(ISTE) Pierre Saguet - Numerical Analysis in Electromagnetics The TLM Method-Wiley-ISTE (2012)
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Numerical Analysis in Electromagnetics
Numerical Analysis
in Electromagnetics
The TLM Method
Pierre Saguet
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
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© ISTE Ltd 2012
The rights of Pierre Saguet to be identified as the author of this work have been asserted by /him in
accordance with the Copyright, Designs and Patents Act 1988.
____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data
Saguet, Pierre.
Numerical analysis in electromagnetics : the TLM method / Pierre Saguet.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-391-3
1. Electromagnetism--Mathematical models. 2. Time-domain analysis. 3. Numerical analysis. 4.
Electrical engineering--Mathematics. I. Title.
TK454.4.E5S34 2012
537.01'515--dc23
2012008582
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-391-3
Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapter 1. Basis of the TLM Method: the 2D TLM
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1. Historical introduction. . . . . . . . . . . . . . . . . . . .
1.2. 2D simulation . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1. Parallel node . . . . . . . . . . . . . . . . . . . . . . .
1.2.2. Series node . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3. Simulation of inhomogeneous media with losses
1.2.4. Scattering matrices . . . . . . . . . . . . . . . . . . .
1.2.5. Boundary conditions . . . . . . . . . . . . . . . . . .
1.2.6. Dielectric interface passage conditions . . . . . . .
1.2.7. Dispersion of 2D nodes. . . . . . . . . . . . . . . . .
1.3. The TLM process . . . . . . . . . . . . . . . . . . . . . . .
1.3.1. Basic algorithm. . . . . . . . . . . . . . . . . . . . . .
1.3.2. Excitation . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3. Output signal processing . . . . . . . . . . . . . . . .
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1
5
5
8
9
11
14
15
17
22
22
23
24
Chapter 2. 3D Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.1. Historical development . . . . . . . . . . . . . .
2.1.1. Distributed nodes . . . . . . . . . . . . . . .
2.1.2. Asymmetrical condensed node (ACN) . .
2.1.3. The symmetrical condensed node (SCN)
2.1.4. Other types of nodes . . . . . . . . . . . . .
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29
29
30
31
33
vi
Numerical Analysis in Electromagnetics
2.2. The generalized condensed node . . . . . . . .
2.2.1. General description . . . . . . . . . . . . . .
2.2.2. Derivation of 3D TLM nodes . . . . . . .
2.2.3. Scattering matrices . . . . . . . . . . . . . .
2.3. Time step. . . . . . . . . . . . . . . . . . . . . . .
2.4. Dispersion of 3D nodes. . . . . . . . . . . . . .
2.4.1. Theoretical study in simple cases . . . . .
2.4.2. Case of inhomogeneous media. . . . . . .
2.5. Absorbing walls . . . . . . . . . . . . . . . . . .
2.5.1. Matched impedance . . . . . . . . . . . . .
2.5.2. Segmentation techniques . . . . . . . . . .
2.5.3. Perfectly matched layers . . . . . . . . . .
2.5.4. Optimization of the PML layer profile . .
2.5.5. Anisotropic and dispersive layers . . . . .
2.5.6. Conclusion . . . . . . . . . . . . . . . . . . .
2.6. Orthogonal curvilinear mesh . . . . . . . . . .
2.6.1. 3D TLM curvilinear cell. . . . . . . . . . .
2.6.2. The TLM algorithm . . . . . . . . . . . . .
2.6.3. Scattering matrices for curvilinear nodes
2.6.4. Stability conditions and the time step . .
2.6.5. Validation of the algorithm . . . . . . . . .
2.7. Non-Cartesian nodes . . . . . . . . . . . . . . .
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37
37
41
46
54
55
56
60
60
61
62
62
65
67
70
70
70
73
75
78
79
81
Chapter 3. Introduction of Discrete Elements and Thin
Wires in the TLM Method. . . . . . . . . . . . . . . . . . . . . . . .
85
3.1. Introduction of discrete elements. . . . . . . . . . . .
3.1.1. History of 2D TLM . . . . . . . . . . . . . . . . . .
3.1.2. 3D TLM . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Application example: modeling of a p-n diode.
3.2. Introduction of thin wires . . . . . . . . . . . . . . . .
3.2.1. Arbitrarily oriented thin wire model . . . . . . .
3.2.2. Validation of the arbitrarily oriented thin
wire model . . . . . . . . . . . . . . . . . . . . . . . . . . .
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85
85
89
100
105
106
....
119
Chapter 4. The TLM Method in Matrix Form and the Z
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Table of Contents
4.2. Matrix form of Maxwell’s equations. . . . . . . . .
4.3. Cubic mesh normalized Maxwell’s equations . . .
4.4. The propagation process . . . . . . . . . . . . . . . .
4.5. Wave-matter interaction. . . . . . . . . . . . . . . . .
4.6. The normalized parallelepipedic mesh Maxwell’s
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7. Application example: plasma modeling . . . . . . .
4.7.1. Theoretical model . . . . . . . . . . . . . . . . . .
4.7.2. Validation of the TLM simulation . . . . . . . .
4.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
vii
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124
125
127
130
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133
136
136
139
144
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Appendix A. Development of Maxwell’s Equations using
the Z Transform with a Variable Mesh . . . . . . . . . . . . . . . 147
Appendix B. Treatment of Plasma using the Z Transform
for the TLM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Introduction
There are a number of modeling methods that are suitable for
solving problems in electromagnetism and analyzing the behavior of
certain media. In order to apply these methods the type of problem
must be specified and the boundary conditions must be clearly
determined and defined. Numerical or analytical solutions are then
carried out.
Analytical solutions, which are already well established, were the
first to be applied. They enabled an efficient resolution of all problems
relating to the majority of electromagnetic wave guiding systems.
However, these analytical methods remain limited, since, in these
cases, it is only possible to analyze structures with simple geometries
and which, in the majority of cases, have a certain degree of
symmetry.
For more realistic modeling of geometries and more complex
materials (indeed, complexity leaves little room for any analytical
resolution), we have numerical methods, which have become an
important element in the analysis of the behavior of various industrial
products. They have progressed in parallel with technology and enable
electronic systems developers to have at their disposal all of the
necessary characteristics and data, which were difficult to obtain
through testing, in order to ensure the reliability of device operation
without any accompanying performance degradation.
x
Numerical Analysis in Electromagnetics
In the specific case of electromagnetism, there are various differing
numerical techniques, whose effectiveness depends on the problem
and on the desired results. These techniques can be classified
according to different criteria.
Classification based on the type of equation
Firstly, we can classify numerical methods based on equation type.
Indeed, most models under consideration lead to differential or
integral mathematical equations. If the problem deals with
electromagnetic wave propagation, the equations which describe its
behavior (such as Maxwell’s and wave equations) can be expressed
using two methods: differential or integral.
In order to solve these equations at any point in a finite space,
differential or integral methods are used to determine the values
required.
Classification based on the application domain
A second classification which may be taken into account is the
domain within which the equations to be solved are defined. In theory,
equations express the space and time variations in the scale of the
problem to be resolved (electromagnetic fields or potentials). Here we
are working in the time domain and the methods used are known as
“time-domain numerical methods”.
However, in the study of certain problems (notably in the area of
telecommunications), it is field cartography varying sinusoidally over
time or from a combination of multiple sinusoids, which is of interest.
In these cases, the electromagnetic characteristics of the majority of
materials can be expressed in a much simpler form, based on the
frequency of these sinusoidal signals. These equations are therefore
expressed frequentially and so the methods used to solve them are
known as “frequency-domain numerical methods”.
The advantage of frequency methods is that they give rise to
equations which are more flexible and easier to simplify.
Introduction
xi
Nevertheless, they are also limited, as they rely on signals always
being sinusoidal or based on a sinusoidal combination.
In all cases a frequency representation can be obtained from a
signal by using a Fourier transform of the time signal.
In this book, we are going to look at the TLM (transmission line
matrix) method, which is one of the “time-domain numerical
methods”. These methods are reputed for their significant reliance on
computer resources. However, they have the advantage of being
highly general. We will focus our attention on the TLM method
which, since the pioneering article on TLM by P.B. Johns and
R.L. Beurle in 1971, has been intensively studied and developed by
many researchers. It has, therefore, acquired a reputation for being a
powerful and effective tool by numerous teams and still benefits today
from significant theoretical developments. In particular, in recent
years, its ability to simulate various situations, including complex
materials, with excellent precision has been demonstrated.
This book consists of an introduction and four chapters.
Chapter 1 describes the basis of the TLM method in two
dimensions and enables different aspects of the method to be tackled,
as well as the errors resulting from space and time sampling.
Chapter 2 is dedicated to a 3D analysis of the method. It maps out
the main types of nodes currently used by pointing out their respective
advantages and disadvantages. This chapter also features the problem
of open structure simulation and the necessity of implementing
absorbing boundaries, including PMLs, which nowadays are used
universally.
Chapter 3 describes techniques which enable the simulation of
structures comprising passive and active discrete elements, as well as
thin metallic wires without the need to mesh these structures, which
would lead to memory problems. These techniques, as well as 3D
node and mesh flexibility, enable the simulation of a wide range of
problems where the properties of the surrounding medium are not
dependent on frequency and are therefore not dispersive.
xii
Numerical Analysis in Electromagnetics
Chapter 4 demonstrates how to simulate dispersive media using the
Z transform within the TLM method in matrix form. This rigorous and
unconditionally stable method makes the use of the TLM method
possible in virtually all cases.
Application examples are included in the last two chapters,
enabling us to draw conclusions regarding the performance of the
implemented techniques and, at the same time, to validate them.
Multi-scale problems which require the TLM method to be
combined with other methods will not be dealt with in this book in
spite of their undeniable interest. There are many papers dedicated to
this which would require collation into a single publication.
Chapter 1
Basis of the TLM Method:
the 2D TLM Method
1.1. Historical introduction
Historically, this method is based on the Huygens–Fresnel
principle, which states that each point of a wave front can be
considered to be an isotropic, spherical, a secondary source, and the
energy is isotropically distributed in all directions, as illustrated in
Figure 1.1.
Diffracting object
Figure 1.1. Huygens’ principle
Screen
2
Numerical Analysis in Electromagnetics
The 2D TLM method was formulated by Johns in 1971, based on
the principles outlined below.
Space is modeled using a Cartesian matrix of points or nodes
separated by a gap ∆l, where ∆t represents the time taken for a pulse to
travel from one node to another.
l
x
l
y
Figure 1.2. Transmission line network in the XY plane
Johns [JOH 71] applied this principle in order to solve
electromagnetic problems and create secondary sources by connecting
transmission lines, regularly spaced in a series or in parallel.
If a Vo=1V Dirac pulse is applied to any branch of this parallel
node, propagating to node A, we obtain:
3
2
A
4
1V
1
Basis of the TLM Method
3
If every branch of the node has the same characteristic impedance
Zc, then the impedance “seen” by the incident pulse will be equal to
Zc/3 and will be reflected with a coefficient:
Γ = 1/3 - 1 / 1/3+1 = -1/2
[1.1]
The voltages transmitted to the other branches will be equal to:
V = Vo 1+ Γ = +1/2
[1.2]
From this, we obtain a new distribution of energy over each branch
of the transmission line:
3
2
1/4
1/4
1/4
44
1/4
1
The node is the secondary source from the Huygens’ principle.
Simulation of propagation is therefore based on the existing
analogy between the propagation equations deduced from the
Kirchhoff circuit laws, which are the voltages and currents over the
branch of a node of the network, and Maxwell equations, which link
the components of the electrical and magnetic field.
The principle of the TLM method consists of applying one or more
Dirac voltage (or current) pulses at a given point of the structure (the
excitation point), then spreading it to the next node. From there it will
4
Numerical Analysis in Electromagnetics
be divided over the various branches of the node, based on a matrix
[S] called the scattering matrix.
The matrix [S] is determined from the equivalent electrical node
network. At each moment it serves as a link between the pulses
arriving at the node (incident pulses) and those which spread
(reflection pulses), based on the following relationship:
V ref S V inc
[1.3]
Each of the pulses created in this way in the four directions will
propagate towards adjacent nodes in a time Δt = Δl/c. Figure 1.3
shows the process for the first iterations. At the chosen output point, a
sequence of pulses representing the time response of the system is
obtained. Clearly, a simple Fourier transform enables results in the
frequency domain to be obtained.
1st iteration
Initial pulse
2nd iteration
Figure 1.3. Huygens’ principle within a transmission line network
Basis of the TLM Method
5
1.2. 2D simulation
In a TLM network, there are two types of connections for
transmission lines:
– nodes connected in parallel;
– nodes connected in series.
1.2.1. Parallel node
The parallel node consists of the parallel connection of two
transmission line sections of length l. An equivalent scheme
corresponds to this parallel connection (Figure 1.4).
Figure 1.4. Equivalent electrical network of the parallel node
We will assume, for the study which follows, that the characteristic
impedance of the transmission lines is equal to 1 (Z0 1).
From Figure 1.4, the variation in voltage Vz in the Ox direction,
using the first-order-limited Taylor formula, is expressed by:
ΔVz ΔI
Vz
x
[1.4]
6
Numerical Analysis in Electromagnetics
If Ix is the current in the Ox direction, we can then say:
ΔVz 2L
Δl I x
2 t
[1.5]
We also obtain:
Vz
I
L x
x
t
[1.6]
Using the same reasoning in the Oy direction gives us:
I y
Vz
L
y
t
[1.7]
The current conservation law at the nodes enables us to write:
Ix I y
V
2C z
x y
t
[1.8]
Therefore, from these three relationships linking currents and
voltages at this node, we see that:
Vz
I
L x
t
x
V
Iy
z
L
t
y
I
I
V
x y 2C z
t
x y
[1.9]
where L and C are the inductance and capacitance per unit length of
the transmission lines.
Basis of the TLM Method
7
We can deduce the following relationship from these equations:
2 Vz
x 2
2 Vz
y 2
2 LC
2 Vz
t 2
[1.10]
The electromagnetic wave propagation equation for a medium of
permittivity and permeability is given by:
2
x 2
2
y 2
2
t 2
[1.11]
Comparing equations [1.10] and [1.11] clearly shows that equation
[1.9] enables the simulation of wave propagation in a medium for
which the propagation velocity is:
V
1
1
2 LC
[1.12]
Depending on whether we are considering the propagation of a TE
wave or a TM wave, the Maxwell equations can be expressed as
follows.
For a TM wave:
H x
E z
y μ t
E z
H y
μ
t
x
H y H x
E
ε z
t
x
y
[1.13]
8
Numerical Analysis in Electromagnetics
For a TE wave:
E x
H z
y ε t
H z
E y
ε
t
x
E y E x
H
μ z
y
t
x
[1.14]
From two equations, [1.9] and either [1.13] or [1.14], the
equivalences between the voltages and currents of the node on one
hand, and the components of the electromagnetic field from the TM
and TE modes on the other, can be established.
These equivalences are given by:
TM: Vz E z
I x H y
Iy Hx
2C
L
[1.15]
TE: Vz H z
Ix E y
I y E x
2C
L
1.2.2. Series node
The series node can be represented by its equivalent node network,
as with the parallel node.
Figure 1.5. Equivalent electrical network of the series node
Basis of the TLM Method
9
Based on the approach used for the parallel node, transmission line
theory gives us:
I C Vy
x
t
Vx
I
C
y
t
Vx Vy 2 L I
x
t
y
[1.16]
where L and C still represent the lineic inductance and capacitance of
the transmission lines.
Comparing these equations with Maxwell’s equations for a TE and
TM wave gives us the following equivalences:
TM: I E z
Vx H x
Vy H y
2L
C
[1.17]
TE: I H z
Vx E y
Vy E x
2L
C
1.2.3. Simulation of inhomogeneous media with losses
A homogeneous medium is characterized by a relative permittivity
r and a relative permeability r. Simulation of a medium with a
relative permittivity r occurs through the introduction of an additional
capacity at the parallel node. For this, an open stub of length l/2 and
normalized characteristic admittance Ys is inserted (Figure 1.6b). This
stub brings about a capacity Cs at the node, given by:
Cs
0 Ys
2
[1.18]
10
Numerical Analysis in Electromagnetics
The total capacity of the junction “2 Ct” is the sum 2 C0 + Cs and
should simulate a medium of permittivity 0r, since 2 C0 is the
capacity for which a permittivity of 0 can be simulated.
Therefore we have:
2 C0 Cs 2 0
0 Ys
2
2 0 r
[1.19]
from which we obtain:
Ys 4 r 1
[1.20]
Ys is the characteristic admittance which simulates the relative
permittivity r.
In an analogous way, in order to simulate the relative permeability
r, a short-circuit stub of length l/2 and characteristic impedance Zs
is introduced into the series node, such that:
Zs 4 r 1
[1.21]
In order to simulate the losses in a medium of conductivity , a
matched (semi-infinite) stub of characteristic conductance G0 is
introduced.
G 0 σ Zair Δl
where Zair
0
: characteristic impedance of air.
0
[1.22]
Basis of the TLM Method
CO
4
5
CC
4
1
11
5
V
3
2
(a)
1
I
2
3
(b)
Figure 1.6. Series and parallel nodes with permittivity, permeability and
loss stubs: a) parallel node; b) series node
Simulation of losses is only possible if they are low enough for the
pulse distortion due to losses to be negligible.
1.2.4. Scattering matrices
The scattering matrix characterizes the node by linking the incident
pulses and those reflected over its branches, according to the
following relationship:
V ref S V inc
[1.23]
For a parallel node (Figure 1.6), the total admittance presented by
the node is:
Yt 4 Ys G 0
[1.24]
where Ys is the normalized characteristic admittance of the branches
with a permittivity r and G0 is the normalized conductance which
simulates losses due to a conductivity of .
The other four branches have unitary characteristic admittances.
12
Numerical Analysis in Electromagnetics
Let us consider a pulse of characteristic admittance Yi occurring
over a branch i; this will be reflected over the branch with a reflection
coefficient:
Γr
2Yi Yt
Yt
[1.25]
and will be transmitted to the other branches j, with characteristic
admittance Yj and a transmission coefficient:
T
2Yj
[1.26]
Yt
As the loss stub is semi-infinite, it does not return pulses. The
scattering matrix of the parallel node is therefore limited to the other
five branches and is given by:
S
2 Yt
2
1
2
Yt
2
2
2
2
2
2 Yt
2
2
2
2 Yt
2
2
2
2
2
2 Yt
2
2Ys
2Ys
2Ys
2Ys
2Ys Yt
[1.27]
The reflected pulses will therefore be given by:
V1
V1
V
V
2
2
V3 S V3
V4
V4
V5 ref
V5 inc
[1.28]
Basis of the TLM Method
13
Similarly, we calculate the scattering matrix [S] of a series node
(Figure 1.6). A pulse arriving from arm 1 of a node toward arm 4 of
the following node sees an impedance Ze = 3 + Zs, which corresponds
to a reflection coefficient:
Γ
2 Zs
4 Zs
[1.29]
where Zs is the normalized characteristic impedance of the stub with
permeability r.
This pulse will be transmitted over the other branches, with a
value:
1 Γ
2
4 Zs
[1.30]
In the case of a pulse coming from the series stub, the reflection
coefficient takes the value:
Γs
4 Zs
4 Zs
[1.31]
It will therefore be transmitted to the other branches with a value:
1 Γs
2Zs
4 Zs
[1.32]
However, note that the voltages at arms 2 and 3 should be taken,
with signs respecting the equivalences given above.
14
Numerical Analysis in Electromagnetics
For the series node, the scattering matrix is as follows:
S
2 Zs
2
1
2
4 Zs
2
2Zs
2
2 Zs
2
2
2
2
2
2 Zs
2
2
2
2 Zs
2Zs
2Zs
2Zs
2
2
2
2
4 Zs
[1.33]
At each iteration during simulation, all of the pulses arriving at a
node will be transmitted on all of the arms, based on the scattering
matrix of the considered node.
1.2.5. Boundary conditions
Of course, meshed space is not infinite. It can be enclosed by
metallic walls, magnetic walls (which are generally magnetic planes
of symmetry of the structure), or even lead to an open space to which
absorbing conditions should be applied. Sometimes, the space will be
enclosed by an ordinary wall which can easily be simulated if its
impedance is not dispersive.
The walls should be perpendicular to the transmission lines and be
simply simulated by applying an appropriate reflection coefficient to
the incident pulses to the wall. The equivalences established in [1.15]
and [1.17] enable the reflection coefficient value to be established.
If we simulate TM modes using parallel nodes, the voltage is
equivalent to an electrical field. Thus, over a perfect conductive wall,
the field is null and therefore the voltage should also be null, which
leads to a reflection coefficient which is equal to –1. A magnetic
wall will give a coefficient equal to +1, and, generally, a normalized
impedance z will lead to a reflection coefficient:
TM
z 1
z 1
Basis of the TLM Method
15
If we simulate TM modes with series nodes, the sign of will be
inverted.
Simulating the propagation of TE modes simply results in a sign
change for the reflection coefficient, and will therefore result in:
TE
z 1
z 1
The walls should be placed mid-way between two nodes, in order
to ensure synchronism of the reflected pulses on the wall with the
other pulses.
The problem of open media will be dealt with in detail in
Chapter 2.
1.2.6. Dielectric interface passage conditions
The addition of reactive stubs to 2D nodes enables the slowing
down of wave propagation over the TLM network. Furthermore the
passage conditions between two separate dielectric media must be
respected.
If the parallel node is used, the stub adds a capacity to the node,
reflecting a relative dielectric constant r. For TM modes, the
tangential component of the electrical field is represented by the
voltage Vz.. As the tangential field is continuous at the interface, the
pulse transmitted between media is also continuous. The propagation
constant, wave impedance and continuity are therefore perfectly
simulated and no further action is required. It should be emphasized
that the interface between the dielectrics must be placed at the midpoint between two nodes.
