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THEORY AND APPLICATIONS OF MULTICONDUCTORTRANSMISSION LINE ANALYSIS FOR SHIELDED SIEVENPIPER AND

RELATED STRUCTURES

by

Francis Elek

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Copyright c© 2010 by Francis Elek

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 analyt-

ical 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 multi-

conductor transmission line analysis accurately captures. It is shown that under a continuous

change of geometrical parameters, the dispersion curves for the shielded structure are trans-

formed 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 struc-

ii

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 assis-

tance 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 particu-

lar 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.

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

1.2.5 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 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

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

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

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

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

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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

4.4 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) . . . . . . . . . . . . . 89

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) . . . . . . . . . . . . . . . . . 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 139

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

vi

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

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

7.4 Antenna pattern results and discussion . . . . . . . . . . . . . . . . . . . . . . . 146

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Conclusions 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

L′

Per-unit-length inductance

C′

Per-unit-length capacitance

Zo Transmission line characteristic impedance

V Voltage

I Current

Z Impedance

Y Admittance

ε Permittivity

µ Permeability

εo 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

L′ Per-unit-length inductance matrix

C′ 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

Q′ Per-unit-length conductor charge vector

Ψ′ Per-unit-length flux-linkage vector

Z′ Per-unit-length longitudinal impedance matrix

Y′ Per-unit-length transverse admittance matrix

I′ Identity matrix

C′u Upper-region per-unit-length capacitance

L′u Upper-region per-unit-length inductance

C′l Lower-region per-unit-length capacitance

L′l 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

Γ′ Propagation constant matrix

Z′w Characteristic wave impedance matrix

Y′w Characteristic wave admittance matrix

vφ Phase velocity

vg Group velocity

λ wavelength

x

List of Tables

3.1 Comparison of the numerical (FEM) and analytic C′ (capacitance) matrices for:

(a) hu = 18 mm, (b) hu = 6, (c) hu = 0.5 mm. The analytic C′ matrix is

calculated for two different values of the effective width, weff = 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, (γad) and (γbd), along with the modal coefficients

a+m, a−m, b+m and b−m for the 4-port scattering theory: column 1 excitation (lower

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


a+m, a−m, b+m and b−m 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


a+m, a−m, b+m and b−m 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Dispersion curve of the Sievenpiper mushroom structure using the surface impedance

approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Dispersion diagram of the Sievenpiper mushroom structure generated from a

FEM simulation (from [25], c© IEEE 2006). . . . . . . . . . . . . . . . . . . . . . 8

1.6 Two-dimensional loaded microstrip grid. . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Typical dispersion relation described by (1.2) for on-axis propagation with βyd =

0 (fixed), and βxd varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Full wave FEM simulation of an NRI grid for on-axis propagation with βyd = 0

(fixed), and βxd 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, εr1 = 1, εr2 = 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 propa-

gation with βyd = 0 (fixed), and βxd 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 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 Sieven-

piper 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, εr1 = 1, εr2 = 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,

εr1 = 1, εr2 = 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 = 12E×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 dis-

persion 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 = 12E×H∗ on the same transverse cut, but with view rotated. . . . . . 35

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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 = 12E×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 as-

suming 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 define the per-unit-length capacitance and inductance matrices, C′

and

L′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Boundary value problems used to determine C′11 and L

′11. . . . . . . . . . . . . . 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 εr1 = εr2 = 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 con-

stants, γa = jβ + α and γb = −jβ + α. . . . . . . . . . . . . . . . . . . . . . . . 75

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, εr1 = 1, εr2 = 2.3. . . . . . . . . . 78



3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3. (cont’d) . . . . 79



3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 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-

lustration of the gap capacitive fringing field, ~E (dashed lines) and the current

distribution (solid lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.11 Field patterns corresponding to f1, (βd)x = π. (a) large hu; (b) small hu. The

electric field, ~E (dashed lines) and the current distribution (solid lines) are shown. 95

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, εr1 = 1, εr2 = 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, εr1 = 1, εr2 = 2.3 . . . . . . 101

4.15 Boundary conditions corresponding to the two degenerate modes at f2: (a) Trans-

verse 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

fixed: hl = 1 mm; d = 2 mm; w = 1.9 mm; via radius = 0.1 mm (from [11], c©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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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, εr1 = 1, εr2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . 124


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, εr1 = 1, εr2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . 125

xvi


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, εr1 = 1, εr2 = 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,

ε1r = 1, hu = 18 mm, ε2r = 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,

εr1 = 1, εr2 = 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.1 Unit cell underlying the proposed slot antenna; hua = 1.54 mm, ε1a−rel = 4.5,

hub = 1.5 mm, ε1b−rel = 1, hl = 3.1 mm, ε2−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 geom-

etry (with the mushroom structure replaced with a solid ground plane), for the

purpose of measuring the transmission, S21; Measured S21 for the shielded struc-

ture, and for the flat conductor backed parallel-plate structure for: (c) the Γ−Xdirection; (d) the Γ−M direction. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.5 (a) Ring slot antenna fed by a CPW line, with the shielded structure’s placement

shown as a dotted line (from [9], c© IEEE 2005). (b) Cross-sectional view of the

geometry with approximate size of the slot’s ground plane and the overall height

given in terms of free space wavelengths. . . . . . . . . . . . . . . . . . . . . . . . 146

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

7.7 Measured and FEM simulated normalized radiation patterns of the reference

ring-slot antenna; f = 3.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xvii


ring-slot antenna backed with a conductor at one quarter wavelength; f = 3.7

GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149


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 his-

tory, 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 per-

turbing 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 addi-

tion 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 peri-

odic boundary conditions a dispersion equation is obtained which characterizes the propagation

constant as a function of frequency. The resulting dispersion contains frequency bands support-

ing 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 ra-

tio, while for the switching noise application the modal suppression prevents signal degradation


2D patcharray

2D viaarray

Shielding plane

Ground plane

(a) Shielded Sievenpiper structure. (b) Dispersion curve corresponding to (a).

Throughvias

(c) Suppression of switching noise.

Slotantenna

(d) Uni-directional slot antenna.

Stackedlayers

(e) 3D-stacked metamaterials.

Port 1 Port 2

Port 3 Port 4

(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 relatedstructures for which the theory developed in this thesis can be applied.


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 propagation-

like 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.


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 Zsgiven by the parallel combination of an inductance, L and a capacitance, C:

Zs =jωL

1− ω2LC(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, ω2 = 1LC , 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-


x

y

y

z

hl, ǫ2

d

d 2D patch grid

ground plane

vias

(a) Side view (b) Top view

Figure 1.2: Geometry of the Sievenpiper structure.

L

C

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


0

5

10

15

20

25

30

Resonance frequency

π 2 π 3 πβd

Fre

quen

cy (

GH

z)

TM wavesTE waves

Light εr1

=1

Figure 1.4: Dispersion curve of the Sievenpiper mushroom structure using the surfaceimpedance 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. Concen-

trating 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.


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

1.2.2 Two dimensional loaded microstrip grids

A two-dimensional grid-like structure, which is topologically related to the Sievenpiper struc-

ture, 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 el-

ements 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 L′ and C ′, re-

spectively, with an associated characteristic impedance, Zo =√

L′

C′ , and propagation constant,

βo = ω√L′C ′. 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)


(i) (ii) (iii)

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

x

x

y

y

y

z

n n + 1

n(x)

n + 1(x)n(y)

n + 1(y)

L

2C 2C

2C 2C

Zo Zo

d

2d

2

d

2D grid from (a)

ground plane

(a) Unit cell (dashed): top view (b) Unit cell (dashed): side view

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

Figure 1.6: Two-dimensional loaded microstrip grid.


boundary conditions between nodes n and n+1. Bloch’s Theorem [23] states that the field vari-

ables separated by the periodicity of the unit cell, d, are related by the Bloch propagation con-

stants, βx and βy. Along the x-direction, Vn+1(x) = Vn(x) e−jβxd and In+1(x) = In(x) e−jβxd,

with analogous relations holding along the y-direction. By transforming the voltage and cur-

rent 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 (βxd) + cos(βyd)

= −[2 sin

(βod

2

)− 1ZoωC

cos(βod

2

)][2 sin

(βod

2

)− Zo

2ωLcos(βod

2

)]+ 2 (1.2)

A qualitative understanding of the dispersion equation may be obtained by assuming propa-

gation along the x direction, with βxd varying and βyd = 0 fixed, for which (1.2) takes the

form:

cos(βxd) = F (ω,L′, C ′, L, C) (1.3)

where F is a function of frequency (ω), the host TL parameters (L′, C ′), 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 ≤ βxd ≤ 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 ≤ βxd ≤ 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 ≤ βxd ≤ π, a unique solution for βxd 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 fre-

quency f1, the TL model yields a stop-band, in which the mode is evanescent, with complex

propagation constant given by γxd = αxd+ jβxd = 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, βxd (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, αxd→ 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 dωdβx

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

, is negative.


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

0 0

1

2

3

4

5

6

7

βd

Fre

qu

en

cy (

GH

z)

π

Surface Wave

Backward Wave

FEM simulation

TL Model (BW only)

Light Line

Stopband

f3

f2

f1

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


However, the phase velocity vφ = ωβx

, 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 sup-

ported, 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 stop-

band, 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 βxd = 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, βxd 1 and βyd 1, and these ap-

proximations, with the additional assumption that the interconnecting microstrip TL segments

are also electrically short, βod 1, result in the exact TL dispersion, (1.2) reducing to:

β2x + β2

y = ω2

(L′ − 1

ω2Cd

)(2C ′ − 1

ω2Ld

)(1.4)

In [28] it was shown that (1.4) may be written as:

β2 = ω2µeff εeff (1.5)

with

µeff =(L′ − 1

ω2Cd

)(1.6)

εeff =(

2C ′ − 1ω2Ld

)(1.7)

where µeff and εeff are the effective permeability and permittivity of the structure. The


dispersion equation (1.5) shows that under the conditions βxd 1, βyd 1, and βod 1,

the structures appears homogenous and isotropic. At low enough frequencies it is clear that

both µeff and εeff 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), βxd 1, βyd 1,

are not satisfied at very low frequencies. However as frequency increases and approaches f2,

both µeff and εeff remain negative and the conditions βxd 1, βyd 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 µeff or εeff 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, L′ and C ′, and are given by:

ω22 = min

1

C(L′d),

1L(2C ′d)

(1.8)

ω23 = max

1

C(L′d),

1L(2C ′d)

(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), L′d, 2C ′d. At

this point it is possible to justify the conditions under which the short TL approximation,

βod 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 βod 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 (βod 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).


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 ε2 and an upper region of height hu and permittivity ε1. 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, εr1 = 1, εr2 = 2.3.

For the larger value of hu = 18 mm, the FEM generated dispersion curve shown in Fig-

ure 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.


hu

hl

ǫ1

ǫ2

Patch conductor; w

Shielding conductor; d

Ground conductor

via; r

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 simu-

lations. 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 de-

veloped. 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 pre-

dicting 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


0

2

4

8

10

12

←f1

f2→

f3→

←fTE

←f4

Stop-band

π(βd)x

Fre

quen

cy (

GH

z)

FEMTLLight ε

r1=1

(a) hu = 18 mm

0

2

4

8

10

12

←f1

f2→

f3→

←fTE

←f4

Stop-band

π(βd)x

Fre

quen

cy (

GH

z)

FEMTLLight ε

r1=1

(b) hu = 0.5 mm

Figure 1.10: Dispersion curves for the shielded Sievenpiper structure with two upper regionheights: (a) hu = 18 mm, (b) hu = 0.5 mm. All of the other physical parametersare fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3.Also shown are the curves for the TL(BW) model of Section 1.2.2 (the unshieldedstructure), and the free space light line.


n

n

n + 1

n + 1

Y YL (via)

C =ǫ1w

hud

hu, ǫ1

hl, ǫ2

Zo =

√L′

C ′

Zo Zo

w

d

2d

2

d

Shielding plane

ground plane

(a) Unloaded 2 conductor TL (b) Transformation into a loaded TL

(c) Equivalent TL circuit (d) Composition of Y

Figure 1.11: Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-platetransmission line (TL) upon which the patches and vias act as loading elements.The underlying unloaded TL consists of the shielding plane and the ground planeas 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 elementshown 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,


Figure 1.12: Typical dispersion diagram as predicted by the model in [8], for on-axis propaga-tion with βyd = 0 (fixed), and βxd 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 dis-

appearance 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 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 dual-

mode 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 gen-

erated 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.


hu, ǫ1

hl, ǫ2

shielding plane

ground plane

(i) Side view (ii) Top view (below shield)(a) Negative refractive index (NRI) medium. This structure is topologically related to the

shielded structure, but with an additional inductive element.

hl, ǫ2

ground plane

Coupledmicrostrip lines

(i) Side view (ii) Top view(b) Microstrip coupled-line coupler

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


Res. freq.

