Table of Contents

Cover

Title

Preface

1 General Principles of the Wave Concept Iterative Process

1.1. Introduction
1.2. The iterative wave method
1.3. General definition of waves
1.4. Application to planar circuits
1.5. Applications to quasi-periodic structures
1.6. Circuits with localized components
1.7. General principles of quasi-periodic circuits
1.8. The significance of using auxiliary sources
1.9. Unidimensional circuits
1.10. Application: transmission line
1.11. Comparison of current density for different cell lengths
1.12. Bi-dimensional circuits
1.13. Two-source bi-dimensional circuits
1.14. Three-source bi-dimensional circuits
1.15. Validation examples
1.16. Lenses and meta-materials
1.17. Conclusion

2 Formulation and Validation of the WCIP Applied to the Analysis of Multilayer Planar Circuits

2.1. Introduction
2.2. WCIP formulation
2.3. Real and ideal polarizers within planar structures using WCIP
2.4. Amplifier structure of compact micro-waves

3 Applications of the WCIP Method to Frequency Selective Surfaces (FSS)

3.1. Introduction
3.2. Formulation of the iterative WCIP method
3.3. Application of the iterative WCIP method to different FSSs
3.4. Anisotropic FSS
3.5. Measurement system
3.6. Conclusion
3.7. Acknowledgments

4 WCIP Applied to Substrate Integrated Circuits: Substrate Integrated Waveguide (SIW) and Substrate Integrated Non-Radiative Dielectic (SINRD) Circuits

4.1. Introduction
4.2. Formulation of WCIP for SIC circuits
4.3. Results for SIW circuits
4.4. Results for the SINRD circuits
4.5. Conclusion

5 WCIP Convergence

5.1. Introduction
5.2. Summary of WCIP
5.3. Improvement of WCIP by mathematical techniques
5.4. Improvement of WCIP by physical considerations
5.5. Conclusions

6 Application of WCIP to Diffraction Problems

6.1. Introduction
6.2. Application
6.3. Coupling simulation
6.4. Conclusion

Bibliography

List of Authors

Index

End User License Agreement

List of Tables

1 General Principles of the Wave Concept Iterative Process

Table 1.1. Possible different dual variables

5 WCIP Convergence

Table 5.1. Choice of the initial value

4 WCIP Applied to Substrate Integrated Circuits: Substrate Integrated Waveguide (SIW) and Substrate Integrated Non-Radiative Dielectic (SINRD) Circuits

Table 4.1. Cut-off frequencies and relative error between the HFSS and WCIP for different LSE-LSM waveguide core widths

6 Application of WCIP to Diffraction Problems

Table 6.1. The various diffraction operators in the case of a single fictitious cylinder
Table 6.2. The various diffraction operators in the case of two coaxial cylinders

List of Illustrations

1 General Principles of the Wave Concept Iterative Process

Figure 1.1. Iterative wave diagram
Figure 1.2. Features of half-spaces
Figure 1.3. Quasi-periodic univariate structure
Figure 1.4. The elementary cell
Figure 1.5. Use of an auxiliary source
Figure 1.6. Quasi-periodic unidimensional circuits in the state ap
Figure 1.7. The elementary cell
Figure 1.8. Dividing up the cell configuration (three cells including periodic walls)
Figure 1.9. Current according to cell number for a length λ/50
Figure 1.10. Comparison of current density for different cell lengths for a length of 30 cm. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.11. Comparison of current density for different cell lengths measured according to wavelength λ, For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.12. Elementary cell
Figure 1.13. Elementary cell with three sources E₁, E₃ and E₅
Figure 1.14. Current density for the central cell activated in Y
Figure 1.15. Current density for a structure made up of 20 *20 cells with a central cell activated in Y
Figure 1.16. Five-cell configuration with a source activated in Z
Figure 1.17. Five-cell configuration with a source activated in Z, shown in two-dimensions
Figure 1.18. Five-cell configuration with an activated source in Z shown in three-dimensions. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.19. Diagrammatic representation of the structure being studied. The source cells in red (lower line) are active and those in blue (above) are impediment cells. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.20. Diagrammatic representation of the studied structure. The active source cells are in red (lower line) and the impediment cells are in blue (above). For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.21. E_z as a function of the number of cells in X and Y directions for a structure made up of 64x32 cells
Figure 1.22. Behavior of the E_z field on the 15^th line
Figure 1.23. Diagrammatic representation of the structure studied. The active source cells are shown in red (lower line) and the impediment cells are shown in blue (above). For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.24. E_z as a function of the cells number in X and Y directions for structure made up of 64 × 32 cells and shown in Figure 23. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.25. Behavior of the E_z field on the 15^th line
Figure 1.26. Description of the configuration 64 *64 cells
Figure 1.27. Field E_z as a function of the cells number in X and Y directions for a structure made up of 64*64 cells and shown in Figure 1.26 with dl = λ/20
Figure 1.28. Behavior of field Ez at the level of the central source in X (red boxes) and in Y (blue crosses). For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 1.29. Field E_z as a function of the number of cells in X and Y directions for a structure made up of 64*64 cells and shown in Figure 1.26 with dl = λ/100
Figure 1.30. Comparison of the Ez field’s behavior for a central source which is active in Z, and the Hankel function
Figure 1.31. Elementary cell
Figure 1.32. Meta-material lens
Figure 1.33. Structure of the Veselago lens
Figure 1.34. Transmission line elementary cells
Figure 1.35. Configuration of the lens
Figure 1.36. Behavior of the field |Ez|. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip

2 Formulation and Validation of the WCIP Applied to the Analysis of Multilayer Planar Circuits

Figure 2.1. Example of the multilayer planar structure. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.2. Behavior of waves A and B, the transformation spheres within the iterative process for a single interface
Figure 2.3. Transfer matrix method, waves and the source within a multilayer structure
Figure 2.4. 3-D view of the multilayer structure
Figure 2.5. Behavior of current density on the second interface at 4 GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.6. Absolute value of the reflection coefficient, |S₁₁| and transmission coefficient |S₂₁| in dB according to the frequency in GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.7. 3-D view of the patch antenna structure containing slit interfaces
Figure 2.8. Cross-section of the antenna structure
Figure 2.9. Behavior of current density across the second 10 GHz interface. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.10. Behavior of the current density upon the third interface at 10 GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.11. Absolute value of the input reflection coefficient, |S₁₁| in dB expressed as a function of frequency in GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.12. 3-D view of a real polarizer across the axis x, O_x
Figure 2.13. a) Multilayer structure, line; b) the feed line across the first interface; c) the resonator on the second interface; d) behavior of current density across the second interface at 5 GHz
Figure 2.14. a) Multilayer structure; b) behavior of the current density across the third interface at 4.1 GHz
Figure 2.15. a) Multilayer structure; b) behavior of current density across the third interface at 3.9 GHz
Figure 2.16. Block diagram of an amplifier
Figure 2.17. Example of an active circuit
Figure 2.18. a) Analysis of the passive structure; b) analysis of the active structure
Figure 2.19. 3-D view of the auxiliary feed structure. For a color version of this figure, www.iste.co.uk/baudrand/waves.zip
Figure 2.20. Amplification according to the impedance of the amplifying structure

