Formula Sheets: Basics

 Page 1


Formula Sheet for Basics of Electrical Machines
(Electrical Engineering) – GATE
1. Basic Concepts
• Electrical Machine: Device that converts electrical energy to mechanical energy
or vice versa.
• Types: Transformers (stationary), DC machines, AC machines (induction, syn-
chronous).
• EMF Equation: Induced voltage due to magnetic ?ux change.
• Torque: Rotational force produced by electromagnetic interaction.
2. Magnetic Circuits
• Magnetic Flux: ? =BA, where B: Magnetic ?ux density (T), A: Area (mš).
• MagnetomotiveForce(MMF):F =NI,whereN: Numberofturns,I: Current
(A).
• Magnetic Field Intensity: H =
F
l
=
NI
l
, where l: Magnetic path length (m).
• Reluctance: R =
l
µA
, where µ : Permeability (H/m).
• Flux-MMF Relation: ? =
F
R
.
3. Transformers
• EMF Equation:
E = 4.44fN? m
where f: Frequency (Hz), N: Number of turns, ? m
: Maximum ?ux (Wb).
• Voltage Ratio:
V
1
V
2
=
N
1
N
2
• Current Ratio:
I
1
I
2
=
N
2
N
1
• Power Conservation: V
1
I
1
cos?
1
=V
2
I
2
cos?
2
(ideal transformer).
• Impedance Transformation:
Z
'
2
=Z
2
(
N
1
N
2
)
2
4. DC Machines
• EMF Equation:
E =
?ZNP
60A
1
Page 2


Formula Sheet for Basics of Electrical Machines
(Electrical Engineering) – GATE
1. Basic Concepts
• Electrical Machine: Device that converts electrical energy to mechanical energy
or vice versa.
• Types: Transformers (stationary), DC machines, AC machines (induction, syn-
chronous).
• EMF Equation: Induced voltage due to magnetic ?ux change.
• Torque: Rotational force produced by electromagnetic interaction.
2. Magnetic Circuits
• Magnetic Flux: ? =BA, where B: Magnetic ?ux density (T), A: Area (mš).
• MagnetomotiveForce(MMF):F =NI,whereN: Numberofturns,I: Current
(A).
• Magnetic Field Intensity: H =
F
l
=
NI
l
, where l: Magnetic path length (m).
• Reluctance: R =
l
µA
, where µ : Permeability (H/m).
• Flux-MMF Relation: ? =
F
R
.
3. Transformers
• EMF Equation:
E = 4.44fN? m
where f: Frequency (Hz), N: Number of turns, ? m
: Maximum ?ux (Wb).
• Voltage Ratio:
V
1
V
2
=
N
1
N
2
• Current Ratio:
I
1
I
2
=
N
2
N
1
• Power Conservation: V
1
I
1
cos?
1
=V
2
I
2
cos?
2
(ideal transformer).
• Impedance Transformation:
Z
'
2
=Z
2
(
N
1
N
2
)
2
4. DC Machines
• EMF Equation:
E =
?ZNP
60A
1
where ? : Flux per pole (Wb), Z: Total conductors, N: Speed (RPM), P: Number
of poles, A: Number of parallel paths.
• Torque Equation:
T =
?ZI
a
P
2pA
where I
a
: Armature current (A).
• Generated Voltage (Generator):
V =E-I
a
R
a
where R
a
: Armature resistance.
• Back EMF (Motor):
E
b
=V -I
a
R
a
• Power Equation:
P
developed
=E
b
I
a
, P
mechanical
=T?
where ? =
2pN
60
(rad/s).
5. AC Machines (General)
• Synchronous Speed:
N
s
=
120f
P
where f: Frequency (Hz), P: Number of poles.
• Slip (Induction Machine):
s =
N
s
-N
r
N
s
where N
r
: Rotor speed (RPM).
• Rotor Frequency:
f
r
=sf
6. Induction Machines
• Torque Equation:
T =
3
?
s
V
2
sR
2
/(R
2
2
+(sX
2
)
2
)
v
R
2
1
+(X
1
+X
2
)
2
where V: Applied voltage, R
2
: Rotor resistance, X
2
: Rotor reactance, ?
s
: Syn-
chronous speed (rad/s).
• Maximum Torque:
s
max
=
R
2
X
2
, T
max
?
V
2
X
2
• Power Transfer:
P
air-gap
= 3I
2
2
R
2
/s, P
developed
= (1-s)P
air-gap
2
Page 3


