Table Of ContentThe Wave Concept in Electromagnetism and Circuits
Series Editor
Daniel Pasquet
The Wave Concept in
Electromagnetism
and Circuits
Theory and Applications
Edited by
Henri Baudrand, Mohammed Titaouine,
Nathalie Raveu
First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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ISBN 978-1-84821-959-5
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter 1. General Principles of the
Wave Concept Iterative Process . . . . . . . . . . . . . . . . . . . . . . . . . 1
Henri BAUDRAND, Med Karim AZIZI, Mohammed TITAOUINE
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. The iterative wave method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. General definition of waves . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4. Application to planar circuits . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5. Applications to quasi-periodic structures . . . . . . . . . . . . . . . . . . 6
1.6. Circuits with localized components . . . . . . . . . . . . . . . . . . . . . . 7
1.7. General principles of quasi-periodic circuits . . . . . . . . . . . . . . . . 7
1.8. The significance of using auxiliary sources . . . . . . . . . . . . . . . . . 8
1.8.1. Description of the environment . . . . . . . . . . . . . . . . . . . . . . 9
1.9. Unidimensional circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.10. Application: transmission line . . . . . . . . . . . . . . . . . . . . . . . . 14
1.11. Comparison of current density for different cell lengths . . . . . . . . . 14
1.12. Bi-dimensional circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.13. Two-source bi-dimensional circuits . . . . . . . . . . . . . . . . . . . . . 16
1.14. Three-source bi-dimensional circuits . . . . . . . . . . . . . . . . . . . . 22
1.15. Validation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.16. Lenses and meta-materials . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.17. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 2. Formulation and Validation of the WCIP
Applied to the Analysis of Multilayer Planar Circuits . . . . . . . . . . 43
Alexandre Jean René SERRES and Georgina Karla DE FREITAS SERRES
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vi The Wave Concept in Electromagnetism and Circuits
2.2. WCIP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1. Multilayer formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3. Real and ideal polarizers within planar structures using WCIP . . . . . 52
2.3.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4. Amplifier structure of compact micro-waves . . . . . . . . . . . . . . . . 57
2.4.1. Formulation of the amplifier interface . . . . . . . . . . . . . . . . . . 57
2.4.2. The simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 3. Applications of the WCIP Method to
Frequency Selective Surfaces (FSS) . . . . . . . . . . . . . . . . . . . . . . 63
Mohammed TITAOUINE and Henri BAUDRAND
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2. Formulation of the iterative WCIP method . . . . . . . . . . . . . . . . . 65
3.2.1. Determining the diffraction operator . . . . . . . . . . . . . . . . . . 68
3.2.2. Determining the reflection operator . . . . . . . . . . . . . . . . . . . 70
3.2.3. The fast modal transform FMT and its inverse FMT−1 . . . . . . . . 72
3.2.4. FSS multilayer devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.5. Multi-level plated FSSs . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3. Application of the iterative WCIP method to different FSSs . . . . . . . 74
3.3.1. Dielectric short-circuited FSS rings . . . . . . . . . . . . . . . . . . . 74
3.3.2. FSSs charged by lumped elements and active FSSs . . . . . . . . . 76
3.3.3. Multi-frequency band FSSs . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.4. Double-layer FSS plating . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.5. Triple-layer plating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.6. Thick FSSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4. Anisotropic FSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5. Measurement system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.7. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Chapter 4. WCIP Applied to Substrate
Integrated Circuits: Substrate Integrated
Waveguide (SIW) and Substrate Integrated
Non-Radiative Dielectic (SINRD) Circuits . . . . . . . . . . . . . . . . . . . 99
Nathalie RAVEU and Ahmad ISMAIL ALHZZOURY
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2. Formulation of WCIP for SIC circuits . . . . . . . . . . . . . . . . . . . . 100
4.2.1. The definition of (cid:2175)(cid:3553). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.2. The definition of (cid:3553) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Contents vii
4.3. Results for SIW circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3.1. Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3.2. Bandpass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4. Results for the SINRD circuits . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4.1. Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.2. Bandpass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter 5. WCIP Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Nathalie RAVEU