If series nodes are used to simulate inhomogeneous dielectric
media, then the situation is different. Simulation of TE nodes enables
field passage conditions to be respected. For these nodes, the stub
enables medium permeability, and not permittivity, to be increased.
This does not disrupt the propagation velocity (r and r intervene in
16
Numerical Analysis in Electromagnetics
the same way in the velocity calculation). Wave impedances in the
media, on the other hand, will not be correctly simulated.
Some reflection and transmission coefficients must be introduced
to the interface in order to correct this problem, as indicated in
Figure 1.7.
Medium 2
Medium 1
Ai
Node i+1
Node i
T12Ai
+
11Ai
22Ai+1
+
T21Ai+1
Ai+1
Figure 1.7. Application of reflection and transmission coefficients to the interface
Let Z1 be the characteristic impedance of medium 1, and Z2 that of
medium 2.
If r = Z1/Z2 , then for a transmission line crossing the interface:
11
1 r
;
1 r
22
r 1
;
1 r
T12
2
;
1 r
T21
2r
1 r
[1.34]
The problem will be identical if inhomogeneous magnetic media
are simulated using parallel nodes.
We will see that it is possible to overcome these problems by using
3D nodes, neutralizing one of the dimensions.
Basis of the TLM Method
17
1.2.7. Dispersion of 2D nodes
The propagation constant in the TLM network is dependent on
frequency and the network is therefore said to be dispersive. This
phenomenon due to discretization is present for all types of 2D and 3D
nodes and causes the velocity error in the TLM method.
1.2.7.1. Case of nodes without stubs
There is a perfect analogy between the field and the mesh
parameters, so long as the meshing is extremely fine compared to the
wavelength. However, the use of very small spatial step values leads
to prohibitive memory sizes and highly significant CPU times during
simulation. Moreover, if the cell size is increased and becomes of the
order of the wavelength, the TLM meshing can no longer be
considered to be a continuum; it must then be treated as an anisotropic
periodic structure. It is therefore important to evaluate the slow wave
properties of the mesh, in order to evaluate its limits.
Firstly, let us consider the propagation of a diagonal wave front at
45° to the mesh axes. The pulses along the branches have identical
amplitudes and phases in the two axial directions. Two of these
identical waves converging towards a network node still see a
matched load, whatever their wavelength. Consequently, the
propagation velocity in the mesh is independent of frequency and is
equal to 1
times the propagation velocity along the transmission
2
lines.
The axial propagation of a plane wave front is, however, dependent
on frequency. Indeed, a pulse propagating along an axis encounters all
l, two open-circuited stubs of length l/2 connected in parallel, thus
simulating the equivalent of a TEM wave.
We therefore have a periodic structure over which to calculate
current and voltage at the input of a cell, based on the current and
voltage at the input of the preceding cell. This is expressed by the
following equation:
18
Numerical Analysis in Electromagnetics
Vi cos / 2
I j sin / 2
i
j sin / 2
1
0 cos / 2
cos / 2 2 j tan / 2 1 j sin / 2
j sin / 2 Vi 1
cos / 2 I i 1
[1.35]
where 2 l / .
If the waves for the structure have a propagation constant
n = n + n, then:
Vi e n
I
i 0
0 Vi 1
e n Ii 1
[1.36]
Comparing these equations gives:
cosh n l cos tan sin
2
[1.37]
For relatively low frequencies, the previous equation reduces to:
sin
n l
2
2 sin
l
2
[1.38]
which can also be expressed as:
l /
1
n sin 2 sin l /
[1.39]
where and n are the phase constants over the transmission lines and
the network of lines, respectively. Figure 1.8 shows their relationship
based on the normalized l network step, relative to the wavelength
over the lines. The first cut-off frequency occurs for l/ = ¼. For
very low frequencies (i.e. very low values of l/), the propagation
Basis of the TLM Method
19
velocity for the network is, as expected, equal to c / 2 . For arbitrary
propagation directions for the network, the constant ratio is
somewhere between the axial and diagonal propagation.
It should be noted that for higher frequencies, the network once
more becomes propagative, which signifies that the parasitic “high
frequency” modes may be excited within the structure.
/n
Diagonal propagation
0.7
Arbitrary angle
Axial propagation
l/
0
0.1
0.25
Figure 1.8. Propagation constant dispersion in the 2D TLM without stubs
In order to estimate the propagation constant for an arbitrary angle,
a modification to equation [1.39] can be used, according to [SAG 85]:
d n l /
1
n sin d n 2 sin l /
[1.40]
where dn = Dn/D. Dn is the distance traveled by a wave between two
mesh nodes, while D is the minimum distance traveled in order to join
these two points along the network axes. We see that dn = 1 (axial
20
Numerical Analysis in Electromagnetics
propagation) and
dn
1
2
(diagonal propagation) provide the
expected results. Knowledge of these expressions enables the eventual
correction of the velocity error, if the direction of propagation of the
wave is known.
1.2.7.2. Case of parallel nodes with permittivity and loss stubs
This time, the nodes of the periodic structure include a
supplementary open circuit stub of length l/2 with normalized
characteristic admittance Ys and a semi-infinite stub with normalized
conductance G0. For an axial propagation, this leads to the equation:
Vi cos / 2
I j sin / 2
i
cos / 2
j sin / 2
1
0
j sin / 2
G j 2 Y tan / 2 1
cos / 2 0
s
j sin / 2 Vi 1
cos / 2 Ii 1
[1.41]
where G0 is the loss stub conductance and Ys is the permittivity stub
admittance 2 l / .
If the waves over the structure have a propagation constant
n = n + n, then:
Vi e n
I
i 0
0 Vi 1
e n Ii 1
[1.42]
By combining [1.43] and [1.44], for l and nl 1 , we obtain:
l /
n sin 1 2 1 Ys / 4 sin l /
[1.43]
Basis of the TLM Method
21
1
2
2
1 2 1 Ys / 4 sin l /
n
2 1 Ys / 4 cos l /
[1.44]
where:
G0
4l 1 Ys / 4
=
2
The first cut-off frequency occurs for:
1
1
l
sin 1
cut off
2 1 Ys / 4
[1.45]
For low frequencies, we obtain:
n 2 1 Ys / 4
[1.46]
where γ = ∝+jβ. It should be noted that when Ys increases, the useful
frequency band reduces. For diagonal propagation, the propagation
constant ratio is independent of frequency, as we have seen
previously. For propagation in an arbitrary direction the following
approximate formula can be used:
d n l /
n sin 1 d n 2 1 Ys / 4 sin l /
[1.47]
where: dn = Dn/D, Dn and D are defined as in section 1.2.7.1.
1.2.7.3. Case of series nodes with permeability stubs
This time, the nodes of the periodic structure include a
supplementary open circuit stub of length l/2 with normalized
characteristic admittance Zs and a semi-infinite stub with normalized
conductance R0. For an axial propagation, this leads to the equation:
22
Numerical Analysis in Electromagnetics
l /
n sin 1 2 1 Z s / 4 sin l /
[1.48]
2
1 2 1 Z s / 4 sin l /
n
2 1 Z s / 4 cos l /
1
2
[1.49]
where:
R0
4l 1 Z s / 4
=
2
The first cut-off frequency occurs for:
1
1
l
sin 1
cut off
2 1 Z s / 4
[1.50]
Note that the introduction of losses into the series node does not
influence the phase constant in the network.
1.3. The TLM process
1.3.1. Basic algorithm
All of the algorithms of the TLM method are derived from the
algorithm presented in Figure 1.9. The meshing and description of the
structure are performed before the iterative process, as well as the
determination of the pulse scattering matrix/matrices on the nodes.
Excitation can be uniquely applied to any node of the structure before
the iterations begin, but may also change over time and therefore be
applied during the process. This is stopped after N iterations, which
represents a simulation time equal to N t. A Fourier transform or
other signal processing then enables the frequency response of the
structure to be obtained.
Basis of the TLM Method
23
Data input:
meshing and description of the structure
Calculation of the scattering matrix
Input of excitation
Propagation
Reflection at boundaries
and metallic walls
N iterations
Scattering
Output: Fourier transform or other
signal processing
Figure 1.9. Basic algorithm of the TLM method
1.3.2. Excitation
Excitation of the TLM network is classically performed by
applying a Dirac pulse along one or more branches of one or more
nodes of the network. This procedure enables an infinite simulated
24
Numerical Analysis in Electromagnetics
frequency range to be obtained. However, working in the time
domain, all excitation combinations are possible; for example, an
infinite sequence of periodic pulses, whose amplitudes vary
sinusoidally, enables the excitation of a pure wave with a given
period. Indeed, it is possible to excite a range of given frequencies,
from an infinite number to a single pure frequency. An infinite
frequency range may pose problems, insofar as high frequency
parasitic modes are also excited within the structure. These modes are
not generally disruptive, except in the case of imperfect absorbing
boundaries, where these modes may be reflected and create
instabilities and divergence from the TLM process.
In order to avoid this problem, if a single frequency band is
necessary, it is better to use a Gaussian or a sinusoid modulated by a
Gaussian. For example, a typical time signal is:
f (t )
2 t t0
tw
t t
e 0
2
/ tw2
[1.51]
It is then very simple to adjust the desired frequency range.
1.3.3. Output signal processing
A series of pulses, spaced by a time t, representing the time
response of the structure to the excitation imposed, occurs for each
chosen output node. This is expressed in the form:
F (t ) Ak t k t
k 1
[1.52]
The frequency response is obtained using a simple Fourier
transform of the time response. As F(t) is a series of Dirac functions,
the Fourier transform becomes a sum for which the real and imaginary
parts are given by:
Basis of the TLM Method
l N
l
Re F Ak cos 2 k
k 1
l
l N
Im F Ak sin 2 k
k 1
25
[1.53]
In the above equations, F(l/) is the frequency response, Ak is the
pulse value read at the instant kt and N is the number of iterations
performed. This number N is finite, which implies a truncation of the
time signal. This truncation is equivalent to the application of a “gate”
function over the time response. This function, which is equal to 1
between instants t = 0 and t = Nt, is 0 between t = NT and infinity.
A resonance line at a given frequency will be affected by the Gibbs
phenomenon, i.e. by a response of sin(x)/x centered on this resonance.
This phenomenon may be extremely disruptive in the case where
multiple resonance frequencies existing in the analyzed structure are
very close to each other. The presence of secondary lobes of nonnegligible amplitudes may make reading the response difficult or even
impossible. It is therefore necessary to apply windows other than the
rectangular window to the time response in order to reduce the error
from truncation. Among the various windows which exist (triangular
window, Hamming’s window, Blackman’s window, etc.) Hann’s (or
Hanning’s) window provides quite suitable results [SAG 80]. We see
in Figure 1.10 that the secondary lobes are considerably attenuated by
applying this window. The time response must be multiplied by the
following expression:
1
k
f hann 1 cos
2
N
[1.54]
The longer the time duration of the chosen window, the narrower it
will be in the frequency domain. Thus, in taking an infinitely long (in
terms of time) window, then the one Dirac frequency limit is reached,
which is the neutral element of the convolution product. For an
infinitely long window, the “real” spectrum of the analyzed signal is
found, effectively corresponding to the TFD of a signal of infinite
duration.
26
Numerical Analysis in Electromagnetics
a) Rectangular window
b) Hann (Hanning) window
Figure 1.10. Spectral density in dB based on the sampling
frequency centered on a resonance line
In addition to the Fourier transform, it is of course possible to use
traditional signal processing which is appropriate to the problems
being handled. In particular, in all problems where specific
frequencies are enhanced, methods such as the Prony–Pisarenko
method [DUB 92] may provide an appreciable gain in computation
Basis of the TLM Method
27
time by significantly reducing the number of iterations necessary to
obtain a sufficiently precise spectrum. Essentially, the TLM method
upsamples the signal. It is therefore possible to apply error prediction
and estimation methods with much smaller samples than a Fourier
transform. Dubard notes a ratio between the number of samples
required equal to 9 between Prony–Pisarenko and the FFT (fast
Fourier transform) in order to obtain equivalent results for an antenna
problem.
Chapter 2
3D Nodes
2.1. Historical development
2.1.1. Distributed nodes
The nodes used in 2D [JOH 71, HOE 91] were combined into a
distributed node which would enable 3D simulation. This node, as
presented by Akhtarzad and Johns [AKH 74], is achieved by the
interconnection of three series nodes and three parallel nodes, which
thus form a cube with edges of length l/2 (Figure 2.1). The
equivalent electrical scheme of the distributed node is given by
[AKH 75]. The theory and the whole package of applications of these
schemes are brought together in a review paper proposed by Hoefer
[HOE 85]. The diverse components of the electromagnetic field are
thus available at the corners of the cube, at the parallel nodes for the
electrical field and at the series nodes for the magnetic field. The
connection of multiple nodes of this type enables the simulation of a
3D medium.
The topology of the distributed node is similar to that of the FDTD
for the Yee cell [YEE 66]. The advantage of the TLM approach is in
the fact that we are making use of three out of six field components at
each scattering point (the point where the transmission lines intersect
on the scheme) against just one for FDTD. However, the distributed
30
Numerical Analysis in Electromagnetics
node requires twice as many variables as the FDTD cell. Furthermore,
the major disadvantage of this node is the complexity of its numerical
scheme [JOH 87]; the fields calculated at the scattering points are
spatially separated and are therefore not instantly updated. This makes
arbitrary wall modeling difficult [HOE 89]. Finally, programming of
the variable meshing (i.e. having a dimension variation in space)
becomes laborious.
Figure 2.1. Distributed 3D node
2.1.2. Asymmetrical condensed node (ACN)
This node was proposed by P. Saguet and E. Pic [SAG 82]. It is
obtained from the original 3D node by suppressing the line sections
between the elementary series and parallel nodes (Figure 2.2a). This is
done through the transfer of the elementary cell from the original T
node to an elementary half-T cell. The interconnection between the
three parallel and three series nodes is then achieved at a point, hence
the name “node”. The asymmetrical condensed node in its simplest
form (Figure 2.2b) is made up of 12 arms. Treatment of
inhomogeneous and lossy media leads to a node containing 21 arms.
3D Nodes
31
Figure 2.2. Asymmetric condensed node (ACN)
The advantage of this node is its ability to carry out scattering
operations at a unique point in space. Thus the six field components
are calculated simultaneously at this point. The walls can be applied at
the center of the node, or more sensibly, at the mid-points between the
nodes (i.e. on the cube faces). However, with the image of the
distributed node, there remains an asymmetry; along the direction of
propagation, the first connection encountered is either parallel or
series. This implies that a wall seen in any direction has properties
which are appreciably different from those under another incidence,
especially at high frequencies. From this phase, development of 3D
TLM nodes goes beyond the classic 2D node. The first step in this
direction was proposed by Johns [JOH 86a, JOH 86b] with the
symmetrical condensed node (SCN).
2.1.3. The symmetrical condensed node (SCN)
The configuration of this node (without permeability, permittivity
and loss stubs) is shown in Figure 2.3. It preserves the advantages of a
condensed scheme, while overcoming the defects from the ACN.
On each arm of this node, two pulses propagate, corresponding to
wave cross-polarizations in the considered direction (Figure 2.4). The
scattering process is carried out at a single point, or “connection”. We
32
Numerical Analysis in Electromagnetics
place this term in quotation marks as the development of this node is
no longer based on an equivalent circuit of lines branching. In order to
calculate its scattering matrix, Johns used Maxwell’s equations and
the energy conservation law [JOH 87]. The node thus obtained has a
scattering matrix of size 12 × 12 [HOE 89] for a cubic cell and an
isotropic inhomogeneous medium.
Y
X
Z
V 12
V7
V4
V2
V3
V10
V6
V 11
V9
V8
V5 V1
Figure 2.3. The symmetrical condensed node
Figure 2.4. SCN and central connection
3D Nodes
33
For simulating variable, non-cubic meshing and other more general
media, only the stubs technique has been used with this node; six
reactive arms are added to the center of the cell. The scattering matrix
is thus a matrix of size 18 × 18. The incident and reflected voltages for
each input are linked by this scattering matrix [HOE 89].
Fields E and H can be calculated at the center of any cell, at each
iteration, from a linear combination of the incident voltages. For the
simulation of electric and magnetic losses of a medium, infinite
reactive arms can be added to the center of the cell, without changing
the size of the S matrix [NAY 90, GER 90]. Indeed, it is sufficient to
include these parameters in the calculation of matrix elements, since
the reflected voltages on each reactive arm never return. Details of
SCN theory and application can be found in [CHR 95], however we
will study them more specifically in the following sections.
2.1.4. Other types of nodes
2.1.4.1. HSCN and SSCN nodes
Memory storage (18 voltages) as well as the time step are the main
disadvantages of the SCN node. Indeed, if the meshing is not cubic,
the time step is proportional to the ratio of the smallest dimensions to
the largest. This constraint has the effect of significantly reducing the
time step for a variable meshing, for instance. As a result, the number
of iterations must be increased according to the same law. Two new
TLM nodes minimize these problems: the HSCN [SCA 90, BER 94]
(Hybrid SCN) and the SSCN [TRE 94] (Super SCN).
The basis for the improvements brought about by these two new
nodes is the following idea: it is not necessary for all transmission
lines linking these nodes to have the same characteristic impedance
(Z0). Using this possibility, it is possible to free either the inductive
reactive arms [SCA 90], or the capacitive reactive arms [BER 94].
This formulation defines the new hybrid node. It therefore enables a
reduction in memory (15 voltages), while enabling a time step of
around 1.7 times that of the SCN in more favorable cases. However, it
is necessary to deal with the interfaces between media, since line
34
Numerical Analysis in Electromagnetics
impedances are no longer identical. On the other hand, based on the
same principle, the SSCN node enables the reactive arms (12 voltages)
to be freed from the structure, as well as the time step to be, in certain
cases double that of the SCN [SCA 90, TRE 95a, TRE 95b]. For this
node, there are six different arm impedances rather than three
(HSCN). Conversely, the SSCN dispersion characteristics are not as
good as those of the SCN and HSCN [TRE 95b, BER 95] in media
other than a vacuum.
The properties of the SCN, HSCN and SSCN nodes can be
deduced from the properties of a general node (GSCN), which we will
study in section 2.2.
2.1.4.2. Alternating scheme (ATLM)
Note that in the best case (SSCN), the TLM algorithm requires up
to 12 voltages per node, which is due to the apparent redundancy in
the TLM method. We should add that this is without a doubt the
reason for the appearance of parasitic (purely numeric) modes
[RUS 95a], which are highly disruptive in the implementation of
absorbing boundaries [CHE 93]. The ATLM scheme proposes
overcoming this redundancy.
TLM
Node
Figure 2.5. The ATLM scheme: white cells indicate the cells calculated in nt; black
cells indicate those calculated in (n+1)t
3D Nodes
35
Imagine a set of TLM cells within a homogeneous medium
(Figure 2.5). In the white cells, we calculate the reflected voltages at
time (n+l/2)t from the incident voltages at (n-l/2)t. These reflected
voltages then become the incident voltages for the adjacent nodes. We
then calculate the reflected voltages of the black cells at (n+l/2)t, and
so on. This scheme was initially developed for the SSCN [RUS 95b].
It demands an alternative meshing philosophy. Indeed the boundaries
must out of necessity be at a distance of l (and not l/2) from the
center of the nodes, to insure synchronism. To that end, these walls
need to be positioned at the center of the TLM nodes. This technique
enables a 50% decrease in memory and makes the TLM method
particularly cost effective in terms of computer resources.
Furthermore, it enables parasitic modes to be eliminated [CHE 93]
without any loss of precision. It is also suitable for the SCN node
[CHE 93, KRU 96, BAD 96], with reactive arms of length l to
preserve synchronism. This latter scheme represents a memory saving
of 25% compared with the classic SCN. However, the ATLM scheme
imposes a significant number of constraints. In order to respect TLM
synchronism, we have indicated that with the ATLM model it is
advisable to position boundaries at the center of the nodes. The direct
impact of this constraint is that a new node needs to be created for
each situation. The consequence is that the trivial issue of TLM
meshing, which constitutes one of its forces and makes the adoption of
this scheme difficult in situations where there are a number of
different materials and walls, is suppressed. This assessment is equally
valid in the case of variable meshing, where the boundary between
areas of different density is situated at the center of the cells, which
implies the creation of particularly “exotic” cells.
2.1.4.3. Split step scheme
The final 3D TLM scheme presented could equally be described as
being alternating, even though it is quite different in nature [NAM 99].
This is an algorithm which aims to suppress the notion of a maximum
time step in time methods. Indeed, this parameter is crippling when
significant contrasts in dimensions exist within an analyzed structure.
The coexistence of meshing zones of different contrasts implies the
use of a very small time step and thus of an inversely proportional
number of iterations. This type of scheme enables freedom from this.
36
Numerical Analysis in Electromagnetics
The first studies in this area were conducted in FDTD under the
name ADI-FDTD (alternate direction implicit FDTD) [NAM 99,
ZHE 00]. Then, a scheme for TLM formalism was created: SS-TLM
[LEM 04], which relies upon an “exploded” derivation of Maxwell’s
equations. If, for instance, we depart from one of the projections of the
Maxwell-Ampère equations along the Ox axis of the Cartesian system:
E x
H z H y
ex E x
t
y
z
[2.1]
where is the permittivity of the medium and ex is the conductivity of
the medium along the Ox axis, we can split this into two as follows:
1 E x
H z
ex E x
2 t
y
[2.2]
H y
1 E x
ex E x
y
2 t
[2.3]
The two equations will be solved sequentially within an iterative
scheme. One TLM scheme arising from this type of equation results in
an implied scheme, which no longer has any upper time step limit,
other than the obvious Nyquist limit. However, the numerical error
generated by this scheme forces the time step to be limited. If this
does not present any advantage within a regular meshing
configuration, in more difficult situations (i.e. with high contrast
levels), the increase in calculation time is typically significant.
