Lecture EE333 - lecture 11 - Clarkson University › ~lwu › ee333 › Lectures... · Lecture 11 Transmission Line Parameters Reading: 4.1 – 4.6 ; 4.8 – 4.10 Homework 3 is due
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Lecture 11
Transmission Line Parameters
Reading: 4.1 – 4.6 ; 4.8 – 4.10
Homework 3 is due on Feb. 20th.
Dr. Lei Wu
Department of Electrical and Computer Engineering
EE 333
POWER SYSTEMS ENGINEERING
Outline
� Develop simple model for transmission lines
� Line resistance (R)
� Line conductance (G)
� Line inductance (L)
� Line capacitance (C)
� Analyze how the geometry of the transmission lines will affect the model parameters
2
R L
CG
,T
dc T
lR
A
ρ=
i0 ln
2
GMDL
GMR
µπ
=
( )( ) ( )2
11 12 1 21 22 2 1 2n
n n n n nnGMR D D D D D D D D D= ⋯ ⋯ ⋯ ⋯
( )( ) ( )11' 12 ' 1 21' 22 ' 2 1' 2 'nm
xy m m n n nmGMD D D D D D D D D D= ⋯ ⋯ ⋯ ⋯
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing in a
bundle is 45cm. Find La.
3
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find La.
� GMR of one stranded conductor
( ) ( )67*711 12 13 14 15 16 17 71 72 73 74 75 76 77
2.1767 3.9181
sGMR D D D D D D D D D D D D D D
r cm
=
= =
12 16 17 2D D D r= = =13 15 2 3D D r= = 14 4D r=
71 72 73 74 75 76 2D D D D D D r= = = = = =
11 77 ' 0.7788D D r r= = =
4
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find La.
� GMR of one bundle
� GMR of each phase
44
34
2
1.091 26.6567
b s
s
GMR GMR GMR dd d
GMR d cm
= =
= =
( )2 21 1,1' 0.266567 * 18 20 2.678bGMR GMR D m= = + =
2 2,2 ' 0.266567 * 24 2.5293bGMR GMR D m= = =
( )2 23 3,3 ' 0.266567 * 18 20 2.678bGMR GMR D m= = + =
5
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find La.
� GMR of the three-phase double circuit
31 2 3 2.6275GMR GMR GMR GMR m= =
6
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find La.
� GMD of the three-phase double circuit
312 23 13 16.5572GMD GMD GMD GMD= =
412 12 12 ' 1'2 1'2 '
4 2 2 2 2 2 2 2 210 4 10 20 10 22 10 2 15.4718
GMD D D D D
m
=
= + + + + =
423 23 23 ' 2 '3 2 '3 ' 15.4718GMD D D D D m= =
413 13 13 ' 1'3 1'3 '
4 2 2 2 220 2 16 20 2 20 18.9619
GMD D D D D
m
=
= + + =7
Line inductance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find La.
70 ln 3.6814 *10 /2a
GMDL H m
GMR
µπ
−= =
2X fLπ=
8
Line capacitance
� Transmission line conductors exhibit capacitance with respect
to each other due to the potential difference between them.
� The relationship between charge q and potential V is
represented by the capacitance
212
0 1
-120
ln2
where ε =8.85*10 Farad/m,
permittivity of free space
DqV
Dπε=
V
qC =
9
Line capacitance
� Transmission line conductors exhibit capacitance with respect
to each other due to the potential difference between them.
� The relationship between charge q and potential V is
represented by the capacitance
� Assuming that
V
qC =
10
10
ii
1ln
2
where D
nkj
ij kk ki
i
DV q
D
r
πε =
=
=
∑
0n
ii
q =∑
Line capacitance
� For a single-phase two-wire line
11
1 2 0q q+ =
12 2212 1 2
0 11 21
0 0
1ln ln
2
1 1ln ln 2 ln
2 2
D DV q q
D D
D r Dq q q
r D r
πε
πε πε
= +
= − =
012
2
2 lnC
D
r
πε=
Line capacitance
� General formula for calculating capacitance
0
c
2
ln
where GMR is similar to GMR except that r is used instead of r'
c
CGMD
GMR
πε=
12
Line capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing in a
bundle is 45cm. Find Ca and admittance.
13
Line capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find Ca and admittance.
� GMD of the three-phase double circuit
312 23 13 16.5572GMD GMD GMD GMD= =
14
Line capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find Ca and admittance.
� GMR of one stranded conductor
( ) ( )67*711 12 13 14 15 16 17 71 72 73 74 75 76 77
2.2558 4.0605
sGMR D D D D D D D D D D D D D D
r cm
=
= =
12 16 17 2D D D r= = =13 15 2 3D D r= = 14 4D r=
71 72 73 74 75 76 2D D D D D D r= = = = = =
11 77D D r= =
15
Line capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find Ca and admittance.
� GMR of one bundle
� GMR of each phase
44
34
2
1.091 26.9079
b s
s
GMR GMR GMR dd d
GMR d cm
= =
= =14 4D r=
( )2 21 1,1' 0.269079 * 18 20 2.6908bGMR GMR D m= = + =
2 2,2 ' 0.269079 * 24 2.5412bGMR GMR D m= = =
( )2 23 3,3 ' 0.269079 * 18 20 2.6908bGMR GMR D m= = + =
16
Line capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
Find Ca and admittance.
� GMR of the three-phase double circuit
� Capacitance
31 2 3 2.6400GMR GMR GMR GMR m= =
17
11023.03*10 /
lna
c
C F mGMD
GMR
πε −= =
11 82 2 * 60 *3.03*10 1.14 *10 /aY j fC j j S mπ π − −= = =
Line inductance & capacitance
� Example: three-phase double circuit, bundled conductors. Each
conductor is stranded, r=1.8cm and conductor spacing is 45cm.
18
Single circuit Double circuit
Inductance (H/m) 7.75*10-7 3.68*10-7
Capacitance (F/m) 1.44*10-11 3.03*10-11
Additional Transmission Topics
� Multi-circuit lines: Multiple lines often share a common transmission right-of-way. This DOES cause mutual inductance
and capacitance, but is often ignored in system analysis.
� Cables: There are about 3000 miles of underground ac cables in U.S. Cables are primarily used in urban areas. In a cable the
conductors are tightly spaced, (< 1ft) with oil impregnated paper
commonly used to provide insulation
� inductance is lower
� capacitance is higher, limiting cable length
19
Additional Transmission topics
� DC Transmission: Because of the large fixed cost necessary to convert ac to dc and then back to ac, dc transmission is only
practical for several specialized applications
� long distance overhead power transfer (> 400 miles)
� long cable power transfer such as underwater
� providing an asynchronous means of joining different power
systems (such as the Eastern and Western grids).
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