Related Structures and Their Application to Anten - TSpace

Characterization of the Reflection And DispersionProperties of ‘Mushroom’-Related Structures and Their

Application to Antennas

by

Shahzad Raza

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Electrical And Computer EngineeringUniversity of Toronto

Copyright c© 2012 by Shahzad Raza

Abstract

Characterization of the Reflection And Dispersion Properties of ‘Mushroom’-Related

Structures and Their Application to Antennas

Shahzad Raza

Master of Applied Science

Graduate Department of Electrical And Computer Engineering

University of Toronto

2012

The conventional mushroom-like Sievenpiper structure is re-visited in this thesis and a

relationship is established between the dispersion and reflection phase characteristics of

the structure. It is shown that the reflection phase frequency at which the structure

behaves as a Perfect Magnetic Conductor (PMC) can be predicted for varying angles

of incidence from the modal distribution in the dispersion diagrams and corresponds

to the supported leaky modes within the light cone. A methodology to independently

tune the location of the PMC frequency point with respect to the surface wave band-gap

location is then presented. The influence of having said PMC frequency point located

inside or outside the surface wave band-gap on a dipole radiation pattern is then studied

numerically. It is demonstrated that the antenna exhibits a higher gain when the PMC

frequency and band-gap coincide versus when they are separated. Two design cases are

then presented for when the aforementioned properties coincide and are separated and a

gain improvement of 1.2 dB is measured for the former case.

ii

To the pursuit of knowledge

iii

Acknowledgements

I would like to acknowledge the invaluable advice and encouragement of my supervisor,

Professor George V. Eleftheriades. His insight and guidance shaped my work and it

would not have been possible without his support.

I would also like to thank Marco Antoniades for his assistance, patience and the many

stimulating discussions on all matters as well as Francis Elek for his insight and advice

on many topics related to my research. In addition, I would like to extend a warm thank

you to the other students in the Electromagnetics Group at the University of Toronto

who made my graduate school experience intellectually satisfying as well as amusing.

Finally, I would like to thank my family, my fiancee and my friends who were encour-

aging and loving throughout my time in graduate school.

Shahzad Raza

University of Toronto, 2012

iv

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The ‘mushroom-like’ Sievenpiper Structure . . . . . . . . . . . . . . . . . 3

1.2.1 Scattering Characteristics . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Dispersion Characteristics . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Other High Impedance Surfaces . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Relating the Scattering & Dispersion Characteristics 24

2.1 Full Wave Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.3 Field Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Scanning the Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Equivalent Circuit Model Analysis . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 The Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.2 Critical Frequencies of the Dispersion Relation . . . . . . . . . . . 43

2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

v

3 Study of Band Gap Effects on Antenna Performance 48

3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Unit Cell Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Variation of Surface Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Variation of Dipole Length . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Evaluation of the ‘PMC effect’ . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Comparison of Radiation Patterns . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Design Cases: Simulated & Measured Results 69

4.1 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Feed Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Simulated Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Reference Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.2 AMC Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 AMC-BG Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.4 Comparison of Simulated Results . . . . . . . . . . . . . . . . . . 78

4.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Conclusion 86

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A Parameter Extraction 91

A.1 Transmission Line Parameters . . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Loading Element Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi

B Notes on the HFSS Eigenmode Solver 95

B.1 Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.1.1 Fundamental Modes . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.1.2 Higher Order Modes . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.2 Mode Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

C Parametric study of dipole antenna on EBG ground plane 101

C.1 Unit Cell Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

C.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

C.1.2 Scattering & Dispersion Characteristics . . . . . . . . . . . . . . . 102

C.2 Dipole performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Bibliography 105

vii

List of Acronyms

AMC Artificial Magnetic Conductor

AMC-BG Artificial Magnetic Conductor with Band-Gap

AUT Antenna under test

CPS Coplanar Strip

EBG Electromagnetic Band-Gap

FBR Front-to-Back Ratio

FEM Finite Element Method

FDTD Finite-Difference Time-Domain

GPS Global Positioning System

HFSS High Frequency Structure Simulator (Ansoft)

HIS High Impedance Surface

MIMO Multiple Input Multiple Output

NRI Negative Refractive Index

PMC Perfect Magnetic Conductor

PML Perfectly Matched Layer

PBC Periodic Boundary Conditions

RF Radio Frequency

TL Transmission-Line

TE Transverse Electric to x-direction

TM Transverse Magnetic to x-direction

viii

UWB Ultra-wideband

VNA Vector network analyzer

WLAN Wireless local area network

ix

List of Symbols

β Bloch propagation constant

(βd)x Phase shift in x-direction

(βd)y Phase shift in y-direction

εr Relative permittivity

εo Permittivity of free space

φx Tangential phase shift in free space

φrefl Reflection phase of incident tangential electric field

ψ Transmission-line phase shift

µo Permeability of free space

µr Relative permeability

ω Angular frequency

ωo Angular design frequency

c Speed of light in vacuum

fgap Resonance frequency of capactive patch gaps with host TL inductance

fvia Resonance frequency of inductive via with host TL capacitance

fPMC Frequency at which reflection phase of incident tangential electric field is 0o

fc1, fc2 Critical frequencies of the NRI-TL dispersion relation where (βd)x = 0

k Free space propagation constant

tanδ Dielectric loss tangent

vφ Phase velocity

x

vg Group velocity

C Series loading capacitance in NRI-TL unit cell

C ′ TL capacitance per unit length

L Shunt loading inductance in NRI-TL unit cell

L′ TL inductance per unit length

~S Poynting Vector

Zo Transmission-line characteristic impedance

Zin Input Impedance

xi

List of Tables

2.1 Geometry Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Resonant frequencies and band gap range. . . . . . . . . . . . . . . . . . 29

2.3 Extracted TL and loading parameters. . . . . . . . . . . . . . . . . . . . 41

3.1 Unit cell dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Resonance locations for antenna on 5x5 surfaces. . . . . . . . . . . . . . . 65

3.3 Resonance characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Summary of Measured and Simulated Results . . . . . . . . . . . . . . . 84

B.1 Eigenmode solutions for Cases 1 through 3 . . . . . . . . . . . . . . . . . 99

B.2 Eigenmode solutions for Case 1 with varying airbox sizes . . . . . . . . . 100

C.1 Unit cell geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xii

List of Figures

1.1 Geometry of the mushroom-like structure . . . . . . . . . . . . . . . . . . 3

1.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Lumped element model reflection phase profile . . . . . . . . . . . . . . . 5

1.4 TE and TM polarized plane-wave incidence on mushroom-like structure . 6

1.5 TL circuit model for calculating equivalent input impedance, ZL . . . . . 7

1.6 Homogenization Model reflection phase profile for TE and TM Incidence 8

1.7 Dispersion diagram as predicted by the lumped element model . . . . . . 9

1.8 Complete dispersion diagram of the mushroom-like structure using Ansoft

HFSS eigenmode solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9 Reduced surface-wave radiation in monopole antenna with mushroom-like

ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.10 Reduced mutual coupling in patch arrays using mushroom-like structures. 13

1.11 Microstrip patch phased array with mushroom-like structure to eliminate

scan blindess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.12 Ultra Wideband monopole with mushroom-like structure used as a band-

stop filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.13 Bent monopole utilizing in-phase reflection characteristics of mushroom-

like structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.14 Parametric study of dipole response over mushroom-like structure. . . . . 16

1.15 Broadband radiating elements over mushroom-like structure. . . . . . . . 17

xiii

1.16 Various HIS unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.17 Dispersion diagrams for (a) grounded patch and (b) Jerusalem cross unit

cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.18 Folded diple on a dog-bone HIS. . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Geometry of the mushroom-like structure. . . . . . . . . . . . . . . . . . 25

2.2 Simulation setups for determining the reflection phase characteristics and

dispersion diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Dispersion diagrams and reflection phase charactersitics of the three ge-

ometry cases. The parameter varied is the patch width, w, as shown in

Figure 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Eigenmode and scattering electric field and Poynting vector profiles for

Case 3 at its resonance points. PBCs are applied on all boundaries of the

unit cell with (βd)x = βd)y = 0. The PBCs correspond to virtual E or H

walls at the unit cell boundaries as shown. . . . . . . . . . . . . . . . . . 31

2.5 Case 1 Reflection Phase Response for varying angles of incidence. . . . . 34

2.6 Inclined angle of incidence on surface . . . . . . . . . . . . . . . . . . . . 35

2.7 Mapping the scattering resonances to the eigenmode dispersion diagram . 37

2.8 NRI-TL unit cell modelling the mushroom-like structure. . . . . . . . . . 38

2.9 Dispersion characteristics of the unit cell of Figure 2.8 with a Zo = 50Ω

TL of electrical length 60o at 2.5 GHz and loading elements L = 2nH and

C = 0.3pF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.10 Comparison of NRI-TL dispersion to FEM eigenmode dispersion. . . . . 42

2.11 Boundary conditions at resonances. . . . . . . . . . . . . . . . . . . . . . 45

3.1 Unit cell designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Unit cell reflection phase profiles and dispersion diagrams. . . . . . . . . 52

3.3 Dipole antenna on grounded substrate . . . . . . . . . . . . . . . . . . . 53

xiv

3.4 Grid of 7x7 unit cells for each case. 3x3 and 5x5 grids (not shown) are

also simulated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 |S11| and input impedance responses for dipole antenna on 3x3, 5x5 and

7x7 AMC-BG and AMC grids. . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Electric field distributions of 5x5 AMC-BG and AMC surfaces at resonance. 58

3.7 Input impedance responses for varying dipole lengths on 5x5 AMC-BG

and AMC grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Dipole response on 5x5 AMC-BG and AMC surfaces . . . . . . . . . . . 62

3.9 Input impedance responses for 26 mm dipole length on ground, 5x5 AMC-

BG and 5x5 AMC grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.10 Comparison of ‘PMC effect’ of various surfaces. . . . . . . . . . . . . . . 63

3.11 Relative surface-wave power density for 5x5 AMC-BG and AMC surfaces 66

3.12 AMC-BG and AMC radiation patterns for the various resonances at the

frequencies listed in Table 3.3 . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Co-axial fed dipole through ground plane with grounded second arm. . . 70

4.2 Matching network implemented with co-axial feed through ground. . . . 71

4.3 Side-fed antennas with integrated and external baluns. . . . . . . . . . . 72

4.4 Co-axial feed extending through ground plane. The outer conductor serves

as the via for the center patch. . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Reference case characteristics . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6 5x5 AMC response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 5x5 AMC radiation patterns. . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.8 5x5 AMC-BG response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.9 5x5 AMC radiation patterns. . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.10 Comparison of |S11| for the AMC-BG and AMC cases. . . . . . . . . . . 78

4.11 Radiation pattern comparison at dipole resonance and surface resonance

for AMC-BG and AMC cases. . . . . . . . . . . . . . . . . . . . . . . . . 79

xv

4.12 Fabricated AMC-BG and AMC surface protoypes. . . . . . . . . . . . . . 81

4.13 Final fabricated prototypes. AMC-BG board with dipole antenna shown

on left and AMC board with dipole antenna on right. . . . . . . . . . . . 82

4.14 Comparison of measured and simulated |S11| responses. . . . . . . . . . . 82

4.15 Comparison of measured and tuned simulated |S11| responses. . . . . . . 83

4.16 Comparison of measured and simulated radiation patterns for AMC and

AMC-BG cases. — Simulated - - - Measured. Patterns are plotted at

frequencies listed in Table 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1 Performance of 33.5 mm dipole on a 5x5 AMC-BG grid at a height of 3 mm. 89

5.2 Folded NRI-TL Monopole and MIMO antenna . . . . . . . . . . . . . . . 90

A.1 Transverse H-walls used for on-axis propagation . . . . . . . . . . . . . . 92

A.2 Extraction of host TL parameters . . . . . . . . . . . . . . . . . . . . . . 93

A.3 Extraction of lumped capacitance . . . . . . . . . . . . . . . . . . . . . . 94

B.1 TM0 mode for Case 1 showing erroneous eigenmode solution at (βd)x = 0 96

B.2 Case 1 dispersion profile showing 6 higher order modes . . . . . . . . . . 97

B.3 Case 1 dispersion profile showing first 3 higher order modes . . . . . . . . 97

B.4 E-field and Poynting vector distributions at eigenmode solutions . . . . . 99

C.1 (a)Reflection-phase and (b)dispersion profile of AMC-BG unit cell. . . . . 103

C.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

C.3 Dipole responses over 5x5 and 7x7 grids . . . . . . . . . . . . . . . . . . 104

xvi

Chapter 1

Introduction

1.1 Background

Over the last decade, an explosion of new ideas and services have been witnessed in

the communications industry that have greatly influenced the way humans interact with

each other. These include the evolution of social media and networking, the availability

of high speed internet on mobile devices such as smart phones and an emerging culture of

constant connectivity. This evolving culture coupled with increasing mobile penetration

has placed unprecedented bandwidth and reliability demands on existing network infras-

tructure. As a result, engineers working in the wireless industy are faced with new and

immediate challenges to deliver enhanced services at faster bitrates while maintaining

network reliability.