Furthermore, the implied formulation can no longer be expressed in
terms of voltage at the surfaces, but must be expressed in terms of
fields at the center of the cell. The required memory is no more than 6
components of the field per cell (against 18 voltages for the SCN).
Finally, this technique removes parasitic modes from the TLM
method.
This formulation appears promising. However, as it is so new, it
will not be developed in this book.
3D Nodes
37
2.2. The generalized condensed node [TRE 95b]
2.2.1. General description
2.2.1.1. Notations
In general, on a particular transmission line, a total voltage pulse is
defined as the sum of the incident and reflected pulses: V = Vi + Vr.
Similarly, the total current is given by I = (Vi – Vr)/Z, where Z is the
characteristic impedance of the line.
If the coordinate origin is situated at the center of the node, a
voltage propagating in direction i polarized along j will be denoted by
Vinj for the negative side of the node and Vipj for the positive side:
i,j x,y,z i j .
On open stubs, the incident voltage is denoted by Voi and on shortcircuit stubs by Vsi. The voltages on electric and magnetic loss
matched stubs will be denoted by Vei and Vmi respectively.
For each transmission line the capacities and inductances per unit
length in the i direction and polarized along j will be denoted by Cij
and Lij. Similarly, the characteristic impedances and admittances will
be Zij and Yij.
The total capacity and inductance of the stubs in the i direction will
be denoted by Coi and Lsi, the characteristic impedances and
admittances of these stubs are denoted by Zsi and Yoi respectively.
Although a general study is possible, here we are only interested in
so-called “balanced” nodes, i.e. where the impedances of the lines
from both sides of the node (in the same direction and for the same
polarity) are identical.
2.2.1.2. Constitutive relationships in generalized nodes
Each cell has dimensions x, y and z and contains a nondispersive anisotropic material, defined by its permittivity tensor
and its permeability tensor :
38
Numerical Analysis in Electromagnetics
x
0
0
y
0
0
x
0
y
0
0
rx
0
0 0 0
z
0
0
rx
0
0 0 0
0
z
0
ry
0
0
0
rz
0
[2.4]
ry
0
0
0
rz
[2.5]
The total capacity of the block is given by:
EdS
Q
C s
V
[2.6]
Edl
which gives, in the 3 directions:
Cty y
z x
y
[2.7]
Ctx x
y z
x
[2.8]
Ctz z
y x
z
[2.9]
Similarly, the total inductance of the block is given by:
HdS
s
L
I
Hdl
[2.10]
3D Nodes
39
which leads to:
Ltx x
y z
x
[2.11]
Lty y
z x
y
[2.12]
Ltz z
xy
z
[2.13]
These are the components of the total capacity Ct and total
inductance Lt vectors of the block of matter, modeled by the TLM
cell. This cell must therefore reproduce the corresponding capacity
and inductance in each direction.
For example, let us take the total capacity of the cell in the y
direction. This consists of the lineic capacities of the two polarized
transmission lines along y, of length x and z, as well as the capacity
of the open stub, which gives us:
Cty Cxy x Czy z Coy
[2.14]
Similarly, for the total inductance in the z direction, for example:
Ltz Lxy x Lyx y Lzs
[2.15]
Following the same process, six equations can be obtained which
can be compared with equations [2.7] to [2.9] and [2.11] to [2.13].
The following six equations are thus obtained:
C yx y C zx z Cox x
y z
x
[2.16]
40
Numerical Analysis in Electromagnetics
C zy z C xy x Coy y
x z
y
[2.17]
C xz x C yz y Coz z
y x
z
[2.18]
L yz y Lzy z Lxs x
y z
x
[2.19]
Lzx z Lxz x Lsy y
xz
y
[2.20]
Lxy x L yx y Lzs z
y x
z
[2.21]
These six equations form the basis for modeling the medium using
TLM networks including any 3D node.
There are therefore 18 parameters to be determined:
Six lineic capacities, six lineic inductances, three total capacities of
open stubs and three total inductances of short-circuit stubs. There
therefore remains 12 degrees of freedom for determining these
parameters.
In TLM schemes, time synchronism must be maintained for the
whole network. The pulses must arrive at the center of the node
simultaneously, following a propagation time t (time step).
Let us take, for example, the propagation velocity on a
transmission line directed along x and polarized along y. We
have: vxy
1
C xy Lxy
and additionally vxy
x
.
t
Time synchronism therefore imposes six new equations:
t x C xy Lxy
[2.22]
3D Nodes
41
t x Cxz Lxz
[2.23]
t y C yz Lyz
[2.24]
t y C yx Lyx
[2.25]
t z Czx Lzx
[2.26]
t z Czy Lzy
[2.27]
Taking these six constraints into account, there still remains six
degrees of freedom.
Additional constraints can therefore be applied which will lead to
different versions of the 3D TLM node.
Of course, modeling losses is always possible, as we have already
shown in Chapter 1 (see 1.2.3). Infinitely long stubs or stubs
terminated by a matched load are inserted. If we consider the electrical
and “magnetic” effective conductivities ek and mk in the k direction,
the loss elements in the 3D TLM node are defined by:
Gek ek
i j
k
Rmk mk
i j
k
[2.28]
[2.29]
2.2.2. Derivation of 3D TLM nodes
2.2.2.1. Nodes equipped with stubs: the SCN node
As stated in the previous section, we have six degrees of freedom
which enable the imposition of new constraints. Six additional
42
Numerical Analysis in Electromagnetics
constraints can easily be obtained by using transmission lines, all with
the same characteristic impedance Z0. This is the situation with the
traditional SCN node.
Therefore: Zij = Z0 (Yij = Y0= 1/Z0) and the system [2.16] to [2.21]
is reduced to:
2Y0
Yok
i j
k
k t
2
[2.30]
2Z 0
Z sk
i j
k
k t
2
[2.31]
i,j,k pertain to { x,y,z } and are not identical.
Therefore the only parameters to be determined are Y0k and Zsk.
We obtain:
i j
Yok 2Y0 rk
2
c t k
[2.32]
i j
Z sk 2 Z 0 rk
2
c
t
k
[2.33]
c 1 0 0 being the velocity of light.
In order for the process to be stable, all of the stubs must have
positive or null impedance or admittance and we must therefore have:
t rk
i j
2ck
[2.34]
t rk
i j
2 c k
[2.35]
i,j,k x,y,z
i j,k .
3D Nodes
43
If the network is cubic, then I = j = k = l and we then find:
tmax
l
.
2c
From equations [2.34] and [2.35] it follows that if the mesh step is
increased in one direction in order to decrease the required memory,
the maximum time step will decrease proportionally, which signifies
that the computing time will also increase. What is gained on one side
is lost on the other.
2.2.2.2. Hybrid nodes
2.2.2.2.1. Type I HSCN nodes
The previous disadvantage may be avoided if the SCN node
constraint (all of the characteristic impedances being identical) is
relaxed. This idea has been explored and implemented for the SCN
and described in [CHR 95, TRE 95b, BER 95] under the name HSCN
(hybrid symmetrical condensed node). Let us assume, for example,
that the transmission lines linking the nodes are sufficient to model the
inductance of the block. This being the case, the short-circuit stubs
become useless and Zsk is therefore null. The transmission lines used
to model the same magnetic component must have the same
characteristic impedance, and there are therefore three new conditions,
which are written as: Zij=Zji for i,j x,y,z i j .
By applying these conditions to equations [2.30] and [2.31], we
obtain:
ij
ij
Z ij Z ji k
Z 0 rk
2k t
2ck t
[2.36]
or:
Yij Y ji Y0
2ck t
rk ij
[2.37]
44
Numerical Analysis in Electromagnetics
Carrying these values forward into expression [2.32], the value for
the characteristic admittance of the open stub in the k direction is
obtained:
2 i j 4ct i
j
Yok Y0 rk
c t k
k ri j rj i
[2.38]
Equation [2.36] shows that if the simulated medium is
inhomogeneous in terms of permeability, or if the meshing varies in a
particular direction, the characteristic impedances of the transmission
lines in this direction will not be identical at the interface between two
nodes of different regions. Reflection and transmission coefficients
should be applied as indicated in 2D in Chapter 1 (Figure 1.7). We
note also that for a uniform meshing, there is no correction to be made
if the medium presents different permittivities but the same
permeability throughout.
2.2.2.2.2. Type II HSCN hybrid nodes
What we have just done for inductances can also be done for
capacities. If the transmission lines linking the nodes are sufficient to
model the total capacity of the simulated matter block, the open stubs
are no longer necessary and we then have: Yok = 0 and the
characteristic admittances of the lines should be such that:
Yik = Yjk i,j,k x,y,z i j,k . By inserting these values into expression
[2.30], then:
Zik Z jk Z 0
2 c k t
rk i j
[2.39]
These values are carried forward into expression [2.31], leading to
the following value for the characteristic impedances of short-circuit
stubs:
2 i j 4ct i
j
Z sk Z 0 rk
k ri j rj i
ct k
[2.40]
3D Nodes
45
Equation [2.39] shows that if the simulated medium is
inhomogeneous in terms of permeability, or if the meshing is variable
in a particular direction, then the characteristic impedances of the
transmission lines in this direction will not be identical at the interface
between two nodes of different regions. Reflected and transmission
coefficients should be applied as indicated in 2D in Chapter 1
(Figure 1.7). We note also that for a uniform meshing, there is no
correction to be made if the medium presents different permeabilities
but the same permittivity throughout.
The choice between the two types of nodes could be dictated by
these considerations.
2.2.2.2.3. Maximum time step of HSCN nodes
The maximum time step for the HSCN node can be determined in
the same way as for the SCN node, by enforcing a positive or null
value at the stubs.
For example, for the type I node, the following condition should be
respected:
t
2 rk
1
2c 1 i 2 j 2
ri
rj
where i,j,k x,y,z
[2.41]
i j,k
Following the values for i and j, the maximum time step will be
such that:
l
l 2
tmax <
2c
2c
where l is the smallest of the dimensions.
[2.42]
46
Numerical Analysis in Electromagnetics
We see that the value for the time step in hybrid nodes is greater
than for the SCN node and is not strictly dependent on the smallest
dimension of the node.
2.2.3. Scattering matrices
2.2.3.1. Line scattering
The scattering of incident pulses on a node constitutes the core of
the TLM algorithm. In order to establish this scattering matrix, let us
consider a polarized voltage along the y axis propagating along the x
axis. Generally, the voltages of the two sides of the node are different,
i.e. Vxny differs from Vxpy. However, the conservation of charge
enables the definition for the total equivalent voltage Vy at the centre
of the node, as the average of the voltages on both sides of the node
(in the case studied where the characteristic impedances of the lines
from the two sides of the node are identical). Therefore:
Vy
Vxny Vxpy
[2.43]
2
Similarly, a total equivalent current Iz can be defined and the
magnetic flux continuity across the node dictates that:
Iz
I xpy I xny
[2.44]
2
By introducing incident and reflected voltages on the node, we
obtain:
Vy
1 i
1 i
r
r
Vxny Vxny
Vxpy
Vxpy
2
2
Iz
1
1
i
r
i
r
Vxny
Vxny
Vxpy
Vxpy
2Z xy
2Z xy
[2.45]
[2.46]
3D Nodes
47
From these two expressions, we deduce:
r
i
Vxny
Vy I z Z xy Vxpy
[2.47]
r
i
Vxpy
Vy I z Z xy Vxny
[2.48]
The reflected pulses for the other directions and polarizations will
be obtained from equations of the same type.
In order to determine the scattering matrix, the values of Vy and Iz,
based on the incident voltages on the node, must be known.
2.2.3.2. Scattering into the stubs
The total voltage on the open and matched stubs is determined by
the component of the corresponding electrical field. Simply, therefore:
r
i
Voy
Vy Voy
[2.49]
Veyr Vy
[2.50]
there is no incident voltage on the electrical loss stub.
In the same way, for the short-circuit and the magnetic (matched)
loss stubs:
Vszr Vszi I z Z sz
[2.51]
r
Vmz
I z Rmz
[2.52]
All of these equations will enable us to determine the value of the
total equivalent voltage as well as the equivalent total current.
2.2.3.3. Total equivalent voltage
The total equivalent voltage Vy is obtained from the conservation
of charge relative to the component Ey of the electrical field:
48
Numerical Analysis in Electromagnetics
Qy Qn, y 0
[2.53]
I n, y 0
[2.54]
n
or:
n
where In,y are the currents in all of the polarized lines along y, which
gives us:
i
r
Vzny
Yzy Vzny
Yzy Vzpyi Vzpyr
i
r
Voy
Yoy Voy
GeyVeyr 0
i
r
i
r
Vxny
Yxy Vxpy
Vxpy
Yxy Vxny
[2.55]
A similar equation can be written for the other two polarizations, x
and z.
By using equation [2.43], the reflected pulses on the polarized lines
along y and propagating along z can be expressed as:
r
r
i
i
Vxny
Vxpy
2Vy Vxny
Vxpy
[2.56]
By symmetry, for the pulses directed along z, we obtain:
r
r
i
i
Vzny
Vzpy
2V y Vzny
Vzpy
[2.57]
Inserting equations [2.56] and [2.57] into expression [2.55], the
value for the polarized equivalent voltage along y, uniquely based on
the incident voltages on the node, can be obtained:
Vy 2
i
i
i
i
i
Yxy Vxny
Vxpy
Yzy Vzny
Vzpy
YoyVoy
2Yxy 2Yzy Yoy Gey
[2.58]
3D Nodes
49
The total voltages for the other two polarizations are obtained in
the same way.
2.2.3.4. Equivalent total current
The total equivalent current Iz can be calculated from the
conservation of magnetic flux for transmission lines coupled with the
magnetic field component Hz. For example:
z n, z 0
n
[2.59]
which is equivalent to:
Vn, z 0
n
[2.60]
By using an identical approach to that used for the total equivalent
voltage, we reach a total equivalent current value which is only
dependent on incident voltages on the node:
Iz
i
i
i
i
V ynx
V ypx
Vxpy
Vxny
Vszi
2
2 Z xy 2 Z yx Z sz Rmz
[2.61]
The total currents Ix and Iy are obtained in the same way.
2.2.3.5. Pulse scattering equations
From equations [2.47], [2.48] and other similar equations, it is now
possible to obtain all of the expressions which give reflected pulses,
and to therefore deduce the scattering matrix. This can be summarized
by:
r
i
Vinj
V j I k Zij Vipj
[2.62]
r
i
Vipj
V j I k Zij Vinj
[2.63]
The + sign is applied for (i,j,k) ϵ {(x,y,z), (y,z,x), (z,x,y)} while the
– sign is applied for (i,j,k) ϵ {(x,z,y), (y,x,z), (z,y,x)}.
50
Numerical Analysis in Electromagnetics
The reflected voltages on the stubs are given by:
Voir Vi Voii
[2.64]
Veir Vi
[2.65]
Vsir Vsii Ii Z si
[2.66]
r
Vmi
Ii Rmi
[2.67]
where i ∈ {x,y,z}.
2.2.3.6. Generalized condensed node scattering matrix
By using equations to [2.67], the S scattering matrix can
now be constructed from the generalized condensed node.
This matrix has 18 columns and 24 rows. The 18 columns
correspond to the 12 transmission lines, 3 permittivity stubs
(13-14-15) and 3 permeability stubs (16-17-18). The numbering of the
transmission lines from 1 to 12 corresponds to the historical scheme of
the SCN node (Figure 2.3).
To obtain the 24 rows, the three electrical loss stubs and the three
magnetic stubs without incident pulses (matched stubs) are added.
The elements of the scattering matrix are given by:
aij Q j bij dij
bij Q j Cˆ kj
cij Q j bij dij 1
ij
dij Pk L
f k 2 1 Pk U k
eij bkj
g j 2 1 Q j W j
iij dij
h j g j 1
jk 1 f k
kij eij
lj gj
mk 2U k
nk mk
[2.68]
3D Nodes
51
where:
kj
C
ij
L
Zij
[2.69]
Zij Z kj
Z ij
[2.70]
Z ij Z ji
Yoj Gej
Q j 1
2 Yij Ykj
1
Z Rmk
Pk 1 sk
2 Zij Z ji
Wj
Uk
[2.71]
1
Gej
Yij Ykj Yoj Gej
2
2
Rmk
Zij Z ji Z R
sk mk
[2.72]
[2.73]
[2.74]
This general matrix will enable us to deal with problems of
dispersion in later chapters. From this the classic SCN node scattering
matrix can be obtained by enforcing that all characteristic impedances
of the transmission lines are identical and equal to Z0.
If, on the other hand, the medium has no losses, then rows 19 to 24
disappear, whereas if there are no permittivity or permeability stubs,
then rows and columns 13 to 18 no longer exist. Figure 2.6, for
example, gives the scattering matrix of the cubic SCN node for
52
Numerical Analysis in Electromagnetics
propagation in air. Of course the original matrix provided by Johns
[JOH 87] is derived.
The scattering matrix of the type I HSCN node is obtained by
imposing that the short-circuit stubs have null impedance, whereas the
type II HSCN node matrix will be obtained by canceling all the open
stub admittances.
For example, the scattering matrix of the type I cubic HSCN node
for an isotropic medium with relative permittivity r and permeability
µ0 is given by Figure 2.8, where:
a = (1 – r )/2 r ; b = 1/2 r ; c = a ; d = ½ ; e = b ; g = 2 (r-1)/ r ;
h = (r -2) /
1
2
3
4
5
6
7
8
9
10
11
12
1
0
1/ 2
1/ 2
0
0
0
0
0
1/ 2
0
-1/2
0
2
1/ 2
0
0
0
0
1/ 2
0
0
0
-1/2
0
1/ 2
3
1/ 2
0
0
1/ 2
0
0
0
1/ 2
0
0
0
-1/2
4
0
0
1/ 2
0
1/ 2
0
-1/2
0
0
0
1/ 2
0
5
0
0
0
1/ 2
0
1/ 2
0
-1/2
0
1/ 2
0
0
6
0
1/ 2
0
0
1/ 2
0
1/ 2
0
-1/2
0
0
0
7
0
0
0
-1/2
0
1/ 2
0
1/ 2
0
1/ 2
0
0
8
0
0
1/ 2
1/ 2
-1/2
0
1/ 2
0
0
0
0
0
9
1/ 2
0
0
0
0
-1/2
0
0
0
1/ 2
0
1/ 2
10
0
-1/2
0
0
1/ 2
0
1/ 2
0
1/ 2
0
0
0
11
-1/2
0
0
1/ 2
0
0
0
1/ 2
0
0
0
1/ 2
12
0
1/ 2
-1/2
0
0
0
0
0
1/ 2
0
1/ 2
0
Figure 2.6. Scattering matrix for the cubic SCN node in air
4
5
0
byz
dzy
ayz
azy
0
0
0
0
-dzx
cyx
eyx
0
0
0
0
fz
kyx
0
0
0
0
mz
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
-my
0
0
0
kzx
0
0
0
0
0
ezx
byx
0
-dxz
0
czx
bzx
0
bzy
0
0
0
0
mx
0
kzy
0
0
0
fx
0
ezy
0
0
bzx
0
0
0
-mx
kyz
0
0
0
0
-fx
eyz
0
0
0
0
bxz
0
my
0
kxz
0
0
0
fy
0
exz
0
0
0
0
cxz
7
0
0
mx
kyz
0
0
0
0
fx
eyz
0
0
0
0
bxz
0
dzy
axz
bxz
cyz
-d z y
0
0
0
8
0
0
-mx
0
kzy
0
0
0
-fx
0
ezy
0
0
bzx
0
0
azy
dxz
0
-dyz
czy
bxy
0
0
9
0
my
0
0
0
kzx
0
fy
0
0
0
ezx
byx
0
dxz
azx
0
0
-dxz
0
0
0
czx
byx
10
0
-my
0
kxz
0
0
0
-fy
0
exz
0
0
0
0
axz
dzx
0
bxz
cxz
byz
0
0
-d z x
0
11
mz
0
0
0
kxy
0
fz
0
0
0
exy
0
dyx
azx
0
0
bzy
0
0
0
bzy
cxy
0
-dyx
12
-mz
0
0
0
0
kyx
-fz
0
0
0
0
eyx
ayx
dzx
0
bzx
0
0
0
0
0
-d x y
bzx
cyx
0
0
0
0
0
lx
0
0
0
0
0
hx
gz
0
0
gx
0
0
0
0
0
0
gx
gx
13
0
0
0
0
ly
0
0
0
0
0
hy
0
0
gy
0
0
g
0
0
0
g
g
0
0
y
y
y
14
15
0
0
0
lz
0
0
0
0
0
hz
0
0
0
0
gz
0
0
gz
gz
gz
0
0
0
0
Figure 2.7. Scattering matrix of the generalized condensed node
-mz
0
0
0
kxy
0
-fz
-fy
0
0
exy
0
-dyx
czx
0
-dzx
0
-dzy
czy
0
bxz
cxz
0
axz
bxz
0
-dxz
0
0
0
dxz
0
0
dyz
0
bzy
bxy
dzx
0
0
0
0
axy
azx
0
6
0
0
0
dxy
0
bzx
3
dyx
2
byx
1
ayx
10
9
8
7
6
5
4
3
2
1
16
0
0
nx
0
0
0
0
0
jx
0
0
0
0
0
0
0
-izy
iyz
0
-iyz
izy
0
0
0
17
0
ny
0
0
0
0
0
jy
0
0
0
0
0
0
-ixz
izx
0
0
ixz
0
0
0
-izx
0
18
nz
0
0
0
0
0
jz
0
0
0
0
0
-iyx
izx
0
0
0
0
0
0
0
-ixy
0
iyx
3D Nodes
53
54
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Numerical Analysis in Electromagnetics
1
a
b
d
0
0
0
0
0
b
0
-d
c
e
0
0
2
b
a
0
0
0
d
0
0
c
-d
0
b
e
0
0
3
d
0
a
b
0
0
0
b
0
0
c
-d
0
e
0
4
0
0
b
a
d
0
-d
c
0
0
b
0
0
e
0
5
0
0
0
d
a
b
c
-d
0
b
0
0
0
0
e
6
0
d
0
0
b
a
b
0
-d
c
0
0
0
0
e
7
0
0
0
-d
c
b
a
d
0
b
0
0
0
0
e
8
0
0
b
a
-d
0
d
a
0
0
b
0
0
e
0
9
b
c
0
0
0
-d
0
0
a
d
0
b
e
0
0
10
0
-d
0
0
b
c
b
0
d
a
0
0
0
0
e
11
-d
0
c
b
0
0
0
b
0
0
a
d
0
0
0
12
c
b
-d
0
0
0
0
0
b
0
d
a
e
0
0
13
g
g
0
0
0
0
0
0
g
0
0
g
h
0
0
14
0
0
g
g
0
0
0
g
0
0
g
0
0
h
0
15
0
0
0
0
g
g
g
0
0
g
0
0
0
0
h
Figure 2.8. Scattering matrix of the type I cubic HSCN node in a medium
2.3. Time step
The time step must be maximum in order to limit the number of
iterations. The algorithm remains stable if all of the TLM network
impedances and admittances are positive or null. This condition
determines the maximum step that can be used. This step is calculated
for each block of space with given properties (basic mesh dimension,
permittivity, etc.). The smallest of the maximum steps obtained is
therefore the upper limit of t. For the SCN node, the equations
therefore give:
t rk
i j
2ck
[2.75]
t rk
i j
2 c k
[2.76]
3D Nodes
55
whereas for the type I HSCN node, equation [2.41] would give us:
t
2 rk
1
2c 1 i 2 j 2
ri
rj
[2.77]
Figure 2.9. Evolution of the maximum time step in
a vacuum with varying cell dimensions
The maximum time step is highly dependent on the ratio of the cell
dimensions, as we see in Figure 2.9, which gives the maximum steps
for the SCN, HSCN and SSCN nodes. For a cubic cell within the
vacuum, the three node types give the same time step t0, but the SCN
node becomes poor from this point of view compared to the HSCN
and SSCN (the best in this case).