π 2 π 3 πβd

freq

uenc

yTMTELight

(i) Surface imp. (Zs) model

(ii) Top view

hl, ǫ2

air

(iii) Side view(a)

(i) TL (BW) model

(ii) Top view

hl, ǫ2

air

(iii) Side view(b)

(i) TL-PP model

(ii) Top view (below shield)

hu, ǫ1

hl, ǫ2

shielding plane

(iii) Side view(c)

(d)

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


1.3 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 multicon-

ductor 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, L′ and C ′, are transformed into n by n matrices, L′ and C′,

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 L′

and C′ 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


the matrices L′ and C′ taking on extremely simple forms, their components related to simple

parallel-plate type geometries. The fact that L′ and C′ 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 L′

and C′ 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 super-

position of Bloch modes will be excited and by examining the scattering parameters along with


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 dual-

mode 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 εr = 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

εr1 = 1 and εr2 = 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


0≤(βd)x≤π

(βd)y = 0(fixed)

x

y

y

z

hu

hl

ǫ1

ǫ2

Patch conductor; w

Shielding conductor; d

Transverse fieldsto be shown

on this yz cut

Ground conductor

via; r

(a) Side view(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 ofpropagation (x), is varied. Modal field plots on the transverse plane at the cell edgeto 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


0

2

4

8

10

12

←f1

f2→

f3→

←fTE

←f4

Stop-band

π(βd)x

Fre

quen

cy (

GH

z)

FEMFEM (TE)TLLight ε

r1=1

(a) hu = 18 mm

1

2

3

4

5

←f1

f2→

f3→

↑[(βd)1 , fmax

]

↑[(βd)2 , fmin

] Stop-band

π(βd)x


r1=1

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

0

2

4

8

10

12

←f1

f2−→f3→

←fTE

←f4

Stop-band

π(βd)x

Fre

quen

cy (

GH

z)


r1=1

(c) hu = 6 mm

1

2

3

4

5

FW 1

BW 1

BW 2

TE

FW 2

←f1

f2→f3→

↑[(βd)1 , fmax

]

↑[(βd)2 , fmin

]Stop-band

π(βd)x


r1=1

(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 arefixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3. Alsoshown are the curves for the TL(BW) model of the unshielded structure, and thefree space light line. Field plots corresponding to the points labeled in (d) will beshown later in this chapter.


0

2

4

8

10

12

FW 1

FW 2←f1

f2→

f3→

←fTE

←f4

Stop-band

π(βd)x

Fre

quen

cy (

GH

z)


r1=1

Figure 2.3: Dispersion curves for the shielded structure with with hu = 0.5 mm. The otherphysical parameters are: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm,εr1 = 1, εr2 = 2.3. Also shown are the curves for the TL(BW) model of theunshielded structure, and the free space light line. Field plots corresponding to thelabeled 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 contra-

directional 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 co-

directional 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.


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 stop-

band 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 = π9 (20), f = 1.33 GHz, (2) A BW1 mode at

(βd)x = 8π9 (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.


y

z

(i) E transverse

x

z

(ii) H transverse (iii) Net Power: +ve

y

z

(a) FW1 mode at f = 1.33 GHz; (βd)x = π9

(20)

y

z

(i) E transverse

x

z

(ii) H transverse (iii) Net Power: -ve

y

z

(b) BW1 mode at f = 3.17 GHz; (βd)x = 8π9

(160)

y

z

(i) E transverse

x

z

(ii) H transverse (iii) Net Power: Zero

y

z

(c) Peak of 1st passband at f = 3.36 GHz; (βd)x ' π9

(88)

Figure 2.4: Transverse modal field plots for the 1st passband of the structure with dispersioncurve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cutat the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 1

2E×H∗

on the same transverse cut, but with view rotated.


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− zplane 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 polar-

ization. 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-


y

z

(i) FW 1 (ii) BW 1 (iii) Band peak

y

z

y

z

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.

y

z

(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


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 byπ2 , 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 = π9 (20), f = 5.50

GHz, (2) A FW TE mode at at (βd)x = π9 (20), f = 5.74 GHz, and (3) A FW2 mode at at

(βd)x = π9 (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 com-

pletely 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 trans-

verse 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


y

z

(i) E transverse

x

z

(ii) H transverse (iii) Net Power: -ve

y

z

(a) BW2 mode at f = 5.50 GHz; (βd)x = π9

(20)

y

z

(i) E transverse

x

z


y

z

(b) FW TE mode at f = 5.74 GHz; (βd)x = π9

(20)

y

z

(i) E transverse

x

z


y

z

(c) FW2 mode at f = 6.21 GHz; (βd)x = π9

(20)

Figure 2.7: Transverse modal field plots for the upper passbands of the structure with dispersioncurve from Figure 2.2d (hu = 6 mm). (i) E and (ii) H viewed on a transverse cutat the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 1

2E×H∗



y

z

(i) E transverse

x

z


y

z

(a) FW1 mode at f = 0.61 GHz; (βd)x = π9

(20)

y

z

(i) E transverse

x

z


y

z

(b) FW2 mode at f = 2.27 GHz; (βd)x = 8π9

(160)

Figure 2.8: Transverse modal field plots for the 1st passband of the structure with dispersioncurve from Figure 2.3 (hu = 0.5 mm). (i) E and (ii) H viewed on a transverse cutat the unit cell edge (y-z plane); (iii) Time averaged Poynting vector, S = 1

2E×H∗


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 = π9 (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 = π9 (20),

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


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 = 8π9 (160), will be compared

with BW1, Figure 2.4b (for hu = 6 mm), which also occurred at (βd)x = 8π9 (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 huvalues. 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 prop-

erties 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. Typ-

ically the loading elements are calculated by considering appropriate scattering simulations,

which usually involve two-port configurations. However, due to the presence of the top shield-

ing 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 neces-

sary, 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 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 transfor-

mation 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) array of strips,

placed between the upper shielding conductor and the ground plane. The direction of propa-

gation is assumed to be along x, 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 y, 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, 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) 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


Side view; dashed lines represent H walls

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

z

x

y

y

(a) Unloaded (b) Loaded

(c) Unloaded (d) Loaded

Figure 3.1: Transformation of an infinite 1-D periodic array of strips, (a) and (c), into an infinite2-D periodic array of isolated patches (b) and (d). Vias connected from the centerof each patch to ground for (b) and (d). The transverse boundary conditions areassumed to be H-walls for the case of on-axis propagation in the MTL model.


xx

11

p

p

n

n

00

I1

Ip

In

Io

V1

Vp

Vn

(a) (b)

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

as:

Vp(x, t) =∫po

E(x, y, z, t) · ds (3.1)

and the current on the p− th conductor becomes:

Ip(x, t) =∮sp

H(x, y, z, t) · ds (3.2)

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]:

− ∂

∂xV(x, t) = L′

∂

∂tI(x, t) (3.3)

− ∂

∂xI(x, t) = C′

∂

∂tV(x, t) (3.4)

where V and I are n component column vectors which define the voltages and currents on

the n lines. L′ and C′ 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

system. The matrix C′ relates the charge per-unit length, Q′p, on conductor p linearly to the


y

z

hu

hl

ǫ1

ǫ2

w

d

Conductor 1; V1, I1

Conductor 2; V2, I2

Ground

Figure 3.3: Parameters defining the unloaded MTL geometry for on-axis propagation assumingtransverse 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 todefine the per-unit-length capacitance and inductance matrices, C

′and L

′.

voltages, Vj on all of the other conductors:

Q′p =

n∑j=1

C′pjVj (3.5)

or in matrix form:

Q′ = C′V (3.6)

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

1 through n. The matrix, L′ relates the per-unit length flux, Ψ′p, linking conductor p to the

ground conductor, linearly to the currents, Ij on all of the other conductors:

Ψ′p =

n∑j=1

L′pjIj (3.7)

or in matrix form:

Ψ′ = L′I (3.8)

where Ψ′ 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

dxV(x) = jωL′ I(x) (3.9)

− d

dxI(x) = jωC′V(x) (3.10)

where the longitudinal impedance matrix, Z′ , and the transverse admittance matrix, Y′, are


y

z

huhu

hlhl

dd

V1

V2 = 0

I1

I2 = 0

(a) C′11

(b) L′11

Figure 3.4: Boundary value problems used to determine C′11 and L

′11.

defined by Z′ = jωL′ and Y′ = jωC′. For an unloaded 2 + 1 conductor system such as that

considered here, C′ and L′ are given explicitly by:[Q′1

Q′2

]= C′

[V1

V2

]=

[C′11 C

′12

C′21 C

′22

][V1

V2

](3.11)

and [Ψ′1

Ψ′2

]= L′

[I1

I2

]=

[L′11 L

′12

L′21 L

′22

][I1

I2

](3.12)

The components of C′ and L′ are obtained by solving a set of boundary value problems. For

example the C′11 is obtained by solving the following electrostatic boundary value problem:

C′11 =

Q′1

V1; V2 = 0 (3.13)

while L′11 is obtained by solving the following magnetostatic problem:

L′11 =

Ψ′1

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

C′11 =

ε1weffhu

, where weff = w + ∆w is the effective width which is used to account for the

fringing fields, and∆ww 1 is small. The remaining components of C′ may be similarly


calculated, resulting in:

C′ =

ε1weffhu

−ε1weffhu

−ε1weffhu

ε1weffhu

+ε2weffhl

. (3.15)

It is interesting to note that the components of C′ are given in terms of the simple expressions,

C′u =

ε1weffhu

and C′l =

ε2weffhl

, which are the capacitance-per-unit-length formulas for parallel-

plate waveguides of width, weff , and heights hu and hl, respectively; that is parallel-plate

waveguides of a structure consisting of the upper and lower regions guides alone.

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 εr1 = 1 and εr1 = 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, weff = 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, weff = 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 weff = d, (3.15) becomes:

C′(weff = d) =

ε1d

hu−ε1dhu

−ε1dhu

ε1d

hu+ε2d

hl

=

[C′u −Cu ′

−Cu ′ Cu′ + Cl

′

](3.16)

The components of L′ may be similarly calculated, as is the L′11 component in (3.14). However,

there exists a relationship between L′ and C′. Recalculating C′ with all of the permittivities

replaced by that of free space, results in C′ε=ε0 , with this matrix related to L′ as follows [32]:

L′C′ε=ε0 = ε0µ0I (3.17)

where I is the identity and L′ is obtained by inverting (3.17), resulting in:

L′ = ε0µ0

[C′ε=ε0

]−1(3.18)


Table 3.1: Comparison of the numerical (FEM) and analytic C′ (capacitance) matrices for: (a)hu = 18 mm, (b) hu = 6, (c) hu = 0.5 mm. The analytic C′ matrix is calculated fortwo different values of the effective width, weff = 10.0 and 9.6 mm.