These trends have been the driving force behind much of the research conducted in RF

front end subsystems. In particular, the demand for multiple connectivity options and

services such as cellular frequencies, GPS, Bluetooth and WLAN has challenged antenna

engineers to revisit existing norms and come up with creative solutions to address these

issues. Antenna engineers are now forced to develop smaller and more efficient antennas

that are in close proximity to ground planes and can easily be integrated on to circuit

1

Chapter 1. Introduction 2

boards and not cause interference with other subsystem functionality. One research area

that has generated considerable interest for several years and has shown great potential

in addressing some of these challenges is that of planar electromagnetic band gap (EBG)

structures.

Planar EBG structures support a two dimensional bound surface-wave stop band over

a frequency bandwidth which is commonly referred to as the surface-wave band gap. The

existence of a band-gap has enabled these structures to be effectively used in minimizing

surface wave radiation [1] and reducing mutual coupling between antenna array elements

[2]. In addition, an incident plane wave reflected off planar EBG structures will experience

in-phase reflection, |φrefl| ≤ 90o, over a certain bandwidth. This bandwidth is referred to

as the in-phase reflection bandwidth and has earned this class of structures the additional

label of Artificial Magnetic Conductors (AMCs). This property is of particular interest to

antenna engineers since it allows an antenna to be closely spaced to the surface of the EBG

structure without inducing out-of-phase image currents that reduce the driving point

impedance and the radiated power of the antenna. Moreover, due to their planar nature,

such structures are easily integrated with existing microwave circuits. The ‘Sievenpiper’

Structure [1] is one such 2D EBG structure that has recieved considerable attention due

to its surface-wave band gap and scattering characteristics, however, certain gaps exist in

the understanding of how these properties relate to each other and their ultimate effect

on antenna performance characteristics such as impedance match bandwidth, gain and

front-to-back ratio.

The ‘Sievenpiper’ structure will be the central focus of this thesis and its dispersion

and scattering properties will now be examined in greater detail.


1.2 The ‘mushroom-like’ Sievenpiper Structure

The Sievenpiper structure, henceforth referred to as the mushroom-like structure, in its

most basic form consists of square metallization patches etched on a conductor-backed

dielectric substrate. The center of each patch is connected to the ground plane by means

of a via to create a shunt inductive loading. The patches are periodic in two dimensions in

the plane of the substrate and are seperated from each other by dielectric gaps to create

a series capacitive loading. Various modifications to the structure have been reported

in literature such as rectangular patches and multiple or offset vias [3] to enhance or

modify the dispersion and scattering properties of the structure. For simplicity, the basic

form of the structure will be considered in this thesis, although, the approach used in

Chapter 2 can be easily extended to analyze the characteristics of the modified structures.

Figure 1.1 shows a side view and top view of the geometry of the mushroom-like structure

and its geometrical parameters.

d

z

x 2D patch grid

Ground Plane

viast 2r

(a) Side View

d

w

x

y

(b) Top View

Figure 1.1: Geometry of the mushroom-like structure


1.2.1 Scattering Characteristics

The scattering characteristics of the structure shown in Figure 1.1 were originally derived

using a lumped element model [4]. A resonance is created in the structure by the capac-

itance due to the fringing gap fields between adjacent patches and the inductance due to

the current path created by the patches, vias and ground plane. This can be represented

as a parallel combination of an inductor and capacitor as shown in Figure 1.2 and the

structure can then be assigned a surface impedance given by Equation 1.1 and a resonant

frequency ωo given by Equation 1.2

Zs =jω L

1− ω2LC(1.1)

ωo =1√LC

(1.2)

The inductance and capacitance in the equation above are then derived analytically

in [4] and are given by the equations:

L = µoµr t, C =w(1 + εr)

πcosh−1

d

d− w(1.3)

where εr and µr are the relative permittivity and permeability of the substrate. The

reflection coefficient is then derived by considering the transmission line analogy with a

system impedance of Zo = η, where η is the wave impedance in free space, terminated

with a load impedance of ZL. Since the load is modeled as a purely reactive load,

the magnitude of the reflection coefficient is unity and the phase can be determined by

C

L

Figure 1.2: Equivalent Circuit Model


2 4 6 8 10−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Lumped ElementSimulated

Figure 1.3: Lumped element model reflection phase profile

Equation 1.5.

Γ =ZL − ηZL + η

(1.4)

φ = Im

ln

(ZL − ηZL + η

)(1.5)

The inductance and capacitance are calculated for an arbitrary geometry of d = 6 mm,

w = 5.7 mm, εr = 4.4, t = 1.6 mm and r = 0.2 mm using Equation 1.3. The resulting

reflection phase profile is plotted in Figure 1.3 alongside the reflection phase obtained by

an FEM simulation using Ansoft HFSS for comparison.

The reflection phase profile shown in Figure 1.3 reveals that the mushroom-like struc-

ture behaves as an AMC supporting in-phase reflection for an incident plane wave over a

certain bandwidth. The reflection phase varies from 180o to −180o around the resonant

frequency of the structure and an in-phase reflection bandwidth is defined from +90o

to −90o. It is seen that although the lumped element model provides physical intuition

regarding the resonance of the structure, its accuracy is limited. Moreover, the lumped

element model fails to capture the response of the structure for oblique incidence of TE


kθ

z

x

E

H

(a) TE Incidence

k

E

θ

z

xH

(b) TM Incidence

Figure 1.4: TE and TM polarized plane-wave incidence on mushroom-like structure

and TM polarized waves. The resonance frequency, ωo, varies as the angle of incidence is

swept for a TE/TM polarized plane-wave. In addition, for a TM polarized plane-wave,

dual resonances are observed in the scattering response. Figure 1.4 shows a TE and TM

plane-wave incidence on the mushroom-like structure.

The angle-dependent scattering properties of the mushroom-like structure can be ac-

curately predicted using a homogenization model such as the one proposed in [5]. For

the TE case, the surface impedance of an array of patches is calculated using averaged

boundary conditions derived from the approximate Babinet principle [6] and added in

shunt to the TE surface impedance of a conductor-backed dielectric. The assumption

made here is that the TE polarized wave does not excite the vias and hence their contri-

bution to the reflection phase properties can be neglected [7]. For a TM polarized wave,

the vias embedded in the dielectric are modelled as an effective wire medium whose sur-

face impedance is calculated in [6] and added in shunt to the TM surface impedance for

an array of patches. The impedances for the TE and TM cases from [5] are shown in

Equations 1.6 to 1.9 where Zg refers to the input impedance for a grid of patches and

Zv refers to the input impedance for a vias embedded in a conductor-backed dielectric.

Figure 1.5 describes the equivalent TL circuit.


,TE TMvZ,TE TM

gZoZ

LZ

Figure 1.5: TL circuit model for calculating equivalent input impedance, ZL

ZTEg = jωµo

tan(kz√εrt)

kz√εr

(1.6)

ZTEv = −j ηeff

2α(

1− k2z2k2eff

) (1.7)

ZTMg = jωµo

tan(γTM t)

γTM

k2 − k2x − k2pk2 − k2p

(1.8)

ZTMv = −j ηeff

2α(1.9)

where

ηeff = ηo/εeff , keff = ko√εeff , εeff = (1 + εr)/2

α =keffd

πln

(1

sin(π(d− w)/2d)

)kp = 1/a

√1

2πln

d2

4r(d− r)

γ2TM = ω2εoεrµo −εrεnk2x

εn = εt

(1−

k2pk2εr

)(1.10)

Once the input impedance has been calculated, the reflection coefficient can be calcu-

lated using Equation 1.5. The resulting reflection phase profile is plotted for TE and TM

incidences as shown in Figures 1.6(a) and 1.6(b) for the geometry described earlier. This

model accurately describes the angular dependant nature of the mushroom-like struc-

ture’s resonant frequency. In addition, it correctly predicts the dual resonance behaviour


2 4 6 8 10−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Simulated 0o

Analytical 0o

Simulated 30o

Analytical 30o

(a) TE Incidence at 0o and 30o

2 4 6 8 10−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Simulated 0o

Analytical 0o

Simulated 30o

Analytical 30o

(b) TM Incidence at 0o and 30o

Figure 1.6: Homogenization Model reflection phase profile for TE and TM Incidence

of the structure under oblique TM polarization as shown in Figure 1.6(b). However, it

does not provide the same level of physical insight into the resonance characteristics of

the structure as the lumped element model.

In the next chapter, the resonances associated with the TM polarizations under vary-

ing angles of incidence will be explained from a different perspective. First, the fre-

quencies of the various resonances will be related to the dispersion characteristics of the

structure and will then be explained using a simple Negative Refractive Index - Transmis-

sion Line (NRI-TL) equivalent circuit. It will be seen that the dispersion characteristics

of the equivalent NRI-TL circuit within the leaky-wave region accurately describe the

angular dependence of the resonant frequency and also capture the dual mode resonances

observed in Figure 1.6(b). The NRI-TL model is, however, limited to capturing only the

resonant frequency of the structure for a given angle of incidence (which corresponds to

the zero reflection phase frequency) and not the entire frequency varying profile of the

reflection phase as described by the model in [5]. It is also unable to predict the reso-

nant frequencies of the TE mode since the equivalent circuit can model only quasi-TEM

modes.


0 30 60 90 120 150 1800

2

4

6

8

10

12

14

Phase Shift [Degrees]

Fre

qu

ency

[G

Hz]

TMTELight Line

Resonance Frequency

Figure 1.7: Dispersion diagram as predicted by the lumped element model

1.2.2 Dispersion Characteristics

The dispersion relations for TM and TE surface waves are originally derived in [4] and are

repeated in Equations 1.11 and 1.12 respectively. The surface impedance, ZL, derived

from a lumped element model in Equation 1.2 is used in the dispersion equations and the

resulting dispersion curves for the geometry under discussion are plotted in Figure 1.7.

kTM =ω

c

√1− Z2

L

η2(1.11)

kTE =ω

c

√1− η2

Z2L

(1.12)

It is observed from Figure 1.7 that the structure supports TM surface wave modes at

low frequencies which are cut off above the resonant frequency predicted by the lumped

element model. Above the resonant frequency, the propagation of TE surface modes is

predicted. Since this model assumes a uniform surface impedance, its accuracy is limited

to regions where the phase shift per unit cell is much smaller than unity, βd 1 rad.

As the phase shift per unit cell increase and becomes of the order of unity or greater,


the effects of the structure’s periodicity become important and the model breaks down.

Additionally, it is seen that this model does not predict a surface-wave band gap which

is known to exist through experiments and Finite Element Method (FEM) simulations.

Indeed, any homogenization model would break down in the small wavelength limit since

the periodic nature of the structure must be taken into account.

Advances in computational power over the last decade have enabled numerical analysis

of the dispersion properties of the mushroom like structure. An eigenmode simulation of

the geometry under consideration is carried out using Ansoft HFSS [8] and the dispersion

characteristics are studied by plotting the Bloch-wave vectors along the edges of the

irreducible Brillouin zone [9]. The resulting band structure is shown in Figure 1.8. The

eigenmode simulation shows the existence of a fundamental forward mode starting at DC

with a propagation constant very close to that of light, suggesting the mode primarily

exists in the air region above the surface. A backward wave mode (labeled TM0) then

emerges at low frequencies and the interaction of the forward and backward wave modes

gives rise to the lower edge of the band gap [10]. A higher order TM mode is then observed

(labeled TM1) which has a solution at the Γ point and consists of a small backward-wave

section within the light line. A degeneracy is then noted at the Γ point which consists

of a TE and TM mode (labeled TE and TM2). Conventionally, the bound surface-wave

band gap is defined from the upper edge of the fundamental mode (TM0) to the light

line crossing of the first upper order mode (the TE mode in this case) which occurs in

the Γ→ X region.

In Chapter 2, FEM simulations will be used to analyze the dispersion curves of

various geometries to draw insight into the relation between the dispersion and scattering

characteristics of the mushroom-like structure. An equivalent NRI-TL circuit model

will then be used to calculate the on-axis dispersion properties and determine the zero

phase shift resonant frequencies. All subsequent dispersion diagrams will be plotted from

Γ→ X for simplicity since the location of the surface-wave band gap can be determined


from this boundary of the irreducible Brillouin Zone.