2.4. Dispersion of 3D nodes
We have already studied the dispersion (or velocity error) of 2D
nodes. In this section, we will look at how regular 3D nodes behave.
The dispersion relation is an implicit function of the propagation
constant in the three network directions, the frequency and the
constitutive parameters of the media. The rigorous procedure for
56
Numerical Analysis in Electromagnetics
determining the dispersion relation in a homogeneous medium for
generalized 3D nodes is given by different authors, but it currently
seems impossible to study an inhomogeneous structure analytically.
The comparison between the different types of nodes will, in this case,
be given by using these different nodes to simulate a specific
structure.
2.4.1. Theoretical study in simple cases [TRE 95b]
The dispersion relation can be written in matrix form according to
[KRU 93]:
det PS e j I 0
[2.78]
NOTE.– For the rest of this book, matrices and vectors will be
indicated in bold.
Θ represents the phase variation along the transmission lines which
make up the network and is defined by t , I is the unit matrix, S
the scattering matrix of the considered node and P is a matrix
connecting two nodes. This connection matrix stems from the Floquet
theorem which links voltage pulses on a node to the voltage pulses on
neighboring nodes. It involves components kx, ky and kz of the wave
vector k, which enables the phase variations in the three directions to
be described. The elements of the P matrix are null, apart from:
P3,11 P6,10 e jk x x
P1,12 P5,7 e
jk y y
P2,9 P4,8 e jk z z
P10,6 P11,3 e jk x x
P12,1 P7,5 e
jk y y
P9,2 P8,4 e jk z z
P13,13 P14,14 P15,15 1
P16,16 P17,17 P18,18 1
[2.79]
3D Nodes
57
The notations used correspond to those used in Figure 2.3, which
represented the SCN node. Of course, rows and columns 13 to 15
and/or 16 to 18 disappear if the corresponding stubs are not present in
the considered node. Relation [2.78] is based on the time step in the
mesh and the properties of the medium (through the S matrix).
Therefore, for nodes without the same scattering matrix, the dispersion
relations are different and the properties of the nodes are not the same.
The solution to relation [2.78] is an eigenvalue problem which may
be complex. Here we will settle for providing some results, but
demonstrations of these solutions can be found in [TRE 95b].
2.4.1.1. Dispersion relation for nodes without stubs
The solution to equation [2.78] in the case of a 12 arm node leads
to the following expression (uniform mesh, unit length l in the three
directions):
4 cos 2 t cos k x l cos k y l cos k y l cos k z l
cos k z l cos k x l 1
[2.80]
2.4.1.2. Analysis of the solution
The dispersion relation for Maxwell’s equations within an
isotropic, non-dispersive medium is given by:
k 2 2
[2.81]
where the amplitude of the propagation constant k is:
k k x2 k y2 k z2
[2.82]
If we use the Taylor expansion of the cosine for small arguments
cos(x) ≈ 1-x2/2, then equation [2.80] is reduced to:
2
2t
2
2
2
kx k y kz
l
[2.83]
which actually corresponds to equation [2.81], with a propagation
velocity equal to l/(2t).
58
Numerical Analysis in Electromagnetics
However, the dispersion relation [2.80] does have other solutions
which do not correspond to the dispersion relation for Maxwell’s
equations. Indeed relation [2.80] only makes use of the products of
cosines, such as:
[2.84]
cos( x) cos( y ) cos( x ) cos( y )
The result of this is that for every low frequency solution ω of
relation [2.80] there is a corresponding high frequency solution
. Therefore, for all LF solutions (0≤ω∆t≤π/2) resulting from
t
[2.80]:
1 cos k x l cos k y l cos k y l cos k z l
2 cos k l cos k l 1
z
x
t arccos
[2.85]
there is an HF solution (π/2≤ω∆t≤π) given by:
1 cos k x l cos k y l cos k y l cos k z l
2 cos k z l cos k x l 1
t arccos
[2.86]
[2.85] is a physical propagation solution which corresponds to
Maxwell’s equations, whereas [2.86] is a parasitic solution which is,
however, easy to eliminate through a low-pass time filter.
Unfortunately, it is not possible, for high values of k, to filter parasitic
mode solutions from relation [2.85]. However, these high spatial
frequency values are not easily excited, although they may
nevertheless be present. The various solutions are presented in
Figure 2.10, where the dispersion diagram is given for direction
[1,1,1].
3D Nodes
High frequency
Parasitic mode
Physical model
Parasitic mode
Low frequency
Figure 2.10. Dispersion diagram of the node
without a stub in direction [1,1,1], [PEN 97]
Figure 2.11. Relative velocity error of the HSCN node
for an axial propagation [CHU 04]
59
60
Numerical Analysis in Electromagnetics
Analytical solutions for SCN nodes with stubs and for HSCN
nodes can be obtained in the same way [TRE 95b]. However, these
solutions are highly complex and it is undoubtedly better to solve
dispersion relation [2.78] using numerical methods. Additionally,
analytical solutions are only valid for homogeneous media, which is
quite restrictive.
Figure 2.11 gives an overview of the velocity error due to
dispersion for the HSCN node, for propagation in an axial direction.
When the relative permittivity is equal to 1, there is no dispersion of
the HSCN (or SCN) node. This dispersion increases significantly
when permittivity increases.
2.4.2. Case of inhomogeneous media
Le Maguer [LEM 98] carried out tests on partially filled resonant
dielectric cavities, with the aim of studying the dispersion of various
nodes by determining the resonance frequency of these cavities. The
TLM has been successively used with SCN (with stubs), HSCN and
SSCN nodes. Several meshes have also been implanted.
The results obtained have shown that, in every case studied, the
HSCN node was the best performer, followed by the SCN node, while
the SSCN node was the most dispersive. Other authors have reached
identical conclusions [PEN 97] [GER 96].
From this point of view, the HSCN node therefore possesses a
certain advantage. Let us not forget, however, that the SSCN node
clearly requires less memory. Sometimes, the computer resources
guide the choice of one node over another.
2.5. Absorbing walls
During the use of volume numerical methods, it is essential to be
able to restrict the computation domain. Indeed, when the structure
studied is of a significant size (several times the wavelength of
interest), the computation volume may restrict, or even prevent the
study of structures, as a consequence of prohibitive required computer
3D Nodes
61
resources. The problem is generally crucial when the computation
domain is open over free space, or when a hyperfrequency circuit
(infinite guide) needs to be matched. The issue is much less simple,
working in the time domain, when it becomes necessary to be able to
terminate the computation domain for a very wide frequency band. In
this case, absorbing boundaries must be inserted into codes at the
limits of the computation domain, which, in the frequency band being
studied, must present a very low reflection, whatever the incidence of
the wave.
Historically, multiple absorbing boundaries have been used, the
simplest of these being matched impedance.
2.5.1. Matched impedance
The fact that all of the tangential fields at the faces of the TLM cell
are being used enables an impedance condition to be imposed. This is
inserted into the TLM algorithm using a reflection coefficient, which
links the reflected and incident voltages of the node placed against the
boundary. The chosen impedance is the characteristic impedance of
the medium. This boundary is then perfect under normal incidence.
However, for arbitrary incidences, their performances degrade
significantly. Indeed, for a wave arriving at the wall with incidence
angle θ, the reflection coefficient is:
R( ) ( 1 cos ( )) / (1 cos ( ))
[2.87]
The quality of the reflection coefficient is dependent on the chosen
value of the impedance, and the layer will only be perfect for a given
incidence and therefore for a given frequency.
This type of boundary is principally used to terminate the TLM
mesh in simulations where the boundary conditions are of little
importance (such as the absence of radiation for the structures).
62
Numerical Analysis in Electromagnetics
2.5.2. Segmentation techniques
This technique consists of dividing the computation domain into
sub-domains, which are handled separately, and then reassembled.
This method is particularly attractive for simulations where a section
of the computation volume remains unchanged. This technique was
first developed by Kron [KRO 63] and then intensively used with
frequency methods. It was later adapted to a time domain method
(TLM) by Johns [JOH 81] in 1981. This is a natural adaptation, when
the tangential field components (or equivalently, the voltages) are
available to the interfaces. However, as the pulse response of the subdomains (which may be interpreted as a numerical Green function)
must be calculated and stored, the computer resources quickly
constitute a limit. The pulse response is also known as the Johns
matrix. More technically, this matrix enables the simulation of a subregion. It enables incident voltages to be obtained in the domain of
interest through the use of a spatiotemporal convolution, calculated
from the series of reflected voltages at the previous incidences.
2.5.3. Perfectly matched layers
Perfectly matched layers (PML) are layers introduced by Bérenger
[BÉR 94] in order to simulate infinite space with the FDTD (finitedifference time-domain) method. PML layers were implemented into
the TLM method for the first time by using an interface between the
TLM and FDTD meshes (Figure 2.12), called a non-unified (hybrid)
algorithm.
The results obtained are of very good quality (-50 dB reflection for
the TE10 mode in a rectangular guide). However, problems due to field
interpolation may occur. Indeed, if the zone where coupling takes
place has a very high field gradient, then it causes an error in
interpolations, which is no longer negligible with regard to the
reflection performance demanded. These problems have prompted
authors to turn to a unified algorithm. A 2D TLM algorithm has been
developed, and a new SCN node has been proposed [PEN 97] so that
Bérenger’s PML layers can be implemented into the 3D TLM method.
3D Nodes
TLM volume
63
FDTD volume
Figure 2.12. Interface between the TLM mesh and the FDTD mesh
2.5.3.1. PML Layer theory
The PML technique consists of enclosing the structure with
matched layers. Within these layers, the components of the field are
divided into two sub-components, as follows:
[2.88]
Ei E ji Eik
where (i, j, k) ∈ {(x, y, z), (y, z, x), (z, x, y)}.
The same expression is valid for magnetic field components.
Bérenger considers the form of Maxwell’s equations which
describes the TE mode in PML layers as follows:
i 0
Eij
t
ej Eij
1 H k
j j
[2.89]
64
Numerical Analysis in Electromagnetics
i 0
Eik
1 H j
ek Eik
t
k k
[2.90]
where:
– σei is the electrical conductivity in the Oi direction;
– αi > 1, is the doping of evanescent waves in the Oi direction.
The same expressions are valid for magnetic field components.
If the structure is terminated by a PML layer in the Oi direction the
following relation (impedance matching) must be verified:
ei
mi
i 0 i 0
[2.91]
where:
– μi is the relative permeability of the medium;
– ei is the electrical conductivity in the Oi direction;
– σmi is the magnetic conductivity in the Oi direction.
The simulation results show that numerical reflections appear
between two media with different conductivities. These reflections are
proportional to the ratio of the conductivities. In order to overcome
this, the PML layer is enlarged and a conductivity profile is imposed
on it. The same treatment is applied to the αi coefficients.
2.5.3.2. PMLs for the SCN node [LEM 01a]
PML layers for the SCN node are described by modifying the
Maxwell equations, with the six electromagnetic field components
divided into 12 sub-components and with anisotropic magnetic and
electrical conductivities.
In the TLM method, each plane wave, which penetrates the cells in
the x, y or z direction, is associated with a pulse propagating from the
center of the cell through one of the twelve transmission lines which
3D Nodes
65
link the node to its six neighbors. According to Huygens’ principle,
these pulses are scattered at the center of the node and the reflected
waves are associated with them.
2.5.4. Optimization of the PML layer profile [DUB 00, KON 03]
In numerical simulation, it is possible, in order to optimize the
reflection coefficient, to use media without any physical reality. This
particularity enables concepts which cannot be utilized in testing.
Let us consider a medium with electrical (e) and magnetic (m)
losses, a permittivity r and the permeability of the vacuum µo. The
following condition is raised:
e
m S
r 0 0
[2.92]
The impedance Z of the medium is the same as the medium
without losses:
Z
0
0 r
We now terminate the computation volume by using this type of
layer (Figure 2.13). (Under normal incidence, a wave can penetrate the
lossy layer of thickness “e” without reflection). For a wave under
oblique incidence, we can calculate the apparent reflection coefficient
(ARC), i.e. the reflection after a round-trip in the lossy layers:
S
2e
e c
[2.93]
assuming that the lossy layer is terminated by a perfectly conducting
wall.
66
Numerical Analysis in Electromagnetics
Traditional medium
Matched layer
Incident wave
Reflection
Figure 2.13. Geometry of the matched layer and medium of interest
An adjustment in the value of coefficient theoretically enables
the desired absorption to be obtained. However, simulations show that
a numerical reflection appears when a wave passes from one medium
to another with different losses (different conductivities). This
reflection is proportional to the ratio of the respective losses of each
medium. In order to compensate for this disadvantage, the layer
should be matched in such a way as to impose a profile describing a
growing change in the losses. Thus, it is possible to minimize the
effect of discontinuity at the interface, while preserving an apparently
identical absorption. In this case the ARC is calculated as follows:
e
2 z1e
S ( z ) dz
c z1
[2.94]
The wave is propagated in the Oz direction, with z1 being affixed
to the interface. Finally, from the technical point of view, each layer
3D Nodes
67
(with a given loss value) is the thickness of one cell. Optimization of
the layer therefore consists of choosing the depth and type of its
conductivity profile for a given ARC.
However, the matched layer is limited in its application by the
presence of evanescent modes. Indeed, for evanescent modes the layer
is not matched and it is therefore necessary to position it sufficiently
far from the discontinuities.
Theoretically, a wave arriving under any incidence over the PML
layer does not produce any reflection, is not subject to any refraction
and is attenuated during its propagation in the layer. In reality,
however, the PML layer is subject to the same constraints as the
matched layer. The layer must therefore be optimized in terms of
depth and profile.
On the other hand, the PML layer is selective regarding the nature
of the wave which will be attenuated. Indeed, propagative waves are
absorbed into the layer while evanescent modes are not disturbed.
They penetrate the layer, but no attenuation is added to their natural
decrease. A supplementary phase term appears.
In their original formulation, PML layers must sometimes be very
thick because of the presence of evanescent modes, which they must
enable to be naturally attenuated.
In the following section, we propose studying the generalized
PML, presented by Jiayuan Fang and Zhonghua Wu [FAN 96], using
anisotropic and dispersive absorbing layers.
2.5.5. Anisotropic and dispersive layers
In this profile, we define new variables Sx, Sy and Sz. PML layers
can be described in terms of an anisotropic, dispersive medium. It is
then reduced to a “Maxwellian” medium, with unconventional
permittivity and permeability tensors, as follows:
68
Numerical Analysis in Electromagnetics
S y Sz
0
0
0
Sz Sx
Sy
0
0
0
Sx
[2.95]
Sx S y
Sz
where:
ei
Si i 1
j r 0
For example, in the z direction:
( z)
S z ( z ) S z 0 ( z ) 1 z
j r 0
The terms Sz0(z) and z(z) are functions of z and must be selected in
order to avoid numerical reflections (in theory, the higher the values,
the faster the attenuation of fields in the PML layers).
Thus, in order to avoid numerical reflections, these parameters
must continually increase as and when we advance through the PML
layers.
The aim is to find a profile (a choice between Sz0(z) and z(z))
which enables propagative and evanescent modes to be absorbed
simultaneously.
A possible profile is as follows:
z
S z 0 ( z ) 1 Sm ( )n
[2.96]
3D Nodes
z ( z ) m sin 2 (
z
)
2
69
[2.97]
being the thickness of the PML layer.
The product Sz0(z) z(z) gradually changes with space, which
enables numerical reflections to be eliminated. It varies uniformly,
like a parabolic function, across the whole PML layer region.
However, if the value of Sz0(z) is very high, then there is a risk of
having numerical reflections at high frequencies. A necessary and
sufficient condition to remedy this problem is to choose Sm such that
2 to 3 , where is the shortest useful wavelength of fields
1 Sm
within the computation volume.
The value of m is determined from the apparent reflection
coefficient (ARC). For example, for n = 2, the ARC is expressed as:
Rth exp[
2
S z 0 ( z) m ( z)dz ]
c 0
[2.98]
1 2 m
)]
]
Rth exp[[1 Sm (
3 2 c
[2.99]
and the value of m is chosen as follows:
m
c
1 Sm (1 3 2 2 )
[2.100]
ln Rth
As we can see in the expression S z ( z ) S z 0 ( z ) 1
z ( z)
, the
j r 0
implementation of Sz(z) is dependent on the frequency through the
term j
70
Numerical Analysis in Electromagnetics
See Chapter 4 for the use of the Z transform to handle the problem
in the time domain.
2.5.6. Conclusion
Absorbing walls are a real problem when simulating open media. It
is quite feasible to obtain reflection coefficients below –60 dB over
the whole frequency band, but beyond this, instabilities may appear.
We should note, however, that such values are largely sufficient for
the great majority of problems.
2.6. Orthogonal curvilinear mesh [YOU 08]
2.6.1. 3D TLM curvilinear cell
The curvilinear node [YOU 08] can cover all regions and arbitrary
variation in mesh size. The mesh is also equally flexible for complex
physical forms. It is therefore advantageous compared with the
Cartesian equations seen previously when boundary curves exist
within the structure being simulated, which implies a mesh which is
very fine in order to describe them.
Of course, curvilinear coordinates (u1, u2, u3) are used and the
Maxwell equations are written using this coordinate system. They are
integrated into a parallelepiped volume for which dimensions in the
three directions are u1, u2 and u3. Figure 2.14 shows the basic cell
of the 3D curvilinear node.
The basic cell therefore consists of six curvilinear surfaces which
can be grouped into pairs: surface u1 perpendicular to u1, surface u2
perpendicular to u2 and surface u3 perpendicular to u3.
The field components perpendicular to ui are referenced by the
corresponding exponent i.
3D Nodes
71
Figure 2.14. Basic cell of a 3D curvilinear node
At the center of each surface, four tangential field components are
defined, two electrical and two magnetic, hence the total of 12
components of each type in order to consider all possible
polarizations. The numbering of voltage pulses is the same as that
initially proposed by Johns for the SCN node, and, by convention,
magnetic fields of the same number are located at the same central
surface points, but perpendicular to the corresponding electrical fields.
The center of the cell will be defined as follows: in each of the
three curvilinear coordinate directions (u1, u2, u3), the three curvilinear
surfaces of symmetry of the cell intersect in pairs forming three
central curvilinear curves, which in turn cross at a single point
(barycenter), which is defined as being the center or node of the cell.
Furthermore, the electrical field components, as well as those of the
magnetic field at the center of the cell are necessary for the SCN node.
The ai incident voltage vectors and the ar reflected voltage vectors
presented at the cell surfaces are given in [2.101].
