Extended Summary pp.

254–262

An Improvement of a Conductor Subdivision Method for Calculating

the Series Impedance Matrix of a Transmission Line
Considering the Skin and Proximity Effects
Toru Miki Member (CRIEPI, torumiki@ieee.org)
Taku Noda Member (CRIEPI, takunoda@ieee.org)

Keywords: line series impedance, arbitrary cross-section conductors, conductor-subdivision-based method, skin effect, proximity
effect, overhead transmission lines and underground cables

1. Abstract Figure 3 illustrates two cylindrical conductors in a rectangular

This paper describes an improvement of a conductor- subdivi- conductor duct. Such conductor arrangement cannot be dealt with
sion-based method for calculating the frequency-dependent series by existing analytical formulas. It is assumed that the currents
impedance matrix of arbitrary cross-section conductors consider- through the inner conductors return back through the duct. The
ing both skin and proximity effects. Analytical skin-effect for- calculated result of the resistance and inductance parts is shown in
mulas are available, but they are only for some simple conductor Fig. 4. Due to the proximity effect, the impedance of the conductor
shapes and the proximity effect is ignored. Conductor-subdivision- #1 is smaller than that of #2. This is because the conductor #1 is
based approaches can take into account both skin and proximity ef- much closer to one of the side walls. Figure 5 shows the calculated
fects and are applicable to arbitrary cross-section conductors. How- current distribution at 5 Hz when a unit sinusoidal voltage is applied
ever, at high frequencies where skin depth becomes fairly small, to the bonded inner conductors with respect to the duct. This shows
the subdivision-based approaches require an unrealistic number of the proposed improvement accurately reproduces the skin and prox-
subdivision. The proposed improvement assumes a current distri- imity effects.
bution obtained by an analytical skin-effect formula at surface sub-
conductors for accurately representing the skin effect at high fre-
quencies. Numerical examples show that the proposed improve-
ment accurately reproduces the skin and proximity effects with a
small number of subdivision even at high frequencies.
2. Main Results
Figure 1 shows the arrangement of two parallel plate conductors.
It is assumed that the current through the upper conductor returns
back through the lower one. In Fig. 2, the calculated results ob-
tained by the conductor-subdivision-based method with and without
the proposed current-distribution correction are shown. A reference Fig. 3. Conductor arrangement of two cylindrical con-
result, obtained using an approximate theoretical formula which ne- ductors in a rectangular conductor (Example B)
glects the edge effect, is also superimposed. The result with the
proposed correction shows a good representation of the skin effects
at high frequencies despite the coarse subdivision.

Fig. 4. Calculated result of the line series impedance for

Example B
Fig. 1. Parallel plate conductors (Example A)

Fig. 2. Calculated result of the resistance part of the line Fig. 5. Calculated result of a current distribution at 5 Hz
series impedance for Example A for Example B

– 30 –
 ∗ ∗

An Improvement of a Conductor Subdivision Method for Calculating the Series Impedance

Matrix of a Transmission Line Considering the Skin and Proximity Effects
Toru Miki∗ , Member, Taku Noda∗ , Member

This paper describes an improvement of a conductor-subdivision-based method for calculating the frequency-
dependent series impedance matrix of arbitrary cross-section conductors considering both skin and proximity effects.
Analytical skin-effect formulas are available, but they are only for some simple conductor shapes and the proximity ef-
fect is ignored. Conductor-subdivision-based approaches can take into account both skin and proximity effects and are
applicable to arbitrary cross-section conductors. However, at high frequencies where skin depth becomes very small,
the subdivision-based approaches require an unrealistic number of subdivision. The proposed improvement assumes a
current distribution obtained by an analytical skin-effect formula at surface subconductors for accurately representing
the skin effect at high frequencies. Numerical examples show that the proposed improvement accurately reproduces
the skin and proximity effects with a small number of subdivision even at high frequencies.

Keywords: line series impedance, arbitrary cross-section conductors, conductor-subdivision-based method, skin effect, proximity
effect, overhead transmission lines and underground cables

1.