1.2.3 Applications

Surface-wave Suppression

The surface-wave suppression properties of the mushroom-like structure have resulted in

its implementation in numerous applications. In Sievenpiper’s original work [1], a patch

antenna with improved radiation characteristics is proposed by making the antenna res-

onant in the surface-wave band gap of the mushroom-like structure. Reduced ripple is

observed in the radiation pattern since the surface-waves are now attenuated as they

propagate towards the edges of the ground plane. Since then, various other designs have

been proposed exploiting the surface-wave suppression characteristics of the mushroom-

like structure. A monopole antenna embedded in a mushroom-like structure is proposed

in [11] which also utilizes the surface-wave suppression characteristics to improve radi-

ation patterns. Figure 1.9 shows a comparison of surface-wave propagation with and

without the mushroom like structure.

0

2

4

6

8

10

12

14

16

Γ X M Γ

Fre

qu

ency

[G

Hz]

TM0TM1TETM2Light Line

Figure 1.8: Complete dispersion diagram of the mushroom-like structure using AnsoftHFSS eigenmode solver.


(a) Monopole embedded in mushroom-like ground plane

(b) Surface-waves on conventionalground

(c) Surface-waves on EBG ground

Figure 1.9: Reduced surface-wave radiation in monopole antenna with mushroom-likeground. From [11] c© 2009 PIERS

A low mutual coupling design for microstrip antenna arrays is proposed in [2] which

utilizes the mushroom-like structure to minimize coupling in patch antennas due to prop-

agating surface waves. Figures 1.10(a) and 1.10(b) show the geometry, return loss and

mutual coupling of the patch array with and without the mushroom-like structure. Mu-

tual coupling effects for three patch sizes (2mm to 4mm) are investigated while keeping

the gap distance between the patches constant. When the surface-wave band gap coin-

cides with the resonant frequency of the patch array and a mutual coupling reduction of

8 dB is achieved.

Another useful application of the mushroom-like structure is the elimination of scan

blindness in phased antenna arrays [12]. Scan blindness is caused by interference between

Floquet modes of the array and surface-wave modes of the same propagation constant.

This limits the useful scan range of the phased array and results in a lower efficiency.


(a) Patch array with mushroom-like structure (b) Measured return loss and mutual couplingof patch array

Figure 1.10: Reduced mutual coupling in patch arrays using mushroom-like structures.From [2] c© IEEE 2003

Scan blindless can be eliminated by incorporating the mushroom-like EBG in the unit

cell of the array and hence suppressing the surface-waves. Figure 1.11(a) shows a unit cell

of a microstrip patch phased array and Figure 1.11(b) shows the scanned characteristics

of the patch array. The solid lines represent the conventional case and the dashed lines

represent the effects of including the mushroom-like structure. Scan blindness is observed

in the conventional occuring at approximately 50o where the magnitude of the reflection

co-efficient is almost unity, however, no scan blindness is observed in the case where the

mushroom-like structure is included in the unit cell.

Finally, in [13], the surface-wave band gap characteristic of the mushroom-like struc-

ture is used to implement a rejection filter in an Ultra Wide Band (UWB) monopole an-

tenna. Interference from undesired frequencies can be rejected by designing the surface-

wave band gap to lie within the appropriate frequency range. The authors propose a

UWB monopole with a bandwidth from 3.1 − 10.6 GHz with a rejection bandwidth of

0.7 GHz around 5.5 GHz. Figures 1.12(a) and 1.12(b) show the geometry and VSWR

response of the UWB monopole.


(a) Phased array unit cell withEBG

(b) Scan characteristics with and withoutEBG

Figure 1.11: Microstrip patch phased array with mushroom-like structure to eliminatescan blindess. From [12] c© IEEE 2004

In-phase Reflection

Several authors have also utilized the in-phase reflection characteristics of the mushroom-

like structure to design antennas in close proximity to ground planes. A bent monopole

on a ground plane [14] is shown in Figure 1.13(a). The monopole operates within the in-

phase reflection bandwidth of the mushroom-like structure and hence maintains a good

impedance match. In addition, it is observed that the pattern is skewed in the H-plane

as shown in Figure 1.13(c). A switching mechanism is then implemented to create two

(a) UWB monopole geometry (b) Measured and simulated VSWR with andwithout the mushroom-like structure

Figure 1.12: Ultra Wideband monopole with mushroom-like structure used as a bandstopfilter. From [13] c© IEEE 2011


dimensional beam switching as shown in Figure 1.13(d).

x

z

Cross View EBG surface

wire

(a) Bent monopole over EBG

2 3 4 5 6−20

−15

−10

−5

0

Freq. (GHz)

S 11

(dB

)

(b) S11 response

−5 dB

0 dB

5 dB

10 dB 30°

60°

−150°

−120°

90°−90°

−60°

120°

150°

−30°

180°

0°

φ= 0°, co–pol.

φ= 90°, co–pol.

(c) Radiation patterns

x

y

z

1 2

3

4

1 23 4

(d) Pattern diversity

Figure 1.13: Bent monopole utilizing in-phase reflection characteristics of mushroom-likestructure. Pattern diversity is created by introducing a switching mechanism to changethe direction of the main lobe. From [14] c© IEEE2004

One of the key challenges in using the mushroom-like structure as a High Impedance

Surface (HIS) that provides an in-phase reflection bandwidth was to determine the op-

timum operating point of an antenna placed close to the surface. A detailed parametric

study of a dipole antenna on top of a mushroom-like EBG surface was conducted in [15].

Numerous FDTD simulation were carried out where a dipole antenna was kept at a fixed

height above the mushroom-like structure and its length was varied as shown in Fig-

ure 1.14(a). The frequency region where the best impedance match for the dipole was

then compared to the reflection phase profile of the mushroom-like structure. It was


concluded that the frequency region where the dipole is best matched corresponds to

the 90 ± 45o region of the reflection phase profile. The S11 results are summarized in

Figure 1.14(b) where the length of each dipole is normalized to λ12GHz. The reflection

phase profile is shown in Figure 1.14(c) with the 90± 45o region shaded.

Dipole EBG surface

(a) Dipole over mushroom-like structure

10 12 14 16 18−40

−35

−30

−25

−20

−15

−10

−5

0

Freq. (GHz)

S 11

(dB

)

0.60

0.54

0.48 0.42 0.36

0.32

0.26

(b) S11 responses for varied dipole length

10 12 14 16 18−50

0

50

100

150

200

Freq. (GHz)

Ref

lect

ion

phas

e (D

egre

es)

(c) Reflection Phase profile of unit cell

Figure 1.14: Parametric study of dipole response over mushroom-like structure. From[15] c© IEEE 2003

It is noted in Figure 1.14(b) that the operating bandwidths of the dipole are nar-

rowband. However, since that work was published, several authors have obtained a

broad bandwidth for antenna operation by using broadband antenna elements instead

of dipoles. Two configurations are shown in Figures 1.15(a) through 1.15(d) depicting

a UWB monopole [16] and a folded-bowtie antenna [17] along with their respective S11

responses.

Multiple resonances are observed in the S11 response for the two cases presented

above indicating that the mushroom-like structure is acting as a resonant surface as

well and not just as an in-phase reflector. It has been noted in previous works [18, 19]


L2

L1

W1

Artificial Magnetic Conductor

Ultra-widebandMonopoleAntenna

SMAConnector

(a) UWB Monopole (b) UWB Monopole S11 Response

(c) Folded bowtie antenna

Frequency [MHz]200 300 400 500

0

-5

-10

-15

-20

-25

S11

[dB

]

(d) Folded bowtie antenna S11 response

Figure 1.15: Broadband radiating elements over mushroom-like structure. UWBmonopole from [16] c© IEEE 2011. Folded bowtie antenna from [17] c© IEEE 2008

that multiple resonances may be excited on a mushroom-like surface by a closely spaced

antenna. In [18], an optimization algorithm is used to design a dipole on a mushroom-

like structure and a dual resonant behaviour is observed. Similarly, in [19], it is stated

that coupling between the dipole and mushroom-like surface results in a dual resonance.

However, the exact nature of these resonances has yet to be investigated in literature and

will be addressed in this thesis. The case of a dipole over a mushroom-like structure will

be revisited in Chapter 3 and it will be shown that there are at least two useful resonances

that occur when a dipole is placed close to the surface of the structure. One of these will

be shown to be the dipole resonance as observed in [15], the second will be attributed

to the structure and will be shown to be independent of the length of the dipole or the


number of unit cells of the structure. It will then be shown that it is possible to merge

these responses to obtain a broadband response as in [18] and [19].

1.3 Other High Impedance Surfaces

In addition to the mushroom-like structure, several other HIS designs have also been

successfully used as antenna ground planes. Figure 1.16 shows the geometry of a subset

of the unit cell designs available in literature including a capacitive surface of square

patches backed by a ground plane [20], a ‘jerusalem cross’ structure [21], an open ring [22]

and a ‘dog-bone’ structure [23]. Various other unit cells have also been proposed utilizing

genetic optimization techniques [24] and fractal geometries [25].

wg

(a) Grounded Patch

w

a

l

d

(b) Jerusalem Cross

θ wg

a

(c) Open Ring

w

a

l

d

(d) Dog-bone

Figure 1.16: Various HIS unit cells

Reflection characteristics similar to those of the mushroom-like structure are exhibited

by each of these structures. However, none of these structures support a surface-wave

band gap at low frequencies similar to that of the mushroom-like structure due to a


0

5

10

15

0 30 60 90 120 150 180

TM0

TE0

TM1

(a)

0

5

10

15

0 30 60 90 120 150 80

TM0

TE0

TM1

(b)

Figure 1.17: Dispersion diagrams for (a) grounded patch and (b) Jerusalem cross unitcell from [21] c© PIER 2011

lack of vias. The absence of such a band-gap is attributed to the inability of these

surfaces to host backward wave pass bands which is responsible for the creation of the

band gap in mushroom-like structures. Figure 1.17(a) and Figure 1.17(b) describe the

dispersion properties of the grounded patch and ‘Jerusalem cross’ structure respectively.

It is noted that no surface-wave band gap is observed in the transition between TM and

TE surface-wave modes.

Nevertheless, these surfaces have also been successfully used as ground planes for

antennas. In [23], the ‘dogbone’ structure is used as the ground plane for a folded dipole

antenna. Figures 1.18(a) and 1.18(b) show the top and middle layers of the antenna. A

grounded substrate (not shown) is placed underneath the antenna. Figure 1.18(c) shows

the measured and simulated return loss characteristics of the antenna.

These works show measured improvements in antenna radiation patterns as a result of

HIS that do not possess a surface-wave band gap. It is then relevant to ask at this point

whether there is any additional benefit of the surface-wave band gap of the mushroom-

like structure when it is used as an HIS for antenna ground plane applications. It is well

understood that the surface-wave band gap plays an important role in the suppression of

surface waves when the structure is in the same plane as the radiating element but the

role of the surface-wave band gap when the structure is acting as an HIS is not clear.


(a) Top Layer (b) Middle Layer

4 4.5 5 5.5 6 6.5 7-50

-40

-30

-20

-10

0

Frequency (GHz)

|S1

1|

MeasurementsIE3D thick metal + infty GNDIE3D thick metal + fnt GNDCST

(c) Measured and Simulated S11 response

Figure 1.18: Folded diple on a dog-bone HIS. From [23] c© IEEE 2009

Intuitively, it is expected that the band gap would result in mitigation of surface-waves

even for this scenario but the improvement, if any, has yet to be quantified. It will be

shown in Chapter 3 that this is indeed the case and that the surface-wave band gap plays

a significant role in reducing radiation from the edges of the substrate compared to a

case when the band gap is not present.


1.4 Motivation

Planar EBG structures have thus far played an important role in improving antenna

performance and mitigating the effects of a ground plane. To further utilize their po-

tential and enhance their performance, their properties must be fully understood. Al-

though the presence of the surface wave band-gap and the PMC frequency have been

well documented, studied and even implemented in applications, there exists a gap in

the understanding of the exact relationship between these two fundamental properties of

the structure. It was originally assumed in [4] that the PMC frequency point coincided

with the location of the bound surface-wave band gap. However, it was later shown

in [26] and [27] that the two properties do not necessarily coincide. Since then, various

attempts have been made to relate the two properties of the structure. In [28], a detailed

parametric study is conducted for various geometries of the mushroom-like structure but

no conclusive result is provided. Similarly, in [29], homogenization models are used to

calculate analytical dispersion diagrams for the mushoom-like structure and a hybrid

analytical-numerical technique is proposed to relate the scattering and dispersion prop-

erties. Finally, in [30], a pole-zero matching method is used to investigate the relationship

between the scattering and dispersion properties for a dipole FSS but no physical insight

or intuition is provided.

