IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 4, NOVEMBER 2009 1051
Comparison Between Exact and Quasi-Static Methods for
HF Analysis of Horizontal Buried Wires
Leonid Grcev and Solza Grceva
Abstract—The validity domain of the quasi-static method for computa-
tion of high frequency and transient characteristics of horizontal buried
wires can be established by comparison with an exact analytical method.
Usually, limitations of the quasi-static method are derived for practical
characteristics, but these are strongly dependent on the specific case and
computed quantities. This paper presents an analysis of the differences in
the application of the exact and quasi-static Green’s function in a method
of moments approach for two important cases: distribution of currents in
directly fed wires and induced currents in passive wires. It is concluded
that the validity domains of the quasi-static method in these two cases are
very different.
Index Terms—Circuit modeling, Green’s functions, grounding, light-
ning, method of moments (MoM), transient analysis, transmission line
theory.
I. INTRODUCTION
High frequency (HF) and transient analysis related to lightning,
faults, or other electromagnetic interferences in buried conductors or
networks of conductors that are part of power, telecommunication, or
railway systems is of interest in electromagnetic compatibility (EMC)
studies [1]. Classical modeling approaches are based on circuit theory
with lumped [2] or distributed parameters [3], which is also the case
in many modern approaches, e.g., [1], [4], and [5]. Since circuit theory
approaches are based on the quasi-static approximation, their validity
is limited to a certain upper frequency [1]. On the other hand, full-
wave methods have been recently introduced, e.g., [6] and [7], based
on the solution of the Maxwell’s equations by the method of moments
(MoM) [8]. However, since electromagnetic MoM models are based on
an exact mathematical solution by Sommerfeld [9], they might serve
as a standard for comparison of more approximate models [10].
The validity domain of the circuit theory approaches has been re-
cently studied in [10] and [11]; however, considering different metrics
for comparison, Olsen and Willis [10] consider the touch and step volt-
ages in the frequency domain, and Theethayi et al. [11] consider the
transient currents and voltages in the time domain (both for directly
fed wires). Both studies [10], [11] suggest different limits of the va-
lidity domain of the considered circuit theory approaches. However,
although both studies give a direct insight into some practical charac-
teristics, conclusions are related to the specific choice of the metrics,
the system under study, and the methodology of the solution of the
complex mathematical models. Another recent publication [12, p. 335]
also considers “the classical transmission line approach to be relevant
for practical use” for coupling to buried wires. It is, therefore, of in-
terest to investigate more thoroughly the validity domain of the circuit
approaches.
Classical circuit models with both distributed and lumped parame-
ters are based on the quasi-static approximation. As a first step toward a
better understanding of their limitations, we look at the most basic case
of the horizontal elemental electric dipole in a conducting half-space,
Manuscript received February 24, 2009; revised July 29, 2009. First published
October 30, 2009; current version published November 18, 2009.
L. Grcev is with the Faculty of Electrical Engineering and Information Tech-
nologies, Ss. Cyril and Methodius University, Skopje 1000, Macedonia (e-mail:
leonid.grcev@ieee.org).
S. Grceva is with the Faculty of Informatics, Goce Delcev University, Stip
2000, Macedonia (e-mail: solza.grceva@gmail.com).
Digital Object Identifier 10.1109/TEMC.2009.2033468
Fig. 1. Coordinates for evaluation of fields caused by buried source.
for which there is a full-wave exact solution [13]. When the Green’s
function for this elementary dipole is obtained, the familiar solution to
the problem of extended wires involves integrating over sources using
Green’s theorem [8]. In this paper, we analyze differences in the appli-
cation of the exact and quasi-static Green’s function in a MoM-based
electromagnetic model [11] for two important cases: distribution of
currents in directly fed wires and induced currents in passive wires. In
both cases, we consider bare wires.
II. FULL-WAVE SOLUTION
Sommerfeld [9] first published the exact solution of the electromag-
netic field for an electric dipole near an interface. The geometry of
the problem considered is illustrated in Fig. 1. The horizontal electric
dipole is in the direction of the x-axis. The dipole and the field evalu-
ation point are both below the boundary between the air and the earth.
The designations of the coordinates and the characteristics of the earth
and the air are given in Fig. 1.
We consider a dipole with harmonic current moment p = Iℓ with
angular frequency ω. The time variation exp(jωt) is suppressed. The
wavenumbers of the earth k
1
and the air k
2
are
k
2
1
= ω
2
µ
0
ε
1
−
jσ
1
ω
k
2
2
= ω
2
µ
0
ε
0
(1)
The complete set of field equations in cylindrical coordinates from
Banos [13] is given here for reference
E
ρ
=
−pjωµ
0
4πk
2
1
cos φ
∂
2
∂ρ
2
(
g
1
−g
2
+k
2
1
V
1
)
+ k
2
1
(g
1
− g
2
+ U
1
)
E
φ
=
pjωµ
0
4πk
2
1
sin φ
1
ρ
∂
∂ρ
(
g
1
− g
2
+ k
2
1
V
1
)
+ k
2
1
(g
1
− g
2
+ U
1
)
E
z
=
−pjωµ
0
4πk
2
1
cos φ
∂
2
∂ρ∂z
(
g
1
+ g
2
− k
2
2
V
1
)
(2)
H
ρ
=
p sin φ
4π
∂
∂z
(g
1
− g
2
+ U
1
) −
1
ρ
∂W
1
∂ρ
H
φ
=
p sin φ
4π
∂
∂z
(g
1
− g
2
+ U
1
) −
∂
2
W
1
∂ρ
2
H
z
=
−p sin φ
4π
∂
∂ρ
(g
1
− g
2
+ U
1
)
(3)
where
V
1
=2
∞
0
exp [γ
1
(h − z )]
k
2
1
γ
2
+ k
2
2
γ
1
J
0
(λρ) dλ
U
1
=2
∞
0
exp [γ
1
(h − z )]
γ
1
+ γ
2
J
0
(λρ) dλ
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