HIGH-FREQUENCY EXTENSION OF THE TRANSMISSION-LINE
THEORY FOR AN OPEN LINE
A. Cozza †, ‡
F. Canavero †
B. Démoulin ‡
† Dipartimento di Elettronica, Politecnico di Torino (Italy)
‡ Laboratoire TELICE, Université des Sciences et Technologies de Lille (France)
Contact : [email protected]
Abstract
A high-frequency extension of the transmission line theory (TLT) for describing a uniform scalar-line is presented.
The main effects not included into the TLT are addressed, extending the model validity over a wider frequencyspan. The radiation losses are taken into account thanks to a new effective definition of the radiation resistance,
yielding load-independent results. The proposed model is then validated by comparisons with the experimental
results for a prototype.
Introduction
The transmission lines theory (TLT) is one of the most effective tool for predicting the evolution of an electric signal
propagating along a wave-guiding structure. Its main advantage is the possibility to yield analytical expressions for
the voltages and currents along a transmission line, whereas other purely numerical methods require the solving
of big linear equation systems, thus demanding a higher price in computation time and resources. Nevertheless,
the frequency-span over which the TLT is valid is limited by the line geometry, especially when the line is of the
kind which is often referred to as open, such as an unshielded conductor hanging above a metallic ground-plane.
The cause of this limitation is due to considering only the TEM mode, thus neglecting the higher modes of the
line, and in particular the radiation ones, which become rather important as the frequency gets higher. On the
other hand, even the discontinuities which are usually regarded as negligible, or approximated as a lumped circuit,
may modify the line behaviour, resulting in an increasingly rough description of the real system by means of the
TLT, whenever the guided wavelength gets comparable with the dimensions of the system devices. Furthermore,
the propagation losses due to the increasing radiation efficiency of the line may lead to an overestimation of the
amplitude of the signals along a transmission-line. The main concern of this paper is to propose a set of operations
that when applied to a uniform scalar line (such as a single-wire above a ground-plane or a two-wire line embedded
into a homogeneous medium), can extend the validity of the TLT far beyond its usual upper-frequency limit.
Furthermore, the proposed model is applied for the evaluation of the input impedance of a single-wire line above
a metallic ground-plane, while validating these results comparing them to the experimental ones obtained with a
prototype.
The system
The physical system here considered is a uniform scalar line constituted by a cylindrical metallic conductor at a
distance h above a metallic ground-plane (see figure 1), embedded into a homogeneous medium. The overhead
conductor is kept hanging by two vertical risers (VRs), providing a mechanical support and introducing two 90
degrees bends at the line ends. The line ends are connected to a couple of N-type connectors, placed below the
ground-plane, in order to reduce the effects of these discontinuities, and thus highlighting the line behaviour. One
end was connected to a network analyzer (NA) Agilent 8753, whereas the other one was connected to a standard
load taken from the NA calibration-set. The NA was calibrated as a one-port device, in order to measure the input
impedance of the line.
This kind of structure is effectively described by a TLT model merely considering the TEM mode, applying it
to the simplified system shown in figure 2a: the horizontal part of the line is regarded as a uniform transmission line
with a characteristic impedance ζh and a complex propagation constant γh , whereas the propagation along the VRs
is considered negligible, approximating them as ideal short-circuits. In the same way, discontinuities such as the
connectors are merely neglected. The electromagnetic analysis of an open guiding structure yields the hypothesis
x
1.5
m
1m
1m
h
z
transitions
towards
the NA
copper
ground−plane
y
coaxial
cable
N−type
standard load
N−type
connector
Figure 1: The system considered.
under which this representation (i.e. just considering the TEM mode) is valid, i.e. when the transversal dimensions
of the discontinuities (both the VRs and the line ends) are negligible with respect to the guided wavelength. To this
end, a simple way for estimating the suitability of the TLT model is to introduce the index ν
D
λg
ν=
(1)
defined as the ratio of the maximum dimension D considered over the guided wavelength. Considering the different
indexes ν which could be defined for each element constituting the system, a more conservative definition would
involve the maximum transversal dimension, which is supposed to be the line height h. Hence, the simplified
model is valid just in the frequency-range satisfying the condition
νmax =
h
λmin
¿ 1.