Furthermore, it is not yet evident whether the location of the PMC frequency relative

to the surface wave band-gap plays an important role in improving antenna performance.

As mentioned earlier, the use of high impedance surfaces without a surface wave band-

gap to improve the radiation characteristics of an antenna placed close to a ground plane

has also been shown in literature. This inspires the question whether there is a significant

benefit of using a HIS that also supports a surface-wave band gap as an antenna ground

plane. A quantitative analysis of the benefits of the PMC frequency of the mushroom-like

EBG coinciding with the location of the band-gap has yet to be conducted.

This thesis will describe the relationship between the PMC frequency point and the


surface wave band-gap with the aid of full-wave simulations and NRI-TL theory for the

mushroom-like structure. It will be shown through the use of dispersion diagrams that it

is not necessary for the PMC frequency and surface-wave band gap of the mushroom-like

structure to coincide and certain design choices have to be made to ensure that this is

the case. An explanation of the structural resonances that give rise to the multiple PMC

frequencies observed for inclined angles of incidence will be offered and it will be shown

that the PMC frequency points for all angles of incidence except grazing can be mapped

directly on to the dispersion diagrams.

Next, the effects of having the PMC frequency coincide with the location of the

surface wave band-gap will be studied numerically to obtain an understanding of the

various resonances that may be excited in the mushroom-like structure by a closely placed

dipole antenna. It will then be shown that there are at least two resonances created

in the impedance match bandwidth of the dipole/mushroom-like structure system and

that these resonances can be merged to create a broadband response. The influence

of having the surface-wave band gap and PMC frequency coincide on the impedance

match bandwidth, radiation patterns and front-to-back ratios will then be quantified.

Finally, simulated and measured results will be presented for dipole located on top of a

mushroom-like structure for a case where the PMC frequency and band gap coincide and

for a case where they are seperated.

1.5 Thesis Outline

This thesis is divided as follows. Chapter 2 presents full wave simulations that are

used to determine a relationship between the scattering and dispersion properties of

the mushroom-like structure. Field distributions are examined for the dispersion and

scattering simulations at resonance and NRI-TL theory is used to obtain a physical

understanding of the resonances associated with the structure.


In Chapter 3, two unit cells are proposed for examination. The surface-wave band

gap and PMC frequency coincide for one of the unit cells whereas for the second, the two

properties are seperated. A dipole antenna is then placed close to a surface composed of

each unit cell and the various resonances that are excited are then examined for varying

dipole lengths and surface sizes. Additionally, the dipole antenna’s radiation properties

are contrasted for each unit cell case to quantify improvements in radiation patterns as

a result of aligning the two properties.

Practical designs for each of the surface cases are presented in Chapter 4 and the

fabrication and measurement process is described for each of the designs. A comparison

of simulated and measured results is then presented.

Finally, in Chapter 5, the thesis is concluded with a summary of the results and

potential future directions.

Chapter 2

Relating the Scattering &

Dispersion Characteristics

This chapter provides a theoretical understanding of the location of the PMC frequency

point on the dispersion diagram of the mushroom-like structure. An explanation of the

two distinct zero-phase, (βd)x,y = 0, resonance points is also provided. To that end,

full-wave simulations discussing the role of the (βd)x,y = 0 resonances in the location of

the PMC frequency point and the surface-wave band-gap are presented, followed by an

equivalent circuit model analysis emphasizing the boundary conditions at the ends of the

unit cell.

2.1 Full Wave Simulations

2.1.1 Simulation Setup

Figure 2.1 describes the physical parameters of the mushroom-like structure. The elec-

tromagnetic properties of such a periodic structure can be investigated by studying the

properties of a single unit cell composing the structure. Three unit cell geometries are

examined as a part of this study. All of the geometric parameters of the structure are

24

Chapter 2. Relating the Scattering & Dispersion Characteristics 25

Table 2.1: Geometry Cases.Case w [mm]

1 5.92 5.73 5.2

fixed, except for the patch width which is varied: w = 5.9, 5.7 and 5.2 mm as shown in Ta-

ble 2.1.1. The fixed parameters are the substrate thickness, t = 1.6 mm and permittivity,

εr = 4.4, unit cell periodicity, d = 6 mm, and via radius, r = 0.2 mm.

Two types of simulations are carried out using Ansoft HFSS: the first is a scattering

simulation to determine the reflection phase characteristics of the unit cell [31] and the

second is an eigenmode simulation to determine the natural resonances and dispersion

properties of the unit cell. Figure 2.2 shows the simulation setups for both cases. Since

an open structure is being considered, the airbox located above the structure must be

appropriately terminated. For the eigenmode analysis, the airbox is terminated with

a Perfectly Matched Layer (PML) which absorbs all incoming radiation and prevents

reflection. In addition, linked Periodic Boundary Conditions (PBCs) are applied in both

the x and y directions. Propagation along the x direction is considered for the eigenmode

d

z

x 2D patch grid

Ground Plane

viast 2r

(a) Side View

d

w

x

y

(b) Top View

Figure 2.1: Geometry of the mushroom-like structure.


Floquet Port

PB

C

PB

C

z

x

De-

embe

ddin

g

(a) Scattering Simulation Setup

PML

PBC

PBC

z

x

(b) Eigenmode Simulation Setup

Figure 2.2: Simulation setups for determining the reflection phase characteristics anddispersion diagram.

analysis due to the symmetry of the structure. The phase shift transverse to the direction

of propagation, (βd)y = 0 is kept fixed while the phase shift along the direction of

propagation is varied as 0 ≤ (βd)x ≤ π. For the scattering simulation the airbox is

excited with a Floquet port de-embedded to the top surface of the unit cell [31] and

linked PBCs are applied in the x-direction in the y-direction with a zero degree angle

of incidence. This simulates an infinite periodic structure illuminated by a normally

incident plane-wave.

2.1.2 Simulation Results

The resulting dispersion diagrams and normal incidence scattering plots are shown in

Figure 2.3. From the dispersion diagrams, it is seen that the structure supports a funda-

mental TM mode, followed by one TE mode and two TM modes in the frequency range

of interest. The determination of whether a mode is TE or TM is made by studying

the field distributions of the mode in question. For a TE mode, electric fields are ob-

served to be predominantly transverse to the direction of propagation whereas for a TM


mode, the electric fields are observed to be predominantly longitudinal to the direction

of propagation. Detailed field distributions will be shown later in this chapter.

In all of the cases, an initial dual-mode pass band is observed that consists of both a

forward-wave TM mode and a backward-wave TM mode. The beginning of the surface-

wave band gap is due to the contra-directional coupling of these two modes [10]. The

upper edge of the band gap is determined by the first of the leaky modes to cross the

light line. For Case 1, the lower edge of the band gap occurs at 4.5 GHz and the upper

edge is determined by the TE mode light line crossing at 6.3 GHz. For Case 2, the

lower edge is at 5.0 GHz and the upper edge is at 7.5 GHz and is once again determined

by the TE mode light line crossing. However, for Case 3, the lower edge occurs at 5.6

GHz but the upper edge is determined by the TM mode light line crossing at 7.6 GHz.

A multi-mode upper passband then follows which consists of both TM and TE modes.

Two distinct (βd)x = (βd)y = 0 resonances are observed in each dispersion diagram and

are labeled fgap and fvia. It will be shown later that fgap corresponds to the resonance

created by the distributed inductance of the TL and the gap capacitance whereas fvia

corresponds to the resonance created by the distributed capacitance of the TL and the via

inductance. It is interesting to note at this point, that the TM1 mode always supports

a small backward-wave band whether it is associated with the fgap or fvia resonance.

In addition, the TE mode is always associated with the fgap resonance. Finally, the

resonance at fgap is always a doubly degenerate resonance consisting of a TM and a TE

mode.

The scattering plots describe the phase of the electric field that is reflected by the

surface as a function of frequency for two orthogonal polarizations that correspond to

normally incident illumination: (i) x-polarization (TM) and (ii) y-polarization (TE). As

expected, the TM and TE scattering responses are identical due to the symmetry of the

structure. For all cases, a 0o reflection phase is observed as expected at a single frequency

point labeled as fPMC for both TM and TE normally incident waves.


0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]

Band Gap: 4.5 − 6.3 GHz


fvia

fgap

(a) Case 1 (5.9 mm): Dispersion

2 4 6 8 10 12 14−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Ban

d G

ap

TM (x−pol)TE (y−pol)

fPMC

(b) Case 1 (5.9 mm): Scattering

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]



fgap

fvia

(c) Case 2 (5.7 mm): Dispersion

2 4 6 8 10 12 14−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Ban

d G

ap


fPMC

(d) Case 2 (5.7 mm): Scattering

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]


TM0TM1TETM2Light Linef

gap

fvia

(e) Case 3 (5.2 mm): Dispersion

2 4 6 8 10 12 14−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

Ban

d G

ap


fPMC

(f) Case 3 (5.2 mm): Scattering

Figure 2.3: Dispersion diagrams and reflection phase charactersitics of the three geometrycases. The parameter varied is the patch width, w, as shown in Figure 2.1


Table 2.2: Resonant frequencies and band gap range.Case fvia [GHz] fgap [GHz] fPMC [GHz] Band Gap Range [GHz]

1 6.45 5.72 5.83 4.5− 6.32 6.46 6.80 6.92 5.0− 7.53 6.60 8.45 8.60 5.6− 7.6

For each one of the three cases in Figure 2.3, it is observed that fgap, the doubly

degenerate (βd)x = (βd)y = 0 resonance, corresponds to fPMC . Hence, the eigenmodes

at fgap consisting of a TE (y-polarized) and a TM (x-polarized) mode are equivalent to

the resonances at fPMC obtained by illumination of the surface by a normally incident

TE or TM wave respectively. At this point, this determination is made by observing the

close proximity of fgap to fPMC and by noting that both fgap and fPMC are degenerate

resonances. It will be shown that the field distributions and unit cell boundary conditions

at these two resonances are identical in Sections 2.1.3 and 2.3 respectively to verify

that fgap and fPMC are indeed identical resonances. It is also noted that the second

(βd)x = (βd)y = 0 resonance corresponding to fvia is not excited by a normally incident

TE or TM wave. Since both fgap and fvia are (βd)x = (βd)y = 0 resonances, it would

appear as if both resonances should be excited by a normally incident wave; however,

an examination of the field profiles in Section 2.1.3 will demonstrate why this is not the

case. In addition, it is seen that depending on the geometry of the structure, fPMC can

lie inside the band gap as with cases 1 and 2 or outside the band-gap as with case 3.

Table 2.2 summarizes the values of fgap, fvia, fPMC and the edges of the band gap for

each of the three cases.

2.1.3 Field Distributions

The field profiles for Case 3 from both the eigenmode analysis and scattering simulation

are now examined to understand the nature of the doubly degenerate resonances at fgap

and fPMC , the single resonance at fvia and to determine why fvia is not excited by a


normally incident TE or TM wave. It is noted that the field profiles for Cases 1 and 2

at fgap and fvia are qualitatively identical to those of Case 3 and one geometry is picked

only for brevity. To study the field distributions, electric field vectors are plotted on

the ground plane and on the boundary walls of the unit cell in the vicinity of the gap

region whereas the Poynting vector is plotted at various elevations above the surface.

The transverse (y-z) and longitudinal (x-z) cut planes are shown for both TM2 and TE

modes at fgap and the TM1 mode at fvia. The corresponding field profiles from the

scattering simulation at fPMC are also presented for comparison. The results are shown

in Figure 2.4.

Figure 2.4(a) shows the longitudinal (x-z) and transverse (y-z) cut planes for the TE

mode field distributions at fgap obtained from the eigenmode simulation. It is observed

that the capacitors along the transverse cut planes are strongly excited for this mode

and that electric field on the ground plane is null at the edges and the central bisecting

plane. The central null implies that the via is not excited for this mode and the same

field distributions at this frequency would be obtained in the absence of a via. In fact,

it has been shown [32] that the via can be removed from the mushroom-like structure

without affecting its scattering properties. From these observations, it is clear that the

boundaries along the transverse direction and the central bi-secting plane are acting as

virtual E-walls due to the lack of tangential E-fields. From the longitudinal (x-z) cut

plane, it is observed that the boundaries along the longitudinal direction are acting as

virtual H-walls since the capacitors along this direction are not excited and the E-fields

are tangential to the boundary. Finally, it is noted that this mode is a radiating leaky

mode since the Poynting vector demonstrates that energy is radiating away from the

surface in the normal direction.

