This document describes a methodology for simulating signal propagation on multiconductor transmission lines directly in the time domain using the Transmission Line Modeling (TLM) method combined with Z-transform techniques. The approach models a two-conductor transmission line using telegrapher's equations converted to the TLM format. It defines quantities such as incident and total voltages/currents and applies a bilinear Z-transform to obtain a numerical algorithm to calculate the total line quantities from the incident quantities. As an example, it simulates propagation of a short pulse on a hypothetical two-channel carbon nanotube transmission line.

1. Time-Domain Simulation of
Multiconductor Transmission-Lines
Reprint (with corrections) of a paper presented at the Proceedings of Frontiers
in Applied Computational Electromagnetics (FACE 2006), University of
Victoria, Canada, June 2006.
John Paul∗
, Christos Christopoulos and David W. P. Thomas
George Green Institute for Electromagnetics Research,
School of Electrical and Electronic Engineering,
University of Nottingham, Nottingham, NG7 2RD, U.K.
∗
Current E-mail (2016): john.derek.paul@gmail.com
Abstract
This paper describes a methodology for the simulation of multicon-
ductor transmission-lines in the time-domain. The approach is based
on the Transmission-Line Modelling (TLM) method combined with
Z-transform techniques. The scheme is applied to the simulation of
signal propagation on a carbon nanotube transmission-line.
Keywords: Multiconductor transmission lines, interconnections, time-
domain analysis, transmission line matrix methods, Z-transforms.
1 Introduction
The requirement to simulate signal transfer along multiconductor transmission-
lines using numerical time-domain codes is motivated by a number of branches
of applied electrical science. Typical examples are in the investigation of electro-
magnetic compatibility issues in high speed digital interconnects, the prediction
of fault behaviour of electrical power transmission systems and recently the pos-
sibility of using carbon nanotubes as nano-transmission lines and nano-antennas.
This paper outlines an approach applicable to the numerical simulation
of multiconductor transmission lines directly in the time-domain. In previ-
ous work, we have described a technique for interfacing single conductors with
field solutions [1]. This method has been applied to the simulation of overhead
lines over lossy grounds [2] and wires loaded with linear and nonlinear lumped
components [3]. Other related work in the TLM modelling of multiconductor
transmission-lines was an extension of the single conductor algorithm developed
by Trenkic et al. [4] by Wlodarczyk et al. [5]. The major difference between the
approach of [4, 5] and the method presented here is that in [4, 5], only inductive
stubs are used in the wire to obtain the correct propagation speed, whereas the
approach developed here allows both inductive and capacitive behaviour in the
1

2. wire and the inductance and capacitance may be balanced to achieve the cor-
rect propagation speed without dispersion. In addition, the method presented
here is based on using Z-transforms in TLM as pioneered by de Menezes and
Hoefer [6] whereas the approach described in [1] is based on stub loading rather
than Z-transforms. Note that the two approaches are identical when the bilin-
ear Z-transform is used. However, in the multiconductor case, we argue that
the Z-transform approach gives a simpler formulation because the coupling be-
tween wires is handled in a straightforward manner. This advantage of the
Z-transform formulation over the traditional stub method has been previously
exploited by us in the development of methods for simulation of EM propagation
in chiral materials [7, 8].
An alternative approach based on modal expansions which can locate wires
anywhere within a two-dimensional TLM node was described by Choong et
al. [9] and the method was extended to multiconductor configurations by Bi-
wojno et al. [10]. This technique is based on coupling the modal solution for
wire(s) to the surrounding mesh describing the fields.
There is much interest currently in the prediction and modelling of carbon
nanotube based transmission-lines [11, 12, 13] and nanotube antennas [14, 15].
These nano-transmission lines offer exciting properties such as ballistic electron
transport. Although to date much theoretical work has been carried out, e.g. see
the references in [11]–[15], there still remain difficult practical challenges to be
solved before nanotubes can be incorporated into applications, for example,
methods for obtaining consistent ohmic contacts have not yet been developed.
These will be required to connect, for example, generators or loads to nanotubes.
This paper is organized as follows, in section 2 the TLM model of a multicon-
ductor transmission-line is developed for the two conductor case and in section
3, the model is applied to the simulation of the charge-mode (common-mode)
and the spin-mode (differential-mode) in a hypothetical metallic nanotube hav-
ing two channels. The paper is concluded in Section 4. Note that the aspects of
the coupling of the multiconductor to an external electric field or multiconduc-
tor junctions is not treated here, but these modifications may be incorporated
into this formulation without too much difficulty.
2 Theory
The theory developed here is analogous to the methods described in [16, 7] for
the simulation of general linear materials in TLM. Considering a multiconductor
transmission-line directed in x, the Telegrapher’s equations in the time-domain
are:
−
∂Vw
∂x
= L ·
∂Iw
∂t
+ R · Iw (1)
−
∂Iw
∂x
= C ·
∂Vw
∂t
+ G · Vw (2)
In (1) and (2), Vw is the vector of wire potentials with respect to the common
ground conductor, Iw is the vector of wire currents and {L, R, C, G} are the
inductance, resistance, capacitance and conductance matrices per unit (p.u.)
length. For simple conductor configurations, expressions for the p.u. matrices
may be derived analytically. For general conductor configurations, the p.u. ma-
trices are evaluated numerically.
2