3 Applications of the WCIP Method to Frequency Selective Surfaces (FSS)

Figure 3.1. Some examples of FSS; a) FSS examples with circular metallic rings, and b) quasi-open square rings
Figure 3.2. a) Geometry of an FSS with dielectric rings taking an arbitrary shape; b) unit cell within the FSS structure; c) section of the unit cell of; b) along the y axis
Figure 3.3. Flow diagram of the WCIP method
Figure 3.4. FSS double-layer plating a) section of unitary cell; b) corresponding diagram
Figure 3.5. Circular ring FSS with short circuits, a) FSS with two short circuits; b) FSS with four short circuits
Figure 3.6. Reflection coefficient variation according to the FSS frequency with two short ring circuits with a normally incident source in the x polarization and in the y polarization [TIT 08]
Figure 3.7. Variation of the reflection coefficient according to frequency, with different values for the angular position φ, the four FSS circular ring short circuits with a normal incidence, that is polarized in y [TIT 08], a) φ = 20°, φ = 40° (b) φ = 30°, φ = 60°
Figure 3.8. Reflection coefficient of an uncharged dielectric FSS circular ring [TIT 08]
Figure 3.9. Reflection coefficient for FSS devices charged by passive elements [TIT 08], a) the geometry of the charged FSS structure; b) FSS charged by capacitance; c) FSS charged by inductances
Figure 3.10. FSS unitary cell with circular rings and the FSS with the two rectangular rings: a) FSS reflection coefficient with dielectric circular rings charged with PIN iodes [TIT 08]; b) FSS geometry with two metallic concentric discs [TIT 08]; c) variation of the transmission coefficient according to the operating frequency [TIT 08]
Figure 3.11. FSS with an open quasi-square ring [TIT 09]: a) geometry of the unitary cell; b) transmission coefficient
Figure 3.12. Metallic pattern U-shaped FSS with dimensions of: A = 10 mm, d₁ = 2.5 mm, d₂ = 5 mm, W = 2 mm, ε_r = 4 and h = 1mm [TIT 09]: a) unitary cell; b) transmission coefficient according to the operating frequency
Figure 3.13. FSS double-layer plating: a) FSS geometry at levels 1 and 2; b) section of FSS unitary cell; c) power transmitted according to operating frequency
Figure 3.14. Metallic configuration FAN type FSS [ERD 02] with double-layer plating: a) FSS 1; b) FSS 2; c) FSS Reflection coefficient
Figure 3.15. Triple-layer plated FSS with identical square slits: a) unitary cell of a square-slit FSS; b) triple-layer plated FSS section; c) reflection coefficient with dielectric rings triple-layer plating FSS obtained by WCIP
Figure 3.16. FSS thick slit resonant FSSs
Figure 3.17. Section of the unitary cell of the thick FSS device
Figure 3.18. The three-layer media WCIP issue containing resonant FSS thick slits
Figure 3.19. Wave direction h_i and _i in the structure when doubled a) according to the x axis direction; b) according to the y axis direction
Figure 3.20. Power both transmitted and reflected according to frequency in the case of a thick FSS with slits filled with air
Figure 3.21. Power outputs transmitted and reflected according to frequency in the case of a thick FSS with slits filled with a dielectric medium ε_r = 4
Figure 3.22. Power transmitted through a thick metal sheet with rectangular slots sandwiched between the substrate and a superstrate both being identical and having ε_r = 4 and h = 0.79 mm
Figure 3.23. Power transmitted through thick metal perforated sheets with rectangular slits sandwiched between a substrate and a superstrate both the same having ε_r = 4 and h = 1.585 mm
Figure 3.24. Power transmitted through a thick metal perforated sheet with rectangular holes and sandwiched between the substrate and a superstrate both identical with ε_r = 4 and h = 3.17 mm
Figure 3.25. Powers both transmitted and reflected according to the frequency of a thick FSS in the absence of an insulator in the holes for different thickness values of the perforated metal sheet
Figure 3.26. Powers which are both transmitted and reflected according to the frequency of a thick FSS with an insulator in the holes with ε_r = 4 for different thickness values for the perforated metal sheet
Figure 3.27. Transmission line on an anisotropic substrate with a system of coordinates
Figure 3.28. Powers reflected and transmitted according to the operating frequency of a rectangular FSS patch etched on an air-like layer sandwiched between the patch and a dielectric anisotropic layer [TIT 07a, TIT 13]
Figure 3.29. Measurement system and the FSS with metallic rectangular rings

4 WCIP Applied to Substrate Integrated Circuits: Substrate Integrated Waveguide (SIW) and Substrate Integrated Non-Radiative Dielectic (SINRD) Circuits

Figure 4.1. Configuration of the SIC circuits across a regular grid
Figure 4.2. The SIW waveguide with a fundamental mode TE₁₀
Figure 4.3. Effective width of the waveguide for the TE₁₀ mode according to frequency
Figure 4.4. Second-order SIW bandpass filter dimensions, w₁ = 34 mm, h₁ = 10 mm, w₂ = 12 mm, h₂ = 8 mm, w₃ = h₃ = 20 mm, p = 4 mm, a = 0.625 mm
Figure 4.5. Measurements and simulations (WCIP-HFSS) results of the S parameters of the second-order filter shown in Figure 4.4
Figure 4.6. Waveguide of fundamental mode TEM through the SIRND zone
Figure 4.7. Determination of effective relative permittivity
Figure 4.8. LSE-LSM waveguides; a) WCIP; b) equivalent guide for HFSS; D_x = 40mm; D_y = 68 mm
Figure 4.9. Second-order SINRD filter dimensions w₁ = 22 mm, w₂ = 12 mm, w₃ = 14 mm, w₄ = 68 mm, d₁ = 20 mm, d₂ = 14 mm, d₃ = 10 mm, d₄ = 40 mm
Figure 4.10. Measurement and simulation (both HFSS and WCIP) results of the S parameters of the second-order filter shown in Figure 4.9
Figure 4.11. Simulation (WCIP) and measurement results of the S parameters of the non-homogeneous SINRD filter

5 WCIP Convergence

Figure 5.1. Configuration of the circuit where fields are solved using the WCIP
Figure 5.2. Tested circuit
Figure 5.3. a) Solution using Richardson’s procedure; b) solution using GMRES. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 5.4. Comparison of the number of iterations between two initial conditions at each frequency
Figure 5.5. Comparison of the number of iterations with so-called mixed conditions
Figure 5.6. Comparison of S parameters obtained using HFSS and the WCIP with two formulations
Figure 5.7. Choice of Z₀