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2. Summary of WCIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1. Representation of homogeneous materials
around the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.2. Description of boundary conditions at the interface . . . . . . . . . . 118
5.2.3. System to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3. Improvement of WCIP by mathematical techniques . . . . . . . . . . . . 119
5.3.1. Number of modes/number of meshes . . . . . . . . . . . . . . . . . . 120
5.3.2. GMRES/Richardson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.3. Selecting the initial value . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4. Improvement of WCIP by physical considerations . . . . . . . . . . . . 124
5.4.1. Simplification of waves at the interface . . . . . . . . . . . . . . . . . 124
5.4.2. Choice of reference impedance . . . . . . . . . . . . . . . . . . . . . . 125
5.4.3. Boundary conditions on the metallic mesh . . . . . . . . . . . . . . . 126
5.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Chapter 6. Application of WCIP to Diffraction Problems . . . . . . . . 129
Noemen AMMAR, Taoufik AGUILI and Henri BAUDRAND
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1.1. Diffraction by multilayer cylindrical structures . . . . . . . . . . . . 130
6.1.2. Descriptors for spectral components
of reflection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.3. The modal coefficients Γext and Γint . . . . . . . . . . . . . . . . . 133
n n
6.1.4. Modal coefficients Γpass . . . . . . . . . . . . . . . . . . . . . . . . . 134
n
6.1.5. Spatial diffraction operator . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.6. Excitation source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.7. Iterative process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.1. Dielectric cylinder diffraction . . . . . . . . . . . . . . . . . . . . . . 139
6.2.2. Diffraction by metallic strips . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.3. Coaxial multi-strip structure . . . . . . . . . . . . . . . . . . . . . . . 148
viii The Wave Concept in Electromagnetism and Circuits
6.2.4. Diffraction by two dielectric co-axials . . . . . . . . . . . . . . . . . 156
6.2.5. Diffraction by structures of any shape . . . . . . . . . . . . . . . . . . 159
6.3. Coupling simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3.1. Different operators involved . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.2. The case of two pixels on a single fictitious cylinder . . . . . . . . . 163
6.3.3. The case where the two pixels are part of two
coaxial cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3.4. Spatial descriptors of diffraction operators . . . . . . . . . . . . . . . 167
6.3.5. The iterative process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3.6. Computation of the remote location electric field . . . . . . . . . . . 169
6.3.7. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Preface
Summary
The Wave Concept Iterative Procedure (WCIP) was first published in
1995 to model planar circuits and antennas. The first method nearest to the
WCIP was harmonic balance (1975) that introduced a successive Fourier
Transform and its inverse. The balance between two states, linked by these
transforms, is now fundamental to elaborate a model for non-linear
electromagnetism problems and high frequency circuits.
After Chapter 1, which introduces the expression and the history of the
WCIP, the structures studied are decomposed in elements that link incident
and reflected waves, as in the Transmission Line Matrix (TLM) method. The
WCIP is quite similar to an integral form of the TLM. One can show that the
WCIP is always convergent. After this description of the WCIP and the
introduction of its applications in the following chapters, the general
principles of the WCIP are presented. They are suitable for the simulation of
Meta materials, Substrate Integrated Circuits (SIC), and more generally
almost-periodic circuits. As a consequence, dual quantities have to be
defined (for instance E and H, V and I, E and J) and two complementary
domains (interior and exterior, spatial and spectral) are presented, as in
harmonic balance. The study of nano-transistors is also possible with this
method.
x The Wave Concept in Electromagnetism and Circuits
In Chapter 2, the WCIP is extended to multilayer structures. New
relations between waves are introduced for the transformation between two
adjacent interfaces. This formulation is obtained from the impedance matrix
of a transmission line, and can be used to model real and ideal polarizer in
planar structure. It can be demonstrated that an ideal polarizer interface can
be expressed directly in the modal domain, reducing significantly the
computational time. One of the applications of a multilayer structure with
ideal polarizer is a compact microwave amplifier structure. The WCIP has
therefore been modified to characterize the amplifier composed of passive
and active elements.
