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Class 8
Carsons Equations
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Work Exam 1
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General Flux Linkages
D in
2
D i2
D i1
O1
On
O2
Oi
On
O2
O1
1
i
n
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General Flux Linkage
Equation
1 2 0i nI I I I
71 2
1 2
1 1 1 12 10 ( ln ln ln ln )i i n
i i i in
I I I ID D GMR D
W-T/m
where: Din = Distance between conductor i and conductor n(ft.)
GMRi = Geometric Mean Distance of conductor i (ft.)
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General Self and Mutual
Inductance (H/m)
7
7
1
2 10 ln
12 10 ln
ii
iii i
in
in n in
L I GMR
LI D
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Transposed Three-Phase Lines
1. Assumptions Transposition Balanced three-phase currents
Series Impedance: 0.12134 lneq
i ii
Dz r j
GMR /mile
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Self and Mutual Impedance
10.12134 ln
10.12134 ln
ii ii
ij
ij
z r j
GMR
z j
D
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Two Conductors with
Dirt Return
I i
I d
Ij
+
-
Vig +
-
Vjgground
z ii
zjj
z
ij
z idzjdz dd
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Primitive Circuit and
Impedances
I i
Ij
+
-
Vig +
-
Vjgground
z ii
z
jj
z ij
V'ig
+
+V'jg
--
10.12134 ln ln
10.12134 ln ln
id diii d i
i d
di jd ij d
ij d
D Dz r r j
GMR GMR
D Dz r j
D GMR
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Carsons Full Equations
S nnSnm
D nm m
m'
n
n'
-0nm
4 2 ln 4
4 2 ln 4
iiii i ii i ii
i
ijij ij ij
ij
Sz r P G j X G Q G
RD
Sz P G j G Q G
D
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= 2f = system angular frequency in radians per second
G = 0.1609347 x 10-3
/mile
RDi = radius of conductor i in feet
GMRi = Geometric mean radius of conductor i in feet
f = system frequency in Hertz
= resistivity of earth in -meters
Dij = distance between conductors i and j in ft. (see Figure4.4)
Sij = distance between conductor i and image j in ft. (see Figure 4.4)
ij = angle between a pair of lines drawn from conductor i to its own
image and to the image of conductor
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2 ln
i
i i
RD
X G GMR /mile
2
1 2cos cos 2 0.6728 ln
8 163 2
ijij ij ij ij
ij
kP k
k
1 2 1
0 03862 3 2
ij ij ijij
Q . ln k cosk
48.565 10ij ijfk S
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Assumptions
2 ln iii
RDX G
GMR /mile
8ijP
1 20 0386
2ij
ij
Q . lnk
48.565 10ij ijf
k S
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Modified Carsons Equations
(after lots of algebra)Assuming:
f = frequency = 60 Hertz
= earth resistivity = 100 Ohm-meter
10.09530 0.12134 ln 7.93402ii i
i
z r jGMR
/mile
10.09530 0.12134 ln 7.93402ij
ij
z jD
/mile
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Comparing Equations
10.12134 ln ln
10.12134 ln ln
1
0.09530 0.12134 ln 7.93402
10.09530 0.12134 ln 7.93402
id djii i d
i d
di jd ij d
ij d
ii ii
ijij
D Dz r r j
GMR GMR
D Dz r j
D GMR
z r j GMR
z jD
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Four-Wire Grounded Wye
V'
bg
V'ag
V'cg
Vag
V
bg
Vcg
+
+
+
+
+
+
Vng
+
-- - -
V'ng
+
- - - -
zaa
zbb
zcc
znn
zab
zbn zanzbc
zcn
zacIa
Ib
In
Ic
'
'
'
'
ag ag aa ab ac an a
bg bg ba bb bc bn b
ccg cg ca cb cc cn
n
ng ng na nb nc nn
V V z z z z I
V V z z z z I
IV V z z z z
IV Vz z z z
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Kron Reduction
1
0 0
abc anabcg abcg abcn m
nna nn
abcg abcg abc an nn na abcn m
z zV V I
Iz z
V V z z z z I
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General Three-Phase Line
Zca
Zbc
Zab
Zaa
Zbb
Zcc
Ia
Ib
Ic
+
+
---
+
Vbg
Vcg
Vag
+
+
+
- - -Vcg
Vbg
Vag
Node n Node m
n
n
n
m
m
m
ag ag aa ab ac a
bg bg ba bb bc b
ca cb cc ccg cg n m
V V Z Z Z I
V V Z Z Z I
Z Z Z IV V
ij ijZ z length
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Work Carsons Example
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Homework #5
4.1
4.2
4.3