(4) (5)
Maxwell

Carson (1)
Pollaczek (2)

Schelkunoff (3)

(6) (10)

240-0196 2-6-1
Electric Power Engineering Reseach Lab., CRIEPI
2-6-1, Nagasaka, Yokosuka 240-0196


c 2008 The Institute of Electrical Engineers of Japan. 254
 Lii i j Li j
(8)

µ D2ik
Lii = ln · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)
2π Dii Dkk
µ Dik D jk
3 Li j = ln · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (3)
2π Dkk Di j
Dii i Di j
i j µ
Dii
ln 2 + π 25
ln Dii = ln a + − · · · · · · · · · · · · · · · · · · · · (4)
3 12
Di j
(11)
i j di j
di j = a
2.
π 25 7
ln Di j = ln 2a − − + ln 5
3 12 24
2 8
+ arctan 2 + arccot 2· · · · · · · · · · · · · · · · (5)
3 3

2 1 Di j  di j · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (6)
n

a
1 3
3 i j k 2 2
k (1) (3)
i j
k V1 V2 . . . Vn
I1 I2 . . . In
Ri ρ a2 −∆Vn

⎡ ⎤ ⎡ ⎤⎡ ⎤
ρ ⎢⎢⎢∆V1 ⎥⎥⎥ ⎢⎢⎢Z11 Z12 . . . Z1n ⎥⎥⎥ ⎢⎢⎢I1 ⎥⎥⎥
Ri = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (1) ⎢⎢⎢ ⎥ ⎢ ⎥⎥ ⎢⎢ ⎥⎥
a2 ⎢⎢⎢∆V2 ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢Z21 Z22 . . . Z2n ⎥⎥⎥⎥ ⎢⎢⎢⎢I2 ⎥⎥⎥⎥
− ⎢⎢⎢⎢ . ⎥⎥⎥⎥ = ⎢⎢⎢⎢ . .. ..
⎥⎢ ⎥
.. ⎥⎥⎥⎥ ⎢⎢⎢⎢ .. ⎥⎥⎥⎥ · · · · · · · · · · (7)
i ⎢⎢⎢ .. ⎥⎥⎥ ⎢⎢⎢ .. . . . ⎥⎥⎥ ⎢⎢⎢ . ⎥⎥⎥
⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦
∆Vn Zn1 Zn2 . . . Znn In

V =
[V1 V2 . . . Vn ] T
I = [I1 I2 . . . In ] T

Z (7)

−∆V = ZI · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (8)

(7) (8)

