Formula Sheets: Single Phase Transformers | Electrical Machines - Electrical Engineering (EE) PDF Download

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Electrical Mac hines: Single Phase T ransformers
F orm ula Sheet for Electrical GA TE
T r ansformer Basics
• EMF Equation :
E =4.44fN?
m
(V)
where f is frequency (Hz ), N is n um b er of turns, ?
m
is m axim um flux ( Wb ).
• T urns Ratio (a ) :
a=
N
1
N
2
,
V
1
V
2
=
N
1
N
2
,
I
1
I
2
=
N
2
N
1
whereN
1
, N
2
are primary and secondary turns, V
1
, V
2
are v oltages, I
1
, I
2
are curren ts.
• Ideal T ransformer : No losses, V
1
I
1
=V
2
I
2
(p o w er conserv ation).
Equiv alen t Circuit
• Referred to Primary :
R
eq1
=R
1
+R
2
a
2
, X
eq1
=X
1
+X
2
a
2
whereR
1
,R
2
are primary and secondary resistances (? ),X
1
,X
2
are leakage reactances
(? ), a=
N
1
N
2
.
• Referred to Secondary :
R
eq2
=R
1
/a
2
+R
2
, X
eq2
=X
1
/a
2
+X
2
• Appro ximate Equiv alen t Circuit : Magnetizing branc h (core loss R
c
, magnetizing
reactance X
m
) in parallel with primary .
Losses and E?iciency
• Copp er Loss :
P
cu
=I
2
1
R
1
+I
2
2
R
2
=I
2
1
R
eq1
=I
2
2
R
eq2
(W)
• Core Loss : P
core
=P
h ysteresis
+P
eddy
, constan t for fixed v oltage and frequency .
• E?iciency :
? =
P
out
P
out
+P
cu
+P
core
×100 (%)
where P
out
=V
2
I
2
cos?
2
(output p o w er, W ).
• Maxim um E?iciency : Occurs when P
cu
=P
core
.
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Electrical Mac hines: Single Phase T ransformers
F orm ula Sheet for Electrical GA TE
T r ansformer Basics
• EMF Equation :
E =4.44fN?
m
(V)
where f is frequency (Hz ), N is n um b er of turns, ?
m
is m axim um flux ( Wb ).
• T urns Ratio (a ) :
a=
N
1
N
2
,
V
1
V
2
=
N
1
N
2
,
I
1
I
2
=
N
2
N
1
whereN
1
, N
2
are primary and secondary turns, V
1
, V
2
are v oltages, I
1
, I
2
are curren ts.
• Ideal T ransformer : No losses, V
1
I
1
=V
2
I
2
(p o w er conserv ation).
Equiv alen t Circuit
• Referred to Primary :
R
eq1
=R
1
+R
2
a
2
, X
eq1
=X
1
+X
2
a
2
whereR
1
,R
2
are primary and secondary resistances (? ),X
1
,X
2
are leakage reactances
(? ), a=
N
1
N
2
.
• Referred to Secondary :
R
eq2
=R
1
/a
2
+R
2
, X
eq2
=X
1
/a
2
+X
2
• Appro ximate Equiv alen t Circuit : Magnetizing branc h (core loss R
c
, magnetizing
reactance X
m
) in parallel with primary .
Losses and E?iciency
• Copp er Loss :
P
cu
=I
2
1
R
1
+I
2
2
R
2
=I
2
1
R
eq1
=I
2
2
R
eq2
(W)
• Core Loss : P
core
=P
h ysteresis
+P
eddy
, constan t for fixed v oltage and frequency .
• E?iciency :
? =
P
out
P
out
+P
cu
+P
core
×100 (%)
where P
out
=V
2
I
2
cos?
2
(output p o w er, W ).
• Maxim um E?iciency : Occurs when P
cu
=P
core
.
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V o ltage Regulation
• V oltage Regulation :
VR=
V
2, no-load
-V
2, full-load
V
2, full-load
×100 (%)
• Appro ximate V oltage Regulation :
VR˜
I
2
R
eq2
cos?
2
±I
2
X
eq2
sin?
2
V
2
×100
where + for lagging p o w er factor,- for leading p o w er factor.
Op e n and Short Circuit T ests
• Op en Circuit T est (on L V side):
P
o c
=P
core
, R
c
=
V
2
o c
P
o c
, X
m
=
V
o c
I
o c
sin?
o c
where V
o c
, I
o c
, P
o c
are op en-circuit v oltage, cur ren t, and p o w er.
• Short Circuit T est (on HV side):
R
eq1
=
P
sc
I
2
sc
, Z
eq1
=
V
sc
I
sc
, X
eq1
=
v
Z
2
eq1
-R
2
eq1
where V
sc
, I
sc
, P
sc
are short-circuit v oltage, curren t, and p o w er.
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), P o w er (W ), F requency (Hz ).
• Op en circuit test determines core loss and magnetizing parameters; short circuit test
determines copp er loss and equiv alen t imp edance.
• F or GA TE, fo cus on equiv alen t circuit parameters, e?iciency , and v oltage regulation
calculations.
• Use RMS v alues for A C quan tities unless sp ecified otherwise.
• Chec k p o w er factor sign for v oltage regulation (lagging/leading).
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