In Chapter 3, Frequency Selective Surfaces (FSS) are widely used as low
pass filters, high pass filters, stop band filters, pass band filters, etc. Loaded
FSSs allow flexibility in the adjustment of transmission characteristics.
However, electronically adjusted FSSs – also called active FSSs are very
important in compensating for the errors of fabrication and for the matching
of FSS filters for wireless communication systems. The use of short circuits
in the dielectric ring FSS ensure the possibility of producing an FSS with a
given resonant frequency only by varying the short circuit positions in the
ring. When seeking FSS for multiband applications, FSS with concentric
rings and multilevel FSSs can be used. The right choice of the FSS circuit
can offer the possibility of exploiting the two perpendicular polarisations
with distinct multiband. For enhanced band FSS with better filtering
characteristics when switching from pass band to stop band, multilevel FSS
is used. Inhomogeneous FSS can be accounted for when a need for a specific
non extant dielectric substrate is necessary. In integrated circuits the
inhomogeneous case corresponds to physical reality. Anisotropic FSS
provides filtering behavior with less power loss, thus its study is of great
importance. In the quasi periodic FSS, FSS with loaded short circuits,
passive elements and active elements are considered as FSS with defects the
coupling between all the elements constituting the defected FSS is accounted
for.
In Chapter 4, the circuits in substrate integrated circuits (SIC) technology
also present the advantage of being integrable with other technologies such
as planar circuits or waveguides. To design such circuits, usual numerical
techniques may be inappropriate: the Finite Element Method implies
important memory storage while the Integral Method is not able to solve
Preface xi
dielectric vias integration. The WCIP is therefore an efficient method for
characterizing SIC when vias are placed on a periodic lattice in metallic and
dielectric cases. Several example circuits are analysed in this chapter and the
computation time is very low.
In Chapter 5, the convergence of the WCIP is proved. Convergence is
ensured as the eigenvalues of the WCIP operators are bounded by one;
however the convergence may be reached very slowly. This method, well
adapted to several configurations of circuits, has been improved in terms of
its computation time by two ways: physical parameter modification and
mathematic development. Physical parameters, such as the reference
impedance in expression of waves and metallic loss insertion, help the
convergence; however, performance is not drastically changed. Mathematic
development of other numerical methods is well-known such as Krylov
solution instead of point fixed solution, physical pre-conditioners etc. They
are implemented and proved to be efficient. Thanks to these improvements
the computation time obtained in Chapter 5 has been improved by almost
95% compared to the usual solution for same tolerance on results.
The electromagnetic waves scattered by metallic or inhomogeneous
objects is of interest in several areas, such as radar and CEM. In Chapter 6,
the formulation of the WCIP in the cylindrical coordinate system is used to
study scattering problems and the penetration of electromagnetic waves into
multilayer cylindrical structures. In this chapter, the operators inside and
outside a cylinder, and the operator ensuring the wave transmission between
two coaxial cylinders defined in spectral domain, are used to calculate the
incident and reflected waves at each interface of the multilayered cylindrical
structure. Finally, the WCIP method is extended to model the scattering of a
structure of arbitrary shape in an open environment. The development of the
diffraction operator is expanded in cylindrical modal basis in the spectral
domain. In fact, the basis functions should be created in the most appropriate
coordinate system, suitable to the local geometry of the arbitrary diffraction
structure. For a planar circuit, the scattering problem is easily solved by the
WCIP method in the Cartesian coordinate system. However, the Cartesian
coordinate system becomes inefficient when dealing with non-regularly
shaped bodies. A cylindrical coordinate formulation is presented in this
section, as a solution to extend the WCIP to scattering of structures of
arbitrary shapes.