(a) hu = 18mm

Capacitance matrix components(pFm

)C′11 C

′12 C

′21 C

′22

FEM 4.92 -4.91 -4.91 70.51analytic (weff = 10 mm) 4.92 -4.92 -4.92 70.58

% difference (FEM & weff = 10 mm) 0 0.2 0.2 0.3analytic (weff = 9.6 mm) 4.72 -4.72 -4.72 67.75

% difference (FEM & weff = 9.6 mm) 3.5 3.8 3.8 4.2

(b) hu = 6mm


)C′11 C

′12 C

′21 C

′22




(c) hu = 0.5mm


)C′11 C

′12 C

′21 C

′22





Using (3.16) and (3.18) results in:

L′ =

µohld +µohud

µohld

µohld

µohld

=

[L′l + L′u L′l

L′l L′l

](3.19)

where the components of L′, L′u =µohud

and L′l =µohld

are again related to those of parallel-

plate guides composed of the upper and lower regions alone. Thus the components of both C′

and L′ 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 C′

will require a numerical solution.

Having obtained expressions for C′ and L′ the system of equations (3.9) and (3.10) may be

solved. Expanding these equations results in:

−dV1

dx= Z1I1 + ZmI2 (3.20)

−dV2

dx= ZmI1 + ZmI2 (3.21)

−dI1dx

= Y1V1 + YmV2 (3.22)

−dI2dx

= YmV1 + Y2V2 (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:

d2V1

dx2− a1V1 − b1V2 = 0 (3.24)

d2V2

dx2− a2V2 − b2V1 = 0 (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:

d4V1

dx4− (a1 + a2)

d2V1

dx2+ (a1a2 − b1b2)V1 = 0 (3.26)


with a similar equation for V2. The coefficients are evaluated to be:

a1 = Y1Z1 + YmZm = −ω2C′uL′u (3.27)

a2 = Y2Z2 + YmZm = −ω2C′l L′l (3.28)

b1 = Z1Ym + Y2Zm = −ω2(C′l L′l − C

′uL′u) (3.29)

b2 = Z2Ym + Y1Zm = 0 (3.30)

Assuming solutions of the form Voe−γz for V1 and V2, the eigenvalue equation for (3.26) is

obtained:

γ4 + (−a1 − a2) γ2 + (a1a2 − b1b2) = 0 (3.31)

which is quadratic equation in γ2, with solutions

γ2a,b =

a1 + a2

2±√

(a1 + a2)2 − 4(a1a2 − b1b2)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:

γa = jω√L′lC′l (3.34)

γb = jω√L′uC

′u (3.35)

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

phase velocities, vl = 1qL′lC′l

and vu = 1√L′uC

′u

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

V2

I1

I2

=

1

Ra

Ya1

RaYa2

Vo and

V1

V2

I1

I2

=

1

Rb

Yb1

RbYb2

Vo (3.36)


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 =V2

V1=γ2a,b − a1

b1(3.37)

The characteristic admittances for conductors 1 and 2, Ya1 and Ya2, are given by:

Ya1 = γaZ2 − ZmRaZ1Z2 − Z2

m

(3.38)

Ya2 =γaRa

Z1Rc − ZmZ1Z2 − Z2

m

(3.39)

with similar formulas for Yb1 and Yb2. Substituting the relevant terms for the geometry under

consideration the modal eigenvectors are given by:

γa = jω√L′lC′l ⇒

V1

V2

I1

I2

=

1

1

01Zl

Vo and γb = jω√L′uC

′u ⇒

V1

V2

I1

I2

=

1

01Zu

− 1Zu

Vo (3.40)

where Zu =√

L′uC ′u

and Zl =√

L′l

C′l

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 C′ and L′ 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.

For the eigenvector corresponding to γa = jω√L′lC′l 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 = VoZl

(Hl 6= 0 and Hu = 0). In a similar manner

it can be shown that for the mode corresponding to γb = jω√L′uC

′u, the upper region fields

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


0

2

4

6

8

π(βd)x

f (G

Hz)

FEM: H−wallsMTL lower: γ

a

MTL upper: γb

Figure 3.5: Dispersion curves of the unloaded geometry.

y

z

(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.


P1+,− P2−

TLTL

Radiation to free space

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 TL 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 de-

embedding 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 Fig-

ure 3.7, the calculated equivalent capacitance, C will consist of both a real part (the capaci-

tance), 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 TL 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


2 4 6 8 10−0.1

0

0.1

0.2

0.3

0.4

f(GHz)C

apac

itanc

e (p

F)

Real(C)imag(C)

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 nu-

merical 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 Fig-

ure 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


M1+,−

M2−

M3−

M4−

MTL(l)MTL(l)

Physicalport plane 1:

x = −l


x = l0− 0+

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

M1+,−

M2−

M3−

M4−

MTL(l)MTL(l)


x = −l


x = l0− 0+

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

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


M1 and the upper region mode, labeled as M2. The plane x = l is designated as the physical

port plane 2, which also supports two modes: the lower region mode labeled as M3 and the

upper region mode labeled as M4.

For the situation considered in Figure 3.9 the lower region mode, M1 at x = −l is excited,

with the other three modes, M2, M3, and M4 matched. The quantities describing each of

the port modes M1, M2, M3, and M4 are given by 2 component voltage and current column

vectors. These column vectors are simply the lower region (M1 and M3) and upper region (M2

and M4) eigenvectors, as given in (3.40). The total mode 1 voltage and current vectors, VM1

and IM1 are comprised of incident, +, and reflected, − components as given by:

VM1 = V +M1 + V −

M1 IM1 = I +M1 + I −M1 (3.41)

where V +M1 =

[1

1

]V +M1 , I +

M1 =

01Zl

V +M1 , and V −

M1 =

[1

1

]V −M1 , I −M1 =

0

− 1Zl

V −M1 . The

scalar variables V +M1 and V −

M1 are incident and reflected amplitudes. The upper region mode

at x = −l, M2 is matched, so that only a reflected component is present. Thus the total mode

2 voltage and current vectors VM2 and IM2 are given by:

VM2 = V −M2 IM2 = I −M2 (3.42)

where VM2 =

[1

0

]V −M2 , and I −M2 =

− 1Zu1Zu

V −M2 . The total voltage, V(x=−l) = V −

M1 + V +M1 +

V −M2, and current, I(x=−l) = I −M1 + I +

M1 + I −M2, at physical port plane 1 is given in component

form as: V1

V2

I1

I2

(x=−l)

=

V +M1 + V −

M1 + V −M2

V +M1 + V −

M1

−V−

M2

ZuV +M1

Zl− V −

M1

Zl+V −M2

Zu

(3.43)

For the physical port plane 2 at x = l only the reflected lower and upper region modes are

present, with amplitudes, V −M3 and V −

M4 , respectively. The total voltage, V(x=l) = V −M3+V −

M4,


and current I(x=l) = I −M3 + I −M4 is given in component form as:

V1

V2

I1

I2

(x=l)

=

V −M3 + V −

M4

V −M3

V −M4

ZuV −M3

Zl− V −

M4

Zu

(3.44)

The transfer matrix, TC , which relates the voltage and current across the series capacitance,

C is given by: V1(0−)

V2(0−)

I1(0−)

I2(0−)

= TC

V1(0+)

V2(0+)

I1(0+)

I2(0+)

=

1 0 0 0

0 1 0 ZC

0 0 1 0

0 0 0 1

V1(0+)

V2(0+)

I1(0+)

I2(0+)

(3.45)

where ZC = 1jωC .

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:

Sij =V −i√Zj

V +j

√Zi

∣∣∣∣∣V +k =0 for k 6=j

(3.46)

By solving the linear system obtained from (3.45) the generalized scattering parameters are

found and given by:

S11 =V −M1

V +M1

=Zu

Zl + 2jωCZlZu + Zu(3.47)

S21 =V −M2

V +M1

√ZlZu

= − ZuZl + 2jωCZlZu + Zu

√ZlZu

(3.48)

S31 =V −M3

V +M1

=(2ωCZu − j)Zlj

Zl + 2jωCZlZu + Zu(3.49)

S41 =V −M4

V +M1

√ZlZu

=Zu

Zl + 2jωCZlZu + Zu

√ZlZu

(3.50)


2 4 6 8 10−0.05

0

0.05

0.1

0.15

0.2

0.25

f(GHz)

Cap

acita

nce

(pF

)

Real(C)imag(C)

(a) hu = 6 mm

2 4 6 8 10−0.05

0

0.05

0.1

0.15

0.2

0.25

f(GHz)

Cap

acita

nce

(pF

)

Real(C)imag(C)

(b) hu = 0.5 mm;

Figure 3.10: The calculated series gap capacitance, C, for (a) hu = 6 mm, and (b) hu = 0.5mm.

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 C′ and L′. 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


hu

hl

ǫ1

ǫ2

V1 = +V V2 = −V

Vshield (arbitrary)

Ground

g

Figure 3.11: Two-dimensional electrostatic boundary value problem used to obtain the chargeaccumulation 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), nearthe 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) >


(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 = QnetV1−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 =V −M1

V +M1

=−jZl

Zlj − 2ωL(3.51)

S21 =V −M2

V +M1

√ZlZu

= 0 (3.52)

S31 =V −M3

V +M1

= − 2ωLZlj − 2ωL

(3.53)

S41 =V −M4

V +M1

√ZlZu

= 0 (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.


M1+,−

M2−

M3−

M4−

MTL(l)MTL(l)


x = −l


x = l0− 0+

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

2 4 6 8 10

0

0.2

0.4

0.6

0.8

f (GHz)

Indu

ctan

ce (

nH) real(L)

imag(L)

(a) hu = 6 mm

2 4 6 8 10

0

0.2

0.4

0.6

0.8

f (GHz)

Indu

ctan

ce (

nH) real(L)

imag(L)

(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 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, C′ and

L′ 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 C′ and L′ 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 induc-

tance, 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 propa-

gation, 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 multi-

conductor 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 respec-

tively 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


n n + 12 n + 1

V1,n, I1,n

V2,n, I2,n

V1,n+1, I1,n+1

V2,n+1, I2,n+1

2L2L 2C2C

MTL(

d2

)MTL

(d2

)

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 =

1 0 0 0

0 1 0 Z2C

0 0 1 0

0 0 0 1

T2L =

1 0 0 0

0 1 0 0

0 0 1 0

0 Y2L 0 1

(4.1)

where Z2C = 1jω2C and Y2L = 1

jω2L . The values of C and L are determined by appropriate

scattering simulations as explained in Chapter 3. The transfer matrix, TMTL [32] of a section

of a multiconductor transmission line of length l is characterized completely in terms of the

per-unit-length matrices, C′ and L′, with TMTL given by:

TMTL =

[cosh(Γl) sinh(Γl) Zw

Yw sinh(Γl) Yw cosh(Γl) Zw

](4.2)

where [V(0)

I(0)

]= TMTL

[V(l)

I(l)

]. (4.3)

The matrix, Γ, given by

Γ =√

Z′Y′ (4.4)

is defined so that the following holds:

Γ2 = Z′Y′ (4.5)


The computation of Γ involves diagonalizing Z′Y′, and is not obtained by simply taking the

square root of each individual component of Z′Y′. The result is:

Γ =

jω√L′uC ′u −jω√L ′uC

′u + jω

√L′lC′l

0 jω√L′lC′l

(4.6)

Writing Γ in diagonalized form gives:

Γ = P

jω√L′uC ′u 0

0 jω√L′lC′l

P−1 (4.7)

where

P =

[1 1

0 1

]and P−1 =

[1 −1

0 1

](4.8)

The quantities, cosh(Γl) and sinh(Γl) are hyperbolic trigonometric functions of a matrix argu-

ment, 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−1Z′ = Γ(Y′)−1 (4.9)

and

Yw =(Z′)−1Γ = Y′Γ−1 (4.10)

