THEORY AND APPLICATIONS OF MULTICONDUCTOR ... - TSpace

THEORY AND APPLICATIONS OF MULTICONDUCTOR TRANSMISSION LINE ANALYSIS FOR SHIELDED SIEVENPIPER AND RELATED STRUCTURES

by

Francis Elek

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto

c 2010 by Francis Elek Copyright

Abstract THEORY AND APPLICATIONS OF MULTICONDUCTOR TRANSMISSION LINE ANALYSIS FOR SHIELDED SIEVENPIPER AND RELATED STRUCTURES Francis Elek Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2010 This thesis focuses on the analytical modeling of periodic structures which contain bands with multiple modes of propagation. The work is motivated by several structures which exhibit dual-mode propagation bands. Initially, transmission line models are focused on. Transmission line models of periodic structures have been used extensively in a wide variety of applications due to their simplicity and the ease with which one can physically interpret the resulting wave propagation effects. These models, however, are fundamentally limited, as they are only capable of capturing a single mode of propagation. In this work multiconductor transmission line theory, which is the multi-mode generalization of transmission line theory, is shown to be an effective and accurate technique for the analytical modeling of periodically loaded structures which support multiple modes of propagation. Many results from standard periodic transmission line analysis are extended and generalized in the multiconductor line analysis, providing a familiar intuitive model of the propagation phenomena. The shielded Sievenpiper structure, a periodic multilayered geometry, is analyzed in depth, and provides a canonical example of the developed analytical method. The shielded Sievenpiper structure exhibits several interesting properties which the multiconductor transmission line analysis accurately captures. It is shown that under a continuous change of geometrical parameters, the dispersion curves for the shielded structure are transformed from dual-mode to single-mode. The structure supports a stop-band characterized by complex modes, which appear as pairs of frequency varying complex conjugate propagation constants. These modes are shown to arise even though the structure is modeled as lossless. In addition to the periodic analysis, the scattering properties of finite cascades of such strucii

tures are analyzed and related to the dispersion curves generated from the periodic analysis. Excellent correspondence with full wave finite element method simulations is demonstrated. In conclusion, a physical application is presented: a compact unidirectional ring-slot antenna utilizing the shielded Sievenpiper structure is constructed and tested.

iii

Acknowledgements I must begin by thanking my supervisor Prof. George V. Eleftheriades, who has supported me tremendously throughout this long process. Prof. Eleftheriades provided intellectual, moral and financial support throughout my studies, especially during some of the more difficult times. As a scientist who is passionate and dedicated to his research, but at the same time a warm human being, he will always be an individual whom I deeply respect. It has been a great pleasure to work with you over the course of my degree. I would like to thank Prof. Costas D. Sarris, Prof. Seav V. Hum, Prof. Raviraj Adve, all from the University of Toronto, and Prof. Lotfollah Shafai from the University of Manitoba for being members of my Ph.D. examination committee and for providing me with valuable feedback on this thesis. Thanks also to Prof. Sergei Dmitrevsky for numerous stimulating discussions on a wide variety of topics throughout the years. I would like to acknowledge our lab managers Gerald Dubois and Tse Chan for their assistance over the course of my studies. Thanks are also due to all of my fellow graduate students in the Electromagnetics group who have helped create a stimulating environment. In particular I would like to sincerely thank Dr. Marco Antoniades who was there the whole time and provided much encouragement, especially in the final stages of this endeavour - thanks dude! I would also like to acknowledge the financial support that I have received from the Natural Sciences and Engineering Research Council Scholarship and the Ontario Graduate Scholarship in Science and Technology. Of course none of this could have been possible without the support of my family. Thanks mom and dad for providing unlimited support and encouragement throughout the years. And to my sister Melisa, thanks for the long distance motivation you provided - it was very inspiring and I appreciated it greatly.

iv

Contents List of Acronyms

viii

List of Symbols

ix

1 Introduction

1

1.1

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Sievenpiper mushroom structure . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.2

Two dimensional loaded microstrip grids . . . . . . . . . . . . . . . . . . .

8

1.2.3

Shielded Sievenpiper structure . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.4

Some other related geometries

1.2.5

Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3

. . . . . . . . . . . . . . . . . . . . . . . . 19

Thesis Contributions and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Analytical Motivation: Finite Element Method Simulations

25

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2

Numerical Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3

FEM simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4

2.3.1

Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.2

Modal Field Profiles: hu = 6 mm . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3

Modal Field Profiles: hu = 0.5 mm . . . . . . . . . . . . . . . . . . . . . . 36

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Multiconductor analysis: Building Blocks

38

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2

Unloaded MTL Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3

Determination of loading elements . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 v

4 Multiconductor analysis: Dispersion analysis

61

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2

MTL analysis of the shielded structure (a): Periodic unit cell and dispersion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3

4.4

MTL analysis of the shielded structure : Simplified analysis . . . . . . . . . . . . 67 4.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.2

Dispersion: Simplified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

MTL analysis of the shielded structure (b): Comparison of full periodic dispersion with FEM simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5

Analytical formulas, equivalent circuits, and modal field structure defining the resonant frequencies at (βd)x = 0 and (βd)x = π . . . . . . . . . . . . . . . . . . 81 4.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5.2

Analytical Formulas for f1 through f4 . . . . . . . . . . . . . . . . . . . . 82

4.5.3

Equivalent Circuits for f1 through f4 . . . . . . . . . . . . . . . . . . . . . 85

4.5.4

Modal field structure for f4 and f5 (at (βd)x = π) . . . . . . . . . . . . . 87

4.5.5

Modal field structure for f3 and f6 (at (βd)x = 0)

4.5.6

Modal field structure for f2 (at (βd)x = 0) . . . . . . . . . . . . . . . . . . 92

4.5.7

Modal field structure for f1 (at (βd)x = π)

. . . . . . . . . . . . . 89

. . . . . . . . . . . . . . . . . 94

4.6

Design considerations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.7

Comparison of the MTL model with the TL-PP model . . . . . . . . . . . . . . . 98

4.8

Modal degeneracy at f2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.9

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Slow Wave Analysis

106

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2

MTL model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Scattering Analysis

115

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2

Four-Port Scattering Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3

Application to 2D microstrip grid excitation . . . . . . . . . . . . . . . . . . . . . 128

6.4

Two-Port Scattering Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Shielded structure based slot antenna 7.1

139

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 vi

7.2

Design of the underlying shielded geometry . . . . . . . . . . . . . . . . . . . . . 141

7.3

Antenna Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.4

Antenna pattern results and discussion

7.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Conclusions

. . . . . . . . . . . . . . . . . . . . . . . 146

151

8.1

Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.2

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A Shielded structure based antenna compared with a cavity-backed antenna 155 Bibliography

159

vii

List of Acronyms TL

Transmission Line

MTL

Multiconductor Transmission Line

TM

Transverse Magnetic

TE

Transverse Electric

FEM

Finite Element Method

BW

Backward-Wave

FW

Forward-Wave

NRI

Negative Refractive Index

PP

Parallel-plate

HFSS

High-Frequency Structure Simulator by Ansoft Corporation

E-wall

Perfect Electric Conductor boundary condition

H-wall

Perfect Magnetic Conductor boundary condition

EBG

Electromagnetic Band-gap

CPW

Coplanar Waveguide

viii

List of Symbols

ω

Angular frequency

C

Capacitance

L

Inductance

d

Unit cell periodicity

Zs

Surface Impedance

0

Per-unit-length inductance

0

Per-unit-length capacitance

L

C

Zo

Transmission line characteristic impedance

V

Voltage

I

Current

Z

Impedance

Y

Admittance

Permittivity

µ

Permeability

o

Permittivity of free space

µo

Permeability of free space

w

Patch width

hu

Upper-region height

hl

Lower-region height

r

Via radius

g

Gap width

L0

Per-unit-length inductance matrix

C0

Per-unit-length capacitance matrix ix

γ

Complex propagation constant

β

Propagation constant

α

Attenuation constant

E

Electric field vector

H

Magnetic field vector

D

Electric displacement field vector

S

Poynting vector

JD

Displacement current vector

V

Voltage vector

I

Current vector

Q0

Per-unit-length conductor charge vector

Ψ0

Per-unit-length flux-linkage vector

Z0

Per-unit-length longitudinal impedance matrix

Y0

Per-unit-length transverse admittance matrix

I0

Identity matrix 0

Upper-region per-unit-length capacitance

0

Upper-region per-unit-length inductance

0

Lower-region per-unit-length capacitance

0

Ll

Lower-region per-unit-length inductance

Zu

Upper-region characteristic impedance

Zl

Lower-region characteristic impedance

θu

Upper-region electrical length

θl

Lower-region electrical length

Sij

ij-component of the generalized scattering matrix

T

Transfer matrix

Γ0

Propagation constant matrix

Z0w

Characteristic wave impedance matrix

0 Yw

Characteristic wave admittance matrix

vφ

Phase velocity

vg

Group velocity

λ

wavelength

Cu Lu Cl

x

List of Tables 3.1

Comparison of the numerical (FEM) and analytic C0 (capacitance) matrices for: (a) hu = 18 mm, (b) hu = 6, (c) hu = 0.5 mm. The analytic C0 matrix is calculated for two different values of the effective width, wef f = 10.0 and 9.6 mm. 46

4.1

Boundary conditions and analytical formulas corresponding to the resonance frequencies at (βd)x = 0 and (βd)x = π. . . . . . . . . . . . . . . . . . . . . . . . 88

6.1

Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients − + − a+ m , am , bm and bm for the 4-port scattering theory: column 1 excitation (lower

region) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2

Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients − + − a+ m , am , bm and bm for the 4-port scattering theory: column 2 excitation (upper

region) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3

Comparison of the port mode impedance, Zpp , calculated using the analytical (static) formula (6.14), with FEM simulated results. . . . . . . . . . . . . . . . . 134

6.4

Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients − + − a+ m , am , bm and bm for the two port scattering results depicted in Figure 6.8. . . 138

xi

List of Figures 1.1

(a) The shielded Sievenpiper structure. (b) A typical dispersion curve. (c) and (d): Applications of the shielded Sievenpiper structure. (e) and (f): Two related structures for which the theory developed in this thesis can be applied. . . . . . .

3

1.2

Geometry of the Sievenpiper structure.

6

1.3

Origin of the the inductance, L, and capacitance, C for the surface impedance

. . . . . . . . . . . . . . . . . . . . . . .

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4

Dispersion curve of the Sievenpiper mushroom structure using the surface impedance approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

6 7

Dispersion diagram of the Sievenpiper mushroom structure generated from a c IEEE 2006). . . . . . . . . . . . . . . . . . . . . . FEM simulation (from [25],

8

1.6

Two-dimensional loaded microstrip grid. . . . . . . . . . . . . . . . . . . . . . . .

9

1.7

Typical dispersion relation described by (1.2) for on-axis propagation with βy d = 0 (fixed), and βx d varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8

Full wave FEM simulation of an NRI grid for on-axis propagation with βy d = 0 (fixed), and βx d varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9

Unit cell of the shielded Sievenpiper structure. . . . . . . . . . . . . . . . . . . . 15

1.10 Dispersion curves for the shielded Sievenpiper structure with two upper region heights: (a) hu = 18 mm, (b) hu = 0.5 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of Section 1.2.2 (the unshielded structure), and the free space light line. . . . . . . . . . . . . . . . . . . . . . . . 16 1.11 Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-plate transmission line (TL) upon which the patches and vias act as loading elements. The underlying unloaded TL consists of the shielding plane and the ground plane as depicted in (a), which is transformed to the actual (loaded) structure in (b). Equivalent circuit for this point of view is shown in (c). Reactive loading element shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 xii

1.12 Typical dispersion diagram as predicted by the model in [8], for on-axis propagation with βy d = 0 (fixed), and βx d varied. . . . . . . . . . . . . . . . . . . . . . 18 1.13 Two structures which are related to the shielded Sievenpiper structure. . . . . . . 20 1.14 Three related structures with dispersion curves obtained from approximate singlemode models: (a) the unshielded Sievenpiper structure (effective surface impedance model), (b) the 2-D microstrip gird (TL-BW model), and (c) the shielded Sievenpiper structure (TL-PP model). In general all three structures exhibit dual-mode behaviour as shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1

(a) Unit cell of the shielded Sievenpiper structure. (b) For on-axis propagation, (βd)y = 0 is fixed, while the phase shift per-unit-cell, (βd)x , along the direction of propagation (x), is varied. Modal field plots on the transverse plane at the cell edge to be shown later in this chapter. . . . . . . . . . . . . . . . . . . . . . . 27

2.2

Dispersion curves for the shielded structure with varying upper region height: (a), (b) hu = 18 mm; (c), (d) hu = 6 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of the unshielded structure, and the free space light line. Field plots corresponding to the points labeled in (d) will be shown later in this chapter. . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

Dispersion curves for the shielded structure with with hu = 0.5 mm. The other physical parameters are: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of the unshielded structure, and the free space light line. Field plots corresponding to the labeled points will be shown later in this chapter. . . . . . . . . . . . . . . . . 29

2.4

Transverse modal field plots for the 1st passband of the structure with dispersion curve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 12 E×H∗ on the same transverse cut, but with view rotated. . . . . . . . . . . . . . . . . . 31

2.5

Longitudinal current on the upper shield and ground plane for the three modes, FW1, BW1, and [(βd)1 , fmax ] of Figure 2.4. . . . . . . . . . . . . . . . . . . . . . 33

2.6

Longitudinal D of the x-directed gap excitation for the three modes, FW1, BW1, and [(βd)1 , fmax ] of Figure 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7

Transverse modal field plots for the upper passbands of the structure with dispersion curve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 21 E × H∗ on the same transverse cut, but with view rotated. . . . . . 35 xiii

2.8

Transverse modal field plots for the 1st passband of the structure with dispersion curve from Figure 2.3 (hu = 0.5 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 21 E×H∗ on the same transverse cut, but with view rotated. . . . . . . . . . . . . . . . . . 36

3.1

Transformation of an infinite 1-D periodic array of strips, (a) and (c), into an infinite 2-D periodic array of isolated patches (b) and (d). Vias connected from the center of each patch to ground for (b) and (d). The transverse boundary conditions are assumed to be H-walls for the case of on-axis propagation in the MTL model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2

Generic multiconductor transmission-line configuration for an n + 1 conductor system. Propagation is along the x axis; Ik and Vk denote conductor k’s current and voltage. (a) Longitudinal view. (b) Cross-sectional view. . . . . . . . . . . . 42

3.3

Parameters defining the unloaded MTL geometry for on-axis propagation assuming transverse H-walls (dashed lines). Conductors 1 and 2 have voltages, {V1 , V2 }, defined with respect to ground, along with currents {I1 , I2 }, which are 0

used to define the per-unit-length capacitance and inductance matrices, C and 0

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 0

0

3.4

Boundary value problems used to determine C11 and L11 . . . . . . . . . . . . . . 44

3.5

Dispersion curves of the unloaded geometry. . . . . . . . . . . . . . . . . . . . . . 50

3.6

E field profiles for the two modes of the unloaded geometry. . . . . . . . . . . . . 50

3.7

Two-port scattering setup used to determine the series capacitance, C. . . . . . . 51

3.8

Real and imaginary parts of C obtained from the two-port scattering setup. . . . 52

3.9

Four-port scattering setup used to determine the series capacitance, C, depicted for (a) large hu and (b) small hu . For a lower region excitation a larger quantity of energy leaks to the upper region when hu is small. . . . . . . . . . . . . . . . . 53

3.10 The calculated series gap capacitance, C, for (a) hu = 6 mm, and (b) hu = 0.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.11 Two-dimensional electrostatic boundary value problem used to obtain the charge accumulation at the patch edges. The dashed lines denote H-walls. . . . . . . . . 57 3.12 Surface charge density [C/m2 ] on the conductor at V1 = +V (Figure 3.11), near the plate edges for (a) hu = 6 mm and (b) hu = 0.5 mm. . . . . . . . . . . . . . . 57 3.13 Streamline plots of the electric field for (a) hu = 6 mm and (b) hu = 0.5 mm. . . 58 3.14 Four-port scattering setup used to determine the shunt inductance, L. . . . . . . 59 3.15 The calculated shunt via inductance, L, for (a) hu = 6 mm, and (b) hu = 0.5 mm. 59 4.1

MTL based equivalent circuit for on-axis propagation. . . . . . . . . . . . . . . . 63 xiv

4.2

Dispersion curves obtained using the simplified dispersion equation (4.30), with varying upper region height. (a) hu = 10 mm; (b) hu = 3 mm; (c) hu = 0.75 mm. All other parameters are fixed: the lower region height, hl = 3 mm; the upper and lower region relative permittivities are r1 = r2 = 4; the loading inductance, L = 1.0 nH; the loading capacitance, C = 0.5 pF. . . . . . . . . . . . . . . . . . . 71

4.3

Plot of the function DiscL , which is negative between fc1 and fc2 and otherwise positive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4

Power flow profiles for complex modes with complex-conjugate propagation constants, γa = jβ + α and γb = −jβ + α.

4.5

. . . . . . . . . . . . . . . . . . . . . . . 75

Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. . . . . . . . . . 78

4.5

Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. (cont’d) . . . . 79

4.5

Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. (cont’d) . . . . 80

4.6

Transfer matrix relationships for a symmetric unit cell. Voltages and currents on each of the 1 through k lines defined at nodes n, n + 12 , and n + 1. Voltages defined with respect to ground. Arrows denote current flow convention. . . . . . 83

4.7

The four resonant circuits corresponding to f1 through f4 for the shielded structure. 87

4.8

Field patterns corresponding to f4 and f5 ; (βd)x = π. . . . . . . . . . . . . . . . 90

4.9

Field patterns corresponding to f6 and f3 ; (βd)x = 0. . . . . . . . . . . . . . . . . 91

4.10 Field patterns corresponding to f2 , (βd)x = 0. (a) large hu ; (b) small hu . Il~ (dashed lines) and the current lustration of the gap capacitive fringing field, E distribution (solid lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.11 Field patterns corresponding to f1 , (βd)x = π. (a) large hu ; (b) small hu . The ~ (dashed lines) and the current distribution (solid lines) are shown. 95 electric field, E 4.12 Dispersion curve for a structure with a via radius of 1.5 mm, corresponding to L = 0.17 nH. All other geometric and electrical parameters are as for the structure of Figure 4.5d: d = 10 mm, hu = 1 mm, hl = 3.1 mm, r1 = 1, r2 = 2.3. 98 xv

4.13 Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-plate transmission line (TL) upon which the patches and vias act as loading elements. The underlying unloaded TL consists of the shielding plane and the ground plane as depicted in (a), which is transformed to the actual (loaded) structure in (b). Equivalent circuit for this point of view is shown in (c). Reactive loading element shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.14 Comparison of the TL-PP model dispersion curves with FEM simulations. (a) hu = 0.2 mm; (b) hu = 1 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3 . . . . . . 101 4.15 Boundary conditions corresponding to the two degenerate modes at f2 : (a) Transverse boundary conditions for the mode described by MTL theory. (b) Transverse boundary conditions for the TE mode. (c) Boundary conditions at the transverse (y) walls, and longitudinal (x) walls for the MTL mode. (d) Boundary conditions for the TE mode are switched compared with (c) . . . . . . . . . . . . . . . . . . 104 5.1

Field structure of the commensurate two conductor geometry with both the entire patch layer and via removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2

Low frequency dispersion with hu = 0.2, 1, and 6 mm; All other parameters are c fixed: hl = 1 mm; d = 2 mm; w = 1.9 mm; via radius = 0.1 mm (from [11], IEEE 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3

Low frequency FW mode voltage and current distribution for the shielded structure.110

5.4

MTL unit cells with one of the loading elements removed at a time. . . . . . . . 111

5.5

Eigenvectors corresponding to the MTL unit cell with one of L or C removed. . . 113

6.1

Four-port scattering: (a) Circuit schematic for the four-port scattering analysis with lower region excitation; (b) Power flow for lower region excitation; (c) Power flow for upper region excitation.

6.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 6 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3

Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 1 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . 125 xvi

6.4

Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 0.2 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. . . . . . . . . . . . . . . . . . . . . . 126

6.5

Dispersion and corresponding four-port scattering curves obtained using MTL analysis for a case where the BW bandwidth is large: L = 10 nH, C = 4 pF, 1r = 1, hu = 18 mm, 2r = 5, and hl = 3.1 mm. . . . . . . . . . . . . . . . . . . . 129

6.6

Two-port scattering: (a) Circuit schematic; (b) Power flow

. . . . . . . . . . . . 131

6.7

Transverse cut used to define the port variables for the investigated two-port scattering situation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.8

Dispersion and corresponding two-port scattering curves comparing the MTL analysis with FEM simulations for a N = 5 cell structure: (a) & (b) hu = 6 mm; (c) & (d) hu = 1 mm; (e) & (f) hu = 0.2 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.1

Unit cell underlying the proposed slot antenna; hua = 1.54 mm, 1a−rel = 4.5, hub = 1.5 mm, 1b−rel = 1, hl = 3.1 mm, 2−rel = 2.3, r = 0.25 mm, w = 9.6 mm, wb = 8.8 mm, d = 10 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2

Comparison of MTL theory with FEM generated dispersion curves for on-axis propagation for the geometry of Figure 7.1. . . . . . . . . . . . . . . . . . . . . . 143

7.3

FEM simulated Brillouin diagram for the shielded structure of Figure 7.1 showing a complete omni-directional band-gap between approximately 2.5 and 5 GHz. . . 143

7.4

Coaxial excitation of: (a) the shielded structure, and (b) a parallel-plate geometry (with the mushroom structure replaced with a solid ground plane), for the purpose of measuring the transmission, S21 ; Measured S21 for the shielded structure, and for the flat conductor backed parallel-plate structure for: (c) the Γ − X direction; (d) the Γ − M direction. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.5

(a) Ring slot antenna fed by a CPW line, with the shielded structure’s placement c IEEE 2005). (b) Cross-sectional view of the shown as a dotted line (from [9], geometry with approximate size of the slot’s ground plane and the overall height given in terms of free space wavelengths. . . . . . . . . . . . . . . . . . . . . . . . 146

7.6

c IEEE 2005). . . . . 147 S11 of the shielded structure-based slot antenna (from [9],

7.7

Measured and FEM simulated normalized radiation patterns of the reference ring-slot antenna; f = 3.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xvii

7.8

Measured and FEM simulated normalized radiation patterns of the reference ring-slot antenna backed with a conductor at one quarter wavelength; f = 3.7 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.9

Measured and FEM simulated normalized radiation patterns of the reference ring-slot antenna backed with the EBG; f = 3.9 GHz. . . . . . . . . . . . . . . . 149

A.1 Normalized radiation patterns for the EBG-backed antenna compared with two cavity-backed antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.2 S11 for the EBG-backed antenna compared with two cavity-backed antennas. . . 158

xviii

Chapter 1

Introduction 1.1

Motivation

The study of electromagnetic wave propagation in non uniform media is one with a long history, which continues to the present time. A particularly useful class of non uniform media are periodic structures, which are created by starting with a uniform structure, and then perturbing it periodically [1–4]. The perturbations act to alter the propagation of electromagnetic waves traveling through the structure. In particular, frequency bands which do not support propagating modes, referred to as stop-bands, will develop for periodic structures, in addition to frequency bands where wave propagation is allowed, referred to as pass-bands. The structures may be one-dimensional guiding media, such as transmission lines or waveguides, two-dimensional structures, or bulk three-dimensional structures. The propagation properties are analyzed by solving Maxwell’s equations, either through numerical techniques, or analytical solutions. Numerical solutions, although important in the precise characterization of a given problem, may be computationally time consuming, and additionally it may be difficult to extract physical intuition on the nature of the underlying mechanisms leading to the resulting propagation effects. In general, it is not possible to obtain exact analytical solutions, and approximation techniques need to be employed to reduce the complexity of the problem. Within the analytical realm there is often a trade-off between ease of solution, and the information contained within a particular solution. Analytical solutions which are close to the exact behaviour described by Maxwell’s equations are often complex, and again difficulties in the physical interpretation of the solutions may arise. On the other hand, overly simplified approximate solutions, while providing a rough understanding, often miss crucial qualitative and quantitative details, which lead to a lack of insight into the true underlying mechanisms of the wave propagation. Transmission line (TL) theory has been used extensively to model periodic structures due 1

Chapter 1. Introduction

2

to its ease of implementation, accuracy, and the resulting physical intuition one can obtain into the origin of the wave propagation effects. By considering a single unit cell and applying periodic boundary conditions a dispersion equation is obtained which characterizes the propagation constant as a function of frequency. The resulting dispersion contains frequency bands supporting propagating modes and bands supporting evanescent modes. A fundamental limitation of TL models is that they are inherently single mode and hence are incapable of capturing the dispersion properties of structures which contain multi-mode propagation bands. In recent years a class of periodic structures which are characterized by such multi-mode dispersion curves have been investigated. A prominent example is the shielded Sievenpiper structure [5], a periodic multilayered geometry, which is depicted in Figure 1.1a, along with a typical dispersion curve which characterizes the resulting wave propagation in Figure 1.1b. The dispersion curve shows the propagation constant, βd over a frequency band. It is observed that from DC up to the frequency f1 the structure supports a single mode, with the propagation constant increasing with increasing frequency. At low frequencies the dispersion of this mode is nearly linear and it is related to the quasi-TEM (transverse electromagnetic) parallel-plate mode that would exist if both the via and patch arrays were absent. As frequency increases the presence of the via/patch array leads to the excitation of another mode at f = f1 . Above the frequency f1 the structure supports two modes of propagation: the dual-mode nature of the structure above f = f1 is a specific example of the limitation of standard transmission line models, which cannot capture such bands. The two modes which exist above f1 coalesce at the frequency fc1 , which defines the beginning of a stop-band, and within this stop-band the quasi-TEM parallel-plate mode is suppressed. Between the frequencies fc1 and fc2 two modes are supported, defined by frequency-varying complex-conjugate propagation constants, αd ± j βd, where for clarity only the one with positive value of βd is depicted. These modes are referred to as complex modes and are a distinct from the modes with complex propagation constants which occur in lossy structures. Complex modes are unusual in that they have some properties of both standard evanescent modes (attenuation effects) and standard propagating modes (phase accumulation effects). It is important to note that such modes will be shown to arise in the structure even though losses are not considered. Finally, above the frequency fc2 the structure enters another dual-mode pass-band, with the frequency f2 defining the transition from a dual-mode to a single-mode band. This structure has been shown to be useful in the suppression of switching noise in digital circuits [6–8] (Figure 1.1c) and in the creation of unidirectional slot antennas [9] (Figure 1.1d). Both of these applications rely on the operation of the structure within the stop-band: for the slot antenna the suppression of the parallel-plate mode results in an improved front-to-back ratio, while for the switching noise application the modal suppression prevents signal degradation

3


Shielding plane 2D patch array 2D via array

Ground plane

(a) Shielded Sievenpiper structure.

(b) Dispersion curve corresponding to (a).

Slot antenna

Through vias

(c) Suppression of switching noise.

(d) Uni-directional slot antenna.

Port 3

Port 4

Stacked layers

Port 1 (e) 3D-stacked metamaterials.

Port 2

(f) A compact directional coupler.

Figure 1.1: (a) The shielded Sievenpiper structure. (b) A typical dispersion curve. (c) and (d): Applications of the shielded Sievenpiper structure. (e) and (f): Two related structures for which the theory developed in this thesis can be applied.


4

due to mode conversion. This structure is also capable of producing a slow-wave effect and thus may be thought of as an artificial medium with enhanced effective relative permittivity [10, 11]. Closely related geometries have been shown to be useful in the creation of 3D stacked artificial media (Figure 1.1e) which are characterized by negative effective permittivity and permeability [12–14]. Such structures have been referred to as metamaterials. When the structure is used as an artificial medium it is the pass-band propagation which is of primary concern. Additionally, another topologically related geometry, a coupled-line microstrip configuration [15, 16], where one of the lines is loaded periodically with series capacitors and shunt inductors (Figure 1.1f), has been shown to yield a compact directional coupler. This application also relies on the operation of the structure in the stop-band, between fc1 and fc2 , however in a manner which is different from both the antenna and switching noise applications which are described above. The frequency regime between fc1 and fc2 defines an unusual stop-band in which a propagationlike behaviour exists if spatially separated regions of the structure are excited in isolation. In the case of the depicted coupler, the excitation of port 1 leads to the transmission of power to port 2, with very little power at ports 3 and 4, for sufficiently long lines. This effect is due to the continuous leakage of power from line 1 to line 2, and is intimately related to the unusual nature of the complex modes.

Transmission line theory is incapable of capturing the dual-mode dispersion behaviour of the shielded Sievenpiper structure and its derivatives, and this provides the primary motivation for this work, which is the development of an analytical method which extends the standard TL model by allowing for multiple modes of propagation. It will be shown that multiconductor transmission line (MTL) theory, which is the multi-mode generalization of TL theory, is capable of modeling the dispersion behaviour of the shielded Sievenpiper structure and its derivative structures in a compact manner [5, 11, 17]. The shielded Sievenpiper structure will be examined in depth and will provide a canonical example of the analytical method developed in this work. Due to the relative simplicity of its geometry, the theory yields compact analytical formulas for critical points on its dispersion curve, f1 , f2 , f3 , and f4 (not shown in Figure 1.1b), along with fc1 and fc2 . The developed analytical formulation will provide one with an enhanced physical understanding of the shielded Sievenpiper structure’s operation, and additionally allow for intuition on the operation of the related geometries.

In the following section a review of some of the models which have been previously used to characterize the shielded Sievenpiper structure and other related structures will be presented.

5


1.2

Background

The analytical approach which will be developed in this thesis can be best appreciated by examining a series of structures which are related, both in terms of their geometry, and in terms of the wave propagation effects they exhibit. Various models describing wave propagation in these structures will be reviewed. Although each of the models will be seen to describe the propagation phenomena within restricted regimes, they will be shown to be overly restrictive in terms of developing an overall analytical and intuitive picture of the observed propagation effects.

1.2.1

Sievenpiper mushroom structure

The Sievenpiper mushroom structure [18], which is depicted in Figure 1.2, is composed of a square grid of isolated metallic (microstrip) patches which are connected to a solid ground plane with vias. This structure has been used to reduce mutual coupling in microstrip antenna arrays [19], perform two-dimensional beam steering [20], and to create low profile wire antennas [21, 22]. It was initially modeled in [18] as a uniform surface, with the surface impedance Zs given by the parallel combination of an inductance, L and a capacitance, C: Zs =

jωL 1 − ω 2 LC

(1.1)

The origin of the inductance and capacitance is depicted in Figure 1.3, where it is observed that the inductance is due to the circulation of current along a path defined by the ground plane, the patches and the vias, and the capacitance is due to the fringing fields between the patches. Dispersion curves, which describe the wave propagation derived from this model, are shown in Figure 1.4, where it is observed that at low frequencies a TM (transverse magnetic) surface wave is supported. At the resonance frequency, ω 2 =

1 LC ,

the surface impedance becomes

infinite (Zs → ∞), and above it TM surface waves are cut off. However, TE (transverse electric) surface waves are supported above the resonant frequency. As the above model assumes that the surface impedance is uniform, the dispersion curves generated from it can only be accurate when the electrical phase shift per-unit-cell, βd is much smaller than unity (βd 1). The condition βd 1 corresponds to an effective wavelength which is much greater than the period, d, and in this long wavelength limit, the effect of the periodicity may in a sense be averaged out, resulting in the surface impedance given by (1.1). Such models are also referred to as homogenization models, as the physical periodic (and hence non-homogenous) structure is assigned an effective homogenous parameter (the surface impedance, Zs ) to describe it. However, when βd is of the order of magnitude of unity or greater, the effects of the period-

6


y x

z y

d

2D patch grid hl , ǫ2

vias d

ground plane

(b) Top view

(a) Side view

Figure 1.2: Geometry of the Sievenpiper structure.

C

L

Figure 1.3: Origin of the the inductance, L, and capacitance, C for the surface impedance model.

7

Chapter 1. Introduction 30

Frequency (GHz)

25

TM waves TE waves Light εr1=1

20 Resonance frequency 15 10 5 0

π

βd

2π

3π

Figure 1.4: Dispersion curve of the Sievenpiper mushroom structure using the surface impedance approximation. icity become important, and the uniform surface impedance model breaks down. A full wave finite element method (FEM) simulation of the structure is shown in Figure 1.5. The dispersion curves are plotted for wave vectors along the edge of the irreducible Brillouin zone [23]. The part of the dispersion curve from Γ to X corresponds to propagation along one of the principle axes of the structure, with the phase shift (βd)x varying, (Γ) 0 ≤ (βd)x ≤ π (X), while the phase shift, (βd)y = 0 fixed. Between X and M (βd)x = π is fixed, with (X) 0 ≤ (βd)y ≤ π (M). Finally, between M and Γ both (βd)x and (βd)y vary, with (Γ) 0 ≤ (βd)x = (βd)y ≤ π (M). It is observed that a stop-band exists for surface waves between the TM and TE modes. Concentrating on the dispersion curves between Γ and X, it is seen that the first pass-band supports two propagating modes. One of the modes has a dispersion curve which tracks just below the light line and extends to DC. The other mode is a high-pass mode, having a cut-off frequency associated with it, and begins to propagate at the X-point of the dispersion curve. An improved homogenized surface impedance model, which is able to capture the dual-mode behaviour of the structure, was given in [24]. In this model a homogenized surface impedance is formed from the parallel connection of the capacitive patch grid surface impedance and the impedance of the via region, which is approximated as an effective uniaxial wire medium composed of infinitely long wires. This approximation is possible since the ground plane acts as one of the image planes for the vias, while the capacitive grid acts approximately as the other image plane. However, as in [18], this model cannot account for the periodicity of the structure, as it is obtained under the assumption that βd 1. Thus this model is not capable of accurately accounting for the dispersion of the high-pass mode which begins to propagate at the X point, where (βd)x = π.


8

Figure 1.5: Dispersion diagram of the Sievenpiper mushroom structure generated from a FEM c IEEE 2006). simulation (from [25],

1.2.2

Two dimensional loaded microstrip grids

A two-dimensional grid-like structure, which is topologically related to the Sievenpiper structure, is depicted in Figure 1.6. This structure has been shown [26–28], in the long wavelength limit, to behave as a medium with effective permittivity and permeability both negative. For structures with both effective permeability and permittivity negative, the refractive index, n, is also negative [29], and hence such structures have been referred to as negative refractive index (NRI) media. The transformation of the original mushroom structure into the new one, which will be referred to as a two-dimensional (2D) loaded microstrip grid is depicted in Figure 1.6c. In [27, 28] this periodic structure was analyzed using transmission line (TL) theory, with the unit cell depicted in Figure 1.6d. A brief review of some of the key points of that work will be presented now, as the analysis developed in this thesis can be viewed as an extension of their TL model. Indeed, many of the salient features of [27, 28] will reappear in a generalized and extended context for the modeling procedure developed in this thesis. The unit cell is composed of both distributed and lumped elements. The distributed elements are the metal traces along the x and y directions, which are modeled as microstrip transmission lines, with per-unit length inductance and capacitance given by L0 and C 0 , req 0 L spectively, with an associated characteristic impedance, Zo = C 0 , and propagation constant, √ βo = ω L0 C 0 . The lumped elements are the series capacitors, 2C, due to the fringing fields between adjacent gaps, and the shunt inductance, L, due to the via. Both C and L may be enhanced by using discrete components. The voltages and currents along both the x and y directions are related by periodic (Bloch)

9


y x z n

y

n+1 2D grid from (a)

ground plane

d (a) Unit cell (dashed): top view

(i)

(b) Unit cell (dashed): side view

(ii)

(iii)

(c) Transformation of the Sievenpiper patch grid (i) to the 2D-microstrip grid (iii).

n(x )

n+

d 2

2C

d 2

Zo

) 1(y

y

2C Zo

L ) n(y

n+

2C

2C

1(x )

x (d) Unit cell for the equivalent 2D-transmission line circuit.

Figure 1.6: Two-dimensional loaded microstrip grid.

