transformer
TRANSCRIPT
![Page 1: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/1.jpg)
Transformer
Professor Mohamed A. El-Sharkawi
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El-Sharkawi@University of Washington
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Why do we need transformers?
• Increase voltage of generator’s output– Transmit high power at low current– Reduce cost of transmission system
• Adjust voltage to a usable level
• Create electrical isolation
• Match load impedance
• Filters
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220kV-750kV220kV-750kV
15 kV- 25kV15 kV- 25kV
208V- 416V208V- 416V
TransmissionTransmissionTransformerTransformer
DistributionDistributionTransformerTransformer
ServiceServiceTransformerTransformer
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Transmission Transformer
![Page 5: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/5.jpg)
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Distribution Transformer
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Distribution Transformer
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Service Transformer
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Service Transformer bank
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Service Transformer bank
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Service Transformer
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Service Transformer
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Service Transformer
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Service Transformer
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Service Transformer
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Low power Transformer
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Basic Components
Iron Core Insulated Copper Wire
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Basic Components
Laminated iron core
Insulated copper wire
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![Page 21: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/21.jpg)
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i1 i2
N1e1 N2
e2
dteN
1dt
dNe
11
11
dt
dNe
22
Primary Secondary
![Page 22: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/22.jpg)
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2
1
2
1
2
1
N
N
dtd
N
dtd
N
te
te
2
1
2
1
N
N
E
E
2
2
1
1
N
E
N
E
Basic Analysis:Voltagei1
i2
N1e1+_
N2e2
+_
• Volts/turn is constant• Voltages are in phase (no phase shift)• Voltage magnitudes vary with turns ratio.
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Basic Analysis: Power and current
21 SS
*22
*11 IEIE
1
2
1
2*2
*1
N
N
E
E
I
I
1
2
2
1
N
N
I
I 2211 ININ
i1i2
N1e1+_
N2e2
+_
• Currents are in phase.
• Current ratio is opposite to the voltage ratio
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Basic Analysis: Reflected impedance
I1I2
N1E1 N2
E2
Primary Secondary
Source
Load
Flux
Zload
2
2
I
EZload
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Basic Analysis: Reflected impedance
'loadZ
I1
E1
Primary
Source
1
1
I
EZ '
load
2
2
1'
2
2
1
1
2
2
1'
N
NZZ
N
N
I
I
E
E
Z
Z
loadload
load
load
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Single-Phase, Ideal Transformer Ratings
2 KVA, 120/240 V
Apparent Power
Primary Voltage Secondary Voltage
N2N1
+
-
V1
I1 I2
+
-V2
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Rated Values
• Rated voltage: The device can continuously operate at the rated voltage without being damaged due to insulation failure
• Rated current: The device can continuously operate at the rated current without being damaged due to thermal destruction
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ExampleTransformer rating:
2 KVA, 240/120 V
Compute the currents
KVAIVIVS 2 2211
A 33.8V240
KVA 2
11
V
SI
A 67.16V 120
KVA 2
22
V
SI
N2N1
+
-
V1
I1 I2
+
-V2
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Multi-Secondary Transformer
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Multi-secondary windings
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I1
I2N1E1
N2E2
Primary
I3
N3E3
3
1
3
1
2
1
2
1
N
N
E
E
N
N
E
E
![Page 32: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/32.jpg)
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Current ratio: superposition
1
2212 N
NII I12
I2N1E1
N2E2
Primary
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Current ratio: superposition
1
3313 N
NII I13
N1E1
Primary
I3
N3E3
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Superposition
I1
I2N1E1
N2E2
Primary
I3
N3E3
332211
1
33
1
2213121
NININI
N
NI
N
NIIII
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Superposition
I1
I2N1E1
N2E2
Primary
I3
N3E3
331 SSS *33
*22
*11 IEIEIE
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Example
• The transformer consists of one primary winding and two secondary windings. The number of turns is each winding is
