Formula Sheets: Three Phase Transformers and Autotransformers | Electrical Machines - Electrical Engineering (EE) PDF Download

 Page 1


Electrical Mac hines: Three Phase T ransformers and
A utotransformers F orm ula Sheet f or Electrical
GA TE
Three-Phase T ransformer Basics
• EMF p er Phase :
E = 4.44fN?
m
(V)
wheref is frequency (Hz ),N is n um b er of turns p er phase,?
m
is maxim um flux ( Wb ).
• Line and Phase V oltages :
– Star (Y) Connection : V
L
=
v
3V
p
, I
L
=I
p
.
– Delta (?) Connection : V
L
=V
p
, I
L
=
v
3I
p
.
• Three-Phase P o w er :
P =
v
3V
L
I
L
cos? (W)
S =
v
3V
L
I
L
()
Q =
v
3V
L
I
L
sin? (-reactive)
where ? is the p o w er factor angle.
Three-Phase T ransformer Connections
• Common Configurations : Y-Y, ?-?, Y-?, ?-Y.
• V oltage and Curren t Ratios (P er Phase) :
V
1
V
2
=
N
1
N
2
,
I
1
I
2
=
N
2
N
1
where N
1
, N
2
are turns p er phase on primary and secondary .
• Line V oltage Ratio :
Y-Y or ?-? :
V
L1
V
L2
=
N
1
N
2
Y-? :
V
L1
V
L2
=
v
3
N
1
N
2
, ?-Y :
V
L1
V
L2
=
N
1
v
3N
2
• Phase Shift : Y-? or ?-Y in tro duces a 30° phase shift.
Equiv alen t Circuit (P er Phase)
• Referred to Primary :
R
eq1
=R
1
+R
2
(
N
1
N
2
)
2
, X
eq1
=X
1
+X
2
(
N
1
N
2
)
2
where R
1
, R
2
are resistances, X
1
, X
2
are leakage reactances p er phase (? ).
1
Page 2


Electrical Mac hines: Three Phase T ransformers and
A utotransformers F orm ula Sheet f or Electrical
GA TE
Three-Phase T ransformer Basics
• EMF p er Phase :
E = 4.44fN?
m
(V)
wheref is frequency (Hz ),N is n um b er of turns p er phase,?
m
is maxim um flux ( Wb ).
• Line and Phase V oltages :
– Star (Y) Connection : V
L
=
v
3V
p
, I
L
=I
p
.
– Delta (?) Connection : V
L
=V
p
, I
L
=
v
3I
p
.
• Three-Phase P o w er :
P =
v
3V
L
I
L
cos? (W)
S =
v
3V
L
I
L
()
Q =
v
3V
L
I
L
sin? (-reactive)
where ? is the p o w er factor angle.
Three-Phase T ransformer Connections
• Common Configurations : Y-Y, ?-?, Y-?, ?-Y.
• V oltage and Curren t Ratios (P er Phase) :
V
1
V
2
=
N
1
N
2
,
I
1
I
2
=
N
2
N
1
where N
1
, N
2
are turns p er phase on primary and secondary .
• Line V oltage Ratio :
Y-Y or ?-? :
V
L1
V
L2
=
N
1
N
2
Y-? :
V
L1
V
L2
=
v
3
N
1
N
2
, ?-Y :
V
L1
V
L2
=
N
1
v
3N
2
• Phase Shift : Y-? or ?-Y in tro duces a 30° phase shift.
Equiv alen t Circuit (P er Phase)
• Referred to Primary :
R
eq1
=R
1
+R
2
(
N
1
N
2
)
2
, X
eq1
=X
1
+X
2
(
N
1
N
2
)
2
where R
1
, R
2
are resistances, X
1
, X
2
are leakage reactances p er phase (? ).
1
• Referred to Secondary :
R
eq2
=R
1
(
N
2
N
1
)
2
+R
2
, X
eq2
=X
1
(
N
2
N
1
)
2
+X
2
• Core Losses : Represen ted b y R
c
(core loss resistance) and X
m
(magnetizing reac-
tance) in parallel.
Losses and E?iciency
• Copp er Loss (P er Phase) :
P
cu
=I
2
1
R
1
+I
2
2
R
2
= 3I
2
1
R
eq1
= 3I
2
2
R
eq2
(W)
• Core Loss : P
core
= 3
V
2
p
Rc
, constan t for fixed v oltage and frequency .
• E?iciency :
? =
P
out
P
out
+P
cu
+P
core
×100 (%)
where P
out
=
v
3V
L
I
L
cos? .
A utotransformers
• V oltage Relationship :
V
H
=V
L
+V
se
,
V
H
V
L
=
N
1
+N
2
N
2
where V
H
is high v oltage, V
L
is lo w v oltage, N
1
is common turns, N
2
is series turns.
• Curren t Relationship :
I
H
(N
1
+N
2
) =I
L
N
2
, I
se
=I
L
-I
H
where I
H
, I
L
are high and lo w side curren ts, I
se
is series winding curren t.
• P o w er A dv an tage :
S
auto
=
N
1
+N
2
N
1
S
load
where S
auto
is apparen t p o w er through trans former, S
load
is load p o w er.
• E?iciency : Higher than t w o-winding transformer due to lo w er copp er losses.
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), P o w er (W , , -reactive ),
F requency (Hz ).
• F or three-phase transformers, analyze p er-phase equiv alen t circuit in Y-connection for
simplicit y .
2
Page 3