10


boundary conditions between nodes n and n+1. Bloch’s Theorem [23] states that the field variables separated by the periodicity of the unit cell, d, are related by the Bloch propagation constants, βx and βy . Along the x-direction, Vn+1 (x) = Vn (x) e−jβx d and In+1 (x) = In (x) e−jβx d , with analogous relations holding along the y-direction. By transforming the voltage and current variables at the unit cell edges, to the central connecting node, and applying Kirchhoff’s voltage and current laws, a homogenous system of equations is obtained. Requiring that the determinant of said system be zero, which is required for non-trivial solutions, the dispersion equation for the structure is obtained, and given by: cos (βx d) + cos(βy d) βo d 1 βo d Zo βo d βo d = − 2 sin − 2 sin − + 2 (1.2) cos cos 2 Zo ωC 2 2 2ωL 2 A qualitative understanding of the dispersion equation may be obtained by assuming propagation along the x direction, with βx d varying and βy d = 0 fixed, for which (1.2) takes the form: cos(βx d) = F (ω, L0 , C 0 , L, C)

(1.3)

where F is a function of frequency (ω), the host TL parameters (L0 , C 0 ), and the loading elements (L and C). The periodicity of the cosine function implies that the dispersion equation has a period 2π and thus can be restricted to the interval −π ≤ βx d ≤ π, which is referred to as the Brillouin zone [23]. Due to the even symmetry of the cosine function, the dispersion may be plotted in the range 0 ≤ βx d ≤ π. For lossless structures, as will be considered here, the function F is purely real, and can have an absolute value greater than or less than 1, and by restricting the interval to 0 ≤ βx d ≤ π, a unique solution for βx d is obtained. When |F | ≤ 1 the solution represents a purely propagating mode, and otherwise it is an evanescent mode. A typical dispersion diagram generated from (1.2) is shown in Figure 1.7. Below the frequency f1 , the TL model yields a stop-band, in which the mode is evanescent, with complex propagation constant given by γx d = αx d + jβx d = αx (ω)d + jπ, indicating that the real part of the propagation constant, αx (ω)d (dashed line) is varying as a function of frequency, while the imaginary part, βx d (solid line) is fixed and equal to π. In Figure 1.7 the regions described by evanescent modes are shaded, while the regions with propagating modes are not. Approaching the frequency f1 from below, αx d → 0, and at f = f1 the propagation constant is purely imaginary and given by γx (f1 )d = jπ. Between f1 and f2 the propagation constant remains purely imaginary indicating that a propagating mode is supported, and hence the region between f1 and f2 is a pass-band. It is noted that between f1 and f2 the slope of the dispersion curve

dω dβx ,

is negative, and hence the group velocity, given by vg =

dω dβx ,

is negative.

11


Figure 1.7: Typical dispersion relation described by (1.2) for on-axis propagation with βy d = 0 (fixed), and βx d varied.

7 6

FEM simulation

f3 5

TL Model (BW only)

Frequency (GHz)

f2

Light Line

4

Stopband

3 2 1

f1 Backward Wave Surface Wave

0 0

π

βd Figure 1.8: Full wave FEM simulation of an NRI grid for on-axis propagation with βy d = 0 (fixed), and βx d varied.

12

Chapter 1. Introduction However, the phase velocity vφ =

ω βx ,

is positive, indicating that the band between between f1

and f2 supports a backward wave (BW) mode [30]. Between f2 and f3 , the second stop-band is encountered, with an evanescent mode supported, while the second pass-band resides between f3 and f4 . In the region between f3 and f4 the group and phase velocities are both positive, indicating that a forward wave (FW) mode is supported. This alternating sequence of stop-bands and pass-bands subsequently repeats itself. A FEM simulation for such a structure is shown in Figure 1.8. The BW mode predicted by the TL model is captured by the FEM simulation, but in addition, a FW surface wave mode, whose dispersion is just below the light line, is also supported, which the TL model does not account for. Contra-directional coupling between the FW and BW modes yields a stopband, which was also observed for the original Sievenpiper structure, as shown in Figure 1.5. Qualitatively the dispersion curves for the 2D-grid and the Sievenpiper structure are identical; however due to the use of discrete components to enhance C and L, the 2D-grid typically has a larger BW bandwidth. From the FEM simulations the field structure of each mode is obtained. For the BW mode, the fields are largely concentrated in the substrate (between the traces and the ground). For the FW mode, the fields are largely concentrated in the air above the substrate. Additionally, the FEM simulations reveal that the TL model is accurate at the Bragg resonance (f1 at βx d = π), which is out of the range of applicability of the previously discussed homogenization models [18, 24]. A simplified understanding of the dispersion for this structure may be obtained by assuming that the Bloch phase shifts across a unit cell are small, βx d 1 and βy d 1, and these approximations, with the additional assumption that the interconnecting microstrip TL segments are also electrically short, βo d 1, result in the exact TL dispersion, (1.2) reducing to: βx2

+

βy2

= ω L0 − 2

1 2 ω Cd

2C 0 −

1 2 ω Ld

(1.4)

In [28] it was shown that (1.4) may be written as: β 2 = ω 2 µef f ef f

(1.5)

with

µef f ef f

1 = L − 2 ω Cd 1 = 2C 0 − 2 ω Ld 0

(1.6) (1.7)

where µef f and ef f are the effective permeability and permittivity of the structure. The

13


dispersion equation (1.5) shows that under the conditions βx d 1, βy d 1, and βo d 1, the structures appears homogenous and isotropic. At low enough frequencies it is clear that both µef f and ef f are less than zero, indicating that the effective medium parameters are negative. As ω → 0, both parameters approach −∞ and the approximate dispersion equation (1.4) predicts that the structure supports a propagating mode. This is inconsistent with the results of the exact dispersion equation (1.2), which predicts that the BW mode is cut-off below f1 . This inconsistency arises because the approximations which lead to (1.4), βx d 1, βy d 1, are not satisfied at very low frequencies. However as frequency increases and approaches f2 , both µef f and ef f remain negative and the conditions βx d 1, βy d 1 are satisfied. Thus, in the region just below f2 , the structure behaves in a homogenous and isotropic manner, with negative material parameters. Within this region the structure supports a BW mode, and hence the effective negative material parameters are associated with a BW band. It is observed from (1.6) and (1.7) that the existence of the BW band is reliant upon the presence of both L and C, and if either of these loading elements were removed from the structure the BW band would be eliminated as well. The frequency f2 is obtained by setting one of µef f or ef f equal to zero, with f3 determined by setting the excluded case equal to zero. These frequencies depend on the loading elements, L and C, and the distributed parameters, L0 and C 0 , and are given by:

1 1 = min , C(L0 d) L(2C 0 d) 1 1 2 , ω3 = max C(L0 d) L(2C 0 d)

ω22

(1.8) (1.9)

Both f2 and f3 describe resonances occurring between one of the loading elements, L, C, and one of the distributed TL parameters (multiplied by the periodicity, d), L0 d, 2C 0 d. At this point it is possible to justify the conditions under which the short TL approximation, βo d 1, could be made in obtaining (1.4). Each of f2 and f3 contain one of the loading elements, L and C individually. By making L and C large enough it is possible to reduce f2 and f3 to arbitrarily low values, ensuring that βo d 1 is satisfied. In [31], it was noted that homogenization models of the mushroom structure are only accurate when the gap spacing between patches, g is sufficiently small, and the substrate height, hl is sufficiently large. In terms of the present analysis such conditions correspond to a large series capacitance, C, and a large shunt inductance, L, and thus the TL model described here provides an elegant explanation of conditions under which homogenization is accurate. However, in the case that L and C are not large, so that the short TL approximation (βo d 1) can’t be made for the interconnecting microstrip lines, the full dispersion equation (1.2), remains accurate and should be used rather than the approximate one, (1.4).


14

The operation of such a structure as a NRI medium relies on the utilization of the BW mode band, as was explained earlier. However the FEM simulation (Figure 1.8) revealed that the BW mode bandwidth was reduced due to the stop-band formed by the contra-directional coupling of the FW and BW modes. Additionally, in regions where the BW is propagating, a FW mode is also supported, so that the structure is fundamentally a dual-mode structure. In [26] it was demonstrated that as long as the operating frequency is away from the stop-band, the FW mode has a negligible impact on the analysis, as long as the excitation mechanism of the structure is such that the source is situated between the microstrip grid layer and the ground plane. This is consistent with the fact that the field structure of the BW mode is largely confined to the substrate, while the field structure of the FW mode is largely confined to the air region above the substrate. However, as the BW and the FW modes coalesce and form a stop-band, the TL model breaks down, and excitations with frequencies close to, or within this stop-band, cannot be modeled with a simple TL analysis. Finally it is noted that a similar TL analysis can be applied to the original mushroom structure, with a fundamental BW mode predicted. However, the fact that the FW mode is not accounted for would again be an obvious major deficiency in the completeness of such a model.

1.2.3

Shielded Sievenpiper structure

In the previous two subsections the Sievenpiper structure, and the topologically related 2D microstrip grid were examined. Another structure, which is related to these two structures is the shielded Sievenpiper structure, which is simply the original Sievenpiper structure from Figure 1.2, with an additional conducting shielding plane above the mushroom layer. A unit cell of the shielded structure is depicted in Figure 1.9. The structure consists of a lower region of height hl and permittivity 2 and an upper region of height hu and permittivity 1 . The shielded structure has been shown to be useful in the suppression of switching noise in digital circuits [6–8], and in the creation of unidirectional slot antennas [9]. Several analytical models for this structure have been proposed, but before describing them it will be interesting and useful to compare full wave FEM simulations of the shielded structure with the TL analysis of Section 1.2.2, which predicted an initial high pass BW band. In Figure 1.10 dispersion curves corresponding to two sets of simulations with varying upper region height, (a) hu = 18 mm, and (b) hu = 0.5 mm are shown. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. For the larger value of hu = 18 mm, the FEM generated dispersion curve shown in Figure 1.10a bears a strong resemblance to that of both the unshielded structure (Figure 1.5) and the 2D microstrip grid (Figure 1.8). The first band is dual mode, with a FW and a BW mode.

15

Chapter 1. Introduction Shielding conductor; d

hu

ǫ1 Patch conductor; w

hl

ǫ2

via; r Ground conductor

Figure 1.9: Unit cell of the shielded Sievenpiper structure. The fields of the FW mode are largely concentrated in the upper region, while the fields of the BW mode are largely concentrated in the lower region. The TL model dispersion is accurate away from the light line, with the resonances, f1 , f2 , f3 and f4 captured by the FEM simulations. However the TL model does not capture the low frequency FW mode and hence is incapable of accounting for the stop-band, which is due to contra-directional coupling of the FW and the BW modes. For the smaller value of hu = 0.5 mm, shown in Figure 1.10b the dispersion is qualitatively altered. The lowest pass-band becomes single mode, with the BW mode eliminated. The FW mode has a significantly smaller slope than for the larger (hu = 18 mm) value, indicating that a strong slow wave effect is achieved. Additionally, the stop-band bandwidth is substantially increased. The resonant frequency f1 is shifted down, while f2 is shifted up, and neither corresponds to those of the TL model. However, the frequencies f3 and f4 predicted by the TL model are captured by the FEM simulation. The fact that the f3 and f4 seem to be invariant as the upper region height, hu , is altered, is an interesting phenomenon which will be explained by the theory developed in this thesis. Several analytical models for the dispersion analysis of the shielded structure have been developed. In [6] the surface impedance model of [18] was used in conjunction with the transverse resonance technique to determine the lowest modes of the structure. This technique predicts a low frequency band with a single FW TM mode, followed by a stop-band, and then an upper TE mode. The use of the surface impedance model precludes the possibility of accurately predicting the dispersion near the Brillouin zone boundary (βd = π). However, it was found that as long as the upper region height, hu is relatively small, such a model provides a reasonable estimate for the edges of the pass-band. In [7, 8] the structure was modeled as a loaded transmission line (TL). These TL models are different than the one described in Section 1.2.2, as they attempt to incorporate the effect of the upper shielding conductor. The model introduced in [8] will be examined below, and from here

16


10

12 FEM TL Light εr1=1

←f4

4

8

Stop-band

Stop-band

←f1 2

0

←f4

FEM TL Light εr1=1

f3→

f3→ f2→

10

←fT E

8

←fT E

f2→

(βd)x (a) hu = 18 mm

π

Frequency (GHz)

Frequency (GHz)

12

4

←f1

2

0

(βd)x

π

(b) hu = 0.5 mm

Figure 1.10: Dispersion curves for the shielded Sievenpiper structure with two upper region heights: (a) hu = 18 mm, (b) hu = 0.5 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of Section 1.2.2 (the unshielded structure), and the free space light line.

17

Chapter 1. Introduction n

Shielding plane hu , ǫ1

r Zo =

hl , ǫ2

w

L′ C′

ground plane

d

(a) Unloaded 2 conductor TL

n

n+1

(b) Transformation into a loaded TL

n+1 C= Zo

Y d 2

Zo

Y

ǫ1 w d hu

L (via)

d 2

(c) Equivalent TL circuit

(d) Composition of Y

Figure 1.11: Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-plate transmission line (TL) upon which the patches and vias act as loading elements. The underlying unloaded TL consists of the shielding plane and the ground plane as depicted in (a), which is transformed to the actual (loaded) structure in (b). Equivalent circuit for this point of view is shown in (c). Reactive loading element shown in (d).

on in it will be referred to as the TL-PP model, with PP designating the parallel-plate nature of the underlying geometry. Figure 1.11 shows the conception of this model, with the underlying transmission line (TL) being formed from the parallel-plate geometry of the shielding conductor and the ground plane. The patches and vias act as loading elements. The TL-PP model predicts that the first pass-band supports a single FW mode, with a typical dispersion diagram shown in Figure 1.12. The first pass-band extends from DC to f1 . The second pass-band begins at the frequency f3 . The frequency f3 will later be shown to correspond to that predicted by the high-pass TL model of Section 1.2.2, in the limit that the upper region height goes to zero (hu → 0). Any model which uses standard TL theory is only capable of predicting a single mode of propagation, and hence is incapable of modeling dual-mode bands. An examination of the results provided in [7, 8] show that the for cases considered therein hu was on the order of magnitude of, or smaller than hl . For such geometries FEM simulations indeed confirm that the first band contains only a single FW mode, corresponding to a situation as in Figure 1.10b. However, for larger values of hu , corresponding to Figure 1.10a, the structure is dual-mode,


18

Figure 1.12: Typical dispersion diagram as predicted by the model in [8], for on-axis propagation with βy d = 0 (fixed), and βx d varied. which cannot be captured by a TL model. The transition of the shielded structure’s dispersion from dual-mode to single-mode, as hu is decreased from large to small values, is an interesting phenomenon, which raises many questions: • Does the TL-PP model for relatively small values of hu accurately describe the attenuation in the stop-band? • Assuming that a model which captures the dual-mode behaviour for relatively large values of hu is developed, can it be shown to collapse to a single-mode model for relatively small values of hu ? • Using such a hypothetical dual-mode model, is it possible to physically explain the disappearance of the BW mode as hu is decreased from large to small values? The answers to these questions would give one more physical intuition into the operation and analytical characterization of the shielded Sievenpiper structure. Additionally, they would yield insight into the dispersion of both the unshielded Sievenpiper structure and the loaded 2D microstrip grid, as these two structures are characterized by similar dual-mode bands.


1.2.4

19

Some other related geometries

Examples of other related geometries for which the theory developed in this thesis has been applied to are shown in Figure 1.13. The first of these structures has the topology of the shielded structure, but with the addition of an extra inductive element connecting the patch plane to the upper shielding plane. By adding this inductive element it is possible to eliminate the FW mode of the shielded structure, while simultaneously increasing the BW bandwidth. Hence this geometry has been shown to be useful in the design of large bandwidth NRI media as shown in [12, 13], with a related geometry given in [14]. The theory developed in this thesis can be used to model the dispersion of these modified shielded structures, and also in physically explaining the conception of such geometries. The second structure is a microstrip coupled-line geometry, where one of the lines has been loaded with series capacitors and shunt inductors. The dispersion of this structure is also dualmode in the lowest band. Interestingly, the operation of such a coupler is intimately related to the nature of the modes which exist in the first stop-band [15, 16], as will be described later in this thesis.

1.2.5

Commentary

The unshielded Sievenpiper structure, the 2D-loaded microstrip grid, and the shielded Sieveniper structure all exhibit dual-mode dispersion curves in their lowest bands. Dispersion curves generated by the previously described approximate models are shown in Figure 1.14, along with a typical dual-mode dispersion curve which all three geometries exhibit. Although the dual-mode behaviour may be accounted for by homogenization models, such models are not accurate near the Bragg resonance at βd = π. Additionally, as was demonstrated in [31], the condition βd 1 is not sufficient for such models to be accurate, and in general they are restricted to low frequencies, where both the guided and free space wavelengths are much larger than the periodicity. The transmission line (TL) models, on the other hand, are capable of accounting for the periodicity of the structure, and hence are accurate at the Bragg resonance condition, βd = π. Additionally, TL models provide for a simple and intuitive understanding of the wave propagation, and yield compact formulas for band-edges. However TL models are deficient in that they inherently only account for a single mode, and hence are incapable of capturing dual-mode behaviour. However the physical intuition obtained from TL models makes them highly appealing, and this aspect would be desirable in any enhanced analysis which takes into account dual-mode, or in general multi-mode propagation bands.

20


shielding plane

hu , ǫ1 hl , ǫ2 ground plane (ii) Top view (below shield)

(i) Side view

(a) Negative refractive index (NRI) medium. This structure is topologically related to the shielded structure, but with an additional inductive element.

Coupled microstrip lines

hl , ǫ2 ground plane (i) Side view

(ii) Top view

(b) Microstrip coupled-line coupler

Figure 1.13: Two structures which are related to the shielded Sievenpiper structure.

21

frequency


TM TE Light Res. freq.

π

βd

2π

3π

(i) Surface imp. (Zs ) model

(i) TL (BW) model

(i) TL-PP model

(ii) Top view

(ii) Top view

(ii) Top view (below shield)

air

air

shielding plane hu , ǫ1

hl , ǫ2

hl , ǫ2

(iii) Side view

(iii) Side view

(a)

(b)

hl , ǫ2 (iii) Side view

(c)

(d) Figure 1.14: Three related structures with dispersion curves obtained from approximate single-mode models: (a) the unshielded Sievenpiper structure (effective surface impedance model), (b) the 2-D microstrip gird (TL-BW model), and (c) the shielded Sievenpiper structure (TL-PP model). In general all three structures exhibit dual-mode behaviour as shown in (d).


1.3

22

Thesis Contributions and Outline

This thesis attempts to bridge the gap between the simplicity of TL models, and the accuracy and generality achievable by full wave numerical techniques. It will be shown that multiconductor transmission line (MTL) theory can be used to model both multi-mode behaviour and periodicity, in a coherent, compact manner. Multiconductor transmission line theory [32] is the generalization of TL theory to the case where the number of parallel conductors is greater than two. For such geometries the per-unit length inductance and capacitance, L0 and C 0 , are transformed into n by n matrices, L0 and C0 , characterizing the coupling of the n + 1 conductors, the case n = 1 being described by standard TL theory. MTL theory characterizes the quasi-TEM modes in a system of n + 1 conductors, showing that such geometries support n such modes. In the quasi-static limit the matrices L0 and C0 are functions of the geometry and the permeability and permittivity of the surrounding medium alone. The theory has also been applied to situations where these matrices have more complicated (frequency dependent) terms, modeling structures which exhibit dispersive effects, due to the continuous reactive loading of the lines [33–35]. Such treatments are related to MTL-homogenization approximations, as will be shown in this work. A literature search revealed that the consideration of MTL geometries in which the loading is modeled in a discrete manner (as in the case of the TL models described previously) has not been extensively examined. In [36] a system in which n uniform uncoupled transmission lines are periodically reactively loaded in a discrete manner was examined. It was shown that such a configuration yields a total of n modes, some of which are propagating and some of which are evanescent; i.e. of the exact type predicted by standard TL theory. The theory developed in this work characterizes the more realistic case in which the n + 1 lines are coupled. This feature will in turn be shown to have a critical effect on the nature of the derivable propagation constants, with a new class of modes, which are separated from standard propagating and evanescent modes, becoming possible. The MTL model will be developed explicitly and in considerable detail for the shielded Sievenpiper structure [5, 11, 17]. This is an attractive structure to study for several reasons. As mentioned previously this geometry has been shown to be useful in the suppression of switching noise in digital circuits [6–8] and in the creation of unidirectional slot antennas [9]. The structure is also capable of producing a strong slow-wave effect due to an enhanced effective relative permittivity [10, 11]. Other closely related shielded geometries have been shown to be relevant in the characterization of 3D stacked NRI metamaterials [12–14]. Another topologically related coupled-line microstrip geometry [15, 16], has been shown to yield a compact directional coupler. From an analytical perspective, the geometry of the shielded structure lends itself to


23

the matrices L0 and C0 taking on extremely simple forms, their components related to simple parallel-plate type geometries. The fact that L0 and C0 can be described by simple closed form expressions for a realistic geometric configuration will aid greatly in the interpretation of the analytical results. The remainder of the thesis is organized as follows. In Chapter 2 finite element method (FEM) simulations of the shielded Sievenpiper structure will be presented. Both dispersion curves and modal field profiles will be shown, and by examining these, several insights into the structure of the sought after multi-mode model will be obtained. Chapter 3 builds on the insights obtained in the previous chapter and will be focused on developing fundamental building blocks which will subsequently be used to model the shielded structure. These building blocks are comprised of both distributed elements, the matrices L0 and C0 which describe propagation along uniform multiconductor transmission lines, and the reactive loading elements L and C, which describe the discontinuities due to the vias and gaps between the patches. In Chapter 4 a periodic multiconductor transmission line unit cell of the shielded structure is presented and a corresponding dispersion equation will be derived from it. A comprehensive account of the salient features described by the dispersion equation will be given. In particular, the MTL model is readily able to handle the dual-mode behaviour of the structure. The nature of the modes in the stop-band will be determined, where it will be shown that the first stop-band is characterized by unusual modes: complex modes [37, 38], which are generated in pairs defined by complex-conjugate propagation constants, γ1,2 = α(ω) ± jβ(ω). Both α(ω) and β(ω) are functions of frequency, and critically, such modes are shown to arise even though the structure is modeled as lossless. The MTL model will also be able to provide an explanation for the changing character of the dispersion as hu is decreased. In particular, as was noted previously, for small values of hu , the first pass-band is single mode, with the BW band completely eliminated. Using the developed theory, analytical formulas for several critical points on the dispersion curves will be derived, with these formulas revealing the mechanism behind this qualitative change in behaviour (dual to single-mode). The MTL model will be compared with FEM simulations, with excellent correspondence demonstrated. In Chapter 5 the low frequency response of the shielded structure will be obtained. By examining both the dispersion and the modal eigenvector in the low frequency limit an elegant physical explanation of the resulting slow wave effect will be given. In Chapter 6 excitations of a finite cascade of unit cells of the shielded structure will be examined, with generalized scattering parameters derived. The dispersion analysis of Chapter 4 corresponds to the Bloch modes of an (infinite) periodic structure. In a finite structure a superposition of Bloch modes will be excited and by examining the scattering parameters along with


24

the related modal excitation strengths additional insights into the operation of the structure will be obtained. Excellent agreement between the MTL model and FEM simulations will be shown in pass-bands and both complex mode and evanescent mode bands, thus confirming the existence of complex modes in the structure. In Chapter 7 a physical application which utilizes the stop-band property of the shielded structure will be shown. A uni-directional slot antenna, which resonates within the stop-band of the shielded structure, will be constructed and tested, demonstrating the usefulness of the structure in suppressing the undesirable back radiation inherent in slot radiators. In Chapter 8 a summary of the thesis contributions is presented and publications associated with this work are listed.

Chapter 2

Analytical Motivation: Finite Element Method Simulations 2.1

Introduction

In the preceding chapter several examples of structures with dual-mode dispersion curves were presented. Two were open structures, the unshielded Sievenpiper structure, and the loaded NRI 2-D transmission line grid, while one was a closed structure, the shielded Sievenpiper structure. Although the dual-mode behaviour could be explained using homogenization approximations, these models could not account for the periodicity of the structure. Transmission line models suffered from the opposite defect; they could account for the periodicity, but not for the dualmode behaviour. In this chapter we will investigate more closely the full electromagnetic dispersion behaviour of the shielded Sievenpiper structure using finite element method (FEM) simulations. The main purpose of the simulations will be to provide motivation for the analytical model which will be developed in the following chapters. FEM generated dispersion curves will be shown for a range of values of the geometric parameters of the shielded structure. In order to develop an intuitive understanding of the structure, a sequence of dispersion curves with varying upper region height, hu will be presented. Previously it was seen that for relatively large hu , the structure exhibited lowest band dual-mode behaviour, while for small enough hu the structure had a single-mode lowest pass-band. Along with the FEM generated dispersion curves, two additional sets of curves will be shown. These will be the TL-theory curves for the unshielded Sievenpiper structure, which exhibit a high-pass BW pass-band, and the light line for free space (as r = 1 for the upper region of the simulated structure). However, the dispersion curves alone do not in themselves provide adequate insight into the development of the sought after dual-mode model. To this end, field profiles for modes 25

Chapter 2. Analytical Motivation: Finite Element Method Simulations

26

corresponding to specific points on the dispersion curves will be shown. By examining the field profiles and polarizations for both the electric and magnetic fields, E and H, in the plane transverse to the direction of propagation, several insights into the structure of the model which could account for the dual-mode behaviour will be arrived at. The hints provided by these insights will provide a solid starting point for the model to be developed in the following chapters.

2.2

Numerical Set-up

The geometry of the shielded structure, along with the boundary conditions to be implemented in the FEM simulations are shown in Figure 2.1. The software package HFSS was used for all of the simulations presented in this thesis, unless stated otherwise. The structure is periodic along both the x and y directions as shown. Propagation along one of the principal axes, x, will be considered. For such propagation, the phase shift transverse to the direction of propagation, (βd)y = 0 is fixed, while the phase shift along the direction of propagation is varied, 0 ≤ (βd)x ≤ π, and hence periodic boundary conditions are implemented along both the x and y directions. The FEM simulations are performed by sweeping the (βd)x values, while holding (βd)y = 0 fixed. The dispersion curves are generated by solving for a fixed number of modes (frequencies) at each pair [0 ≤ (βd)x ≤ π, (βd)y = 0]. Additionally, in Figure 2.1b, the transverse plane at the edge of the unit cell is marked. Later in this chapter modal field profiles will be shown on this transverse y − z cut plane.

2.3

FEM simulations

2.3.1

Dispersion Curves

Initially a sequence of dispersion curves will be presented. The dispersion curves correspond to a series of simulations in which the upper region height, hu , is given three values: hu = 18, 6, and 0.5 mm, with the lower region height fixed, hl = 3.1 mm. All of the other electrical and geometric parameters of the structure are fixed. The upper and lower region permittivities are r1 = 1 and r2 = 2.3, respectively, while the periodicity and patch width are d = 10 mm and w = 9.6 mm, respectively. The via radius is r = 0.5 mm. The FEM generated curves for hu = 18 and 6 mm are shown in Figure 2.2. As discussed in the previous section, the FEM curves are obtained by varying the phase along x, (βd)x , while (βd)y = 0 is fixed. Even though the FEM curves are obtained with a fixed transverse phase shift, (βd)y = 0, it will later be shown that the modes thus obtained can be divided into two separate classes with different field polarizations at the transverse boundaries. In anticipation

Chapter 2. Analytical Motivation: Finite Element Method Simulations z

27

y y

Transverse fields to be shown on this yz cut

x

Shielding conductor; d

hu

ǫ1

0 ≤ (βd)x ≤ π Patch conductor; w

hl

ǫ2

via; r Ground conductor

(a) Side view

(βd)y = 0 (fixed) (b) Top view, as seen below the Shielding conductor

Figure 2.1: (a) Unit cell of the shielded Sievenpiper structure. (b) For on-axis propagation, (βd)y = 0 is fixed, while the phase shift per-unit-cell, (βd)x , along the direction of propagation (x), is varied. Modal field plots on the transverse plane at the cell edge to be shown later in this chapter. of this, the simulated dispersion curves have been marked with squares, (FEM), and circles, ◦ (FEM-TE). Excluding the FEM-TE points on the dispersion curves, the remainder of the dispersion of the shielded structure will be seen to be formed, at least for large hu , from a union and deformation of the two other sets of curves which are plotted: the TL- theory of the unshielded structure, and the light line for free space. The resonance points of the shielded structure defined by (βd) = 0, (f2 and f3 ), and (βd) = π, (f1 and f4 ), are also shown, with a comparison of the the location of these points with the corresponding ones of the TL model providing additional insight. Figure 2.2a corresponds to hu = 18 mm, with a zoomed in version shown in Figure 2.2b. It is observed that within the first pass-band, at low frequencies, a forward wave (FW) mode with dispersion just below the light line is supported. Above the frequency, f1 , the structure becomes dual-mode, with a backward wave (BW) mode being excited. It is noted that the resonance f1 corresponds closely with the resonance of the TL-model BW mode. As frequency increases above f1 , the dispersion curves of the two distinct modes approach each other and eventually coalesce at a maximum frequency, fmax . Above this point on the dispersion curve, [(βd)1 , fmax ], the contra-directional coupling of the two modes results in a stop-band. It is interesting to note that the peak of the first pass-band (or commencement of the first stop-band) occurs at a value

28


Frequency (GHz)

12

10

FEM FEM (TE) TL Light εr1=1

f3→


←f4 5

←fT E

8

f2→ h

↑

4 (βd)2 , fmin

i

f3→ f2→ 4

h

3

Stop-band

Stop-band

↑

(βd)1 , fmax

← f1

i

←f1 2

2

0

(βd)x

π

1

(a) hu = 18 mm

(b) hu = 18 mm; zoom in of (a) FW2

Frequency (GHz)

12

10

π

(βd)x


←f4

f3→ f2→

FEM FEM (TE) TL Light ε =1

TE

BW 2 ↑ h i 5 (βd)2 , fmin

r1

←fT E

8

Stop-band

4

f3→ f2−→

4

h

3

Stop-band

↑

(βd)1 , fmax

i

BW 1

←f1

←f1 2

2

FW1

0

(βd)x (c) hu = 6 mm

π

1

(βd)x

π

(d) hu = 6 mm; zoom in of (c)

Figure 2.2: Dispersion curves for the shielded structure with varying upper region height: (a), (b) hu = 18 mm; (c), (d) hu = 6 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of the unshielded structure, and the free space light line. Field plots corresponding to the points labeled in (d) will be shown later in this chapter.


29

12

f2→ 10

8

←fT E ←f4

FEM FEM (TE) TL Light ε =1 r1

f3→ Frequency (GHz)

Stop-band

4

2

0

FW2 FW1

(βd)x

←f1

π

Figure 2.3: Dispersion curves for the shielded structure with with hu = 0.5 mm. The other physical parameters are: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. Also shown are the curves for the TL(BW) model of the unshielded structure, and the free space light line. Field plots corresponding to the labeled points will be shown later in this chapter. of (βd) 6= 0 or π, which is an indication that the stop-band is not of the type more typically encountered for periodic structures. The dispersion in this first band is seen to be roughly a combination of the FW light line and the BW of the TL model, except in the region where these two curves coalesce, above which a stop-band is formed. In the second pass-band, it is observed that f2 is a point of modal degeneracy, with both a FW (TE) and a BW mode emerging from f2 . Disregarding the FW (TE) mode, it is again observed that the dispersion curve is formed from a combination of the BW emerging from f2 and the light line, with the peak of the stop-band at [(βd)2 , fmin ] formed due to contradirectional coupling of the BW and the FW modes. It is noted that the BW mode emerging from f2 is close to, but shifted up slightly, relative to that of the TL theory curve. The phase shifts at the commencement and conclusion of the stop band, (βd)1 and (βd)2 , are not 0 or π, but 0 ≤ (βd)2 < (βd)1 < π, which again is an indication that the stop-band is not of the type more typically encountered for periodic structures. Continuing with Figure 2.2a, at f3 , which is close to the TL resonance, a FW mode emerges, which again follows the TL dispersion closely, until it reaches the light line, at which point it veers above the TL dispersion and begins to track the light line dispersion, demonstrating codirectional coupling. The BW mode which initially emerges from f2 , and with increasing phase shift, (βd)x becomes a FW tracking just above the light line experiences a similar situation.

30


It tracks just above the light line, until it is about to intersect the upper TL FW band, upon which it follows the TL curve up to the resonance f4 . The FEM simulations for the case hu = 6 mm are similar to those presented for hu = 18 mm. One notable difference is the increase in the stop-band bandwidth, which is an indication that the contra-directional coupling effect is stronger for smaller hu values. The first band remains dual-mode, with f1 still corresponding closely with the TL dispersion. The frequency f2 , though still a point of modal degeneracy, has shifted up significantly relative to that of the TL dispersion curve. However a BW and FW mode still emerge from f2 . The peak of the first pass-band and the minimum of the second pass-band retain the qualitative features they had for the hu = 18 mm case. In the upper bands the resonance frequencies f3 and f4 still correspond to those given by the TL model. The simulation results for the case hu = 0.5 mm are shown in Figure 2.3. The stopband bandwidth has increased substantially compared to the previous two cases, which is an indication of an even stronger contra-directional coupling effect. Additionally, a seemingly fundamental qualitative change has occurred in the dispersion. The first band is no longer dual-mode, with only a single FW mode supported, while the BW mode has been eliminated. The resonance f1 is shifted down significantly from that of the TL resonance, and f1 is no longer the initial frequency of a BW band (as in the hu = 18 and 6 mm cases), but the termination point of the first pass-band. The fact that the initial point of the stop-band occurs at βd = π for this geometry could lead one to believe that the nature of the stop-band is like that of more typically encountered periodic structures. This statement will be addressed more thoroughly in the following chapters with the developed analytical model. Above the first pass-band, it is observed that f2 has been altered substantially, as it occurs at a much higher frequency than f3 . However, as in the previous case f2 represents a point of modal degeneracy, with two modes emerging from it. The second pass-band now commences at the frequency f3 . Interestingly f3 and f4 still coincide with the TL theory resonances and the invariance of these two frequencies as hu is varied will be addressed upon developing the analytical model.

2.3.2

Modal Field Profiles: hu = 6 mm

In Figure 2.2d, for hu = 6 mm, three points in the first pass-band have been marked with solid markers: (1) A FW1 mode, at (βd)x = (βd)x =

8π 9

π 9

(20◦ ), f = 1.33 GHz, (2) A BW1 mode at

(160◦ ), f = 3.17 GHz, and (3) the peak of the first pass-band at (βd)x '

π 2

(88◦ ),

f = 3.36 GHz. Modal field plots for these three points are shown in Figure 2.4. The field plots show the transverse components of E and H on the transverse plane at the unit cell edge as shown previously in Figure 2.1. The transverse components are focused on initially since these are responsible for the Poynting vector, and hence the modal power flow.


z

z

z y

y

(i) E transverse

x

(ii) H transverse (a) FW1 mode at f = 1.33 GHz; (βd)x =

(iii) Net Power: +ve π 9

◦

(20 )

z

z

z y

y

(i) E transverse

x

(ii) H transverse

(b) BW1 mode at f = 3.17 GHz; (βd)x =

z

(iii) Net Power: -ve 8π 9

(160◦ )

z y

z y

(i) E transverse

31

(ii) H transverse

(c) Peak of 1st passband at f = 3.36 GHz; (βd)x '

x

(iii) Net Power: Zero π 9

(88◦ )

Figure 2.4: Transverse modal field plots for the 1st passband of the structure with dispersion curve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 12 E × H∗ on the same transverse cut, but with view rotated.