72
Numerical Analysis in Electromagnetics
Each vector consists of 18 voltages, the first 12 of which
correspond to the node inputs and the last 6 to the voltages of stubs
situated at the center of the cell.
u1 E '11 Z 0 u3 H '13
u1E '12 Z 0 u2 H '22
u2 E '32 Z 0 u3 H '33
2
1
u2 E '4 Z 0 u1H '4
3
1
u3 E '5 Z 0 u1 H '5
3
2
u3 E '6 Z 0 u2 H '6
3
1
u3 E '7 Z 0 u1 H '7
u E '2 Z u H '1
2 8
0 1 8
1
2
1 u E ' Z u H '
ai 1 9 0 2 9
2 u E '3 Z u H '2
3 10
0 2 10
2
3
u2 E '11 Z 0 u3 H '11
1
3
u1E '12 Z 0 u3 H '12
u1E '113
2
'
u
E
2 14
1
u3 E '15
Z 0 u1 H '116
Z 0 u2 H '117
Z 0 u3 H '118
u1 E '11 Z 0 u3 H '13
u1E '12 Z 0 u2 H '22
u2 E '32 Z 0 u3 H '33
2
1
u2 E '4 Z 0 u1H '4
3
1
u3 E '5 Z 0 u1H '5
3
2
u3 E '6 Z 0 u2 H '6
3
1
u3 E '7 Z 0 u1H '7
u E '2 Z u H '1
2 8
0 1 8
1
2
1 u E ' Z u H '
a r 1 9 0 2 9 [2.101]
2 u E '3 Z u H '2
3 10
0 2 10
2
3
u2 E '11 Z 0 u3 H '11
1
3
u1E '12 Z 0 u3 H '12
1
u1 E '13
2
'
u
E
2 14
u3 E '115
Z 0 u1 H '116
1
Z 0 u2 H '17
Z 0 u3 H '118
In curvilinear coordinate form, Maxwell’s integral equations are
expressed as:
E'
1
1
1
H ' d dl t d dS e E' d dS J e dS
S
S
S
[2.102]
3D Nodes
73
H'
1
1
1
E ' d dl t d dS m H' d dS J m dS
S
S
S
[2.103]
where:
0 diag (11 , 22 , 33 )
diag ( , , )
0
11 22 33
e diag ( e1 , e 2 , e3 )
E' d E
m diag ( m1 , m 2 , m3 )
and H' d H
2
2
x y z
hi 2
u i u i u i
dS hi h j dui du j uk
d diag (h1 , h2 , h3 )
2
hi = scale factor
dl hi dui ui
u
li 0 i hi dui
2.6.2. The TLM algorithm
In order to develop the TLM algorithm from Maxwell’s equations
in integral form, firstly six relations between incident voltages and the
field at the center of the cell are obtained by integration of the
equations for the three main curvilinear surfaces.
Figure 2.15. Contour and surface for approximating the
Maxwell–Ampère law for the curvilinear surface surf-u1
74
Numerical Analysis in Electromagnetics
Assuming constant field samples along each side, it is possible to
approximate the flow of magnetic and electrical fields by integrating
equation [2.102] over the time between (n-1/2)t and (n+1/2)t. From
this we obtain:
1 T
2
1 T '1
u1Y 1 E
H '3 H1'3 u3 H 2' 2 H 9' 2 u2
12
Z0
1 T '1
+
u1 G e1 E13 Ve1
2Z 0
[2.104]
T is the backshift operator for a t.
In the above expression:
Y1 11
ct
1
du1
u
0 1
h2 h3
du2 du3
u u h1
2
G
e1
3
Z 0 e1
du1
u
0 1
h2 h3
du2 du3
u u h1
2
3
Ve1 I 2 I3 J e1
'1 is calculated based on an average of the field
The flux E
components at the surface edge, to which the contribution of the stub
at the center is added.
1
1
1
1
1
'1 E '1 E '12 E '2 E '9 Y s1E '13
E
4 Y s1
[2.105]
3D Nodes
75
in which Y s1 is the normal admittance of the reactive branch (shortcircuit stub) in the u1 direction, supporting voltage V13. Its value is
given by: Y s1 2 Y1 2 .
Taking into account [2.101], we finally obtain:
r
r
i
i
i
i
i
a1r a2r a9r a12
Y
s1 G e1 a13 a1 a2 a9 a12 Ys1 G e1 a13 Vei
[2.106]
Repeating this process for the magnetic field flux and then for the
other curvilinear surfaces [YOU 08] ultimately leads to the matrix
form and to scattering matrices.
2.6.3. Scattering matrices for curvilinear nodes
The scattering matrix of the node [S] and the [Ss] matrix, which
takes into account the fictive sources Je and Jm, enable the reflected
and incident pulses to be linked at each iteration:
( n 1/2) a
r
S ( n 1/2) ai S S n u
,
where:
Ve1
V
e2
Ve3
where Vei l j lk J ei
nu
Vm1
Vm 2
V
n m3
and
Vmi l j lk J mi
76
Numerical Analysis in Electromagnetics
The [S] matrix is given in Figure 2.16, where:
Z
1
u
0 0 j
Rmj MJ
du j
hi hk
h dui duk
j
ui uk
aij
ei
Y i G
ei 4 2
2 Y i G
j R
mj
Z
j R
mj 4
Z
cij
ei
Y i G
ei 4 2
2 Y i G
j R
mj
Z
j R
mj 4
Z
bi ei
gi
hi
4
ei 4
2 Yi G
ei
4 Y i G
ei 4
2 Y i G
ei 4
Y i G
ei 4
Y i G
;
d j ij
;
fj
;
jj
4
mj 4
2 Zj R
j R
mj
4 Z
j R
mj 4
2 Z
j R
mj 4
Z
j R
mj 4
Z
The [Ss] matrix is given in Figure 2.17, where:
;
AH i
1
mi
4 Zi R
Y i G ei 2
CEi
ei 4
2 Y i G
;
BH i
i R
mi
Z
i R
mi
2 4Z
;
CH i
i R
mi 2
Z
i R
mi
2 4Z
AEi
BEi
1
ei 4
2 Y i G
Y i G ei
ei 4
2 Y i G
[2.107]
u j du j
and
Figure 2.16. The [S] matrix of the curvilinear node
3D Nodes
77
78
Numerical Analysis in Electromagnetics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
AE1
AE1
0
0
0
0
0
0
AE 1
0
0
AE 1
AE 1
0
0
0
0
0
2
0
0
AE2
AE2
0
0
0
AE2
0
0
AE2
0
0
AE2
0
0
0
0
3
0
0
0
0
-AE3
AE3
AE3
0
0
AE3
0
0
0
0
AE3
0
0
0
4
0
0
0
AH1
AH1
0
AH1
-AH1
0
0
0
0
0
0
0
AH1
0
0
5
0
-AH2
0
0
0
AH2
0
0
AH2
-AH2
0
0
0
0
0
0
AH2
0
6
A H 3
0
-AH3
0
0
0
0
0
0
0
AH3
-AH3
0
0
0
0
0
AH3
Figure 2.17. The [Ss] matrix of the curvilinear node
Of course, in a volume without sources, the [Ss] matrix is useless.
2.6.4. Stability conditions and the time step
In order to ensure the stability of the process by keeping the time
step at a maximum value, impedances must remain positive or null in
value. This step is calculated for each set of blocks of similar
properties. These conditions are written:
Y si 0 t ii
2c ui
0
u j uk
1
dui
h j hk
hi
[2.108]
du j duk
3D Nodes
si 0 t ii
Z
2c ui
0
u j uk
1
dui
h j hk
hi
79
[2.109]
du j duk
where (i ,j, k) ∈ {(1, 2, 3), (2, 3, 1), (3, 1, 2)}.
2.6.5. Validation of the algorithm
Validation is performed on a cylindrical curvilinear mesh fitted
with a coaxial resonator (Figure 2.18).
z
Figure 2.18. Coaxial resonator
The resonance frequencies are given by the formula f n
nc
.
2h
The first three resonance modes are found by varying the height h
of the cavity.
80
Numerical Analysis in Electromagnetics
Error %
TEM1 Error(%)
Error %
TEM2 Error(%)
Error %
TEM 3 Error(%)
Figure 2.19. Respective errors over resonance frequencies
of the first three TEM modes [YOU 08]
3D Nodes
81
The errors committed are equivalent for the three modes. They do
not exceed 1.5% when l/ is less than the generally accepted limit of
0.1. The results are therefore reasonable and the curvilinear node
algorithm is validated.
This error over the resonance frequencies comes from the velocity
error due to dispersion of the node. Youssef [YOU 08] studied the
changes in the velocity error based on different strains which could be
applied to the cylindrical cell. His conclusion was that dispersion
remains generally low and remains perfectly adequate. He did,
however, emphasize that this error is magnified when approaching
singularity.
Finally, we note that the insertion of absorbing TLM walls is quite
feasible with curvilinear nodes.
2.7. Non-Cartesian nodes
In all time-domain 3D methods, such as the TLM and FDTD, it is
often assumed that the cell represents a volume delimited by a
rectangular parallelepiped. Generally, users of such algorithms arrange
for the structure that they are analyzing to correspond to this
description. However, structures which do not include interfaces or
slanted or curved boundaries are rare. In this case a stepped mesh must
be adopted, which requires a mesh with a significant density, for a
good-quality boundary description (i.e. providing highly precise
results). Consequently, the rapid increase in computer costs (i.e. CPU
time, as well as memory) often becomes prohibitive and prevents
analysis, under acceptable conditions, of the device.
It is possible to develop a cell which is able to take this type of
boundary into consideration, with a view to obtaining a sufficient level
of precision without the need to refine the mesh. Historically, the first
cells were developed in 2D. Initial testing on this subject was
proposed by Al-Mukhtar and Sitch [ALM 81] and creative works in
this direction have been conducted by Johns with a curvilinear cell
[MEL 88], whose surfaces follow the benchmark. However, this cell is
not entirely general, since it does not enable the meshing of any
82
Numerical Analysis in Electromagnetics
objects. A second 2D approach consists of creating a nonperpendicular cell (whose surfaces are always flat) from a formulation
of the finite element method in the time domain [SIM 95]. However,
these two approaches have never been extended to 3D in space. A
significant contribution, called AS-TLM (Axi-Symmetric TLM) was
recently produced by S. Le Maguer [LEM 01b], regarding the specific
classification of structures with axial symmetry (cylindrical or coaxial
wave guides, conic antennas) and no discontinuity. AS-TLM proposes
a completely new arrangement of the TLM for this type of geometry,
thus leading to a significant improvement in computation costs and
reducing the size of the 3D SCN node problem (18 voltage
components: 12 real and six stubs) to a 2D special AS-TLM node
(14 voltage components: eight real and six stubs). In addition,
perfectly matched layers which are appropriate to this type of problem
have been developed.
Figure 2.20. Hein’s non-orthogonal node
The last solution was finally proposed by Hein, who laid
the mathematical foundations for a non-orthogonal 3D TLM
algorithm [HEI 94]. This is naturally compatible with the traditional
3D Nodes
83
SCN-TLM node network, from where its formulation is derived [HEI
93]. It can thus be considered to be a generalization of the SCN node.
Its interest is in the fact that it enables a local non-orthogonal mesh
(Figure 2.20), which allows the use of the traditional algorithm over a
large part of the computation domain.
A complete and detailed presentation on this algorithm has recently
been implemented by Z. Li [LI 05]. This presentation additionally
contains the algorithm, an analysis of the numerical dispersion of this
node network, validations of this theory using several examples and a
performance evaluation compared with the traditional mesh. A big
advantage of this node is the potential to only use it locally, which
enables a high level of precision to be obtained without also unduly
increasing the computation time.
Chapter 3
Introduction of Discrete Elements and
Thin Wires in the TLM Method
3.1. Introduction of discrete elements [BIS 99]
3.1.1. History of 2D TLM
The idea of introducing localized nonlinear elements into the TLM
was initially demonstrated by Johns and O’Brien [JOH 80], who used
line sections to connect the element to the nodes of the mesh.
Kosmopoulos [KOS 89] and Voelker and Lomax [VOL 90] modeled
nonlinearities by using variable (with voltage) admittance line
sections, whose value was updated at each time step.
Russer et al. [RUS 91] linked nonlinear elements to the TLM
network using transmission lines, which were matched at the
connection point of the admittance to the node, which led to an
implied integration of the nonlinear equations defining the device. In
all of these approaches, the time step used to update the coefficient
values of the nonlinear equations were linked to the TLM time step. In
order to ensure a high level of precision and stability in the solving of
differential equations, the time step of the TLM had to be quite fine.
However, as the “rigidity” of differential equations varies over time,
this may lead to periods where the update time for the values of the
86
Numerical Analysis in Electromagnetics
coefficients is excessive and therefore the algorithm is not efficient.
On the other hand, relaxing the sampling interval may be the cause of
erroneous solutions and computation instabilities. The approach
presented in this chapter avoids these limitations since it ensures the
uncoupling of the discrete model from the nonlinear device and the
TLM node.
3.1.1.1. Mono-port circuit with parallel node
The connection of localized circuit elements into the TLM2D mesh
has already been presented in a number of papers and books [JOH 80,
RUS 91, ISE 92, ALA 94]. In the technique used by Russer, nonlinear
active regions are connected to parallel TLM2D nodes via line
sections and the reflection coefficient for these sections is updated in
accordance with the integration of the nonlinear differential equation
which defines the localized element.
In a second approach [ALA 94] the connection of a localized
element to the parallel node occurs directly. The two possibilities are
illustrated in Figure 3.1. The consequences of the second approach
(Figure 3.1b) are that the scattering matrix of the node is modified at
each time step (each iteration), and that, in the case of memory
circuits, for example, not only should the latest data be stored, but also
data concerning old reflected and incident voltage values in the
connection lines of the device. The first approach (Figure 3.1a)
requires the connection matrix to be modified at each time step, but
implies less regarding variables (only the voltages on the two
connection lines), which is why this is the preferred solution. In this
case, the voltages generated at the interface with the localized element
will be incident on the adjacent node, but a part of these voltages will
be reflected and return over the connection line. When a nonlinear
element is modeled, these reflections will create an unwanted coupling
between the linear and nonlinear parts of the circuit [RUS 91, ISE 92].
There is a third approach, in which (Figure 3.1c) line sections are
added to the parallel 2D TLM node in order to thereby perform the
connection with the localized circuit element. The length of the
section is set at l/2 in order to ensure the synchronism of pulses
across the whole TLM network. This technique forces the scattering
Introduction of Discrete Elements and Thin Wires
87
matrix to be modified in order to take into account the new line
section, but it will not be changed for each iteration. On the other
hand, the reflection coefficient of the added line section will be
recomputed for each iteration, but this implies a single variable (the
voltage in the section).
Figure 3.1. A localized circuit element connected to the TLM2D meshing:
a) directly connected to a transmission line, b) directly over the node,
c) connected to the TLM node by a line section
The voltage V(t) on the localized circuit at time t is given by
superimposing the incident and reflected voltages propagating along
the transmission line section, whereas the current i(t) circulating
towards the device is given by their difference, weighted by the
admittance Ys of the section:
v(t ) Vsr (t ) Vsi (t )
i (t ) Ys Vsr (t ) Vsi (t )
[3.1]
The admittance of the line section may be randomly chosen, but a
wise choice will enable the computations to be simplified. The value
of this admittance is determined in accordance with the scattering
matrix of the TLM node loaded with the line section (Figure 3.2). The
coefficient S55 of the matrix is equal to (Ys-4)/ (Ys+4). For normalized
admittance, Ys = 4, the S55 element of the scattering matrix cancels out
and therefore the reflected voltage Vsr is not dependent on the incident
voltage Vsi, which is equivalent to decoupling the linear part of the
nonlinear part of the device.
88
Numerical Analysis in Electromagnetics
Figure 3.2. Localized circuit element connected to a parallel TLM2D node
The total voltage V(t) and the current i(t) must also satisfy the
current-voltage relation which defines the circuit. In general, for a
mono-port device, this is expressed by:
i t f v t , dv / dt
[3.2]
v t f i t , di / dt
[3.3]
To simplify, we will only use expression [3.2]. All of the signals
are considered to be completely determined by discrete sampling at
time intervals t, within a limited signal range. The following
notations are used:
k
Vsr Vsr k t and
k
Vsi Vsi k t
In a centered finite difference node network, the following
substitutions are given for the time half-step (k-1/2) t:
/2
k 1 2 I k I k 1 I / 2
k 1 2V
kV
k 1V
dV
kV k 1V / t
k 1 2 dt
dI
k I k 1 I / t
k 1 2 dt
[3.4]
Introduction of Discrete Elements and Thin Wires
89
By substituting the expressions for current and voltage [3.1] into
the equation defining the localized circuit and by applying the
discretization node network previously defined, a differential equation
is obtained which is solved for the unknown kVsi. For the first order
circuits, this recursive formula can be expressed in the form:
i
kVs
F
kVsr , k 1Vsi , k 1Vsr
[3.5]
For first order circuits, values for Vsi and Vsr will be stored at the
previous time step. The computation of new incident pulses in the case
of modeling a nonlinear device therefore passes through the solution
of a nonlinear equation (F in [3.5] is nonlinear).
The main advantage of this approach is the fact that in choosing
appropriate values for the impedance of the added line section, the
linear and nonlinear parts of the circuit can be decoupled. For this
reason, and also because the interaction of the TLM mesh and the
nonlinear circuit (the element) is limited to a single TLM voltage
pulse, it is feasible to link the TLM algorithm to other circuit
simulators, such as SPICE (simulation program with integrated circuit
emphasis).
One disadvantage of this technique is the fact that the line sections
added are introducing an additional capacity into the equivalent model
of the device, but this error can be offset.
3.1.2. 3D TLM
Here we present one of the two methods in use for the introduction
of localized elements into a 3D TLM network, derived from the 2D
developments and presented in the previous section, assuming firstly
that the localized element occupies a single TLM cell, which does not
distort the physical conditions of the problem so long as the
dimensions of this circuit element are small in comparison with the
other dimensions of the general electromagnetic problem. On the other
hand, we are only concerned with the insertion into the SCN node.
90
Numerical Analysis in Electromagnetics
3.1.2.1. Mono-port element
3.1.2.1.1. Mono-port element of a volume equal to the TLM cell
The standard SCN node can be fitted with three capacitive line
sections (connected in parallel), which modify the three components
Ex, Ey and Ez, and three inductive line sections (connected in series),
which influence the three components of the magnetic field
(Chapter 2). In general, these line sections enable inhomogeneous
media to be simulated. If the sub-volume to be modeled consists of an
isotropic material, the characteristic admittances and impedances of
the line sections are identical.
The developments below are made on the hypothesis that the
connection of a localized circuit element produces an anisotropic field
modification (corresponding to the direction in which the load has
been used) and that the element’s dimensions enable it to be
assimilated into the volume of a single node. In this case, the
modification of the TLM cell will also be anisotropic. In the case
described in Figure 3.3, the localized element will only influence
component Ey, or even, in terms of TLM “voltages”, the voltage
pulses V3, V4, V8 and V11.
Figure 3.3. Connection of a mono-port device to a TLM node: the presence of the
device only modifies the voltages in the y direction
Introduction of Discrete Elements and Thin Wires
91
As in the 2D case, the technique proposed for modifying these
voltages is to add a line section, connected in parallel in the y
direction, which, for the general scattering matrix (Chapter 2),
corresponds to the case where all line section admittances are null,
with the exception of Yy. As in 2D, the choice of appropriate values
for normal admittance of the line section enables the incident voltage
to be decoupled from that reflected onto the transmission line which
joins the device with the SCN node. In the same way that a
transmission pulse of 14 in the scattering matrix corresponds with the
line section relative to the Ey component, so decoupling will occur at
S14,14 = 0, which implies that Yy = 4.
Subsequently, the voltage and current in the circuit are defined
respectively as the sum and difference between the incident voltage
and the voltage reflected onto the transmission line (see [3.1]). The
current-voltage relation which defines the circuit will be replaced by a
reflected-incident voltage relation, which is discretized and which,
through a recursive relation, enables a new incident voltage to be
obtained, which is propagated towards the TLM node (see [3.5]).
3.1.2.1.2. Mono-port element with a higher volume than a TLM cell
The remarks above, regarding anisotropy from the modification of
nodes connected to the localized circuit element, remain valid for
devices with dimensions greater than a TLM cell. The extent of the
device may be in the direction of the voltage drop at the device
boundaries or even in the plane perpendicular to the voltage drop at
these boundaries.
Let us consider, firstly, the second situation, where the width of the
localized circuit element is equivalent to several TLM cells. In this
case, the localized element can be considered to be a parallel
combination of equivalent identical sub-elements, whose properties
can be obtained by the simple application of Kirchhoff’s equations.
Each of these elements will be connected to one of the TLM nodes
contained in the device volume. This situation is illustrated in
Figure 3.4.
92
Numerical Analysis in Electromagnetics
Ys
Figure 3.4. Equivalent model of the device, containing multiple TLM cells in the
plane perpendicular to the voltage drop (xz plane)
With this approach, the behavior of each sub-element is only
dependent on the local field corresponding to the TLM cell. This
hypothesis is acceptable for most situations and offers the advantage
of being easy to implement.
If the length of the component is greater than a TLM cell, the
component will be placed over multiple nodes in the direction of the
voltage drop at its boundaries and will be decomposed into multiple
identical elements connected in series (Figure 3.5). The series
decomposition may be induced from the instabilities in the case of
strongly nonlinear devices, such as the Gunn or tunnel diodes
[CAS 97]. Another disadvantage is the fact that for nonlinear devices,
in order to determine the Vmi voltage, a system of nonlinear equations
at each time step must be solved.
It is therefore preferable to avoid series element decomposition.
In order to avoid this type of behavior, a new technique must be
used for the nonlinear active elements in the 3D TLM mesh. This
technique models the active region as a whole, such as a single device
in the voltage drop direction, and is dealt with for the other directions
by a combination of parallel sub-elements.