The component form of TMTL for the shielded Sievenpiper structure is given by:

TMTL =

cos(θu) − cos(θu) + cos(θl) j

(sin(θu)Zu + sin(θl)Zl

)j sin(θl)Zl

0 cos(θl) j sin(θl)Zl j sin(θl)ZljYu sin(θu) −jYu sin(θu) cos(θu) 0

−jYu sin(θu) j(Yu sin(θu) + Yl sin(θl)

) − cos(θu) + cos(θl) cos(θl)

(4.11)

where Zu = 1Yu

=√

L′uC ′u

and Zl = 1Yl

=√

L′l

C′l

are the characteristic impedances of parallel-

plate waveguides consisting of the upper and lower regions alone, as determined in the previous

chapter. The arguments of cos and sin are θu = ω√L′uC

′u l, and θl = ω

√L′lC′l 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


those at node n+ 1, Vn+1 and In+1, through a propagation constant γd:[Vn+1

In+1

]= e−γd

[Vn

In

](4.12)

The transfer matrix of the unit-cell is given by:

Tunit−cell−MTL = T2C TMTL T2L T2L TMTL T2C =

[Af Bf

Cf Df

](4.13)

where [Vn

In

]= Tunit−cell−MTL

[Vn+1

In+1

](4.14)

For TMTL, 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−MTL

[Vn+1

In+1

]= eγd

[Vn+1

In+1

](4.15)

Thus the eigenvalues of Tunit−cell−MTL 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−MTL from (4.13), (4.15) yields:

[Af − Ieγd Bf

Cf Df − Ieγd

][Vn+1

In+1

]=

[0

0

](4.16)

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 DfCtf = CfDt

f and from [40] this commutation property allows the block determinant to

be simplified as follows:

DfCtf −CfDt

f = 0⇒ det

[Af Bf

Cf Df

]= det

(AfDt

f −BfCtf

)(4.18)

For lossless networks, AfDtf −BfCt

f = I, which establishes that det(TMTL−unit−cell

)= 1. For


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 sin2

(θu2

)]+

−2 + 4

sin(θl2

)Zl −

cos(θl2

)2ω C

sin(θl2

)Zl

−cos(θl2

)2ω L

− 2 sin

(θu2

)cos(θu2

)ω C Zu

(4.21)

g(ω) =[−2 + 4 sin2

(θu2

)] ·−2 + 4

sin(θl2

)Zl −

cos(θl2

)2ω C

sin(θl2

)Zl

−cos(θl2

)2ω L

+4 sin

(θu2

)cos(θu2

) (1− 2 sin2

(θl2

))ω C Zu

+4Zl sin

(θu2

)cos(θu2

)sin(θl2

)cos(θl2

)ω2C LZu

(4.22)

The dispersion equation, (4.20) describes propagation for the periodic MTL unit cell of Fig-

ure 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 proper-

ties. 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.


4.3 MTL analysis of the shielded structure : Simplified analysis

4.3.1 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

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

and

L′, or alternatively the admittance and impedance per-unit-length matrices, Y

′= jωC

′and

Z′

= jωL′. By incorporating the effects of L and C, two augmented matrices, Y

′Loaded and

Z′Loaded 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(θu2

) → θu2 ,

sin(θl2

)→ θl

2 , cos(θu2

) → 1 , cos(θl2

)→ 1 are substituted into f(ω) (4.21) and g(ω) (4.22),


which simplify to:

fa(ω) = −4 + (ωd)2L′uCu

′ + (ωd)2[L′l −

1ω2dC

] [C′l −

1ω2dL

]− dC

′u

C(4.23)

ga(ω) = −2(ωd)2[L′l −

1ω2dC

] [C′l −

1ω2dL

]+(ωd)4L

′uC

′u

[L′l −

1ω2dC

] [C′l −

1ω2dL

]

+ 4− 2(ωd)2L′uC

′u + 2

C′ud

C− C

′ud

C(ωd)2Ll ′Cl ′ +

Ll′Cu

′

LCd2 (4.24)

If in addition it is assumed that the Bloch phase shift, γd is electrically small, then 2 cosh(γd) =

2(

2 sinh2(γd2 ) + 1)→ (γd)2 + 2, where the approximation sinh(γd2 ) → γd

2 has been applied.

With these approximations, the exact dispersion equation (4.20) simplifies to:

γ4 +(

4 + fa(ω)d2

)γ2 +

(4 + 2fa(ω) + ga(ω)

d4

)= 0 (4.25)

which is a quadratic equation in γ2. 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 ap-

proximations 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 = 1jωLd to the (2, 2) component of Y

′= jωC

′, a new loaded

version of the admittance matrix, Y′Loaded is obtained, resulting in:

Y′Loaded = Y

′+

0 0

01

jωLd

(4.26)

In a similar manner the loaded impedance matrix, Z′Loaded is given by:

Z′Loaded = Z

′+

0 0

01

jωCd

. (4.27)


with the explicit forms of Y′Loaded and Z

′Loaded given by:

Y′Loaded = jω

C′u −C ′u

−C ′u(C′u + C

′l −

1ω2Ld

) =

[Y L

1 Y Lm

Y Lm Y L

2

](4.28)

and

Z′Loaded = jω

L ′u + L′l L

′l

L′l

(L′l −

1ω2Cd

) =

[ZL1 ZLm

ZLm ZL2

](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:

γ4 + (−aL1 − aL2 ) γ2 + (aL1 aL2 − bL1 bL2 ) = 0 (4.30)

The two solutions of (4.30) are:

γ2a,b =

12

[(aL1 + aL2

)±√DiscL] (4.31)

where

DiscL =(aL1 + aL2

)2 − 4(aL1 a

L2 − bL1 bL2

)(4.32)

is the discriminant of the quadratic in γ2, (4.30). The discriminant, DiscL will be shown to be

important in the appearance of complex modes. The parameters, aL1 , bL1 , aL2 , and bL2 are given

by

aL1 = −ω2CuLu (4.33)

bL1 = −ω2

[−LuCu +

(Cl − 1

ω2Ld

)Ll

](4.34)

aL2 = −ω2

[− Cuω2Cd

+(Cl − 1

ω2Ld

)(Ll − 1

ω2Cd

)](4.35)

bL2 = −ω2

[Cuω2Cd

](4.36)

Using (4.33) through (4.36) it can be shown that

− aL1 − aL2 =4 + fa(ω)

d2(4.37)

and

aL1 aL2 − bL1 bL2 =

(4 + 2fa(ω) + ga(ω)

d4

). (4.38)


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

εr1 = εr2 = 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 ≤ minf2, 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:

dγ

dω=γ2 d

dω

(aL1 + aL2

)− d

dω

(aL1 a

L2 − bL1 bL2

)2γ(

2γ2 − (aL1 + aL2 )) (4.39)


(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), withvarying upper region height. (a) hu = 10 mm; (b) hu = 3 mm; (c) hu = 0.75mm. All other parameters are fixed: the lower region height, hl = 3 mm; the upperand lower region relative permittivities are εr1 = εr2 = 4; the loading inductance,L = 1.0 nH; the loading capacitance, C = 0.5 pF.


from which it is observed that dωdγ = 0 when the denominator of (4.39) is set equal to zero. The

critical points are thus given by:

γ = 0 ⇒ f2, f3 (4.40)

and

γ2 =12

(aL1 + aL2 ) ⇒ fc1, fc2 (4.41)

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

aL1 aL2 − bL1 bL2 = ω4C

′l L′lC′uL′u

(1− 1

ω2L(C ′l d)

)[1− 1

ω2C

(1L′l d

+1L′l d

)]. (4.42)

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

ω22 =

1C

(1L′l d

+1L ′ud

)(4.43)

and

ω23 =

1L(C ′l d)

. (4.44)

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

parallel combination of two inductances, Leff1 = L′l d and Leff2 = L

′ud. The inductances Leff1

and Leff2 are due to the distributed MTL per-unit-length inductive parameters of the upper

and lower regions, L′u, and L

′l . 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

lower region height, hl, as hu is decreased, L′u = µohu

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

loading inductance, L and Ceff = C′l d, which is due to the lower region distributed capacitance.

Thus f3 is invariant as the upper region height is altered, as C′l = ε2w

hlis a function of the lower

region height, hl, solely.

The frequencies fc1 and fc2 are obtained when (4.41), γ2 = 12(aL1 + aL2 ) is satisfied. Substi-

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

− 14

(aL1 + aL2

)2+ (aL1 a

L2 − bL1 bL2 ) = 0 (4.45)

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

DiscL =1ω4

(A1ω

8 +B1ω6 + C1ω

4 +D1ω2 + E1

)(4.46)


with coefficients:

A1 =(C′uL′u − C

′l L′l

)2(4.47)

B1 =(C′uL′u − C

′l L′l

)[2L′l

Ld− 2

(C′u − C

′l

)Cd

](4.48)

C1 =(C′u)2(

Cd)2 +

2C′uC

′l(

Cd)2 +

(C′l )2(

Cd)2 − 2C

′uL′u(

Cd)(Ld) − 2C

′uL′l(

Cd)(Ld) +

4C′l L′l(

Cd)(Ld) +

(L′l )

2(Ld)2 (4.49)

D1 = − 2(Cd)(Ld) (C ′u

Cd+C′l

Cd+L′l

Ld

)(4.50)

E1 =1(

Ld)2(

Cd)2 (4.51)

In general, the zeroes of (4.46) will reduce to the solution of a quartic equation in ω2. 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

themselves. However, when the condition ε1 = ε2 is satisfied it can be shown that C′uL′u = C

′l L′l ,

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 ω2. The two solutions are given by:

ω2c1 =

1

LC′l d+

(√LC ′ud+

√CL

′l d)2 (4.52)

and

ω2c2 =

1

LC′l d+

(√LC ′ud−

√CL

′l d)2 (4.53)

From (4.53) it is observed that:

ω2c2 ≤ ω2

3 =1

LC′l 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:

ω2c2 =

1

CL′uL′l d

L ′u + L′l

+

(√LC

′l L′ud+

√LC ′uL

′l d−

√C(L ′l )

2d)2

L ′u + L′l

(4.55)

which shows that

ω2c2 ≤ ω2

2 =1C

(1L′l d

+1L ′ud

)(4.56)

Thus fc2 is bounded from above by both f2 and f3 (fc2 ≤ minf2, f3).


f2

↑f3

↑↑fc1

↑fc2

Dis

cL

0

Figure 4.3: Plot of the function DiscL, which is negative between fc1 and fc2 and otherwisepositive.

The behaviour of DiscL will now be examined. From (4.52) and (4.53) it is clear that

ω2c2 > ω2

c1 represents the larger of two real positive frequencies representing the solutions of

DiscL = 0. The asymptotic behaviour of DiscL as ω → 0 is determined by E1ω4 , and as E1 is

positive, Disc→∞ as ω → 0. The asymptotic behaviour of DiscL 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:

γ2a,b = p(ω)± jq(ω) (4.57)

where p(ω) = 12

(aL1 + aL2

)and q(ω) = 1

2

√|DiscL| are purely real numbers. The solutions of

(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)


f2 →

f3 →

ւ fc1

← fc2

βdαd

γa = jβaγb = jβb

γa = jβ + αγb = −jβ + α

jβa jβb

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

f3 →

ւ fc1

← fc2

βd

αd

γa = jβa γb = jβb

γa = jβ + αγb = −jβ + α

γa = αa γb = αb

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

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


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 stop-

band 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 exam-

ining 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 εr1 = 1 and εr2 = 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


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 (βxd) value,

while the upper band modes required appoximately 5 minutes for a single (βxd) value. The

dispersion curves were obtained by sweeping over the (βxd) 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 reso-

nance 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.


(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 gen-erated dispersion curves. All of the other physical parameters are fixed: hl = 3.1mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3.