32

For the FW1 mode, Figure 2.4a, it is observed that in the region between the shield and the patch layer, the electric field, E is nearly uniform, with a small amount of fringing near the patch edges. However, in the region between the patch layer and the ground plane E is nearly zero. The magnetic field, H, on the other hand is nearly uniform throughout both regions. The polarizations of both E and H indicate that the side boundaries are acting as perfect magnetic conductors, or H-walls. Due to the relatively simple nature of the fields in the regions above and below the patch plane, the structure can be thought of as consisting of an upper region and a lower region. For this mode Eupper is nearly uniform, while Elower ≈ 0. For the magnetic field, Hupper ≈ Hlower , both being nearly uniform throughout. Due to the fact that Elower ≈ 0, the modal power flow is confined to the upper region, as is also shown. The Poynting vector is depicted on the same transverse cut used for the fields, but the axes have been rotated, so that the direction of the power flow is clearly observed. The H field appears to be identical to that of a parallel-plate waveguide formed from the shielding conductor and the ground conductor alone, with the patches and vias removed. This would seem to indicate that there is a longitudinal x directed surface current on the shield, and an equal and oppositely directed return current on the ground plane. However the E field seems to be that of a parallel-plate waveguide formed from the shield and the patch layer, and so this mode, at least at first glance appears to be a combination of two different waveguide modes. Field plots of the surface current distributions will be shown later on in this section, in order to address these issues. Turning now to the BW1 mode, in Figure 2.4b it is observed that Eupper is nearly uniform, but now Elower is non-zero, also nearly uniform, but with polarization opposite of Eupper . The magnetic field, H is again non-zero in both regions, and nearly uniform in each of the upper and lower regions, but with differing magnitudes now. In the lower region, Hlower is large and nearly constant, while Hupper is smaller and nearly constant. Due to the significant field strengths in both the upper and the lower regions, power flow occurs in both regions. The Poynting vectors in the upper region are small and directed in the +x direction, while in the lower region they are large and directed in the −x direction. The net power, integrated over the transverse y − z plane is in the −x direction, consistent with the fact that the mode as a whole is a BW mode. The side boundaries are acting as H-walls, and hence for both the FW mode and the BW mode the transverse boundaries are H-walls. At the peak of the first pass-band, [(βd)1 , fmax ], Figure 2.4c shows that the field profile is qualitatively similar to that of the BW1 mode. The fields Eupper and Elower are nearly constant (with differing magnitudes), in their respective regions, but with opposite polarizations, while Hupper and Hlower are each nearly constant (with differing magnitudes), and the same polarization. However, for this point on the dispersion, the Poynting vectors in the upper region are larger, and the net integrated power becomes zero. This is unlike the more typically encoun-


z

33

z

y

y

(i) FW 1

(ii) BW 1

y

(iii) Band peak

Figure 2.5: Longitudinal current on the upper shield and ground plane for the three modes, FW1, BW1, and [(βd)1 , fmax ] of Figure 2.4. z y

(i) D longitudinal

Figure 2.6: Longitudinal D of the x-directed gap excitation for the three modes, FW1, BW1, and [(βd)1 , fmax ] of Figure 2.4. tered situation, in which the commencement of a stop-band is defined by a standing wave. For a standing wave field, not only is the net integrated power equal to zero, but additionally the power at each point on a transverse cut is equal to zero as well. This is the second clue that the stop-band encountered for the shielded structure is atypical, the first being that the edges of the initial stop-band were not at βd = 0 or π. Before examining the field profiles for the upper bands, the surface conduction current distributions on the shielding and ground conductor, corresponding to the three points on the dispersion curve of Figure 2.2d, FW1, BW1, and [(βd)1 , fmax ], will be examined. The surface currents are responsible for magnetic fields in the structure. As was noted previously, for FW1, H is virtually constant throughout the entire region, with Hupper ≈ Hlower , which would seem to imply that the shielding conductor and the ground conductor have nearly equal and opposite current distributions. Due to the fact that H is polarized in the y direction, these surface current densities flow along the x direction. However, for both BW1 and the band peak, Hupper differs significantly from Hlower , which implies that the current densities are not equal. FEM generated surface conduction current distributions are shown in Figure 2.5, verifying these observations. For the FW1 mode, the surface current densities on shielding and ground conductor are nearly identical, but oppositely directed. For both the BW1 mode and the band peak ([(βd)1 , fmax ]) these current densities are of unequal magnitude. Such unbalanced currents would be a source


34

of radiation, but as the structure is closed this is not possible, and there must be a compensating current source which prevents this from occurring. Recalling that the field plots are shown at the edge of the unit cell (Figure 2.1), it seen that only the shielding conductor and the ground conductor are continuous there, while the patch layer conductor is at a gap. In the gap region there is no conduction current, but a displacement current, JD =

∂D ∂t ,

may exist. This displacement current is due to the largely longitudinal x

directed D fields, which are formed in the gaps. These gap fields have nearly the same profile for all three points, FW1, BW1, and [(βd)1 , fmax ], as shown in Figure 2.6. However, the gap field magnitudes, relative to the vertical (transverse) components of E vary as a function of frequency and position on the dispersion curve. In the limit ω → 0, the displacement current approaches zero, and this is what occurs for the FW1 mode. For both the BW1 mode and the band peak, the displacement current is not negligible and for these two points the displacement current produced is such that total current through the transverse cut (shield + ground + displacement current) is zero. This non-zero displacement current allows Hupper to differ from Hlower , and is also consistent with the fact that the closed structure does not radiate. The FEM simulations also revealed that the gap fields are out of phase with the transverse fields by π 2,

indicating that the gap fields are reactive and do not contribute to power flow. Field profiles for the three upper band modes from the dispersion curves of Figure 2.2d are

shown in Figure 2.7. The three points are: (1) A BW2 mode at (βd)x = GHz, (2) A FW TE mode at at (βd)x = (βd)x =

π 9

π 9

π 9

(20◦ ), f = 5.50

(20◦ ), f = 5.74 GHz, and (3) A FW2 mode at at

(20◦ ), f = 6.21 GHz. For the point BW2, Figure 2.7a, the polarization of the fields

in the upper and lower regions have been altered relative to those of BW1. For BW2, Eupper and Elower have the same polarizations, while Hupper and Hlower are oppositely polarized. As the field strengths are significant in both regions there exist non-zero Poynting vectors in both regions, but as for BW1, the net integrated power is in the −x direction. The field polarizations at the transverse boundaries are consistent with H-walls for this mode. Skipping up in frequency to FW2, Figure 2.7c, again there are significant fields in both regions, but the polarization of both E and H are directed oppositely in both the upper and lower regions. Thus for this mode the Poynting vectors in both the upper and lower regions are directed in the +x direction, and the mode is a FW. Again, the transverse boundaries are acting as H-walls for this mode. Returning to the mode labeled TE, Figure 2.7b, the transverse field profiles are of a completely different character than those of the previously examined cases. For this mode E is strongly confined to the patch edges, and highly non-uniform, corresponding to a strong transverse gap excitation. The transverse H fields are also non-uniform, and strongest near the patch layer. These field polarizations are consistent with the transverse boundaries acting as


z

z

z y

y

(i) E transverse

x

(ii) H transverse (a) BW2 mode at f = 5.50 GHz; (βd)x =

(iii) Net Power: -ve π 9

◦

(20 )

z

z

z y

y

(i) E transverse

x

(ii) H transverse

(iii) Net Power: +ve

(b) FW TE mode at f = 5.74 GHz; (βd)x =

π 9

(20◦ )

z

z y

(i) E transverse

35

z y

x

(ii) H transverse (c) FW2 mode at f = 6.21 GHz; (βd)x =

(iii) Net Power: +ve π 9

(20◦ )

Figure 2.7: Transverse modal field plots for the upper passbands of the structure with dispersion curve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 12 E × H∗ on the same transverse cut, but with view rotated.

36


z

y

y

(i) E transverse

z x


(ii) H transverse (a) FW1 mode at f = 0.61 GHz; (βd)x =

π 9

◦

(20 )

z

z

y

y

(i) E transverse

z x


(ii) H transverse (b) FW2 mode at f = 2.27 GHz; (βd)x =

8π 9

(160◦ )

Figure 2.8: Transverse modal field plots for the 1st passband of the structure with dispersion curve from Figure 2.3 (hu = 0.5 mm). (i) E and (ii) H viewed on a transverse cut at the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 12 E × H∗ on the same transverse cut, but with view rotated. E-walls for this mode. Although not shown, there exists a strong longitudinally directed H field, and thus the mode is a TE (transverse electric) mode. The frequency f2 is a point of modal degeneracy, with two modes, one with transverse H-walls and the other with transverse E-walls, emerging from it. Although the theory developed in subsequent chapters will not be able to account fully for the complete dispersion of the E-wall (TE) mode, it will explain the physical origin of the modal degeneracy at f2 .

2.3.3

Modal Field Profiles: hu = 0.5 mm

Modal field profiles corresponding to the dispersion curve for hu = 0.5 mm (Figure 2.3), are shown in Figure 2.8. For hu = 0.5 mm a qualitative change in the dispersion occurs, with the first band becoming single mode. For this reason our attention will be focused on field plots in this band. Two points on the dispersion curve have been marked: 1) A FW1 mode, at (βd)x =

π 9

(20◦ ), f = 0.61 GHz and (2) A FW2 mode at (βd)x =

8π 9

(160◦ ), f = 2.27 GHz.

For FW1, Figure 2.8a, the field profiles are virtually identical to those of FW1 for the case of hu = 6 mm. The electric field, E is confined to the upper region, while the magnetic field, H is uniform throughout both upper and lower regions. The resulting Poynting vectors are virtually null in the lower region and directed in the +x direction in the upper region. The principal difference between the two cases hu = 0.5 and 6 mm, for the FW mode at (βd)x =

π 9

(20◦ ),

is that for hu = 0.5 mm, the mode has been slowed significantly, as the group velocity, vg ,


37

has been reduced substantially. This slowing effect will be examined and explained using the developed theory in Chapter 5. The point labeled FW2, Figure 2.8b, which occurs at (βd)x =

8π 9

(160◦ ), will be compared

with BW1, Figure 2.4b (for hu = 6 mm), which also occurred at (βd)x =

8π 9

(160◦ ). The

fields for FW2 are only slightly different from those of FW1, for the case hu = 0.5 mm. The principal difference is that for FW2, a small oppositely polarized E has been generated in the lower region. However significantly, the magnetic fields in both regions are nearly equal, Hupper ≈ Hlower , unlike for BW1 (where hu = 6 mm). The fact that the magnetic field remains nearly equal throughout both regions suggests that little displacement current is generated along the longitudinally directed gaps, suppressing the BW mode in this case. The suppression of the BW mode results in the structure losing its dual-mode character for relatively small hu values. These observations will be addressed and explained in Chapters 4 and 5.

2.4

Summary

From the dispersion simulations the following points are noted. Disregarding the FW (TE) mode emerging from f2 , for the relatively large values of hu = 18 and 6 mm, the dispersion of the shielded structure is seen to be a combination and deformation of the TL model BW dispersion curve and the light line. For smaller values of hu , the lowest band dual-mode behaviour is eliminated, and an identification of the dispersion as a combination of the two previously mentioned curves becomes more difficult. It is interesting to note that the resonance frequencies f3 and f4 remain invariant with respect to those of the TL model for any upper region height, hu . From the modal field profiles the following points are observed. All of the modes, excluding those denoted by TE, are characterized by field strengths which are nearly uniform in each of the two regions, upper and lower. The polarizations for these modes are such that the transverse boundaries are acting as H-walls. However, in general for a given mode, the Poynting vectors in the upper and lower regions are oppositely directed, so that the modes as a whole are characterized as forward waves or backward waves by considering the net integrated power on a transverse cross section. The fact that the modes appear to be formed as the result of combinations of field excitations in two distinct regions, with transverse H-wall boundary conditions, will be used as a motivation for the analytical model developed in the following chapters. Using the developed analytical model, the observations obtained from the FEM simulations will be re-examined and explained.

Chapter 3

Multiconductor transmission line analysis: Building blocks 3.1

Introduction

In the previous chapter, FEM simulations for the shielded Sievenpiper structure were performed, with both dispersion curves and modal field profiles examined. Several insights into the type of model which could account for the observed dual-mode behaviour were obtained. The case of on-axis propagation, (βd)y = 0 was examined, which was seen to be general enough to reveal interesting dispersion and modal behaviour, yet it allows simplification in the development of the analytical model to be initiated in this chapter and fully presented in the next chapter. The key features of the dispersion curves were the dual-mode behaviour for relatively large values of hu , with the dispersion curves in this case being formed from a slight perturbation of the light line and the unshielded Sievenpiper TL model. Additionally, the peak of the first stop-band occurred at a point away from the Brillouin zone center (βd = 0) or edge (βd = π). The forward wave (FW) mode in the first band was seen to have strong upper region forward direction power flow, and weak negative direction power flow in the lower region, with the opposite being true for the backward wave (BW) mode. For the peak of the first band, the net integrated power flow was zero. For relatively small values of hu the dispersion curve within the first band became completely single mode, and could seemingly no longer be identified with a slight perturbation of the light line and the TL model curves. Significantly, the low frequency FW mode was severely slowed down, while the BW band was completely eliminated. The modal power flow profiles in the first band, which contained only a forward wave mode, had strong upper region forward direction power flow, with minimal negative lower region power flow. Finally, the peak of the first band occurred at the Brillouin zone edge, βd = π. 38

Chapter 3. Multiconductor analysis: Building Blocks

39

The field polarizations for all of the modes in the lowest band, and all, except one mode in the upper bands, were seen to correspond to H-walls on the transverse boundaries. For the lone upper band mode referred to above, which did not correspond to transverse H-walls, the field polarization on the transverse boundary was an E-wall. In this chapter, and the next, some of the features described above will be used initially to aid in the development of the proposed model, and subsequently will be explained with the derived analytical model. It will be demonstrated that wave propagation of the shielded structure can be analyzed using multiconductor transmission line (MTL) theory. In the remainder of this chapter the following points will be addressed. Standard MTL theory describes propagation for systems which are uniform along the propagation direction. However, the MTL model of the shielded structure will be shown to correspond to a periodic (and hence non-uniform along the direction of propagation) MTL geometry. In order to understand the propagation in the actual structure, which is periodically loaded, it will be useful to define and analyze the propagation properties of the underlying unloaded MTL geometry, which is one of the building blocks in the fully loaded periodic model. It will be determined that for the defined underlying unloaded geometry, two independent modes of propagation, which correspond essentially to plane waves propagating in the upper and lower regions of the structure, are supported. Both the dispersion curves and modal field profiles of the unloaded MTL geometry will be analyzed. That these modes form a basis to describe wave propagation in the actual shielded structure is anticipated due to the field profiles observed in the previous chapter. Subsequently, the determination of the loading elements, which alter the propagation properties of the unloaded geometry, will be undertaken. The gaps along the direction of propagation will be shown to correspond to equivalent series capacitances, C, while the vias between the patch layer and ground will be shown to correspond to equivalent shunt inductances, L. Typically the loading elements are calculated by considering appropriate scattering simulations, which usually involve two-port configurations. However, due to the presence of the top shielding conductor, a simple two-port scattering analysis will be shown to be insufficient in the calculation of C, and a more thorough four-port scattering analysis will be shown to be necessary, especially in the case where hu is relatively small. The calculation of L will proceed in a similar manner. This analysis will subsequently be used in the next chapter to characterize the propagation properties of the actual periodic geometry of the shielded structure, with dispersion curves, modal field profiles, and important resonant frequencies derived and explained.


3.2

40

Unloaded MTL Geometry

In correspondence with the FEM simulations of the previous chapter, on-axis propagation of the shielded structure, with (βd)y = 0 will be examined. It was observed that for the lowest band and most of the upper bands, the transverse boundaries were characterized by H-walls. Initially, the assumption of transverse H-walls will be made, although later it will be shown that (βd)y = 0 implies that the transverse boundaries are either H-walls or E-walls for symmetric structures. In fact, a stronger condition will be shown to be true in Chapter 4: due to symmetry, the condition (βd)y = 0 implies that the central bisecting plane of the unit cell has the same boundary condition as the edge walls. The present section will focus on defining and then analyzing propagation in the unloaded MTL geometry, which will be a key building block in the analysis of the shielded structure. The simplest way to introduce the unloaded geometry, though, will be to show the transformation which takes the unloaded structure into the actual loaded (shielded) structure. This transformation is depicted in Figure 3.1. The unloaded geometry is depicted from a side view, Figure 3.1a and a top view, Figure 3.1c, where the top view is taken just below the upper shielding plane. The unloaded geometry is seen to be an infinite (along x ˆ) array of strips, placed between the upper shielding conductor and the ground plane. The direction of propagation is assumed to be along x ˆ, and hence waves propagating along the length of the infinite strips do not experience any discontinuities. The side boundaries (dashed lines) are assumed to be H-walls. Figures 3.1b and d depict the transformed, loaded geometry, which is seen to be generated by periodically cutting, along yˆ, gaps in the infinite strips, thereby creating islands of patches and simultaneously placing vias from the center of each patch to the ground plane. The gaps and the vias provide discontinuities to waves propagating along x ˆ, and thus will be referred to as the loading elements. In Figure 3.2 a system of n+1 parallel conductors is depicted, with conductor 0 taken as the reference conductor. The system is assumed uniform along the x axis. Such a system represents a generic multiconductor transmission line (MTL) geometry, and the lowest order modes of such a configuration are quasi-TEM in general (purely TEM if the surrounding dielectric medium is uniform) [32]. A transverse cut of the unloaded geometry for the specific case of the shielded Sievenpiper structure is shown in Figure 3.3. The upper shielding conductor is labeled 1, the patch layer conductor is labeled 2, and the ground plane is left unlabeled. Thus the unloaded shielded Sievenpiper structure is a 2 + 1 conductor MTL geometry with n = 2. Returning to the generic MTL geometry shown in Figure 3.2, under the assumption that the currents on the conductors are purely longitudinal (along x ˆ) the resulting magnetic fields are purely transverse. One is then able to uniquely define the voltage on the p − th conductor with respect to ground

41


Side view; dashed lines represent H walls z y

(b) Loaded

(a) Unloaded

Viewed looking down (-z), below the top (shielding) plane y x

(c) Unloaded

(d) Loaded

Figure 3.1: Transformation of an infinite 1-D periodic array of strips, (a) and (c), into an infinite 2-D periodic array of isolated patches (b) and (d). Vias connected from the center of each patch to ground for (b) and (d). The transverse boundary conditions are assumed to be H-walls for the case of on-axis propagation in the MTL model.

42

Chapter 3. Multiconductor analysis: Building Blocks x

x 1

1 p n 0

I1

p

Ip

V1

n

Vp In Vn Io

0

(b)

(a)

Figure 3.2: Generic multiconductor transmission-line configuration for an n + 1 conductor system. Propagation is along the x axis; Ik and Vk denote conductor k’s current and voltage. (a) Longitudinal view. (b) Cross-sectional view.

as:

Z Vp (x, t) =

E(x, y, z, t) · ds

(3.1)

po

and the current on the p − th conductor becomes: I Ip (x, t) =

H(x, y, z, t) · ds

(3.2)

sp

where po and sp are integration paths lying on the transverse y −z plane; po connects conductor p to the reference conductor, while sp encircles conductor p. Using linearity and the principle of superposition, Maxwell’s equations can be transformed into the following system [32]: ∂ ∂ V(x, t) = L0 I(x, t) ∂x ∂t ∂ ∂ − I(x, t) = C0 V(x, t) ∂x ∂t

(3.3)

−

(3.4)

where V and I are n component column vectors which define the voltages and currents on the n lines. L0 and C0 are the inductance and capacitance per-unit length matrices, which are the generalizations of the scalar per-unit length parameters defining a standard 2-conductor 0

system. The matrix C0 relates the charge per-unit length, Qp , on conductor p linearly to the

43

Chapter 3. Multiconductor analysis: Building Blocks d

z

Conductor 1; {V1 , I1 }

y hu

ǫ1 w

hl

ǫ2

Conductor 2; {V2 , I2 } Ground

Figure 3.3: Parameters defining the unloaded MTL geometry for on-axis propagation assuming transverse H-walls (dashed lines). Conductors 1 and 2 have voltages, {V1 , V2 }, defined with respect to ground, along with currents {I1 , I2 }, which are used to 0 0 define the per-unit-length capacitance and inductance matrices, C and L . voltages, Vj on all of the other conductors: 0

Qp =

n X

0

Cpj Vj

(3.5)

j=1

or in matrix form: Q0 = C0 V

(3.6)

where Q0 is an n component column vector containing the charge per-unit-lengths on conductors 0

1 through n. The matrix, L0 relates the per-unit length flux, Ψp , linking conductor p to the ground conductor, linearly to the currents, Ij on all of the other conductors: 0

n X

0

(3.7)

Ψ0 = L0 I

(3.8)

Ψp =

Lpj Ij

j=1

or in matrix form:

where Ψ0 is an n component column flux linkage vector. Under the assumption that the fields are time harmonic, with angular frequency ω the equations (3.3) and (3.4) reduce to: d V(x) = jωL0 I(x) dx d − I(x) = jωC0 V(x) dx

−

(3.9) (3.10)

where the longitudinal impedance matrix, Z0 , and the transverse admittance matrix, Y0 , are

44

Chapter 3. Multiconductor analysis: Building Blocks d

d

z

V1

I1

y

hu

hu V2 = 0

I2 = 0

hl

hl ′

′

(b) L11

(a) C11

0

0

Figure 3.4: Boundary value problems used to determine C11 and L11 .

defined by Z0 = jωL0 and Y0 = jωC0 . For an unloaded 2 + 1 conductor system such as that considered here, C0 and L0 are given explicitly by: "

0

Q1

# =C

0

Q2 and

"

0

Ψ1 0

Ψ2

" 0

V1

# =

V2

# 0

=L

" # I1 I2

" 0 #" # 0 C11 C12 V1 0

" =

0

C21 C22 0

0

0

0

L11 L12 L21 L22

V2

#" # I1 I2

(3.11)

(3.12)

The components of C0 and L0 are obtained by solving a set of boundary value problems. For 0

example the C11 is obtained by solving the following electrostatic boundary value problem: 0

0

C11 =

Q1 ; V1

V2 = 0

(3.13)

0

while L11 is obtained by solving the following magnetostatic problem: 0

0

L11 =

Ψ1 ; I1

I2 = 0.

(3.14)

The approximate field structures which are obtained in the solutions of these two problems are depicted in Figure 3.4. Assuming that the plate width, w is approximately equal to the periodicity so that, w ≈ d, it is anticipated that the fields will be approximately those of a multiple parallel-plate geometry. Due to the presence of an H-wall at the transverse boundary the fringing fields are assumed to be small. With these approximations it can be shown that 1 wef f 0 C11 = , where wef f = w + ∆w is the effective width which is used to account for the hu ∆w fringing fields, and 1 is small. The remaining components of C0 may be similarly w

45

Chapter 3. Multiconductor analysis: Building Blocks calculated, resulting in:



 1 wef f  h 1 wef f u2 wef f  . + hu hl

1 wef f  hu 0 C =  1 wef f − hu

−

(3.15)

It is interesting to note that the components of C0 are given in terms of the simple expressions, 2 wef f 1 wef f 0 0 and Cl = Cu = , which are the capacitance-per-unit-length formulas for parallelhu hl plate waveguides of width, wef f , and heights hu and hl , respectively; that is parallel-plate waveguides of a structure consisting of the upper and lower regions guides alone.

0

In order to verify these assumptions FEM simulations calculating C were performed, with the results shown in Table 3.1. The simulations were carried out using the software package COMSOL Multiphysics. Three different geometries were simulated, with varying upper region height, hu = 18 mm, hu = 6 mm, and hu = 0.5 mm. The lower region height was fixed at hl = 3.1 mm. The permittivities of the upper and lower regions were r1 = 1 and r1 = 2.3, respectively and the patch width was w = 9.6 mm. The FEM results were compared with the analytical formulas, with two different values for the effective plate width, wef f = 9.6 and 10 mm, corresponding to ∆w = 0 and 0.4 mm respectively. From the table it is observed that when the effective width, wef f = 10 mm, which is equal to the periodicity, d, the numerically calculated components differ by at most 0.3 %, and hence this approximation will be employed from here on in. With wef f = d, (3.15) becomes: 

C0 (wef f

1 d  hu = d) =  d 1 − hu

 1 d " # 0 0 C −C u  u hu 1 d 2 d  = −C 0 C 0 + C 0 u u l + hu hl −

(3.16)

0

The components of L0 may be similarly calculated, as is the L11 component in (3.14). However, there exists a relationship between L0 and C0 . Recalculating C0 with all of the permittivities 0

replaced by that of free space, results in C=0 , with this matrix related to L0 as follows [32]: 0

L0 C=0 = 0 µ0 I

(3.17)

where I is the identity and L0 is obtained by inverting (3.17), resulting in: h 0 i−1 L0 = 0 µ0 C=0

(3.18)

46


Table 3.1: Comparison of the numerical (FEM) and analytic C0 (capacitance) matrices for: (a) hu = 18 mm, (b) hu = 6, (c) hu = 0.5 mm. The analytic C0 matrix is calculated for two different values of the effective width, wef f = 10.0 and 9.6 mm. (a) hu = 18mm Capacitance matrix components

pF m

FEM analytic (wef f = 10 mm) % difference (FEM & wef f = 10 mm) analytic (wef f = 9.6 mm) % difference (FEM & wef f = 9.6 mm)

0

0

C21

C22

4.92 4.92 0 4.72 3.5

-4.91 -4.92 0.2 -4.72 3.8

-4.91 -4.92 0.2 -4.72 3.8

70.51 70.58 0.3 67.75 4.2

14.75 14.75 0 14.16 3.5

(c) hu = 0.5mm 0 Capacitance matrix components pF C11 m FEM analytic (wef f = 10 mm) % difference (FEM & wef f = 10 mm) analytic (wef f = 9.6 mm) % difference (FEM & wef f = 9.6 mm)

0

C12

(b) hu = 6mm 0 Capacitance matrix components pF C11 m FEM analytic (wef f = 10 mm) % difference (FEM & wef f = 10 mm) analytic (wef f = 9.6 mm) % difference (FEM & wef f = 9.6 mm)

0

C11

176.45 177.0 0 169.9 3.5

0

0

0

C12

C21

C22

-14.73 -14.75 0.2 -14.16 3.8

-14.73 -14.75 0.2 -14.16 3.8

80.32 80.41 0.3 77.19 4.2

0

0

0

C12

C21

C22

-176.21 -177.0 0.2 -169.9 3.8

-176.21 -177.0 0.2 -169.9 3.8

241.58 242.7 0.3 233.0 4.2


47

Using (3.16) and (3.18) results in: 

µo hl µo hu  d + d 0 L = µo hl d

 " # µ o hl 0 + L0 0 L L  u l l d µ o hl  = L0l L0l d

(3.19)

µo hu µo hl and L0l = are again related to those of paralleld d plate guides composed of the upper and lower regions alone. Thus the components of both C0

where the components of L0 , L0u =

and L0 are given in terms of simple expressions involving parallel-plate geometries when w ≈ d. When w is significantly less than d these expressions will not be valid and the evaluation of C0 will require a numerical solution.

Having obtained expressions for C0 and L0 the system of equations (3.9) and (3.10) may be solved. Expanding these equations results in: dV1 dx dV2 − dx dI1 − dx dI2 − dx

−

= Z1 I1 + Zm I2

(3.20)

= Zm I1 + Zm I2

(3.21)

= Y1 V1 + Ym V2

(3.22)

= Ym V1 + Y2 V2

(3.23)

The above first order system of equation can be transformed into a second order system involving only the voltages by eliminating the current in (3.20) through (3.23) resulting in: d2 V1 − a1 V1 − b1 V2 = 0 dx2 d2 V2 − a2 V2 − b2 V1 = 0 dx2

(3.24) (3.25)

Solving for V2 in (3.24) and inserting it into (3.25), the above coupled system of equations is reduced to an ordinary differential equation: d4 V1 d2 V1 − (a + a ) + (a1 a2 − b1 b2 )V1 = 0 1 2 dx4 dx2

(3.26)

48

Chapter 3. Multiconductor analysis: Building Blocks with a similar equation for V2 . The coefficients are evaluated to be: 0

0

0

0

a1 = Y1 Z1 + Ym Zm = −ω 2 Cu Lu

(3.27)

a2 = Y2 Z2 + Ym Zm = −ω 2 Cl Ll 0

(3.28) 0

0

0

b1 = Z1 Ym + Y2 Zm = −ω 2 (Cl Ll − Cu Lu )

(3.29)

b2 = Z2 Ym + Y1 Zm = 0

(3.30)

Assuming solutions of the form Vo e−γz for V1 and V2 , the eigenvalue equation for (3.26) is obtained: γ 4 + (−a1 − a2 ) γ 2 + (a1 a2 − b1 b2 ) = 0

(3.31)

which is quadratic equation in γ 2 , with solutions 2 γa,b

a1 + a2 ± = 2

p (a1 + a2 )2 − 4(a1 a2 − b1 b2 ) 2

(3.32)

The solutions of (3.32) represent two independent modes, propagating in the positive and negative directions: γ1,2 = ±γa

and γ3,4 = ±γb

(3.33)

Substituting (3.27) through (3.30) into (3.32) yields the propagation constants in term of the MTL parameters: q 0 0 Ll Cl q γb = jω L0u Cu0

γa = jω

(3.34) (3.35)

The two solutions (3.34) and (3.35) represent purely propagating dispersion-free modes, with phase velocities, vl =

q 1 0 0 Ll Cl

and vu = √

1 L0u Cu0

completely determined by the electrical properties

of the upper and lower regions alone, and which are independent of the heights of the lower and upper regions. Dispersion curves for the two modes obtained using MTL analysis are compared with FEM simulated results in Figure 3.5, which confirm that the two independent modes are indeed virtually dispersion free, and match well the FEM simulated results. The modal eigenvectors, corresponding to the propagation constants are given by:     V1 1     V   R   2  a    =  V  I1   Ya1  o     I2 Ra Ya2

and

    V1 1     V   R   2  b    =  V  I1   Yb1  o     I2 Rb Yb2

(3.36)


49

where Ra and Rb represent the ratio of the voltages on the two conductors. From (3.24) and (3.25) the ratios are determined to be: Ra,b =

2 −a γa,b 1 V2 = V1 b1

(3.37)

The characteristic admittances for conductors 1 and 2, Ya1 and Ya2 , are given by: Z2 − Zm Ra 2 Z1 Z2 − Zm γa Z1 Rc − Zm = 2 Ra Z1 Z2 − Zm

Ya1 = γa

(3.38)

Ya2

(3.39)

with similar formulas for Yb1 and Yb2 . Substituting the relevant terms for the geometry under consideration the modal eigenvectors are given by:     V1 1     q V  1 0 0  2   γa = jω Ll Cl ⇒   =   Vo  I1  0     1 I2 Zl r where Zu =

L0u Cu0

r and Zl =

0

Ll 0 Cl

    V1 1     q V   0   2   and γb = jω L0u Cu0 ⇒   =   Vo  I1   1     Zu  − Z1u I2

(3.40)

are the characteristic impedances of parallel-plate waveguides

consisting of the upper and lower regions alone. The eigenvectors for the negative traveling waves, −γa,b are identical, except that the current components are negative of those in (3.40). It is recalled that the components of the matrices C0 and L0 were derived analytically under the assumption of parallel-plate type fields with minor fringing, and that these were verified numerically. Under these assumptions the upper region the electric field, Eu is proportional to (V1 − V2 ), while in the lower region, the electric field, El is proportional to (V2 − 0). For the magnetic fields, the upper region magnetic field, Hu is proportional to I1 , while the lower region magnetic field, Hl is proportional to I1 + I2 . q 0 0 For the eigenvector corresponding to γa = jω Ll Cl it is seen that V1 − V2 = Vo − Vo = 0, while V2 − 0 = Vo − 0, and hence E is confined to the lower region ( El 6= 0 and Eu = 0). The magnetic field in the upper region is proportional to I1 = 0, and hence zero, while in the lower region the magnetic field is non-zero as I1 + I2 =

(Hl 6= 0 and Hu = 0). In a similar manner p it can be shown that for the mode corresponding to γb = jω L0u Cu0 , the upper region fields Vo Zl

are non-zero, Eu 6= 0, Hu 6= 0, while the lower region fields are zero, El = 0, Hl = 0. Modal field plots obtained from FEM simulations depicting E confirm these conclusions as shown in Figure 3.6. Each of the two modes are confined to one of the lower or upper regions, with minimal fringing into the other region. This is consistent with the fact that the propagation

50


8

FEM: H−walls MTL lower: γa MTL upper: γb

f (GHz)

6

4

2

0

π

(βd)x

Figure 3.5: Dispersion curves of the unloaded geometry.

z y

(a) Mode 1: γa = jβa ; E field

(b) Mode 2: γb = jβb ; E field

Figure 3.6: E field profiles for the two modes of the unloaded geometry.

51

Chapter 3. Multiconductor analysis: Building Blocks Radiation to free space

P 2−

P 1+,− TL

TL

Figure 3.7: Two-port scattering setup used to determine the series capacitance, C. constants of each of the modes depends only on the electrical properties of one region and are independent of the other.

3.3

Determination of loading elements

Without the presence of the top shielding conductor, a two-port full wave scattering simulation would be sufficient to calculate the equivalent series capacitance, C, due to the gap, and the equivalent shunt inductance, L, due to the via. The depiction of such a scenario for the determination of C is shown in Figure 3.7. The transmission lines, labeled T L on either side of the gap are simply the geometry of the lower region of the shielded structure, with transverse H-walls. At each port only a single quasi-TEM mode, which is essentially the parallel-plate mode of the lower region, is considered. By performing a full wave simulation and then deembedding the two-port scattering parameters to the location of the discontinuity, C may be calculated. However, in attempting this, two problems arise. First, for the situation depicted in Figure 3.7, the calculated equivalent capacitance, C will consist of both a real part (the capacitance), and an imaginary part, which is due to radiation leakage into free space. Figure 3.8 shows the results for a two port scattering simulation used to determine C. The real part of C is approximately 0.28 pF from 1 to 7 GHz and then increases and finally dips slightly at 10 GHz. The imaginary part of C is non-zero and represents a radiation leakage component. The shielded structure is a closed geometry, and therefore no energy can be lost to free space. Thus, the values of C obtained for the shielded structure should be purely real. Although the values of C obtained using the T L setup of Figure 3.7 consist of both real and imaginary parts, one might assume that by taking the real part of this number, a reasonable approximation to the actual value of C for the shielded structure could be obtained. The second problem encountered with this approach is as follows. In comparing the dispersion curves of the shielded structure obtained using MTL analysis (to be shown in the next chapter), which incorporate C values obtained as above, and the FEM generated dispersion curves, it was observed that the two sets

52

Chapter 3. Multiconductor analysis: Building Blocks Real(C) imag(C)

Capacitance (pF)

0.4 0.3 0.2 0.1 0 −0.1

2

4

6 f(GHz)

8

10

Figure 3.8: Real and imaginary parts of C obtained from the two-port scattering setup. of curves didn’t match well, for relatively small values of hu . When hu was relatively small, it was determined through a parametric study, that smaller values of C were required to match the MTL and FEM results. Apparently the top shielding cover had a significant effect on the value of the series gap capacitance, and thus needed to be included in the analytical and numerical set-up to determine C. This led to a reconsideration of the use of the above described two-port scattering analysis in favor of that described next. The modified scattering analysis, which includes the top shielding cover is depicted in Figure 3.9, where it is observed that there are still only two physical port planes. However, due to the fact that the physical geometry at the port planes is that of the underlying unloaded MTL geometry, both MTL modes (upper region and lower region) need to be taken into account, and hence a four-port analysis is required. Figure 3.9a depicts a situation were hu is relatively large, and in this case there is relatively little leakage of energy into the upper region of the guide, when the lower region port mode at x = −l is excited. In fact the values of C obtained in such cases are very close to the real part of C obtained using the two-port set-up. The situation with small hu is quite different, as shown in Figure 3.9b. For small values of hu , there is substantial energy leakage into the upper region of the guide, and hence if one ignores the energy delivered to the upper region ports, the calculation of C will be physically inconsistent. Thus it is not possible to ignore any of the four ports for small hu . By determining the four-port scattering parameters for the situation depicted in Figure 3.9 the value of the capacitance, C can be calculated. This is accomplished by analytically solving for the scattering parameters, which will be functions of C and the MTL geometry. Equating these analytically derived expressions with those derived from full wave simulations will yield C. The four-port scattering problem that will be considered is one in which the lower region mode is excited, as is depicted in Figure 3.9. The plane x = −l is designated as the physical port plane 1. At physical port plane 1, two modes are supported: the lower region mode, labeled as

53


Physical port plane 1: x = −l

Physical port plane 2: x=l

0− 0+

M 2−

M 4−

M 1+,−

M 3− M T L(l)

M T L(l)

(a) Large hu : Small energy leakage to upper region



0− 0+

M 2−

M 4−

M 1+,−

M 3− M T L(l)

M T L(l)

(b) Small hu : Large energy leakage to upper region

Figure 3.9: Four-port scattering setup used to determine the series capacitance, C, depicted for (a) large hu and (b) small hu . For a lower region excitation a larger quantity of energy leaks to the upper region when hu is small.

54


M 1 and the upper region mode, labeled as M 2. The plane x = l is designated as the physical port plane 2, which also supports two modes: the lower region mode labeled as M 3 and the upper region mode labeled as M 4.

