14th PSCC, Sevilla, 24-28 June 2002
Session 35, Paper 1, Page 1
RESONANCE EFFECTS DUE TO CONDUCTOR TRANSPOSITION
IN THREE-PHASE POWER LINES
J. A. Brandão Faria and M. V. Guerreiro das Neves
Centro de Electrotecnia Teórica e Medidas Eléctricas – Instituto Superior Técnico
Lisbon, Portugal
brandao.faria@ieee.org gneves@alfa.ist.utl.pt
Abstract — This paper is concerned with the effects
produced by conductor transposition in long overhead
three-phase power lines. We conduct a frequency domain
analysis that shows for the first time to our knowledge that
the modal attenuation constants and modal phase velocities characterizing the transmission line structure exhibit a
repetitive resonant behavior for frequencies such that the
overall transposition cycle length gets close to an integer
multiple of one half wavelength. Consideration of these
resonance phenomena is of major importance and should
be taken into account in a variety of situations, PLC communications, EMI, and line transient studies.
Keywords: Transposed conductors, multiconductor
transmission lines, modal analysis, nonuniform lines,
modeling and simulation, resonance
1
INTRODUCTION
For balancing purposes many overhead power lines
have their phase conductors periodically transposed
along the line length, a complete transposition cycle
consisting of three cascaded line sections differing on
the conductor sequence order —see scheme in Fig. 1.
The resulting global structure is thus to be classified as a
nonuniform line since its per-unit-length seriesimpedance Z and shunt-admittance Y matrices are not
constant along the line longitudinal coordinate z.
The theme of nonuniform transmission lines is an
important actual research topic with applications ranging from microwave systems [1-2] to power systems [34]. This paper is a new contribution to the latter area.
The frequency-domain multiconductor transmission
line equations for nonuniform lines are
d
d
V = −Z (z ) I ;
I = −Y (z ) V
dz
dz
(1)
where V and I are complex column vectors gathering
the conductor voltages and conductor current phasors.
(A)
1
(B)
3
(C)
2
2
3
1
2
3
1
lB
lC
lA
0
The line structure in Fig. 1 includes three line sections A, B, and C, of length lA, lB, and lC respectively.
Assuming that each individual line section is uniform,
the series-impedance Z(z) and shunt-admittance Y(z) in
(1) are stepwise functions:
for 0 < z < l A
ZA

−1
for l A < z < l A + l B
Z ( z ) =  Z B = PZ A P
 Z = P −1 Z P for l + l < z < l + l + l
A
A
B
A
B
C
 C
(2)
for 0 < z < l A
 YA

Y ( z ) =  YB = PY A P −1 for l A < z < l A + l B
 Y = P −1Y P for l + l < z < l + l + l
A
A
B
A
B
C
 C
where ZK and YK (with K = A, B, C) denote the seriesimpedance and shunt-admittance matrices for the Kth
line section. The permutation matrix operator P, in (2),
defined as
0 1 0 
P = 0 0 1 
(3a)
1 0 0
obeys the following special properties:
P 2 = P −1 = P t ; P 3 = E
(3b)
where E is the identity matrix, and superscript t denotes
transposition.
The standard approach to deal with the line structure
depicted in Fig. 1 employs a homogenization technique
leading to a uniform line model described by balanced
per-unit-length series-impedance <Z> and shunt-admittance <Y> matrices:
Z p

1
< Z > = (Z A + Z B + Z C ) =  Z m
3
Z
 m
Zm
Zp
Zm
Zm 

Zm 
Z p 
(4)
z
L
Figure 1: Transposition cycle for a single-circuit three-phase
line structure.
Y p