The field distributions obtained from a normally incident, TE polarized plane-wave

excitation at fPMC are shown in Figure 2.4(b). It is seen that the field profile in the

longitudinal and transverse direction is identical to that obtained from an eigenmode


simulation in Figure 2.4(a). The main difference observed is the lack of net power flow

away from the surface in the scattering simulation. This is due to the fact that the

simulation space is terminated in a port, rather than a PML, which results in a standing

wave being set up between the port and the surface as opposed to power leaking from

(a) TE Eigenmode at fgap(8.45 GHz)

z

y

z

yPlaneWave

PlaneWave

PB

C (

E-w

all)

PB

C (

E-w

all)

PB

C (

H-w

all)

PB

C (

H-w

all)

E

(b) TE Scattering at fPMC(8.60 GHz)

(c) TM2 Eigenmode at fgap(8.45 GHz)

z

y

z

xPlaneWave

PlaneWave

EP

BC

(E

-wal

l)

PB

C (

E-w

all)

PB

C (

H-w

all)

PB

C (

H-w

all)

EH

k

. H E

k

.

(d) TM2 Scattering at fPMC(8.60 GHz)

z

y

z

x

PB

C (

H-w

all)

PB

C (

H-w

all)

PB

C (

H-w

all)

PB

C (

H-w

all)

Non-radiating

mode

E

Non-radiating

mode

(e) TM1 Eigenmode at fvia(6.60 GHz)

z

y

z

xPlaneWave

PlaneWave

E

PB

C (

E-w

all)

PB

C (

E-w

all)

PB

C (

H-w

all)

PB

C (

H-w

all)

EH

k

. H E

k

.

(f) TM1 Scattering at fvia(6.60 GHz)

Figure 2.4: Eigenmode and scattering electric field and Poynting vector profiles for Case3 at its resonance points. PBCs are applied on all boundaries of the unit cell with(βd)x = βd)y = 0. The PBCs correspond to virtual E or H walls at the unit cellboundaries as shown.


the structure and being absorbed the PML.

Similar eigenmode and scattering field distributions are seen for the TM2 mode at fgap

in Figure 2.4(c) with the only difference being the locations of the virtual E-walls and H-

walls interchanged. The TM mode sees H-walls along the transverse (y-z) direction and

E-walls along the longitudinal (x-z) direction. Once again, the field distributions at fPMC

for a TM polarized normally incident plane-wave excitation, as shown in Figure 2.4(d),

are identical to those obtained from the corresponding eigenmode simulation.

As a result, it is apparent that the degenerate resonance occuring at fPMC for TE

and TM plane-wave excitations corresponds to the degenerate resonance obtained at fgap

from the eigenmode simulations as seen through analysis of the field distributions. This

resonance is attributed to the electric fields fringing across the gaps resonating with the

magnetic fields between the patch and the ground plane, hence the label fgap, and will

be confirmed later in this chapter through the use of an equivalent circuit model.

The field profiles corresponding to the TM1 eigenmode at fvia in Figure 2.4(e) show

strong electric fields between the patch and the ground plane that remain approximately

constant throughout the unit cell and have the same profile along both the x and y

directions. The capacitive gaps are not excited in this case and all four side boundaries

act as virtual H-walls since the E-fields are tangential to all of the unit cell boundaries.

Additionally, it is noted that the Poynting vector profile for the region above the structure

is non-existent and hence this mode does not radiate. This appears unusual as the mode

is located within the light cone, as shown in Figure 2.3(e), and such modes generally

correspond to leaky radiating modes for open structures. However, it is important to

note that the leaky nature of the mode at fgap is due to the excitation of the fields

in the capacitive gap, and since these fields are not excited at fvia the mode does not

radiate. The resonance associated with this mode is due to the capacitance between

the patch and ground plane and the inductance of the via. The excitation of the via is

confirmed by plotting the current distribution on the via (not shown) and will also be


shown through the use of an equivalent circuit model in Section 2.3. For comparison, the

field profile corresponding to a TM plane-wave excitation at fvia is shown in Figure 2.4(f).

It is apparent that this field profile does not correspond to the eigenmode field profile

of Figure 2.4(e). In fact, the scattering field profile at fvia is similar to that of the

TM plane-wave excitation at fgap (see Figure 2.4(d)). However, since the field profile of

the incident TM plane-wave excitation at fvia(longitudinal E-walls/ transverse H-walls)

does not correspond to the eigenmode field profile (four H-walls), the structure does not

respond in a resonant manner.

The similarities between the field profiles of the eigenmode simulations and scattering

simulations can be interpreted as follows. The eigenmode resonances correspond to the

source-free natural modes of the mushroom-like structure, whereas the scattering simu-

lations correspond to the same structure being driven to resonance by an external source

which, in this case, is an impinging plane-wave. Therefore, for the driven simulations,

the structure responds in a resonant manner at the frequency defined by its eigenmode

frequency as long as the field profile of the excitation is consistent with that of the eigen-

mode. From this point of view, it is clear that the eigenmode and scattering simulations

have a natural dual relationship with regards to the resonant behaviour at fgap. At fvia,

however, the field distribution of an incident plane-wave excitation does not correspond

to that of the eigenmode and hence the structure does not respond in a resonant manner.

Additionally, it is seen that by controlling the order in which the two resonances occur,

fPMC can be tuned to be either inside or outside the band gap. In other words, by

ensuring that fgap occurs at a lower frequency than fvia, the PMC frequency is located

inside the band gap (Case 1) since the band gap is created between the TM0 and TE

modes and fgap occurs at a frequency lower than the TE mode light line crossing (See

Figure 2.3(a)). However, if fvia occurs at a frequency significantly lower than fgap, the

PMC frequency can be tuned to lie outside the band gap (Case 3) since the band-gap

is formed between the TM0 and TM1 modes and fgap occurs above the TM1 mode light


2 4 6 8 10−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

0o

30o

60o

(a) TE Polarization

2 4 6 8 10−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]

0o

30o

60o

(b) TM Polarization

Figure 2.5: Case 1 Reflection Phase Response for varying angles of incidence.

line crossing (see Figure 2.3(e)).

2.2 Scanning the Angle of Incidence

Now that the resonance occuring at fPMC for a mushroom-like structure has been iden-

tified as the degenerate eigenmode occuring at fgap, a natural extension is to attempt

to map the resonant frequencies at inclined angles of incidence for both TE and TM

polarized wave to the dispersion diagram. Figure 2.5(a) and Figure 2.5(b) show the re-

flection phase characteristics of Case 1 for various angles of incidence under TE and TM

polarizations respectively. A shift in the location of fPMC , where the reflection phase is

zero degrees, is seen under varying angles of incidence for TE polarization and multiple

resonances are observed under TM polarization that are also dependent on the angle of

incidence. It is observed that under TE polarization the resonant frequency, fPMC , in-

creases as the angle of incidence is increased. Under TM polarization, it is observed that

fPMC decreases as the angle of incidence is increased in addition to a second resonant

frequency being introduced at a higher frequency point which increases as the angle of

incidence is swept. To obtain a better understanding of the variation of these resonances,


kx

kz θ

z

x

Figure 2.6: Inclined angle of incidence on surface

it would be beneficial to map them back to the dispersion diagram by calculating the

tangential phase shift induced by an inclined plane-wave on the structure at its reso-

nance frequency. Assuming the structure is uniformly illuminated in the y-direction, an

incoming plane-wave of spatial frequency, k, can be resolved into tangential and normal

components, kx and kz, as shown in Figure 2.6. Next, by enforcing phase matching at

the interface between air and the surface, the tangential phase shift across the unit cell,

βd, can be calculated for a known angle of incidence, θ, as shown in Equation 2.1. Here,

Φx represents the tangential phase shift in air.

Φx = kxd sin θ

βd = Φx =2πf

cd sin θ (2.1)

The unit cell phase shift, βd, is calculated for each resonant frequency obtained from

the reflection phase characteristics for TE and TM polarized incident plane-waves as

the angle of incidence is swept from normal to grazing. The resulting information is

then superimposed on the dispersion diagram for Case 1 as shown in Figure 2.7. It is

observed that the resonant frequencies obtained from the scanning the angle of incidence

map on to the various eigenmodes of the structure. The TE resonances observed in

Figure 2.5(a) correspond to the leaky TE eigenmodes of the structure whereas the TM


resonances in Figure 2.5(b) correspond to the leaky TM eigenmodes. It is important to

note that the higher-order resonance introduced in Figure 2.5(b) corresponds to the TM2

mode in the dispersion diagram for this particular case. However, if the same simulation

had been conducted for Case 3 for example, the on-axis scattering resonance would be

associated with the TE and TM2 modes at fgap whereas the additional TM -polarization

scattering resonance introduced would be associated with the TM1 mode and would

occur at a frequency lower than fgap (see Figure 2.3(e)). Moreover, it is seen that the

mode corresponding to fvia which was not initially captured by the normal incidence

scattering analysis is captured when the the incidence angle goes off normal. This can be

understood by commenting on the boundary conditions on the unit cell. For an on-axis

excitation, the eigenmode at fvia has four H-walls as its boundaries and hence is not

excitable by a plane-wave excitation. However, for an off-axis excitation, since there is a

finite phase shift across the unit cell, the boundaries in the longitudinal (x-z) direction

are no longer H-walls and hence the eigenmode is now susceptible to excitation by a

plane-wave. In fact, the longitudinal boundaries will have a resistive component to them

as the angle of incidence moves away from normal since there is a net power flow away

from the unit cell. Once again, the reflection phase response and eigenmode dispersion

diagram show a natural duality. It is observed that the leaky eigenmodes of the structure

in question are responsible for both its on-axis and off-axis scattering responses. The only

eigenmode that is not captured by a scattering simulation is fvia, which is associated with

four H-wall boundaries and is hence not excitable by a plane-wave. It is also interesting

to note the variation in the TM1 resonance in Figure 2.5(b). As mentioned earlier, the

resonant frequency of this mode decreases as the angle of incidence, and hence phase

shift, is increased verifying the existence of a backward wave in the structure.


0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]


TM0TM1TETM2Light LineScattering Resonance

fvia

fgap

Figure 2.7: Mapping the scattering resonances to the eigenmode dispersion diagram

2.3 Equivalent Circuit Model Analysis

2.3.1 The Dispersion Relation

Transmission Line (TL) theory has been successfully used to calculate the disperion rela-

tions of the mushroom-like and other closely related structures [33,34]. For propagation

along the x-direction with a zero phase shift along the y-direction, i.e. 0 ≤ (βd)x ≤ π,

(βd)y = 0, the transverse boundaries can be treated as H-walls [34] and hence a 1D

equivalent unit cell can be used as shown in Figure 2.8. The unit cell consists of a loading

capacitor, C, due to the gaps, followed by a short section of transmission line of length

d2

and a shunt inductance, L, due to the via [35]. The shunt inductance is treated as

a point of symmetry and hence the loading inductance is split into two. The transfer


matrix for this unit cell can be represented as:

Tunit−cell = Th1Th2

=

Ah Bh

Ch Dh

Dh Bh

Ch Ah

=

AhDh +BhCh 2AhBh

2ChDh AhDh +BhCh

=

Af Bf

Cf Df

(2.2)

where the h subscript refers to the matrix element for the half unit cell and the f subscript

refers to the matrix element for the full unit cell. Next, by using the Bloch theorem [36]

for lossless periodic structures, the voltages and currents at the terminals of an nth unit

cell can be related to those of the (n+1 )th by a propagation factor of e−jβd where β is the

bloch propagation constant and d is the periodicity of the unit cell [37]. In conjunction

with the transfer matrix of Equation 2.2, the currents and voltages at the terminals of

the unit cell can be related as:VnIn

=

Af Bf

Cf Df

Vn+1

In+1

= ejβd

Vn+1

In+1

(2.3)

( )2

dTL ( )

2

dTL

2L2L2C 2C

nV 1n+V

1n+InIn

1

2n+ n+1

oZ oZ

Figure 2.8: NRI-TL unit cell modelling the mushroom-like structure.


Next, by re-arranging the equation, taking the determinant of the matrix and selecting

the non-trivial solution, the following relationship is obtained:

cos(βd) =Af +Df

2= Af (2.4)

where the fact that A = D for a symmetric unit cell has been used in Equation 2.4.