(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 gen-erated dispersion curves. All of the other physical parameters are fixed: hl = 3.1mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3. (cont’d)


(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 gen-erated dispersion curves. All of the other physical parameters are fixed: hl = 3.1mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3. (cont’d)


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 stop-

band 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 huis 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

the upper and lower distributed inductances, L′u and L

′l, while f3 was a function of the loading

inductance, L and the lower region distributed capacitance, C′l . However, the simplified model

could not account for the band edges occurring at (βd)x = π, which were out of its range


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 = AhDth + BhCt

h (4.59)

Combining (4.59) with the matrix identity AhDth − BhCt

h = I allows one to decompose Af

into two different forms:

Af = 2AhDth − I (4.60a)

Af = 2BhCth + 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(2AhDth − I− cosh(Γd) I) = 0 (4.61a)

det(2BhCth + I− cosh(Γd) I) = 0 (4.61b)


Th1 =[Ah Bh

Ch Dh

]Th2 =

[D t

h B th

C th A t

h

]V1,n; I1,n

Vk,n; Ik,n

000

V1,n+ 12; I1,n+ 1

2

Vk,n+ 12; Ik,n+ 1

2

V1,n+1; I1,n+1

Vk,n+1; Ik,n+1

Figure 4.6: Transfer matrix relationships for a symmetric unit cell. Voltages and currents oneach of the 1 through k lines defined at nodes n, n+ 1

2 , and n+ 1. Voltages definedwith 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(2AhDth) = 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 sym-

metric loaded MTL structures. For the specific case of the shielded structure the block (matrix)

components of Th1 are given by:

Ah =

cos(θu2

) − cos(θu2

)+ cos

(θl2

)+

sin(θl2

)Zl

2ω L

− sin(θu2

)2ω C Zu cos

(θl2

)+

sin(θu2

)2ω C Zu

+sin(θl2

)2ω C Zl

+sin(θl2

)Zl

2ω L−

cos(θl2

)4ω2C L

(4.65)


Bh =

j

(Zu sin

(θu2

)+ Zl sin

(θl2

))j Zl sin

(θl2

)j

Zl sin(θl2

)+

cos(θu2

)2ω C

−cos(θl2

)2ω C

j

Zl sin(θl2

)−

cos(θl2

)2ω C

(4.66)

Ch =

j Yu sin

(θu2

) −j Yu sin(θu2

)−j Yu sin

(θu2

)j

Yu sin(θu2

)+ Yl sin

(θl2

)−

cos(θl2

)2ω L

(4.67)

Dh =

cos(θu2

)0

− cos(θu2

)+ cos

(θl2

)cos(θl2

) (4.68)

with the corresponding determinants given by:

det(Ah) = cos(θu2

)cos(θl2

)+

cos(θu2

)sin(θl2

)2ωCZl

−cos(θu2

)cos(θl2

)4ω2LC

+cos(θu2

)sin(θl2

)Zl

2ωL+

cos(θl2

)sin(θu2

)2ωCZu

+sin(θu2

)sin(θl2

)4ω2LC

ZlZu︸︷︷︸

f1

(4.69)

det(Bh) = − sin(θu2

)sin(θl2

)+

sin(θu2

)cos(θl2

)2ωCZl

+sin(θl2

)cos(θu2

)2ωCZu︸︷︷︸

f2

(4.70)

det(Ch) =sin(θu2

)Zu︸︷︷︸f6

sin(θl2

)Zl

−cos(θl2

)2ωL

︸︷︷︸

f3

(4.71)

det(Dh) = cos(θu2

)︸︷︷︸f5

cos(θl2

)︸︷︷︸

f4

(4.72)

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


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 huis 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(θu2

)cos(θl2

)(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 = λu2 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 + 12) 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 = Ineγd = 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 + 12 is


short circuited, Vn+ 12

= 0, then the loading inductance, L, will be shorted out and hence not

excited.

From the above discussion it is expected that the boundary conditions, In = In+1 = 0,

along with Vn+ 12

= 0 correspond to det(Dh) = 0, and hence f4 (and f5). To show that this is

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

n+ 12 and n+ 1, shown in Figure 4.6 are required:

[Vn

In

]=

[Ah Bh

Ch Dh

]Vn+ 12

In+ 12

(4.73)

and Vn+ 12

In+ 12

=

[Dth Bt

h

Cth At

h

][Vn+1

In+1

]. (4.74)

Substituting In = 0, Vn+ 12

= 0 into (4.73) yields:

DhIn+ 12

= 0 (4.75a)

Vn = BhIn+ 12

(4.75b)

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+ 12

= 0 into (4.74) yields:

DthVn+1 = 0 (4.76a)

In+ 12

= CthVn+1 (4.76b)

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+ 12

= 0 imply that

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+ 126= 0. Substituting the solution of

(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 = Vneγd =

Vnejπ = −Vn. Thus, due to the symmetry of the unit cell the physical structure of the non-

zero 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+ 12) nodes are zero, while the

other three are in general non-zero. The non-zero variables are in fact the modal eigenvectors,


f1

f2

f3

f4 OpenOpen

OpenOpenOpen

Open

Short

ShortShortShort

ShortShort

2L2L 2C2C

MTL(

d2

)MTL

(d2

)

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 + 12 , then the terminal boundary conditions corresponding to f4

would be the dual(In = In+1 = 0 → Vn = Vn+1 = 0 and Vn+ 1

2= 0 → In+ 1

2= 0

)of those

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


Table 4.1: Boundary conditions and analytical formulas corresponding to the resonance fre-quencies at (βd)x = 0 and (βd)x = π.

Boundary Conditions Formula Non-Zero Variables phase shift / frequency

Vn = Vn+1 = 0In+ 1

2= 0 ⇒ det(Ah) = 0

AhVn+ 12

= 0In = ChVn+ 1

2

In+1 = −In

(βd)x = π (f1 or f4)

Vn = Vn+1 = 0Vn+ 1

2= 0 ⇒ det(Bh) = 0

BhIn+ 12

= 0In = DhIn+ 1

2

In+1 = In

(βd)x = 0 (f2 or f3)

In = In+1 = 0In+ 1

2= 0 ⇒ det(Ch) = 0

ChVn+ 12

= 0Vn = AhVn+ 1

2

Vn+1 = Vn

(βd)x = 0 (f3 or f2)

In = In+1 = 0Vn+ 1

2= 0 ⇒ det(Dh) = 0

DhIn+ 12

= 0Vn = BhIn+ 1

2

Vn+1 = −Vn

(βd)x = π (f4 or f1)

of vector variables at node n, Vn, In, and one of pair at node n + 12 , Vn+ 1

2, In+ 1

2. The

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+ 12

= 0. The corresponding nonzero variables, In+ 12, and Vn are obtained by solving the

complimentary homogenous system, DhIn+ 12

= 0, and finally Vn is obtained from Vn =

BhIn+ 12, as in the last row of Table 4.1. Since det(Dh) = cos

(θl2

)cos(θu2

)can by factored,

its zeroes occur at two different frequencies, corresponding to f4 and f5. The frequency f4

corresponds to cos(θl2

)= 0. Solving for the frequency yields:

ω4 =π

d√L′lC′l

(4.77)

When cos(θl2

)= 0 the homogeneous system DhIn+ 1

2= 0, with Dh given in (4.68) reduces to:

[cos(θu2

)0

− cos(θu2

)0

]I (1)

n+ 12

I(2)

n+ 12

=

[0

0

](4.78)

with solution:

In+ 12

=

I (1)

n+ 12

I(2)

n+ 12

=

[0

Io

], (4.79)


where Io is an arbitrary constant. The corresponding solution for Vn is given by:

Vn =

[V

(1)n

V(2)n

]=

[Vo

Vo

](4.80)

where Vo is an arbitrary constant. Thus the modal eigenvectors at nodes n and n + 12 corre-

sponding to f4 are given by:

f4 ⇒[Vn

In

]=

Vo

Vo

0

0

and

Vn+ 12

In+ 12

=

0

0

0

Io

(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, λl2 = 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, λl4 .

In a similar manner, f5 corresponds to cos(θu2

)= 0, with the solution:

ω5 =π

d√L′uC

′u

(4.82)

and corresponding modal eigenvectors given by:

f5 ⇒[Vn

In

]=

Vo

0

0

0

and

Vn+ 12

In+ 12

=

0

0

Io

−Io

(4.83)

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

wave pattern with λu2 = 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+ 12

for the condition det(Ch) = 0, (4.71), also occur at

two different frequencies corresponding to upper and lower region field concentration, and are

depicted in Figure 4.9. The frequency f6 corresponds to sin(θu2

)= 0⇒ θu = 2π, with solution


open: n short: n + 12

d

2

V(1)n = Vo

V(2)n = Vo

I(1)

n+ 12

= 0

I(2)

n+ 12

= Io

No upper region fields

(a) f4: Lower region standing wave

open: n short: n + 12

d

2

V(1)n = Vo

V(2)n = 0

I(1)

n+ 12

= Io

I(2)

n+ 12

= −Io

No lower region fields

(b) f5: Upper region standing wave

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


open: nopen: n

d

2

V(1)n = Vo

V(2)n = 0

V(1)

n+ 12

= −Vo

V(2)

n+ 12

= 0

No lower region fieldsNo lower region fields

(a) f6: Upper region standing wave

open: nopen: n

d

2

V(1)n = Vo

V(2)n = Vo

V(1)

n+ 12

= Vo

V(2)

n+ 12

= Vo

No upper region fields

(b) f3: Resonance between L and C′l d

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

given by:

ω6 =2π

d√L′uC

′u

(4.84)

which is one full wavelength fitting within the unit cell, with field structure depicted in Fig-

ure 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 interconnect-

ing 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(θl2

)→ θl

2 , cos(θl2

)→ 1 can be made.

With these substitutions, the equation det(Ch) = 0 may be solved for ω3 yielding:

ω23 =

1L(C ′l d)

=1

2L(C′l d

2

) (4.85)

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


region capacitance multiplied by the periodicity, C′l 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 C′l d2 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(θu2

) → θu2 , sin

(θl2

)→ θl

2 , cos(θu2

) → 1 ,

cos(θl2

)→ 1 to det(Bh) and solving for ω2 yields:

ω 22 =

12C

(2

dL′u+

2dL

′l

), (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, L′ud2 and L

′l d2 . Approximate solutions

for the non-zero fields, In and In+ 12

are given by:

In = In+ 12

=

1

−1− huhl

Io =

[1

−1

]︸︷︷︸

Hu

Io +

0

−huhl

︸︷︷︸

Hl

Io (4.87)

The current vector, In = In+ 12, which in the short MTL approximation is the same at nodes n

and n + 12 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.


short: n short: n + 12

d

2

I(1)n = Io

I(2)n = −

(1 + hu

hl

)Io

I(1)

n+ 12

= Io

I(2)

n+ 12

= −(1 + hu

hl

)Io

weak Hupper

strong Hlower

(a) Field structure for large hu

short: n short: n + 12

d

2

I(1)n = Io

I(2)n = −

(1 + hu

hl

)Io

I(1)

n+ 12

= Io

I(2)

n+ 12

= −(1 + hu

hl

)Io

strong Hupper

weak Hlower

(b) Field structure for small hu

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


4.5.7 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:

ω 21 =

14LC + LC ′ud+ LC

′l d+ CL

′l d+ 1

4d2L′lC′u

(4.88)

If in addition it is assumed that the loading inductance, L is large, (4.88) can be simplified

further, resulting in:

ω 21 =

1

2L(

2C + C ′ud2

) (4.89)

Approximate solutions for the non-zero fields, In and Vn+ 12

are given by:

In =

12C(C′ud2

) Io =

[1

−1

]︸︷︷︸

Hu

Io +

0

1 +2C(C′ud2

)

︸︷︷︸Hl

Io (4.90)

Vn+ 12

=

1

− 4LL′l d

Vo (4.91)

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

capacitance, C′ud2 , such that 2C C

′ud2 , and (4.90) may be approximated as:

In =

12C(C′ud2

) Io =

[1

−1

]︸︷︷︸

Hu

Io +

[0

1 +N

]︸︷︷︸

Hl

Io (4.92)

where N =2C(C′ud2

) 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:

Eupper ∝V

(1)

n+ 12

− V (2)

n+ 12

hu=

1 +4LL′l d

hu(4.93)


short: n open: n + 12

d

2

I(1)n = Io

I(2)n =

2C(C′

ud2

)Io

V(1)

n+ 12

= Vo

V(2)

n+ 12

= − 4L

L′l d

Vo

weak Eupper

(a) Field structure for large hu

short: n open: n + 12

d

2

I(1)n = Io

I(2)n =

2C(C′

ud2

)Io

V(1)

n+ 12

= Vo

V(2)

n+ 12

= − 4L

L′l d

Vo

strong Eupper

(b) Field structure for small hu

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


is relatively weak, and hence both the magnetic and electric fields are largely confined to the

lower region. When 2C C′ud2 (4.89) can be further approximated, resulting in:

ω 21 =

14LC

(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.