The connection between the TLM nodes and the device is always
fulfilled with parallel line sections, the voltage at the boundaries of the
device being equal to the sum of voltage drops over each line section
Introduction of Discrete Elements and Thin Wires
93
in the active region. This model is physically equivalent to the
arrangement of lines in series, as shown in Figure 3.5. In the interests
of simplicity, we are, firstly, going to assume that the width of the
device is that of a single TLM node and we will not take its length into
account.
Volume of the element
Node N
I(t)
Ys
V(t)
Node 2
Ydevice
Ys
Node 1
Δl
Ys
Figure 3.5. Device occupying multiple cells in the direction of the
voltage drop at the boundaries of the device (y direction)
The procedure used to connect the device explained using [3.5]
must take into account the new line section configuration. Equation
[3.1] becomes:
v(t ) =
M
Vmr (t ) + Vmi (t )
m =1
M
i (t ) = Ys Vmr (t ) − Vmi (t )
m =1
m=1,…, M
[3.6]
94
Numerical Analysis in Electromagnetics
where M is the number of TLM cells used by the device, m indicates
the node to which the line section pertains and Ys represents the line
section admittance (which is the same for all line sections in the active
region). In the example given, Ys = YyY0, when Y0 is the admittance
of the TLM lines.
As a result of the series connection of the lines, the same current
i(t) flows through the device and the line sections added, which
enables all of the incident voltages Vmi(t) to be expressed based on a
single one, for example V1i(t). Thus, from the second system equation
[3.6]:
Vmi t Vm r t V1r t V1i t
m= 2,…, M
[3.7]
By substituting this relation into the first equation [3.6], the voltage
and current over the localized circuit are expressed is such a way as to
be dependent on a single unknown V1i t :
M
v (t ) M .V1i t (2 M ).V1r t 2 Vm r t
m2
i (t ) Ys V1r t V1i t
[3.8]
When these relations are substituted into the current-voltage
relation which characterizes the device, they are subsequently
discretized. The finite difference equation obtained thus is solved for
the unknown V1i t : incident pulse in the first node used to connect
the element into the TLM mesh. The other incident pulses will be
determined by using relation [3.7].
For first order circuits, a general recursive formula for calculating
the new incident voltage based on the reflected pulses can be
expressed as follows:
Introduction of Discrete Elements and Thin Wires
M
M
r
i
i
r
r
r
V
F
V
V
V
V
,
,
,
,
k 1
k 1 k 1 1 k 1 1
k m
k 1Vm
m é
m é
i
kVm
kVmr kV1r kV1i
95
m=2,… M
[3.9]
[3.10]
Only one nonlinear equation F needs to be solved in order to
determine the incident pulses in the nodes connected to the localized
elements for each iteration.
In general, when the volume of the device is greater than that of a
single TLM cell, a combination of the techniques described in this
section is used. The device is firstly divided into equivalent parallel
sub-elements which are defined by the same relations as the original
device, but with impedances and admittances divided in relation to the
number of nodes “covered” by the device volume. Each of the subelements is interfaced with a series of nodes, in the voltage drop
direction at the device boundaries.
3.1.2.1.3. Mono-port elements in inhomogeneous media
The hypothesis that the connection of a localized element to a SCN
node affects a single component of the field (anisotropic modification
of the TLM cell) remains valid. In order for the device to be oriented
in any Cartesian direction, the SCN node is supplied with three
supplementary appropriate admittance lines, a single one of which is
effectively used to connect the device, whereas the other two have a
null admittance (eventually we can use sections 13–18 if they have not
already been used to describe the medium).
The scattering matrix of the new node, which now takes the
interaction between the 21 lines (12 transmission lines and nine line
sections) into account, is determined in accordance with the charge
conservation law. By using the line sections of the localized elements,
an incident voltage pulse in the TLM node coming from the TLM
96
Numerical Analysis in Electromagnetics
node is added to the other voltage pulses from the node. For this, a
21 × 21 matrix is required to model the properties of the medium.
The coefficients of the new matrix are given by:
a
Y Ys
Z
2 4 Y Ys 2 4 Z
b
4
2 4 Y Ys
c
Y Ys
Z
2 4 Y Ys 2 4 Z
d
4
24 Z
eb
g Y .b
id
m
f Z .d
Y Ys 4
h
4 Y Ys
j
Ys Y 4
4 Y Ys
[3.11]
4Z
4Z
n Ys .b
where Y is the normalized admittance of the line section for modeling
the permittivity of the material, Z is the normalized impedance of the
line section for modeling the permeability and Ys is the normalized
admittance of the line section of the localized element, which differs
based on the device direction (Ysx, Ysy, Ysz). The form of the scattering
matrix, modified by the presence of line sections of the localized
element, is given in Figure 3.6.
In order to decouple the linear from the nonlinear parts in the
behavior of the circuit, the admittance value of the line sections of the
device is chosen such that the elements (m) of the scattering matrix,
which transfer the reflected voltage into the TLM node and which
come back onto the device, are null (Sii = 0, for i = 19 – 21). For
example, for a device positioned in the y direction (Figure 3.5), the
m20,20 element should be cancelled, which implies that:
Ysy Yy 4
[3.12]
Introduction of Discrete Elements and Thin Wires
97
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1
a
b d
2
b
a
3
d
b
d
c -d
a b
4
b
b a d
5
-d c
d a b c -d
6
d
b a b
7
b c -d
9 b
c
10
-d
11 -d
b c b
e
e
e e
17
18 f
19 e
20
21
e
n
i
n
-i
n
g
-i
g
d
a g
-i n
e h
n
-i
n
h
n
h
f -f
n
j
f -f
e
j
-e g
e
e e e
n
d
j
-f
e e
n
a
f -f
e
n
i
e
f
n
i
b g
e
e e e
-f
g
d a
13 e
n
g -i
b
b
n
i
g
b
f -f
g
g
b -d
16
b
n
-i
b
12 c
15
-i
g
a d
c b
14
b g
c -d
-d c
d a
-d
i
b
-d c b a d
8
-d c g
e
e
m
g
m
g
m
Figure 3.6. Scattering matrix for the SCN with a localized element
The procedure used to determine the new incident voltages, as
explained in the previous sections, is the same in the case of line
sections with the same characteristic admittance. For situations in
which the localized element is placed in a medium with different
permittivities, equations [3.7]–[3.8] become:
Y
Vmi t Vmr t s1 Vmr t Vmi t
Ysm
m=2,…,M
[3.13]
98
Numerical Analysis in Electromagnetics
M
M
Y
v t V1r t V1i t s1 2 Vmr t
m1 Ysm
m 1
i t
Ys1 V1r
t
V1i
t
[3.14]
where Ysm is the characteristic admittance of the mth line section of the
device.
3.1.2.1.4. Two-port elements
For devices such as transistors, the procedure described above can
be extended to the modeling of two-port nonlinear devices. In order to
simplify the analysis of this procedure, we will assume that the device
occupies two rows of adjacent nodes in the z direction and M in the y
direction (Figure 3.7). As in the previous cases, the device is modeled
as a single element in the voltage drop direction at its boundaries. The
device will thus be linked to the row of network nodes which are
connected in series in the y direction. In order to be assured that the
interaction of the two rows of nodes is only due to the presence of the
device connected between them, their line sections in the z direction
are terminated by open circuits.
Figure 3.7. Two-port connection of a device in a 3D TLM node network
Introduction of Discrete Elements and Thin Wires
99
The localized circuit variables are v1(t), i1(t), v2(t) and i2(t);
generally, the behavior of the device is represented by a system of two
coupled, nonlinear equations:
i1 f1 v1 t , v2 t
i2 f 2 v1 t , v2 t
[3.15]
The currents and voltages can also be expressed in terms of the
incident and reflected voltages in the arms (the line sections) of the
device. Equation [3.9] can be re-written:
v1,2 (t ) M .V1i1,2 t (2 M ).V1r1,2 t 2
i1,2 (t )
Ys V1r1,2
t
V1i1,2
M
Vm r1,2 t
m2
t
[3.16]
where the indices 1, 2 refer to the two rows of nodes for which these
equations are used.
Using the same method for mono-port devices, system [3.16] is
substituted into [3.15], which links the voltages and currents in the
device; and we use the same principle, as described in the previous
section, in order to discretize the resulting equation. In this way, also
using relation [3.7] imposed by the series connection of the line
sections, we obtain a system of coupled finite difference equations
which will describe the incident voltages in the two rows of line
sections of the device. For first order circuits, the following
expressions are obtained:
i
k 1V11,2
M
M
i
i
, k 1V11
, kV1r , kVmr1 , k 1Vmr1 , kV1r 2 ,
F kV11
m2
m2
, k 1V1i 2 , k 1V2r ,
i
k 1Vm
M
M
m2
m2
kVmr 2 ,
i
kVmr1,2 kV1r1,2 k 1V11,2
r
k 1Vm 2
[3.17]
100
Numerical Analysis in Electromagnetics
Z0
Yf
[3.18]
R R ' 2 Z m Z s
3.1.3. Application example: modeling of a p-n diode
3.1.3.1. Physical model of the diode
The p-n diode is the simplest nonlinear semi-conductor device,
consisting of a junction between type p and type n materials
(Figure 3.8). When the junction has been polarized, the potential
barrier is modified and, as a result, the scattering of carriers between
regions. In order to determine the current in the junction based on the
bias voltage, the carrier continuity equation, which needs to consider
drift currents and electron and hole diffusion currents, must be used.
Id
+
Vd
Junction
-
Metallic contact
Id
n region
p region
Vd
+ -
Figure 3.8. Diode at the p-n junction and its physical mode
By firstly considering the low injection level hypothesis (where the
density of majority carriers in each region is not affected by bias) we
show that the ideal current-voltage relation governing the behavior of
the diode is expressed in the following form:
I d I s eqVd /( kT ) 1
[3.19]
where Is is the reverse saturation current, q is the electron charge, k is
the Boltzman constant and T is the device temperature in Kelvin.
Under reverse polarization conditions the current tends towards –Is.
Introduction of Discrete Elements and Thin Wires
101
When the low injection level hypothesis is no longer valid, the
difference between the two operating regions is solely the increase in
amplitude of the current circulating in the diode. A modification in its
variation distribution with the bias voltage of the diode is added to this
increase. In order to obtain the model of the diode in the high injection
hypothesis, the effects of the storage of the charge in the junction,
which transfers the inertia of the diode during a change of direction of
the current, will need to be considered. The variation in the space
charge relative to the variations in bias voltage results in a dynamic
differential capacity, called the “junction transition capacity” and
which is given by the following expression:
C j (Vd )
C j (0)
1
C j (Vd )
for Vd FC 0
Vd
0
C j (0)
1 m
(1 FC )
mVd
1 FC (1 m)
0
for Vd FC 0
[3.20]
where Cj(0) and 0 are the capacity and potential of the junction at
zero bias, m is a coefficient which depends on the junction type (for a
linear junction m = 0.33 and for an abrupt junction m = 0.5) and FC is
a coefficient used for the change in computation formula for the
forward bias capacity (by default FC = 0.5).
When the diode is forward biased, the dominant process is the
diffusion of free carriers and the storage of charge is proportional to
the total current injected into the junction:
Qd d I d Vd
[3.21]
where d is the transit time of the diode, which represents the
minimum time required to store or free the charge. When a time-
102
Numerical Analysis in Electromagnetics
variable voltage is applied, the diffusion capacity Cd, given below,
must be added to the diode model:
Cd (Vd )
d (Qd )
q
d I s eqVd /( kT )
d (Vd ) kT
[3.22]
Other effects may make the diode’s performance different from its
ideal behavior. It is incorrect to presume that the carriers are not lost
in the depletion region and the actual diode may produce a lower
current than that given by [3.19] for low forward bias voltage values.
If the voltage applied through reverse bias is too high, then very
significant reverse currents may result which may destroy the diode.
This results in the consideration of an ohmic resistance due to the
voltage drop in the neutral region of the junction, which is necessary
for forward bias at a high voltage level. This theory justifies the diode
model illustrated in Figure 3.9.
IIDD
IIDD
RRs s
VVDD
+
IdId
C
Cdd ++CCj j
Figure 3.9. Equivalent model of the high level p-n junction diode
3.1.3.2. Diode implantation
The incident pulses in the line sections added to nodes connected
to the device (Figure 3.10) are obtained starting from formula [3.9],
which is the result of the solution of a system of nonlinear equations
which depends on the discrete model chosen.
Introduction of Discrete Elements and Thin Wires
103
For the diode model defined above, this is expressed as:
ID
V1r
V1i
Yc
Cd Vd'
VD n V1i 2 n V1r 2
C j Vd'
qVd'
dVd'
I s e kT 1 [3.23]
dt
n
Vmr Rs I D Vd'
[3.24]
m2
where:
– Is is the reverse saturation current of the diode;
– V’d is the voltage applied to the p-n junction;
– q is the electronic charge, k is the Boltzmann constant;
– T is the temperature in Kelvin;
– Cj(V’d) and Cd(V’d) are the junction and diffusion capacities
(given by equations [3.20] and [3.21]);
– Rs is the series resistance of the diode (Figure 3.9).
A nonlinear system of two equations and two unknowns (Vd’ and
V1i) are therefore obtained, which can be solved using traditional
software.
Ground plane
Metallic strip
Diode connection
Equivalent circuit
ID
V1i
r= 4.8
V1r
Id
V’D
V2
Absorbing walls
Rs
VD
Cd + Cj
i
V2r
Figure 3.10. Simulation of a micro-strip line loaded by a p-n diode: connection of the
diode in the TLM mesh and equivalent circuit of the diode
104
Numerical Analysis in Electromagnetics
3.1.3.3. Simulation results
Validations have been made in the configuration described in
Figure 3.11, using a diode model present in the SPICE library
compared with a time regime SPICE simulation, with a
1 GHz sinusoidal excitation signal. The length of the micro-strip
line is L = 75 mm. The mesh step is l = 0.8 mm and
t = 1.33 ps. Figures 3.12 and 3.13 compare the results obtained by
SPICE and TLM. The graphs are virtually identical, which validates
the approach proposed by [BIS 99]
Current (A)
Figure 3.11. Modeled SPICE scheme
Time (ps)
Figure 3.12. Current in the diode parallel connected to a 50 Ohm micro-strip line
105
Current (A)
Introduction of Discrete Elements and Thin Wires
Time (ps)
Figure 3.13. Voltage on the diode parallel connected to a 50 Ohm micro-strip line
3.2. Introduction of thin wires [LAR 06]
The modeling of thin wires in electromagnetic computation
software, and more specifically the TLM method, is part of the wider
area of the modeling of the effect of geometric details on
electromagnetic waves. These scenarios are described as being “multiscale” problems and modeling them may be arduous.
Indeed, the use of a very fine mesh which fits the shape of every
conductor requires a high number of cells and then leads to a largely
prohibitive computation time. Thus multiple numerical processes have
been proposed in order to model the interaction between the wires and
the surrounding electromagnetic field.
Consideration of the presence of the wire takes place by means of a
node incorporating a wider structure. This node transfers the influence
of the thin wire onto the surrounding fields, without recourse to a fine
mesh. In thin wire modeling, multiple numerical methods have been
106
Numerical Analysis in Electromagnetics
suggested, whether in FDTD ([HOL 81, UMA 87, MAK 02, BÉR 00,
EDE 03]) or TLM ([NAY 90, POR 92, WLO 92, CHO 0l]).
The various TLM models have seen the wire placed between two
adjacent cells [WLO 92], or at the center of one of them [NAY 90].
Another TLM school [SEW 03] favors thin wire node development,
based on rigorous field theory.
In this paper, we present the arbitrarily oriented thin wire model,
developed by Larbi [LAR 06], using the HSCN node, which is the
most complete model available today.
3.2.1. Arbitrarily oriented thin wire model
The wire is oriented in any direction defined using the director
vector x y z , where are the director cosines
(Figure 3.14).
The length of the wire within the volume of the cell is equal to .
The system of equations for the wire is given by:
v
i
C v
t
inc
i
v
L einter v R fil i
t
C
[3.25]
i indicates the current along the wire and v indicates the voltage on the
inc
is a voltage
wire; einter is the electrical field along the wire and v
source per unit length situated at any point along the wire. L, C and Rfil
are the inductance, capacitance and resistance respectively of the wire.
Introduction of Discrete Elements and Thin Wires
ξ
107
z
w
∆
y
u
x
v
Figure 3.14. Randomly oriented thin wire in the cell
3.2.1.1. Current and voltage on the wire
By applying the following normalizations: t
equation system [3.25] becomes:
l
T ; = , the
2c
v
1 l i
l
v
T C 2c
2c
inc
v
2c i
Einter V R fil i
L
l T
[3.26]
The normalized impedances and a coefficient A are defined:
L l
1 l
Z m C 2c 2c
1 l
2c
Z s L
Z m r r 2
2
l
C
c
l
A l
2c
2
1
[3.27]
108
Numerical Analysis in Electromagnetics
System [3.26] becomes:
v
i
Zm
Av
T
inc
i
v
i
Zs
Einter V R fil i
Zm
T
T
[3.28]
The propagation of the current and voltage on the wire is
characterized by two pseudo-lines on which voltage pulses V16 and
V17, defined as follows, propagate:
i,r
2 nV16
i,r
2 nV17
1V
n
2
n
1V
2
i , j , k Zm
2
2
2
i , j , k Zm
2
2
2
1 I i
n
2
n
1 I i
2
2
2
, j
, j
,k
2
2
2
,k
2
[3.29]
In order to take into account the delays in current, a short-circuit
stub is connected to the pseudo-line network. This stub supports the
V18 pulses:
i,r
Zs
2V18
n
1I
2
i, j , k
[3.30]
Of course, on this short-circuit stub, we have:
r
nV18
i
n1V18
[3.31]
Introduction of Discrete Elements and Thin Wires
109
When the finite difference method is applied to system [3.28] at
point (n+1/2, i, j, k), after several manipulations [LAR 06] we arrive
at the voltage and current on the wire:
nV
2
i
i
nV17 nV16
A 2
i, j, k
inc
inter
n E i, j , k nV i, j , k
1
,
,
I
i
j
k
n
i
i
i
R fil 2 Z s Z m 2 nV16 nV17 2 nV18
[3.32]
inc
The source voltage on the wire is expressed as: V E ' R ' I .
System [3.32] becomes:
nV
2
i
i
nV17 nV16
A 2
i, j, k
n Einter i, j , k n E ' i, j , k
1
n I i, j , k
i
i
i
R fil R ' 2 Z s Z m 2 nV16
2
V
V
n 17
n 18
[3.33]
3.2.1.2. Thin wire scattering
The scattering of pulses V16, V17, V18 is obtained by discretizing the
voltage and current using the finite difference method, centered at
point (n, i, j, k). We obtain:
r
nV16
r
nV17
r
nV18
i
Vn i, j , k Z m I n i, j , k nV17
i
Vn i, j , k Z m I n i, j , k nV16
i
nV18
Z s I n i, j , k
[3.34]
110
Numerical Analysis in Electromagnetics
The combination of [3.33] and [3.34] gives reflected pulses based
on the incident pulses:
r
nV16
inter
n E
r
nV17
Zm
2
i
i
V
V
n 16 n 17
A 2
R fil R ' 2 Z s Z m
i, j , k n E ' i, j, k 2 nV16i nV17i 2 nV18i
i
n 17
Zm
2
i
i
V
V
n 16 n 17
A 2
R fil R ' 2 Z s Z m
inter
n E
i, j , k n E ' i, j, k 2 nV16i nV17i 2 nV18i
Zs
i
r
V
V
n 18
n 18
R fil R ' 2 Z s Z m
V
inter
n E
i, j , k n E ' i, j, k 2 nV16i nV17i 2 nV18i
V
[3.35]
i
n 16
3.2.1.3. Wire inductance and interpolated electrical field
In system [3.35], the lineic inductance of the field and the
interpolated electrical field are not defined.
3.2.1.3.1. Wire inductance
The inductance L of the wire defines the interaction between the
external electrical field and the current on the wire. For a cylindrical
wire of radius a, the inductance L is given by:
L
R
ln
2 a
[3.36]
R is the radius of the sphere of influence of the electrical field on
the field. The proposition put forward by [EDE 03] is used here:
Introduction of Discrete Elements and Thin Wires
111
R= (r0+a)/2, where r0 =√3.max(u,v,w). This distance is therefore
dependent on the mesh. We therefore have:
L
2
r a
ln 0
2a
[3.37]
3.2.1.3.2. Interpolated electrical field on the wire
We assume that the influence of the electrical field on the wire is
null beyond the limit r0. The field on the wire is computed from
components Ex, Ey, Ez using a weighted interpolation method:
Einter
1
V Ei g (r ).dV
i
[3.38]
where:
– x y z is the director vector of the wire;
– Ei are the computed electrical field components at the center of
the TLM cell;
– g(r) is a weight function;
– the volume V is a sphere of radius r0 encircling the wire.
For continuity conditions, the function g(r) must satisfy the
following relations:
g (r ) 0 for r a and r r0
g (r ) is continuous in r0
g (r ) varies by 1/r
g (r )2 r.dr 1
r a
[3.39]
112
Numerical Analysis in Electromagnetics
A function fulfilling these conditions is:
0
g (r )
0
ra
r02
a
2
r
1 cos
r0
2
a a
a
2r
0 1 cos
sin
r0 r0
r0
a r r0
r r0
[3.40]
The interpolated field is written:
Einter
1
E x E y Ez g (r ).dV
V
[3.41]
The elementary volume is: V dV 0 sin d 02 d ar0 r 2 dr and the
interpolated field is:
Einter
Ex E y Ez
[3.42]
where:
a
a r0 2r02
r03 a3 2r03 2ar02
2
2 2 cos
2 a sin
3
r0
r0
4
a a
a
sin
r02 a 2 1 cos
r0 r0
r0
[3.43]
3.2.1.4. Coupling between the electromagnetic field and the current
The influence of the wire on the surrounding electromagnetic field
is expressed in Maxwell’s equations by a supplementary term: the
current density.