For the situation considered in Figure 3.9 the lower region mode, M 1 at x = −l is excited, with the other three modes, M 2, M 3, and M 4 matched. The quantities describing each of the port modes M 1, M 2, M 3, and M 4 are given by 2 component voltage and current column vectors. These column vectors are simply the lower region (M 1 and M 3) and upper region (M 2 and M 4) eigenvectors, as given in (3.40). The total mode 1 voltage and current vectors, VM 1 and IM 1 are comprised of incident, +, and reflected, − components as given by: VM 1 = VM+1 + VM−1

IM 1 = IM+1 + IM−1



 0 where VM+1 = VM 1+ , IM+1 =  1  VM 1+ , and VM−1 = 1 Zl − + scalar variables VM 1 and VM 1 are incident and reflected " # 1

(3.41) 

 0 VM 1− , IM−1 =  1  VM 1− . The − 1 Zl amplitudes. The upper region mode " # 1

at x = −l, M 2 is matched, so that only a reflected component is present. Thus the total mode 2 voltage and current vectors VM 2 and IM 2 are given by: VM 2 = VM−2

where VM 2 =

" # 1 0

IM 2 = IM−2

(3.42)

  1 −   =  1Zu  VM 2− . The total voltage, V(x=−l) = VM−1 + VM+1 +

VM 2− , and IM−2

Zu VM−2 , and current, I(x=−l) = IM−1 + IM+1 + IM−2 , at physical port plane 1 is given in component form as:

  V  1 V   2    I1    I2

(x=−l)



VM 1+ + VM 1− + VM 2−

    =   V



  VM 1+ + VM 1−  −  VM 2  −  Zu  + − − VM 1 VM 2 M1 − + Zl Zl Zu

(3.43)

For the physical port plane 2 at x = l only the reflected lower and upper region modes are present, with amplitudes, VM 3− and VM 4− , respectively. The total voltage, V(x=l) = VM−3 +VM−4 ,


55

and current I(x=l) = IM−3 + IM−4 is given in component form as:   V  1 V   2    I1    I2



(x=l)

VM 3− + VM 4−



  VM 3−  −  VM 4   Zu  − − V M3 − M4 Zl Zu

    =   V

(3.44)

The transfer matrix, TC , which relates the voltage and current across the series capacitance, C is given by:

where ZC =

     1 V1 (0+ ) V1 (0− )      V (0+ ) 0 V (0− )     2  2 = TC  =    I1 (0+ )  0  I1 (0− )       0 I2 (0+ ) I2 (0− )

  V1 (0+ )   V (0+ ) 1 0 ZC  2     +   0 1 0   I1 (0 )   I2 (0+ ) 0 0 1 0 0

0

(3.45)

1 jωC .

After the numerical simulation is performed, the reference planes used to calculate the S parameters are transferred just to the right and the left of the discontinuity, with the original port 1 reference plane de-embedded from x = −l → x = 0− , while the port 2 reference plane is de-embedded from x = l → x = 0+ . With the new reference planes defined as such, the total voltage and current definitions at x = −l and x = l may be substituted into (3.45), and the resulting linear system of equations can be solved for the multiport scattering parameters. Due to the fact that the impedances, Zl and Zu of lower and upper modes are not equal, generalized scattering parameters are needed, which are defined by: p Zj Sij = + √ Vj Zi Vi−

(3.46) Vk+ =0

for k6=j

By solving the linear system obtained from (3.45) the generalized scattering parameters are found and given by:

VM 1− Zu + = Z + 2jωCZ Z + Z VM 1 u l l u r r VM 2− Zl Zu Zl = =− + Zl + 2jωCZl Zu + Zu Zu V M 1 Zu S11 =

S21

VM 3− (2ωCZu − j)Zl j + = Z + 2jωCZ Z + Z VM 1 u l l u r r VM 4− Zl Zu Zl = = + Zl + 2jωCZl Zu + Zu Zu V M 1 Zu S31 =

S41

(3.47) (3.48) (3.49) (3.50)

56


0.2

Real(C) imag(C)

0.15 0.1 0.05 0 −0.05

Real(C) imag(C)

0.25 Capacitance (pF)

Capacitance (pF)

0.25

0.2 0.15 0.1 0.05 0

2

4

6 f(GHz) (a) hu = 6 mm

8

10

−0.05

2

4

6 f(GHz) (b) hu = 0.5 mm;

8

10

Figure 3.10: The calculated series gap capacitance, C, for (a) hu = 6 mm, and (b) hu = 0.5 mm. By equating any one of the expressions in (3.47) through (3.50) with the full wave simulation results, corresponding values of C are determined. Two sets of results for hu = 6 mm and hu = 0.5 mm are shown in Figure 3.10. From (3.47) through (3.50) it is apparent that in general all four S parameters are non-zero. However for large hu , and in particular in limit hu → ∞, then Zu → ∞ and from (3.48) and (3.50) S21 , S41 → 0. Thus, a two-port scattering analysis is sufficient to obtain a reasonable approximation for the real part of C when hu is large, but otherwise the full four-port parameters are necessary. The value for C obtained for hu = 6 mm, shown in Figure 3.10a is nearly purely real, and is approximately equal to 0.24 pF at 1 GHz, and increases slightly to just below 0.25 pF at 10 GHz. This is quite close to the real part of C obtained in the two-port simulation, where C = 0.28 pF. For hu = 0.5 mm, C is again nearly purely real, but its value has decreased significantly to approximately 0.16 pF over the same frequency range, and its imaginary part is again close to zero. Although the four-port scattering analysis shows that the value of the series capacitance, C, decreases as the upper region height, hu decreases, the physical origin of this fact is not immediately clear. To this end a series of electrostatic simulations were performed with the FEM package COMSOL Multiphysics. The simulation set-up is depicted in Figure 3.11. It is observed that two adjacent patch layer conductors are assigned two different voltages, V1 and V2 , which for convenience are set as V1 = +V and V2 = −V . This is in contrast to the H-wall boundary condition between patches (V1 = V2 = +V ), which was used in the derivation of the MTL parameters C0 and L0 . In addition, the upper shielding conductor is set to a voltage, Vupper , which is arbitrary. The simulation solves the resulting electrostatic boundary value problem, from which the surface charge density on the patches is obtained. The surface charge densities for the patch

57

Chapter 3. Multiconductor analysis: Building Blocks Vshield (arbitrary) hu

ǫ1 V1 = +V

hl

ǫ2

V2 = −V g Ground

Figure 3.11: Two-dimensional electrostatic boundary value problem used to obtain the charge accumulation at the patch edges. The dashed lines denote H-walls.

(a) hu = 6 mm

(b) hu = 0.5 mm;

Figure 3.12: Surface charge density [C/m2 ] on the conductor at V1 = +V (Figure 3.11), near the plate edges for (a) hu = 6 mm and (b) hu = 0.5 mm. at potential V1 = 1 V, with the two upper region heights, hu = 6 mm and hu = 0.5 mm are shown in Figure 3.12. The resulting charge distributions accumulate near the patch edges, as expected, with the total charge (which is proportional to the area under the charge density curve), Qtotal = Qbase + Qnet , decomposed into a superposition of two contributions: (1) a constant baseline value, Qbase and (2) a component, Qnet , which is the difference of the total charge and the constant baseline charge. For the smaller upper region height, hu = 0.5 mm, the total charge is larger, Qtotal (hu = 0.5 mm) > Qtotal (hu = 6 mm), which is due to the fact that Qbase (hu = 0.5 mm) > Qbase (hu = 6 mm). However, Qbase is simply the contribution to the parallel-plate capacitances between the patch layer and the upper conductor and ground conductor, respectively, and hence does not contribute to the series capacitance, C. The series capacitance, C is due to the net excess charge and the simulations reveal that Qnet decreases as hu decreases, with Qnet (hu = 6 mm) >

58


(a) hu = 6 mm

(b) hu = 0.5 mm;

Figure 3.13: Streamline plots of the electric field for (a) hu = 6 mm and (b) hu = 0.5 mm.

Qnet (hu = 0.5 mm). The series capacitance, C is given by C =

Qnet V1 −V2

and hence C decreases

as hu decreases. The excess charge accumulation is due to the potential difference between the patches, but as the upper region height decreases a larger density of field lines which begin on one patch will terminate on the shielding conductor rather than the adjacent patch, as seen in Figure 3.13, resulting in reduced series capacitance as hu decreases.

The determination of the equivalent via inductance, L as depicted in Figure 3.14 proceeded in a similar manner to that for C, with a four-port scattering setup. The calculation of the four-port scattering parameters for the equivalent via inductance resulted in: S11 =

S21

VM 1− −jZl = Zl j − 2ωL VM 1+ VM 2− = VM 1+

r

Zl =0 Zu

VM 3− 2ωL + = − Z j − 2ωL VM 1 l r VM 4− Zl S41 = =0 VM 1+ Zu

S31 =

(3.51)

(3.52) (3.53) (3.54)

from which it is apparent that when a lower region port is excited, there is no energy leakage to the upper region, and hence L is not dependent on the upper region height, hu , unlike C. This also shows that in order to calculate L, a two-port simulation would be sufficient. The values for L obtained from simulations with hu = 6 mm and hu = 0.5 mm are virtually identical, as shown in Figure 3.15, confirming that L is not dependent on hu . It is also noted that L is determined to be very nearly a purely real number, as no energy leakage is possible again.

59




0− 0+

M 2−

M 4−

M 1+,−

M 3− M T L(l)

M T L(l)

Figure 3.14: Four-port scattering setup used to determine the shunt inductance, L.

0.8 real(L) imag(L)

0.6

Inductance (nH)

Inductance (nH)

0.8

0.4 0.2 0

real(L) imag(L)

0.6 0.4 0.2 0

2

4

6 f (GHz)

(a) hu = 6 mm

8

10

2

4

6 f (GHz)

8

10

(b) hu = 0.5 mm;

Figure 3.15: The calculated shunt via inductance, L, for (a) hu = 6 mm, and (b) hu = 0.5 mm.


3.4

60

Summary

In this chapter the fundamental analytical building blocks which describe the shielded structure were developed. By examining the transformation of an unloaded geometry (one without vias and gaps along the direction of propagation) into the geometry of the shielded structure, four parameters which describe the propagation were obtained. The first two parameters, the per-unit length capacitance and inductance matrices, C0 and L0 are distributed elements, which describe the quasi-TEM propagation along an unloaded multi-layer strip geometry. Analytical formulas for the components of these matrices were obtained and compared with FEM simulated results with excellent correspondence between the two shown. The propagation described by the C0 and L0 matrices consisted of two quasi-TEM modes, one concentrated in the upper region of the structure and the other concentrated in the lower region of the structure. The discontinuities due to the vias and the gaps along the direction of propagation were characterized by lumped elements, with the via corresponding to an equivalent shunt inductance, L and the gap corresponding to a series capacitance, C. By examining specific four-port scattering situations the values of the lumped components L and C were obtained by comparing FEM simulations with an analytical formulation of the scattering geometry. It was shown that for a fixed gap width the series capacitance C varied as the upper region height varied, with this variation due to the alteration of the net edge-charge accumulation as the upper region height was varied. For the shunt inductance no such effect occurred, with L dependent solely on the via diameter and length.

Chapter 4

Multiconductor transmission line analysis: Dispersion analysis 4.1

Introduction

In Chapter 3, building blocks were developed which will be used in this chapter to define a unit cell for the shielded Sievenpiper structure. The building blocks consist of distributed elements, the sections of unloaded multiconductor transmission lines, and lumped elements, a series capacitance, C, and a shunt inductance, L. These building blocks will be shown to be sufficient to derive many of the dispersion properties of the shielded structure. Initially, a periodic unit cell, which describes the shielded structure will be presented, with the resulting dispersion equations obtained using periodic Bloch analysis. However, before examining the results given by the periodic Bloch analysis, a simplified (approximate) analysis, in which the loading elements are incorporated smoothly within the unit cell, is undertaken. This analysis is accurate when the electrical lengths of the MTL sections comprising the unit cell are small, or when the loading elements are large enough so that they produce the dominant electrical effects in the structure. From this approximate analysis a tremendous amount of insight into the parameters which affect the dispersion behaviour can be easily obtained. In particular, it will be shown that within a band of frequencies defined by fc1 and fc2 the modes are characterized by complex conjugate propagation constants. Simple approximate formulas, defining the band transitions, f2 and f3 , will be established. It was observed that f2 varied as the upper region height varied, while f3 remained constant as long as the lower region height was fixed, and these observations will be validated with the approximate model. However, this simplified analysis does not capture the periodic nature of the structure, and hence is not accurate near the Brillouin zone boundary at βd = π. Having established several useful results with the approximate model, the fully periodic 61

Chapter 4. Multiconductor analysis: Dispersion analysis

62

MTL model will be revisited. Quantitatively, the fully periodic model is more accurate than the simplified model, and this is particularly true when the loading elements are relatively weak, which is typically the case in an actual Sievenpiper structure, where discrete components are generally not used. The most significant qualitative feature which is completely missed by the simplified model is the absence of a BW mode within the first pass-band, when the upper region height, hu is relatively small, and this will be explained with the full model. Additionally, the fully periodic MTL model is able to capture the resonant frequencies, f1 and f4 , which occur at βd = π, and are out of the range of applicability of the approximate model. Using the fully periodic MTL model analytical expressions for the resonance frequencies f1 , f2 , f3 , and f4 will be derived, with the expressions for f2 and f3 reducing to those provided by the simplified model in the appropriate limit. In addition to the analytical expressions for f1 , f2 , f3 , and f4 , equivalent circuits corresponding to these frequencies will be derived. These equivalent circuits correspond to those of the unit cell, with specific terminal boundary conditions. Using the analytical expression for f1 along with its equivalent resonant circuit, an explanation for the absence of the BW band for relatively small values of hu will be obtained. Additionally, the equivalent resonant circuit corresponding to f2 will be seen to provide an explanation of the modal degeneracy which occurs at this frequency. Excellent quantitative agreement between the MTL model and FEM simulations will be demonstrated over a broad range of physical parameters. The power of the MTL analysis will be seen to reside in its ability to yield relatively simple closed form expressions for the dispersion curves, and various critical points of these curves, leading to greater physical insight into their composition.

4.2

MTL analysis of the shielded structure (a): Periodic unit cell and dispersion equation

In Chapter 2, it was determined that the lowest order modes of the shielded structure, and in particular the lowest dual-mode band, corresponded to transverse H-walls for on-axis propagation, where (βd)y = 0. With transverse H-walls, it was determined in Chapter 3 that the underlying transmission medium comprising the shielded structure is characterized by multiconductor transmission line theory (MTL). The shielded structure is created by starting with said sections of MTLs and periodically cutting gaps in the middle layer conductor, and adding vias from the middle layer conductor to ground. The gaps and vias were characterized respectively by a series capacitance, C and shunt inductance, L. A unit cell representing on-axis propagation is depicted in Figure 4.1. It will later be shown that due to the symmetry of the unit cell, additional simplifications are possible, and to that end the central inductance, L has

63

Chapter 4. Multiconductor analysis: Dispersion analysis n MTL

n+

d

1 2

MTL

2

d 2

n+1

V1,n , I1,n V2,n , I2,n

V1,n+1 , I1,n+1

2C

2C

2L 2L

V2,n+1 , I2,n+1

Figure 4.1: MTL based equivalent circuit for on-axis propagation.

been split into a parallel combination of two inductances of value 2L. The transfer matrices, T2C and T2L , of the series capacitance, 2C, and the shunt inductance, 2L, respectively, are given by:

T2C

where Z2C =

1 jω2C

 1  0  = 0  0

and Y2L =

0 0

0



 1 0 Z2C    0 1 0   0 0 1 1 jω2L .

T2L

 1 0  0 1  = 0 0  0 Y2L

0 0



 0 0   1 0  0 1

(4.1)

The values of C and L are determined by appropriate

scattering simulations as explained in Chapter 3. The transfer matrix, TM T L [32] of a section of a multiconductor transmission line of length l is characterized completely in terms of the per-unit-length matrices, C0 and L0 , with TM T L given by: " TM T L = where

cosh(Γl)

sinh(Γl) Zw

Yw sinh(Γl)

Yw cosh(Γl) Zw

" # V(0) I(0)

" = TM T L

The matrix, Γ, given by Γ=

√

V(l) I(l)

# (4.2)

# .

(4.3)

Z0 Y 0

(4.4)

Γ2 = Z0 Y0

(4.5)

is defined so that the following holds:

64


The computation of Γ involves diagonalizing Z0 Y0 , and is not obtained by simply taking the square root of each individual component of Z0 Y0 . The result is:  q  p p 0 0 0 0 0 0 jω Lu Cu −jω Lu Cu + jω Ll Cl  q Γ= 0 0 0 jω Ll Cl

(4.6)

Writing Γ in diagonalized form gives:   p jω L0u Cu0 0 q  P−1 Γ = P 0 0 0 jω Ll Cl where

" P=

1 1

#

" and P

0 1

−1

=

1 −1 0

(4.7)

# (4.8)

1

The quantities, cosh(Γl) and sinh(Γl) are hyperbolic trigonometric functions of a matrix argument, Γl, and are calculated using the diagonalized form of Γ from (4.7), and not by taking cosh and sinh of each element of Γ individually. The matrices, Zw and Yw are given by: Zw = Γ−1 Z0 = and Y w = Z0

−1

Γ Y0

Γ=

−1

(4.9)

Y0 Γ−1

(4.10)

The component form of TM T L for the shielded Sievenpiper structure is given by:  TM T L

cos(θu )

− cos(θu ) + cos(θl )

  0 cos(θl )  =  jYu sin(θu ) −jYu sin(θu )  −jYu sin(θu ) j Yu sin(θu ) + Yl sin(θl ) r

where Zu =

1 Yu

=

L0u Cu0

r and Zl =

1 Yl

=

0

Ll 0 Cl

j sin(θu )Zu + sin(θl )Zl j sin(θl )Zl cos(θu ) − cos(θu ) + cos(θl )

j sin(θl )Zl



 j sin(θl )Zl     0  cos(θl ) (4.11)

are the characteristic impedances of parallel-

plate waveguides consisting of the upper and lower regions alone, as determined in the previous q p 0 0 0 0 chapter. The arguments of cos and sin are θu = ω Lu Cu l, and θl = ω Ll Cl l, which are the electrical lengths corresponding to the propagation constants of the two independent modes of the unloaded MTL geometry.

Bloch’s theorem [23] relates the voltage and current vectors at node n, Vn and In , with

65

Chapter 4. Multiconductor analysis: Dispersion analysis those at node n + 1, Vn+1 and In+1 , through a propagation constant γd: # " Vn+1 In+1

" = e−γd

Vn

# (4.12)

In

The transfer matrix of the unit-cell is given by: " Tunit−cell−M T L = T2C TM T L T2L T2L TM T L T2C = where

"

Vn

# = Tunit−cell−M T L

In

Af

Bf

Cf

Df

#

# " Vn+1

(4.13)

(4.14)

In+1

For TM T L , the length of the sections of MTL are half of the unit cell length, l = d2 . Combining (4.12) and (4.14) yields the following: Tunit−cell−M T L

" # Vn+1 In+1

" = eγd

Vn+1

# (4.15)

In+1

Thus the eigenvalues of Tunit−cell−M T L yield the Bloch propagation constant(s), γd, with the eigenvectors from (4.15) yielding information on the relative modal field concentrations. Using the block form of Tunit−cell−M T L from (4.13), (4.15) yields: " Af − Ieγd

Bf

Cf

Df − Ieγd

#" # Vn+1 In+1

" =

0

# (4.16)

0

Non-trivial solutions require that the determinant of the above system equal zero: " det

#

Af − Ieγd

Bf

Cf

Df − Ieγd

=0

(4.17)

The determinant (4.17) can be simplified by using the commutation properties satisfied by the individual k × k component blocks (with k = 2 for the shielded structure). In [39] it is shown that Df Ctf = Cf Dtf and from [40] this commutation property allows the block determinant to be simplified as follows: Df Ctf − Cf Dtf = 0 ⇒ det

" Af

Bf

Cf

Df

# = det Af Dtf − Bf Ctf

(4.18)

For lossless networks, Af Dtf − Bf Ctf = I, which establishes that det TM T L−unit−cell = 1. For

66


symmetric networks, as is the case for the structure under consideration, Ctf − Cf = 0 , and combining this with (4.18), the determinant from (4.17) simplifies to: det

" Af − Ieγd

Bf

Cf

Df − Ieγd

#

= det Af − cosh(γd)I = 0

(4.19)

For the shielded Sievenpiper structure this results in a quadratic equation in the variable cosh(γd): 4 cosh2 (γd) + 2 cosh(γd)f (ω) + g(ω) = 0

(4.20)

and hence describes two independent modes. The functions f (ω) and g(ω) are given by: f (ω) = " −2 + 4 sin

2

θu 2



#



+ −2 + 4 sin

θl 2

Zl −

cos

  θl 2

2ωC



sin

" −2 + 4 sin

2

+

4 sin

θu 2

θu 2

cos

# 



· −2 + 4 sin

θu 2

1 − 2 sin2

ω C Zu

θl 2

Zl −

θl 2

+

cos

  θl 2

2ωC

4 Zl sin

θu 2



cos

θl 2

2 sin

θl 2

Zl

θu 2

sin

ω 2 C L Zu

 θl 2

cos ω C Zu

−

cos

θl 2



2ωL

θu 2

sin

cos

−

Zl −

g(ω) =

θu 2

(4.21)

 θl 2



2ωL

cos

θl 2

(4.22)

The dispersion equation, (4.20) describes propagation for the periodic MTL unit cell of Figure 4.1. Before examining its full implications, an approximation to (4.20) will be examined in the next section. The simplified, approximate analysis of Section 4.3 will yield much insight into the dispersion of the structure. However it will not be sufficient to explain all of its properties. To that end, in Section 4.4 dispersion curves obtained from (4.20) without any additional approximations will be revisited, which will supplement and enhance the results obtained from the approximate model.

67


4.3 4.3.1

MTL analysis of the shielded structure : Simplified analysis Introduction

An approximate analysis of the dispersion equation (4.20) will be examined in this section. This simplified analysis will be obtained under the assumption that the electrical lengths of the MTL sections, θu and θl are small. Alternatively, this analysis is accurate, within a certain frequency range, if the loading elements, L and C are large, which shifts the dispersion curves down in frequency. Under such an approximation it will be shown that the effect of the loading elements, L and C can be included directly into the two principal quantities which define the underlying 0

multiconductor geometry, the capacitance and inductance per-unit-length matrices, C and 0

0

0

L , or alternatively the admittance and impedance per-unit-length matrices, Y = jωC and 0

0

0

Z = jωL . By incorporating the effects of L and C, two augmented matrices, YLoaded and 0

ZLoaded will be obtained. Subsequently, with these new matrices, the analysis will proceed as it did for the uniform (unloaded) MTL geometry, given in Section 3.2. It will be demonstrated analytically that there exists a band of frequencies within the first stop-band for which the propagation constants of the structure are given by pairs of complex conjugate numbers. Analytical formulas for the limits of the corresponding complex band, given by fc1 and fc2 will be derived. Such modes are referred to as complex modes, and their properties will be reviewed. It will be shown that the first stop-band is not necessarily comprised solely of complex modes. For relatively large values of hu the first stop-band contains only complex modes. However, as hu is decreased, the stop-band contains two regions: initially a complex mode band, followed by a second band composed of two independent evanescent modes. Additionally, analytical formulas for the resonances f2 and f3 which occur at βd = 0 will be derived. Previously, it was observed that as hu varied from large to small values (with hl fixed), f3 remained constant, while f2 increased, and this will be verified with the analytical formulas. It will also be demonstrated that the frequency peak of the complex band, fc2 is bounded from above by f2 and f3 . The analysis will provide a solid basis for understanding the dispersion of the shielded structure, which will be supplemented and enhanced in the remaining sections with the exact periodic analysis.

4.3.2

Dispersion: Simplified

An approximate form of the dispersion equation (4.20) will now be derived. Assuming that the interconnecting MTL sections are electrically short, the approximations sin θ2u → θ2u , sin θ2l → θ2l , cos θ2u → 1 , cos θ2l → 1 are substituted into f (ω) (4.21) and g(ω) (4.22),

68

Chapter 4. Multiconductor analysis: Dispersion analysis which simplify to: 0

fa (ω) = −4 + (ωd) Lu Cu 2

ga (ω) = −2(ωd)

2

0

0 + (ωd) Ll −

1 2 ω dC

2

1 Ll − 2 ω dC 0

0 Cl −

0

0 1 dCu − ω 2 dL C

(4.23)

1 1 1 0 0 4 0 0 +(ωd) Lu Cu Ll − 2 Cl − 2 ω 2 dL ω dC ω dL 0

0

0 Cl −

+ 4 − 2(ωd)2 Lu Cu + 2

0

Cu d Cu d Ll 0 Cu 0 2 − (ωd)2 Ll 0 Cl 0 + d (4.24) C C LC

If in addition it is assumed that the Bloch phase shift, γd is electrically small, then 2 cosh(γd) = γd → (γd)2 + 2, where the approximation sinh( γd 2 2 sinh2 ( γd ) + 1 2 2 ) → 2 has been applied. With these approximations, the exact dispersion equation (4.20) simplifies to: γ + 4

4 + fa (ω) d2

γ + 2

4 + 2fa (ω) + ga (ω) d4

=0

(4.25)

which is a quadratic equation in γ 2 . The dispersion equation for the uniform (unloaded) MTL structure, (3.31) has the same basic functional form as (4.25). It thus appears that the approximations made to arrive at (4.25) allow one to construct a model of the shielded structure in which the loading parameters are incorporated into the underlying MTL geometry in a continuous (smooth) manner.

In fact, by assuming that the shunt inductor, L between conductor 2 and ground is adding an additional admittance, YL =

1 jωLd

0

0

to the (2, 2) component of Y = jωC , a new loaded

0

version of the admittance matrix, YLoaded is obtained, resulting in: 0

YLoaded

 0 1  jωLd

 0 0 =Y + 0

(4.26)

0

In a similar manner the loaded impedance matrix, ZLoaded is given by: 0

ZLoaded

 0 =Z + 0 0

 0 1 . jωCd

(4.27)


69

0

0

with the explicit forms of YLoaded and ZLoaded given by:  0

YLoaded

ZLoaded



0

−Cu

 = jω  0 0 0 −Cu Cu + Cl −

and 0

0

Cu

 0 0 L + Ll  u = jω  0 Ll

1 ω 2 Ld

0

Ll −

(4.28)



0

Ll

" # Y1L YmL  = YmL Y2L

1

ω 2 Cd

# " L Z1L Zm  = L ZL Zm 2

(4.29)

Using the loaded matrices, (4.28) and (4.29) the analysis proceeds as it did for the uniform MTL geometry, with the dispersion equation given by: L 2 L L L L γ 4 + (−aL 1 − a2 ) γ + (a1 a2 − b1 b2 ) = 0

(4.30)

The two solutions of (4.30) are: 2 γa,b =

i √ 1h L L Disc a1 + aL ± 2 2

where L DiscL = aL 1 + a2

2

(4.31)

L L L − 4 aL 1 a2 − b1 b2

(4.32)

is the discriminant of the quadratic in γ 2 , (4.30). The discriminant, DiscL will be shown to be L L L important in the appearance of complex modes. The parameters, aL 1 , b1 , a2 , and b2 are given

by 2 aL 1 = −ω Cu Lu L 2 b1 = −ω −Lu Cu + Cl −

aL 2

= −ω

2

1 ω 2 Ld

Ll

(4.33) (4.34)

Cu 1 1 − 2 + Cl − 2 Ll − 2 ω Cd ω Ld ω Cd Cu 2 bL 2 = −ω ω 2 Cd

(4.35) (4.36)

Using (4.33) through (4.36) it can be shown that L − aL 1 − a2 =

and

L aL 1 a2

−

L bL 1 b2

=

4 + fa (ω) d2

4 + 2fa (ω) + ga (ω) d4

(4.37) .

(4.38)


70

This proves that (4.25) is identical to (4.30), and hence in the limit of short electrical lengths for the MTL sections, the exact (fully periodic) dispersion equation, (4.20), reduces to an equation for which the loading is incorporated in a continuous manner. Typical dispersion curves obtained using (4.30) are shown in Figure 4.2. These figures show a sequence of dispersion curves with varying upper region height: hu which runs through (a) 10 mm, (b) 3.0 mm, and (c) 0.75 mm. All of the other electrical and geometric parameters are constant. The lower region height, hl = 3 mm; the upper and lower region relative permittivities r1 = r2 = 4; The loading inductance, L = 1.0 nH; the loading capacitance, C = 0.5 pF. The first band, in all three cases contains one FW mode and one BW mode. As hu is decreased the FW mode becomes slower, as was previously observed in the FEM simulations, while the bandwidth of the BW mode becomes smaller. The BW mode does not posses a cut-off frequency in this simplified analysis, as βd → ∞ as ω → 0. However in deriving the simplified dispersion it was assumed that γd 1 and hence βd 1. Thus the simplified dispersion (4.30) is not accurate for large phase shifts. The critical points of the dispersion curve which occur at γ = 0 (equivalently βd = 0), are labeled f2 and f3 . It is observed that f3 does not vary as the hu is decreased. However, f2 , which for hu = 10 mm occurs below f3 , increases as the upper region height hu is decreased. For all three curves, it will be shown that in the frequency range between fc1 and fc2 , which is lightly shaded, the solutions of (4.30) are given by pairs of complex conjugate propagation constants, γa = α + jβ and γb = α − jβ. For convenience the figures only show the solution γa . As hu is decreased from 10 to 3 mm, the complex mode bandwidth increases. However in decreasing hu to 0.75 mm, the stop-band develops a more complicated structure, with only its initial part, from fc1 to fc2 composed of complex modes. In the frequency range from fc2 to f3 the structure supports two independent standard evanescent modes, with γa = α1 + 0j, γb = α2 + 0j and α1 6= α2 . However the overall stop-band bandwidth, given by the union of the complex mode bandwidth and the evanescent mode bandwidth increases monotonically as hu is decreased, as was observed in the FEM simulations. It is also observed that the upper frequency limit of the complex mode band, fc2 appears to be bounded above by both f2 and f3 (fc2 ≤ min{f2 , f3 }), which will be verified later in this section. Returning now to the analysis, it is observed that the slope of the dispersion curve is zero at the critical frequencies f2 and f3 , and additionally at the complex mode band edges, fc1 and fc2 . Differentiating (4.30) with respect to ω results in: d L d L L L γ2 a1 + aL a1 a2 − bL 2 − 1 b2 dγ dω dω = dω L) 2 2γ 2γ − (aL + a 1 2

(4.39)


71

(a) hu = 10 mm

(b) hu = 3 mm

(c) hu = 0.75 mm

Figure 4.2: Dispersion curves obtained using the simplified dispersion equation (4.30), with varying upper region height. (a) hu = 10 mm; (b) hu = 3 mm; (c) hu = 0.75 mm. All other parameters are fixed: the lower region height, hl = 3 mm; the upper and lower region relative permittivities are r1 = r2 = 4; the loading inductance, L = 1.0 nH; the loading capacitance, C = 0.5 pF.

72

Chapter 4. Multiconductor analysis: Dispersion analysis from which it is observed that

dω dγ

= 0 when the denominator of (4.39) is set equal to zero. The

critical points are thus given by:

and

γ = 0 ⇒ {f2 , f3 }

(4.40)

1 γ 2 = (aL + aL 2 ) ⇒ {fc1 , fc2 } 2 1

(4.41)

Substituting γ = 0 (4.40) into the dispersion equation (4.30) results in the following expression: L aL 1 a2

−

L bL 1 b2

= ω Cl Ll Cu Lu 1 − 4

0

0

0

0

1 0 2 ω L(Cl d)

1−

1 ω2C

1 1 + 0 0 Ll d Ll d

.

(4.42)

The zeroes of (4.42) correspond to the critical frequencies f2 and f3 : ω22

1 = C

and ω32 =

1 1 + 0 0 Ll d Lu d

(4.43)

1 . 0 L(Cl d)

(4.44)

The frequency f2 (ω2 ) corresponds to a resonance between the loading capacitance, C and a 0

0

f f f parallel combination of two inductances, Lef = Ll d and Lef = Lu d. The inductances Lef 1 2 1 f are due to the distributed MTL per-unit-length inductive parameters of the upper and Lef 2 0

0

and lower regions, Lu , and Ll . From this expression it is seen that f2 depends on the MTL parameters of both the upper and lower regions, and hence f2 varies as hu is varied. For a fixed 0

lower region height, hl , as hu is decreased, Lu =

µo hu d

decreases, corresponding to an increase in

f2 as is observed in Figure 4.2. On the other hand, the frequency f3 is a resonance between the 0

loading inductance, L and C ef f = Cl d, which is due to the lower region distributed capacitance. 0

Thus f3 is invariant as the upper region height is altered, as Cl =

2 w hl

is a function of the lower

region height, hl , solely.

L The frequencies fc1 and fc2 are obtained when (4.41), γ 2 = 12 (aL 1 + a2 ) is satisfied. Substi-

tuting (4.41) into the dispersion equation (4.30) results in: −

2 1 L L L L a1 + aL + (aL 1 a2 − b1 b2 ) = 0 2 4

(4.45)

which is zero when DiscL = 0 (4.32). The discriminant DiscL is given by: DiscL =

1 A1 ω 8 + B1 ω 6 + C1 ω 4 + D1 ω 2 + E1 4 ω

(4.46)

73

Chapter 4. Multiconductor analysis: Dispersion analysis with coefficients:

0 0 0 0 2 A1 = Cu Lu − Cl Ll

(4.47)

# " 0 0 2L0 2 C − C u l l B1 = Cu Lu − Cl Ll − Ld Cd

0

0

0

0

0

0

0

0

0

0

0

0

(4.48) 0

0

0

2Cu Cl (Cl )2 2Cu Ll 4Cl Ll (Ll )2 (Cu )2 2Cu Lu − + + C1 = 2 + 2 + 2 − 2 Cd Ld Cd Ld Cd Ld Cd Cd Cd Ld ! 0 0 0 Cl Ll 2 Cu D1 = − + + Cd Cd Ld Cd Ld E1 =

Ld

1 2

(4.49)

(4.50) (4.51)

2 Cd

In general, the zeroes of (4.46) will reduce to the solution of a quartic equation in ω 2 . Although closed form solutions for quartic equations exist, the formulas are quite lengthy, especially for a situation like the case considered here where the coefficients are not numbers, but variables 0

0

0

0

themselves. However, when the condition 1 = 2 is satisfied it can be shown that Cu Lu = Cl Ll , so that from (4.47) and (4.48) A1 = B1 = 0 and thus the zeroes of (4.46) are reduced to the solutions of a quadratic in ω 2 . The two solutions are given by: 2 ωc1 =

1 q p 0 2 LCl d+ LCu0 d + CLl d

(4.52)

2 ωc2 =

1 q p 0 2 LCl d+ LCu0 d − CLl d

(4.53)

and

0

0

From (4.53) it is observed that: 2 ωc2 ≤ ω32 =

1 0 LCl d

(4.54)

thus proving that the peak of the complex mode band at fc2 is bounded above by f3 . After some algebra (4.53) can be rewritten into another form: 2 ωc2 =

q 0

0

CLu Ll d 0 + Lu0 + Ll

1 q q 2 0 0 0 LCl Lu d + LCu Ll d − C(Ll )2 d 0

(4.55)

0

Lu0 + Ll

which shows that 2 ωc2

0

≤

ω22

1 = C

1 1 + 0 0 Ll d Lu d

Thus fc2 is bounded from above by both f2 and f3 (fc2 ≤ min{f2 , f3 }).

(4.56)

74

DiscL


0

↑ fc1

↑ ↑ ↑ fc2 f2 f3

Figure 4.3: Plot of the function DiscL , which is negative between fc1 and fc2 and otherwise positive. The behaviour of DiscL will now be examined. From (4.52) and (4.53) it is clear that 2 > ω 2 represents the larger of two real positive frequencies representing the solutions of ωc2 c1

DiscL = 0. The asymptotic behaviour of DiscL as ω → 0 is determined by positive, Disc → ∞ as ω → 0. The asymptotic behaviour of

DiscL

E1 , ω4

and as E1 is

as ω → ∞ is determined by

C1 which can also be shown to be positive, and hence Disc → C1 as ω → ∞. Thus DiscL ≤ 0 for ωc1 ≤ ω ≤ ωc2 , and Disc ≥ 0 otherwise, as is shown in Figure 4.3. For the finite frequency range when DiscL < 0, the roots of the dispersion equation are given by: 2 γa,b = p(ω) ± jq(ω)

where p(ω) =

1 2

(4.57)

p 1 L aL |DiscL | are purely real numbers. The solutions of 1 + a2 and q(ω) = 2

(4.57) are given by: γa = ± α(ω) + jβ(ω) . γb = ± α(ω) − jβ(ω)

(4.58)

One pair of complex modes corresponds to exponential decay, γa,b (1) = α(ω) ± jβ(ω) while the other pair corresponds to exponential increase, γa,b (2) = −α(ω) ± jβ(ω). Both the real, α(ω) and imaginary parts, β(ω) of γa,b have frequency variation within the complex mode band. Each complex mode has a real power flow, which is oppositely directed in the upper and lower regions, in such a manner that the net integrated power over a transverse cross-section is zero [38]. In addition, there is no net reactive energy associated with a single complex mode, unlike the case for standard evanescent modes. In Figure 4.4 the stop-band is depicted for the two cases: (a) relatively large hu and (b)

75


f3 → f2 →

jβa

βd αd

jβb

← fc2

γb = −jβ + α

γa = jβ + α

ւ

fc1

γb = jβb

γa = jβa

(a) Relatively large hu : stop-band consists solely of complex modes.

f3 → γa = αa

γb = αb

← fc2

γb = −jβ + α

βd αd

γa = jβ + α

ւ fc1 γa = jβa

γb = jβb

(b) Relatively small hu : stop-band consists of both complex modes (light shading) and evanescent modes (dark shading).