Z
k V I
1 k
Fig. 1. Subconductor arrangement. n n = n − 1

B 128 1 2008 255

 3 a b c g Yreduced
3 Zreduced

(7) (8) Z −1
Zreduced = Yreduced · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)
V I a b c g
a b c g
(7) ⎡ ⎤ ⎡ ⎤⎡ ⎤
⎢⎢⎢∆Va ⎥⎥⎥ ⎢⎢⎢Zaa Zab Zac Zag ⎥⎥⎥ ⎢⎢⎢Ia ⎥⎥⎥
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥
⎢⎢⎢[∆Va ]⎥⎥⎥ ⎢⎢⎢[Zaa ] [Zab ] [Zac ] [Zag ]⎥⎥⎥ ⎢⎢⎢[Ia ]⎥⎥⎥ ⎢∆Vb ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢Zba Zbb Zbc Zbg ⎥⎥⎥⎥ ⎢⎢⎢⎢Ib ⎥⎥⎥⎥
⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎢ ⎥ − ⎢⎢⎢⎢ = ⎥⎥ ⎢⎢ ⎥⎥ · · · · · · · · (15)
⎢⎢⎢[∆Vb ]⎥⎥⎥ ⎢⎢⎢[Zba ] [Zbb ] [Zbc ] [Zbg ]⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢[Ib ]⎥⎥⎥⎥⎥ ⎢⎢⎢∆Vc ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢Zca Zcb Zcc Zcg ⎥⎥⎥⎥ ⎢⎢⎢⎢ Ic ⎥⎥⎥⎥
− ⎢⎢ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎥⎦ ⎢⎣ ⎥⎦
⎢⎢⎢[∆Vc ]⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢[Zca ] [Zcb ] [Zcc ] [Zcg ]⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢[Ic ]⎥⎥⎥⎥⎥ ∆Vg Zga Zgb Zgc Zgg Ig
⎣⎢ ⎦⎥ ⎣⎢ ⎥⎦ ⎢⎣ ⎥⎦
[∆Vg ] [Zga ] [Zgb ] [Zgc ] [Zgg ] [Ig ]
· · · · · · · · · · · · · · · · · · · · (9)
Zreduced
a b c
a b c g
Ia + Ib + Ic + Ig = 0 · · · · · · · · · · · · · · · · · · · · · · · · · (16)
Z sorted
(15)
−1 Zreduced 3×3
Y sorted = Z sorted · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (10)
Zreduced ij
(9)
⎡ ⎤ ⎡ ⎤⎡ ⎤ Zi j + Zgg − Zig − Zg j · · · · · · · · · · · · · · · · · · · · · · · · · (17)
⎢⎢⎢[Ia ]⎥⎥⎥ ⎢⎢⎢[Yaa ] [Yab ] [Yac ] [Yag ]⎥⎥⎥ ⎢⎢⎢[∆Va ]⎥⎥⎥
⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥ j=a b c
⎢⎢⎢[Ib ]⎥⎥⎥ ⎢⎢[Yba ] [Ybb ] [Ybc ] [Ybg ]⎥⎥⎥ ⎢⎢⎢[∆Vb ]⎥⎥⎥⎥ i
⎢⎢⎢⎢ ⎥⎥⎥⎥ = − ⎢⎢⎢⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥
⎢⎢⎢[Ic ]⎥⎥⎥ ⎢⎢⎢[Yca ] [Ycb ] [Ycc ] [Ycg ]⎥⎥⎥⎥ ⎢⎢⎢⎢[∆Vc ]⎥⎥⎥⎥ N
⎣ ⎦ ⎣ ⎥⎦ ⎢⎣ ⎥⎦
[Ig ] [Yga ] [Ygb ] [Ygc ] [Ygg ] [∆Vg ]
· · · · · · · · · · · · · · · · · · (11) 3.

a
[∆Va ]
[∆Va ] ∆Va 3 1
[∆Vb ] [∆Vc ] [∆Vg ] a
a 2
[Ia ]
Ia y
[Ib ] [Ic ] [Ig ] n × 1
V 4×1
 T
∆Vreduced = ∆Va ∆Vb ∆Vc ∆Vg · · · · · · · · · (12)

n × 1 I 4×1
 T
Ireduced = Ia Ib Ic Ig · · · · · · · · · · · · · · · · · · · · (13)

n × n
Y sorted 4×4

Fig. 2. Skin effect at surface subconductors.

256 IEEJ Trans. PE, Vol.128, No.1, 2008

 Hy = H0 e jωt K = S Rdc /l2 (22)
Maxwell 
µρ
(12) jω · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (25)
a2
jωµ
− x
By = µH0 e ρ
e jωt · · · · · · · · · · · · · · · · · · · · · · · · · (18) (24) (25) K
ρ
Jz z K= · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (26)
a2
(23)
jωµ
− x 1 µρ
Jz = −J0 e ρ
e jωt · · · · · · · · · · · · · · · · · · · · · · · · · · (19) Rdc + jω 2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (27)
2Rdc a
J0
jωµ/(8π) Rdc
∆V
Rdc
∆V
J0 = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (20) 4πρ
ρ Rdc = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (28)
a2
2

µ/(8π) (27)
(22)
(20)
a a jωµ
− x
I skin = J0 e ρ
dxdy ⎧  ⎫
⎪
⎪ ⎪
⎪ 4πρ µa2 ⎪⎪
0 0
1 ⎨ ⎬
a L skin  Im ⎪
⎪ 1 + jω ⎪ · · · · · · · · · · · (29)
16π ρ ⎪
jωµ
−
= ∆V √ 1−e ρ a
· · · · · · · · · · · · · · · (21) ω ⎪
⎩ a 2 2 ⎪
⎭
jωµρ
(1) (22)
Ri (2)

∆V
R skin = Re D2ik
I skin µ µ
⎧ ⎫ Lii = ln − + L skin · · · · · · · · · · · · · · · · (30)
⎪
⎪
⎪ ⎪
⎪
⎪
2π Dii Dkk 8π
⎪
⎪ √ ⎪
⎪
⎪
⎨ jωµρ ⎪
⎬ L skin (29)
= Re ⎪
⎪ ⎪
⎪ · · · · · · · · · · · · · · · · · (22)
⎪
⎪
⎪ − jωµ ⎪
⎪
⎪
⎪
⎩a 1 − e ρ a ⎪
⎭
Ri
Lii