(2)
The inadequacy of the basic model is shown in figure 5, where the input impedance of a line as in figure 1
is shown, in modulus and phase, for h = 30 cm, over the frequency-range 1-300 MHz. The line per-unit-length
(p.u.l.) parameters are evaluated by means of the classical approximated expressions for a wire (radius r) above a
perfect metallic ground-plane:
µ ¶
60
2h
ζh =
ln
(3)
nr
r
ω
(4)
γh = j nr
c0
where nr is the refraction index of the medium surrounding the line, while c0 is the light-speed in free-space. For
this line, analyzed over the range 1-300 MHz νmax = 0.3, thus the hypothesis of a “closed” line is not fulfilled. In
1m
ζh
γ
Z load
h
Zin
Ct
h
1m
ζ v γv
ζ h γh
h
ζ v γv
Ct
Z load
Zin
Figure 2: The basic model (a) and the proposed model (b).
ground
plane
vertical
riser
coaxial
cable
Figure 3: The transition.
fact, the model predicts the first resonance frequency at about 75 MHz, whereas the actual resonance takes place
around 42 MHz.
Line-end discontinuities
Even when hypothesis (2) is satisfied, the propagation along the VRs should be taken into account. It is important
to realize that the VRs do not modify the line behaviour just when their length is comparable with the guided
wavelength (which accounts for further high-frequency effects), but rather when it is comparable with the horizontal line length. In order to include their effects (for higher-frequency too) into the basic model, it is worthwhile
trying to describe the VRs as two additional transmission lines, thus including them in a very straight way. This
approach was sought in [1, 2], approximating a cylindric vertical conductor above a perfectly electric conductive
(PEC) ground-plane as a conical line. Approximating a cylinder as a cone with varying angle and evaluating its
average characteristic impedance, as proposed in [3], yields:
· µ ¶
¸
60
2h
ζv =
ln
−1 .
(5)
nr
r
whereas the propagation constant is given by equation (4), being the VRs embedded into the same homogeneous
medium as the horizontal line, being the TEM mode considered. This model describes the line as the series of a
VR, the horizontal line and the other VR. The results obtained through this representation are shown in figure 5,
and approximate the experimental ones in a better way, providing the first resonance frequency at 46 MHz.
In order to enhance the model, the effects of the transitions from the VRs to the coaxial connectors should
be taken into account. To this end, the transitions can be described as in figure 3. An equivalent model for this
sub-structure was developed in [4], yielding an estimation of about 1 pF for the equivalent parallel capacitance to
add at the line ends (see 2b). The results obtained with this model are shown in figure 5, providing a far better
description of the line behaviour. Anyway, this model performs poorly around the structure resonances, especially
as νmax increases.
x
Γ3
Γ2
Γ1
z
Γ4
Figure 4: The four regions the system has been divided into.
5
|Zin| (Ω)
10
3
10
1
10
0
50
100
150
200
250
300
0
50
100
150
200
Frequency (MHz)
250
300
∠Zin (degrees)
100
50
0
−50
−100
Figure 5: Model validation: (•) experimental data, (−) proposed model w/ def. (7), (− ¦ −) proposed model w/
def. (6), (−−) prop. model for Rrad = 0, (− · −) basic model plus VRs, (− ◦ −) basic model.
Radiation losses
The reduced sharpness of the peaks in the modulus and the smoother phase transitions are clues of losses neglected
within the model, which happens to be due to the line radiation. Indeed, the effects of these kind of losses are more
important around the structure resonances, corresponding to a current distribution providing the greatest radiation
effectiveness, as one can notice comparing the experimental and the theoretical results. Several authors [5, 6]
have proposed an extended formulation for the TLT which includes the radiation losses into the p.u.l. definition.
Though providing a very general tool, this approach tends to complicate the simple structure of the standard p.u.l.
computation. In this paper the radiation losses will be included into the p.u.l. as in a two-step approach, adopting
the procedure proposed in [7]. Substituting the ground-plane (supposed to be perfect) with the corresponding
image of the overhead current distribution, the electromagnetic field radiated by the system was estimated, by
numerically convolving the currents with the Green’s free-space function. Subsequently, the active power radiated
by the line is estimated by integrating the Poynting vector all over a closed surface, actually an hemisphere, being
the field identically zero below the ground-plane. Having estimated the active power Prad radiated by the line, it
is now possible to define the radiation resistance Rrad as:
Prad = |Iin |2 Rrad
(6)
where Iin is the current at the line’s input port. Then, the radiation effects can be included (at least approximately)
into the TLT model by “spreading” this resistance all over the entire line length, i.e. dividing it by the line length
and adding it to the p.u.l. series-impedance, and subsequently computing the new current distribution, repeating
this procedure all over the frequency spectrum. Actually, this procedure should be iterated in order to converge
to a better estimation of the radiation losses. Though yielding good results, this procedure performs very poorly
when applied to a resonant structure . Let consider the second resonance of a line (i.e. the line length coinciding
with half a wavelength) terminated by a short-circuit (standard N-load set). In this case the line radiates very
effectively, whereas the input current is nearly zero, due to the short-circuit, thus the radiation resistance diverges.