< Y > = 1 (Y A + YB + YC ) = Ym
3
Y
 m
Ym
Yp
Ym
Ym 

Ym 
Y p 
14th PSCC, Sevilla, 24-28 June 2002
The preceding matrices being used for the evaluation of
the propagation parameters characterizing the normal
modes of the structure —one ground mode and two degenerate aerial modes.
As is well known, the modal attenuation constants α
and modal phase velocities v obtained with this approach are smoothly varying functions of the frequency,
Ground mode (g):
α g (ω ) + j
ω =
v g (ω )
ω =
va (ω )
d
h
(a)
60º
d
h
h
(b)
(c)
(5a)
Figure 2: Line configurations showing conductor arrangement
and dimensions, h = 25 m, d = 10 m, conductor radii r=16 mm.
(a) horizontal line, (b) vertical line, (c) triangular line.
( Z p − Z m )(Y p − Ym )
(5b)
2.1 Analysis of one Line Section
The entries of the complex symmetric matrices Z A
and Y A depend on the frequency and on the geometrical arrangement of the line conductors. Because the
commonly used procedure for computing the per-unitlength series-impedance and shunt-admittance matrices
has been established elsewhere [6-7] details will not be
gone into here and, thus, only a brief summary is presented. For section A, Z A and Y A are obtained via
A first flaw with the standard approach can be immediately pointed out by noting that equivalent uniform
overhead lines with Z = <Z> and Y = <Y> can never be
found [5]. Even, in the simplest case of losseless lines,
the product <Z><Y> does not yield a diagonal matrix,
modal phase velocities obtained from (5) being not
equal to the speed of light in a vacuum, v ≠ c.
The main problem with the standard approach is that
it is not based on a correct model, as the real structure is
not uniform along its entire length due to the presence
of transposition interconnects.
The discontinuities associated with conductor transpositions generate forward and backward propagating
waves that strongly interfere at certain critical frequencies, thus making the wave propagation parameters to
vary not smoothly with the frequency.
In this paper we will take into account the nonuniformities originated by the presence of conductor transpositions. A typical single-circuit three-phase overhead
line configurations with frequency dependent parameters, consisting of three uniform line sections 100 km
long each, will be used as an example.
By cascading the line sections, the overall 6 × 6 transmission matrix characterizing the structure will be
determined and from it the corresponding propagation
constants will be extracted. The modal attenuation constants and modal phase velocities will then be obtained
and plotted against the frequency, plots showing that
resonance manifestations repeatedly occur for frequencies that are a multiple of 500 Hz.
The magnitude and width of the resonance peaks are
influenced by several factors. In this paper we will discuss the influence of line losses (wire losses and ground
losses) as well as the influence of some line geometrical
parameters, e.g., section lengths and conductor arrangement (cross-sectional views of typical line configurations showing conductor distances and heights are
depicted in Fig. 2).
2
d
( Z p + 2 Z m )(Y p + 2Ym )
Aerial modes (a):
α a (ω ) + j
Session 35, Paper 1, Page 2
MODAL ANALYSIS EQUATIONS
The multiconductor transmission line matrix equations in (1) should be solved separately for each line
section domain A, B, and C. However, since the solutions regarding sections B and C can be obtained (using
the permutation operator) from the one for section A, it
will suffice to pay attention to the latter.
Z A = jωL + Z g + Z sk
;
Y A = j ω2 L−1
c
(6)
where L and Zg are, respectively, the external-inductance matrix and the ground return impedance matrix
(see Appendix I). Zsk, the phase conductor internal impedance, is a frequency-dependent complex diagonal
matrix whose elements can be determined by using
skin-effect theory results for cylindrical conductors, involving Bessel and Neumann functions.
The solution to the propagation equations in (1) can
be presented in various forms [8-9]. However, bearing
in mind the problem we have to solve, the solution in
the form of a transmission matrix is the most adequate
one.
The transmission matrix for section A, TA, yields the
relationship between line currents and voltages measured at ports z = 0 and z = lA
A A
V (l A )
V (0)
 I (0)  = T A  I (l )  ; T A = C