(1) (3) (22) (29) (30)

(13)

 ACSR
jωµS
Zapprox  Rdc 1+
Rdc l2
 3 2
µS Rdc
= R2dc + jω 2 · · · · · · · · · · · · · · · · · · · (23)
l
S l
Rdc = ρ/S (23)
(22)
(23) µ/(8π)
(23) 3

(23)

µS Rdc 
jω 2

jωµK · · · · · · · · · · · · · · · · · · · · · · · · (24)
l

B 128 1 2008 257

 2 2

3 1
3

4.

3
a
ρ = 2.5 × 10−8 Ωm

4 1 1 4(a)
a 1 2
1 1 cm 300
d a × d
(21) (22) (29)
⎧ ⎫ ⎫ 0.1 Hz 10 MHz
⎪
⎪
⎪ ⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪ √ ⎪
⎪ ⎪
⎪
⎪
⎨ jωµρ ⎪
⎬ ⎪
⎪
⎪ 50
Rskin = Re ⎪
⎪ ⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎪ − jωµ ⎪
⎪
⎪ ⎪
⎬ 5 “with correction”
⎪ 
⎩a 1 − e ρ d
⎪
⎭ ⎪
⎪ · · · · · · · · · (31)
⎪
⎪
⎪ “without correc-
 ⎪
⎪
1 4πρ µa d ⎪
⎪
⎪ tion”
Lskin  Im  1 + jω ⎪
⎪
ω ad 16π2 ρ ⎭ “without correction”
(1) (3)
“with correction” (1) (3)
(30) (31)
4(b)
2
(3)
Schelkunoff

1 4 Schelkunoff
4 TLCP Transmission Line Constants Program
(14) (15)
Ver. 2 4(c)
1 3

3 (b) Two hollow conductors (c) Part of hollow conductors

Fig. 3. Conductor surface length assigned to a surface 4 1

subconductor and its length in the skin-depth direction. Fig. 4. Conductor arrangement of Example 1.

258 IEEJ Trans. PE, Vol.128, No.1, 2008

 “ref.
result tlcp ” 5

Schelkunoff 1 1 cm 300

0.1 Hz 10 MHz
50 7(a)
(b)
4(a) 1 “with
correction”

Schelkunoff

300

4 2 2
6(a)

(a) Resistance

5
1 (b) Inductance
Fig. 5. Calculated result of the resistance part of the line
series impedance (Example 1).

(c) Current distribution (at 5 kHz)

(a) Eccentric and rectangular (b) Coaxial conductors
7
conductors
2
6 2 Fig. 7. Calculated result of the line series impedance
Fig. 6. Conductor arrangement of Example 2. and a current distribution (Example 2).

B 128 1 2008 259

 “without correction” 0.1 Hz 10 MHz
7(a) 50 9(a) (b)

2
2×2
1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢⎢⎢Z11 Z12 ⎥⎥⎥⎥ ⎢⎢⎢⎢R11 R12 ⎥⎥⎥⎥ ⎢⎢L11 L12 ⎥⎥⎥⎥
⎢⎢⎣ ⎥=⎢ ⎥⎦ + jω ⎢⎢⎢⎣ ⎥ · · · · · · (33)
1 Z12 Z22 ⎦ ⎣R12 R22 L12 L22 ⎦
R11 R12 R22 L11
L12 L22
6(b) 6(a)
1 2
6(b)
Z11 Z22

µ0 0.10
L= ln = 0.139 µH.· · · · · · · · · · · · · · · · · · · (32)
2π 0.05

7(b)
(32)

7(c) 5 kHz 3
1m
1V
(a) Resistance

7(c)

4 3 3 2

8
2
(b) Inductance
1
0.8 cm 3626

(c) Current distribution (at 5 Hz)

9
3
8 3 Fig. 9. Calculated result of the line series impedance
Fig. 8. Conductor arrangement of Example 3. and a current distribution (Example 3).

260 IEEJ Trans. PE, Vol.128, No.1, 2008

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