Figure 4.4: Power flow profiles for complex modes with complex-conjugate propagation constants, γa = jβ + α and γb = −jβ + α.


76

relatively small hu . It is observed that for large hu the stop-band is characterized completely by complex modes. The real part of the power flow for the two independent modes, characterized by complex conjugate propagation constants, is depicted in the figure. It is seen that for the two modes characterized by exponential decay, the real power flow in the upper and lower regions are reversed. This would seem to suggest that the possibility of exciting a single complex mode would depend on the region in which the structure is excited, either upper or lower. In fact it will be shown in Chapter 6 that it possible to strongly excite a single complex mode with a properly confined excitation. However, if the structure is excited with a source which is not confined to a single region it will be shown that both exponentially decaying complex modes are strongly excited. For the small hu case, Figure 4.4b shows that only the initial part of the stopband is composed of complex modes. Above the complex mode band, there exists a stop-band in which two independent evanescent modes, with γa = αa and γb = αb are supported. It will be shown in Chapter 6 that the signature of the complex stop-band has a different character than that of a stop-band formed from standard evanescent modes, when a finite cascade of unit cells of the structure is excited.

4.4

MTL analysis of the shielded structure (b): Comparison of full periodic dispersion with FEM simulations

Having derived the dispersion equation for the periodic unit cell in Section 4.2, and then examining it using a simplifying approximation in Section 4.3, the dispersion curves generated using the fully periodic MTL dispersion equation (4.20) will now be compared with FEM simulated results. The simulation software could not readily calculate the propagation constants within the stop-band. However, as the MTL model is able to obtain the propagation constants within stop-bands, they are included for convenience and will be validated in Chapter 6, where the scattering properties of finite cascades of unit cells are considered. The FEM generated dispersion curves correspond to a series of simulations in which the upper region height, hu , is given six values: hu = 18, 12, 6, 1, 0.5 and 0.2 mm, and are shown in Figure 4.5 (a) through (f). All of the other electrical and geometric parameters of the structure are fixed. The upper and lower region permittivities are r1 = 1 and r2 = 2.3, respectively, while the periodicity and patch width are d = 10 mm and w = 9.6 mm, respectively. The via radius is r = 0.5 mm. It is recalled that the series capacitance, C, used in the MTL model is dependent on the upper region height, and thus for each upper region height a different value for C was used, with C varying from 0.28 pF for hu = 18 mm, to 0.16 pF for hu = 0.2 mm, as calculated using the four port scattering set-up described in Chapter 3. The inductance L, is independent of the upper region height and its value is calculated to be 0.75 nH, again


77

using the scattering analysis described in Chapter 3. The FEM generated dispersion curves are obtained by varying the phase along x, (βd)x , while the phase shift transverse to the direction of propagation is fixed, (βd)y = 0. The simulations were performed on a PC using an Intel(R) Xeon(R) CPU 5160, with a clock rate of 3.00 GHz, and with 16.0 GB of RAM. The simulation was terminated when the difference in modal frequency between successive passes was below 0.2 %. In order to obtain the dispersion diagram two sets of simulations were performed; first the lower band modes, and then the remaining upper band modes. The reason for this was that the eigenmode frequency extraction requires one to set a minimum frequency for a given simulation, and it was found that since the upper band modes occur much higher in frequency, it was advantageous to split up the simulation into two sets, so as to allow for faster convergence of the upper band modes. The simulation time for a lower band mode was approximately 2 minutes for a single (βx d) value, while the upper band modes required appoximately 5 minutes for a single (βx d) value. The dispersion curves were obtained by sweeping over the (βx d) values in 5◦ increments resulting in 37 points from 0 to 180◦ (π radians), for a total simulation time of approximately 4.5 hours per geometry. Even though the FEM curves are obtained with a fixed transverse phase shift (βd)y = 0, it was shown in Chapter 2 that the modes thus obtained could be divided into two classes with different field polarizations at the transverse boundaries. The FEM simulated dispersion curves defining these two classes have been marked with squares, (FEM), and circles, ◦ (FEM-TE). For large values of hu , corresponding to hu = 18, 12, and 6 mm, the general form of the dispersion curves are similar to those obtained using the simplified analysis, with the exception of a cut-off frequency, f1 , at βd = π for the lowest backward wave mode, due to the structure’s periodicity. This is in contrast to the simplified analysis were there was no limit on βd, which asymptotically approached infinity as frequency approached zero. Additionally, another resonance for βd = π occurs at f4 , which is again due to the periodicity. It is observed that the MTL generated curves match extremely well within the pass-band of the structure with the FEM curves, with the exception of the FEM (TE) mode. The FEM(TE) mode is not captured by the MTL model, as it is not a quasi-TEM mode as was seen in Chapter 2. There is a modal degeneracy at f2 , with one mode emerging from f2 captured by the MTL model, and the other mode being the TE mode. This degeneracy will be explained in Section 4.8. As in the simplified approximation, for large values of hu , the stop-band is composed entirely of complex modes, with the complex mode band situated between the frequencies, fc1 and fc2 . As hu is decreased from 18 to 6 mm, the complex mode bandwidth increases, as does the value of α within the complex mode band, indicating a stronger interaction between the upper and lower regions as hu is decreased. The resonance frequencies f3 and f4 are invariant as well.


78

(a) hu = 18 mm; C = 0.28 pF

(b) hu = 12 mm; C = 0.26 pF

Figure 4.5: Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3.


79

(c) hu = 6 mm; C = 0.24 pF

(d) hu = 1 mm; C = 0.20 pF

Figure 4.5: Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. (cont’d)


80

(e) hu = 0.5 mm; C = 0.18 pF

(f) hu = 0.2 mm; C = 0.16 pF

Figure 4.5: Sequence of MTL derived dispersion curves with varying hu , along with FEM generated dispersion curves. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3. (cont’d)


81

When hu is decreased to a sufficiently small value (hu = 1 mm for the geometry considered) the BW band is completely eliminated, with the first pass-band consisting of a single FW mode. For hu = 1 mm, the resonance frequency f1 is equal to the commencement frequency of the complex mode band, f1 = fc1 . Thus the frequency f1 defines the peak of a single mode first pass-band, and not the commencement of a dual-mode band consisting of a FW and BW mode, as it does for hu = 18, 12, and 6 mm. Additionally, the stop-band now consists of two regions with different modal behaviour. Initially there is a complex mode band, for fc1 ≤ f ≤ fc2 , which is followed by a band determined by two evanescent modes, for fc2 ≤ f ≤ f3 . As hu is decreased further to hu = 0.5 and 0.2 mm, the frequency f1 , which defines the peak of the first pass-band decreases, but the commencement of the next pass-band remains fixed at f = f3 . For hu = 0.5 and 0.2 mm a final qualitative change in behaviour occurs. The initial part of the stop-band now consists of two evanescent modes. At the commencement of the stopband one of the evanescent modes has αd = 0 (corresponding to the FW of the first pass-band which is at it’s upper cut-off point) and the other evanescent mode has αd > 0. As frequency increases the two curves of the individual evanescent modes approach one another and coalesce to a maximum value, upon which the complex mode band begins. A similar situation occurs at the upper edge of the complex mode band, where two evanescent modes coalesce to a minimum. Thus the complex mode is sandwiched between two bands containing two evanescent modes, with the stop-band formed as a combination of three regions. It is also noted that as hu is decreased from 1 to 0.2 mm, the portion of the stop-band determined by complex modes, fc1 ≤ f ≤ fc2 , decreases relative to the portion determined by standard evanescent modes, fc2 ≤ f ≤ f3 .

4.5

Analytical formulas, equivalent circuits, and modal field structure defining the resonant frequencies at (βd)x = 0 and (βd)x = π

4.5.1

Introduction

From the dispersion analysis of the shielded structure it was revealed that the band structure was largely determined by the resonant frequencies occurring at (βd)x = 0 (f2 and f3 ), and (βd)x = π (f1 and f4 ). Approximate formulas for f2 and f3 were obtained from the simplified dispersion analysis, where it was seen that f2 was a function of the loading capacitance, C, and 0

0

the upper and lower distributed inductances, Lu and Ll , while f3 was a function of the loading 0

inductance, L and the lower region distributed capacitance, Cl . However, the simplified model could not account for the band edges occurring at (βd)x = π, which were out of its range


82

of applicability. One of the consequences of this was that the simplified model displayed the qualitatively incorrect behaviour that the backward-wave mode had no cut-off frequency. The fully periodic MTL dispersion curves, from Figure 4.5, showed that for small values of hu , the first band contained no BW mode, and in fact the frequency f1 became the peak of a single-mode FW pass-band. In this section exact analytical formulas for the frequencies f1 through f4 will be derived. The formulas for f1 and f4 have no counterpart within the simplified analysis, while the formulas for f2 and f3 will be shown under appropriate limits to correspond to those obtained using the simplified model. In addition to the analytical formulas, equivalent circuits and modal eigenvectors, corresponding to each of f1 through f4 will be derived and examined. This information will aid in developing further intuition into the nature of the dispersion for the shielded structure.

4.5.2

Analytical Formulas for f1 through f4

Analytical formulas for the frequencies f1 through f4 will be obtained by factoring the fully periodic dispersion relation (4.19), given by det(Af − cosh(γd) I) = 0. The shielded Sievenpiper structure represents a specific case of a general, loaded, symmetric k + 1 conductor MTL unit cell, which is depicted in Figure 4.6. It is observed that the unit cell is composed of a cascade of two matrices, Th1 and Th2 which relate the voltage, V and current, I vectors between nodes n and n + 12 , and nodes n +

1 2

and n + 1, respectively. The k × k block matrices, Ah , Bh , Ch ,

and Dh , which comprise Th1 and Th2 are related through matrix transpositions [39] as shown in Figure 4.6. The block matrix Af can then be written in terms of the block components of Th1 and Th2 , resulting in: Af = Ah Dth + Bh Cth

(4.59)

Combining (4.59) with the matrix identity Ah Dth − Bh Cth = I allows one to decompose Af into two different forms: Af = 2Ah Dth − I

(4.60a)

Af = 2Bh Cth + I

(4.60b)

Substituting (4.60a) and (4.60b) individually into the dispersion equation (4.19), allows it to be written in two different forms: det(2Ah Dth − I − cosh(Γd) I) = 0

(4.61a)

det(2Bh Cth + I − cosh(Γd) I) = 0

(4.61b)

83


V1,n+ 21 ; I1,n+ 12

V1,n ; I1,n

Vk,n ; Ik,n

A Th1 = h Ch

Bh Dh

V1,n+1 ; I1,n+1 t D Th2 = ht Ch

Vk,n+ 12 ; Ik,n+ 21

0

Bht Aht

Vk,n+1 ; Ik,n+1

0

0

Figure 4.6: Transfer matrix relationships for a symmetric unit cell. Voltages and currents on each of the 1 through k lines defined at nodes n, n + 12 , and n + 1. Voltages defined with respect to ground. Arrows denote current flow convention.

Analytical formulas for f1 and f4 are obtained by noting that at these frequencies, γd = jπ, which when plugged into (4.61a), gives: det(2Ah Dth ) = 0.

(4.62)

Using the identities det(PQ) = det(P) det(Q), and det(Pt ) = det(P), results in the factored form of (4.62): det(Ah ) = 0 or det(Dh ) = 0 ⇒ {f1 , f4 }

(4.63)

Thus f1 corresponds to either det(Ah ) = 0 or det(Dh ) = 0, with f4 given by the excluded case. Formulas for f2 and f3 are obtained by noting that at these frequencies γd = j 0, which when plugged into (4.61b), gives: det(Bh ) = 0 or det(Ch ) = 0 ⇒ {f2 , f2 }

(4.64)

In this case f2 corresponds to either det(Bh ) = 0 or det(Ch ) = 0, with f3 given by the excluded case. It is possible that f2 = f3 , in which case the two determinants are zero at the same frequency. The expressions for the resonant frequencies, (4.63) and (4.64), are valid for general symmetric loaded MTL structures. For the specific case of the shielded structure the block (matrix) components of Th1 are given by:   cos θu  2   Ah =     sin θu  2 − 2 ω C Zu

 − cos

cos

θl 2

θu 2

+ cos

θl 2

+

sin

θl 2

2ωL

Zl







        cos θ2l 

sin θ2l sin θ2l Zl sin θ2u + + + − 2 ω C Zu 2 ω C Zl 2ωL 4 ω2 C L

(4.65)

84




j Zu sin + Zl sin     Bh =  cos θu cos θ2l  2 j Zl sin θl +  − 2 2ωC 2ωC θu 2



θl 2

j Yu sin

θu 2

  Ch =   −j Yu sin

θu 2

−j Yu sin

 j Yu sin

 Dh =  − cos

cos θu 2

θu 2

θl 2

   cos θ2l  θl    j Zl sin 2 − 2ωC 

j Zl sin



θu 2

+ Yl sin

θu 2



θl 2

(4.66)

  θl  2  2ωL

cos

−

(4.67)



+ cos

cos

θl 2

0 

(4.68)

θl 2

with the corresponding determinants given by:

det(Ah ) = cos

|

+

θu 2

cos

cos θu 2

θl 2

sin

cos

+ θl 2

2ωL

θu 2

sin

cos

θl 2

θu 2

cos

θl 2

− 2ωCZl 4ω 2 LC Zl cos θ2l sin θ2u sin θ2u sin θ2l Z l + + (4.69) 2 2ωCZu 4ω LC Zu {z } f1

det(Bh ) = − sin |

θu 2

sin

θl 2

+

sin

θu 2

cos

2ωCZl {z

θl 2

+

sin

θl 2

cos

2ωCZu

θu 2

}

(4.70)

f2

det(Ch )

=

  θl θl sin cos sin 2 2   − Zu Zl 2ωL | {z } | {z } f6 θu 2

(4.71)

f3

det(Dh )

=

cos θ2u cos θ2l | {z } | {z } f5

(4.72)

f4

From (4.71) and (4.72) it is observed that the expressions for det(Ch ) and det(Dh ) can be factored into a product of two terms, which are functions of the upper region or lower

85


region parameters alone. The factors corresponding to f3 and f4 involve only terms containing the lower region MTL geometry, which verifies the previous observation that both f3 and f4 remained constant as long as the lower region height, hl was constant. For the sequence of dispersion curves shown in Figure 4.5 only hu was varied with hl remaining fixed, and indeed it is observed that both f3 and f4 are invariant. The remaining factors of these two determinants yield two additional resonances, f5 and f6 which occurred at higher frequencies than depicted in Figure 4.5. However, both f5 and f6 involve only the upper region geometry. Conversely, the expressions for det(Ah ) and det(Bh ) cannot be factored as a product of functions involving the upper and lower regions separately, and hence when either hl or hu is varied, both f1 and f2 will vary, as seen in sequence of curves shown in Figure 4.5. The resonance f2 involves the loading series capacitor, C and the MTL geometry of both regions, while the resonance f1 involves both loading elements L and C and the MTL parameters of both regions.

4.5.3

Equivalent Circuits for f1 through f4

Having obtained analytical formulas for f1 through f4 , equivalent circuits corresponding to these frequencies will now be established. Although the equivalent circuits presented will be valid for general symmetric loaded MTL structures, intuition on how to obtain these is most easily provided by looking at the analytical formulas for the resonant frequencies of the shielded structure. In particular, the expression for det(Dh ) = cos

θu 2

cos

θl 2

(4.72), corresponding to f4

and f5 , involves neither of the loading elements, L or C. Additionally, the individual factors of det(Dh ) are zero when θu = π and θl = π, corresponding to half wavelengths of the upper and lower region modes fitting in one unit cell, d =

λu 2

and d =

λl 2,

respectively. It thus

appears that f4 and f5 correspond to frequencies where standing-wave patterns develop in the upper and lower regions, respectively. Intuitively, one would expect that associated with such standing wave patterns, the terminal boundary conditions would correspond to either open of short circuits. However, such boundary conditions would also have to correspond to the possible excitation (or lack thereof) of the loading elements, 2C, and 2L which are located at the edge (n, n + 1) and central (n + 21 ) nodes, respectively. From the symmetric cell of the shielded structure, depicted in Figure 4.1, it is observed that if the outer nodes, n, n + 1 are left open circuited, with In = In+1 = 0, then the loading capacitance, 2C will not have current passing through it and hence will not be excited. From Bloch’s theorem if In = 0, then In+1 = In eγd = 0, and thus a homogeneous boundary condition at node n implies the same condition at node n + 1. In fact, the physical structure of the resonance is identical, up to sign, in both halves of the symmetric unit cell. In a similar manner, if the central node, n +

1 2

is

86


short circuited, Vn+ 1 = 0, then the loading inductance, L, will be shorted out and hence not 2

excited. From the above discussion it is expected that the boundary conditions, In = In+1 = 0, along with Vn+ 1 = 0 correspond to det(Dh ) = 0, and hence f4 (and f5 ). To show that this is 2

indeed the case, the transfer matrices relating quantities between nodes n and n + 12 , and nodes n+

1 2

and n + 1, shown in Figure 4.6 are required: " # Vn In

and

" =

Ah Ch

 # V 1 Bh  n+ 2  In+ 1 Dh

(4.73)

2

  " # #" t t Vn+ 1 V D B n+1 h h 2 =  . In+1 In+ 1 Cth Ath

(4.74)

2

Substituting In = 0, Vn+ 1 = 0 into (4.73) yields: 2

Dh In+ 1 = 0

(4.75a)

Vn = Bh In+ 1

(4.75b)

2

2

A non-trivial solution for (4.75a) (and hence (4.75b)) requires det(Dh ) = 0. This establishes the desired result for the network between nodes n and n + 12 . For the other symmetric half of the unit cell plugging In+1 = 0, Vn+ 1 = 0 into (4.74) yields: 2

Dth Vn+1 = 0

(4.76a)

In+ 1 = Cth Vn+1

(4.76b)

2

with a non-trivial solution for (4.76a) (and hence (4.76b)) requiring det(Dth ) = det(Dh ) = 0. Thus, the homogeneous boundary conditions, In = In+1 = 0 and Vn+ 1 = 0 imply that 2

det(Dh ) = 0, corresponding to f4 and f5 . The non-zero variables at resonance are obtained by solving the homogenous system, (4.75a), resulting in In+ 1 6= 0. Substituting the solution of 2

(4.75a) into (4.75b) gives Vn 6= 0. The variable Vn+1 may be obtained from the set of equations, (4.76a) and (4.76b), but due to Bloch’s theorem this is not necessary as Vn+1 = Vn eγd = Vn ejπ = −Vn . Thus, due to the symmetry of the unit cell the physical structure of the nonzero variables in either half of the unit cell are identical up to sign (symmetric/antisymmetric), and hence only the variables in one half of the unit cell need to be calculated. In summary, 3 of the 6 vector variables at the outer (n and n + 1) and central (n + 21 ) nodes are zero, while the other three are in general non-zero. The non-zero variables are in fact the modal eigenvectors,

87

Chapter 4. Multiconductor analysis: Dispersion analysis f4

Open

Short

Open

f3

Open

Open

Open

f2

Short

Short

Short

f1

Short

Open

Short

MTL 2C

d 2

MTL 2L

d 2

2C

2L

Figure 4.7: The four resonant circuits corresponding to f1 through f4 for the shielded structure. and will be examined in Sections 4.5.4 through 4.5.7, yielding further insight into the dispersion curves. In a similar manner, equivalent circuits corresponding to the four resonance frequencies, f1 through f4 , of the shielded structure, may be determined, and are shown in Figure 4.7. Table 4.1 compiles the boundary conditions, analytical formulas, and homogenous systems (which yield the modal eigenvectors) for general symmetric loaded MTL structures. It is observed that a given set of boundary conditions implies a specific zero determinant, but that the resulting frequency may alternate: (f1 or f4 ) and (f2 or f3 ). This is most easily seen by examining the case worked out above for the shielded structure. Had the unit cell of the shielded structure been chosen so that the inductors, 2L, were located at nodes n and n + 1, with the two series capacitors, 2C, located at n + 21 , then the terminal boundary conditions corresponding to f4 would be the dual In = In+1 = 0 → Vn = Vn+1 = 0 and Vn+ 1 = 0 → In+ 1 = 0 of those 2

2

shown in Figure 4.7.

4.5.4

Modal field structure for f4 and f5 (at (βd)x = π)

In the previous two sections analytical formulas, and equivalent circuits, corresponding to the band-edge frequencies f1 through f4 were derived. Two additional resonances, at f5 and f6 , which occurred out of the frequency range considered in previous simulations, were also shown. The results of Sections 4.5.2 and 4.5.3 will now be used to calculate the modal eigenvectors at these critical frequencies, which will provide further intuition into the nature of, and parameters affecting, the dispersion behaviour of the shielded structure. In Section 4.5.3 it was shown that the modal eigenvectors at resonance could be determined by considering one half of the symmetric unit cell, with homogeneous boundary conditions associated with one of the pair

88


Table 4.1: Boundary conditions and analytical formulas corresponding to the resonance frequencies at (βd)x = 0 and (βd)x = π. Boundary Conditions Formula Non-Zero Variables phase shift / frequency Ah Vn+ 1 = 0 2 Vn = Vn+1 = 0 In = Ch Vn+ 1 ⇒ det(Ah ) = 0 (βd)x = π (f1 or f4 ) In+ 1 = 0 2 2 In+1 = −In Bh In+ 1 = 0 2 Vn = Vn+1 = 0 In = Dh In+ 1 ⇒ det(Bh ) = 0 (βd)x = 0 (f2 or f3 ) Vn+ 1 = 0 2 2 In+1 = In Ch Vn+ 1 = 0 2 In = In+1 = 0 Vn = Ah Vn+ 1 (βd)x = 0 (f3 or f2 ) ⇒ det(Ch ) = 0 In+ 1 = 0 2 2 Vn+1 = Vn Dh In+ 1 = 0 2 In = In+1 = 0 Vn = Bh In+ 1 (βd)x = π (f4 or f1 ) ⇒ det(Dh ) = 0 Vn+ 1 = 0 2 2 Vn+1 = −Vn

of vector variables at node n, {Vn , In }, and one of pair at node n + 21 , {Vn+ 1 , In+ 1 }. The 2

2

complimentary non-zero variables are obtained by solving the homogenous systems shown in Table 4.1. The frequencies f4 and f5 are characterized by the homogenous conditions In = 0, and Vn+ 1 = 0. The corresponding nonzero variables, In+ 1 , and Vn are obtained by solving the 2

2

complimentary homogenous system, Dh In+ 1 = 0, and finally Vn is obtained from Vn = 2 Bh In+ 1 , as in the last row of Table 4.1. Since det(Dh ) = cos θ2l cos θ2u can by factored, 2

its zeroes occur at two different frequencies, corresponding to f4 and f5 . The frequency f4 corresponds to cos θ2l = 0. Solving for the frequency yields: π ω4 = q 0 0 d Ll Cl When cos

θl 2

(4.77)

= 0 the homogeneous system Dh In+ 1 = 0, with Dh given in (4.68) reduces to: 2

"

cos − cos

θu 2 θu 2

with solution: In+ 1

2

#  (1)  " # 0 In+ 21 0 =  (2) 0 In+ 1 0

(4.78)

2

 (1)  " # In+ 1 0 , =  (2)2  = In+ 1 Io 2

(4.79)

89

Chapter 4. Multiconductor analysis: Dispersion analysis where Io is an arbitrary constant. The corresponding solution for Vn is given by: Vn =

" (1) # Vn (2)

Vn

" =

Vo

# (4.80)

Vo

where Vo is an arbitrary constant. Thus the modal eigenvectors at nodes n and n +

1 2

corre-

sponding to f4 are given by:

f4 ⇒

  Vo " #     Vn Vo  =  0 In   0

    0 0 V 1   n+ 2  =      In+ 1 0 2 Io

and

(4.81)

The fields described by (4.81) are only non-zero in the lower region and represent a standing wave pattern with a half wavelength,

λl 2

= d, fitting within a unit cell, as shown in Figure 4.8a.

As was mentioned previously, due to the symmetry of the unit cell only one half of the cell is considered and hence the standing wave pattern depicted in Figure 4.8a shows only a quarter wavelength,

λl 4.

In a similar manner, f5 corresponds to cos

θu 2

= 0, with the solution:

π ω5 = p 0 0 d Lu Cu

(4.82)

and corresponding modal eigenvectors given by:

f5 ⇒

  Vo " #     Vn 0 =  0 In   0

 and

0



     0  Vn+ 1   2 =      I In+ 1 o   2 −Io

(4.83)

which has non-zero fields only in the upper region of the structure and represents a standing wave pattern with

λu 2

= d, as shown in Figure 4.8b. This resonance occurred above the frequency

range of the previously shown dispersion curves.

4.5.5

Modal field structure for f3 and f6 (at (βd)x = 0)

The non-zero variables, Vn and Vn+ 1 for the condition det(Ch ) = 0, (4.71), also occur at 2

two different frequencies corresponding to upper and lower region field concentration, and are depicted in Figure 4.9. The frequency f6 corresponds to sin θ2u = 0 ⇒ θu = 2π, with solution

90


(1)

Vn

(1)

In+ 1 = 0

= Vo

2

No upper region fields (2)

Vn

(2)

In+ 1 = Io

= Vo

2

d 2 open: n

short: n +

1 2

(a) f4 : Lower region standing wave

(1)

Vn

(2)

Vn

(1)

= Vo

In+ 1 = Io

=0

In+ 1 = −Io

2

(2)

2

No lower region fields d 2 open: n

short: n +

1 2

(b) f5 : Upper region standing wave

Figure 4.8: Field patterns corresponding to f4 and f5 ; (βd)x = π.

91

Chapter 4. Multiconductor analysis: Dispersion analysis (1)

Vn

(2)

Vn

(1)

= Vo

Vn+ 1 = −Vo

=0

Vn+ 1 = 0

2

(2) 2

No lower region fields d 2 open: n

open: n

(a) f6 : Upper region standing wave

(1)

Vn

(1)

Vn+ 1 = Vo

= Vo

2

No upper region fields (2)

Vn

(2)

Vn+ 1 = Vo

= Vo

2

d 2 open: n

open: n 0

(b) f3 : Resonance between L and Cl d

Figure 4.9: Field patterns corresponding to f6 and f3 ; (βd)x = 0. given by:

2π ω6 = p 0 0 d Lu Cu

(4.84)

which is one full wavelength fitting within the unit cell, with field structure depicted in Figure 4.9a. This resonance was not observed in the previous dispersion plots, as it occurred out of their range. Setting the other factor equal to zero results in f3 . The resonances considered up to this point, f4 , f5 , and f6 occurred at frequencies for which the electrical lengths of the interconnecting MTL sections were not negligible, and have no counterparts from the simplified analysis. However, a simplified formula for f3 may be derived under the assumption that the MTL sec tions are electrically short, so that the substitutions sin θ2l → θ2l , cos θ2l → 1 can be made. With these substitutions, the equation det(Ch ) = 0 may be solved for ω3 yielding: ω32 =

1 = 0 L(Cl d)

1 2L

0

Cl d 2

(4.85)

which represents a resonance between the loading inductance, L, and the distributed lower

92

Chapter 4. Multiconductor analysis: Dispersion analysis 0

region capacitance multiplied by the periodicity, Cl d, and this is the same formula that was obtained in the simplified analysis (4.44). The second form of the equation with 2L and

0

Cl d 2

is

given for convenience, and corresponds to what is occurring in one half of the unit cell. With the short MTL approximation the lower region voltage between the patch conductor and ground is nearly constant over the unit cell, with the field patterns depicted in Figure 4.9b.

4.5.6

Modal field structure for f2 (at (βd)x = 0)

The resonances examined up to this point had field concentrations restricted to either the lower or upper regions alone, and the determinants corresponding to these frequencies could be factored into terms dependent on the lower or upper region alone. However the expressions for det(Ah ) (f1 ) and det(Bh ) (f2 ) given in (4.69) and (4.70), could not be further factored in that manner, and hence these two resonances are functions of both the upper and lower region geometries. Applying the approximations sin θ2u → θ2u , sin θ2l → θ2l , cos θ2u → 1 , cos θ2l → 1 to det(Bh ) and solving for ω2 yields: ω22

1 = 2C

2 2 0 + d Lu d L0l

,

(4.86)

which is identical to that obtained in the simplified analysis (4.43), and represents a resonance between the loading capacitance, 2C and a parallel combination of two inductances determined by the upper and lower regions distributed inductances,

0

Lu d 2

and

0

Ll d 2 .

Approximate solutions

for the non-zero fields, In and In+ 1 are given by: 2

 0 1 = Io +  hu  Io hu  Io = − −1 − −1 hl h | {z } | {z l } 

In = In+ 1 2

1



"

#

Hu



(4.87)

Hl

The current vector, In = In+ 1 , which in the short MTL approximation is the same at nodes n 2

and n +

1 2

is decomposed into two components labeled Hu and Hl , which are the sources for

the upper and lower region magnetic fields (and hence inductances). Physically, the excited capacitance, 2C provides a displacement current which must divide between the upper and lower regions, as depicted in Figure 4.10. When hu > hl , as in Figure 4.10a the upper regions inductive impedance is much larger than that of the lower region, resulting in the majority of the current shunting off into the lower region. However when hu < hl , as in Figure 4.10b, the majority of the current is shunted off into the upper region, corresponding to an increase of f2 , as hu is decreased.

93


(1)

(1)

In+ 1 = Io

In = Io

2

weak Hupper

(2) In = − 1 +

hu hl

(2) In+ 1 = − 1 +

Io

2

hu hl

Io

strong Hlower

d 2 short: n

short: n +

1 2

(a) Field structure for large hu

(1)

(1)

In = Io (2) In = − 1 +

hu hl

In+ 1 = Io strong Hupper

Io

2

(2) In+ 1 = − 1 + 2

hu hl

Io

weak Hlower

d 2 short: n

short: n +

1 2

(b) Field structure for small hu

Figure 4.10: Field patterns corresponding to f2 , (βd)x = 0. (a) large hu ; (b) small hu . Il~ (dashed lines) and the current lustration of the gap capacitive fringing field, E distribution (solid lines).


4.5.7

94

Modal field structure for f1 (at (βd)x = π)

Examining the modal field structure at f1 is of particular interest, as it is recalled that for large values of hu the frequency f1 represented the initial point of a BW mode (and dual-mode band) Figure 4.5 (a) through (c), while for small values of hu it was the peak of a single-mode FW pass-band, Figure 4.5 (d) through (f). Applying the short MTL approximations to det(Ah ) = 0 and solving for ω1 yields: ω12 =

1 0 0 4LC + LCu d + LCl d + CLl d + 41 d2 Ll Cu0 0

0

(4.88)

If in addition it is assumed that the loading inductance, L is large, (4.88) can be simplified further, resulting in: ω12 =

1

2L 2C +

Cu0 d 2

(4.89)

Approximate solutions for the non-zero fields, In and Vn+ 1 are given by: 2



   " # 1 0     1 2C  2C  In =  Io +   0  Io = 1 + 0  Io −1 Cu d Cu d | {z } 2 2 {z } | Hu

(4.90)

Hl

 1   =  4L  Vo − 0 Ll d 

Vn+ 1

2

(4.91)

For large values of hu , the loading capacitance, 2C dominates the upper region distributed capacitance,

0

Cu d 2 ,

such that 2C

0

Cu d 2 ,

and (4.90) may be approximated as:

 " # " # 1   1 0 2C  Io + Io In =   0  Io = −1 1+N Cu d | {z } | {z } 2 

Hu

where N =

2C Cu0 d 2

(4.92)

Hl

is a large number, and thus the magnetic field is largely confined to the

lower region. The upper region electric field, which is given by: (1)

Eupper ∝

2

2

hu

4L 0 Ll d hu

1+

(2)

Vn+ 1 − Vn+ 1

=

(4.93)

95


weak Eupper (1)

(1)

Vn+ 1 = Vo

In = Io

(2) In =

2C ′ Cu d 2

2

(2)

Io

Vn+ 1 = − 2

d 2 short: n

open: n +

4L Vo ′ Ll d

1 2

(a) Field structure for large hu

strong Eupper (1)

(1)

In = Io (2) In =

2C ′ Cu d 2

Vn+ 1 = Vo 2

(2)

Io

Vn+ 1 = − 2

d 2 short: n

open: n +

4L Vo ′ Ll d

1 2

(b) Field structure for small hu

Figure 4.11: Field patterns corresponding to f1 , (βd)x = π. (a) large hu ; (b) small hu . The ~ (dashed lines) and the current distribution (solid lines) are shown. electric field, E

96


is relatively weak, and hence both the magnetic and electric fields are largely confined to the lower region. When 2C

0

Cu d 2

(4.89) can be further approximated, resulting in: ω12 =

1 4LC

(4.94)

which represents a resonance between the loading inductance, 2L, and the loading capacitance, 2C and is indicative of the commencement of a BW band, as noted in Section 1.2.2. The approximate field structure for this case (relatively large hu ) is shown in Figure 4.11a. 0 2C Conversely for small hu , C2u d dominates 2C, so that 0 → 0, allowing In to be approximated as:

Cu d 2



 " # " # 1   0 1 2C  Io + Io In =   0  Io = 1 −1 Cu d |{z} | {z } 2 Hu

(4.95)

Hl

from which it is apparent that the magnetic field permeates both regions with equal strength. Additionally, for small hu , upper region electric field, (4.93) is large, causing a significant displacement current to be generated in the upper region, as depicted in Figure 4.11b. Thus for small upper region height, hu , the impedance due to the upper region distributed capacitance, 0

Cu d 2

is smaller than that due to the loading capacitance, 2C, and a majority of the displacement

current travels through the upper region, rather than through 2C. In this case the resonant frequency, (4.89) may be approximated as: ω12 =

1 LCu0 d

(4.96)

That a BW mode does not commence at f1 for small hu is due to the fact that the fields are no longer confined to the lower region of the geometry, and hence the fringing capacitance, 2C, which is associated with the backward wave mode, is weakly excited relative to the upper region parallel plate capacitance,

Cu d 2 .

Additionally, from (4.96) it is apparent that as hu → 0,

f1 → 0 indicating that the bandwidth of the lowest pass-band becomes arbitrarily small.

4.6

Design considerations

Up until this point the alteration of the dispersion curves of the shielded Sievenpiper structure were observed under the variation of a single parameter alone, hu the upper region height, with all other geometric and electrical parameters held fixed. This was done so as to simplify the presentation, although to achieve a desired response many other parameters may be altered. Tailoring of the dispersion can be achieved by examining the frequencies f1 through f4 , which

97

Chapter 4. Multiconductor analysis: Dispersion analysis largely determine the band structure. 0

0

0

0

Substituting the analytical expressions for Cu , Cl (3.16), and Lu , Ll (3.19), into ω1 (4.89), ω2 (4.86), ω3 (4.85), and ω4 (4.77), results in the following expressions: ω12 =

ω22

1 1 d2 2L 2C + 2hu

1 1 = 2C µo ω32 =

2 2 + hu hl

1 2 d2 2L 2hl

π ω4 = √ d µo 2

(4.97)

(4.98) (4.99)

(4.100)

The periodicity, d appears in the denominator of (4.97), (4.99), and (4.100), and thus by increasing\decreasing the periodicity a commensurate decrease\increase of ω1 , ω3 , and ω4 is achieved. Although the periodicity does not appear in the denominator of (4.98) it is noted that the series capacitance, C is directly proportional to d (C ∝ d), as the capacitance is due to the fringing fields across the gap. From this it is concluded that there is also an inverse relationship between ω2 and d, and thus all four frequencies ω1 through ω4 may be shifted by altering the periodicity. Although altering the periodicity allows one to shift the resonant frequencies and hence the dispersion curve, this approach may not be the simplest to implement in practice. If one wanted to shift the dispersion curve up in frequency a decrease in periodicity would be an option. However, a decrease in periodicity requires that the density of vias be increased resulting in a more complicated fabrication. Another option would be an alteration of the loading elements L and C. The inductance L is inversely related to the via radius, r, while the gap capacitance, C is inversely related to the gap spacing, g, with the equivalent lumped component values obtained from scattering analysis as in Chapter 3. For example, if one wanted to change the dispersion curve for the structure investigated in Figure 4.5d, without changing either the periodicity (d = 10 mm), or the upper (hu = 1 mm; 1 = 1) or lower (hl = 3.1 mm; 2 = 2.3) region substrates, then the two remaining free parameters would be the gap spacing, g and the via radius, r. From Figure 4.5d it is observed that f3 occurs below f2 and thus if a larger stop-band bandwidth is desired than f3 must be increased. An increase of f3 may be achieved by decreasing L, and from scattering simulations it was determined that a via with a radius of r = 1.5 mm yields an inductance of 0.17 nH, down from 0.75 nH for a via radius of r = 0.5 mm (Figure 4.5d). The FEM simulated dispersion


98

Figure 4.12: Dispersion curve for a structure with a via radius of 1.5 mm, corresponding to L = 0.17 nH. All other geometric and electrical parameters are as for the structure of Figure 4.5d: d = 10 mm, hu = 1 mm, hl = 3.1 mm, r1 = 1, r2 = 2.3. results, along with those of the MTL model are shown in Figure 4.12. It is observed that the frequency ω3 increases to approximately 8.44 GHz (MTL model), with the new stop-band extending from approximately 3.34 to 8.44 GHz, whereas previously it was from 2.66 to 5.93 GHz (Figure 4.5d). It is noted that there is a slight discrepancy between the MTL model and the FEM simulated results, which is due to the fact that the larger via radius takes up a substantially larger fraction of the unit cell, and hence the lumped component approximation is not as accurate, resulting in the observed frequency shifts. In a similar manner one may vary the gap spacing, g, or simultaneously both g and r. By adjusting these parameters one is able to achieve a broad range of responses, with insight into the starting point of the design process provided by the analytical expressions for the resonant frequencies ω1 through ω4 .