 A 
 A
BA

DA
(7)
where the square complex sub-matrices AA, BA, CA, and
DA, can be obtained from ZA and YA as follows, [8-9]:
A A = D tA = COSH[S A l A ]
(8a)
B A = B tA = SINH[S A l A ] S A Y A−1
(8b)
[S A l A ]
(8c)
C A = C tA = Z −A1S A
SINH
In (8), the matrix SA represents the square root of the
ZAYA product. The determination of
S A = SQRT(Z AYA )
involves the solution of a matrix eigenvector-eigenvalue
problem concerning the ZAYA matrix product (see Appendix II).
14th PSCC, Sevilla, 24-28 June 2002
2.2 The Transmission Matrix of the Structure
The transmission matrix T characterizing the overall
line structure is obtained by multiplying the individual
transmission matrices pertaining to the three line sections A, B, and C.
A B 
T = T A TB TC = 

 C D
(9)
Matrices TB and TC can be evaluated following the
guidelines outlined in 2.1 for TA. However, such a job
can be greatly simplified by making use of the existing
relationships (2) among ZA, ZB, ZC and YA, YB, YC,
which also extend to the square-root matrix functions
-1
-1
SA, SB, SC, that is, SB = PSAP and SC = P SAP.
Further simplification arises in the case when the
line sections A, B, and C, have exactly the same length,
lA = lB = lC. For a strictly regular transposition scheme
we find
P 0   P 0 
TB = 
TA 