6 Application of WCIP to Diffraction Problems

Figure 6.1. Inhomogeneous multilayer structure
Figure 6.2. Corresponding diagram shown in waves
Figure 6.3. Reflection operators for two coaxial cylinders
Figure 6.4. Diagram of iterative process
Figure 6.5. Dielectric cylinder
Figure 6.6. Table of simulation constraints
Figure 6.7. Convergence of an electric field
Figure 6.8. Electric field modulus for R = λ
Figure 6.9. Electric field modulus for R = 0.3λ
Figure 6.10. RCS variation for R = λ
Figure 6.11. RCS variation for R = 0.3λ
Figure 6.12. Diffraction by two metallic strips
Figure 6.13. Electric field modulus in p = R for {ψ₁, ψ₂, ψ₃, ψ₄} {25°, 155°, 205°, 335°}
Figure 6.14. Current density modulus in p = R for _{ψ_1, ψ_2, ψ_3, ψ_{4} =} {25°, 155°, 205°, 335°}
Figure 6.15. Electric field modulus in p = R for {ψ₁, ψ₂, ψ₃, ψ₄} = {60°, 120°, 240°, 360°}
Figure 6.16. Current density modulus in p = R for {ψ₁, ψ₂, ψ₃, ψ₄} = {60°, 120°, 240°, 360°}
Figure 6.17. Diffraction by three cylindrical metallic strips
Figure 6.18. The current density modulus
Figure 6.19. The electric field modulus
Figure 6.20. Modulus of the electric field for {ψ₁, ψ₂, ψ₃, ψ₄, ψ₅, ψ₆} = {25°, 155°, 205°, 335°}
Figure 6.21. Modulus of the electric field for {ψ₁, ψ₂, ψ₃, ψ₄, ψ₅, ψ₆} = {30°, 90°, 150°, 210°, 270°, 330°}
Figure 6.22. Convergence of current density across the cylinder of the radius R₁
Figure 6.23. Convergence of the electric field across the cylinder of the beam R₁
Figure 6.24. Convergence of the electric field the cylinder of the beam R₂
Figure 6.25. Electric field for configuration A
Figure 6.26. Current density for configuration A
Figure 6.27. Electric field for configuration B
Figure 6.28. Current density for configuration B
Figure 6.29. Current density for configuration A
Figure 6.30. Electric field for configuration A
Figure 6.31. Current density for configuration B
Figure 6.32. Electric field for configuration B
Figure 6.33. Current density for configuration A using the formula R₁ = 0.9λ
Figure 6.34. Electric field for configuration A using the formula R₁ = 0.9*R₂
Figure 6.35. Two-layer structure
Figure 6.36. Geometric characteristics of Figure 6.36
Figure 6.37. Convergence of the electric field across the exterior interface
Figure 6.38. Convergence of the electric field across the interior interface
Figure 6.39. Computation of infinite field
Figure 6.40. Infinite electric field
Figure 6.41. Cross-section of a structure of any given form
Figure 6.42. Interaction between two pixels of the structure
Figure 6.43. Position of two pixels across two fictitious co-axial cylinders
Figure 6.44. Definition of two-pixel waves on a single cylinder
Figure 6.45. Corresponding wave-based diagram
Figure 6.46. Definition of two-pixel waves placed across two co-axial cylinders
Figure 6.47. Modeling two-pixel waves which are part of two co-axial cylinders
Figure 6.48. Calculation of diffracted fields in the remote zone
Figure 6.49. Right-angled metallic corner resting upon absorbent material
Figure 6.50. Current density for the equation φ₀ = π/4
Figure 6.51. Current density for φ₀ = π/2
Figure 6.52. Electric field in the remote zone for φ₀ = π/4
Figure 6.53. Electric field in the remote zone for φ₀ = π/2
Figure 6.54. Metallic corner with an aperture of α = 120°
Figure 6.55. Standardized current density
Figure 6.56. Transverse section of a square cylinder
Figure 6.57. Standardized current density
Figure 6.58. Variation of the electric field within the remote zone
Figure 6.59. Transverse section of the structure
Figure 6.60. Distribution of current density
Figure 6.61. Diffraction coefficient in the standardized remote zone
Figure 6.62. Structure which generates an electromagnetic jet [AMM 15]
Figure 6.63. Equivalent diagram in waves
Figure 6.64. Diagram of the iterative process
Figure 6.65. Computation of the near field
Figure 6.66. Variation of the electric field across the interface C (see [AMM 15])
Figure 6.67. Electric field modulus for various ε_r values_: a) ε_r = 2, b) ε_r = 4, c) ε_r = 6 and d) ε_r = 8. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip

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.

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, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Henri Baudrand, Mohammed Titaouine and Nathalie Raveu to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016945696

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ISBN 978-1-84821-959-5