Now considering a completely transposed line with bundeled conductors:
L
1
L
2
=
2 10
7
−
⋅
ln
D
GMR
⋅
=
L
a
λ
a
I
a
=
2 10
7
−
⋅
ln
D
GMR
⋅
=
λ
a
2 10
7
−
⋅
I
a
⋅
ln
D
GMR
⋅
=
λ
a
2 10
7
−
⋅
I
a
ln
1
GMR
⋅
I
a
ln
1
D
⋅
−
⋅
=
Using the Fact, that all currents add to zero:
I
a
−
I
b
I
c
+
(
)
=
λ
a
2 10
7
−
⋅
I
a
ln
1
GMR
⋅
I
b
I
c
+
(
)
ln
1
D
⋅
+
⋅
=
λ
a
2 10
7
−
⋅
I
a
ln
1
GMR
⋅
I
b
ln
1
D
⋅
+
I
c
ln
1
D
⋅
+
⋅
=
2 10
7
−
⋅
ln
1
GMR
ln
1
D
ln
1
D
⋅
I
a
I
b
I
c
⋅
=
Assume positive sequence currents whose sum is zero:
ThreePhase, ThreeWire Line with equal Phase Spacing D:
Symmetrical Arrangement of
Conductors without Ground Return:
λ
k
2 10
7
−
⋅
1
M
m
I
m
ln
1
D
km
⋅
∑
=
⋅
=
Total Flux linking conductor
k
in an array of
M
conductors
carrying currents Im, whose sum is zero.
Starting from Equation (4.4.30) on Page 180 Glover, Sarma 4
th
edition:
Dr. M. Giesselmann, Mar032008:
Series Inductance of Transmission Lines:
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L
a
λ
a
I
a
=
2 10
7
−
⋅
ln
D
GMR
⋅
=
L
a
L
1
=
L
2
=
λ
a
I
a
=
2 10
7
−
⋅
ln
D
eq
GMR
Bundle
⋅
=
D
eq
D
12
D
23
⋅
D
31
⋅
(
)
1
3
=
λ
a
2 10
7
−
⋅
I
a
⋅
ln
D
12
D
23
⋅
D
31
⋅
(
)
1
3
GMR
Bundle
⋅
=
λ
a
2 10
7
−
⋅
3
3 I
a
⋅
ln
1
GMR
Bundle
⋅
I
a
ln
1
D
12
D
23
⋅
D
31
⋅
⋅
−
⋅
=
Using Boundary condition
for positive Sequence:
I
a
−
I
b
I
c
+
(
)
=
λ
a
2 10
7
−
⋅
3
3 I
a
⋅
ln
1
GMR
Bundle
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 Spring '08
 GIESSELMANN
 Flux, Volt, Threephase, Mikhail DolivoDobrovolsky, Symmetrical components

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