 0 P  0 P
−1
P 0 
; TC = 

 0 P
−1
P 0 
TA 

 0 P
thus yielding
A B  A A P B A P 
T=

=
 C D  C A P D A P 
3
(10)
The eigenvalues of T are [8] the positive and negative exponential propagation factors:
e
+γ g L
,e
−γ g L
, e +γ a′L , e −γ a′L , e +γ a ''L , e −γ a '' L
(11)
where L = lA + lB + lC is the length of the transposition
cycle and γk is the kth propagation constant of the overall
structure yielding the required information for computing the modal attenuation constants and modal phase
velocities
γ k = α k (ω ) + j ω , for k = g, a’, a”
vk (ω )
(12)
where subscripts g, a’, and a” are used as remainders
for ground and aerial modes.
3
RESULTS
To get a clear understanding of the influence of the
many factors that impact on the propagation properties
of the transposed-line structure a step by step procedure
will be used. First, we consider a commonly used overhead line geometry —a horizontal line configuration—
and analyze the influence of line losses, i.e., wire and
ground losses. Afterwards, by changing the conductor
configuration and section lengths, we will examine the
influence of line geometrical parameters.
In this work attention will focus on certain critical
frequencies for which the structure period gets close to a
multiple of one half wavelength L ≈ mλ / 2 , that is, for
resonant frequencies given by
f res ≈ mc /(2 L)
(13)
Session 35, Paper 1, Page 3
3.1 The Influence of Line Losses
This subsection is concerned with the horizontal line
configuration in Fig. 2a, attention being paid to the influence of line losses on the computation of the modal
propagation constants of the transposed line structure.
To take losses into account typical parameter values
of 57.3 mΩ/km and of 100 Ωm were assigned to the
wires dc resistance and ground resistivity, respectively.
Assuming a regular transposition scheme with lA = lB
= lC = L/3 = 100 km we see from (13) that the first two
resonance events are expected to take place at about 500
and 1000 Hz. Therefore, a frequency scan in the range
400 to 1100 Hz will be suited for assessing the behavior
of the modal propagation constants as a function of the
frequency.
The overall attenuation in dB, α (ω ) L × 20 log 10 e , as
well as the phase velocity in normalized units, v(ω)/c,
were evaluated for all the propagation modes.
To gain insight into the problem the strategy of progressively increasing the degree of complexity was
used. First, the modal propagation constants are evaluated considering an ideal lossless line. Then, wire losses
are added. Finally, the real scenario with wire and
ground losses simultaneously present is examined. Results obtained are shown in Fig. 3. The following conclusions can be drawn:
w Even when losses are absent structural attenuation
bands do indeed show up at around 500 Hz and 1 kHz.
In the lossless line case these bands are identified by
regions of non-null attenuation with linearly increasing
phase velocity (for frequencies inside these bands the
net power flow in the line is zero —electric and magnetic fields have a phase lag of π/2).
w The transitions between attenuation bands and
passing bands, which are very sharp in the lossless case,
become smoother and smoother as losses are included.
w The resonant characteristics of the aerial modes a’
and a”, although identical in magnitude, are frequency
shifted. While mode a’ predominantly resonates at 500
Hz, mode a” predominantly resonates at about 1 kHz.
w The importance of the resonance peaks in the aerial modes deserves special remark. In particular, it must
be emphasized that those resonance peaks are very
weakly affected by line losses, an attenuation peak of
almost 2 dB being observed with or without losses considered. The value reached by the attenuation resonance
peak is about 4 or 5 times greater than the one predicted
in (5b) by the standard approach.
w The ground mode shows identical resonance features either at 500 Hz or 1 kHz. In the lossless case, the
attenuation and phase velocity characteristics of the
ground mode cannot be distinguished from those belonging to one of the aerial modes —they are superposed. However, they immediately split as soon as wire
losses are accounted for.
w Contrary to the aerial modes the ground mode
resonance characteristics are strongly affected by line
losses. When only wire losses are considered the ground
mode attenuation peak still remains relevant, its value
being about three times greater than the one predicted in
14th PSCC, Sevilla, 24-28 June 2002
Session 35, Paper 1, Page 4
NO LOSSES INCLUDED
a’
WIRE LOSSES INCLUDED
a“
a’
TOTAL LOSSES INCLUDED
a“
g
a“, g
a’, g
a’
a“
a“
a’
g
a’
a’
a’
g
g
a“
a“
a“
g
Figure 3: Plot of the attenuation and phase velocity characteristics against frequency for the horizontal line configuration, assuming
a strictly regular transposition cycle. Resonance effects are displayed for three different situations: no losses, wire losses, and total
losses included. Dotted lines represent the attenuation and phase velocity characteristics obtained using equation (5) from the standard homogenization technique. The ground mode is identified by label g, whereas the aerial modes are identified by a’ and a”
(5a) using the standard approach. However, when
ground losses are added, the resonance phenomenon is
seen to lose importance. Resonance marks although still
visible almost disappear diluted with the high attenuation background.
3.2 The Influence of Geometrical Parameters
The preceding results were obtained by considering
a strictly regular transposition cycle with L = 300 km
and lA = lB = lC.. Now, we analyze the influence of transposition cycle length irregularities.
Using a random number generator we defined different values for lA, lB , and lC, with relative deviations to
their mean value L/3 as high as ± 15%. Results regarding the modal attenuation and modal phase velocity
characteristics for regular and irregular transposition
cycles are compared in Fig. 4 (horizontal line configuration with global losses included). For analysis purposes a frequency scan comprising five resonant peaks
from f = 400 Hz to f = 2.6 kHz was considered.
Results obtained lead to the following conclusions
w For strictly regular transposition schemes, 3rd order resonance peaks (1.5 kHz) are absent —indeed,
every mth order resonance with m a multiple of 3 vanishes.
w When length irregularities are taken into account
resonance peaks of order 3, 6, 9, etc, show up. However,
the magnitude of those peaks is rather small (while for
the aerial modes a small disturbance at 1.5 kHz may still
be detected, for the ground mode the effect is absolutely
negligible).
w For strictly regular transposition cycles the aerial
modes are seen to exhibit alternate resonance peaks.
The consideration of length irregularities makes both
aerial modes to resonate at every critical frequency (although with different magnitudes).
Lastly, the question about the influence of conductor’s geometrical arrangement on the modal propagation