4.7

Comparison of the MTL model with the TL-PP model

In Section 1.2.3 the transmission line parallel-plate (TL-PP) model of the shielded Sievenpiper structure, proposed in [7, 8] was briefly described. The model attempts to incorporate the effect of the upper shielding conductor, with the underlying parallel-plate geometry formed from the

99

Chapter 4. Multiconductor analysis: Dispersion analysis n

Shielding plane s

hu , ǫ1 Zo =

hl , ǫ2

w

L′ (TL-PP) C ′ (TL-PP)

ground plane

d (b) Transformation into a loaded TL-PP structure

(a) Unloaded 2 conductor TL-PP

n

n+1

n+1 ′

Zo

Y

Co = Cu d

Zo Y

d 2

d 2

Lo = L

(d) Loading admittance, Y

(c) Equivalent TL-PP circuit

Figure 4.13: Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-plate transmission line (TL) upon which the patches and vias act as loading elements. The underlying unloaded TL consists of the shielding plane and the ground plane as depicted in (a), which is transformed to the actual (loaded) structure in (b). Equivalent circuit for this point of view is shown in (c). Reactive loading element shown in (d). shielding conductor and the ground conductor. This is in contrast to the TL(BW) model of the unshielded Sievenpiper structure, introduced in Section 1.2.2, which does not incorporate the upper shielding conductor. In this section the TL-PP model will be examined and compared with both the FEM simulations and the MTL model. The proposed equivalent circuit for the TL-PP model, along with the underlying parallelplate environment of the shielded structure are shown in Figure 4.13. It is observed that the underlying geometry is that of a parallel-plate TL formed from the shielding conductor and the ground conductor, with per-unit-length parameters, L0 (TL-PP) and C 0 (TL-PP). These parameters can be written in terms of the components of the per-unit-length capacitance and inductance matrices, C0 (3.16) and L0 (3.19), which define the unloaded MTL geometry, and are given by:

0

0

C C C (TL-PP) = 0 u l 0 Cu + Cl 0

0

0

0

L (TL-PP) = Lu + Ll

(4.101) (4.102)

The loading is modeled as a shunt admittance, Y , formed from the series combination of the 0

capacitance, Co = Cu d (due to the parallel-plate capacitance between the patch layer and the


100

shield), and the inductance, Lo = L (due to the via). It is noted that the fringing capacitance, C, of the MTL model, is not accounted for in the TL-PP model. The dispersion equation is given by:

where

with

cosh(γd) = Af

(4.103)

1 Af = cos(θTL-PP ) + jZo Y sin(θTL-PP ) 2

(4.104)

q θTL-PP = ωd L0 (TL-PP) C 0 (TL-PP), s L0 (TL-PP) Zo = C 0 (TL-PP)

and Y =

jωCo 1 − ω 2 Lo Co

(4.105) (4.106)

(4.107)

The dispersion equation (4.103) can only account for a single mode of propagation, and hence unlike the MTL model is incapable of capturing the dual-mode behaviour of the shielded structure for large values of hu . However, even for relatively small values of hu the TL-PP model greatly overestimates the bandwidth of the stop-band. In Figure 4.14, FEM generated dispersion curves are compared with those obtained from (4.103). Two simulations are shown, with upper region height, hu having the values 0.2 mm and 1 mm. All of the other geometric and electrical parameters are identical to those used to generate the dispersion curves from Figure 4.5. The stop-band predicted by the TL-PP model is shaded in these figures. For both hu = 0.2 mm and hu = 1 mm the first pass-band contains a single FW mode. With hu = 0.2 mm the peak of the first pass-band, at f1 = 1.78 GHz is accurately predicted by the TL-PP model; however for hu = 1 mm the frequency f1 (TL-PP) = 3.64 GHz obtained from the TL-PP model is approximately 35% greater than that given by the FEM simulation, f1 (FEM) = 2.66 GHz. The FEM simulations show that the peak of the first stop-band occurs at f3 (FEM) = 5.93 GHz for both hu = 0.2 mm and 1 mm, as was captured by the MTL model (Figures 4.5:d,f). However the TL-PP model overpredicts the peak of the stop-band in both cases; for hu = 0.2 mm, the peak predicted by the TL-PP model is at f3 (TL-PP) = 6.68 GHz (12.5% greater), while for hu = 1 mm the peak occurs at f3 (TL-PP) = 8.83 GHz (48.9% greater). This shows that even for cases where the dispersion is single mode, the TL-PP model does not accurately predict the band-edges. Additionally, it is noted that the TL-PP model cannot capture complex modes within the stop-band, as obtained from the MTL analysis. Approximate analytical formulas for the resonant frequencies ω1 and ω3 obtained from the


101

(a) hu = 0.2 mm

(b) hu = 1 mm

Figure 4.14: Comparison of the TL-PP model dispersion curves with FEM simulations. (a) hu = 0.2 mm; (b) hu = 1 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3


102

TL-PP dispersion (4.103) are given by: ω12 (TL-PP) =

1 0 LCu0 d + 14 Ll d2 Cu0 + 14 L0u d2 Cu0

and ω32 (TL-PP)

1 = L

2 1 + 0 0 Cu d Cl d

(4.108)

(4.109)

The expression for ω12 (TL-PP) (4.108) converges to that obtained from the MTL analysis (4.96), in the limit hu → 0, which shows that the peak of the first pass-band, predicted by the TL-PP model converges to that of the MTL model for small upper region height, hu . However, for larger upper region height, the TL-PP formula (4.108) greatly overestimates that obtained from the MTL model in the same limit, (4.94), ω12 (MTL) =

1 4LC,

which is a function of the series

fringing capacitance, C of the MTL model. As the TL-PP model does not include the fringing capacitance, it is incorrect in this limit. However, even more fundamentally, for large hu , the frequency f1 is the commencement of dual-mode pass-band, which the TL-PP model cannot capture. The expression for ω32 (TL-PP) (4.109) is a function of both the upper and lower region geometries, and does not account for the invariance of f3 when hl is fixed. The expression for ω32 (MTL) (4.85) correctly accounts for the fact that f3 is only a function of the lower region geometry. Comparing (4.109) and (4.85) shows that: ω32 (TL-PP)

1 = L

2 1 + 0 0 Cu d Cl d

≥

1 = ω32 (MTL) 0 LCl d

(4.110)

and hence the peak of the stop-band predicted by the TL-PP model, ω32 (TL-PP), is always larger than that of the MTL model, ω32 (MTL), leading to the overestimation of the stop-band bandwidth. Only in the limit hu → 0 do the two models converge with ω32 (TL-PP) → ω32 (MTL) in that case. Thus, both ω12 (TL-PP) and ω32 (TL-PP) converge to the correct expressions obtained from MTL analysis only in the limit of very small upper region height.

4.8

Modal degeneracy at f2

The FEM dispersion simulations from Figure 4.5 revealed that two modes emerged from the resonance frequency, f2 , which occurs at (βd)x = 0. One mode was captured by the MTL model, and the other was not. In Section 4.5 it was determined that the condition (βd)x = 0 corresponds to the boundary conditions, O:O:O or S:S:S at nodes {n : n + 21 : n + 1}, where O and S stand for open or short circuits. These condition imply H-walls or E-walls at the corresponding nodes: H:H:H or E:E:E.


103

Although these conditions were obtained for the unit cell along the longitudinal (propagation) direction, x, due to the symmetry of the structure it is apparent that the transverse boundary conditions corresponding to the phase shift (βd)y = 0 also would have to conform to the same conditions. The MTL model assumed that the transverse walls were described by H-walls. However due to the symmetry of the structure the fact that the central bisecting plane acted as an H-wall was automatically incorporated, as is shown in Figure 4.15 (a). For the case were the transverse boundary conditions are E:E:E, as shown in Figure 4.15 (b), it is observed that the structure becomes electrically connected. Such a geometry does not support quasi-TEM modes, and thus the MTL model cannot capture the mode corresponding to E:E:E boundary conditions, which was TE. The top view of the transverse and central boundary conditions corresponding to the frequency f2 are shown in Figure 4.15 (c) and (d). Due to symmetry, both configurations correspond to the same frequency. However with (βd)y = 0 fixed, only the boundary conditions shown in Figure 4.15 (c) are captured by the MTL model, for values of (βd)x 6= 0; that is when propagation along the x-direction is assumed. In conclusion, it is noted that even though the MTL model does not account for the dispersion of the TE mode, the fact that a modal degeneracy occurs at f2 is predicted by the MTL analysis.

4.9

Summary

In this chapter a periodic multiconductor transmission line (MTL) model of the shielded Sievenpiper structure was developed. Initially, a periodic unit cell, composed of a cascade of lumped components and sections of MTL lines was presented, with a corresponding dispersion equation derived from it. By applying suitable approximations to the full periodic dispersion equation, a simplified dispersion equation, in which the effect of the loading elements was incorporated directly into the per-unit-length MTL parameters, was obtained and analyzed. The simplified dispersion equation yields simple formulas relating the band-edge frequencies f2 and f3 (which occur at (βd)y = 0) to the geometry of the structure and the loading elements. Additionally, a frequency band between fc1 and fc2 supporting complex modes was shown to exist, with some of the properties of complex modes reviewed. The fully periodic MTL model was subsequently returned to, supplementing the deficiencies of the the simplified model. The fully periodic dispersion equation was shown to have excellent correspondence with FEM simulations, over a broad range of geometric parameters of the structure. Additionally, the fully periodic model was able to capture the two additional band edge frequencies, f1 and f4 , which occur at (βd)x = π. Equivalent circuits, analytical formulas, and modal eigenvectors corresponding to the band-edges were obtained. These results were

104


n

n+

1 2

n

n+1

n+

1 2

n+1

z y

(b) Side view (TE)

(a) Side view (MTL)

y

(βd)x = 0

(βd)x = 0

x

(βd)y = 0

(βd)y = 0

(c) Top view, as seen below the Shielding conductor (MTL)

(d) Top view, as seen below the Shielding conductor (TE) H-walls E-walls

Figure 4.15: Boundary conditions corresponding to the two degenerate modes at f2 : (a) Transverse boundary conditions for the mode described by MTL theory. (b) Transverse boundary conditions for the TE mode. (c) Boundary conditions at the transverse (y) walls, and longitudinal (x) walls for the MTL mode. (d) Boundary conditions for the TE mode are switched compared with (c)


105

particulary important in revealing the physical mechanism underlying the qualitative change in behaviour (dual-mode to single-mode dispersion) of the structure as the upper region height is decreased from large to small values. Finally, using the band-edge equivalent circuits, the modal degeneracy which occurs at f2 was explained.

Chapter 5

Slow Wave Analysis 5.1

Introduction

The dispersion analysis of the shielded structure developed in Chapter 4 revealed that in the limit ω → 0 the structure supported a FW mode commencing at DC. Modal plots from Chapter 2 revealed that the field structure had an asymmetric distribution: The electric field in the upper region was nearly constant, Eu = Eo , while in the lower region the electric field was virtually zero, El ≈ 0. The magnetic field however was virtually constant throughout both the upper and lower regions, Hu ≈ Hl = Ho . For relatively large shield-patch distances, corresponding to large hu the degree of slowing was minimal, but as the shielding conductor was brought closer to the patch layer significant slowing was achievable. In this chapter the low frequency limit for both the propagation constant and the modal eigenvectors will be determined using the MTL analysis. Interestingly, in this limit, the propagation constant, βd will be seen to be independent of the value of the loading elements, L and C, and only depend on a subset of the parameters defining the underlying MTL geometry, 0

0

Cu 0 , Cl 0 , Lu , and Ll . The analysis shows that the degree of the slow-wave effect is controlled by the distance between the shielding conductor and the patch (mushroom) layer, hu , with a significant slowing of the mode possible as the aforementioned distance is decreased. A physical explanation of this phenomenon is arrived at by examining the corresponding low frequency modal eigenvector. The fact that the low frequency dispersion is independent of L and C does not imply that they are not needed to achieve the slow wave effect. To confirm the necessity of the simultaneous presence of both L and C in achieving the effect, two related structures will be considered: First, a structure with the loading capacitance removed (shorted out), corresponding to C → ∞, and subsequently a structure with the loading inductance removed, corresponding to L → ∞. Propagation constants and modal eigenvectors for each of these separate structures will be calculated, showing that the slow wave effect is lost, thus proving 106

107

Chapter 5. Slow Wave Analysis

the necessity of both L and C, and further enhancing the physical picture developed for the shielded structure. The theory is compared with full-wave finite element simulations, with excellent correspondence between the two observed.

5.2

MTL model

The dispersion equation for the shielded structure (4.20) is a quadratic in the variable, cosh(γd), corresponding to two independent modes of propagation. In the low frequency limit a Taylor expansion may be performed on (4.20). One mode is evanescent, which corresponds to a backward wave (BW) mode below cut-off, while the other mode, which is the mode under consideration now, is a forward wave (FW) mode, which extends to DC. In the limit ω → 0 the dispersion of the FW mode is given by: 0 0 0 (βd)2LC = d2 Cu Lu + Ll ω 2 +

0 0 0 0 0 0 d3 C(Ll )2 Cu + LLu (Cu )2 + L(Cu )2 Ll ω 4 + · · · (5.1)

where the subscript LC of (βd)LC denotes the fact the dispersion equation is for the actual shielded structure, which has both a loading L and a loading C, in contrast to related structures which will be analyzed later in this chapter, where one of the loading elements is eliminated. When (βd)LC 1 the dispersion (5.1), is well approximated by considering only the first term in the series expansion, resulting in the following low frequency (linear) dispersion curve: βLC =

q 0 Cu0 (L0u + Ll ) ω

(5.2)

which represents a forward wave with group, vg , and phase velocity, vφ , equal to each other and given by:

1 vφ (LC) = vg (LC) = q 0 0 Cu (L0u + Ll ) 0

0

(5.3)

0

In the limit hu → 0, Cu → ∞, while the term Lu + Ll remains finite, so that vg → 0 as hu → 0. The group velocity is thus bounded below only by zero, and can be made arbitrarily small: 0 ≤ vg (LC) ≤ p

1 Cu0 L0u

(5.4)

From (5.3) the resulting mode has an effective capacitance and inductance per-unit-length given by: 0

0

Cef f (LC) = Cu

(5.5)

108

Chapter 5. Slow Wave Analysis 0

0

0

Lef f (LC) = Lu + Ll

(5.6)

The low frequency phase velocity is independent of the loading elements L and C, and depends only on the electrical and geometric parameters of the upper and lower regions. However, the electrical parameters appear in an asymmetrical manner, with the effective inductance, 0

0

0

Lef f (LC) = Lu + Ll involving both the upper and lower region geometry, while the effective 0

0

capacitance, Cef f (LC) = Cu involves only the upper region geometry. A similar phenomenon was observed in [10], where a slow wave effect was achieved by utilizing a two-dimensional array of metallic posts. The slow wave effect can be understood as being due to a capacitance enhancement, and this is most readily seen by considering the propagation on a commensurate two conductor geometry which is identical to the shielded structure, but with the entire patch layer (Conductor 2) and the via removed. Such a structure is depicted in Figure 5.1, with the figure indicating the equal and opposite currents on the two conductors as is expected for a two conductor geometry, and non-zero fields, in both the upper and lower regions. This structure is a simple two conductor transmission line (TL) with dispersion given by: (βd)2T L

! 0 0 Cu Cl 0 0 =d Lu + Ll ω 2 0 0 Cu + Cl | {z } | {z } L0

(5.7)

2

ef f

0

Cef f 0

0

and whose per-unit length capacitance, C (T L) and inductance L (T L) can be written in terms of the components of the capacitance and inductance matrices characterizing the shielded structure:

0

0

C (T L) =

0

Cu Cl 0 Cu0 + Cl

0

0

(5.8)

0

L (T L) = Lu + Ll 0

0

(5.9)

0

The effective inductance Lef f (LC) = Lu +Ll (5.6), of the FW mode for the shielded mushroom 0

structure is identical to the inductance of the the commensurate (2-conductor) geometry, L (T L) (5.9), consisting of the upper shielding conductor and the ground conductor alone. However, the 0

per-unit length effective capacitance of the shielded structure, Cef f (LC) (5.5) is always greater 0

than that of the commensurate T L geometry, C (T L) (5.8), as can be shown by computing their ratio,

0

rC =

Cef f (LC) C (T L) 0

=1+

1 hl >1 2 hu

(5.10)

which is always greater than 1 and hence a capacitance enhancement has been achieved. Even in the case where the relative permittivities of the upper and lower regions are equal to one, 1 = 2 = o , is the ratio rC (1 = 2 = o ) = 1 +

hl hu

> 1. Thus obtaining the slow wave effect

109


Eu 6= 0

V1 = Vo 1 Vo I1 = Zo

Hu 6= 0

s Zo =

(βd)2T L El 6= 0

′

Lef f ′

Cef f ! ′ ′ ′ ′ C C u l L + L ω2 = d2 ′ Cu′ + Cl | u {z l } | {z } L′ef f ′

Cef f

Hl 6= 0

Ig → −1

Figure 5.1: Field structure of the commensurate two conductor geometry with both the entire patch layer and via removed.

14

Frequency (GHz)

12 10 8

Light εr=1 0.2 mm (FEM) 1.0 mm (FEM) 6.0 mm (FEM) MTL theory

6 4 2 0

0.2

βd (radians)

0.4

0.6

Figure 5.2: Low frequency dispersion with hu = 0.2, 1, and 6 mm; All other parameters are c fixed: hl = 1 mm; d = 2 mm; w = 1.9 mm; via radius = 0.1 mm (from [11], IEEE 2008).

is not dependent on the use of high permittivity substrates, and can be achieved simply by altering the geometric parameters. When the ratio

hl hu

is small, corresponding to large hu , the

degree of slowing is negligible, but as hu → 0, while hl remains finite, the ratio can be made arbitrarily large, corresponding to a strong slow-wave effect. The variation of the dispersion as a function of varying hu (for a fixed value of hl ) is depicted in Figure 5.2, where excellent correspondence between MTL theory and full-wave finite element method (FEM) simulations is demonstrated. The field structure of the FW mode in the low frequency limit is obtained by substituting

110


Eu 6= 0

V1 → Vo 1 I1 → Vo Zt

Hu 6= 0

s Zt =

V2 → 0

′

Lef f ′

Cef f

′ ′ ′ (βd)2LC = d2 Cu Lu + Ll ω 2 + O(ω 4 ) |{z} | {z }

I2 → 0

′

Cef f

El ≈ 0

′

Lef f

Hl 6= 0

Ig → −1

Figure 5.3: Low frequency FW mode voltage and current distribution for the shielded structure. 0

0

0

(βd)2LC = d2 Cu (Lu + Ll ) ω 2 into (4.15), and solving for the resulting eigenvector:     1 V    1  −dCu Lω 2  V     2  Vo + smaller terms 1   =     I1        1 Zt I2 Ll dCω 2 Zt (5.11) s

0

0

Lu + Ll . The above eigenvector expression is valid in the limit ω → 0 where V1 Cu0 V2 and I1 I2 . In this limit only the V1 , I1 terms remain finite, while V2 , I2 → 0 as O(ω 2 ),

where Zt =

resulting in the further simplification:     1 V    1   V  0  2    → 1  Vo as ω → 0  I1       Zt  I2 0 s

0

(5.12)

0

Lu + Ll V1 = Zt = , confirming the earlier observation that the I1 Cu0 0 0 0 mode has an effective inductance and capacitance per-unit-length, Lef f (LC) = Lu + Ll and

From (5.12) the ratio, 0

0

Cef f (LC) = Cu , respectively. The voltage component of the modal eigenvector is given by Vmodal =

V1 V2

=

V o 0 , which

shows that the electric field is confined to the upper region, and is a confirmation of the fact

111

Chapter 5. Slow Wave Analysis d 2

MTL

MTL

d 2

2C

2C (a) MTL with only C present.

MTL

d 2

MTL

d 2

2L 2L (b) MTL with only L present.

Figure 5.4: MTL unit cells with one of the loading elements removed at a time. 0

0

that the effective capacitance Cef f (LC) = Cu involves only the upper region capacitance. A physical explanation for this effect is that the loading inductor, L, provides an effective short circuit at low frequencies, hence shorting out the electric field in the lower region. The modal 1 current eigenvector Imodal = II12 = Zt Vo , shows that there is no net return current flowing 0

on conductor 2 at low frequencies, and hence the ground conductor provides the return path for the current on conductor 1. Physically, this occurs because the capacitive gaps between the patches disrupt the return current, and at low frequencies the impedance due to the capacitive gaps is large Z2C = 2 j 1ω C → ∞ as ω → 0 , and hence the displacement current across the gaps is negligible. The manner in which the return current is established on the ground conductor is depicted in Figure 5.3. The return current, which attempts to establish itself on the upper part of conductor 2, encounters the series gap and simply takes the path of least impedance and flows onto the lower part of conductor 2, travels on the via, and then finally onto the ground conductor. Thus, only the net current on conductor 2 (the patch layer) is zero, due to the equal and opposite currents on the upper part and lower part of the patch layer. However, the magnetic field created by the current distribution from Figure 5.3 is identical to that which would occur if both the patch (Conductor 2) and the via were removed, confirming the formula 0

0

0

for the effective inductance, Lef f (LC) = Lu + Ll , which is identical to the inductance of the commensurate geometry consisting of only the shielding conductor and the ground conductor alone, (5.9). Although the actual values of the loading elements, L and C are not relevant for the low

112


frequency FW mode described by (5.2), they must remain finite for the analysis which lead to them to be valid. However, if the patches are shorted out along the direction of propagation, then C → ∞, and the dispersion equation obtained using a Taylor series approximation, (5.1) is not valid. Similarly if the via is removed, corresponding to L → ∞, (5.1) is not valid. The behaviour of these modified structures is obtained by considering unit cells where the transfer matrices corresponding to C and L are removed as in Figure 5.4. Both of the modified structures also support a single low frequency FW propagating mode, in addition to an evanescent mode. The low frequency modal eigenvectors and propagation constants, (βd)C and (βd)L of the structures with only the loading C and L present, respectively are depicted in Figure 5.5. For the structure with only the C present, the low frequency dispersion equation is given by: ! 0 0 C C 0 0 u l (βd)2C = d2 Lu + Ll ω 2 + O(ω 4 ) 0 0 Cu + Cl | {z } {z } L0 |

(5.13)

ef f

0

Cef f

which is identical to that of the commensurate two conductor geometry with the entire patch layer and via removed, (5.7), except that (5.13) contains higher order corrections, O(ω 4 ), due to its periodic nature and the loading C. From (5.13), and assuming that the upper region permittivity is less than that of the lower region it can be shown that the group velocity is bounded as:

1 1 q ≤ vg (C) ≤ p 0 0 0 0 Cu Lu Cl Ll

(5.14)

and thus can’t be slowed down arbitrarily as for the shielded structure. The field structure for this mode is depicted in Figure 5.5a. It is observed that the current is diverted around the gap in conductor 2 just as it is for the shielded structure, but as there is no via to short out the lower region fields, both the electric and magnetic fields are non-zero everywhere, in this case. For the structure with only the L present, the low frequency dispersion equation is given by: 0

0

(βd)2L = d2 Cu Lu ω 2 |{z} |{z} 0

(5.15)

0

Cef f Lef f

from which it is concluded that the group velocity is constant and given by the upper region mode velocity:

1 vg (L) = p 0 0 Cu Lu

(5.16)

The shorting of the patches results in no coupling between the lower and upper region modes, and hence the fields are completely concentrated in the upper region as depicted in Figure 5.5b. Thus, for the structures where one of the loading elements are missing and one is present,

113


Eu 6= 0

Hu 6= 0

V1 → Vo 1 I1 → Vo Zo V2 → V2 (6= 0) I2 → 0

El 6= 0

(βd)2C

Hl 6= 0

s Zo =

′

Lef f ′

Cef f

! ′ ′ ′ ′ Cu Cl Lu + Ll ω 2 + O(ω 4 ) =d ′ ′ Cu + Cl | {z } {z } L′ef f | 2

′

Cef f

Ig → −1 (a) Voltage and current eigenvectors with only C present.

Eu 6= 0

Hu 6= 0

V1 → Vo 1 Vo I1 = Zu V2 = 0 I2 = −

1 Vo Zu

s Zu =

′

Lef f ′

Cef f ′

′

(βd)2L = d2 Cu Lu ω 2 |{z} |{z} ′

′

Cef f Lef f

El = 0 Hl = 0

Ig = 0 (b) Voltage and current eigenvectors with only L present.

Figure 5.5: Eigenvectors corresponding to the MTL unit cell with one of L or C removed. there is a lack of asymmetry in the modal field profile, and it is this lack of asymmetry which prevents either a capacitive or an inductive enhancement, and thus a slowing of the low frequency FW mode.

5.3

Summary

In this chapter the slow-wave effect produced by the shielded mushroom structure has been demonstrated analytically using MTL theory, and confirmed with full-wave FEM simulations. MTL analysis revealed that the low frequency slow-wave effect was due to an enhanced effective capacitance per-unit-length, which could be made arbitrarily large, while the inductance perunit length remained finite. A physical picture of the mechanism behind the slow-wave effect was also developed. Additionally, even though the low frequency phase velocity was independent


114

of the loading elements, L and C, the necessity of their presence was established by considering commensurate geometries where each of L and C was removed from the structure.

Chapter 6

Scattering Analysis 6.1

Introduction

In Chapter 4 the dispersion properties of the shielded Sievenpiper structure were analyzed using MTL theory. By applying a Bloch analysis to a single MTL unit cell, two propagation constants, (γd){a,b} , corresponding to two independent modes of propagation, were derived. For each propagation constant, which is related to the eigenvalue of the unit cell’s transfer matrix, h it Tunit−cell−M T L (4.13), there exists a corresponding modal eigenvector, (4.15), V, I , which {a,b}

reveals a given modes’ field concentration. The MTL theory analysis was shown to have excellent correspondence with FEM simulations. In this chapter we will consider the excitation of a finite cascade of unit cells of the shielded structure, from which generalized scattering parameters will be derived. The dispersion equation and modal eigenvectors correspond to an (infinite) periodic structure, but in actual physical applications the structure will of course be finite. The scattering analysis thus provides one with an understanding of the operation of the shielded structure under realistic situations. A general excitation will support a superposition of all the Bloch modes, and by examining the scattering parameters in conjunction with the modal excitation strengths a picture of the operation of multimodal structures will be developed. In addition, for some finite element method solvers, it is difficult to obtain the propagation constants in stop-bands (both complex and evanescent bands). However a scattering simulation can always be performed over all frequencies, and can thus provide confirmation of the propagation constants within both evanescent mode and complex mode bands. Thus scattering simulations provide an additional way to test the effectiveness of the MTL analysis. In particular it will be observed that when plotting the S-parameters as a function of frequency, evanescent mode bands and complex mode bands are characterized by distinct shapes. For the shielded structure it was seen that both standard evanescent modes and complex modes are supported, and the distinct signatures of these two 115

Chapter 6. Scattering Analysis

116

types of bands will be observed. Different forms of excitations will be considered, which are related to different applications of the structure. Initially, a four-port scattering scenario will be examined, in which the upper and lower regions of the structure, respectively, are excited separately. Such an excitation is relevant in the understanding of the operation of the coupled mode coupler developed in [15, 16], where a compact directional coupler was demonstrated. The operation of such a coupler is directly related to the existence of complex modes, and in particular the dominant excitation of a single complex mode. Another application of the four-port scattering analysis is related to 2D NRI TL grids as developed in [26–28]. It is recalled from Section 1.2.2, that in that work TL theory was used to model a structure which was fundamentally dual mode, with both a backward wave (BW) and a forward wave (FW) mode supported. The BW mode was largely confined to the substrate, while the FW mode was largely situated above the substrate. Multiconductor transmission line theory can be used to model the dual mode dispersion of such structures. In particular, by using the MTL model of the shielded structure, it will be shown that a lower region excitation strongly excites the BW mode, while an upper region excitation generally excites the FW mode. Additionally, a two-port scattering scenario will be examined. In this case the excitation is between the shielding conductor and the ground conductor. This type of excitation is relevant in the operation of the structure as a noise suppression device in digital circuits [6–8], and as a slot antenna created on the upper shielding conductor [9]. The importance of the scattering analysis may be summarized by the following points: • Confirmation of the applicability of the periodic Bloch analysis to a more realistic finite case. • Confirmation of the propagation constant(s) within the stop-band, including the existence of complex modes. • Relates various forms of excitations to structures which support multi-mode/ complexmode bands, and their applications. As mentioned above, different scattering situations will be considered, corresponding to different input excitations of the cascaded structure. Although the detailed description of the individual scattering situations will be provided in subsequent sections, the general form of each analysis may be summarized as follows: A cascade of N unit cells of the shielded structure is considered, with the Bloch modes obtained from the periodic analysis used as the basis modes for the cascade. The N unit cell cascade is extended at its input and output planes by unloaded waveguide sections, for which appropriate port variables will be defined. The port variables correspond to wave propagation on the unloaded waveguide sections. By

117


applying appropriate boundary conditions at the waveguide-periodic structure transition the complete boundary value problem will be solved. Comparison of the analytically (MTL theory) derived S-parameters with those obtained from FEM simulations will provide confirmation of the applicability of the MTL analysis to finite cascades of multi-mode structures.

6.2

Four-Port Scattering Analysis

The initial scattering geometry to be examined is depicted in Figure 6.1a, in which N unit cells of the shielded Sievenpiper structure are cascaded. At the two ends of the cascade, x = 0, and x = N d, the three conductors, 1, 2, and ground, are extended by a length, l, and at the positions, x = −l and x = N d+l, the ports of the given scattering problem are defined. The port variables are identical to those derived in Section 3.3, where they were used in the calculation of the series gap capacitance, C. Two types of excitations are considered: (a) a lower region excitation, which is the one depicted in Figure 6.1a, and an upper region excitation (which is not shown). For the lower region excitation the voltage source is located between the patch layer conductor and the ground, while for the upper region excitation the voltage source is located between the upper shielding conductor and the patch layer conductor. For a lower + region excitation the lower region incident wave, VM 1 is excited. All of the other ports are − − terminated in matched impedances, and so only the reflected amplitude coefficients, VM 1 , VM 2 , − − VM 3 , and VM 4 are present at the other ports. Figure 6.1b depicts the power flow paths for the

scattering matrix parameters which are obtained from the lower region excitation: S11 , S21 , S31 and S41 , while Figure 6.1c shows the same for the upper region excitation, which yields S12 , S22 , S32 and S42 . On the N unit cell cascade, the Bloch modes calculated from the periodic MTL analysis form the modal basis. The two independent modes are labeled γa and γb , corresponding to phase variations e−(γa )x and e−(γb )x , where x is an integer multiple of d. The eigenvectors for the individual Bloch modes are obtained by computing the eigenvectors of the transfer matrix, Tunit−cell−M T L , (4.15), and those corresponding to γa and γb are labeled by:     V Va  1  1  V  V a   2   (γa );   =  2   I1   I a     1 I2a I2

    V Vb  1  1  V  V b   2   (γb );   =  2   I1   I b     1 I2b I2

(6.1)

where, for the reflected modes, −γa and −γb , the eigenvectors are identical, except that the current components are reversed. At x = N d the voltage/current vectors given in (6.1) are transformed through the propagation constants e∓(γa d)N and e∓(γb d)N . Each Bloch mode and it’s

118


x = −l

x=0

Zu

− VM 2

Zl

− VM 1

vs

+ VM 1

Port modes; modal coefficients: + − − VM 1 , VM 1 , VM 2

d

cell 1

2d

cell 2

Nd

x = Nd + l

cell N

− VM 4

Zu

− VM 3

Zl

Port modes; modal coefficients: − − VM 3 , VM 4

Bloch modes; modal coefficients: − + − a+ m , am , bm , bm

MTL

2C

d 2

MTL

d 2

2C

L

(a)

S21

S11

P2

P4

P1+

P3

S41

S31

(b)

S22

S12

P2+

P4

P1

P3

S42

S32

(c)

Figure 6.1: Four-port scattering: (a) Circuit schematic for the four-port scattering analysis with lower region excitation; (b) Power flow for lower region excitation; (c) Power flow for upper region excitation.

119


+ − − reflected counterpart may be excited, with the amplitudes of excitations given by a+ m , bm , am , bm .

For the total system consisting of the port extensions and the cascade of unit cells there are nine n o + , a− , b− and unknown variables in total to be determined VM 1+ , VM 1− , VM 2− , VM 3− , VM 4− , a+ , b m m m m hence nine equations are needed to uniquely solve the system.