Conversely for small hu, C′ud2 dominates 2C, so that

2C(C′ud2

) → 0, allowing In to be approx-

imated as:

In =

12C(C′ud2

) Io =

[1

−1

]︸︷︷︸

Hu

Io +

[0

1

]︸︷︷︸Hl

Io (4.95)

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,C′ud2 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:

ω 21 =

1LC ′ud

(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, Cud2 . 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


largely determine the band structure.

Substituting the analytical expressions for C′u, C

′l (3.16), and L

′u, L

′l (3.19), into ω1 (4.89),

ω2 (4.86), ω3 (4.85), and ω4 (4.77), results in the following expressions:

ω21 =

1

2L(

2C +ε1d

2

2hu

) (4.97)

ω22 =

12C

1µo

(2hu

+2hl

)(4.98)

ω23 =

1

2L(ε2d

2

2hl

) (4.99)

ω4 =π

d√µoε2

(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; ε1 = 1) or lower (hl = 3.1 mm; ε2 = 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


Figure 4.12: Dispersion curve for a structure with a via radius of 1.5 mm, corresponding toL = 0.17 nH. All other geometric and electrical parameters are as for the structureof Figure 4.5d: d = 10 mm, hu = 1 mm, hl = 3.1 mm, εr1 = 1, εr2 = 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


n

n

n + 1

n + 1

Y

Y

Lo = L

Co = C′ud

hu, ǫ1

hl, ǫ2

Zo =

√L′(TL-PP)C ′(TL-PP)

Zo Zo

w

d

2d

2

d

Shielding plane

ground plane

(a) Unloaded 2 conductor TL-PP (b) Transformation into a loaded TL-PP structure

(c) Equivalent TL-PP circuit (d) Loading admittance, Y

Figure 4.13: Envisioning the shielded Sievenpiper structure as a 2-conductor parallel-platetransmission line (TL) upon which the patches and vias act as loading elements.The underlying unloaded TL consists of the shielding plane and the ground planeas 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 elementshown 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 parallel-

plate 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, L′(TL-PP) and C ′(TL-PP). These

parameters can be written in terms of the components of the per-unit-length capacitance and

inductance matrices, C′ (3.16) and L′ (3.19), which define the unloaded MTL geometry, and

are given by:

C′(TL-PP) =

C′uC

′l

C ′u + C′l

(4.101)

L′(TL-PP) = L

′u + L

′l (4.102)

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

capacitance, Co = C′ud (due to the parallel-plate capacitance between the patch layer and the


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:

cosh(γd) = Af (4.103)

where

Af = cos(θTL-PP) +12jZoY sin(θTL-PP) (4.104)

with

θTL-PP = ωd√L′(TL-PP)C ′(TL-PP), (4.105)

Zo =

√L′(TL-PP)C ′(TL-PP)

(4.106)

and

Y =jωCo

1− ω2LoCo(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 struc-

ture 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 dis-

persion 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


(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, εr1 = 1, εr2 = 2.3


TL-PP dispersion (4.103) are given by:

ω21 (TL-PP) =

1LC ′ud+ 1

4L′ld

2C ′u + 14L′ud

2C ′u(4.108)

and

ω23 (TL-PP) =

1L

(2C ′ud

+1C′l d

)(4.109)

The expression for ω21 (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), ω21 (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 ω23 (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

ω23 (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:

ω23 (TL-PP) =

1L

(2C ′ud

+1C′l d

)≥ 1LC

′l d

= ω23 (MTL) (4.110)

and hence the peak of the stop-band predicted by the TL-PP model, ω23 (TL-PP), is always

larger than that of the MTL model, ω23 (MTL), leading to the overestimation of the stop-band

bandwidth. Only in the limit hu → 0 do the two models converge with ω23 (TL-PP)→ ω2

3 (MTL)

in that case. Thus, both ω21 (TL-PP) and ω2

3 (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+ 12 : 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.


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 Sieven-

piper 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


(βd)x = 0(βd)x = 0

(βd)y = 0(βd)y = 0

x

y

y

z

(a) Side view (MTL) (b) Side view (TE)

(c) Top view, as seenbelow the Shielding conductor (MTL)

(d) Top view, as seenbelow the Shielding conductor (TE)

nn n + 12n + 1

2 n + 1n + 1

H-walls

E-walls

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


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 Chap-

ter 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, corre-

sponding 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 prop-

agation 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,

Cu′, Cl ′, L

′u, and L

′l. 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), cor-

responding 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

Chapter 5. Slow Wave Analysis 107

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:

(βd)2LC = d2C′u

(L′u + L

′l

)ω2 +

d3(C(L

′l)

2C′u + LL

′u(C

′u)2 + L(C

′u)2L

′l

)ω4 + · · · (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 =√C ′u(L′u + L

′l)ω (5.2)

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

given by:

vφ(LC) = vg(LC) =1√

C ′u(L′u + L′l)

(5.3)

In the limit hu → 0, C′u →∞, while the term L

′u +L

′l 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) ≤ 1√C ′uL

′u

(5.4)

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

given by:

C′eff (LC) = C

′u (5.5)


L′eff (LC) = L

′u + L

′l (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,

L′eff (LC) = L

′u + L

′l involving both the upper and lower region geometry, while the effective

capacitance, C′eff (LC) = C

′u 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)2TL = d2

(C′uC′l

C ′u + C′l

)︸︷︷︸

C′eff

(L′u + L

′l

)︸︷︷︸L′eff

ω2 (5.7)

and whose per-unit length capacitance, C′(TL) and inductance L

′(TL) can be written in

terms of the components of the capacitance and inductance matrices characterizing the shielded

structure:

C′(TL) =

C′uC

′l

C ′u + C′l

(5.8)

L′(TL) = L

′u + L

′l (5.9)

The effective inductance L′eff (LC) = L

′u+L

′l (5.6), of the FW mode for the shielded mushroom

structure is identical to the inductance of the the commensurate (2-conductor) geometry, L′(TL)

(5.9), consisting of the upper shielding conductor and the ground conductor alone. However, the

per-unit length effective capacitance of the shielded structure, C′eff (LC) (5.5) is always greater

than that of the commensurate TL geometry, C′(TL) (5.8), as can be shown by computing

their ratio,

rC =C′eff (LC)

C ′(TL)= 1 +

ε1 hlε2 hu

> 1 (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,

ε1 = ε2 = εo, is the ratio rC(ε1 = ε2 = εo) = 1 + hlhu

> 1. Thus obtaining the slow wave effect


V1 = Vo

I1 =1Zo

Vo

Ig → −1

(βd)2TL = d2

(C

′uC

′l

C ′u + C

′l

)︸︷︷︸

C′eff

(L

′u + L

′l

)︸︷︷︸L

′eff

ω2

Zo =

√L

′eff

C′eff

Hu 6= 0

Hl 6= 0

Eu 6= 0

El 6= 0

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

0.2 0.4 0.60

2

4

6

8

10

12

14

βd (radians)

Fre

quen

cy (

GH

z)

Light εr=1

0.2 mm (FEM)1.0 mm (FEM)6.0 mm (FEM)MTL theory

Figure 5.2: Low frequency dispersion with hu = 0.2, 1, and 6 mm; All other parameters arefixed: hl = 1 mm; d = 2 mm; w = 1.9 mm; via radius = 0.1 mm (from [11], c©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 hlhu

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


V1 → Vo

V2 → 0

I1 → 1Zt

Vo

I2 → 0

Ig → −1

(βd)2LC = d2 C′u︸︷︷︸

C′eff

(L

′u + L

′l

)︸︷︷︸L

′eff

ω2 + O(ω4)

Zt =

√L

′eff

C′eff

Hu 6= 0

Hl 6= 0

Eu 6= 0

El ≈ 0

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

(βd)2LC = d2C′u(L

′u + L

′l)ω

2 into (4.15), and solving for the resulting eigenvector:

V1

V2

I1

I2

=

1

−dCuLω2

1Zt

1ZtLldCω

2

Vo + smaller terms

(5.11)

where Zt =

√L′u + L

′l

C ′u. The above eigenvector expression is valid in the limit ω → 0 where V1

V2 and I1 I2. In this limit only the V1, I1 terms remain finite, while V2, I2 → 0 as O(ω2),

resulting in the further simplification:

V1

V2

I1

I2

→

1

01Zt0

Vo as ω → 0 (5.12)

From (5.12) the ratio,V1

I1= Zt =

√L′u + L

′l

C ′u, confirming the earlier observation that the

mode has an effective inductance and capacitance per-unit-length, L′eff (LC) = L

′u + L

′l and

C′eff (LC) = C

′u, respectively.

The voltage component of the modal eigenvector is given by Vmodal =[V1V2

]=[Vo0

], which

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


2C2C

MTL(

d2

)MTL

(d2

)

(a) MTL with only C present.

2L2L

MTL(

d2

)MTL

(d2

)

(b) MTL with only L present.

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

that the effective capacitance C′eff (LC) = C

′u 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

current eigenvector Imodal =[I1I2

]=[ 1ZtVo

0

], shows that there is no net return current flowing

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 = 1

2 j ω 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

for the effective inductance, L′eff (LC) = L

′u + L

′l, 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


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:

(βd)2C = d2

(C′uC′l

C ′u + C′l

)︸︷︷︸

C′eff

(L′u + L

′l

)︸︷︷︸L′eff

ω2 +O(ω4) (5.13)

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(ω4), 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√C′l L′l

≤ vg(C) ≤ 1√C ′uL

′u

(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:

(βd)2L = d2 C′u︸︷︷︸

C′eff

L′u︸︷︷︸

L′eff

ω2 (5.15)

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

mode velocity:

vg(L) =1√C ′uL

′u

(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,


V1 → Vo

V2 → V2(6= 0)

I1 → 1Zo

Vo

I2 → 0

Ig → −1

(βd)2C = d2

(C

′uC

′l

C ′u + C

′l

)︸︷︷︸

C′eff

(L

′u + L

′l

)︸︷︷︸L

′eff

ω2 + O(ω4)

Zo =

√L

′eff

C′eff

Hu 6= 0

Hl 6= 0

Eu 6= 0

El 6= 0

(a) Voltage and current eigenvectors with only C present.

