Download Ideal Transformer

Transformer
Professor Mohamed A. El-Sharkawi
Why do we need transformers?
• Increase voltage of generator output
– Transmit power and low current
– Reduce cost of transmission system
•
•
•
•
Adjust voltage to a usable level
Create electrical isolation
Match load impedance
Filters
2
Basic Components
Iron Core
Insulated Copper Wire
3
4
5
d
e1  N1
dt
1

e1 dt

N1
d
e2  N 2
dt
i1
i2
+
+
e1
_
Primary
N1
N2
e2
_
Secondary
6
Basic Analysis:Voltage
d
N1
e1 t 
N1
dt


e2 t  N d N 2
2
dt

E1 N1
 
E2 N 2
i2
i1
e1 _+
N1
N2
+
_
e2


E1 E2

N1 N 2
• Volts/turn is constant
• Voltages are in phase (no phase shift)
• Magnitudes vary with turns ratio.
7
Basic Analysis: Power and current
S1  S2
EI E I
*
1 1
*
1
*
2
i2
i1
*
2 2
I
E2 N 2


I
E1 N1
e1 _+
I1 N 2

I 2 N1
N1
N2
+
_
e2
N1I1  N 2 I 2
• Currents are in phase.
• Current ratio is opposite to the voltage ratio
8
Basic Analysis: Reflected impedance
I1
+
E1
I1
I2
+
+
N
1
N
2
-
E2
-
Z 2 I 2  E2
Z1I1  E1
Z2
E1
Z1
-
Z 2 E 2 I1 N 2 N 2  N 2 



 
Z1 E1 I 2 N1 N1  N1 
Z2  N2 

 
Z1  N1 
2
9
2
Ideal Transformer: Ratings
I1
Apparent Power
100 MVA, 13.8/138 KV
Primary Voltage
+
V1
-
N1
I2
N2
+
V2
-
Secondary Voltage
10
Ideal Transformer: Ratings
I1
100 MVA, 138/13.8 KV
S  V1I1  V2 I 2  100 MVA
V1 N1 138 kV


 10
V2 N 2 13.8 kV
+
V1
-
N1
I2
N2
+
V2
-
S
100 MVA
I2 

 7250 A
V2
13.8 kV
S
100 MVA
I1  
 725 A
V1
138 kV
11
Actual Transformer
• Windings:
– Resistance
– Inductance
i2
i1
e1 _+
N1
N2
+
_
e2
• Core:
– Eddy Current
– Hysteresis
12
Windings Impedance
R1
X1
N1
R2
X2
N2
Ideal Transformer
13
Core Hysteresis
i
B
+
e
_
N
H
B f
 e dt 
H  f i 
14
e
i
15
i
Core Model
Let
e  Emax sin t
 e dt  
Emax

e
R
e
cos t
e Emax
i 
sin t
R
R
i
16
i
Core Model
Let
e  Emax sin t
 e dt  
Emax

e
cos t
di
eL
dt
Emax
1
i   e dt  
cos t
L
L
Xl
e
i
17
i
e
R
Xl
e
e
i
i
18
Equivalent Circuit
X1
R1
V1
I
'
2
N1 N2
R2
Io
I1
Ro
Xo
E1
N1 V1


E2
N 2 V2
E1
E2
X2
I2
'
I 2 N2
I1


I2
N1 I 2
19
V2