Formula Sheets: Synchronous Machines | Electrical Machines - Electrical Engineering (EE) PDF Download

 Page 1


Electrical Mac hines: Sync hronous Mac hines F orm ula
Sheet for Electrical GA TE
Sync hronous Mac hine Basics
• Sync hronous Sp eed :
n
s
=
120f
P
(), ?
s
=
2pn
s
60
(rads
-1
)
where f is frequency (Hz ), P is n um b er of p oles.
• EMF Equation (P er Phase) :
E =4.44fN? (V)
where N is turns p er phase, ? is flux p er p ole ( Wb ).
• Pitc h F actor : k
p
=cos
(
a
2
)
, where a is pitc h angle.
• Distribution F actor : k
d
=
sin(mß/2)
msin(ß/2)
, where m is slots p er p ole p er phase, ß is slot
angle.
Sync hronous Generator
• V oltage Equation (P er Phase) :
E =V +I
a
(R
a
+jX
s
) (V)
where V is terminal v oltage, I
a
is armature curren t (A ), R
a
is armature resistance
(? ), X
s
is sync hronous reactance (? ).
• P o w er Output (P er Phase) :
P =VI
a
cos? (W)
P
total
=3VI
a
cos? (W)
where ? is p o w er factor angle.
• P o w er Angle Equation :
P =
EV
X
s
sind (W)
where d is p o w er angle (angle b et w een E and V ).
• V oltage Regulation :
VR=
E-V
V
×100 (%)
1
Page 2


Electrical Mac hines: Sync hronous Mac hines F orm ula
Sheet for Electrical GA TE
Sync hronous Mac hine Basics
• Sync hronous Sp eed :
n
s
=
120f
P
(), ?
s
=
2pn
s
60
(rads
-1
)
where f is frequency (Hz ), P is n um b er of p oles.
• EMF Equation (P er Phase) :
E =4.44fN? (V)
where N is turns p er phase, ? is flux p er p ole ( Wb ).
• Pitc h F actor : k
p
=cos
(
a
2
)
, where a is pitc h angle.
• Distribution F actor : k
d
=
sin(mß/2)
msin(ß/2)
, where m is slots p er p ole p er phase, ß is slot
angle.
Sync hronous Generator
• V oltage Equation (P er Phase) :
E =V +I
a
(R
a
+jX
s
) (V)
where V is terminal v oltage, I
a
is armature curren t (A ), R
a
is armature resistance
(? ), X
s
is sync hronous reactance (? ).
• P o w er Output (P er Phase) :
P =VI
a
cos? (W)
P
total
=3VI
a
cos? (W)
where ? is p o w er factor angle.
• P o w er Angle Equation :
P =
EV
X
s
sind (W)
where d is p o w er angle (angle b et w een E and V ).
• V oltage Regulation :
VR=
E-V
V
×100 (%)
1
Sync hronous Motor
• V oltage Equation (P er Phase) :
V =E+I
a
(R
a
+jX
s
) (V)
• T orque :
T =
3EV sind
?
s
X
s
(Nm)
where ?
s
is sync hronous sp eed (rads
-1
).
• P o w er Input (P er Phase) :
P
in
=VI
a
cos? (W)
• P o w er F actor Con trol : V aries with field curren t ( I
f
):
– Under-excited: Lagging p o w er factor.
– Ov er-excited: Leading p o w er factor.
– Normal excitation: Unit y p o w er factor.
Equiv alen t Circuit (P er Phase)
• Comp onen ts : R
a
(armature resistance), X
s
(sync hronous reactance), E (induced
EMF).
• Sync hronous Imp edance : Z
s
=R
a
+jX
s
(?) .
• Curren t :
I
a
=
V -E
Z
s
(A)
Losses and E?iciency
• Losses :
– Copp er Loss: P
cu
=3I
2
a
R
a
(W) .
– Core Loss: Hysteresis and eddy curren t losses (constan t for fixed sp eed).
– Mec hanical Loss: F riction and windage.
• E?iciency :
? =
P
out
P
in
=
P
out
P
out
+P
cu
+P
core
+P
mec h
×100 (%)
where P
out
=3VI
a
cos? (generator) or T?
s
(motor).
2
Page 3


Electrical Mac hines: Sync hronous Mac hines F orm ula
Sheet for Electrical GA TE
Sync hronous Mac hine Basics
• Sync hronous Sp eed :
n
s
=
120f
P
(), ?
s
=
2pn
s
60
(rads
-1
)
where f is frequency (Hz ), P is n um b er of p oles.
• EMF Equation (P er Phase) :
E =4.44fN? (V)
where N is turns p er phase, ? is flux p er p ole ( Wb ).
• Pitc h F actor : k
p
=cos
(
a
2
)
, where a is pitc h angle.
• Distribution F actor : k
d
=
sin(mß/2)
msin(ß/2)
, where m is slots p er p ole p er phase, ß is slot
angle.
Sync hronous Generator
• V oltage Equation (P er Phase) :
E =V +I
a
(R
a
+jX
s
) (V)
where V is terminal v oltage, I
a
is armature curren t (A ), R
a
is armature resistance
(? ), X
s
is sync hronous reactance (? ).
• P o w er Output (P er Phase) :
P =VI
a
cos? (W)
P
total
=3VI
a
cos? (W)
where ? is p o w er factor angle.
• P o w er Angle Equation :
P =
EV
X
s
sind (W)
where d is p o w er angle (angle b et w een E and V ).
• V oltage Regulation :
VR=
E-V
V
×100 (%)
1
Sync hronous Motor
• V oltage Equation (P er Phase) :
V =E+I
a
(R
a
+jX
s
) (V)
• T orque :
T =
3EV sind
?
s
X
s
(Nm)
where ?
s
is sync hronous sp eed (rads
-1
).
• P o w er Input (P er Phase) :
P
in
=VI
a
cos? (W)
• P o w er F actor Con trol : V aries with field curren t ( I
f
):
– Under-excited: Lagging p o w er factor.
– Ov er-excited: Leading p o w er factor.
– Normal excitation: Unit y p o w er factor.
Equiv alen t Circuit (P er Phase)
• Comp onen ts : R
a
(armature resistance), X
s
(sync hronous reactance), E (induced
EMF).
• Sync hronous Imp edance : Z
s
=R
a
+jX
s
(?) .
• Curren t :
I
a
=
V -E
Z
s
(A)
Losses and E?iciency
• Losses :
– Copp er Loss: P
cu
=3I
2
a
R
a
(W) .
– Core Loss: Hysteresis and eddy curren t losses (constan t for fixed sp eed).
– Mec hanical Loss: F riction and windage.
• E?iciency :
? =
P
out
P
in
=
P
out
P
out
+P
cu
+P
core
+P
mec h
×100 (%)
where P
out
=3VI
a
cos? (generator) or T?
s
(motor).
2
Key Notes
– Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), P o w er (W ), T orque (Nm ),
Sp eed ( or rads
-1
).
– Sync hronous mac hines op erate a t constan t sp eed (n
s
).
– F or GA TE, fo cus on p o w er angle, v oltage regulation, and equiv alen t circuit analysis.
– P o w er factor con trol is k ey for s ync hronous motors.
– Use phasor diagrams for v oltage and curren t relationships.
3