V1 → Vo

V2 = 0

I1 =1

ZuVo

I2 = − 1Zu

Vo

Ig = 0

(βd)2L = d2 C′u︸︷︷︸

C′eff

L′u︸︷︷︸

L′eff

ω2

Zu =

√L

′eff

C′eff

Hu 6= 0

Hl = 0

Eu 6= 0

El = 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 fre-

quency 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 per-

unit 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


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,

Tunit−cell−MTL (4.13), there exists a corresponding modal eigenvector, (4.15),[V, I

]ta,b

, which

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 equa-

tion 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 oper-

ation 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 frequen-

cies, 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/ complex-

mode 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


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 = Nd, the three conductors, 1, 2, and ground, are extended by a length, l, and at the

positions, x = −l and x = Nd+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, V +M1 is excited. All of the other ports are

terminated in matched impedances, and so only the reflected amplitude coefficients, V −M1, V −M2,

V −M3, and V −M4 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−MTL, (4.15), and those corresponding to γa and γb are labeled by:

(γa);

V1

V2

I1

I2

=

V a

1

V a2

Ia1

Ia2

(γb);

V1

V2

I1

I2

=

V b

1

V b2

Ib1

Ib2

(6.1)

where, for the reflected modes, −γa and −γb, the eigenvectors are identical, except that the cur-

rent components are reversed. At x = Nd the voltage/current vectors given in (6.1) are trans-

formed through the propagation constants e∓(γad)N and e∓(γbd)N . Each Bloch mode and it’s


Port modes;modal coefficients:V +

M1, V −M1, V −M2

Port modes;modal coefficients:

V −M3, V −M4

Bloch modes;modal coefficients:a+

m, a−m, b+m, b−m

ZuZu

Zl

Zl

vs V +M1

V −M1

V −M2

V −M3

V −M4

cell 1 cell 2 cell N

x = −l x = 0 d 2d Nd x = Nd + l

MTL(

d2

)MTL

(d2

)

L 2C2C

(a)

S11

S21

S31

S41

P1+

P2

P3

P4

(b)

S12

S22

S32

S42

P1

P2+

P3

P4

(c)

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


reflected counterpart may be excited, with the amplitudes of excitations given by a+m, b

+m, a

−m, b

−m.

For the total system consisting of the port extensions and the cascade of unit cells there are nine

unknown variables in total to be determinedV +M1 , V

−M1 , V

−M2 , V

−M3 , V

−M4 , a

+m, b

+m, a

−m, b

−m

and

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:

V1

V2

I1

I2

(x=−l)

=

V +M1 + V −

M1 + V −M2

V +M1 + V −

M1

−V−

M2

ZuV +M1

Zl− V −

M1

Zl+V −M2

Zu

(6.2)

The incident voltage component, V +M1 is solved for by considering the source boundary condi-

tion:

− vs + Zl(I1 + I2) + V2 = 0 (6.3)

Upon substituting the expressions for I1, I2, and V2 from (6.2), V +M1 is solved for from (6.3),

resulting in:

V +M1 =

vs2

(6.4)

The output port plane voltage/current vector (at x = Nd + 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 = Nd+l)

to the beginning and end of the MTL unit cell cascade (at x = 0 and x = Nd) are made using


the lower and upper region mode propagation constants, βl and βu, and yield:

V1

V2

I1

I2

(x=0)

=

V +M1 e

−jβll + V −M1 e

jβll + V −M2 e

jβul

V +M1 e

−jβll + V −M1 e

jβll

−V−

M2

Zuejβul

V +M1

Zle−jβll − V −

M1

Zlejβll +

V −M2

Zuejβul

(6.5)

V1

V2

I1

I2

(x=Nd)

=

V −M3 e

jβll + V −M4 e

jβul

V −M3 e

jβll

V −M4

Zuejβul

V −M3

Zle−jβll − V −

M4

Zuejβul

(6.6)

A system of 8 equations in the 8 unknowns,V −M1 , V

−M2 , V

−M3 , V

−M4 , a

+m, b

+m, a

−m, b

−m

is obtained

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 = Nd, resulting in:

[M11 M12

M21 M22

][VP

BP

]=

[Vs

0

](6.7)

where

M11 =

−ejβll ejβul 0 0

−ejβll 0 0 0

0 ejβul

Zu0 0

ejβll

Zl− ejβul

Zu0 0

M12 =

V a

1 V b1 V a

1 V b1

V a2 V b

2 V a2 V b

2

Ia1 Ib1 −Ia1 −Ib1Ia2 Ib2 −Ia2 −Ib2

(6.8)

M21 =

0 0 −ejβll −ejβul0 0 −ejβll 0

0 0 0 − ejβul

Zu

0 0 − ejβll

Zlejβul

Zu

(6.9)

M22 =

V a

1 e−(γad)N V b

1 e−(γbd)N V a

1 e(γad)N V b

1 e(γbd)N

V a2 e−(γad)N V b

2 e−(γbd)N V a

2 e(γad)N V b

2 e(γbd)N

Ia1 e−(γad)N Ib1e

−(γbd)N −Ia1 e(γad)N −Ib1e(γbd)NIa2 e−(γad)N Ib2e

−(γad)N −Ia2 e(γad)N −Ib2e(γbd)N

(6.10)


Table 6.1: Bloch propagation constants, (γad) and (γbd), along with the modal coefficients a+m,

a−m, b+m and b−m for the 4-port scattering theory: column 1 excitation (lower region)(a) hu = 6 mm;

f (GHz) (γad), |a+m| , |a−m| (γbd), |b+m| , |b−m|

1.0 (0.26 j), 0.50 , 0.03 (4.87 + π j), 0.87 , 0.003.0 (0.95 j), 0.60 , 0.17 (1.19 + π j), 0.78 , 0.005.0 (0.54 + 0.64 j), 0.29 , 0.00 (0.54− 0.64 j), 0.96 , 0.007.0 (1.38 j), 0.73 , 0.19 (0.85 j), 0.65 , 0.079

(b) hu = 1 mm;f (GHz) (γad), |a+

m| , |a−m| (γbd), |b+m| , |b−m| 1.0 (0.44 j), 0.55 , 0.13 (5.09 + π j), 0.82 , 0.003.0 (1.35 + 1.73 j), 0.49 , 0.00 (1.35− 1.73 j), 0.87 , 0.005.0 (1.31 + 0.53 j), 0.53 , 0.00 (1.31− 0.53 j), 0.85 , 0.007.0 (1.28 j), 0.91 , 0.34 (1.33 + 0 j), 0.22 , 0.00

(c) hu = 0.2 mm;f (GHz) (γad), |a+

m| , |a−m| (γbd), |b+m| , |b−m| 1.0 (0.97 j), 0.66 , 0.37 (5.18 + π j), 0.65 , 0.003.0 (2.46 + 0.65 j), 0.66 , 0.00 (2.46− 0.65 j), 0.75 , 0.005.0 (1.13 + 0 j), 0.997 , 0.00 (3.16 + 0 j), 0.07 , 0.007.0 (1.25 j), 0.91 , 0.42 (3.04 + 0 j), 0.05 , 0.00

VM =

V −M1

V −M2

V −M3

V −M4

BP =

a +m

b +m

a −m

b −m

Vs =

vs2 e−jβll

vs2 e−jβll

0vs

2Zle−jβll

(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:

Sij =V −i√Zj

V +j

√Zi

∣∣∣∣∣V +k =0 for k 6=j

(6.12)

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,

the MTL analysis also yields the relative excitation strengths of each Bloch mode,a+m, b

+m, a

−m, b

−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.



a−m, b+m and b−m for the 4-port scattering theory: column 2 excitation (upper region)(a) hu = 6 mm;


1.0 (0.26 j), 0.96 , 0.05 (4.87 + π j), 0.28 , 0.003.0 (0.95 j), 0.95 , 0.26 (1.19 + π j), 0.15 , 0.005.0 (0.54 + 0.64 j), 0.95 , 0.00 (0.54− 0.64 j), 0.32 , 0.007.0 (1.38 j), 0.91 , 0.24 (0.85 j), 0.30 , 0.18

(b) hu = 1 mm;f (GHz) (γad), |a+


(c) hu = 0.2 mm;f (GHz) (γad), |a+


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, andN = 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,


|a+m| = 0.5, while the EW (γb) mode has modal coefficient, |b+m| = 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 |b+m| = 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

inductance, L′eff = L′u + L

′l and capacitance, C ′eff = C

′u, and hence characteristic impedance,

Zt =

√L′u + L

′l

C ′u. The impedance of the upper region (port) mode, on the other hand, is

Zu =

√L′u

C ′u. When hu hl then Zt ≈ Zu and the impedances are approximately equal,

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 (γad)(hu=1mm) = 1.35 + 1.73 j, (γbd)(hu=1mm) = 1.35−1.73 j, and (γad)(hu=0.2mm) = 2.46+0.65 j, (γbd)(hu=0.2mm) = 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


(a)

(b) (c)

(d) (e)

Figure 6.2: Dispersion and corresponding four-port scattering curves comparing the MTL anal-ysis with FEM simulations for an N = 7 unit cell cascade with hu = 6 mm. All ofthe other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm,r = 0.5 mm, εr1 = 1, εr2 = 2.3.


(a)

(b) (c)

(d) (e)

Figure 6.3: Dispersion and corresponding four-port scattering curves comparing the MTL anal-ysis with FEM simulations for an N = 7 unit cell cascade with hu = 1 mm. All ofthe other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm,r = 0.5 mm, εr1 = 1, εr2 = 2.3.


(a)

(b) (c)

(d) (e)

Figure 6.4: Dispersion and corresponding four-port scattering curves comparing the MTL anal-ysis with FEM simulations for an N = 7 unit cell cascade with hu = 0.2 mm. Allof the other physical parameters are fixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm,r = 0.5 mm, εr1 = 1, εr2 = 2.3.


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

γad = 1.13 + j 0 and γbd = 3.16 + j 0. Only the mode, γad, with the smaller decay constant,

αad = 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 co-

efficients S31, S41, S42, and S32 are no longer small. However, the modal structure is more


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

ε1r = 1, hu = 18 mm, ε2r = 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


(a)

(b) (c)

(d) (e)

Figure 6.5: Dispersion and corresponding four-port scattering curves obtained using MTL anal-ysis for a case where the BW bandwidth is large: L = 10 nH, C = 4 pF, ε1r = 1,hu = 18 mm, ε2r = 5, and hl = 3.1 mm.


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 trans-

mission 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 Nd. 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 = Nd + 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

removed. This allows one to write the per-unit-length inductance, L′pp, and capacitance, C

′pp,

of the port mode, in terms of the variables, L′u , L

′l , C

′u , and C

′l , which were used to define the

MTL per-unit-length parameters:

L′pp = L

′u + L

′l C

′pp =

C′uC

′l

C ′u + C

′l

(6.13)


Port modes;modal coefficients:

V +M1, V −M1

Port mode;modal coefficient:

V −M2

Bloch modes;modal coefficients:a+

m, a−m, b+m, b−m

ZppZpp

vs V +M1

V −M1V −M2cell 1 cell 2 cell N

x = −l x = 0 d 2d Nd x = Nd + l

MTL(

d2

)MTL

(d2

)

L 2C2C

(a)

S11S21P1+

P2

(b)

Figure 6.6: Two-port scattering: (a) Circuit schematic; (b) Power flow

with characteristic impedance, Zpp and propagation constant, βpp given by:

Zpp =

√L′ppC ′pp

βpp = ω√L′ppC

′pp (6.14)

Due to the symmetry of the structure the complete scattering parameters can be de-

rived by considering port 1 as the excited port, with port 2 matched. The port variables

are the incident, V +M1 , I

+M1 and reflected, V −

M1 , I−

M1 voltage, current pairs for port 1,

and the corresponding reflected quantities, V −M2 , I

−M2 for port 2. Only 3 of the 6 port

variables are independent as the current and voltage quantities are related through Zpp as

I +M1 =

V +M1

Zpp, I −M1 = −V

−M1

Zpp, I −M2 =

V −M2

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

there are seven unknown variables in total to be determinedV +M1 , V

−M1 , V

−M2 , a

+m, b

+m, a

−m, b

−m

and hence seven equations are needed to uniquely solve the system. The incident voltage, V +

