Download TRANSFORMERS Ideal transformer voltage and current relations Transformer modelling - impedance relations E

TRANSFORMERS
Ideal transformer voltage and current relations
E1  N1  a  1
n
E2
N2
I1  N2  1  n
a
I2
N1
Transformer modelling - impedance relations
secondary winding impedance reflected to primary
2
Z 2  N 1 Z 2  a 2 Z 2
N2
primary winding impedance reflected to secondary
2
Z 1  N 2 Z 1  n 2 Z 1
N1
equivalent impedance seen at primary
Z in  Z 1  a 2 Z 2  a 2 Z load
Impedance Diagram Representation
2
V2
V 22,B
2 V 1,B
and
Z

n
Z 1,B  S1,B
2,B  S
S
B
B
B
Efficiency
  P out  100%  1  P losses  100%
P in
P in
Voltage Regulation
V 1,serving rated load at rated voltage
 1  100%
VR 
V 1,rated
LINE PARAMETERS
Line inductance per phase and unit length


L  o ln GMD
2
GMR b
if transposed
GMD  3 D ab D bc D ac
if bundled
GMR b 
b
r  d 12 d 13  d 1b
r   re  r /4
o
 2  10 7 henrys/meter
2
Line capacitance per phase and unit length
o
C  2
GMD
ln GMR
c
b
GMR cb 
b
rd 12 d 13  d 1b
 o  8. 854  10 12
farads/meter
Series impedance per phase and unit length

z  jL  R
Shunt admittance per phase and unit length
y  jC
CIRCUIT MODELS
Admittance parameters regulating (off-nominal) transformer
I1
I2
Y eq /a
V1
Y eq /a  Y eq /|a| 2
V2
Y eq

Line Characteristics
Zc  z
y

yz
Long transmission line
sinh l
Z   zl
 Z c sinh l
l
Y   yl
tanh
l
2
l
2
2
tanh
Zc
l
2
Medium line
Z  zl and Y  yl
Short line
Z  zl and Y  0
POWER CAPABILITY OF LINES
Power flow on lossless line
for medium line and short line B  jX so
P  V 1 V 2 sin 
X
P max  V 1 V 2
X
POWER FLOW
Admittance Matrix
Y bus  G  jB  Y ij   Y ij  ij
Y ii 
 y ij  Sum of admittances connected to bus i
Y ij  y ij  Admittance between bus i and j
Power flow equations
n
S i  S Gi  S Di  V i  V j Y ij  i   j   ij 
j1
or
n
S i  V i  Y ij V j


j1
or
n
P i  P Gi  P Di  V i  V j G ij cos i   j   B ij sin i   j   f i V, 
j1
n
Q i  Q Gi  Q Di  V i  V j G ij sin i   j   B ij cos i   j   g i V, 
j1