Formula Sheets: Transmission Lines & Line Parameters | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


Power S ystems F ormula Sheet:
Tr ansmission Lines & Line Par ameters
1. Tr ansmission Line Par ameters
• Resistance (R ):
R =
?l
A
?/ km
where? is resistivity ,l is length,A is cross-sectional area.
• Inductance (L ): F or a single conductor:
L =
µ
0
2p
ln
(
D
r
)
H/m
whereµ
0
= 4p×10
-7
H/m,D is geometric mean distance (GMD),r is conductor
r adius.
• Capacitance (C ): F or a single-phase line:
C =
2p?
0
ln
(
D
r
) F/m
where?
0
= 8.854×10
-12
F/m.
• Conductance (G ): Usually negligible for overhead lines, accounts for leakage
current:
G =
2ps
ln
(
D
r
) S/m
wheres is conductivity of the medium.
2. Geometric Mean Radius (GMR) and Distance (GMD)
• Geometric Mean Radius (GMR) : F or a single conductor:
GMR =re
-1/4
˜ 0.7788r
F or bundled conductors (e.g.,n sub-conductors):
GMR
bundle
=
(
GMR
conductor
·d
n-1
)
1/n
whered is spacing between sub-conductors.
1
Page 2


Power S ystems F ormula Sheet:
Tr ansmission Lines & Line Par ameters
1. Tr ansmission Line Par ameters
• Resistance (R ):
R =
?l
A
?/ km
where? is resistivity ,l is length,A is cross-sectional area.
• Inductance (L ): F or a single conductor:
L =
µ
0
2p
ln
(
D
r
)
H/m
whereµ
0
= 4p×10
-7
H/m,D is geometric mean distance (GMD),r is conductor
r adius.
• Capacitance (C ): F or a single-phase line:
C =
2p?
0
ln
(
D
r
) F/m
where?
0
= 8.854×10
-12
F/m.
• Conductance (G ): Usually negligible for overhead lines, accounts for leakage
current:
G =
2ps
ln
(
D
r
) S/m
wheres is conductivity of the medium.
2. Geometric Mean Radius (GMR) and Distance (GMD)
• Geometric Mean Radius (GMR) : F or a single conductor:
GMR =re
-1/4
˜ 0.7788r
F or bundled conductors (e.g.,n sub-conductors):
GMR
bundle
=
(
GMR
conductor
·d
n-1
)
1/n
whered is spacing between sub-conductors.
1
• Geometric Mean Distance (GMD) : F or a three-phase line:
GMD = (D
ab
·D
bc
·D
ca
)
1/3
whereD
ab
,D
bc
,D
ca
are distances between phase conductors.
3. Tr ansmission Line Models
• Short Line (< 80 km):
Z =R+jX
L
, Y ˜ 0
V
s
=V
r
+IZ, I
s
=I
r
whereV
s
,I
s
are sending-end voltage and current,V
r
,I
r
are receiving-end volt-
age and current.
• Medium Line (80-250 km, Nomin al-p model):
Z =R+jX
L
, Y =j?C
V
s
=V
r
+I
r
(
Z +
YZV
r
2
)
, I
s
=I
r
+
YV
r
2
+
YV
s
2
• Long Line (> 250 km):
? =
v
zy, Z
c
=
v
z
y
where ? is propagation constant, Z
c
is char acteristic impedance, z = R +j?L
(series impedance per unit length), y = G + j?C (shunt admittance per unit
length).
V
s
=V
r
cosh(?l)+I
r
Z
c
sinh(?l)
I
s
=I
r
cosh(?l)+
V
r
Z
c
sinh(?l)
4. ABCD Par ameters
• Gener al F orm :
[
V
s
I
s
]
=
[
A B
C D
][
V
r
I
r
]
• Short Line :
A = 1, B =Z, C = 0, D = 1
• Medium Line (Nominal-p ) :
A = 1+
YZ
2
, B =Z, C =Y
(
1+
YZ
4
)
, D = 1+
YZ
2
• Long Line :
A = cosh(?l), B =Z
c
sinh(?l), C =
sinh(?l)
Z
c
, D = cosh(?l)
2
Page 3


Power S ystems F ormula Sheet:
Tr ansmission Lines & Line Par ameters
1. Tr ansmission Line Par ameters
• Resistance (R ):
R =
?l
A
?/ km
where? is resistivity ,l is length,A is cross-sectional area.
• Inductance (L ): F or a single conductor:
L =
µ
0
2p
ln
(
D
r
)
H/m
whereµ
0
= 4p×10
-7
H/m,D is geometric mean distance (GMD),r is conductor
r adius.
• Capacitance (C ): F or a single-phase line:
C =
2p?
0
ln
(
D
r
) F/m
where?
0
= 8.854×10
-12
F/m.
• Conductance (G ): Usually negligible for overhead lines, accounts for leakage
current:
G =
2ps
ln
(
D
r
) S/m
wheres is conductivity of the medium.
2. Geometric Mean Radius (GMR) and Distance (GMD)
• Geometric Mean Radius (GMR) : F or a single conductor:
GMR =re
-1/4
˜ 0.7788r
F or bundled conductors (e.g.,n sub-conductors):
GMR
bundle
=
(
GMR
conductor
·d
n-1
)
1/n
whered is spacing between sub-conductors.
1
• Geometric Mean Distance (GMD) : F or a three-phase line:
GMD = (D
ab
·D
bc
·D
ca
)
1/3
whereD
ab
,D
bc
,D
ca
are distances between phase conductors.
3. Tr ansmission Line Models
• Short Line (< 80 km):
Z =R+jX
L
, Y ˜ 0
V
s
=V
r
+IZ, I
s
=I
r
whereV
s
,I
s
are sending-end voltage and current,V
r
,I
r
are receiving-end volt-
age and current.
• Medium Line (80-250 km, Nomin al-p model):
Z =R+jX
L
, Y =j?C
V
s
=V
r
+I
r
(
Z +
YZV
r
2
)
, I
s
=I
r
+
YV
r
2
+
YV
s
2
• Long Line (> 250 km):
? =
v
zy, Z
c
=
v
z
y
where ? is propagation constant, Z
c
is char acteristic impedance, z = R +j?L
(series impedance per unit length), y = G + j?C (shunt admittance per unit
length).
V
s
=V
r
cosh(?l)+I
r
Z
c
sinh(?l)
I
s
=I
r
cosh(?l)+
V
r
Z
c
sinh(?l)
4. ABCD Par ameters
• Gener al F orm :
[
V
s
I
s
]
=
[
A B
C D
][
V
r
I
r
]
• Short Line :
A = 1, B =Z, C = 0, D = 1
• Medium Line (Nominal-p ) :
A = 1+
YZ
2
, B =Z, C =Y
(
1+
YZ
4
)
, D = 1+
YZ
2
• Long Line :
A = cosh(?l), B =Z
c
sinh(?l), C =
sinh(?l)
Z
c
, D = cosh(?l)
2
5. Performance Par ameters
6. Surge Impedance and Loading
• Surge Impedance :
Z
c
=
v
L
C
?
• Surge Impedance Loading (SIL) :
SIL =
V
2
LL
Z
c
MW
whereV
LL
is line-to-line voltage (in kV , converted to V for calculation).
3