M1 ,


y

z

hu

hl

ǫ1

ǫ2

d

Conductor 1; V1, I1

Ground

Figure 6.7: Transverse cut used to define the port variables for the investigated two-port scat-tering situation.

is solved immediately by considering the source boundary conditions:

− vs + ZppI1 + V1 = 0 (6.15)

Substituting I1 =(V +M1 −V

−M1

Zpp

)and V1 = (V +

M1 + V −M1 ) into (6.15) yields:

V +M1 =

vs2

(6.16)

The transformation of the port variables at x = −l and x = Nd+ l to the beginning and end of

the MTL cascade (at x = 0 and x = Nd) is made through the port TL propagation constant,

βpp yielding:

[V1

I1

](x=0)

=

V +M1 e

−jβppl + V −M1 e

jβppl

V +M1Zpp

e−jβppl − V −M1Zpp

ejβppl

(6.17)

[V1

I1

](x=Nd)

=

V −M2 e

jβppl

V −M1Zpp

ejβppl

(6.18)

Having explicitly solved for the incident voltage V +M1 , 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 = Nd. The final two equations

are obtained by noting that conductor 2 is open circuited at the transition junctions x = 0 and

x = Nd. The resulting (6 by 6) system of equations is given by:

[M11 M12

M21 M22

][VM

BP

]=

[Vs

0

](6.19)


where

M11 =

−ejβppl 0ejβppl

Zpp0

0 0

M12 =

V a

1 V b1 V a

1 V b1

Ia1 Ib1 −Ia1 −Ib1Ia2 Ib2 −Ia2 −Ib2

M21 =

0 −ejβppl

0 −ejβppl

Zpp

0 0

(6.20)

M22 =

V a

1 e−(γad)N V b

1 e−(γbd)N V a

1 e(γad)N V b

1 e(γbd)N

Ia1 e−(γad)N Ib1e

−(γbd)N −Ia1 e(γad)N −Ib1e(γbd)NIa2 e−(γad)N Ib2e

−(γbd)N −Ia2 e(γad)N −Ib2e(γbd)N

(6.21)

VM =

[V −M1

V −M2

]BP =

a +m

b +m

a −m

b −m

Vs =

vs2e−jβppl

vs2Zpp

e−jβppl

(6.22)

and 0 is the 4× 1 zero matrix.

Solving the linear system, (6.19), allows one to obtain the S-parameters, S11 =V −M1

V +M1

and

S21 =V −M2

V +M1

, and the modal coefficients,a+m, b

+m, a

−m, b

−m

. Some typical results for a 5 unit-cell

cascade, with hu = 6, 1, and 0.2 mm, are shown in Figure 6.8. The correspondence of the

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 εr2 = 2.3, and hence the


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


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 (γad)(hu=1mm) = 1.35 + 1.73 j, (γbd)(hu=1mm) =

1.35−1.73 j, and (γad)(hu=0.2mm) = 2.46+0.65 j, (γbd)(hu=0.2mm) = 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| = |b+m| = 0.71, with |a−m| = |b−m| = 0.

However, in practice, due to the exponential dependence of the system on (αd) × (N), the

approximate solution |a+m| = |b+m| = 0.71 holds even for the cases considered where the products,

(αd)(hu=1mm) × (N) = 6.75 and (αd)(hu=0.2mm) × (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|, |b+m| |a−m|, |b−m|, the voltage, V1(Nd), which is proportional to V −

M2 , is given by:

V1(Nd) = a+mV

a1 e

(−α−j β)dN + b+mVb1 e

(−α+j β)dN (6.23)

In general a+m and b+m are related through a complex phase, such that a+

m

b+m= e−2 jφ4 , with a

similar relation holding for the modal voltage coefficients, V a1V b1

= e2jφ1 . Using these relations,

(6.23) becomes:

V1(Nd) = b+mVb1 e

(−αdN−jφ4+jφ1) ×(ej(φ1−φ4−βdN) + e−j(φ1−φ4−βdN)

)= b+mV

b1 e

(−αdN−jφ4+jφ1)[2 cos(φ1 − φ4 − βdN)

] (6.24)

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


(a) hu = 6 mm (b) hu = 6 mm

(c) hu = 1 mm (d) hu = 1 mm

(e) hu = 0.2 mm (f) hu = 0.2 mm

Figure 6.8: Dispersion and corresponding two-port scattering curves comparing the MTL anal-ysis 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 arefixed: hl = 3.1 mm, d = 10 mm, w = 9.6 mm, r = 0.5 mm, εr1 = 1, εr2 = 2.3.


band. The transmission zeroes correspond to frequencies for which (φ1−φ4−βdN) = π2 (2n+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 γad = 1.13 + j 0 and γbd = 3.16 + j 0.

Only the mode, γad, with the smaller decay constant, αad = 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 stop-

bands for the structure.

6.5 Summary

In this chapter the scattering characteristics of a cascade of unit cells of the shielded struc-

ture 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 scat-

tering 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 prop-

agation 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 evanes-

cent modes were shown to be supported by the structure. The existence of complex modes and



a−m, b+m and b−m for the two port scattering results depicted in Figure 6.8.(a) hu = 6 mm


1.0 (0.26 j), 0.997 , 0.05 (4.87 + π j), 0.05 , 0.003.0 (0.95 j), 0.992 , 0.04 (1.19 + π j), 0.12 , 0.005.0 (0.54 + 0.64 j), 0.71 , 0.00 (0.54− 0.64 j), 0.71 , 0.007.0 (1.38 j), 0.98 , 0.14 (0.85 j), 0.10 , 0.13

(b) hu = 1 mmf (GHz) (γad), |a+


(c) hu = 0.2 mmf (GHz) (γad), |a+


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 εr,

and for a slot antenna printed on a hemispheric dielectric lens the ratio is ε3/2r [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 back-

ing 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 environ-

ment 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 electromag-

netic 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 com-

pared 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 two-

dimensional 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 ( λ4 ), 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.


hua

hub

hl

ǫ1a

ǫ1b

ǫ2

w wb

d

d

via; r

(a) Side view(b) Top view, as seen

below the Shielding conductor

Figure 7.1: Unit cell underlying the proposed slot antenna; hua = 1.54 mm, ε1a−rel = 4.5,hub = 1.5 mm, ε1b−rel = 1, hl = 3.1 mm, ε2−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 measure-

ments, 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 ε1a−rel = 4.5;

the other with hub = 1.5 mm, and ε1b−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 con-

stants 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, C′, which is used in the determina-

tion of the MTL dispersion, is composed of elements, C′u and C

′l , which are the per-unit-length


(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 ε1, with the lower region (height hl)

also a uniform medium with dielectric constant, ε2. For the structure proposed in Figure 7.1

the upper region is composed of two uniform layers, and hence the expression for C′u needs to

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

formed from the series combination of two per-unit-length capacitances, C′u(hua) and C

′u(hub),

C′u(hua) =

ε1ad

hua(7.1)

C′u(hub) =

ε1bd

hub(7.2)

with C′u(n.u.) given by

1C ′u(n.u.)

=1

C ′u(hua)+

1C ′u(hub)

(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 deter-

mined 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 cor-

respondence 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 ≤ βxd ≤ π, with βyd = 0 fixed; from X to M , βxd = π is fixed

with 0 ≤ βyd ≤ π; finally for M to Γ, both βxd = βyd 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.


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

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


Port 1 Port 2

8 cm

co-axial probes

(a) Shielded geometry of Figure 7.1.

Port 1 Port 2

8 cm

co-axial probes

(b) Parallel plate geometry.

2 3 4 5 6−100

−80

−60

−40

−20

0

Frequency (GHz)

S21

(d

B)

S21

: EBG

S21

: PP

(c) (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 purposeof measuring the transmission, S21; Measured S21 for the shielded structure, andfor the flat conductor backed parallel-plate structure for: (c) the Γ −X direction;(d) the Γ−M direction.


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 semi-

infinite 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


3.0 mm45 mm 55 mm

70 mm

Central conductor width: 1.3 mmCPW slot width: 0.130 mm



Outer radius: 30 mm

Antenna ground plane:200 mm x 200 mm

Outline of the EBG surface/Flat Conductor backing; both are100 x 100 mm, centered on the slot

(matching network)

24.6 mm

12.3 mm

28.1 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 placementshown as a dotted line (from [9], c© IEEE 2005). (b) Cross-sectional view of thegeometry with approximate size of the slot’s ground plane and the overall heightgiven 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


3.7 3.8 3.9 4.0

−13

−11

−9

−7

−5

Frequency (GHz)

S11

(d

B)

MeasuredFEM

Figure 7.6: S11 of the shielded structure-based slot antenna (from [9], c© IEEE 2005).

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 parallel-

plate 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


2.5 λο

E-plane H-plane

FEM co-pol.

FEM cross-pol.

Exp. co-pol.

Exp. cross-pol.

λο

50

30

210

60

240

270

120

300

150

330

180 0

-10

-20 30

210

60

240

270

120

300

150

330

180 0

-10

--20

Figure 7.7: Measured and FEM simulated normalized radiation patterns of the reference ring-slot 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 conductor-

backed 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


30

210

60

240

270

120

300

150

330

180 0

30

210

60

240

270

120

300

150

330

180 0

E-plane H-plane

FEM co-pol.

FEM cross-pol.

Exp. co-pol.

Exp. cross-pol.

λο

4

2.5 λο

Figure 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.

30

210

60

240

270

120

300

150

330

180 0

-10

-20 30

210

60

240

270

120

300

150

330

180 0

-10

-20

E-plane H-plane

FEM co-pol.

FEM cross-pol.

Exp. co-pol.

Exp. cross-pol.

λο

12

2.5 λο

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


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 config-

uration, 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 ge-

ometries, 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 Sieven-

piper 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, in-

cluding 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 stop-

band 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-unit-

length 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


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 Compo-

nent 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 meta-

material 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 co-

directional 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


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 an-

tenna) 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 im-

proved 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

( λ12) 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 λ12 . 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 λ4 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

larger for the cavity-backed antenna. For the low profile ( λ12) cavity-backed design the front-

to-back ratios are again comparable to the EBG-backed design; however in this case the cross-

polarization 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 λ4 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 com-

pared to both cavity backed designs, and this is achieved while maintaining a low profile.


30

210

60

240

270

120

300

150

330

180 0

-10

-20 30

210

60

240

270

120

300

150

330

180 0

-10

-20

E-plane H-plane

FEM co-pol.

FEM cross-pol.

Exp. co-pol.

Exp. cross-pol.

λο

12

2.5 λο

(a) EBG-backed antenna.

−20

−10

30

210

60

240

270

120

300

150

330

180 0

E-plane

−20

−10

30

210

60

240

270

120

300

150

330

180 0

H-plane

FEM co-pol.

FEM cross-pol.

λο

4

2.5 λο

(b) Cavity-backed antenna 1: cavity depth = λ4.

E-plane H-plane

FEM co-pol.

FEM cross-pol.

−20

−10

30

210

60

240

270

120

300

150

330

180 0

−20

−10

30

210

60

240

270

120

300

150

330

180 0

2.5 λολ

ο

12

(c) Cavity-backed antenna 2: cavity depth = λ12

.

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


3.7 3.8 3.9 4.0

−13

−11

−9

−7

−5

Frequency (GHz)

S11

(d

B)

MeasuredFEM

(a) EBG-backed antenna.

3.7 3.8 3.9 4.0−25

−20

−15

−10

−5

Frequency (GHz)

S11

(d

B)

(b) Cavity-backed antenna 1: cavity depth = λ4

(FEM).

3.7 3.8 3.9 4.0−25

−20

−15

−10

−5

Frequency (GHz)

S11

(d

B)

(c) Cavity-backed antenna 2: cavity depth = λ12

(FEM).

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

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