Formula Sheets: Transformer | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Single-Phase Transformers (Basic Electrical
Technology) – GATE
1. Basic Concepts
• Transformer: Staticdevicethattransferselectricalenergybetweencircuitsviamagneticcoupling.
• Types: Step-up (V
2
>V
1
), Step-down (V
2
<V
1
), single-phase.
• Operation Principle: Faradays law of electromagnetic induction.
2. EMF Equation
E = 4.44fN? m
where E = Induced EMF (V),
f = Frequency (Hz),
N = Number of turns,
? m
= Maximum ?ux (Wb).
3. Voltage and Current Relationships (Ideal Transformer)
• Voltage Ratio:
V
1
V
2
=
N
1
N
2
=a
where V
1
: Primary voltage,V
2
: Secondary voltage,N
1
,N
2
: Primary/secondary turns,a : Turns ratio.Current Ratio :
I1
I2
=
N2
N1
=
1
a
where I
1
: Primary current,I
2
: Secondary current.
• Power Conservation:
V
1
I
1
cos? 1
=V
2
I
2
cos? 2
where ? 1
,? 2
: Power factor angles.
4. Equivalent Circuit Parameters
• Primary Side: R
1
: Resistance, X
1
: Leakage reactance.
• Secondary Side (referred to primary): R
'
2
=R
2
(
N1
N2
)
2
, X
'
2
=X
2
(
N1
N2
)
2
.
• Magnetizing Branch: X
m
: Magnetizing reactance, R
c
: Core loss resistance.
• Total Equivalent Impedance (referred to primary):
Z
eq
=R
eq
+jX
eq
, R
eq
=R
1
+R
'
2
, X
eq
=X
1
+X
'
2
5. Losses
• Copper Loss:
P
cu
=I
2
1
R
1
+I
2
2
R
2
=I
2
1
(R
1
+R
'
2
)
• Core Loss:
P
core
=P
hyst
+P
eddy
, P
hyst
?B
2
f, P
eddy
?B
2
f
2
where B: Flux density, f: Frequency.
• Total Loss:
P
loss
=P
cu
+P
core
1
Page 2


Formula Sheet for Single-Phase Transformers (Basic Electrical
Technology) – GATE
1. Basic Concepts
• Transformer: Staticdevicethattransferselectricalenergybetweencircuitsviamagneticcoupling.
• Types: Step-up (V
2
>V
1
), Step-down (V
2
<V
1
), single-phase.
• Operation Principle: Faradays law of electromagnetic induction.
2. EMF Equation
E = 4.44fN? m
where E = Induced EMF (V),
f = Frequency (Hz),
N = Number of turns,
? m
= Maximum ?ux (Wb).
3. Voltage and Current Relationships (Ideal Transformer)
• Voltage Ratio:
V
1
V
2
=
N
1
N
2
=a
where V
1
: Primary voltage,V
2
: Secondary voltage,N
1
,N
2
: Primary/secondary turns,a : Turns ratio.Current Ratio :
I1
I2
=
N2
N1
=
1
a
where I
1
: Primary current,I
2
: Secondary current.
• Power Conservation:
V
1
I
1
cos? 1
=V
2
I
2
cos? 2
where ? 1
,? 2
: Power factor angles.
4. Equivalent Circuit Parameters
• Primary Side: R
1
: Resistance, X
1
: Leakage reactance.
• Secondary Side (referred to primary): R
'
2
=R
2
(
N1
N2
)
2
, X
'
2
=X
2
(
N1
N2
)
2
.
• Magnetizing Branch: X
m
: Magnetizing reactance, R
c
: Core loss resistance.
• Total Equivalent Impedance (referred to primary):
Z
eq
=R
eq
+jX
eq
, R
eq
=R
1
+R
'
2
, X
eq
=X
1
+X
'
2
5. Losses
• Copper Loss:
P
cu
=I
2
1
R
1
+I
2
2
R
2
=I
2
1
(R
1
+R
'
2
)
• Core Loss:
P
core
=P
hyst
+P
eddy
, P
hyst
?B
2
f, P
eddy
?B
2
f
2
where B: Flux density, f: Frequency.
• Total Loss:
P
loss
=P
cu
+P
core
1
6. E?ciency
• E?ciency:
? =
P
out
P
in
=
P
out
P
out
+P
cu
+P
core
? =
V
2
I
2
cos? 2
V
2
I
2
cos? 2
+P
cu
+P
core
• Maximum E?ciency: Occurs when P
cu
=P
core
.
7. Voltage Regulation
• Voltage Regulation:
VR =
V
2,no-load
-V
2,full-load
V
2,full-load
×100%
• Approximate Voltage Regulation:
VR˜
I
2
(R
eq
cos? 2
+X
eq
sin? 2
)
V
2
×100%
where cos? 2
: Load power factor.
8. Open-Circuit and Short-Circuit Tests
• Open-Circuit Test (determines core loss and magnetizing branch):
P
oc
=P
core
, I
m
=I
oc
sin? oc
, I
c
=I
oc
cos? oc
R
c
=
V
2
oc
P
oc
, X
m
=
V
oc
I
m
• Short-Circuit Test (determines copper loss and equivalent impedance):
P
sc
=P
cu
, Z
eq
=
V
sc
I
sc
, R
eq
=
P
sc
I
2
sc
X
eq
=
v
Z
2
eq
-R
2
eq
9. Impedance Transformation
• Impedance Re?ected to Primary:
Z
'
2
=Z
2
(
N
1
N
2
)
2
=
Z
2
a
2
• Impedance Re?ected to Secondary:
Z
'
1
=Z
1
(
N
2
N
1
)
2
=Z
1
a
2
10. Design Considerations
• Magnetic Saturation: Limit ?ux density (B˜ 1.5-2T).
• Thermal Management: Ensure losses do not exceed cooling capacity.
• Applications: Power distribution, voltage regulation, isolation.
• Testing: Use open-circuit and short-circuit tests to determine parameters.
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