
+jX
1
+
?
?
?
?
1
1
Rc
+
1
jXm
+
1
R
'
2
s
+jX
'
2
?
?
?
?
-1
5. Power Flow
• Input Power:
P
in
= 3V
1
I
1
cos? where V
1
: Phase voltage, I
1
: Stator current, cos? : Power factor.
• Air-Gap Power:
P
ag
= 3I
2
2
R
'
2
s
where I
2
: Rotor current (referred to stator).
• Rotor Copper Loss:
P
cu,r
=sP
ag
= 3I
2
2
R
'
2
• Developed Mechanical Power:
P
mech
= (1-s)P
ag
• Output Power:
P
out
=P
mech
-P
mech,loss
where P
mech,loss
: Friction and windage losses.
6. Torque
• Torque:
T =
P
ag
?
s
=
3I
2
2
R
'
2
/s
?
s
where ?
s
=
2pNs
60
.
• Torque Expression:
T =
3
?
s
V
2
1
sR
'
2
(R
1
+R
'
2
/s)
2
+(X
1
+X
'
2
)
2
• Maximum Torque:
s
max
=
R
'
2
v
R
2
1
+(X
1
+X
'
2
)
2
T
max
=
3V
2
1
2?
s
(
R
1
+
v
R
2
1
+(X
1
+X
'
2
)
2
)
• Starting Torque (s = 1):
T
start
=
3V
2
1
R
'
2
?
s
[(R
1
+R
'
2
)
2
+(X
1
+X
'
2
)
2
]
2
7. E?ciency
• Losses:
– Stator copper loss: P
cu,s
= 3I
2
1
R
1
.
– Rotor copper loss: P
cu,r
= 3I
2
2
R
'
2
.
– Core loss: P
core
?B
2
f
2
.
– Mechanical loss: Friction and windage.
• E?ciency:
? =
P
out
P
in
=
P
out
P
out
+P
cu,s
+P
cu,r
+P
core
+P
mech,loss
8. Starting Methods
• Direct On-Line (DOL): Full voltage applied, high starting current.
• Star-Delta Starter: Reduces starting current by
1
v
3
.
• Autotransformer Starter: Reduces voltage, starting torque?V
2
.
• Rotor Resistance Starter (wound rotor): Increases R
'
2
to improve starting
torque.
9. Performance Characteristics
• Power Factor:
cos? =
P
in
3V
1
I
1
• Torque-Speed Curve: Maximum torque at s
max
, zero at s = 0.
• Starting Current:
I
start
=
V
1
v
(R
1
+R
'
2
)
2
+(X
1
+X
'
2
)
2
Typically 5-7 times full-load current.
10. Design Considerations
• Rotor Design: Squirrel-cage for ruggedness, wound rotor for control via external
resistance.
• Applications: Pumps, fans, compressors, conveyors, industrial drives.
• Thermal Limits: Ensure losses do not exceed cooling capacity.
• Testing: No-load test (core loss, magnetizing current), blocked-rotor test (copper
loss, leakage reactance).
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