A voltage source of 120V is applied to the primary winding, and purely resistive loads are connected across the secondary windings. A wattmeter placed in the primary circuit measures 300W. Another wattmeter placed in the secondary winding N2 measures 90W. Compute the following:
• The voltages of the secondary windings• The currents in N3• The power consumed by the load connected across N3
500;1000;4000 321 NNN
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Solution
304000
1000120
1
212
N
NEE
154000
500120
1
313
N
NEE
5.2120
300
cos 11
11
E
PI
330
90
cos 22
22
E
PI
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Solution
3
332211
5003*10004000*5.2 I
NININI
143 I
3
332211
153*305.2*120 I
IEIEIE
143 I
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Solution
3
321
90300 P
PPP
2103P
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Autotransformer
![Page 41: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/41.jpg)
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I1 I2
N1V1N2
E2
A1
A2
B1
B2
E1 V2
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Autotransformer: Voltage and current
21 III load
I2
N2
V1
N1I1
Is
Iload
V2
A1
A2
B1
B2
E1
E2
211 EEV
![Page 43: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/43.jpg)
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Autotransformer
2211 IEIESA
loadsB IVIVS 21
N2N1
+
-
E1
I1 I2
+
-
E2
I2
N2
V1
N1I1
Is
Iload
V2
A1
A2
B1
B2
E1
E2
![Page 44: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/44.jpg)
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Autotransformer: Power
12
12111211 )(
IESS
IEIEIEEIVS
AB
sB
AB SS
![Page 45: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/45.jpg)
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ExampleRatings of regular transformer: 10 kVA, 400/200 V
New voltage ratio: 600/200 V
Compute the new ratings
Solution
AI
AI
254.0
10
502.0
10
1
2
kVAIVSB 152560011
I2
N2
V1
N1I1
Is
Iload
V2
A1
A2
B1
B2
E1
E2
![Page 46: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/46.jpg)
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VARIC: Variable Auto-Transformer
I2N2
V1
N1
I1
Is
Iload
V2
N3
Z
Y
Sliding terminal
![Page 47: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/47.jpg)
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Output Voltage
21
2
NN
NVV sload
ssload VNN
NNVV
21
21
21
321
NN
NNNVV sload
I2N2
V1
N1
I1
Is
Iload
V2
N3
Z
Y
Sliding terminal
At Y
At Z
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Three-Phase Transformer
![Page 49: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/49.jpg)
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3-phase transformer
![Page 50: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/50.jpg)
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3-phase transformer Y-Y connection. Also known as star-star connection
a
c b
N1
A
BC
N2
2
1
N
N
V
V
AN
an 2
1
3
3
N
N
V
V
V
V
AN
an
AC
ac
n N
Rat
io o
f P
has
e V
olta
ge
Rat
io o
f L
ine
Vol
tage
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3-phase transformer ( -)
A
B
C
N2
a
bc
N1
2
1
N
N
V
V
AC
ac
Rat
io o
f P
has
e V
olta
ge a
nd
lin
e vo
ltag
e
![Page 52: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/52.jpg)
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3-phase transformer (Y-)
Also known as star-delta connectionA
B
C
N2
N1
2
1
N
N
V
V
AC
an 2
133
N
N
V
V
V
V
AC
an
AC
ac
a
c b
n
Rat
io o
f P
has
e V
olta
ge
Rat
io o
f L
ine
Vol
tage
![Page 53: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/53.jpg)
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3-phase transformer bank (Y-)
N2N1Van
N2N1
N2N1
a
c
A
B
C
VABVab
1
2
an
AB
ab
AB
N3
N
V3
V
V
V
1
2
an
AB
N
N
V
V
b
![Page 54: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/54.jpg)
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Ratings of Ideal 3-phase Transformer
100 MVA, 13.8/138 KV
Apparent Power (3-phase)
Primary Voltageline-to-line
Secondary Voltageline-to-line
![Page 56: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/56.jpg)
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Example
• Three single-phase transformers are used to form a three-phase transformer bank. Each single-phase transformer is rated at 10 kVA, 13.8 KV/240 V. One side of the transformer bank is connected to a three-phase, 13.8 kV transmission line. The other side of the transformer is connected to a three-phase residential load of 415.7V, 9kVA at 0.8 power factor lagging. Determine the connection of the transformer bank, the voltage ratio of the transformer bank, and the line current of the bank at the 13.8 kV side
![Page 57: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/57.jpg)
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Solution• Secondary voltage (Low voltage side) should be in
Y to provide the needed residential voltage
• The high voltage side must be Delta-connection
– The line-to-line voltage of the supply is 13.8 kV. Same as the transformer rating of the primary.