The second iteration shall yield a current distribution nearly zero all over the entire line, due to the very strong
attenuation provided by the radiation resistance previously estimated. This new configuration is no more effective
for the radiation, so that the second estimation for the Rrad will be nearly zero. Subsequently the procedure goes
on as for the first iteration in a sort of bistable trend, without converging to a suitable result.
In order to avoid this kind of situation, and yet for pursuing a non-iterative approach (which is quite timeconsuming due to the active power estimation), a rather different definition for the radiation resistance has been
adopted:
¯ ¯2 0
Prad = ¯I¯¯ Rrad
(7)
7
10
5
10
R
rad
(Ω)
3
10
1
10
−1
10
−3
10
0
50
100
150
200
Frequency (MHz)
250
300
Figure 6: Radiation resistance: (− ¦ −) Zload = 0, (− ◦ −) Zload = ∞, (− × −) Zload = 50 Ω, solid lines
correspond to eq. (6), dashed lines to eq. (7); (- -) empirical model.
where I¯ stands for the average current computed over the entire current distribution. The radiation resistance has
been evaluated thanks to the following expressions, derived from equation (7) assuming γ0 R À 1:
0
Rrad
=
Ik
=
∆Rk
=
¯¯
¯¯¯2
4
¯ γ ¯2 Z Z
¯¯ X
¯¯
¯ 0 ¯
¯¯
ζ0 ¯ ¯¯
sin ϑ¯¯
Ik (ϑ, ϕ)¯¯¯ dϑ dϕ
¯
¯
4π I
Σ
k=1
Z
ϕ̂k sin ϑk
Ik (`)e−γ0 ∆Rk d`
Γk

−xx̂ · ρ̂
k=1


−Lẑ · ρ̂ − xx̂ · ρ̂ k = 2
k=3

 −(hx̂ + z ẑ) · ρ̂
−(−hx̂ + z ẑ) · ρ̂ k = 4
(8)
(9)
(10)
where ` stands for x for the VRs and z for the horizontal line; γ0 and ζ0 are the free-space propagation constant
and characteristic impedance, whereas index k refers to the four regions shown in figure 4, identified by the use of
two spherical reference systems along the axis x and z for the VRs and the horizontal line. Surface Σ is the upper
hemisphere of radius R.
Definition (7) provides very good results around the structure resonances, as shown in figure 5. Furthermore,
these results have been obtained in just an iteration, thus reducing the time-consumption implicit in definition (6).
A comparison between the radiation resistance provided by the two definition is in figure 6, where it is shown
that the proposed definition approaches the empirical law Rrad ∼
= f 2 , along with the radiation resistances for
the open-circuited line and for the 50 Ω load. The results obtained through definition (7) show a rather strong
independence from the line load, whereas definition (6) provides very varying results. The explanation for this
result is related with the normalization adopted, which avoid including the current distribution behaviour into the
radiation resistance, in particular into the terms Ik , thus providing a result which is related to the line geometry
rather than its electrical properties.
This result is attractive, especially considering that being able to approximate the radiation resistance as Rrad =
κf 2 , then it is no more necessary to evaluate the active power radiated all over the frequency-spectrum, but rather
just for a few samples (ideally just one), in order to estimate through fitting the parameter κ, operation which could
not be performed using the definition proposed in [7], due to its oscillating behaviour.
Conclusions
A set of improvements have been proposed in order to extend the frequency-range over which a TLT model can be
employed for a uniform scalar line. The VRs are effectively included into the model, along with the line transitions.
Subsequently, the radiation losses are estimated and included into the model, thanks to a new definition of the
radiation resistance, which provides very good results as compared with an experimental prototype. Finally, it is
shown that the time-consuming integration can be strongly reduced avoided thanks to a peculiar property of the
proposed new definition.
References
[1] S.A. Schelkunoff, Electromagnetic Waves, Van Nostrand, New York, 1945
[2] E.F. Vance, Coupling to shielded cables, Wiley Interscience Publications
[3] P. Degauque, A. Zeddam, “Remarks on the transmission-line approach to determining the current induced on
above-ground cables”, IEEE Transactions on electromagnetic compatibility, vol. 30, No.1, February 1988
[4] R. King, Trasmission-line theory, McGraw-Hill Book Company, 1955
[5] A. Maffucci, G. Miano, F. Villone, “Full-wave transmission-line theory”, IEEE Trans. on Magnetics, vol. 39,
No. 3, May 2003
[6] H. Haase, J. Nitsch, “Full-wave transmission-line theory for the analysis of three-dimensional wire like structures”, Proc. EMC, Zurich, Switzerland, 2001
[7] D.O. Wendt, J.L ter Haseborg, “Radiation losses representation in the transmission-line theory”, International
Symposium on Electromagnetic Compatibility, September 13-16, 1994, Rome, Italy