constants of the transposed-line structure is addressed.
Assuming a regular transposition scheme, with wire
and ground losses accounted, we computed the modal
attenuation and modal phase velocity characteristics for
the horizontal, vertical and triangular line configurations
depicted in Fig. 2. Results obtained, considering a frequency sweep from f = 400 Hz to f = 1.1 kHz, are
shown in Fig. 5. Comparison established leads to the
following conclusion:
w While the horizontal and vertical line configurations have very identical characteristics, the triangular
line, with a naturally more balanced geometry, is seen to
be almost immune to the resonance effects due to conductors transposition.
4
CONCLUSIONS
For balancing purposes many overhead power lines
have their phase conductors periodically transposed
along the line length. The modal propagation constants
of the overall line structure are ordinarily evaluated using a simple homogenization technique based on average series-impedance matrices and average shuntadmittance matrices.
14th PSCC, Sevilla, 24-28 June 2002
Session 35, Paper 1, Page 5
REGULAR TRANSPOSITION CYCLE
IRREGULAR TRANSPOSITION CYCLE
g
a’
g
a“
a’
a’ a“
a’
a“
a’
a’
a“
a“
a’
a’
a’
a“
a’
a“
a“
g
g
Figure 4: Plot of the attenuation and phase velocity characteristics against frequency for the horizontal line configuration with total
losses included. Resonance effects are displayed for two cases: regular and irregular transposition cycles. Dotted lines represent results obtained using the standard homogenization technique. Labels g, a’ and a” have the same meaning as in Fig. 3.
HORIZONTAL LINE
VERTICAL LINE
g
TRIANGULAR LINE
g
a’
a“
a’
g
a“
a’
a’
a“
g
a’
a“
a’
a“
a“
g
g
Figure 5: Plot of the attenuation and phase velocity characteristics against frequency for the horizontal, vertical and triangular line
configurations, assuming regular transposition cycles, and considering all losses included. Dotted lines represent results obtained
using the standard homogenization technique. The ground mode is identified by label g, whereas the aerial modes are identified by
a’ and a”.
14th PSCC, Sevilla, 24-28 June 2002
Albeit very appealing this procedure is not a rigorous one as it cannot adequately account for the wave
interference phenomena produced by conductor interconnections occurring at discrete transposition points.
Making use of matrix methods for nonuniform multiconductor line structures, conductor transposition effects in single-circuit three-phase transmission lines
have been analyzed in this work. Modal attenuation and
modal phase velocity characteristics against the frequency have been determined for several transmission
line configurations, the influence of cycle irregularities
and the influence of line losses being examined as well.
The main conclusions of this research can be summarized as follows. Results obtained show that conductor transposition actually gives rise to repetitive
resonance effects at certain critical frequencies —frequencies for which the overall transposition cycle length
approaches an integer multiple of one half wavelength
(e.g., for a cycle 300 km long, resonance peaks of different magnitude can be seen to occur at 0.5 kHz, 1 kHz,
1.5 kHz, etc). The ground and aerial modes of the transposed line structure exhibit peculiar resonant behaviors;
for real lossy transmission lines, the ground mode does
not show significant resonance marks, however, the
aerial modes are highly affected (observed peak values
for the attenuation are about 4 or 5 times greater than
predicted by the standard homogenization technique).
Resonance effects due to conductor transposition are not
independent of the line geometry; while equally important for horizontal or vertical lines, resonance effects
turn out to be practically negligible for triangular lines.
Results herein obtained should find special application in all areas concerned with the propagation of
waves along power lines, namely, PLC communications, EMI, and line transients.
The evaluation of the resonance effects associated
with forward and backward surge impedances is a subject that has not been covered here (a subject that will
involve the analysis of the eigenvectors of the global
transmission matrix T). Further work on such a topic is
still required.
Session 35, Paper 1, Page 6
[4] J. Faria, “High-Frequency Modal Analysis of
Lossy Nonuniform Three-Phase Overhead Lines
Taking into Account the Catenary Effect”, Europ.
Transactions on Electrical Power, vol. 11, pp. 195201, May/June 2001
[5] J. Faria, “On the Realizability of Balanced Untransposed Overhead Line Configurations”, European Trans. Electrical Power, vol. 8, pp. 451-454,
Nov/Dec. 1998
[6] R. Galloway, W. Shorrocks and L. Wedepohl,
“Calculation of Electrical Parameters for Short and
Long Polyphase Transmission Lines”, Proc. IEE,
vol. 111, pp. 2051-2059, Dec. 1964
[7] C. Gary, “Approche Complète de la Propagation
Multifilaire en Haute Frequence par Utilisation des
Matrices Complexes”, EDF Bulletin de la Direction des Études et Recherches, no. 3/4, pp. 5-20,
1976
[8] J. B. Faria, “Multiconductor Transmission-Line
Structures”, New York, Wiley, 1993, ISBN 0-47157443-0
[9] C. Paul, “Analysis of Multiconductor Transmission Lines”, New York, Wiley, 1994, ISBN 0-47102080-X
[10] C. Dubanton, “Calcul Approché des Paramètres
Primaires et Secondaires d’une Ligne de Transport”, EDF Bulletin de la Direction des Études et
Recherches, no. 1, pp. 53-62, 1969
[11] J. Faria and J. Silva, “Irregular Eigenvalues in the
Analysis of Multimodal Propagation”, 8th PSCC
Proceedings, pp. 760-764, Aug. 1984
APPENDIX I
Evaluation of L and Zg matrices
The per-unit-length external-inductance matrix L is
a frequency-independent real symmetric non-singular
matrix whose elements are evaluated according to
REFERENCES
[1] N. Boulejfen, A. Kouki and F. Ghannouchi, “Frequency- and Time-Domain Analyses of Nonuniform Lossy Coupled Transmission Lines with Linear and Nonlinear Terminations”, IEEE Trans. Microwave Theory Tech, vol. 48, pp. 367-379, March
2000
[2] G. Xiao, K. Yashiro, N. Guan and S. Ohkawa, “A
New Numerical Method for Synthesis of Arbitrarily Terminated Lossless Nonuniform Transmission
Lines”, IEEE Trans. Microwave Theory Tech, vol.
49, pp. 369-376, Feb. 2001
[3] M. T. Barros and M. Almeida, “Computation of
Electromagnetic Transients on Nonuniform Transmission Lines”, IEEE Trans. Power Delivery, vol.
11, pp. 1082-1091, April 1996
Lkk =
Lik =
µ o 2hk
ln
r
2π
µ o (hi + h k )2 + ( y i − y k )2
ln
4π (h − h )2 + ( y − y )2
i
k
i
k
where r denotes phase conductor radius, and hk and yk
denote, respectively, the vertical and horizontal coordinates of conductor k.
The per-unit-length ground return impedance matrix
Zg, is a frequency-dependent complex symmetric matrix
whose elements can be determined using either Carson's
approach or Dubanton's complex ground plane approach, the latter yielding [10]
(Z g )kk = jω µ2πo ln1 + hp 