– If the primary is connected in Y, the voltage of the load would be lower than 240 V.
V2403
7.415
![Page 58: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/58.jpg)
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Solution
N2N1Van
N2N1
a
c
A
B
C
VAB= 13.8 kV Vab
b
Van= 240 V
![Page 59: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/59.jpg)
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Solution2403
800,13
3
an
AB
ab
AB
V
V
V
V
N2N1Van
N2N1
a
c
A
B
C
VAB= 13.8 kV Vab
b
Van= 240 V
AI 5.12240
39000
2
Phase current of the load
Phase current of the Transformer primary
AN
NII 2174.0
800,13
2405.12
1
221
Line Current in primary
AII A 377.03 1
![Page 60: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/60.jpg)
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Actual Transformer
• Windings:– Resistance– Inductance
• Core:– Eddy Current– Hysteresis
i1i2
N1e1+_
N2e2
+_
![Page 61: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/61.jpg)
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R1X1
N1 N2
R2 X2
Ideal Transformer
Windings Impedance
![Page 62: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/62.jpg)
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Core Hysteresis
B
H
i
Ne+
_
dtefB
ifH
![Page 63: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/63.jpg)
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i
e
![Page 64: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/64.jpg)
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Core Model
e
i
RLet
tEe sinmax
tE
dte
cosmax
tR
E
R
ei sinmax i
e
![Page 65: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/65.jpg)
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Core Model
LettEe sinmax
tE
dte
cosmax
tL
Edte
Li
dt
diLe
cos1 max
i
e
i
e Xl
![Page 66: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/66.jpg)
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e
i
R Xl
i
e
i
e
![Page 67: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/67.jpg)
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Ro Xo
R1X1 R2 X2
V1
I1
Io
'2I
I2V2E1 E2
N1 N2
Equivalent Circuit
2
1
2
1
2
1
V
V
N
N
E
E
2
1
1
2
2
'2
I
I
N
N
I
I
load
![Page 68: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/68.jpg)
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Referred impedance
Ro Xo
R1X1 R2 X2
V1
I1
Io
'2I
I2E1 E2
N1 N2
V2
22222
12
2
11 VjXRI
N
NE
N
NE
![Page 69: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/69.jpg)
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Referred impedance
Ro Xo
R1X1 R2 X2
V1
I1
Io
'2I
I2E1 E2
N1 N2
V2
222
2
1'2
2
12
2
11 VjXR
N
NI
N
NE
N
NE
![Page 70: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/70.jpg)
El-Sharkawi@University of Washington
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222
2
1'2
2
12
2
11 VjXR
N
NI
N
NE
N
NE
22
1'2
2
2
2
1'2
2
2
2
1'2
VN
NV
XN
NX
RN
NR
Define:
'2
'2
'2
'21 VjXRIE
Then:
22
122
2
2
1'21 V
N
NjXR
N
NIE
![Page 71: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/71.jpg)
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'2
'2
'2
'21 VjXRIE
Ro Xo
R1X1 R2 X2
V1
I1
Io
'2I
I2E1 E2
N1 N2
V2
Ro Xo
R1X1
V1
I1
Io '2I
E1
'2R
'2X
'2V
Equivalent Circuit Referred to Source Side
![Page 72: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/72.jpg)
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o'21 III
1o'2 III
Ro Xo
V1
I1
Io '2I
E1'2V
R1X1
'2R
'2X
'2o1 RRR '2o1 XXX
Practical Considerations
![Page 73: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/73.jpg)
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V1
I1
Ro Xo
Io '2I '
2V
R1X1 '
2R'2X
V1
I1
Ro Xo
Io
'2I
'2V
R1
X1 '2R
'2X
![Page 74: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/74.jpg)