Due to symmetry, only the excitations of the lower region (mode 1), and the upper region (mode 2), are required to completely determine the scattering parameters. The steps required to solve for the scattering parameters for the lower region excitation, S11 , S21 , S31 and S41 are given now. The input port plane voltage/current vector (at x = −l) was derived in Section 3.3, and is given again for convenience:   V  1 V   2    I1    I2



(x=−l)

VM 1+ + VM 1− + VM 2−



  VM 1+ + VM 1−  −  VM 2  −  Zu  + − − V V M1 − M1 + M2 Zl Zl Zu

    =   V

(6.2)

The incident voltage component, VM 1+ is solved for by considering the source boundary condition: − vs + Zl (I1 + I2 ) + V2 = 0

(6.3)

Upon substituting the expressions for I1 , I2 , and V2 from (6.2), VM 1+ is solved for from (6.3), resulting in: VM 1+ =

vs 2

(6.4)

The output port plane voltage/current vector (at x = N d + l) is also identical to that from Section 3.3, given in (3.44). The transformations of the port variables (at x = −l and x = N d+l) to the beginning and end of the MTL unit cell cascade (at x = 0 and x = N d) are made using

120

Chapter 6. Scattering Analysis the lower and upper region mode propagation constants, βl and βu , and yield:   V  1 V   2    I1    I2



VM 1+ e−jβl l + VM 1− ejβl l + VM 2− ejβu l



    =   V

  VM 1+ e−jβl l + VM 1− ejβl l  −  VM 2 jβu l  e −  Zu  + − − VM 1 jβl l VM 2 jβu l  M 1 −jβl l e e e − + (x=0) Zl Zl Zu     VM 3− ejβl l + VM 4− ejβu l V     1  VM 3− ejβl l  V   −   2  V =    M 4 jβu l e   I1   Zu     − V −  V I2 M 3 −jβl l e − M 4 ejβu l (x=N d) Zl Zu

(6.5)

(6.6)

o n + , a− , b− is obtained , b A system of 8 equations in the 8 unknowns, VM 1− , VM 2− , VM 3− , VM 4− , a+ m m m m by imposing continuity of both the voltage and current on conductors 1 and 2, using (6.1), (6.5), (6.6), at x = 0 and x = N d, resulting in: " #" # M11 M12 VP M21 M22 where M11

 −ejβl l  −ejβl l  =  0  ejβl l Zl

M22

ejβu l

0 0

BP



 0 0   M12 ejβu l  0 0 Zu  jβu l − e Zu 0 0  0 0 −ejβl l  0 0 −ejβl l  M21 =  0 0 0  jβ l 0 0 − e Zll 0

 V a e−(γa d)N  1 V a e−(γa d)N  = 2  I a e−(γa d)N  1 I2a e−(γa d)N

" =

Vs

# (6.7)

0

  V1a V1b V1a V1b   V a V b V a V b   2 2 2 2  =   I a I b −I a −I b  1 1 1  1 I2a I2b −I2a −I2b  −ejβu l  0    ejβu l  − Zu  ejβu l Zu

V1b e−(γb d)N

V1a e(γa d)N

V2b e−(γb d)N

V2a e(γa d)N

I1b e−(γb d)N

−I1a e(γa d)N

I2b e−(γa d)N

−I2a e(γa d)N

V1b e(γb d)N

(6.8)

(6.9)



 V2b e(γb d)N    b (γ d)N  b −I1 e  −I2b e(γb d)N

(6.10)

121


Table 6.1: Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients a+ m, − + − am , bm and bm for the 4-port scattering theory: column 1 excitation (lower region) (a) hu = 6 mm; + − f (GHz) (γa d), { |am | , |a− (γb d), { |b+ m| } m | , |bm | } 1.0 (0.26 j), {0.50 , 0.03} (4.87 + π j), {0.87 , 0.00} 3.0 (0.95 j), {0.60 , 0.17} (1.19 + π j), {0.78 , 0.00} 5.0 (0.54 + 0.64 j), {0.29 , 0.00} (0.54 − 0.64 j), {0.96 , 0.00} 7.0 (1.38 j), {0.73 , 0.19} (0.85 j), {0.65 , 0.079} (b) hu = 1 mm; − − f (GHz) (γa d), { |a+ | (γb d), { |b+ m , |am | } m | , |bm | } 1.0 (0.44 j), {0.55 , 0.13} (5.09 + π j), {0.82 , 0.00} 3.0 (1.35 + 1.73 j), {0.49 , 0.00} (1.35 − 1.73 j), {0.87 , 0.00} 5.0 (1.31 + 0.53 j), {0.53 , 0.00} (1.31 − 0.53 j), {0.85 , 0.00} 7.0 (1.28 j), {0.91 , 0.34} (1.33 + 0 j), {0.22 , 0.00} (c) hu = 0.2 mm; − − f (GHz) (γa d), { |a+ (γb d), { |b+ m | , |am | } m | , |bm | } 1.0 (0.97 j), {0.66 , 0.37} (5.18 + π j), {0.65 , 0.00} 3.0 (2.46 + 0.65 j), {0.66 , 0.00} (2.46 − 0.65 j), {0.75 , 0.00} 5.0 (1.13 + 0 j), {0.997 , 0.00} (3.16 + 0 j), {0.07 , 0.00} 7.0 (1.25 j), {0.91 , 0.42} (3.04 + 0 j), {0.05 , 0.00}

VM

  VM 1−   V −   M2  =  − V   M3  VM 4−

  am+   b +   m  BP =   a −   m  bm−

    Vs =   

vs −jβl l 2e vs −jβl l 2e

0 vs −jβl l 2 Zl e

      

(6.11)

and 0 is the 1 × 4 zero matrix. Solving the system (6.7) for the components VP allows one to obtain the generalized scattering parameters: p Zj Sij = + √ Vj Zi Vi−

(6.12) Vk+ =0

for k6=j

Generalized scattering parameters [41] are required due to the fact that the upper and lower mode impedances are typically different. As mentioned previously, in addition to S-parameters, n o + , a− , b− , the MTL analysis also yields the relative excitation strengths of each Bloch mode, a+ , b m m m m which are contained in BP (6.11). In general, all of the Bloch modal coefficients are non-zero, but by examining which coefficients are dominant, one is able to obtain an intuitive, physical understanding of the resulting S-parameters. The modal coefficients are given for the case of lower mode excitation in Table 6.1, while the corresponding results for upper mode excitation are given in Table 6.2. It is noted that the S-parameters corresponding to an upper region excitation are calculated in a similar manner, and will not be shown here.


122

Table 6.2: Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients a+ m, − + − am , bm and bm for the 4-port scattering theory: column 2 excitation (upper region) (a) hu = 6 mm; + − f (GHz) (γa d), { |am | , |a− (γb d), { |b+ m| } m | , |bm | } 1.0 (0.26 j), {0.96 , 0.05} (4.87 + π j), {0.28 , 0.00} 3.0 (0.95 j), {0.95 , 0.26} (1.19 + π j), {0.15 , 0.00} 5.0 (0.54 + 0.64 j), {0.95 , 0.00} (0.54 − 0.64 j), {0.32 , 0.00} 7.0 (1.38 j), {0.91 , 0.24} (0.85 j), {0.30 , 0.18} (b) hu = 1 mm; − − f (GHz) (γa d), { |a+ | (γb d), { |b+ m , |am | } m | , |bm | } 1.0 (0.44 j), {0.68 , 0.16} (5.09 + π j), {0.72 , 0.00} 3.0 (1.35 + 1.73 j), {0.97 , 0.00} (1.35 − 1.73 j), {0.26 , 0.00} 5.0 (1.31 + 0.53 j), {0.82 , 0.00} (1.31 − 0.53 j), {0.57 , 0.00} 7.0 (1.28 j), {0.78 , 0.29} (1.33 + 0 j), {0.55 , 0.00} (c) hu = 0.2 mm; − − f (GHz) (γa d), { |a+ (γb d), { |b+ m | , |am | } m | , |bm | } 1.0 (0.97 j), {0.21 , 0.12} (5.18 + π j), {0.97 , 0.00} 3.0 (2.46 + 0.65 j), {0.88 , 0.00} (2.46 − 0.65 j), {0.48 , 0.00} 5.0 (1.13 + 0 j), {0.78 , 0.00} (3.16 + 0 j), {0.63 , 0.00} 7.0 (1.25 j), {0.76 , 0.34} (3.04 + 0 j), {0.56 , 0.00}

Some typical S-parameter results, for both lower and upper region excitations are shown in Figures 6.2 (hu = 6 mm), 6.3 (hu = 1 mm), and 6.4 (hu = 0.2 mm). The geometries are identical to those used in Chapter 4 to verify the MTL dispersion theory, and N = 7 unit cells are considered. For convenience, each of the figures also have the corresponding dispersion curves calculated from MTL theory, with the band-edges, f1 , f2 , f3 , and f4 , and the complex mode band edges, fc1 and fc2 defined as they were before. It is noted that the MTL theory curves have excellent correspondence with those obtained from FEM simulations. Even for Figure 6.4, where S31 , S41 , S32 , S42 are all ≈ −150 dB from approximately 2 to 3 GHz, which would be out of the range of typical experimental measurements, both sets of curves match exceedingly well. These results, in combination with the excellent correspondence of the MTL and FEM dispersion curves from Chapter 4, provide further evidence of the validity and accuracy of the MTL theory, and its ability to capture the response of the shielded structure. It is observed that for all of the heights, hu = 6, 1, and 0.2 mm, the structures are in a band with one propagating FW (forward wave) mode, and one EW (evanescent wave) mode at 1 GHz. For a lower region excitation, with hu = 6 mm, the transmission coefficients to the opposite side of the structure, S31 ≈ −11.8 dB, and S41 ≈ −7.2 dB, indicate that the FW mode is not well matched. This is borne out from the relative excitation strengths of the Bloch modes, given in Table 6.1, where it is observed that the FW mode (γa ) has a modal coefficient,

123

Chapter 6. Scattering Analysis + |a+ m | = 0.5, while the EW (γb ) mode has modal coefficient, |bm | = 0.87.

However when the structure is excited in the upper region S42 ≈ −2.63 dB, indicating that the majority of the power is transmitted from the near (excited) side upper region to the far side upper region. The corresponding modal coefficient (from Table 6.2) for the FW (γa ) is + |a+ m | = 0.96, while for the EW (γb ) it is |bm | = 0.28, confirming that in this case the FW is

excited in a dominant manner. These results are consistent with the fact that the FW mode has modal power largely confined to the upper region, and hence in order to strongly excite it, one would need an excitation mechanism which encompasses the upper region. Interestingly, though, for both hu = 1 mm and hu = 0.2 mm S42 is not dominant, and most of the power is reflected back into port 2. This can be explained by recalling from Chapter 5 that in the low frequency limit, the FW mode is characterized by an effective per-unit-length 0

0

0

0 inductance, L0ef f = Lu + Ll and capacitance, Cef f = Cu , and hence characteristic impedance, s 0 0 Lu + Ll Zt = . The impedance of the upper region (port) mode, on the other hand, is Cu0 s 0 Lu Zu = . When hu hl then Zt ≈ Zu and the impedances are approximately equal, Cu0 indicating a well matched structure. However when hl is comparable to, or greater than hu ,

there exists a significant impedance mismatch, indicating that even though the structure is in a FW pass-band the FW mode is not well matched. For the structure considered, hl = 3.1 mm, and hence for hu = 6 mm, the matching is reasonably good, while for both hu = 1 mm and hu = 0.2 mm the structures are not as well matched, resulting in lower transmission for S42 . At 3 GHz, the structure with hu = 6 mm is still in a FW pass-band with results qualitatively similar to those obtained at 1 GHz. However, for both hu = 1 mm and hu = 0.2 mm, the structures support complex modes with (γa d)(hu =1 mm) = 1.35 + 1.73 j, (γb d)(hu =1 mm) = 1.35 − 1.73 j, and (γa d)(hu =0.2 mm) = 2.46+0.65 j, (γb d)(hu =0.2 mm) = 2.46−0.65 j. From Table 6.1, for a lower region excitation, for hu = 1 mm, γa has modal coefficient |a+ m | = 0.49, while γb has modal coefficient |b+ m | = 0.87, indicating that both complex modes with exponential decay are excited, but in an asymmetric manner. This is revealed by the fact that S21 = −2.23 dB, indicating that the majority of the power incident on port 1 (lower region) circulates up into port 2 (upper region). For the upper region excitation (Table 6.2), both exponentially decaying complex modes are again excited, but the dominant one now is γa , while γb is more weakly excited. This is consistent with the fact that complex modes with complex-conjugate propagation constants have oppositely directed real power flow at any fixed point on the structure’s cross-section [38], and hence the location of the excitation determines which of the complex-conjugate pairs is dominantly excited. For hu = 0.2 mm the complex modes are again excited in an asymmetric manner, but S21 does not dominate, and in fact the majority of the input power is reflected

124


(a)

(b)

(c)

(d)

(e)

Figure 6.2: Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 6 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3.

125


(a)

(b)

(c)

(d)

(e)

Figure 6.3: Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 1 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3.

126


(a)

(b)

(c)

(d)

(e)

Figure 6.4: Dispersion and corresponding four-port scattering curves comparing the MTL analysis with FEM simulations for an N = 7 unit cell cascade with hu = 0.2 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3.


127

back into port 1, with S11 dominant. This is due to the fact that complex modes, in addition to having complex propagation constants, are characterized by complex impedances, and as α increases (as it does for hu = 0.2 mm relative to hu = 1 mm), the port impedances would need to take on significant real and imaginary parts in order to dominantly excite and match a single complex mode. However, as the port impedances are real, this does not occur. Continuing at 3 GHz, the transmission coefficients S31 , S41 , S42 , and S32 are ≈ −100 dB for hu = 1 mm, and ≈ −150 dB for hu = 0.2 mm, indicating that very little power is transmitted from the near to the far side along any possible four-port power path, confirming that within the complex mode band the structure does indeed support a stop-band for transmission through a cascade of cells. It is also observed that as both exponentially decaying complex modes are excited (although unequally), resonant-like dips appear in the scattering parameters. These dips are observed in the complex mode band for the scattering parameters S31 , S41 , S32 , and S42 as seen in Figures 6.2 through 6.4. More pronounced versions of these dips will be seen and explained in the subsequent two-port scattering analysis. In summary, within the complex mode band, the structure is observed to act in a manner similar to standard band-gap structures as far as energy transmission across a cascade of unit cells is concerned. However, in contrast to standard band-gap structures, the complex mode band allows for the leakage of energy from port 1 to port 2 and vice versa, a property which has been exploited in the construction of a compact directional coupler [15, 16]. At 5 GHz the structure with hu = 6 mm enters a complex band, and the results are qualitatively similar to those obtained for hu = 1 mm at 3 GHz. An examination of Figures 6.2b and 6.2d show that S21 = S12 are clearly dominant over the entirety of the complex band. As the structure with hu = 6 mm has the smallest attenuation, α, in the complex band, it leads to a better match for the transmission coefficients, S21 = S12 , since the port impedances are purely real. For the structure with hu = 0.2 mm at 5 GHz a pair of EW modes are supported, with γa d = 1.13 + j 0 and γb d = 3.16 + j 0. Only the mode, γa d, with the smaller decay constant, αa d = 1.13 is significantly excited. It is also observed from Figures 6.4c and 6.4e that between fc2 (the upper limit of the complex mode band) and f3 (the commencement of a pass-band), the transmission coefficients, S31 , S41 , S42 , and S32 are smooth, which is a signature of the excitation of standard EW modes, and in contrast to the complex mode band. The transmission from the excited side of the structure to the opposite side is small. Hence, regarding energy transmission through a cascade of unit cells, within both the complex mode band and evanescent mode band, a stop-band is confirmed. At 7 GHz all of the structures are again in pass-bands, and hence the transmission coefficients S31 , S41 , S42 , and S32 are no longer small. However, the modal structure is more


128

complicated than at 1 GHz (where the low frequency slow wave was supported), and hence the transmission coefficient S42 for hu = 6 mm is no longer clearly dominant. However, it is again noted that the FEM simulations and the MTL analysis results have excellent correspondence.

6.3

Application to 2D microstrip grid excitation

In Section 1.2.2 the 2D loaded microstrip transmission line grid from [26–28] was reviewed. The structure considered in that work was modeled using standard periodic transmission line analysis, with a fundamental backward wave (BW) mode predicted from the TL model. Full wave FEM simulations of the structure revealed that in addition to the BW mode, a FW mode was also supported, so that the structure was in fact a dual-mode structure. This structure was shown to behave as an effective medium with negative material parameters in the region of the dispersion where the BW is supported. However, the frequency band which supports the BW mode also contains the FW mode. In the work of [26–28] the FW mode was not accounted for; however it was demonstrated that if sources were located in the region between the microstrip lines and the ground plane (akin to the lower region), the resulting excitation could be described well by considering the BW mode alone. It was noted in Section 1.2.3 that the dispersion curves of the shielded Sievenpiper structure were similar to those of the 2D microstrip grid. Subsequently it was seen that the multiconductor transmission line model of the shielded Sievenpiper structure is able to account for both the FW and the BW mode. Thus by using the MTL model of the shielded structure to represent the dispersion of the 2D microstrip grid, an analytical confirmation of the use of standard TL theory, as in [26–28], is obtained. It was observed in [26–28] that a large BW bandwidth could be obtained by increasing the values of the loading elements L and C with the use of discrete components. Subsequently, the analysis of the shielded structure using MTL theory revealed that a large BW bandwidth generally required a large upper region height. To that end, the shielded Sievenpiper geometry which will be used to generate the dispersion has loading elements, L = 10 nH, C = 4 pF, with 1r = 1, hu = 18 mm, 2r = 5, and hl = 3.1 mm, which yields a BW mode with a significant bandwidth (from 0.4 to 0.9 GHz), as shown in Figure 6.5a. The four-port scattering parameters obtained from MTL theory are shown in Figures 6.5b through 6.5e. It is observed from Figures 6.5b, 6.5c, corresponding to a lower region excitation, that the transmission coefficient, S31 is less than -20 dB up until the commencement of the BW band at f1 . However between f1 and fc1 , the commencement of the complex mode band, S31 is dominant, indicating strong transmission, and hence strong excitation of the BW mode. As f → fc1 , S31 dips slightly, which is due to the fact that in this region the dispersion of the

129


(a)

(b)

(c)

(d)

(e)

Figure 6.5: Dispersion and corresponding four-port scattering curves obtained using MTL analysis for a case where the BW bandwidth is large: L = 10 nH, C = 4 pF, 1r = 1, hu = 18 mm, 2r = 5, and hl = 3.1 mm.

130


FW and the BW mode begin to coalesce, and hence the BW mode is not as tightly confined to the lower region alone. For a large portion of the region between f1 and fc1 all of the other S-parameters for the lower region excitation, S11 , S21 , and S41 are less than -8 dB, indicating that not only is the BW mode dominantly excited, it is well matched as well. Thus the MTL analysis justifies the use of the TL model, for a lower region excitation, in the case where the BW mode is confined in the lower region. This is generally true in the frequency range away from fc1 , where the BW and FW modes coalesce. In that region both the BW and FW modes have significant field concentration in both upper and lower regions, and hence a simple TL model is incapable of accurately capturing both the dispersion and scattering properties. For the upper region excitation, observed in Figures 6.5d, 6.5e, it is seen that the transmission coefficient, S42 is dominant from f = 0 to f = fc1 which is an indication that it is the FW mode and not the BW mode that is dominantly excited. Again there is a dip in the transmission as f → fc1 , with the same reason as that observed for the lower region excitation. In conclusion, it is seen that the MTL model provides an elegant explanation of the validity of the TL model in cases where the modal field strength is confined to certain regions of space. However, due to its generality, the MTL model is able to capture a larger set of excitations, including ones within the complex-mode band, that cannot be accounted for with simple TL models.

6.4

Two-Port Scattering Analysis

The next scattering situation to be examined is a two-port set-up as depicted in Figure 6.6a, which shows a cascade of N unit cells of the shielded structure of length N d. At the two ends of the cascade only the upper conductor (Conductor 1) and the ground conductor are extended a distance, l, and again the port variables are defined at x = −l and x = N d + l. The port variables are defined with respect to the parallel-plate geometry formed by the upper (Conductor 1) and ground conductor, as depicted in Figure 6.7, and this parallel-plate geometry supports a quasi-TEM (2-conductor) standard TL port mode. The geometry is identical to that used to define the transverse cut of the MTL from Figure 3.3, with the exception that Conductor 2 is 0

0

removed. This allows one to write the per-unit-length inductance, Lpp , and capacitance, Cpp , 0

0

0

0

of the port mode, in terms of the variables, Lu , Ll , Cu , and Cl , which were used to define the MTL per-unit-length parameters: 0

0

0

Lpp = Lu + Ll

0

0

Cpp

0

C C = 0u l 0 Cu + Cl

(6.13)

131

Chapter 6. Scattering Analysis x = −l

x=0

Zpp

− VM 1

vs

+ VM 1

d

cell 1

2d

Nd

cell 2

Port modes; modal coefficients: + − VM 1 , VM 1

x = Nd + l − VM 2

cell N

Zpp

Port mode; modal coefficient: − VM 2

Bloch modes; modal coefficients: − + − a+ m , am , bm , bm

MTL

d 2

2C

MTL

d 2

2C

L

(a)

S11

P1+

P2

S21

(b)

Figure 6.6: Two-port scattering: (a) Circuit schematic; (b) Power flow with characteristic impedance, Zpp and propagation constant, βpp given by: s Zpp =

L0pp 0 Cpp

βpp = ω

q 0 L0pp Cpp

(6.14)

Due to the symmetry of the structure the complete scattering parameters can be derived by considering port 1 as the excited port, with port 2 matched. The port variables are the incident, {VM 1+ , IM+1 } and reflected, {VM 1− , IM−1 } {voltage, current} pairs for port 1, and the corresponding reflected quantities, {VM 2− , IM−2 } for port 2. Only 3 of the 6 port variables are independent as the current and voltage quantities are related through Zpp as V + V − V − IM+1 = M 1 , IM−1 = − M 1 , IM−2 = M 2 . Zpp Zpp Zpp The Bloch modes are labeled identically as in the four-port scattering analysis and are given by (6.1). For the total system consisting of the port extensions and the cascade of unit cells n o + , a− , b− there are seven unknown variables in total to be determined VM 1+ , VM 1− , VM 2− , a+ , b m m m m and hence seven equations are needed to uniquely solve the system. The incident voltage, VM 1+ ,

132


d

z

Conductor 1; {V1 , I1 }

y hu

ǫ1

hl

ǫ2 Ground

Figure 6.7: Transverse cut used to define the port variables for the investigated two-port scattering situation. is solved immediately by considering the source boundary conditions: − vs + Zpp I1 + V1 = 0 Substituting I1 =

− VM + 1 −VM 1 Zpp

(6.15)

and V1 = (VM 1+ + VM 1− ) into (6.15) yields: VM 1+ =

vs 2

(6.16)

The transformation of the port variables at x = −l and x = N d + l to the beginning and end of the MTL cascade (at x = 0 and x = N d) is made through the port TL propagation constant, βpp yielding: "

V1 I1

#

  VM 1+ e−jβpp l + VM 1− ejβpp l  =V + − −jβpp l − VM 1 ejβpp l M1 e Zpp Zpp (x=0)   " # VM 2− ejβpp l V1  = V − jβpp l M1 I1 e Zpp

(6.17)

(6.18)

(x=N d)

Having explicitly solved for the incident voltage VM 1+ , six remaining equations are required. Four equations are formed by applying the continuity of the voltage and current on the shielding conductor (Conductor 1), at the transition junctions x = 0 and x = N d. The final two equations are obtained by noting that conductor 2 is open circuited at the transition junctions x = 0 and x = N d. The resulting (6 by 6) system of equations is given by: "

M11 M12 M21 M22

#" # VM BP

=

" # Vs 0

(6.19)

133

Chapter 6. Scattering Analysis where 

M11

−ejβpp l  jβpp l  e =  Zpp 0

M22

 0   0  0

M12

 V a V1b  1 a b =  I1 I1 I2a I2b

 V a e−(γa d)N  1 a −(γa d)N =  I1 e I2a e−(γa d)N

VM =

" # VM 1− VM 2−

V1a

V1b



−I1a

 −I1b  

−I2a

−I2b

M21

V1b e−(γb d)N

V1a e(γa d)N

I1b e−(γb d)N

−I1a e(γa d)N

  0 −ejβpp l   ejβpp l   = 0 −   Zpp  0 0

V1b e(γb d)N

(6.20)



 −I1b e(γb d)N  

I2b e−(γb d)N −I2a e(γa d)N −I2b e(γb d)N   am+   vs   b +  e−jβpp l   BP =  m  Vs =  v2s −jβpp l  a −  e  m  2Zpp bm−

(6.21)

(6.22)

and 0 is the 4 × 1 zero matrix. Solving the linear system, (6.19), allows one to obtain the S-parameters, S11

VM 1− = and VM 1+

n o VM 2− + , b+ , a− , b− . Some typical results for a 5 unit-cell , and the modal coefficients, a m m m m VM 1+ cascade, with hu = 6, 1, and 0.2 mm, are shown in Figure 6.8. The correspondence of the S21 =

MTL and FEM results is very good, although for the structure with hu = 6 mm (Figure 6.8b), the two sets of curves show some discrepancies, especially in the frequency range above 5 GHz. The discrepancy between the MTL analysis and FEM simulations for the 2-port case can be explained by examining the dispersion of the port modes. The port mode is the TM0 mode of the cross-sectional geometry from Figure 6.7. The analytical characterization of this quasi-TEM mode in terms of the static parameters from (6.14) is accurate in the limit ω → 0, but will begin to diverge with increasing frequency. The value of the port impedance using (6.14) is compared with FEM simulated results for f = 1 and 10 GHz, in Table 6.3. For both hu = 0.2 and 1 mm the analytical values, Zpp (analytical) = 85.16 and 116.91 ohms, respectively, are very close to the FEM simulated results at 10 GHz of Zpp (FEM) = 85.78 and 115.7 ohms, respectively, which gives an error of less than 1.1 %. However for hu = 6 mm, the discrepancy between the analytical, Zpp (analytical) = 308.13 Ω and the FEM value at 10 GHz, Zpp (FEM) = 259.24 Ω is 15.9 %, indicating that the port mode exhibits significantly more dispersion for this case, leading to the S-parameter discrepancies observed in Figure 6.8b. The dispersion is due to the presence of an inhomogeneous dielectric in the transverse profile of the port geometry, but for the cases hu = 0.2 and 1 mm, the transverse profile is largely determined by the lower region geometry which has hl = 3.1 mm, with r2 = 2.3, and hence the

134


Table 6.3: Comparison of the port mode impedance, Zpp , calculated using the analytical (static) formula (6.14), with FEM simulated results. Zpp (Ω)

hu = 6mm

hu = 1mm

hu = 0.2mm

Analytical

308.13

116.91

85.16

FEM (1 GHz)

307.55

116.87

85.15

FEM (10 GHz)

259.24

115.7

85.78

effects of dispersion are relatively minor. For the case hu = 6 mm, the lower region’s geometry doesn’t dominate over upper region’s, leading to significant port modal dispersion. The fact that such a discrepancy was not observed in the 4-port scattering results for the the case hu = 6 mm (Figure 6.2) reinforces the point that the discrepancy for the 2-port scattering parameters is due to the port modal dispersion, and not an inadequacy of the MTL model of the shielded structure. For the 4-port case, the port modes are the lower and upper region eigenmodes of the cross-sectional geometry of Figure 3.3. The presence of the patch layer conductor in this case lead to each of eigenmodes having field concentration largely confined to distinct regions (upper and lower) with homogeneous permittivities, and hence the modal dispersion was negligible. In conclusion, it is noted that the modal dispersion due to non-homogeneous transverse permittivity profiles can be modeled using coupled-line models [33–35], but this approach was not pursued in this work. The modal coefficients, associated with each of the two independent Bloch modes, are given in Table 6.4, for 1, 3, 5, and 7 GHz. It is observed that for all considered cases, at 1 GHz there is one FW mode and one EW mode. When hu = 6 mm, the FW (γa ) mode is excited in an extremely dominant manner as |a+ m | = 0.998, with the remaining modal strengths being significantly smaller. This is also confirmed from the S-parameters, with S21 = −0.04 dB, S11 = −20.37 dB. The extremely strong excitation of the FW mode in this case (two-port) is contrasted with that observed in the four-port case, for an upper region excitation (Figure 6.2e), where the corresponding transmission coefficient S42 = −2.63 was much smaller. This can be explained by recalling the modal field profile for the low frequency FW mode from Chapter 5, shown in Figure 5.3. In that figure it is observed that the FW mode has a return current established on the ground conductor, and hence a two-port excitation, which has current on both the upper and ground conductors, provides a better field match, relative to the four-port excitation, which is completely confined to the upper region. Continuing at 1 GHz for hu = 1 mm, the FW mode (γa ) is again dominant, with |a+ m | = 0.96, but the EW mode (γb ) excitation is slightly more pronounced, with |b+ m | = 0.18, which results in a slightly smaller transmission as S21 = −0.56 dB, and S11 = −9.15 dB. When hu = 0.2 mm, the FW mode is still excited strongly, with |a+ m | = 0.71. However, the reflected FW mode also

135


has a significant excitation strength as |a− m | = 0.38, and additionally the decaying EW mode, with |b+ m | = 0.59, is also excited strongly. These effects combine in such a manner that the transmission coefficient, S21 = −5.07 dB, is less than the reflection coefficient, S11 = −1.62 dB, indicating that even though the dispersion corresponds to a FW pass-band, the FW mode is not well matched to the port impedance. At f = 3 GHz, the results are qualitatively similar for the structure with hu = 6 mm, with a dominant FW mode excitation and a well matched transmission. However for both hu = 1 and 0.2 mm, the modes are complex with (γa d)(hu =1 mm) = 1.35 + 1.73 j, (γb d)(hu =1 mm) = 1.35 − 1.73 j, and (γa d)(hu =0.2 mm) = 2.46 + 0.65 j, (γb d)(hu =0.2 mm) = 2.46 − 0.65 j. It is observed that the modes corresponding to exponential decay are excited with equal strengths, |a+ m | = 0.71 and |b+ m | = 0.71, while those with exponential increase are negligibly excited, for both hu = 1 mm and hu = 0.2 mm. This results in a stop-band for the structure with S11 ≈ 0 for both hu = 1 and 0.2 mm. The larger αd value for hu = 0.2 mm shows up in the fact that the transmission coefficient, S21 is smaller in that case, which is seen in comparing Figures 6.8d and 6.8f. The equal strength excitation of the complex modes corresponding to exponential decay, for such a two-port geometry, is not a completely general phenomenon. For example, it does not occur if the product (αd) × (N ) is small. An approximate solution to the system, (6.19), in − − + the case that (αd) × (N ) 1 may be shown to yield |a+ m | = |bm | = 0.71, with |am | = |bm | = 0.

However, in practice, due to the exponential dependence of the system on (αd) × (N ), the + approximate solution |a+ m | = |bm | = 0.71 holds even for the cases considered where the products,

(αd)(hu =1 mm) × (N ) = 6.75 and (αd)(hu =0.2 mm) × (N ) = 12.3, are not significantly larger than unity. The resonant-like dips observed in S21 are a signature for the existence of complex modes, and are a more pronounced version of what was observed in the four-port case (compare Figure 6.8 with Figures 6.2 to 6.4 within the complex mode band). Using the approximation − + − − |a+ m |, |bm | |am |, |bm |, the voltage, V1 (N d), which is proportional to VM 2 , is given by: a (−α−j β)dN b (−α+j β)dN V1 (N d) = a+ + b+ m V1 e m V1 e + In general a+ m and bm are related through a complex phase, such that

similar relation holding for the modal voltage coefficients, (6.23) becomes:

V1a V1b

(6.23) a+ m b+ m

= e−2 jφ4 , with a

= e2jφ1 . Using these relations,

b (−αd N −jφ4 +jφ1 ) j(φ1 −φ4 −βd N ) −j(φ1 −φ4 −βd N ) V1 (N d) = b+ V e × e + e m 1 h i b (−αd N −jφ4 +jφ1 ) = b+ V e 2 cos(φ − φ − βd N ) 1 4 m 1

(6.24)

For a given cascade N is fixed, but βd varies as the frequency is swept through the complex mode

136


(a) hu = 6 mm

(c) hu = 1 mm

(e) hu = 0.2 mm

(b) hu = 6 mm

(d) hu = 1 mm

(f) hu = 0.2 mm

Figure 6.8: Dispersion and corresponding two-port scattering curves comparing the MTL analysis with FEM simulations for a N = 5 cell structure: (a) & (b) hu = 6 mm; (c) & (d) hu = 1 mm; (e) & (f) hu = 0.2 mm. All of the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, r1 = 1, r2 = 2.3.

137

Chapter 6. Scattering Analysis band. The transmission zeroes correspond to frequencies for which (φ1 −φ4 −βd N ) =

π 2

(2 n+1).

For small values of hu , βd is swept from the value π to 0, over a narrower band, corresponding to more closely spaced transmission zeros on the S-parameter curves, as is observed in Figure 6.8. It is emphasized again, that these resonant dips occur only in the complex mode band, and are a unique signature of its existence. For f = 5 GHz, the structure with hu = 6 mm has entered a complex mode band, while the hu = 1 mm structure remains within a complex mode band, with results which are qualitatively identical to those discussed above. However for hu = 0.2 mm the structure has entered a band defined by pairs of independent evanescent modes, with γa d = 1.13 + j 0 and γb d = 3.16 + j 0. Only the mode, γa d, with the smaller decay constant, αa d = 1.13 is significantly excited, with |a+ m | = 0.998, which also occurred in the four-port case. The resulting smooth shape of S21 curve reflects that typically encountered when standard evanescent modes are excited, with the resonant dips observed in the complex mode case eliminated. It is also observed that there is a significantly more pronounced variation of S21 in the EW mode band in comparison to the complex mode band. From a qualitative viewpoint, for both the two-port and the four-port scattering analysis, the transmission across a cascade of unit cells is small when exciting the structure within the frequency range spanned by the complex mode band and the evanescent mode band. This confirms that both the complex band and the evanescent bands act as stopbands for the structure.

6.5

Summary

In this chapter the scattering characteristics of a cascade of unit cells of the shielded structure were examined. The scattering analysis extends the applicability of the MTL model by accounting for the more realistic situation were a finite number of unit cells are excited. The scattering analysis is connected with the previously obtained dispersion analysis in multiple ways. It was shown that a general excitation of the structure corresponds to a linear combination of all possible Bloch modes. This confirmed that the dispersion analysis, which describes the propagation properties of a single unit cell, may be extended to model the scattering effects of a cascade of unit cells. The scattering analysis also provides a confirmation of the nature of the modes which exist in the stop-band. Although the MTL model accounted for both pass-bands and stop-bands the FEM eigenmode simulations did not provide the propagation constants within the stop-bands. Driven FEM simulations, on the other hand, which corresponded to various scattering situations, were shown to confirm the modal nature of the stop-band as predicted by the MTL model: both complex modes and pairs of standard evanescent modes were shown to be supported by the structure. The existence of complex modes and


138

Table 6.4: Bloch propagation constants, (γa d) and (γb d), along with the modal coefficients a+ m, − + − am , bm and bm for the two port scattering results depicted in Figure 6.8. (a) hu = 6 mm + − f (GHz) (γa d), { |am | , |a− (γb d), { |b+ m| } m | , |bm | } 1.0 (0.26 j), {0.997 , 0.05} (4.87 + π j), {0.05 , 0.00} 3.0 (0.95 j), {0.992 , 0.04} (1.19 + π j), {0.12 , 0.00} 5.0 (0.54 + 0.64 j), {0.71 , 0.00} (0.54 − 0.64 j), {0.71 , 0.00} 7.0 (1.38 j), {0.98 , 0.14} (0.85 j), {0.10 , 0.13} f (GHz) 1.0 3.0 5.0 7.0

(b) hu = 1 mm − (γa d), |a− (γb d), { |b+ m| } m | , |bm | } (0.44 j), {0.96 , 0.21} (5.09 + π j), {0.18 , 0.00} (1.35 + 1.73 j), {0.71 , 0.00} (1.35 − 1.73 j), {0.71 , 0.00} (1.31 + 0.53 j), {0.71 , 0.00} (1.31 − 0.53 j), {0.71 , 0.00} (1.28 j), {0.94 , 0.35} (1.33 + 0 j), {0.03 , 0.00}

f (GHz) 1.0 3.0 5.0 7.0

(c) hu = 0.2 mm − − (γa d), { |a+ (γb d), { |b+ m | , |am | } m | , |bm | } (0.97 j), {0.71 , 0.38} (5.18 + π j), {0.59 , 0.00} (2.46 + 0.65 j), {0.71 , 0.00} (2.46 − 0.65 j), {0.71 , 0.00} (1.13 + 0 j), {0.998 , 0.00} (3.16 + 0 j), {0.06 , 0.00} (1.25 j), {0.91 , 0.41} (3.04 + 0 j), {0.01 , 0.00}

{ |a+ m| ,

the dominant excitation of a single complex mode, under a particular input, was demonstrated to have relevance in the understanding of a novel directional coupler.