Finally, by substituting in the relations between the half unit cell matrix and full unit

cell matrix from Equation 2.2 along with the relationship AhDh−BhCh = 1 for a lossless

reciprocal network, the following dispersion relation is obtained:

cos(βd) = 1 + 2BhCh (2.5)

The expressions for Bh and Ch can now be evaluated and substitued in to Equation 2.5 to

obtain the complete form of the dispersion relationship for the unit cell as shown below:

Th1 = T2CTTLT2L

=

1 −j2ωC

0 1

cos(ψ/2) jZo sin(ψ/2)

jYo sin(ψ/2) cos(ψ/2)

1 0

−j2ωL

1

=

(1− 14ω2LC

) cos(ψ2) + ( Zo

2ωL+ Yo

2ωC) sin(ψ

2) j(Zo sin(ψ

2)− 1

2ωCcos(ψ

2))

j(Yo sin(ψ2)− 1

2ωLcos(ψ

2)) cos(ψ

2)

=

Ah Bh

Ch Dh

; (2.6)

where ψ is the phase shift associated with a transmission line section of length d and

characteristic impedance Zo. The final expression for the dispersion relation is then given

by [34]:

cos(βd) = (1− 1

4ω2LC)cosψ + (

Zo2ωL

+Yo

2ωC) sinψ − 1

4ω2LC(2.7)


The dispersion relation of Equation 2.7 is then plotted in Figure 2.9 to reveal the

general dispersion characteristics of the unit cell of Figure 2.8. Two distinct transmission

line pass bands are observed, one of which is a backward wave passband, seperated by a

transmission line mode band gap. Two (βd)x = 0 resonant frequencies are also observed

and are labeled fc1 and fc2. At these frequencies, it is observed that the group velocity,

vg, goes to zero and hence purely reactive boundary conditions would be expected, similar

to those of fgap and fvia.

Figure 2.9: Dispersion characteristics of the unit cell of Figure 2.8 with a Zo = 50Ω TLof electrical length 60o at 2.5 GHz and loading elements L = 2nH and C = 0.3pF

Now that the dispersion relation has been derived, case-specific dispersion plots are

generated for each of the geometries outlined in Table 2.1.1 by extracting the relevant TL

and loading element parameters as discussed in Appendix A and substituting them into

Equation 2.7. The extracted parameters are summarized in Table 2.3. The TL dispersion

curves are then overlayed on the FEM eigenmode simulations and the resulting curves

are plotted in Figure 2.10.

It is observed that the NRI-TL circuit model accurately predicts the dispersion char-

acteristics of the guided TM modes of the mushroom-like unit cell away from the light

line. Eigenmodes close to the light line are not captured by the TL model since the


Table 2.3: Extracted TL and loading parameters.Case Zo [Ω] vφ/c L[nH] C[pF ]

1 47.9 0.476 0.52 0.372 48.1 0.476 0.51 0.243 49.0 0.474 0.50 0.14

field distributions for these modes are largely in air. Therefore, these regions show up

as TL band gaps, not to be confused with the surface-wave mode band gap formed by

contra-directional coupling of the forward and backward wave TM0 modes and the light

line crossing of the first higher order TM/TE mode. Additionally, the TE mode is not

captured by the NRI-TL model since it has transverse E-wall boundaries and is not a

quasi-TEM mode.

It is noted that, in addition to the guided modes of mushroom-like structure, the

NRI-TL dispersion relation also accurately models the leaky modes associated with fvia

(the single TM mode). However, a discrepancy is noted between the analytical NRI-

TL dispersion and the eigenmode dispersion properties for the leaky mode resonances

associated with fgap for each of the three cases, particularly at the βdx = 0 resonance

point. The deviation for Cases 1 through 3 is 7.0%, 4.6% and 10.2% respectively. This

discrepancy is attributed to the lossless nature of the unit cell being used to model an

inherently leaky mode. As seen from the field distributions for fgap in Figure 2.4, the

eigenmode is associated with a net power flow away from the unit cell and hence the

loading capacitor is more accurately modeled by a real radiation resistance in series with

the capacitive loading in the leaky wave region. The value of the radiation resistance

can be calculated by either an energy-based parameter extraction as proposed in [38]

or by calculating the leakage constant associated with a large number of unit cells as is

done for the design of leaky wave antennas [39,40]. It is interesting to note that a similar

discrepancy is not observed at the βdx = 0 associated with fvia. This can be understood

in light of the observations made earlier in Section 2.1.3, noting that the leaky resonance


0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]

NRI−TLTM0TM1TETM2Light Line

(a) Case 1 Comparison

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]


(b) Case 2 Comparison

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]


(c) Case 3 Comparison

Figure 2.10: Comparison of NRI-TL dispersion to FEM eigenmode dispersion.


associated with fvia radiates very small amounts of power even for eigenmodes away from

the βdx = 0.

Since the objective here is to demonstrate an equivalency of resonance characteristics

between the TL model and the FEM simulations, no further attempt will be made to

refine the model to include radiation losses. Instead, the boundary conditions at the

walls of the unit cell at frequencies fc1 and fc2 will now be derived and compared to

the boundary conditions from the FEM simulations at fgap and fvia, as discussed in

Section 2.1.3, to show that they are, indeed, equivalent resonances.

2.3.2 Critical Frequencies of the Dispersion Relation

The resonances, fc1 and fc2, occur at (βd)x = 0 which when substituted into Equation 2.5

results in the following two conditions:

1 = 1 +BhCh ⇒ Bh = 0, Ch 6= 0 or Ch = 0, Bh 6= 0 (2.8)

Here, fc1 is given by either Bh = 0 or Ch = 0 with fc2 given by the excluded case. It

is possible for Bh and Ch to be simultaneously zero which implies fc1 = fc2 and results

in a closed transmission line stop band [34]. The conditions Bh = 0 and Ch = 0 can be

written out in full using Equation 2.6 to give the relationships:

Bh = 0⇒ Zo sin(ψ

2) =

1

2ωCcos(

ψ

2) (2.9)

Ch = 0⇒ Yo sin(ψ

2) =

1

2ωLcos(

ψ

2) (2.10)

Now, under the assumption that the host TL sections are electrically small such that

sin(ψ/2) → ψ/2 and cos(ψ/2) → 1, and by using the host TL dispersion relationship,

ψ = ω√L′C ′d, where L′ and C ′ are the distributed inductance and capacitance of the


host TL, these expressions can be simplified to give [34]:

fc1 = min

(1

2π√CL′d

,1

2π√C ′Ld

)(2.11)

fc2 = max

(1

2π√CL′d

,1

2π√C ′Ld

)(2.12)

To establish complete equivalency between the resonant frequencies fc1 and fc2 and

the corresponding resonances obtained from FEM simulations, fgap and fvia, the unit cell

boundary conditions at fc1 and fc2 will now be examined and equated to those of fgap and

fvia. Recall from Section 2.1.3 that the tranverse boundaries of the TM modes at both

fgap and fvia are virtual H-walls whereas the longitudinal boundaries of the TM mode

at fgap are virtual E-walls and those of the TM mode at fvia are virtual H-walls (see

Figure 2.4). The TE mode at fgap was shown to have virtual E-walls as its transverse

boundaries and as shown earlier, is not captured by the transmission line model.

The transverse unit cell boundaries for both fc1 and fc2 act as H-walls since on-axis

(x-directed) propagation is being considered with (βd)y = 0. The longitudinal unit cell

boundary conditions associated with fc1 and fc2 can then be evaluated by substituting

(βd)x = 0 along with the resonance conditions Bh = 0 and Ch = 0 individually in to

Equation 2.3.

For Bh = 0:

VnIn

=

AhDh 0

2ChDh AhDh

Vn+1

In+1

; = ej0

Vn+1

In+1

(2.13)

which simplifies to:

Vn = AhDhVn+1 = Vn+1 ⇒ Vn = Vn+1 and AhDh = 1 (2.14)

In = 2ChDhVn+1 + AhDhIn+1 = In+1 ⇒ In = In+1 (2.15)


x

y

H-wall

E-wall

fvia, Ch = 0 fgap, Bh = 0

Figure 2.11: Boundary conditions at resonances.

Using the results from Equations 2.14 and 2.15 and substituting back into the first

equality from Equation 2.15, the following result is obtained:

In = 2ChDhVn + In ⇒ Vn = 0 for Ch 6= 0 (2.16)

which necessitates Vn+1 = Vn+1/2 = 0. This implies short-circuits or E-wall boundaries at

the beginning, middle and end of the unit cell in the longitudinal direction. Therefore, for

the Bh = 0 resonance, the shunt loading inductor is shorted out and the gap capacitance

then resonates with the distributed inductance of the TL. These boundary conditions

are consistent with the ones observed for the TM mode of fgap in Section 2.1.3. The

boundary conditions for the resonance Ch = 0 are derived in a similar manner to give

the relation:

Vn = 2AhBhIn + Vn ⇒ In = 0 for Bh 6= 0 (2.17)

which necessitates In+1 = In+1/2 = 0 and implies open-circuits or H-wall boundaries

along the longitudinal direction. In this case, the gap capacitance is open circuited and

the shunt inductance of the via resonates with the distributed capacitance of the host

TL. These boundary conditions are consistent with those of fvia. Figure 2.11 summarizes

the derived boundary conditions.


Hence it is seen that the two (βd)x,y = 0 resonant frequencies fc1 and fc2 correspond

to the resonances created by the series loading capacitor and the distributed inductance

of the host TL and the shunt loading inductor and the distributed capacitance of the host

TL. These resonances are identical to those of fgap and fvia as discussed in Section 2.1.2.

Additionally, it is noted that depending on the host TL parameters and magnitudes of

the loading elements, the order of fgap and fvia can be manipulated allowing control over

whether the PMC frequency lies inside or outside the surface-wave band gap as suggested

earlier in the closing paragraph of Section 2.1.3. More specifically, by ensuring fgap occurs

at a lower frequency than fvia, it is guaranteed that the PMC frequency will be located

inside the band gap. However, if fgap occurs significantly above fvia, the PMC frequency

will be at a higher frequency point and will be completely removed from the band gap.

2.4 Chapter Summary

Several key points have been presented in this chapter regarding the scattering and

dispersion characteristics of the mushroom-like structure. Firstly, it has been shown

through the comparison of FEM eigenmode and scattering simulations that there is a

clear relationship between the scattering resonance that produces the PMC effect and

the zero-phase shift resonances that occur on the eigenmode dispersion diagram. This

was then confirmed by comparing the field distribution of the resonances. Additionally,

it was shown that the PMC frequency and surface-wave band gap do not coincide for

all geometries of the mushroom like structure. Next, it was shown that the variation in

the PMC frequency as the angle of incidence is scanned can be mapped directly on to

the leaky modes shown in the dispersion diagram. An NRI-TL circuit model was then

used to describe the on-axis dispersion characteristics of the mushroom-like structure and

approximate the (βd)x,y = 0 resonant frequencies. It was then demonstrated that these

resonances can be independently tuned by adjusting the capacitive and inductive loading


elements to either force the PMC frequency and band gap to coincide or be separated.

The effect on antenna performance of having these properties coincide or separated will

now be investigated in the next chapter.

Chapter 3

Study of Band Gap Effects on

Antenna Performance

3.1 Approach

The theory presented in the previous chapter has provided a recipe for designing unit

cells of the mushroom-like structure where the surface-wave band gap and PMC frequency

can coincide or are forced to be separated. This approach will now be implemented in

this chapter to determine the effect of the band gap on the performance of a dipole

antenna. Two unit cell designs will be presented. In the first case, the in-phase reflection

and surface-wave suppression properties coincide, while in the second case, the two are

separated.

The intention is to draw a parallel between the understanding developed in the pre-

vious chapter for infinite surfaces and the response of a dipole antenna on top of a finite

surface. In particular, attention will be paid to the zero-degree reflection phase frequen-

cies of the unit cells and their influence on the impedance response of the dipole antenna

when placed on a finite surface. An improvement in the input impedance of the dipole

antenna is expected in the vicinity of the zero-degree reflection phase frequency which

48

Chapter 3. Study of Band Gap Effects on Antenna Performance 49

may manifest as a resonance in the impedance response. Once an analogy has been es-

tablished between the responses of the infinite and finite surfaces and the effect of the

band-gap has been quantified, then the reflection phase curves and dispersion diagrams

may be used as tools to design antennas above finite surface sizes.

Two sets of parametric studies will be carried out to understand the interactions

between a dipole and a finite surface composed of the unit cells.The first study will

consider a half-wavelength dipole of a fixed length placed above a surface consisting of

each unit cell. The number of unit cells in each surface will be varied to determine the

nature of the interactions between the antenna and surface and to examine the various

resonances that may be excited. The second study will involve varying the length of the

dipole on a fixed size surface to create a clear distinction between the dipole resonance

and any resonances associated with the surface. The radiation characteristics of the

common resonances will then be examined to determine the effects of the surface-wave

band gap.