Introduction of Discrete Elements and Thin Wires
113
We thus have the system:
E
H t e E J
E H H
m
t
[3.44]
where σe and σm are the electrical and magnetic conductivities of the
medium.
The second equation from the previous system is not modified by
the presence of the wire. The first equation is expanded into:
0 x
0 y
0 z
Ex
H z H y
ex Ex
Jx
t
y
z
E y
t
ey E y
H x H z
Jy
z
x
[3.45]
H y H x
Ez
ez Ez
Jz
t
x
y
The current density J is based on the radial distance separating the
wire from the center of the cell, i.e.:
J r , I ( ) g (r )
[3.46]
g(r) is the weight function defined in section 3.2.1.3.2.
The distributed current density is approached by the relation:
J (r , )N.dV
J dist r , V
uvw
with N = i + j + k the base functions of the Cartesian system.
[3.47]
114
Numerical Analysis in Electromagnetics
By normalizing the coordinates x, y, z and t: x = uX, y = vY, z = wZ
and t = (t/2c)T, (the normalized electrical fields E and normalized
magnetic fields H are the voltage dimensions), [3.45] becomes:
0 x
Ymy H y I ( ) V g (r ).dV
Y
H z
2c Ex ex
Ex mz
u
Z 0 .v.w Y Z 0 .v.w Z
u.v.w
l.u T
0 y
H x
Y
Y
H z I ( ) V g (r ).dV
2c E y ey
mz
E y mx
l.v T
v
Z 0 .u.w Z Z 0 .u.w X
u.v.w
0 z
Ymy H y
H x I ( ) V g (r ).dV
Y
2c Ez ez
mx
Ez
l.w T
w
Z 0 .v.u X
Z 0 .v.u Y
u.v.w
[3.48]
with the following normalized admittances:
Ymx
l.u
x vw
, Ymx
l.v
y uw
and Ymx
l.w
z vu
[3.49]
as well as:
Ysx
Ysy
Ysz
4 x vw
2Ymz 2Ymy
u l
4 y uw
v l
2Ymz 2Ymy
4 z uv
2Ymx 2Ymy
wl
[3.50]
Introduction of Discrete Elements and Thin Wires
115
The equations of [3.48] become:
H y IZ 0
H z
Ysx
E x
2 Ymz Ymy T Gex E x Ymz Y Ymy Z u
Ysy
E y
H x
H z IZ 0
Ymz Ymx
Gey E y Ymx
Ymz
Y
2
T
Z
v
H y
H x IZ 0
Ysz
E z
2 Ymx Ymy T Gez E z Ymy X Ymx Y w
[3.51]
vw
uw
vu
, I is the current
; Gey Z 0 ey
; Gez Z 0 ez
u
v
w
in the wire and is defined by expression [3.43]. Maxwell’s
where: Gex Z 0 ex
equations for the magnetic field are not modified by the current on the
wire and invoke the magnetic conductivities which, once normalized,
occur with the equations in the following form:
Gmx
Ymx vw
;
Z0 u
Gmy
Ymy uw
;
Z0 v
Gmz
Ymz uv
.
Z0 w
By applying the finite difference method to point (n, i, j, k) in
expressions [3.51], following a number of manipulations [LAR 06],
we obtain the expression for reflected voltages, based on the incident
pulses. These are divided into three parts:
– The 12 HSCN node pulses:
r
nV1
i
n E x i, j , k n H z i, j , k V12
r
nV2
n E x i, j , k n H y i, j , k V9i
r
nV3
i
n E y i, j , k n H z i, j , k V11
r
nV4
n E y i, j , k n H x i, j , k V8i
r
nV5
n E z i, j , k n H x i, j , k V7i
[3.52]
116
Numerical Analysis in Electromagnetics
r
nV6
i
n E z i, j , k n H y i, j , k V10
r
nV7
n E z i, j , k n H x i, j , k V5i
r
nV8
n E y i, j , k n H x i, j , k V2i
r
nV9
i
n E x i, j , k n H y i, j , k V12
r
nV10
n E z i, j , k n H y i, j , k V6i
r
nV11
n E y i, j , k n H z i, j , k V3i
r
nV12
n E x i, j , k n H z i, j , k V1i
[3.52(cont.)]
– The 3 open circuit stubs:
r
nV13
r
nV14
i
n E x i, j , k V13
r
nV15
i
n E z i, j , k V15
i
n E y i, j , k V14
[3.53]
– The 2 pulses and the short-circuit stub linked to the wire:
r
nV16
r
nV17
r
nV18
i
Vn i, j , k Z m I n i, j , k nV17
i
Vn i, j , k Z m I n i, j , k nV16
[3.54]
i
nV18
Z s I n i, j , k
The scattering matrix from Figure 3.15 is then obtained. In this
table, the index in the quantities a, b, and d represent values xyz
in the order indicated in the 1st column.
The various coefficients of the matrix are defined as follows:
a αβγ
bαβγ
Ym
Ys Ge 2 Ym Ym
2Ym
Ys Ge 2 Ym Ym
2
Gm 4
[3.55]
[3.56]
Introduction of Discrete Elements and Thin Wires
c
2
Gm 4
d αβγ
eαβγ
[3.57]
2Ym
Ys Ge 2 Ym Ym
2Ys
Ys Ge 2 Ym Ym
Y f
f .k
2
117
2
Gm 4
[3.59]
i
j
or : k x , k y
u
v
[3.58]
k
and k z
w
1
Z
Yf m
g m± 2
Z0
A 2
[3.60]
[3.61]
The coefficients A, Zm, Zs, and Yf are:
– A, the constant, taking into account losses in the medium,
A
l
;
2c
[3.62]
– Zm, the impedance encountered by the pulses on the wire,
Z m L
2c
L l
;
l 2c
[3.63]
– Zs, the characteristic impedance of the short-circuit stubs,
Z s L
2
2c
1 l
2
Z m r r
1 ;
l C 2c
l
[3.64]
– Yf, the normalized admittance of the wire,
Yf
Z0
.
R R ' 2 Z m Z s
[3.65]
b
bzyx
bzxy
byxz
bxzy
b
cy
a
- cy
d -1
b
a
cy
b
d -1
b
-cy
10
11
b
cz
a
b
b
d -1
-cz
12
b
a
cz
b
-cz
b
d-1
13
exyz -1
exyz
exyz
exyz
exyz
eyzx -1
eyzx
eyzx
eyzx
eyzx
14
ezxy -1
ezxy
ezxy
ezxy
ezxy
15
16
-2fz
-2fy
-2fx
-2fx
-2fy
-2fz
-2fx
-2fy
-2fz
-2fz
-2fz
-2fy
-2fy
-2fx
-2fx
Figure 3.15. Scattering matrix of the HSCN node with a thin wire
-2Yfs
byxz
b
9
b
18
b
-cz
b
b
a
cx
- cx
d -1
b
8
gm+-1
15
zxy
b
bxzy
d -1
b
cx
a
b
d -1
-cx
7
17
14
yzx
b
d -1
b
- cy
- cx
d -1
a
b
cy
6
d -1
d -1
- cx
a
cx
b
cx
a
b
5
-cy
b
b
a
4
d -1
cy
a
3
cz
gm-
13
xyz
-cz
b
cz
2
b
16
12
xyz
8
yzx
11
7
zyx
yxz
6
zxy
9
5
zyx
10
4
yzx
zxy
3
yxz
xzy
2
xzy
b
1
a
1
xyz
17
18
-2fz
-2fy
-2fx
-2fx
-2fy
-2fz
-2fx
-2fy
-2fz
-2fz
-2fz
-2fy
-2fy
-2fx
-2fx
-fz
-fy
-fx
-fx
-fy
-fz
-fx
-fy
-fz
-fz
-fz
-fy
-fy
-fx
-fx
Ex
2Yfm
Yfm
2Yfs 1-2Yfs -Yfs
j
gm+-1 -2Yfm -Yfm
2fz
2fy
2fx
-2fx
2fy
2fz
2fx
2fy
2fz
2fz
2fz
2fy
2fy
2fx
2fx
118
Numerical Analysis in Electromagnetics
Introduction of Discrete Elements and Thin Wires
119
3.2.2. Validation of the arbitrarily oriented thin wire model
Validation of the thin wire model is performed on a dipole
consisting of a wire 41 m in length and 1 cm in diameter. Mesh is
uniform, with u = v = w = 10 mm. The computation of the input
impedance of this dipole in various orientations enables the algorithm
previously explained to be tested. This impedance must be
independent of the orientation and of course equal to the theoretical
value. Moreover, the various resonance frequencies are also
independent of orientation. Since the wire is parallel to the Ox axis
(t00), it is subject, in the first instance, to movements in the yOz plane,
with steps 0.25v (t25) and 0.50v+0.50w (t55) (Figure 3.16a). The other
series of transformations consists of making it carry out rotations of
27° (r27) and 45° (r45) in the xOy plane (Figure 3.16b).
y
z
x
y
55
27
00
25
45
(a)
(b)
Figure 3.16. Dipole configurations
This dipole is fed at its center using the localized voltage source,
which is integrated with the wire. The excitation has the appearance of
a Gaussian pulse:
f (t )
2 t t0
t0 12 10
tw
8
t t
e 0
2
/ tw2
s , tw 4 10
[3.66]
8
s
120
Numerical Analysis in Electromagnetics
The TLM
characteristics:
simulations
carried
out
have
the
Computation volume:
27 × 67 × 27 nodes
Cell dimensions l:
1m
Number of cells in the PML:
10
Excitation:
[2 MHz; 24 MHz]
Number of processors:
8
Number of iterations:
2,000
following
Frequency (MHz)
Figure 3.17. Input resistance of the dipole obtained using thin wire modeling
Figure 3.17 enables a visualization of the real part of the input
impedance of the dipole for the five configurations from Figure 3.16.
There is no difference between any of these various configurations.
Moreover, the anti-resonance frequencies and the maximum
amplitudes of the input resistance conform to those from the method
of moments (MoM) used for the purposes of comparison.
Introduction of Discrete Elements and Thin Wires
121
Table 3.1 summarizes the values for the half-wave resonance
frequency, as well as those for the various simulated cases. The
relative error between f0 and ZTLM is computed by taking the
theoretical values as a reference: f0 = 3.66 MHz and Z = 73
Provision
f0 (MHz)
f0
ZTLM ()
ZTLM
t00
3.64
0.54%
72.2
1.10%
t25
3.64
0.54%
72.2
1.10%
t55
3.64
0.54%
72.3
0.96%
t27
3.65
0.27%
72.3
0.96%
T45
3.64
0.54%
72.1
1.23%
Table 3.1. Frequencies and resistances of the half-wave resonance of the dipole
The relative errors obtained for resonance frequencies are less than
1.5%, which is quite acceptable and thus validates the method.
Chapter 4
The TLM Method in Matrix Form
and the Z Transform
4.1. Introduction
In Chapter 2 we saw the various nodes that can be used in the
traditional form of the TLM method. For example, the SCN node
formed of 12 transmission lines, to which open circuit and shortcircuit stubs, representing permittivities and permeabilities
respectively, are added. Matched lines can be added in order to
account for lossy media. This approach is effective for a simple
medium whose parameters are independent of frequency.
Nevertheless, complex media, such as dispersive or anisotropic media,
whose parameters are dependent on frequency, require a different
process in order to account for this variation. An interesting approach
has been presented by Christopoulos [PAU 99], where the propagation
of a wave in a vacuum is separated from the wave-matter interaction.
The propagation is handled by the traditional TLM method [HOE 92],
using 12-input nodes (vacuum), whereas the wave-matter interaction
appears in the form of additional electromagnetic field sources. Since
the interaction is causal, the Z Transform can be used to treat the
problem. The advantage of this application is that convolution
124
Numerical Analysis in Electromagnetics
products can be avoided in the temporal processing of dispersive
media. Moreover, in this case, the algorithm is unconditionally stable.
In this chapter we are going to present the matrix formulation of
Maxwell’s equations, then the application and implementation of the
Z Transform, which will enable the simulation of dispersive media.
4.2. Matrix form of Maxwell’s equations
Maxwell’s equations and the constitutive relations of the matter
can be expressed in a matrix form, which is convenient for
formulating the problem:
Je
H
D
t
E
B
J m
0 e
D
0E
0 H
B
r
c
(a)
(b)
[4.1]
r
E
H
0 m
c
[4.2]
(a) expresses the relation in the vacuum, and (b) the wave-matter
interaction.
The currents are given by:
e
Je
J ef
J
J
m
mf
me
em
E
H
m
[4.3]
The TLM Method in Matrix Form and the Z Transform
125
e and m are the electrical and magnetic susceptibility tensors,
respectively (without dimension); r
and r are the electromagnetic
and magnetoelectrical coupling tensors; Jef and Jmf are the free
electrical and magnetic currents; e and m are the electrical and
magnetic conductivity tensors and em and me reflect an eventual
magnetoelectrical interaction. All of the tensors are (3 3).
The notation “” in equations [4.2] and [4.3] in the remainder of
this chapter expresses convolution in the time domain.
4.3. Cubic mesh normalized Maxwell’s equations
In this section, the mesh of the elementary 3D TLM node cell is
such that x = y = z = l (Figure 4.1).
y
y
x
z
z
x
Figure 4.1. The elementary cell used
The normalizations are as follows:
E
where 0
-V
l
and
0
0
H
-i
l 0
is the free space impedance.
[4.4]
126
Numerical Analysis in Electromagnetics
The normalized current sources Jef and Jmf are also obtained:
J ef
-i f
2
l 0
J mf
and
-V f
[4.5]
l2
which leads to normalization on the electrical and magnetic
conductivity:
e
ge
l 0
; m
g m 0
l
; em
gem
l
; me
g me
l
[4.6]
The spatial and temporal derivatives are also normalized:
x
1
;
l x y
t
1
t t
1
;
l y z
1
;
l z
[4.7]
t is the time step and t is the normalized time.
By applying these normalizations in 3D TLM, with l 2 c
the following expression is obtained:
i - if
- V - Vf
V g
e
-2
t g me
i
V
gem
e
2
t r
g me i
c
t
r
V
c
i
m
[4.8]
The TLM Method in Matrix Form and the Z Transform
127
The first part of equation [4.8] refers to propagation in a vacuum,
and can be solved using the traditional TLM method. The second part
deals with wave-matter interaction, which behaves in the same way as
the sources.
4.4. The propagation process
The propagation process is directly based on the equivalence
between the model deduced from the circuit laws and that computed
by Maxwell’s equations. The formulation for 3D TLM is based on the
symmetrical condensed node (SCN), which is a node with 12 different
polarizations represented by the 12 corresponding voltage pulses
Vi = [V1 V2 …. V12] T on its six arms (see Figure 2.3). The six
electromagnetic field components (Ex, Ey, Ez, Hx, Hy and Hz) are
defined at the center of the cell formed by this node.
The “total field” vector F = [Vx Vy Vz ix iy iz]T is defined by:
Vx V1 V12 V2 V9
V y V4 V8 V3 V11
Vz V6 V10 V5 V7
ix V4 V8 V5 V7
i y V6 V10 V2 V9
[4.9]
[4.10]
iz V1 V12 V3 V11
This total field vector enables the components of the
electromagnetic field to be expressed based on the voltage pulses Vi,
by taking into account the normalizations defined above.
The propagation process occurs in three essential stages:
– Definition of the external excitation vector at node FR from
incident voltage pulses and free excitation sources Ff:
128
Numerical Analysis in Electromagnetics
R
Vx
V
y
V
z
ix
i
y
iz
This
F R R
vector
i fx
i fy
1 i fz
2 V fx
V fy
V fz
i
Vx
V
y
V
z
ix
i
y
iz
is
[4.11]
expressed
in
1
Vi F f , where:
2
R
1
0
0
0
0
1
simple
matrix
1 0
0
0
0
0
0
1
0
0 1
1
0
0
0
1
0
0
0 0
0
1
1
1
0
0
1
0 0 1 1 0 1 1 0 0
1 0 0 0 1 0 0 1 1
0 1
0
0
0
0
0
0
0
form,
0
1
1 0
0 0
0 0 [4.12]
0 0
1 1
– Evaluation of the total fields: the left hand side of relation [4.8]
with these definitions is written 2 FR – 4 F. The relation between the
“total field” vector F and the “excitation” vector FR is obtained by
solving the general equation [4.8]. This is performed by using the Z
transform. It is expressed by the matrix tz , which represents the
matrix of transmission coefficients.
We can then write:
F
by:
t z F R
[4.13]
– Computation of reflected pulses: the reflected pulses are obtained
VR
Rt F - Vci
[4.14]
The TLM Method in Matrix Form and the Z Transform
129
where:
VR = [V2 V9 V1 V12 V3 V11 V4 V8 V5 V7 V6 V10 ] T
Vci = [V9 V2 V12 V1 V11 V3 V8 V4 V7 V5 V10 V6 ] T
and the matrix [Rt]:
1 0 0
1 0 0
1 0 0
1 0 0
0 1 0
Rt 00 11 00
0 1 0
0 0 1
0 0 1
0 0 1
0 0 1
0 1 0
0 1 0
0 0 1
0 0 1
0 0 1
0 0 1
1 0 0
1 0 0
1 0 0
1 0 0
0 1 0
0 1 0
The process is summarized in the following organigram
(Figure 4.2):
Vci
P
V
i
Ff
RT
FR
+
-
t (z)
F
R
+
Vr
0.5
Figure 4.2. Organigram of the propagation process for the TLM with the Z transform
The role of the [P] matrix in this organigram is to arrange incident
voltage pulses Vi in order to obtain the vector Vci . It should also be
130
Numerical Analysis in Electromagnetics
noted that the components of the reflected pulses vector VR are
renumbered.
4.5. Wave-matter interaction
The term on the right-hand side of equation [4.8] involves
convolution products. In order to deal with this problem, the Z
transform is used which enables the products of convolution to be
passed to simple products. A bilinear transformation is used, of the
form:
1 - z -1
2
t 1 z -1
s
[4.15]
If s = j, represents the derivative, we obtain:
t
2
1 - z -1
[4.16]
1 z -1
where z-1 is the time shift t.
Under these conditions, the second part of relation [4.8] can be
written based on the Z transform, as follows:
V
V
1 - z-1
Z
G
G
z F
F
2
z
4
t
1 z-1
i
i
[4.17]
where:
G z
ge
g me
g em
and z e
gm
r
r
.
m
The TLM Method in Matrix Form and the Z Transform
131
Relation [4.8], including these transforms, is written:
1 - z -1
2 FR - 4 F G F 4
1 z 1
[4.18]
F
or:
2 1 z -1 F R 1 z -1 4 F 1 z -1 G F 1 z -1 4 F
[4.19]
In matrices G and there are some constant elements and others
which are dependent on frequency. In order to cause the frequency
dependence, we make use of the causality of elements in order to use
the shift in time.
Thus, the expression z-1 F transfers the value for F at the previous
instant. This is written:
1 z-1 G z
G0 z -1
1 z-1 z
0 z -1
G
1
1
G z
z
[4.20]
[4.21]
where G0 , G1 , 0 and 1 are frequency independent, which then
gives us:
2 1 z 1 F R 1 z 1 4 F 1 z 1 G F 1 z 1 4 F
[4.22]
2 1 z 1 F R 4 G 0 z 1 G1 G z 4 0 - 4 z -1 1 z F
[4.23]
132
Numerical Analysis in Electromagnetics
Following further development of the computations, we arrive at
the following expression:
F T0 2 F R z -1 S
[4.24]
where:
T0 4 G 0 4 0
1
S 2 F R T1 F - G z F 4 z F
T1 - 4 G1 4 1
From these last relations, the organigram of the process is
established, including all of the permittivity, permeability and
conductivity tensors (Figure 4.3).
F
R
2
+
T0
F
+
z 1
S
+
+
T1 - G z 4 z
Figure 4.3. Organigram of the process
In order to model the media under investigation, by knowing the
general tensors G z and z , it only remains for expressions [4.20]
and [4.21] to be re-used in order to determine the terms G z , G 0 ,
G1 , 0 , 1 and z (for example, see section 4.7).
The TLM Method in Matrix Form and the Z Transform
133
4.6. The normalized parallelepipedic mesh Maxwell’s equations
[LOU 04]
A variable step mesh is often used for structures with
inhomogeneities, i.e. the application of a thin mesh in high field
gradient areas and then a coarse mesh in the rest of the structure.
For the development which follows, the following normalizations
are used:
lx = u.l, ly = v.l and lz = w.l,
where l is the spatial step used in the process.
Thus, the following normalized values are obtained:
Ex -
1 Vy
1 Vx
1 Vz
; Ez ; Ey ;
l v
l u
l w
Hx -
iy
ix
i
1
1
1
; Hy .
; Hz - z
u l 0
w l 0
v l 0
where 0
0
0
is the free space impedance.
Time is also normalized:
t
1
t t
The tensor values r , e and m that need to be taken into
account in the process of the method are based on the elementary
dimensions (u, v, w) of the TLM cell.