Chapter 7

Shielded structure based slot antenna 7.1

Introduction

Printed slot antennas have been extensively studied due to their low profile, low cost, light weight and ease of fabrication [42]. However, a single slot antenna printed on a thin dielectric substrate is essentially a bi-directional radiator, with the back radiation being undesirable. There have been several methods which successfully demonstrated reduced back radiation. One method is to print the slots on electrically thick substrates or at the back of dielectric lenses. For an infinitesimal slot printed on a quarter dielectric wavelength thick substrate, the front-to-back power density ratio at broadside is equal to the substrate relative permittivity r , 3/2

and for a slot antenna printed on a hemispheric dielectric lens the ratio is r

[43]. The main

drawback of these techniques is the severe loss due to surface wave excitation in the former case, and the lack of a low-profile character in the latter. For an array of slot radiators the above drawbacks may be alleviated by using phase cancellation techniques [44–46], which utilize the proper spacing of array elements to achieve destructive interference of surface wave modes. A common technique employed to restore the back radiation is backing the slot with a metallic cavity (box) [47]. While a uni-directional pattern may be achieved, a drawback of this method is the additional manufacturing difficulty in machining the cavity, especially for array designs. In addition, spurious resonances may be produced, limiting the resulting bandwidth. Another method commonly employed to reduce the back-radiation is the addition of a backing metallic reflector in order to redirect the back radiation forward [48]. The main drawback in this case is that the geometry of the antenna now is transformed into a parallel-plate environment and hence the excitation of the parallel-plate TEM mode degrades the radiation efficiency as well as the antenna patterns. In addition, the reflector must be placed a quarter-wavelength 139

Chapter 7. Shielded structure based slot antenna

140

away from the slot ground-plane for proper operation. Hence the resulting structure is not of a low-profile nature at lower RF frequencies. In order to mitigate the effects of the excited TEM mode in conductor-backed slot antennas various types of periodically loaded structures, also referred to as electromagnetic band-gap (EBG) structures have been utilized. Within the stop-bands of these structures electromagnetic wave propagation is prohibited [49], and it can be expected that if a slot antenna is designed to resonate in the stop-band of a properly designed EBG structure, with a complete 2D (omni-directional) band-gap, the parallel-plate mode becomes evanescent. Hence, in such an arrangement the radiation front-to-back ratio and the patterns should be improved compared to the case of a simple conductor-backed slot. Such an approach has been utilized in [50, 51]. In [50], a square lattice of holes was drilled in a substrate in order to create an EBG structure. In this case though, the periodic pattern of the substrate does not greatly perturb the underlying parallel-plate environment, and thus the position of the stop-band is essentially determined by the lattice spacing, hence compromising compactness at low microwave and RF frequencies. This becomes evident by noting that the unit cell periodicity in [50] is 1.2 cm when the band-gap lies between 7.5 and 10 GHz. In addition, the achievable bandwidth for the stop-band is smaller than that of the structure proposed in this chapter. In [51] a twodimensional EBG surface was constructed by etching a periodic metallic pattern and utilized to back a microstrip-fed slot antenna. The resulting perforated ground plane is prone to leakage through radiation, and hence an additional ground plane must be added behind the perforated plane thus adding to fabrication complexity and possibly limiting the useful bandwidth. The shielded Sievenpiper geometry has been demonstrated to be a suitable metallic EBG structure which is not prone to radiation leakage, as it has a solid ground plane, and can also maintain a compact geometry when implemented at frequencies in the range of 3 to 5 GHz. The configuration to be studied in this chapter consists of a coplanar waveguide (CPW) fed ring-slot antenna, which is chosen to maintain compactness, and is printed on the upper shielding conductor of the shielded Sievenpiper structure. By operating within the stop-band, it is expected that most of the energy radiated by the slot into the shielded structure will be redirected back into the region above the shielding plane, resulting in a uni-directional antenna. This EBG based antenna may be designed to maintain a low profile, with an overall thickness mush less than a quarter wavelength (

λ 4 ),

which is another advantage when compared to

the standard conductor backed antenna. In the next section the design of the underlying shielded structure will be detailed, followed by the antenna design, with experimental results and conclusions following.

141

Chapter 7. Shielded structure based slot antenna d d hua

ǫ1a

hub

ǫ1b

hl

ǫ2

wb

w

via; r (b) Top view, as seen below the Shielding conductor

(a) Side view

Figure 7.1: Unit cell underlying the proposed slot antenna; hua = 1.54 mm, 1a−rel = 4.5, hub = 1.5 mm, 1b−rel = 1, hl = 3.1 mm, 2−rel = 2.3, r = 0.25 mm, w = 9.6 mm, wb = 8.8 mm, d = 10 mm.

7.2

Design of the underlying shielded geometry

The proposed unit cell of the underlying shielded structure is depicted in Figure 7.1. From Figure 7.1b, it is observed that the patch layer squares are modified, with small square regions cut out at their edges. This was done in anticipation of experimental transmission measurements, which will be detailed later in this chapter. However, as the deformation from the original square geometry is small, such a change was observed to have a negligible effect on the applicability of the MTL analysis developed in Chapter 4. It is noted that the upper region of the structure is comprised of two layers; one with height hua = 1.54 mm, and 1a−rel = 4.5; the other with hub = 1.5 mm, and 1b−rel = 1. The air-filled region is held in place by using low permittivity dielectric spacers at the edges of the constructed structure. Previously it was noted that smaller upper region heights yield larger stop-bands and larger attenuation constants within the stop-band, which would be desirable in the suppression of the back directed radiation. Using this as guidance, the geometry that was initially considered was identical to that of Figure 7.1, but with the hub layer removed. Indeed, such a structure was determined to have both a larger band-gap and attenuation constant, using the MTL analysis. However, it presented a drawback, as it was determined that it was difficult to match the proposed antenna. It was observed that by slightly increasing the upper region height, the attenuation constant remained adequate, and additionally the antenna was easier to match. The structure depicted in Figure 7.1 remains compact, and was used in the construction of the antenna. It is recalled from Chapter 3 that the capacitance matrix, C0 , which is used in the determina0

0

tion of the MTL dispersion, is composed of elements, Cu and Cl , which are the per-unit-length

142


(scalar) capacitances of parallel-plate geometries consisting of the upper and lower regions alone. However, for all previously considered geometries (Figure 3.3), the upper region (height hu ) consisted of a uniform dielectric with permittivity 1 , with the lower region (height hl ) also a uniform medium with dielectric constant, 2 . For the structure proposed in Figure 7.1 0

the upper region is composed of two uniform layers, and hence the expression for Cu needs to 0

be changed appropriately. The modified (non-uniform) upper region capacitance, Cu (n.u.) is 0

0

formed from the series combination of two per-unit-length capacitances, Cu (hua ) and Cu (hub ), 1a d hua 1b d 0 Cu (hub ) = hub 0

Cu (hua ) =

(7.1) (7.2)

0

with Cu (n.u.) given by

1 1 1 = 0 + 0 Cu (n.u.) Cu (hua ) Cu (hub ) 0

(7.3)

With this modification the previously developed MTL analysis may be applied to the present structure. The loading components, calculated to be L = 1.15 nH, and C = 0.23 pF are determined using scattering analysis as shown in Chapter 3. The MTL theory dispersion curves are compared with FEM generated results in Figure 7.2. It is observed that there is excellent correspondence between the two sets of curves, with a band-gap for on-axis propagation extending from approximately 2.59 to 5.09 GHz. In order to verify that an omni-directional stop-band is achieved, a full two-dimensional Brillouin diagram has been generated using FEM simulations, with the results depicted in Figure 7.3. The dispersion diagram is shown for the irreducible Brillouin zone; the portion of the curve from Γ to X has 0 ≤ βx d ≤ π, with βy d = 0 fixed; from X to M , βx d = π is fixed with 0 ≤ βy d ≤ π; finally for M to Γ, both βx d = βy d vary from π to 0. It is observed that a complete omnidirectional band-gap exists from approximately 2.5 to 5.0 GHz. An experimental determination of the stop-band of the designed structure was performed next, with a schematic of the test method shown in Figure 7.4. The constructed shielded structure was ten by ten unit cells (Figure 7.4a). Additionally, a parallel plate geometry, with the mushroom surface replaced by a solid ground plane was also considered (Figure 7.4b). Two holes are drilled into the top conductor so that co-axial probes may be placed about 8 cm apart, forming two (coaxial) ports. The outer conductor of the co-axial probe is connected to the shielding plate and the inner conductor goes through the structure, (and through the gaps cut out of the patch corners) and is connected to the solid ground plane. From this arrangement the transmission parameter S21 was experimentally measured, with the resulting data shown in Figure 7.4c for the Γ − X direction, and Figure 7.4d for the M − Γ direction.


143

Figure 7.2: Comparison of MTL theory with FEM generated dispersion curves for on-axis propagation for the geometry of Figure 7.1.

Figure 7.3: FEM simulated Brillouin diagram for the shielded structure of Figure 7.1 showing a complete omni-directional band-gap between approximately 2.5 and 5 GHz.

144


co-axial probes

Port 1

Port 2

8 cm (a) Shielded geometry of Figure 7.1. co-axial probes

Port 1

Port 2

8 cm (b) Parallel plate geometry. 0

S21 (dB)

−20

−40

−60 S : EBG 21

−80

S21: PP −100

2

3

4 Frequency (GHz)

(c)

5

6

(d)

Figure 7.4: Coaxial excitation of: (a) the shielded structure, and (b) a parallel-plate geometry (with the mushroom structure replaced with a solid ground plane), for the purpose of measuring the transmission, S21 ; Measured S21 for the shielded structure, and for the flat conductor backed parallel-plate structure for: (c) the Γ − X direction; (d) the Γ − M direction.


145

Along the Γ − X direction (Figure 7.4c), for the shielded structure (labeled EBG), S21 is in the range of -10 to -30 dB from 2 to about 2.7 GHz, where it dips to the range of -80 dB. It is seen that the experimentally determined stop-band in the Γ − X direction was measured to lie from about 2.7 to 4.7 GHz, where the end of the stop-band is not as clearly defined as the initial 60 dB dip at the beginning of the stop-band. In comparison, for the parallel plate geometry (labeled P.P.), there is no discernable stop-band with S21 remaining between -10 and -40 dB throughout the frequency range shown. The frequency where the stop-band begins is seen to be very close to the value determined by the MTL model, which was at 2.59 GHz from Figure 7.2. In order to determine the stop-band in the M − Γ direction another shielding conductor was used with two holes drilled diagonally along the M − Γ direction. The results for these measurements are shown in Figure 7.4d. It is observed that the onset of the stop-band is again at about 2.7 GHz, but the end of the stop-band is pushed slightly higher in frequency, to about 5.2 GHz. It is noted that the excitation for the experimentally determined transmission coefficients excites cylindrical waves, and additionally the edges of the structure are not matched; these are the main factors which lead to the slight discrepancies between the FEM simulated dispersion diagram from Figure 7.3 and the experimental results of Figure 7.4.

7.3

Antenna Design

An unbacked CPW-fed ring-slot antenna was designed initially, using FEM simulations. The substrate used in the simulation had a thickness of 1.54 mm (identical to the substrate for the shielded structure-based antenna), with a relative permittivity of 4.5 followed by a semiinfinite region of free-space. This structure was used as the reference antenna. It is again noted that the reference antenna has no backing ground plane. The ring-slot was designed to have its second resonance in the center of the band-gap of the designed shielded structure. The measured return loss of the antenna without any backing (not in the shielded structure environment) exhibited a resonance at 3.8 GHz, which was well within the band-gap of the shielded structure. The antenna which is used to test the effectiveness of the EBG concept is simply the reference antenna backed by the unshielded Sievenpiper structure (with 1.5 mm spacers between as described previously). From here on the unshielded Sievenpiper structure will be referred to as the EBG surface. The ring-slot antenna with its relevant dimensions is shown in Figure 7.5. The antenna was fabricated on a substrate of size 20 x 20 cm with the EBG surface comprising ten by ten unit cells, as mentioned previously. The placement of the EBG surface relative to the antenna substrate is also shown in Figure 7.5. The measured return loss of the antenna with the EBG surface backing exhibited a resonance at 3.9 GHz. FEM simulations were also performed for

146


Antenna ground plane: 200 mm x 200 mm

70 mm Outline of the EBG surface/ Flat Conductor backing; both are 100 x 100 mm, centered on the slot

Outer radius: 30 mm

45 mm

3.0 mm

24.6 mm

55 mm

Central conductor width: 1.3 mm CPW slot width: 0.130 mm

12.3 mm

Central conductor width: 1.34 mm CPW slot width: 0.090 mm (matching network)

28.1 mm

Central conductor width: 1.3 mm CPW slot width: 0.130 mm

(a) Top view.

λο 12

2.5 λο (b) Cross-sectional view.

Figure 7.5: (a) Ring slot antenna fed by a CPW line, with the shielded structure’s placement c IEEE 2005). (b) Cross-sectional view of the shown as a dotted line (from [9], geometry with approximate size of the slot’s ground plane and the overall height given in terms of free space wavelengths. the EBG backed antenna. Figure 7.6 shows the experimentally measured and FEM simulated results for S11 of the EBG-backed antenna. The FEM simulation resonance frequency has been slightly shifted (less than 1.3%) to 3.95 GHz, from 3.90 GHz for the experimental measurement. This small deviation is most likely due to deviations in the substrate’s relative permittivity. It is also noted that the bandwidth referred to the 10 dB return loss is measured and simulated as 5%.

7.4

Antenna pattern results and discussion

As mentioned previously, the goal is the demonstration of a uni-directional single element slot radiator, since on a thin substrate, the slot radiates nearly equally on both sides so that the

147

Chapter 7. Shielded structure based slot antenna −5 Measured FEM

S11 (dB)

−7

−9

−11

−13

3.7

3.8

3.9 Frequency (GHz)

4.0

c IEEE 2005). Figure 7.6: S11 of the shielded structure-based slot antenna (from [9], front-to-back ratio is close to 0 dB. The normalized patterns of the reference ring-slot antenna (the antenna of Figure 7.5 without the backing EBG surface) are shown in Figure 7.7, where it is clearly seen that the radiated power is bi-directional. It is observed that the FEM generated patterns match the measured patterns very well. The lack of a null over the horizontal (ground) plane in the H-plane is attributed to the finite size of the antenna ground plane. Furthermore, the finite size of the ground plane manifests itself in the E-plane ripples, which are due to edge diffraction. A finite metal plate of size 10 x 10 cm was used as a plane reflector, and it was placed a quarter wavelength (20 mm) from the ring-slot’s ground plane so that the reflected fields would add in phase to the forward directed fields. The corresponding normalized patterns are shown in Figure 7.8. It is seen that the front-to-back ratio is improved to approximately 8 dB at broadside. In addition, distinct undulations appear in the back-radiated E-plane pattern. This relatively low front-to-back ratio is most likely due to radiation of the trapped parallel-plate mode which diffracts from the edges of the plates. Suppression of this mode must be achieved in order to increase the front-to-back ratio and smooth out the patterns. The unwanted effect of this parasitic radiation should be eliminated if the trapped parallelplate mode is forbidden from propagating in the underlying geometry, which is achieved with the shielded structure. The corresponding patterns when backing the ring-slot antenna with the EBG surface are depicted in Figure 7.9. It is observed that the experimentally measured and FEM simulated patterns match very well again. As shown, the front-to-back ratio is approximately 20 dB at broadside. The relative gain improvement at broadside in both the E

148

Chapter 7. Shielded structure based slot antenna 2.5 λο

λο 50 60

120

-10

-20

150

60

120

-10 30

150

FEM co-pol.

30

-20 -

FEM cross-pol. 0

180

180

0

Exp. co-pol. 210

330

240

300 270

E-plane

210

330

240

Exp. cross-pol.

300 270

H-plane

Figure 7.7: Measured and FEM simulated normalized radiation patterns of the reference ringslot antenna; f = 3.8 GHz. and H planes was experimentally measured to be between 2.5 and 2.9 dB over the frequency range spanning from 3.8 to 4.0 GHz, which is close to maximum possible 3 dB gain. In addition, the H-plane co-polarization is much smoother than either the reference antenna or the conductor backed antenna. Furthermore, there is less ripple in the E-plane co-polarization when compared to the reference antenna, or the conductor backed antenna (see Figures 7.7 and 7.8). These results constitute a substantial improvement over the patterns of the conductorbacked ring-slot of Figure 7.8. In addition to the considerably improved front-to-back ratio, the EBG-backed antenna structure is much more compact, as its total thickness is only 6.14 mm compared to 21.5 mm for the quarter-wavelength conductor-backed geometry. The E-plane cross-polarization level, which was low for the reference slot, remains low and is in fact reduced, while the cross-polarization level in the H-plane is also reduced. Previous related work [50, 51] had shown front-to-back ratios in the range of 8 to 15 dB, and hence the 20 dB front-to-back ratio observed in this work is a definite improvement. However for fairness it should be pointed out that in [50] the size of the substrate was approximately 4 free space wavelengths and for [51] it was about 0.6 free space wavelengths. For our antenna it is 2.5 free space wavelengths, indicating that a good compromise between compactness and attenuation of the trapped TEM mode has been achieved.

7.5

Conclusions

In this chapter we have demonstrated a uni-directional ring-slot antenna with smooth patterns and improved front-to-back ratio (approximately 20 dB) for both the E and H planes when compared to previous related work. The measured relative gain improvement compared to an

149


2.5 λο

λο 4 120

60

150

60

120

30

150

FEM co-pol.

30

FEM cross-pol. 180

0

180

0

Exp. co-pol. 330

210

300

240

Exp. cross-pol.

330

210

240

270

300 270

E-plane

H-plane

Figure 7.8: Measured and FEM simulated normalized radiation patterns of the reference ringslot antenna backed with a conductor at one quarter wavelength; f = 3.7 GHz.

2.5 λο

λο 12 60

120

-10

-20

150

60

120

-10 30

-20

150

FEM co-pol.

30

FEM cross-pol. 180

0

0

180

Exp. co-pol.

330

210

240

300 270

E-plane

210

330

240

Exp. cross-pol.

300 270

H-plane

Figure 7.9: Measured and FEM simulated normalized radiation patterns of the reference ringslot antenna backed with the EBG; f = 3.9 GHz.


150

unbacked ring-slot antenna was between 2.5 and 2.9 dB over the frequency span of 3.8 to 4.0 GHz. This was achieved by using the shielded Sievenpiper structure as an underlying EBG structure which supports an omni-directional stop-band. Full-wave simulations were shown, which matched well the experimental data for both the radiation patterns and the shielded structure’s stop-band. Compared to the standard quarter wavelength conductor-backed configuration, the EBG-backed ring-slot offers an improved front-to-back ratio (from approximately 8 dB to 20 dB) and compactness of the antenna both laterally and in terms of thickness (75% size reduction). It has thus been demonstrated that the use of a properly designed EBG structure can lead to a compact unidirectional slot antenna at RF frequencies.

Chapter 8

Conclusions 8.1

Summary of Contributions

In this thesis, periodic transmission line (TL) analysis, which is applicable for loaded 2-conductor geometries, was extended and generalized to the case of loaded, coupled (n + 1)-conductor geometries, using multiconductor transmission line (MTL) theory. Standard periodic TL analysis can account for only a single mode of propagation, and hence is inadequate for structures which support bands with multiple coupled modes. Using MTL theory, a concise analytical description and intuitive understanding of the wave propagation properties of such multi-mode structures was obtained. In particular, the shielded Sievenpiper structure was studied in depth. The shielded Sievenpiper structure and several other topologically related structures, both shielded and unshielded, have been shown to be useful in a wide variety of applications, including noise suppression in digital circuits, the creation of uni-directional slot antennas, the analysis of artificial media, including slow-wave structures and negative refractive index media, and in the understanding and analysis of novel compact coupled-line couplers. Although the developed theory is applicable for all of the above mentioned applications, the specific geometry of the shielded Sievenpiper structure admits a particularly simple description in terms of the two distinct regions of the structure: the upper region and the lower region. The underlying electrical parameters which describe this multiconductor system are related to simple parallel-plate geometries, which allows for significant simplification of the analytical model, while still retaining accuracy. The above factors made the shielded Sievenpiper structure an ideal candidate as a canonical example of the developed theory. Multiconductor transmission line theory was used to determine the dispersion behaviour of the shielded Sievenpiper structure under changing geometric parameters. When the height of the upper region was sufficiently large, the structure was observed to support a dual mode 151

Chapter 8. Conclusions

152

initial pass-band, with a stop-band formed from the contra-directional coupling of a forward wave (FW) mode and a backward wave (BW) mode. The nature of the modes in the stopband were not typical: they were described by pairs given by complex conjugate propagation constants, which are referred to as complex modes. The properties of complex modes were reviewed. As the upper region height was decreased, the dispersion of the FW mode became flatter, indicating a slow wave effect, and the bandwidth of the BW mode decreased. For small enough upper region heights, the qualitative nature of the dispersion curves changed, and the BW mode was eliminated, resulting in an initial band which contained only a single FW mode. The theory also revealed that for such cases the stop-band consisted of a union of regions defined by complex modes, in addition to regions described by standard evanescent modes. Critical points, which characterize band transitions, were analytically determined, leading to a deeper understanding of the parameters which controlled the shape of the dispersion curves. It was shown that some of the critical points were dependent on properties of the upper or lower regions alone, while others were dependent on combinations of both. Additionally, a physical explanation of the transition from dual mode to single mode behaviour was obtained by examining these critical points. In the low frequency limit the theory revealed that the structure supported a slow FW mode. By examining the propagation constant of this mode, along with the modal eigenvector, an interesting physical explanation of this effect was suggested. In particular, it was shown that the slow wave effect was due to a per-unit-length capacitance enhancement, with the per-unitlength inductance remaining invariant. These results were related to the simple geometrical parameters which characterize the structure. In addition to the periodic analysis, various forms of scattering analysis were examined. By considering a finite cascade of unit cells of the shielded structure, with different types of excitation mechanisms, generalized scattering parameters were obtained using the MTL analysis. The results were shown to have excellent correspondence with FEM simulations, over a wide frequency range, including pass-bands, complex mode bands, and evanescent wave bands. The different excitations corresponded to different applications, and these were noted. A slot antenna utilizing the shielded Sievenpiper structure was designed and constructed. By operating the antenna in the frequency range defined by the band-gap of the structure, a uni-directional slot antenna was demonstrated. Measurements were made, and compared with those of an un-backed slot antenna, and a conductor-backed slot antenna. The shielded structure based slot antenna was more compact, and observed to have a larger broadside gain than the conductor-backed antenna. Having listed the specific contributions, the overarching contribution of this work will be described now. Periodically loaded transmission line theory has long been a useful tool in


153

the analysis of electromagnetic wave propagation problems. However it is limited in that it can only model a single mode of propagation. The periodic multiconductor transmission line analysis developed in this thesis provides an extension to standard periodic transmission line theory, as it is capable of modeling multi-mode behaviour. However, it was shown that even in cases where the dispersion of the structure is single mode, as it can be for the shielded Sievenpiper structure, a simple TL model is not adequate. In such cases MTL analysis is still needed to accurately account for the dispersion, especially in the stop-band, where complex modes are supported. The analysis method developed in this thesis, in addition to providing an extension to TL analysis which is capable of accurately capturing multi-mode behaviour, will provide researchers with an awareness of the usefulness of the multi-modal viewpoint, even in structures which nominally support single (propagating) mode bands.

8.2

Publications

Parts of the work presented in this thesis have appeared in the publications listed below.

Refereed Journal Papers 1. F. Elek and G.V. Eleftheriades, “Dispersion analysis of the shielded Sievenpiper structure using multiconductor transmission-line theory,” IEEE Microwave and Wireless Component Letters, vol. 14, no. 9, pp. 434-436, Sep. 2004. 2. F. Elek, R. Abhari and G.V. Eleftheriades, “A uni-directional ring-slot antenna achieved by using an electromagnetic band-gap surface,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 1, pp. 181-190, Jan. 2005. 3. F. Elek and G.V. Eleftheriades, “A two-dimensional uniplanar transmission-line metamaterial with a negative index of refraction,” New Journal of Physics; Focused Issue on Negative Refraction, vol. 7, no. 163, pp. 1-18, Aug. 2005. 4. R. Islam, F. Elek and G.V. Eleftheriades, “Coupled-line metamaterial coupler having codirectional phase but contra-directional power flow,” Electronics Letters, vol. 40, no. 5, pp. 315-317, Mar. 2004. 5. M. Stickel, F. Elek, J. Zhu and G. V. Eleftheriades, “Volumetric negative-refractive-index metamaterials based upon the shunt-node transmission-line configuration,” Journal of Applied Physics, vol. 102, p. 094903, Nov. 2007. Refereed Conference Proceedings


154

1. F. Elek and G.V. Eleftheriades, “Simple analytical dispersion equations for the shielded Sievenpiper structure,” IEEE International Microwave Symposium, San Francisco, CA, Jun. 2006, pp. 1651-1654. 2. F. Elek and G.V. Eleftheriades, “On the slow wave behaviour of the shielded mushroom structure,” IEEE International Microwave Symposium, Atlanta, GA, Jun. 2008, pp. 1333-1336.

Appendix A

Shielded structure based antenna compared with a cavity-backed antenna In Chapter 7 a shielded structure based slot antenna (also referred to as an EBG-backed antenna) was demonstrated to have a superior front-to-back ratio when compared to both a conductor-backed antenna and a reference (un-backed) slot antenna. This was confirmed with both pattern measurements and FEM simulations. The EBG-backed antenna achieves the improved front-to-back ratio by suppressing the parallel-plate mode which is supported by the conductor-backed design. Additionally, the EBG-backed design also has a much lower profile λ ( 12 ) when compared to the conductor-backed antenna ( λ4 ).

The parallel-plate mode may also be suppressed by backing a slot antenna with a metallic cavity, which will be investigated in this Appendix with the aid of FEM simulations. Two different cavity-backed designs are considered: for the first one the cavity depth is λ4 , while for the second the cavity depth is

λ 12 .

The FEM simulated results for the cavity-backed radiation

patterns are shown in Figure A.1, along with EBG-backed design. Comparing the EBG-backed antenna with the

λ 4

cavity-backed design it is observed that similar front-to-back ratios are

observed for both the E and H planes. However, the E-plane cross-polarization is significantly λ larger for the cavity-backed antenna. For the low profile ( 12 ) cavity-backed design the front-

to-back ratios are again comparable to the EBG-backed design; however in this case the crosspolarization levels for both the E and H-planes are significantly higher than the EBG-backed antenna. The degradation of the radiation patterns may be understood by examining the S11 curves, which are shown in Figure A.2. The EBG-backed antenna exhibits a smooth resonance centered 155

Appendix A. Shielded structure based antenna compared with a cavity-backed antenna156 at around 3.9 GHz, while the

λ 4

cavity-backed design shows multiple resonances within the

depicted band. These additional resonances are due to the resonant modes of the cavity and are responsible for the pattern degradation. For the low-profile cavity-backed design, the matching bandwidth is much narrower, which limits it usefulness. Thus a low-profile cavity-backed design suffers from both degraded radiation patterns and also a narrow matching bandwidth. In summary it is noted that the EBG-backed antenna has superior performance when compared to both cavity backed designs, and this is achieved while maintaining a low profile.

Appendix A. Shielded structure based antenna compared with a cavity-backed antenna157 2.5 λο

λο 12 60

120

-10

-20

150

60

120

-10 30

-20

150

FEM co-pol.

30

FEM cross-pol. 180

0

0

180

Exp. co-pol.

330

210

240

210

300

Exp. cross-pol.

330

240

270

300 270

E-plane

H-plane

(a) EBG-backed antenna.

2.5 λο

λο 4 60

120

120

150

60 −10

−10 30

−20

150

30

−20

FEM co-pol. 180

0

180

0

FEM cross-pol.

210

330

240

330

210

300

240

270

300 270

E-plane

H-plane

(b) Cavity-backed antenna 1: cavity depth =

2.5 λο

λο 12 120

60

60

120 −10

−10 150

λ . 4

30

−20

−20

150

30

FEM co-pol. 180

0

180

0

FEM cross-pol.

330

210

240

300 270

E-plane

210

330

240

300 270

H-plane

(c) Cavity-backed antenna 2: cavity depth =

λ . 12

Figure A.1: Normalized radiation patterns for the EBG-backed antenna compared with two cavity-backed antennas.

Appendix A. Shielded structure based antenna compared with a cavity-backed antenna158

−5 Measured FEM

S11 (dB)

−7

−9

−11

−13

3.7

3.8

3.9 Frequency (GHz)

4.0

(a) EBG-backed antenna. −5

S11 (dB)

−10

−15

−20

−25 3.7

3.8

3.9 Frequency (GHz)

4.0

(b) Cavity-backed antenna 1: cavity depth =

λ 4

(FEM).

−5

S11 (dB)

−10

−15

−20

−25 3.7

3.8

3.9 Frequency (GHz)

4.0

(c) Cavity-backed antenna 2: cavity depth =

λ 12

(FEM).

Figure A.2: S11 for the EBG-backed antenna compared with two cavity-backed antennas.

Bibliography [1] L. Brillouin, “Wave guides for slow waves,” J. Appl. Phys, vol. 19, pp. 1023–1041, Nov. 1948. [2] L. Stark, “Electromagnetic waves in periodic structures,” MIT Research Laboratory of Electronics, Tech. Rep. 208, 1952. [3] P. Yeh, A. Yariv, and C-S Hong, “Electromagnetic propagation in periodic stratified media. I. general theory,” J. Opt. Soc. Am., vol. 67, no. 4, pp. 423–438, Apr. 1977. [4] A. Grbic and G.V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys, vol. 92, no. 10, pp. 5030– 5935, Nov. 2002. [5] F. Elek and G.V. Eleftheriades, “Dispersion analysis of the shielded Sievenpiper structure using multiconductor transmission-line theory,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 434–436, Sep. 2004. [6] R. Abhari and G.V. Eleftheriades, “Metallo-dielectric electromagnetic bandgap structures for suppression and isolation of the parallel-plate noise in high-speed circuits,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 6, pp. 1629–1639, Jun. 2003. [7] S. Shahparina and O.M. Ramahi, “A simple and effective model for electromagnetic bandgap structures embedded in printed circuit boards,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 621–623, Oct. 2005. [8] S.D. Rogers, “Electromagnetic-bandgap layers for broad-band suppression of TEM modes in power planes,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2495–2505, Aug. 2005. [9] F. Elek, R. Abhari, and G.V. Eleftheriades, “A uni-directional ring-slot antenna achieved by using an electromagnetic band-gap surface,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 181–190, Jan. 2005. 159

BIBLIOGRAPHY

160

[10] J. Machac, “Microstrip line on an artificial dielectric substrate,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 416–418, Jul. 2006. [11] F. Elek and G.V. Eleftheriades, “On the slow wave behaviour of the shielded mushroom structure,” in Proc. IEEE Int. Microw. Symp., Atlanta, GA, Jun. 2008, pp. 1333–1336. [12] M. Stickel, F. Elek, J. Zhu, and G. V. Eleftheriades, “Volumetric negative-refractive-index metamaterials based upon the shunt-node transmission-line configuration,” J. Appl. Phys, vol. 102, p. 094903, Nov. 2007. [13] J. Zhu and G.V. Eleftheriades, “Experimental verification of overcoming the diffraction limit with a volumetric Veselago-Pendry transmission-line lens,” Phys. Rev. Lett., vol. 101, no. 1, p. 013902, Jul. 2008. [14] S.M. Rudolph and A. Grbic, “Modeling of volumetric negative-refractive-index media using multiconductor transmission-line analysis,” in Applied Computational Electromagnetics Society (ACES) Conference, Niagra Falls, Canada, March 30 - April 4 2008. [15] R. Islam, F. Elek, and G.V. Eleftheriades, “Coupled-line metamaterial coupler having codirectional phase but contra-directional power flow,” Electronics Letters, vol. 40, no. 5, pp. 315–317, Mar. 2004. [16] R. Islam and G.V. Eleftheriades, “Printed high-directivity metamaterial MS/NRI coupledline coupler for signal monitoring applications,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 164–166, Apr. 2006. [17] F. Elek and G.V. Eleftheriades, “Simple analytical dispersion equations for the shielded Sievenpiper structure,” in Proc. IEEE Int. Microw. Symp., San Francisco, CA, Jun. 2006, pp. 1651–1654. [18] D. Sievenpiper, Z. Lijun, R.F.J. Broas, N.G. Alexopoulos, and E. Yablonovitch, “Highimpedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [19] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated with electromagnetic band-gap (EBG) structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2936–2946, Oct. 2003. [20] D.F. Sievenpiper, J.H. Schaffner, H. Jae Song, R.L. Loo, and G. Tangonan, “Twodimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2713–2722, Oct. 2003.

161

BIBLIOGRAPHY

[21] F. Yang and Y. Rahmat-Samii, “A low-profile circularly polarized curl antenna over an electromagnetic bandgap (EBG) surface,” Microwave and Optical Technology Letters, vol. 31, no. 4, pp. 264–267, Nov. 2001. [22] ——, “Reflection phase characterization of the EBG ground plane for low profile wire antenna applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2691–2703, Oct. 2003. [23] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals.

Princeton, NJ:

Princeton University Press, 1995. [24] A. B. Yakovlev, O. Luukkonen, C. R. Simovski, S. A. Tretyakov, S. Paulotto, P. Baccarelli, and G. W. Hanson, Metamaterials and Plasmonics: Fundamentals, Modelling, Applications.

Springer Netherlands, 2009, ch. Analytical Modeling of Surface Waves on High

Impedance Surfaces, pp. 239–254. [25] L. Li, B. Li, H.X. Liu, and C.H. Liang, “Locally resonant cavity cell model for electromagnetic band gap structures,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 90–100, Jan. 2006. [26] A.K. Iyer and G.V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in Proc. IEEE Int. Microw. Symp., vol. 2, Seattle, WA, Jul. 2002, pp. 1067–1070. [27] G.V. Eleftheriades, A.K. Iyer, and P.C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [28] A. Grbic and G.V. Eleftheriades, “Periodic analysis of a 2-D Negative Refractive Index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2604– 2611, Oct. 2003. [29] V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of and µ,” Soviet Physics Uspekhi, vol. 10, no. 4, pp. 509–514, Jan.-Feb. 1968. [30] S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed.

New York, NY: John Wiley & Sons, 1994.

[31] K. Inafune and E. Sano, “Theoretical investigation of artificial magnetic conductors using effective medium model and finite-difference time-domain method,” Jpn. J. Appl. Phys., vol. 45, no. 5A, pp. 3981–3987, 2006.

162

BIBLIOGRAPHY

[32] J.A. Brandao Faria, Multiconductor Transmission-Line Structures. New York, NY: John Wiley & Sons, 1993. [33] D.F. Noble and H.J. Carlin, “Circuit properties of coupled dispersive transmission lines,” IEEE Trans. Circuit Theory, vol. 20, no. 1, pp. 56–64, Jan. 1973. [34] H.J. Carlin, “A simplified circuit model for microstrip,” IEEE Trans. Microw. Theory Tech., vol. 21, no. 9, pp. 589–591, Sep. 1973. [35] H.J. Carlin and P.P. Civalleri, “A coupled-line model for dispersion in parallel-coupled microstrips,” IEEE Trans. Microw. Theory Tech., vol. 23, no. 5, pp. 444–446, May 1975. [36] J. Brown, “Propagation in coupled transmission line systems,” Quarterly Journal of Mechanics and Applied Mathematics, vol. 11, no. 2, pp. 235–243, 1958. [37] P. Chorney, “Power and energy relations in bidirectional waveguides,” MIT Research Laboratory of Electronics, Tech. Rep. 396, 1961. [38] M. Mrozowski, Guided Electromagnetic Waves.

Taunton, Somerset, England: Research

Studies Press Ltd., 1997. [39] J. A. Brando Faria, “On the transmission matrix of 2n-port reciprocal networks,” Microwave and Optical Technology Letters, vol. 33, no. 3, May 2002. [40] J.R. Silvester, “Determinants of block matrices,” The Mathematical Gazette, vol. 84, no. 501, pp. 460–467, Nov. 2000. [41] D.M. Pozar, Microwave Engineering, 3rd ed.

Hoboken, NJ: John Wiley & Sons, 2005.

[42] G. V. Eleftheriades and G.M. Rebeiz, “Self and mutual admittance of slot antennas on a dielectric half-space,” Int. J. Infrared Millimeter Waves, vol. 14, no. 10, pp. 1925–1946, Oct. 1993. [43] G.M. Rebeiz, “Millimeter-wave and terahertz integrated circuit antennas,” IEEE Proc., vol. 80, no. 11, pp. 1748–1770, Nov. 1992. [44] R.L. Rogers and D.P. Neikirk, “Use of broadside twin element antennas to increase efficiency on electrically thick dielectric substrates,” Int. J. Infrared Millimeter Waves, vol. 9, no. 11, pp. 949–969, Nov. 1988. [45] G.V. Eleftheriades and M. Qiu, “Efficiency and gain of slot antennas and arrays on thick substrates for mm-wave applications: a unified approach,” IEEE Trans. Antennas Propag., vol. 50, no. 8, pp. 1088–1098, Aug. 2002.

163

BIBLIOGRAPHY

[46] M. Qiu, M. Simcoe, and G.V. Eleftheriades, “High-gain meander-less slot arrays on electrically thick substrates at millimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 2, pp. 517–528, Feb. 2002. [47] J.D. Kraus, Antennas.

New York, NY: McGraw-Hill, 1950.

[48] Y. Yoshimura, “A microstripline slot antenna,” IEEE Trans. Microw. Theory Tech., vol. 20, no. 11, pp. 760–762, Nov. 1972. [49] L. Brillouin, Wave Propagation in Periodic Structures.

New York, NY: Dover, 1953.

[50] J. D. Shumpert, W. J. Chappell, and L. B. Katehi, “Parallel-plate mode reduction in conductor backed slots using electromagnetic bandgap substrates,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2099–2104, Nov. 1999. [51] J. Y. Park, C.C. Chang, Y. Qian, and T. Itoh, “An improved low-profile cavity-backed slot antenna loaded with 2D UC-PBG reflector,” in Proc. IEEE Int. Symp. Antennas and Propag., Boston, MA, Jul. 2001, pp. 194–197.

THEORY AND APPLICATIONS OF MULTICONDUCTOR ... - TSpace

THEORY AND APPLICATIONS OF MULTICONDUCTOR ... - TSpace

Suggest Documents

THEORY AND APPLICATIONS OF MULTICONDUCTOR ...www.researchgate.net › file.PostFileLoader.html

Nucleation Theory and Applications

Microwave Theory and Applications