3.2 Unit Cell Design

Two approaches can be taken towards designing the unit cells that satisfy the reflection

phase and band gap requirements. The first approach is to design a standard mushroom-

like unit cell at a chosen design frequency such that fvia occurs below fgap, resulting in

the PMC frequency occuring outside the surface-wave band gap. The other unit cell

case can then be designed by using an interdigitated capacitive loading to force fgap

below fvia, resulting in the PMC frequency occuring inside the surface-wave band gap.

This approach is, however, computationally infeasible when performing the parametric

studies mentioned above, due to the extremely high memory requirements associated with

meshing a large number of interdigitated capacitors. The alternative, computationally

feasible approach is to design a standard mushroom-like unit cell at a chosen design


w

d

2r

(a) Standard unit cell

w

d

gs

lx

ly 2r

(b) Loaded unit cell

Figure 3.1: Unit cell designs.

frequency such that the PMC frequency and surface-wave band gap coincide. A spiral

inductive loading can then be introduced in the cell to push fvia to a much lower frequency

resulting in seperation of the PMC frequency and the surface-wave band gap. This

problem is mush less costly in terms of computational resources since the number of

fine features is greatly reduced allowing for a coarser mesh to be used when solving the

problem numerically. Since the objective here is to examine the effect of the band gap

on radiation performance of a closely spaced antenna, the second approach is acceptable.

A Rogers TMM4 substrate of thickness 1.524 mm is selected for the design of each unit

cell. The relative permittivity, εr, of the substrate is 4.5 with a loss tangent, tan δ = 0.002.

The design frequency for the on-axis PMC frequency, fPMC , for each unit cell is selected

to be 5 GHz. A standard mushroom-unit cell designed with these substrate parameters

for the chosen design frequency is shown in Figure 3.1(a). An inductive spiral loading is

then introduced in the unit cell to push the band gap down to a lower frequency range.

The second unit cell is shown in Figure 3.1(b). The physical details of each cell are listed

in Table 3.1. The size of the second unit cell had to be slightly reduced to account for

the small series loading as a result of the spiral inductor so that the design frequency of

each cell could be kept constant.

The scattering simulation setup described in Chapter 2.1.1 is then used to determine


Table 3.1: Unit cell dimensions.Case d [mm] w [mm] r [mm] lx [mm] ly [mm] s [mm] g [mm]

Standard 8.6 8.4 0.5 NA NA NA NALoaded 8 7.8 0.2 3 3 0.2 0.2

the on-axis scattering characteristics of each unit cell. The resulting reflection phase

profiles are shown in Figure 3.2(a) with the in-phase reflection bandwidth between ±90o

from 4.66 GHz to 5.38 GHz shaded in grey. It is observed that the in-phase reflection

characteristics of both unit cells are almost identical. An eigenmode simulation is then

carried out to determine the dispersion characteristics of both unit cells. The resulting

dispersion diagrams from Γ to X (0 ≤ (βd)x ≤ π, (βd)y = 0), for each unit cell are

plotted in Figures 3.2(b) and 3.2(c). It is evident from the dispersion diagrams that for

the first case, the PMC frequency at 5 GHz lies within the surface-wave band gap which

ranges from 4 to 5.25 GHz. This case will be referred to as the AMC-BG unit cell since

it possesses both an in-phase reflection bandwidth and surface-wave band gap at the

design frequency. For the second case, it is observed that a surface-wave band gap is now

located from 1.5 GHz to 2.1 GHz due to the spiral loading, whereas the PMC frequency

is still at 5 GHz. This case will be referred to as the AMC unit cell since there is only an

in-phase reflection bandwidth and no surface-wave band gap at the design frequency.

3.3 Variation of Surface Size

The first study to be carried out is an evaluation of the size of each surface to examine

the different resonances that are excited in the surface by a closely spaced antenna. It

has previously been shown that a dual resonance is achievable by placing an antenna in

close proximity to a mushroom-like surface [17–19]. It has been shown that one of these

resonances is strongly associated with the dipole antenna [15]; however, the effect of the

second resonance is not investigated and has not been characterized. It is suspected that


3.5 4 4.5 5 5.5 6 6.5 7−180

−135

−90

−45

0

45

90

135

180

f [GHz]R

efle

ctio

n P

has

e [D

eg.]

AMC−BGAMC

4.66 GHz

5 GHz

5.38 GHz

(a) Reflection phase profiles

0 30 60 90 120 150 1800

1

2

3

4

5

6

7

8

9

βd [Deg.]

f[G

Hz]

TM0TM1TE1TM2Light Line

Band Gap: 4 − 5.25 GHz

(b) AMC-BG Dispersion diagram

0 30 60 90 120 150 1800

1

2

3

4

5

6

7

8

9

βd [Deg.]

f[G

Hz]



(c) AMC Dispersion diagram

Figure 3.2: Unit cell reflection phase profiles and dispersion diagrams.

this second resonance is a zero-order resonance similar to the one investigated in [41].

Zero order resonances in a structure occur at the (βd)x,y = 0 point on the dispersion

curve of the unit cell that comprises the resonator. It has been seen in Chapter 2 that

the unit cell of each surface supports two (βd)x,y = 0 resonances that were labeled fgap

and fvia. Since a dipole antenna is being placed in close proximity to each surface to

exploit its scattering characteristics, it is probable that the surface is acting as a large

resonator, and is being excited by the dipole resulting in its behaviour as a PMC. Zero

order resonances are independent of the size of a resonator [42] and by varying the size

of the surface in the presence of the dipole, such resonances can be identified.


60.2

mm

60.2 mm

13 mm

y

x

z

(a) Top View

3 mm

1.524 mm

(b) Side View

(c) |S11| Response

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Zin

[Oh

ms]

RealImaginary

(d) Input Impedance

Figure 3.3: Dipole antenna on grounded substrate

It is also noted in [17], that additional higher order resonances are also present;

however, the useable bandwidth of the antenna is limited by the pattern degradation

observed at higher frequencies. Additionally, in [17], the size of the surface was also

varied but a detailed analysis of the effects was not carried out.

The objective of this study is to determine the effect of varying the structure size on

the resonances observed in the input impedance response of the dipole, identifying any

zero order resonances and comparing the responses of the AMC-BG and AMC structures

for the various sizes.

The simulation setup involves a rectangular dipole antenna of length 26mm and width


y

x

z

(a) AMC-BG 7x7 Grid (b) AMC 7x7 Grid

Figure 3.4: Grid of 7x7 unit cells for each case. 3x3 and 5x5 grids (not shown) are alsosimulated.

0.5mm placed at a height of 3mm from the surface of a grounded substrate as shown in

Figure 3.3. The substrate parameters are those identified during the unit cell design. The

size of the substrate and ground plane is chosen to be large enough to exactly accomodate

a 7x7 grid of the AMC-BG unit cell resulting in a square ground plane of size 60.2mm.

Since the AMC-BG unit cell is slightly larger than the AMC unit cell, the size of the

ground plane is sufficient to accomodate a 7x7 grid of the AMC unit cell as well. A

lumped source is used to excite the dipole and the |S11| and input impedance responses

are shown in Figure 3.3(c) and Figure 3.3(d). It is observed that the dipole is resonant at

approximately 5GHz and, as expected, the close proximity of the ground plane results in

a reduction in the input impedance at the resonant frequency from 73Ω to 14.5Ω, due to

induced image currents. This case will serve as the reference case for comparison against

the performance of the AMC-BG and AMC surfaces.

Three grid sizes (3x3, 5x5 and 7x7) are then introduced into the grounded substrate

for each unit cell. The dipole position is kept constant in the center of the surface for

each case. Figures 3.4(a) and 3.4(b) show the top view of a 7x7 AMC-BG and 7x7 AMC

surface respectively.


The resulting input impedance and |S11| responses are shown in Figure 3.5. The first

observation made is that the closely spaced narrowband resonances that occur between

3.5GHz and 4GHz for the AMC-BG case do not occur in the AMC case. This frequency

range corresponds to the backward-wave bandwidth of the AMC-BG unit cell and it

is postulated that these resonances occur as a result of interactions between the dipole

and the backward wave band of the AMC-BG surface as shown in Figures 3.5(a), 3.5(c)

and 3.5(e). In the AMC case, the spiral loading shifts the backward-wave band to a

much lower frequency as shown in Figure 3.2(c) and hence these resonances are not

observed and a much cleaner response is observed in Figures 3.5(b), 3.5(d) and 3.5(b)

at lower frequencies. Additionally, it is observed that as the number of cells is increased

in the AMC-BG case, an additional resonance is introduced. This resonance occurs at

approximately 4GHz for the 5x5 case and 4.5GHz for the 7x7 case. This resonance has a

large capacitive component associated with it and is hence poorly matched for the given

design. Moreover, this resonance is not observed in the equivalent AMC surface case and

is, therefore, attributed to the presence of the band gap. Moreover, the location of the

resonance varies signficantly with the size of the structure and hence does not qualify as

a zero order resonance.

The next point of observation is the effect of each surface on the dipole resonance

which occurs at 5GHz for the reference case. For the AMC-BG case, it is observed that

the structure does not greatly influence the resonant frequency of the dipole. In fact,

an observation of Figure 3.5(c) reveals that the dipole resonance remains constant at

5GHz. Furthermore, only a minor increase in the real part of the radiation resistance is

seen in Figure 3.5(a). On the other hand, it is seen for the AMC case in Figure 3.5(d)

that the presence of the surface reduces the resonant frequency of the dipole from 5GHz

to approximately 4.75GHz. More importantly, it is observed in Figure 3.5(b) that the

real part of the input impedance is significantly increased for the dipole on the AMC

case compared to the dipole on the ground case. A more detailed comparison of the


3.5 4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

Frequency [GHz]

Re(

Zin

) [Ω

]

GND3x35x57x7

(a) AMC-BG Real Zin

3.5 4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

Frequency [GHz]

Re(

Zin

) [Ω

]

GND3x35x57x7

(b) AMC Real Zin

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Im(Z

in)

[Ω]

GND3x35x57x7

(c) AMC-BG Imaginary Zin

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Im(Z

in)

[Ω]

GND3x35x57x7

(d) AMC Imaginary Zin

3.5 4 4.5 5 5.5 6 6.5 7−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

GND3x35x57x7

(e) AMC-BG |S11| Response

3.5 4 4.5 5 5.5 6 6.5 7−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

GND3x35x57x7

(f) AMC |S11| Response

Figure 3.5: |S11| and input impedance responses for dipole antenna on 3x3, 5x5 and 7x7AMC-BG and AMC grids.

improvement in input impedance of the AMC-BG and AMC cases versus the dipole on

ground case will be carried out in Section 3.5.


Next, a main resonance is observed that occurs at 5.7GHz for the 3x3 AMC-BG case

(dashed blue line) in Figure 3.5(a) and approximately 5.5 GHz for the 3x3 AMC case

in Figure 3.5(b). A quick comparison of the real and imaginary impedances for each of

the 3x3 cases confirms that these resonances are identical in nature. In addition, both of

them occur in close proximity to the PMC frequency of the unit cell which is by design

5GHz. Moreover, it is noted that as the number of cells increase in each case, this main

resonance shifts down in frequency closer to the PMC frequency to about 5.4 GHz for

the 5x5 AMC-BG case and close to 5GHz for the 5x5 AMC case. Intuitively, this is

expected since a larger structure size would more closely simulate an infinite structure.

A further increase in the number of unit cells from 5x5 to 7x7 does not affect the location

of this resonance and as a result it is concluded that this resonance is indeed the zero

order resonance. A 9x9 case (not shown) was also simulated for both surfaces to confirm

and no further change in the location of this resonance was observed. However, it is

interesting to note that even a 3x3 structure supports this resonance for both the AMC-

BG and AMC surfaces. The electric field distribution at resonance in the capacitive gaps

between the patches is shown in Figure 3.6 for the 5x5 AMC-BG and AMC surfaces. It is

seen that the capacitive gaps are strongly excited as a result of the proximity of the dipole.

The field distributions are also of the same general shape as those shown in Chapter 2

at the fgap resonance, with strong excitation of capacitor gap fields in the direction of

propagation which, in this case, is determined by the polarization of the dipole source.

Moreover, it is seen that the field distributions for the two cases are almost identical,

confirming that the resonances are, indeed, the same.