k 
14th PSCC, Sevilla, 24-28 June 2002
(h + hk + 2 p )2 + ( y i − y k )2
µ
Zg
= jω o ln i
ik
4π
(hi + hk )2 + ( y i − y k )2
( )
where p —the so-called complex depth— simultaneously depends on the ground resistivity ρ and on the
frequency.
p=
ρ
jωµ o
APPENDIX II
Evaluation of the square-root matrix S
Let Z and Y denote the series-impedance and shuntadmittance matrices of a given uniform line section.
Their product ZY is a non-symmetric square complex
matrix.
If, as it usually happens, ZY is a simple structure
matrix [8] then ZY will be brought into diagonal form
by employing a non-singular similarity transformation
defined by the modal matrix M
M −1 ZY M = diag{γ 12 , γ 22 , γ 32 }
The eigenvectors of ZY are the columns of M, the
associated eigenvalues are the squared modal propagation constants.
The square-root matrix S is obtained as
S = SQRT (ZY ) = M diag{γ 1 , γ 2 , γ 3 } M −1
If, however, ZY happens to exhibit irregular eigenvalues [11], a non-singular modal transformation matrix
will never be found. In such a case the computation of S
will require a different more elaborated strategy making
use of Jordan normal forms [8].
ACKNOWLEDGMENT
This research was carried out within the framework
of the PRAXIS XXI Program under the sponsorship of
the FCT — Fundação para a Ciência e a Tecnologia
whose support the authors gratefully acknowledge.
Session 35, Paper 1, Page 7