3. For simplicity in the formulation, consider a two conductor line with a per-
fectly conducting ground return as shown in Fig. 1. The conductors are labelled
A and B and the ground plane is indicated by E. For this case, (1) and (2) are
simplified to:
−
∂
∂x
VA
VB
=
LAA LAB
LBA LBB
·
∂
∂t
IA
IB
+
RA 0
0 RB
·
IA
IB
(3)
−
∂
∂x
IA
IB
=
CAA CAB
CBA CBB
·
∂
∂t
VA
VB
+
GAA GAB
GBA GBB
·
VA
VB
(4)
The equivalent circuit of a 1m segment of the two-conductor transmission
line is in Fig. 2. In this figure, it is assumed that the two lines A and B have
identical p.u. parameters, i.e.,
LAA = LBB = Le + Lm (5)
LAB = LBA = Lm (6)
RA = RB = Rs (7)
CAA = CBB = Ce + Cm (8)
CAB = CBA = −Cm (9)
GAA = GBB = Ge + Gm (10)
GAB = GBA = −Gm (11)
In (5)–(11), the self inductances of the line are Le+Lm, the mutual inductance is
Lm and the series resistances are Rs. As indicated in Fig. 2, the line capacitances
and conductances are described using delta networks where the self capacitances
are Ce +Cm, the mutual capacitance is −Cm, the self conductances are Ge +Gm
and the mutual conductance is −Gm. Using (5)–(11) in (3) and (4) yields
−
∂
∂x
VA
VB
=
Le + Lm −Lm
−Lm Le + Lm
·
∂
∂t
IA
IB
+ RS
IA
IB
(12)
A
B
E
x
Figure 1: A two-conductor transmission-line over a ground plane
3

4. −
∂
∂x
IA
IB
=
Ce + Cm −Cm
−Cm Ce + Cm
·
∂
∂t
VA
VB
+
Ge + Gm −Gm
−Gm Ge + Gm
·
VA
VB
(13)
These equations are converted into the TLM format using:
Le = L0(1 + χLe) , Lm = L0 χLm (14)
Ce = C0(1 + χCe) , Cm = C0 χCm (15)
Z0 = L0/C0 , ∆t = ∆ℓ L0C0 (16)
IA = iA/Z0 , IB = iB/Z0 (17)
∂
∂x
→
1
∆ℓ
∂
∂X
,
∂
∂t
→
1
∆t
∂
∂T
(18)
Rs = rsZ0/∆ℓ (19)
Ge = ge/(Z0∆ℓ) , Gm = gm/(Z0∆ℓ) (20)
In these equations, L0 and C0 are the wire link-line inductance and capacitance
p.u. length, {χLe, χLm, χCe, χCm} are dimensionless positive numbers, Z0 is the
characteristic impedance of the wire link-lines and iA and iB are the normalized
wire currents. The space-step is ∆ℓ and the time-step is ∆t. Note that when
the model is coupled to a numerical field solver, the time-step will in general be
set by the field code. The normalized wire resistance is rs and the normalized
conductances are ge and gm. By substituting (14)–(20) into (12) and (13) gives
−
∂
∂X
VA
VB
−
∂
∂T
iA
iB
=
χLe + χLm χLm
χLm χLe + χLm
·
∂
∂T
iA
iB
+ rs
iA
iB
(21)
−
∂
∂X
iA
iB
−
∂
∂T
VA
VB
=
χCe + χCm −χCm
−χCm χCe + χCm
·
∂
∂T
VA
VB
1m
Ce
Ce
Cm
Ge
Gm
Ge
Rs
Rs
A
B
Le
Le
Lm
Lm
x
Figure 2: Equivalent circuit of a 1m segment of a two-conductor transmission-
line over a lossless ground plane
4