Formula Sheet for Basics of Electrical Machines
(Electrical Engineering) – GATE
1. Basic Concepts
• Electrical Machine: Device that converts electrical energy to mechanical energy
or vice versa.
• Types: Transformers (stationary), DC machines, AC machines (induction, syn-
chronous).
• EMF Equation: Induced voltage due to magnetic ?ux change.
• Torque: Rotational force produced by electromagnetic interaction.
2. Magnetic Circuits
• Magnetic Flux: ? =BA, where B: Magnetic ?ux density (T), A: Area (mš).
• MagnetomotiveForce(MMF):F =NI,whereN: Numberofturns,I: Current
(A).
• Magnetic Field Intensity: H =
F
l
=
NI
l
, where l: Magnetic path length (m).
• Reluctance: R =
l
µA
, where µ : Permeability (H/m).
• Flux-MMF Relation: ? =
F
R
.
3. Transformers
• EMF Equation:
E = 4.44fN? m
where f: Frequency (Hz), N: Number of turns, ? m
: Maximum ?ux (Wb).
• Voltage Ratio:
V
1
V
2
=
N
1
N
2
• Current Ratio:
I
1
I
2
=
N
2
N
1
• Power Conservation: V
1
I
1
cos?
1
=V
2
I
2
cos?
2
(ideal transformer).
• Impedance Transformation:
Z
'
2
=Z
2
(
N
1
N
2
)
2
4. DC Machines
• EMF Equation:
E =
?ZNP
60A
1
where ? : Flux per pole (Wb), Z: Total conductors, N: Speed (RPM), P: Number
of poles, A: Number of parallel paths.
• Torque Equation:
T =
?ZI
a
P
2pA
where I
a
: Armature current (A).
• Generated Voltage (Generator):
V =E-I
a
R
a
where R
a
: Armature resistance.
• Back EMF (Motor):
E
b
=V -I
a
R
a
• Power Equation:
P
developed
=E
b
I
a
, P
mechanical
=T?
where ? =
2pN
60
(rad/s).
5. AC Machines (General)
• Synchronous Speed:
N
s
=
120f
P
where f: Frequency (Hz), P: Number of poles.
• Slip (Induction Machine):
s =
N
s
-N
r
N
s
where N
r
: Rotor speed (RPM).
• Rotor Frequency:
f
r
=sf
6. Induction Machines
• Torque Equation:
T =
3
?
s
V
2
sR
2
/(R
2
2
+(sX
2
)
2
)
v
R
2
1
+(X
1
+X
2
)
2
where V: Applied voltage, R
2
: Rotor resistance, X
2
: Rotor reactance, ?
s
: Syn-
chronous speed (rad/s).
• Maximum Torque:
s
max
=
R
2
X
2
, T
max
?
V
2
X
2
• Power Transfer:
P
air-gap
= 3I
2
2
R
2
/s, P
developed
= (1-s)P
air-gap
2
7. Synchronous Machines
• EMF Equation:
E = 4.44f?N
ph
where N
ph
: Turns per phase.
• Synchronous Reactance:
V =E-I(R
a
+jX
s
)
where X
s
: Synchronous reactance, R
a
: Armature resistance.
• Power Angle Equation:
P =
3VE
X
s
sind
where d: Load angle.
8. Losses and E?ciency
• Types of Losses: Copper (IšR), core (hysteresis, eddy current), mechanical (fric-
tion, windage).
• E?ciency:
? =
P
out
P
in
=
P
out
P
out
+P
losses
• Copper Loss:
P
cu
=I
2
R
• Core Loss:
P
core
?B
2
f
2
9. Voltage Regulation
• Transformer:
VR =
V
no-load
-V
full-load
V
full-load
×100%
• Synchronous Machine:
VR =
E-V
V
×100%
10. Design Considerations
• Magnetic Saturation: Limit ?ux density to avoid core saturation.
• Thermal Limits: Ensure losses do not exceed cooling capacity.
• Applications: Power generation, industrial drives, traction.
• Testing: Open-circuit, short-circuit tests for transformers; load tests for machines.
3