Maxwell’s equations are solved by considering relation [4.2]:
– First equation: rot E
B
.
t
134
Numerical Analysis in Electromagnetics
The curl is written:
rot E
B
r
E 0 m H
0 H
t t
t c
t
[4.25]
After further development (Appendix A), the final expression for
the equation becomes simpler and is written in the following form:
V - Vf - 2
i
2
V m
t
t r
i
[4.26]
and the new tensor parameters are given by:
r 1,1
vw
r 1,1
u
r 1, 2 wr 1, 2
r 1,3 vr 1,3
r 2,1 wr 2,1
r 2, 2
r 2,3 ur 2,3
r 3,1 vr 3,1
r 3, 3
m 1,1
wu
r 2, 2
v
r 3, 2 ur 3, 2
uv
r 3, 3
w
vw
m 1,1 1 1
u
m 1, 2 wm 1, 2
m 1,3 vm 1,3
m 2,1 wm 2,1
m 2, 2
m 2,3 u m 2,3
m 3,1 vm 3,1
m 3,3
wu
m 2, 2 1 1
v
m, 3,2 u m, 3,2
uv
m 3,3 1 1
w
The TLM Method in Matrix Form and the Z Transform
– Second equation: rot H
135
D
.
t
Using equation [4.2], the curl is expressed in the following form:
rot H
D
r
0 e E
0 E
H
t t
t c
t
[4.27]
and, in this case, equation [4.8] is written as:
i - i f - 2
V
V r i
2
t
t e
[4.28]
The new tensor parameters are given by (Appendix A):
e 1,1
vw
e 1,1 1 1
u
e 1, 2 we 1, 2
e 1,3 v e 1,3
e 2,1 we 2,1
e 2, 2
e 2,3 u e 2,3
e 3,1 v e 3,1
e 3, 3
r 1,1
wu
e 2, 2 1 1
v
e 3, 2 u e 3, 2
uv
e 3, 3 1 1
w
vw
r 1,1
u
r 1, 2 wr 1, 2
r 1,3 vr 1,3
r 2,1 wr 2,1
r 2,3 ur 2,3
r 3,1 vr 3,1
r 3,3
r 2, 2
wu
r 2, 2
v
r 3, 2 ur 3, 2
uv
r 3, 3
w
136
Numerical Analysis in Electromagnetics
Once the tensor parameters have been computed, we then have an
identical situation to that of the uniform mesh.
4.7. Application example: plasma modeling [MOU 06]
4.7.1. Theoretical model
By way of example, we will discuss the case of non-magnetized
plasma. Its permittivity tensor is given by:
p
p
0 p 0
0
where p 1
0
p
0
0
0
p
p2
refers to the relative permittivity of
j p
the plasma.
The electrical susceptibility tensor of the plasma is expressed in the
form:
p 1
0
0
e 0
p 1
0
0
p 1
0
Within the technical context of the Z transform for the TLM
method, the plasma can be handled as a purely dielectric medium. Its
conductivity tensor is null and its susceptibility tensor is reduced to:
e
0
0
0
The TLM Method in Matrix Form and the Z Transform
137
The representation of the computation process, which enables the
plasma to be integrated, occurs through solving the ( z ) equation
(equation [4.21]), having applied the Z transform to the expression for
its susceptibility tensor. Indeed, this solution enables the parameters of
the plasma, which must be considered in the algorithm of this Z
transform technique for the TLM method, to be determined. This
computation is developed in Appendix B. To summarize, the
following parameters are obtained:
1 0 ,
0
b0'
p 1 a1
,
b2'
p
p
1
0
1
0
1
0
0
0
0
0
,
b1'
p 1 a2
1
0
0
0
0
0
where:
p
p 2 t 2
2
2
p t
; a1
4
2 p t
; a2
2 p t
2 p t
The solution obtained using the Z transform technique for the TLM
method is written in the following form:
F T0 2 F R z -1 S
138
Numerical Analysis in Electromagnetics
where:
T0 4 G 0 4 0
1
S 2 F R T1 F - G z F 4 z F
T1 - 4 G 0 4 1
For the plasma, the following is obtained:
T0 4 4 0
1
S 2F R T1F + 4 z F
T1 4 4 1
4 z F 4
1
b0' b1' z 1 b2' z 2
1 a1 z 1 a2 z 2
F
In order to evaluate the term 4 z F , we make use of the phasevariable states technique. We obtain:
4 z F 4 b0'
X1
X 2
(b1' b2' z 1 )
where:
X1
z 1
X 2
a1
1
a2
0
X1 1
F
X 2 0
The TLM Method in Matrix Form and the Z Transform
139
This computation process is summarized by the organigram in
Figure 4.4.
F
+
+
+
X2
+
+
X4
+
Figure 4.4. Organigram of the computation of a plasma medium
4.7.2. Validation of the TLM simulation
Validation of the propagation simulation concept is performed on a
structure using a parallelepipedic resonant metallic cavity (Figure 4.5).
Firstly, the resonant modes are determined by simulating an empty
cavity, which is thus considered to be a reference cavity. Then,
cavities filled with plasma are simulated and their responses compared
with that of the reference cavity.
The TLM method has been designed in order to deal with media
with permittivity and permeability such that r 1 and r 1 . When
this is not the case, the convergence of its algorithm is not guaranteed.
In the plasma example, the condition regarding permittivity is not
fulfilled (see [4.29]). In order to compensate for this problem, a
dielectric, or static constant can be added to the plasma, which thus
presents a constant permittivity r 1 and which compensates for the
value of the permittivity brought into play by the plasma, in order to
avoid this type of divergence. This value for static permittivity is
140
Numerical Analysis in Electromagnetics
applied during testing of the reference cavity in order to isolate the
response obtained by the plasma effect.
y
x
z
Figure 4.5. Metallic cavity
In order to achieve simulations which enable the plasma to be
modeled, cavities with the dimensions 23 × 17 × 60 µm3 have been
used. For these cavities, a static permittivity with a value of 2 has been
used. At the same time, several plasma types have been tested. Indeed,
the plasma is characterized by its plasma angular frequency p and the
frequency of collisions between particles p. For the micrometric
dimensions chosen, the plasma samples tested are as follows:
Sample #1: p = 5,640 G rad/s, p = 100 GHz
Sample #2: p = 8,460 G rad/s, p = 100 GHz
Sample #3: p = 11,280 G rad/s, p = 100 GHz
This choice for the value p is justified by the approximation,
which is often used in low-pressure plasma studies, enabling the
The TLM Method in Matrix Form and the Z Transform
141
expression for the relative permittivity of the plasma to be simplified.
It is expressed in the form:
p2
2
p 0 p 0 1
[4.29]
The simulation of these cavities is achieved using a network of
transmission lines with a base mesh l of 1 µm and a time
discretization step t of 1.667 × 10-15 s. Each simulation uses
200,000 iterations. Excitation is a Dirac signal.
For each cavity, the response obtained is compared with the
reference cavity, as well as the expected theoretical values.
For this series of cavities, the resonant modes researched were 101,
102 and 103.
cavity
… Vaccum
Vacuum
constant
+ +constant
__ Plasma
Plasma
Plasma
Reference cavity
The responses from the filled cavities for plasma #1, #2 and #3
samples are given in Figures 4.6 to 4.8. Each response is compared
with the reference cavity corresponding to this series.
Frequency (THz)
Figure 4.6. Resonant frequencies in the filled cavity of a
plasma #1 sample, simulated by the TLM method
142
Numerical Analysis in Electromagnetics
Plasma
Reference cavity
cavity
+ +constant
… Vaccum
Vacuum
constant
__ Plasma
Plasma
Frequency (THz)
Figure 4.7. Resonant frequencies in the filled cavity of a
plasma #2 sample, simulated by the TLM method
Plasma
Reference cavity
Plasma
cavity
… Vaccum
Vacuum
constant
+ +constant
__ Plasma
Frequency (THz)
Figure 4.8. Resonant frequencies in the filled cavity of a
plasma #3 sample, simulated by the TLM method
The TLM Method in Matrix Form and the Z Transform
143
The resonant frequencies of each resonant mode of all of the
cavities are given by the formula:
f mnp
Resonant
modes
101
102
103
c
r
m 2
2a
2
n
2b
Resonant frequency
[in THz] of the
plasma cavity
(theoretical)
5.022
5.881
7.086
p
2l
1
2 2
[4.30]
Resonant frequency
[in THz] of the
plasma cavity using
the TLM method
5.019
5.880
7.086
Relative
error
[%]
0.05
0.02
0.0
Table 4.1. Comparison of results obtained using the TLM method with theoretical
values for a cavity filled with the plasma #1 sample
Resonant
modes
101
102
103
Resonant frequency
[in THz] of the
plasma cavity
(theoretical)
5.133
5.973
7.161
Resonant frequency
[in THz] of the
plasma cavity using
the TLM method
5.121
5.964
7.155
Relative
error
[%]
0.2
0.15
0.08
Table 4.2. Comparison of results obtained using the TLM method with theoretical
values for a cavity filled with the plasma #2 sample
Resonant
modes
101
102
103
Resonant frequency
[in THz] of the
plasma cavity
(theoretical)
5.301
6.110
7.269
Resonant frequency
[in THz] of the
plasma cavity using
the TLM method
5.256
6.081
7.254
Relative
error
[%]
0.85
0.47
0.21
Table 4.3. Comparison of results obtained using the TLM method with theoretical
values for a cavity filled with the plasma #3 sample
144
Numerical Analysis in Electromagnetics
The comparison of resonant frequency values for each cavity,
obtained by simulation, compared to those computed using traditional
formulae, enable the relative errors resulting from these simulations to
be estimated. The value of these errors does not exceed 1%. This low
simulation error rate confirms the good modeling of wave propagation
for the TLM method and the Z transform.
4.8. Conclusion
The application of the Z transform in the TLM method is
extremely useful for two main reasons. On one hand, it enables the
rigorous simulation of physical dispersive media (causality property)
without the need to deal with problems from convolution, which are
always memory hungry. On the other hand, the process is
unconditionally stable. It can, of course, be used for non-dispersive
media, although, historically, the traditional method is always used.
APPENDICES
Appendix A
Development of Maxwell’s Equations
using the Z Transform with a
Variable Mesh
Maxwell’s equations are written (without considering sources) as
follows:
– First equation: rot E
B
t
E
E y
z y
z
Bx
r
0 H x
E
t
t
t c
x
0 m H x
t
E
E z
x z
x
B y
r
0 H y
E
t
t
t c
y
0 m H
t
E y E
x
x
y
Bz
r
0 H z
E
t
t
t c
z
0 m H
t
y
z
148
Numerical Analysis in Electromagnetics
For normalized field values, we obtain:
1
l wv
2
Vz
y
-
1
l wv
2
1 r
E
=
t t c
1
l wu
2
Vx
z
-
1
l uv
2
Vy
x
-
1
l uv
2
1 r
E
t t c
=
-
Vz
1
l wu
2
1 r
E
t t c
=
0
ix
1
t l 0 t u
z
0 m H x
x
t
Vy
-
x
y
t
iy
v
H y
0
1
t l 0 t
0 m
0
iz
1
-
t l 0 t w
y
H
z
0 m
z
t
Vx
-
Knowing that l = 2 c t, and following simplification, we obtain:
rot V
rot V
rot V
x
2
l2
wv
ix
t
t u
r E
t c
x
y
2
l2
uw
iy
t
t v
r E
t c
y
z
2
l2
uv
iz
t
t w
r E
t c
z
t
t
t
0 m H
vw
x
0 m H
uw
y
0 m H
uv
z
Appendix A
149
In order to be able to use the traditional TLM algorithm, the term
2 i x is added to both sides of each equation:
t
rot V
l2
t
rot V
l2
t
rot V
l2
t
x
2
t
ix 2
t
i x -2
wv
ix
t u
r E
0 m
x
t c
t
2
iy 2
iy - 2
y
t
t
t
r E
0 m
y
t c
t
2
iz 2
iz - 2
z
t
t
t
r E
0 m
z
t c
t
H
vw
x
uw
iy
v
H
uw
y
uv
iz
w
H
uv
z
We can see that each part of this set of equations is simplified and
can be integrated into the classic TLM process. It only remains for us
to develop the second part. For example, for the first line, we obtain:
2
r 1,1 Vx
1,2 Vy
wv
i x -2
ix - 2 c
vw r
vw
c
u
c
v
t
t
u t
r 1,3 Vz
c
w
0 m 1,3
vw 0 m 1,1
iz
0
vw
w
i y vw
i x vw
μ 0 χ m 1,2
η0 u
η0 v
150
Numerical Analysis in Electromagnetics
Knowing that
c 0
0
1 , this second term is then expressed in the
following form:
2
t
wv
i x 1 2
u
t
r 1,1
r 1,3 Vz v 0 m 1,1 i x
Vx
vw r 1,2 Vy w
u
vw
μ 0 χ m 1,2 i y w 0 m 1,3 i z v
u
We define the new parameters based on the elementary dimensions
of the TLM cell:
vw
r 1,1 , r 1,2 w r 1,2 ,
u
r 1,1
r 1,3 v r 1,3 ,
m 1,1
vw
vw
vw
m 1,1 -1
m 1,1 1 - 1 ,
u
u
u
m 1,2 w m 1,2 , m 1,3 v m 1,3 .
For the other terms, we follow the same process of development
and obtain:
r 2,1 w r 2,1 , r 2,2
uw
r 2,2 ,
v
r 2,3 u r 2,3 ,
r 3,1 v r 3,1 , r 3,2 u r 3,2 ,
r 3,3
m 2,2
uv
r 3,3 , m 2,1 w m 2,1 ,
w
uw
m 2,2 1 -1 ,
v
m 2,3 u m 2,3 , m 3,1 v m 3,1 ,
m 3,2 u m 3,2 , m 3,3
uv
m 3,3 1 -1 .
w
Appendix A
151
These new terms are taken into consideration in the 3D TLM
algorithm.
In this case, the final version of [4.26] is more simply expressed as:
V - Vf - 2
i
V m i .
2
t
t r
– Second equation: rot H
D
t
H z H y
r
0
H
Ex
0 e E x
y
z
t
t
t c
H x H z
0 e E
0
Ey
z
x
t
t
H y
x
-
y
r
H
t c
H x
r
0
H
Ez
0 e E z
y
t
t
t c
x
y
z
In the same way, the first line is expressed using normalizations:
i y
Vx
i z
1
1
1
0
2
2
t t u l
0 vw l y vw l z
V
0 e 1,1 - Vx 0 e 1, 2 - y + 0 e 1,3 - Vz +
u l
w l
v l
1
=
t t 1,1
iy
r 1, 2
r 1,3
ix
iz
r
w l
u l
v l 0
c
c
c
0
0
152
Numerical Analysis in Electromagnetics
or:
vw
vw
e 1,1 Vx 2 w
e 1, 2 Vy
Vx 2
u t
u t
t
vw
e 1,3 Vz 2 v
e 1,3 Vz 2
r 1,1 i x
+2v
u t
t
t
r 1, 2 i y + 2 v
r 1,3 i z
+2w
t
t
rot i
x
2
As with the first Maxwell equation, a term - 2 Vx is added to
t
each part, which gives:
Vx 2
e 1,1 Vx 2
e 1, 2 Vy 2
e 1,3 Vz
t
t
t
t
r 1,1 i x 2
r 1, 2 i y 2
r 1,3 i z
2
t
t
t
rot i
x
2
Hence we deduce the general parameter values:
r 1,1
vw
r 1,1 , r 1,2 w r 1,2 ,
u
r 1,3 v r 1,3 ,
e 1,1
vw
e 1,1 1 - 1 , e 1,2 w e 1,2 , e 1,3 v e 1,3
u
r 2,1 w r 2,1 , r 2,2
uw
r 2,2 , r 2,3 u r 2,3
v
r 3,1 v r 3,1 , r 3,2 u r 3,2 , r 3,3
e 2,1 w e 2,1 , e 2,2
uv
r 3,3
w
uw
e 2,2 1 -1 , e 2,3 u e 2,3
v
e 3,1 v e 3,1 , e 3,2 u e 3,2 , e 3,3
uv
e 3,3 1 -1
w
Appendix A
and in this instance, equation [4.28] is written:
i - if - 2
V
2
V r i
t
t e
153
Appendix B
Treatment of Plasma using the Z
Transform for the TLM Method
For a non-magnetized plasma, the permittivity tensor is given by:
p
p
0 p 0
0
where p 1
0
0
p
0
p
0
p2
refers to the relative permittivity of
j p
the plasma.
The electrical susceptibility tensor of the plasma is written in the
form:
p 1
e 0
0
0
p 1
0
0
0
p 1
156
Numerical Analysis in Electromagnetics
Within the technical context of the Z transform for the TLM
method, the plasma can be handled as a purely dielectric medium. Its
conductivity tensor is null and its susceptibility tensor is reduced to:
e
0
0
0
The Z transform is applied to this tensor of the form:
j
1 z 1
2
t
1 z 1
This amounts to treating the element:
p 1
p2
j p
It is expressed in the form:
p 1 p2
1
j p
j
With the transform above, this becomes:
p 1 p2
1
1
1
2 1 z 2 1 z
p
t 1 z 1 t 1 z 1
The simplification reduces this element to:
p 1
p 2 t 2
2
.
1 2 z 1 z 2
2 p t 4
z 1 2 p t
z 2
Appendix B
157
It can be expressed in the following form, which is practical for the
following computations:
p 1
p
b0 b1 z 1 b2 z 2
1 a1 z 1 a2 z 2
where:
p
p 2 t 2
2
2 p t
b0 1 , b1 2 , b2 1
and:
a1
4
2 p t
a2
2 p t
2 p t
The equation (see equation [4.21]) corresponding to the
susceptibility tensor z must be solved:
1 z-1 z
0 z -1
1
z
This equation is reduced for the plasma to its electrical
susceptibility tensor:
1 z 1
e z e0 z 1 e1 e z
This solution amounts to identifying
second part of this equation.
1 z 1
p 1 in the
158
Numerical Analysis in Electromagnetics
We set the following:
e0 z
1
e1
e z A0 z
be' 0 be' 1 z 1 be' 2 z 2
B
1
2
0
1 a1 z a2 z
1
By developing this equation, we obtain:
p b0 b1 b0 z 1 b2 b1 z 2 b2 z 3 A0 a1 A0 B0 be' 0 z 1
a2 A0 a1 B0 be' 1 z 2 a2 B0 be' 2 z 3
Hence, by identification, we have:
p b0 A0
a
a
p b1 b0
p b 2 b1
p b2
1
2
A0 B0 be' 0
A0 a1 B0 be' 1
a2 B0 be' 2
This system of equations is solved in order to determine the tensors
A0 , B0 , be' 0 , be' 1 and be' 2 , enabling the plasma to be represented in
the TLM method according to the Z transform technique. We can
choose to keep just the “dispersive” terms in the expression for the
tensor z by taking: B0 0 ( 1 0 ).
Appendix B
The researched solution is thus the following:
A0
p b0
be' 0
p b0 b1 a1 b0
be' 1
p b1 b2 a2 b0
be' 2
p b2
Since b0 1 , b1 2 and b2 1 , we obtain:
0
be' 0
b0'
be' 1
b1'
p
1
0
0
0
p 1 a1 1
p 1 a1
p 1 a2 1
p 1 a2
1
0
be' 2 p 1
b2'
1
0
p
1
0
0
0
0
0
0
0
159
160
Numerical Analysis in Electromagnetics
We recall that the solution (see [4.24]) obtained using the Z
transform technique for the TLM method is expressed in the following
form:
F T0 2 F R z -1 S
T0 4 G 0 4 0
1
S 2 F R T1 F - G z F 4 z F
T1 - 4 G1 4 1
For the plasma, we obtain:
T0 4 4 0
1
S 2F R T1F + 4 z F
T1 4 4 1
4 z F 4
1
b0' b1' z 1 b2' z 2
1 a1 z 1 a2 z 2
F
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Index
2D, 1-2, 5, 15, 17, 19, 29, 31,
44-45, 55, 62, 81, 85-86, 89, 91
3D, 16-17, 29-31, 35, 40-41, 55, 62,
70-71, 81-82, 89, 92, 98, 125-127,
151
curvilinear, 70-71
B
boundary conditions, 14, 61
C
cubic mesh, 33
D
discrete elements
applications, 25, 91, 100, 133,
136
mono-port, 86-91, 95, 99
two-port, 98
dispersion, 19, 34, 51, 55-60, 81,
83
E
excitation, 3, 22-24, 104, 119,
127-128, 141
M
matched impedance, 61
matrix form, 56, 75, 124, 128
Maxwell’s equations, 3, 7, 57, 64,
70, 133
matrix form, 56, 75, 123-124,
128
wave-matter interaction, 130
Z transform, 123, 128-130,
136-137, 144, 147, 155-160
N
normalized Maxwell’s equations
cubic mesh, 125
parallelepipedic mesh, 133
O
output signal, 24
P
parallelepipedic mesh, 133
PML layers, 62-64, 67-68
profile, 65
172
Numerical Analysis in Electromagnetics
S
HSCN, 33-34, 43-45, 52-55,
59-60, 106, 115
non-Cartesian, 81
SCN, 31-36, 41-46, 50-57,
60, 62, 64, 71, 82-83,
89-91, 95, 97, 123, 127
split step, 35
SSCN, 33-35, 55, 60
scattering matrix, 4, 11-14, 22,
32-33, 46-47, 49-52, 56-57, 75,
86-87, 91, 95-96, 116
segmentation, 62
stability conditions, 78, 85
T
thin wires, 105
model, 106, 119
scattering on the wire, 109
time step, 33-36, 40, 43, 45-46,
54-57, 78, 85-89, 92, 126
TLM nodes, 31, 33, 35, 41, 91-92
2D TLM nodes
parallel, 14, 16, 29
series, 15, 21, 30
3D TLM nodes
ACN, 30-31
curvilinear, 70-71, 75-78
distributed, 29
V
velocity error, 17, 20, 55, 59-60,
81
W
wave-matter interaction, 123-124,
127
Z
Z transform, 70, 128-130,
136-137, 144