Electrical Mac hines: Three Phase T ransformers and
A utotransformers F orm ula Sheet f or Electrical
GA TE
Three-Phase T ransformer Basics
• EMF p er Phase :
E = 4.44fN?
m
(V)
wheref is frequency (Hz ),N is n um b er of turns p er phase,?
m
is maxim um flux ( Wb ).
• Line and Phase V oltages :
– Star (Y) Connection : V
L
=
v
3V
p
, I
L
=I
p
.
– Delta (?) Connection : V
L
=V
p
, I
L
=
v
3I
p
.
• Three-Phase P o w er :
P =
v
3V
L
I
L
cos? (W)
S =
v
3V
L
I
L
()
Q =
v
3V
L
I
L
sin? (-reactive)
where ? is the p o w er factor angle.
Three-Phase T ransformer Connections
• Common Configurations : Y-Y, ?-?, Y-?, ?-Y.
• V oltage and Curren t Ratios (P er Phase) :
V
1
V
2
=
N
1
N
2
,
I
1
I
2
=
N
2
N
1
where N
1
, N
2
are turns p er phase on primary and secondary .
• Line V oltage Ratio :
Y-Y or ?-? :
V
L1
V
L2
=
N
1
N
2
Y-? :
V
L1
V
L2
=
v
3
N
1
N
2
, ?-Y :
V
L1
V
L2
=
N
1
v
3N
2
• Phase Shift : Y-? or ?-Y in tro duces a 30° phase shift.
Equiv alen t Circuit (P er Phase)
• Referred to Primary :
R
eq1
=R
1
+R
2
(
N
1
N
2
)
2
, X
eq1
=X
1
+X
2
(
N
1
N
2
)
2
where R
1
, R
2
are resistances, X
1
, X
2
are leakage reactances p er phase (? ).
1
• Referred to Secondary :
R
eq2
=R
1
(
N
2
N
1
)
2
+R
2
, X
eq2
=X
1
(
N
2
N
1
)
2
+X
2
• Core Losses : Represen ted b y R
c
(core loss resistance) and X
m
(magnetizing reac-
tance) in parallel.
Losses and E?iciency
• Copp er Loss (P er Phase) :
P
cu
=I
2
1
R
1
+I
2
2
R
2
= 3I
2
1
R
eq1
= 3I
2
2
R
eq2
(W)
• Core Loss : P
core
= 3
V
2
p
Rc
, constan t for fixed v oltage and frequency .
• E?iciency :
? =
P
out
P
out
+P
cu
+P
core
×100 (%)
where P
out
=
v
3V
L
I
L
cos? .
A utotransformers
• V oltage Relationship :
V
H
=V
L
+V
se
,
V
H
V
L
=
N
1
+N
2
N
2
where V
H
is high v oltage, V
L
is lo w v oltage, N
1
is common turns, N
2
is series turns.
• Curren t Relationship :
I
H
(N
1
+N
2
) =I
L
N
2
, I
se
=I
L
-I
H
where I
H
, I
L
are high and lo w side curren ts, I
se
is series winding curren t.
• P o w er A dv an tage :
S
auto
=
N
1
+N
2
N
1
S
load
where S
auto
is apparen t p o w er through trans former, S
load
is load p o w er.
• E?iciency : Higher than t w o-winding transformer due to lo w er copp er losses.
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), P o w er (W , , -reactive ),
F requency (Hz ).
• F or three-phase transformers, analyze p er-phase equiv alen t circuit in Y-connection for
simplicit y .
2
• Chec k connection t yp e (Y-Y, Y-?, etc.) for correct v oltage/curren t ratios and phase
shifts.
• A utotransformers ha v e higher e?iciency but no isolation b et w een windings.
• F or GA TE, fo cus on p er-phase analysis, p o w er calculations, and star-delta transfor-
mations.
3