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V1
'21 II '
2V
R1
X1 '2R
'2X
Define:'21eq
'21eq
XXX
RRR
![Page 75: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/75.jpg)
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Analysis of Transformer
'21 II
V1'2V
eqR eqX
eqeq'2
'21 jXRIVV
Z
![Page 76: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/76.jpg)
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22
1'2 V
N
NV
TerminologiesTerminologies
2
2
2
1'2
2
2
2
1'2
XN
NX
RN
NR
2I
21
2'2 I
N
NI
2V Load VoltageLoad Voltage
Load Voltage referred to Source sideLoad Voltage referred to Source side
Impedance referred to Source sideImpedance referred to Source side
Load CurrentLoad Current
Load current referred to Source sideLoad current referred to Source side
![Page 77: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/77.jpg)
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Analysis of Transformer
'21 II
V1'2V
eqR eqX
eqeq'2
'21 jXRIVV
Z’
![Page 78: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/78.jpg)
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eqX
'21 II
V1'2V
eqR
Z’
'2I
eq'2 RI
eq'2 XI
![Page 79: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/79.jpg)
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'21 II
V1'2V
eqR eqX
Z
eq'2eq
'2
'21 XIjRIVV
eq'2 RI
eq'2 XI
'2I
eq'2 ZI
'2V
1V
![Page 80: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/80.jpg)
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Ratings of Actual 3-phase Transformer
100 MVA, 13.8/138 KV
Apparent Power (3-phase)
line-to-line
'V2
line-to-line2V
![Page 81: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/81.jpg)
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Example
102
1 N
N
5000;1000;10;1 0'21
'21 XRXXXRRR oeqeq
ol .Z 3050
A transformer has the following parameters:
The rated voltage of the primary winding is 1000V. Compute the load voltage.
![Page 82: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/82.jpg)
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Solution
502
2
1
N
NZZ L
'L
A..jZXjR
VI o
o'Leqeq
' 31387173050101
01000 01
2
'21 II
V1'2V
eqR eqX
Z’
V...ZIV oo'L
'' 022 31888530503138717
V.N
NVV ' 588
10
1885
1
222
![Page 83: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/83.jpg)
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Voltage Regulation VR
loadfull
loadfullloadno
V
VVVR
'2
'21
V
VVVR
Measured at the load side
'21 II '
2V
eqR eqX
V1 Loa
d
![Page 84: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/84.jpg)
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ExampleCalculate the voltage regulation of the transformer in the previous problem.
Solution:
%14.01007200
72005.7209VR
V
VVVR
'2
'21
![Page 85: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/85.jpg)
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Efficiency
V1
I1
Ro Xo
Io'2I '
2V
ReqXeq
in
out
P
P
PowerInput
PowerOutput
LossesPP outin
ironculosses PPP
eq2'
2cu RIP
o
21
iron R
VP
cosIVP '2
'2out
![Page 86: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/86.jpg)
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Example
A 10 kVA, 2300/230 V, single phase distribution transformerhas the following parameters:
k4.69X;k6.75R
05.6R;12XX;8.5R
oo
'2
'211
At full load and 0.8 power factor lagging, compute the efficiencyof the transformer.
![Page 87: Transformer](https://reader033.vdocuments.mx/reader033/viewer/2022061117/5467f367b4af9fe6518b4730/html5/thumbnails/87.jpg)
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Solution
W23.22405.68.5)35.4(RIP 2eq
2'2cu
W
R
V
R
VP
ooiron 70
600,75
2300 22'2
21
kW88.010cosScosIVP '2
'2out
A35.42300
000,10
V
SI
'2
'2
%45.961007023.2248000
8000
P
P
in
out