5. +
ge + gm −gm
−gm ge + gm
·
VA
VB
(22)
The expressions on the left-hand sides of these equations are converted to the
travelling wave format by extending the TLM identities between the normalized
mixed derivatives and the incident wave components [16], i.e.,
−
∂
∂X
VA
VB
−
∂
∂T
iA
iB
≡ 2
V i
A4 − V i
A5
V i
B4 − V i
B5
− 2
iA
iB
(23)
−
∂
∂X
iA
iB
−
∂
∂T
VA
VB
≡ 2
V i
A4 + V i
A5
V i
B4 + V i
B5
− 2
VA
VB
(24)
Fig. 3 shows the definition of the incident voltages {V i
A4, V i
B4, V i
A5, V i
B5} and the
total line quantities {VA, VB, iA, iB} in the TLM model of the multiconductor
transmission-line.
To obtain a compact notation describing the multiconductor system and ease
the extension of the approach to lines having more than two conductors, the
following quantities are defined
V i
A4 − V i
A5
V i
B4 − V i
B5
= −
ir
A
ir
B
= −ir
w (25)
V i
A4 + V i
A5
V i
B4 + V i
B5
=
V r
A
V r
B
= V r
w (26)
VA
VB
= Vw ,
iA
iB
= iw (27)
χLe + χLm χLm
χLm χLe + χLm
= l (28)
+_2VA4
i
1
+_2VB4
i
1
+_2VB5
i
1
+_2VA5
i
1
VA
V
B
iA
iB
0V
Figure 3: Definition of the incident voltages and the total line quantities in the
travelling wave model
5

6. rs 0
0 rs
= r (29)
χCe + χCm −χCm
−χCm χCe + χCm
= c (30)
ge + gm −gm
−gm ge + gm
= g (31)
where {ir
w, V r
w} are combinations of the incident wire voltages, {Vw, iw} are the
total line quantities and {l, r, c, g} are the normalized inductance, resistance,
capacitance and conductance p.u. matrices. Using (23)–(31) in (21) and (22)
gives
−2ir
w = 2 + r · iw + l ·
∂iw
∂T
(32)
2V r
w = 2 + g · Vw + c ·
∂Vw
∂T
(33)
here the matrix 2 = 21 where 1 is the identity matrix. The bilinear transform
is
∂
∂T
= 2
1 − z−1
1 + z−1
(34)
where z−1
represents a delay of one time-step. The numerical algorithm is
obtained by application of (34) to (32) and (33). By defining the coefficient
matrices
2 + r + 2l = Tiw
−1
, − 2 + r − 2l = κiw (35)
2 + g + 2c = Tvw
−1
, − 2 + g − 2c = κvw (36)
and the accumulator vectors
−2ir
w + κiw · iw = Siw (37)
2ir
w + κvw · Vw = Svw (38)
leads to the algorithm for the calculation of the total line quantities
iw = Tiw · −2ir
w + z−1
Siw (39)
V = Tvw · 2V r
w + z−1
Svw (40)
For the two line case considered here, the reflected wire voltage pulses are
V r
A4 = VA − iA − V i
A5
V r
A5 = VA + iA − V i
A4
V r
B4 = VB − iB − V i
B5
V r
B5 = VB + iB − V i
B4 (41)
by analogy with the single wire model described in [1].
6

7. 3 Results
In this section the propagation of a short pulse along a hypothetical two-channel
carbon nanotube transmission-line is calculated. The model parameters were:
LAA = LBB = LK = R0/vF , CQ = 1/(R0vF ) and CE = 2πε0/ cosh−1
(h/a)
where LK is the kinetic inductance, CQ is the quantum capacitance, R0 =
12.91kΩ is the resistance quantum, vF = 0.8Mm/s is the Fermi speed, CE is
the electrostatic capacitance, h = 100nm is the height of the nanotube above
the ground plane and a = 1nm is the nanotube radius. The model capacitances
were obtained from the star to delta transformation, giving CAA = CBB =
CQ(CQ + CE)/(CE + 2CQ) and CAB = CBA = −C2
Q/(CE + 2CQ). The space-
step was ∆ℓ = 100nm, the time-step was ∆t ∼ 14fs and the model consisted
of 1000 cells. The result shown in Fig. 4 was obtained by sourcing the line
with a short Gaussian pulse at left-hand end of the line and allowing the pulse
to propagate for 2000∆t. As shown, the differential-mode (spin-mode) pulse
travels at the Fermi speed and the common-mode (charge-mode) pulse travels
at about 4.4vF . These results are in agreement with references [11, 12, 13]. No
losses have been included in this model because at present there is no consensus
in the literature as to their numerical value. However, loss may be included into
the resistance and conductance p.u. matrices.
-40
-20
0
20
40
60
80
100
0 10 20 30 40 50 60 70 80 90 100
NormalizedVoltage(mV/V)
Distance (um)
Common mode
Differential mode (A)
Differential mode (B)
Figure 4: Propagation of common-mode and differential-mode signals on a nano-
transmission line
4 Conclusions
In this paper we have described a TLM/Z-transform based time-domain model
for simulation of multiconductor transmission-lines. The model was used to
predict the propagation of a pulse along a carbon nanotube. In further work,
this model will be coupled to field solutions and extended to deal with multi-
conductor junctions.
7

8. References
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Modeling of Electromagnetic Wave Interaction with Thin-Wires using
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9