Finally, it is also observed that various higher order resonances are introduced as the

size of the structure is increased for both AMC-BG and AMC cases. The 3x3 AMC-BG

case shows a primary resonance at 5.7GHz, as discussed, and a higher order resonance

is seen at the edge of the graph at 7GHz. As the size of the structure is increased,

the higher order resonance shifts down to 6GHz for the 5x5 case and even further down


(a) 5x5 AMC-BG Resonance (b) 5x5 AMC Resonance

Figure 3.6: Electric field distributions of 5x5 AMC-BG and AMC surfaces at resonance.

to 5.7GHz for the 7x7 case. In fact, it is observed that the resonance begins to merge

with the primary resonance of the AMC-BG structure for the 7x7, case which negatively

affects the matching characteristics. A similar trend is seen for the variation of the AMC

structure size.

To summarize, it has been shown that the presence of the band gap in the AMC-

BG structure results in several narrowband resonances occuring in the backward wave

frequency band of the unit cell. A similar response is not seen in the AMC case. Moreover,

one additional resonance is introduced in the AMC-BG case which varies with the size

of the structure and does not manifest in the AMC case. Next, a zero order resonance is

observed in the surface for both the AMC-BG and AMC cases as a result of excitation

by the dipole antenna. This resonance is responsible for the surface acting as a PMC for

a closely spaced current source and corresponds to the degenerate (βd)x,y = 0 resonance

from the dispersion diagram in Figure 3.2. Another important fact is the observation

of several higher order resonances that are introduced in the impedance response of the

dipole as the structure size is increased. These resonances are a strong function of the

structure size and are capable of merging with the main resonance of the structure that

occurs in the vicinity of the the PMC frequency. Hence, it is critical to be aware of these

higher resonances when designing the antenna/EBG combination to ensure that they

do not disrupt the performance of the antenna at the design frequency. Finally, it was


also observed that the AMC structure increases the input impedance of a dipole antenna

significantly more than the AMC-BG structure.

3.4 Variation of Dipole Length

The size of the AMC-BG and AMC surfaces were varied in the previous section to

obtain an understanding of the structure size on the various resonances that are observed

in the dipole’s impedance response. Now, the length of the dipole will be varied for

a fixed surface size to understand the influence of the dipole length. Based on the

previous analysis, it is expected that only the dipole resonance should shift by varying

its length and that there may be possible side effects on the matching characteristics of

the remaining resonances. An appropriate surface size must first be selected to conduct

this study. From Figures 3.5(a) and 3.5(c) it is observed that the 7x7 AMC-BG case

suffers from interference from higher order resonances and is hence unsuitable. The 3x3

case offers the best isolation of the main resonance from higher order resonances, however,

its impedance characteristics are undesirable. The real impedance peaks at 200Ω at the

main resonance and is unsuitable for matching to a system impedance of 50Ω. The 5x5

case offers the best compromise for both the AMC-BG and AMC surfaces. The higher

order resonance is sufficiently separated from the main resonance and the real impedance

peaks at approximately 100Ω. Moreover, the imaginary part of the impedance at the

main resonance exhibits a favorably flat response around the resonance point. As a

result, the 5x5 case is selected to investigate the effect of the dipole length.

The dipole length is varied from 29mm to 26mm in 1mm increments corresponding

to a frequency range of 4.54 − 5GHz covering the in-phase reflection bandwidth from

+90o to 0o. The dipole length is not varied to cover the range from 0o to −90o, since

strong coupling between the surface and the dipole prevents the dipole resonance from

being observed. The height of the dipole above the surfaces and the ground plane size are


kept fixed at 3mm and 60.2mm respectively. The resulting input impedance responses

for the AMC-BG and AMC surfaces are shown in Figure 3.7. It is clearly observed

from Figures 3.7(a) and 3.7(b) that the location of the surface resonance, given by the

change in slope of the reactance around 5 GHz, remains constant as the dipole length

is varied. On the other hand, it is evident from Figures 3.7(c) and 3.7(d) that variation

in the dipole length causes a clear shift in the dipole resonance, as expected. Moreover,

it is seen that the length of the dipole can be used to adjust the reactive component of

the impedance associated with the surface resonances. If the resonant frequency of the

dipole is tuned to lie in the proximity of a surface resonance, a good impedance match

can be obtained. Figure 3.8 shows the |S11| response of a 26mm dipole, which is tuned

to resonate at approximately 5GHz, at a height of 3mm above the 5x5 AMC-BG and

AMC surfaces. A clear dual-band response is observed in both cases as a result of the

close proximity of the dipole resonance to the surface resonance. A higher order surface

resonance is also observed for both cases.

It is worth repeating at this point the study conducted in [15], where a detailed

parametric study of the effect of the dipole length on the impedance match bandwidth is

considered. Using the on-axis reflection phase profile of a unit cell as a design tool, it was

concluded in that study that the 90± 45o reflection phase bandwidth is the region that

provides optimum matching for a dipole antenna and yields directive radiation patterns.

However, this approach works only for instances where large surfaces are used (the surface

size used in [15] was 7x7). The authors in [15] also investigated a larger surface size and

reached the same conclusion. For smaller surface sizes, such as a 5x5 surface, it is more

practical to design the antenna to resonate close to the PMC frequency of the unit cell

as demonstrated in this section. The study conducted in [15] is revisited in Appendix C

and it is shown that although the 90±45o criteria works well for their chosen surface size,

the matching of the antenna in that bandwidth deteriorates if the size of the structure

is reduced.


3.5 4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

Frequency [GHz]

Re(

Zin

) [Ω

]26mm27mm28mm29mm

(a) AMC-BG Real Zin

3.5 4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

Frequency [GHz]

Re(

Zin

) [Ω

]

26mm27mm28mm29mm

(b) AMC Real Zin

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Im(Z

in)

[Ω]

26mm27mm28mm29mm

(c) AMC-BG Imaginary Zin

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Im(Z

in)

[Ω]

26mm27mm28mm29mm

(d) AMC Imaginary Zin

Figure 3.7: Input impedance responses for varying dipole lengths on 5x5 AMC-BG andAMC grids.

3.5 Evaluation of the ‘PMC effect’

An interesting parameter to quantify is the improvement in the input impedance of a

closely spaced dipole antenna on top of each of the surfaces being examined. From

Figure 3.2(a), it is known that the on-axis reflection characteristics of the two unit cells

are identical, therefore, it is expected that both structures should act as PMCs within


3.5 4 4.5 5 5.5 6 6.5 7

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

AMC−BGAMC

Figure 3.8: Dipole response on 5x5 AMC-BG and AMC surfaces

the in-phase reflection bandwidth. It is well understood that the reduction in input

impedance of a dipole spaced close to a ground plane is attributed to the out-of-phase

image currents induced on the ground plane, therefore, by operating the dipole within the

in-phase reflection bandwidth of the surfaces, in-phase image currents should be induced

and an improvement in the input impedance should be oberved at the terminals of the

dipole. Figure 3.9 compares the input impedance at the terminals of a 26mm dipole over

a ground, 5x5 AMC-BG and 5x5 AMC surface respectively.

A grounded dipole has a real input impedance of 14.5Ω at resonance whereas a dipole

over 5x5 AMC-BG and AMC surfaces has a real input impedance of 29.4Ω and 60.6Ω

respectively. It is observed that the AMC structure provides a substantial improvement

in the real input impedance at resonance for a dipole over the ground case whereas

only a marginal improvment is noted for the AMC-BG case. Additionally, the resonant

frequency of the dipole is shifted down from 5GHz to 4.84GHz by the AMC surface. A

complete picture of this ‘PMC effect’ can be constructed by varying the length of the

the dipole to keep it resonant within the in-phase reflection bandwidth of each structure

and plotting the real part of Zin versus resonant frequency. This is carried out for the

5x5 and 7x7 grid sizes for each unit cell and the results are plotted in Figure 3.10 over

the positive in-phase reflection bandwidth (+90o to 0o) which ranges from 4.66− 5GHz.


3.5 4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

Frequency [GHz]

Re(

Zin

) [Ω

]GNDAMC−BGAMC

(a) Real Zin

3.5 4 4.5 5 5.5 6 6.5 7−500

−400

−300

−200

−100

0

100

200

300

400

500

Frequency [GHz]

Im(Z

in)

[Ω]

GNDAMC−BGAMC

(b) Imaginary Zin

Figure 3.9: Input impedance responses for 26 mm dipole length on ground, 5x5 AMC-BGand 5x5 AMC grids.

The limits of each curve are determined by the ability to distinguish the dipole resonance

from the surface resonances.

It is seen from the resulting curves that the AMC surface provides a significant increase

in the input resistance of the dipole at resonance as compared to the AMC-BG surface or a

grounded substrate. Moreover, only a minor improvement is seen in the input resistance

4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.30

10

20

30

40

50

60

70

Resonant Frequency [GHz]

Re(

Zin

) [Ω

]

GND5x5 AMC−BG7x7 AMC−BG5x5 AMC7x7 AMC

Figure 3.10: Comparison of ‘PMC effect’ of various surfaces.


as the size of the AMC surface is increased from 5x5 to 7x7. An interesting trend is

seen for the AMC-BG surface cases. It is seen that for a smaller surface size (5x5),

the best performance for a dipole is attained at a point slightly higher than the design

frequency, 5GHz. The input resistance of the dipole deteriorates as the dipole resonance

moves away from the design frequency. However, for a larger surface size (7x7), the input

impedance improves as the dipole resonance moves away from the design frequency. This

observation is analogous to the statement made in the previous section regarding [15]; for

a larger surface size, the optimum operating point for a dipole occurs within the 90±45o

frequency range, however, if the surface size is reduced, the optimum operating point

shifts to the vicinity of the design frequency of the unit cell.

It is significant to mention at this point that the improvement in input resistance

of a dipole resonant within the 90 ± 45o frequency range of the AMC-BG unit cell for

the 7x7 case is a direct result of the size-dependant surface resonance that was previ-

ously discussed (see resonance occuring at 4.75GHz for 7x7 case in Figure 3.5(a)). It is

erroneous to consider any improvement in the input resistance due to this resonance to

be a consequence of the PMC effect, since this resonance does not occur for the AMC

case which possesses the same reflection phase characteristics. Nevertheless, this analysis

demonstrates that the optimum operating point for a dipole is strongly dependent on the

surface size and type of surface (AMC-BG vs AMC) being used.

3.6 Comparison of Radiation Patterns

The radiation patterns of the antenna/surface combinations will now be examined at the

dipole resonance to determine the effect of the surface-wave band gap on the radiation

performance of the system. Radiation patterns for the surface resonance and the first

higher order resonances are also included. Table 3.2 lists the locations of the various

resonances at which the patterns are computed for a 26mm dipole at a height of 3mm


above each surface. The corresponding |S11| response was shown in Figure 3.8.

Table 3.2: Resonance locations for antenna on 5x5 surfaces.Case Dipole [GHz] Surface [GHz] Higher [GHz] Band Gap [GHz]

AMC-BG 5.11 5.62 6.36 4− 5.25AMC 4.81 5.65 6.56 1.5− 2.1

First, to demonstrate the effect of the surface-wave band gap, the power density

propagating along the substrate is plotted at the dipole resonance for the 5x5 AMC-BG

and AMC cases. This is done using the fields calculator in Ansoft HFSS. A reference

circle is drawn that is centered on the substrate with a diameter equal to the dimension

of the ground plane and is located at the midpoint of the substrate height. The surface-

wave power density is calculated by taking the scalar product of the real part of the

Poynting vector, ~S, and the normal to the reference circle that lies in the plane of the

substrate. This scalar product represents the power density propagating in the radial

direction away from the dipole and is a superposition of all propagating surface-wave

modes.

The resulting power density is plotted in polar co-ordinates in Figure 3.11 and is

normalized to the maximum power density for the AMC surface case. As expected,

the maximum power density is observed along the axis of the dipole corresponding to

the propagating TMo surface-wave mode for a grounded dielectric as discussed in [43].

The effect of the much weaker TEo mode is seen at broadside. It is observed that the

maximum power density for the AMC-BG case is 3.5 dB/m2 lower than that of the AMC

case since the resonant frequency of the dipole falls within the band gap of the structure.

It can then be expected from these results that the most significant improvement in

radiation patterns should be observed in the E-plane of the dipole. Table 3.3 lists the

performance characteristics of each resonance. The front-to-back ratio is defined as the

ratio of the power radiated in the upper half plane to the power radiated in the lower

half plane.


0 dB/m2

−5 dB/m2

−10 dB/m2

−15 dB/m2

−20 dB/m2

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

Figure 3.11: Relative surface-wave power density for 5x5 AMC-BG and AMC surfaces

The radiation patterns for each resonance are shown below in Figure 3.12 where the

total gain is plotted. A clear improvement is seen for the AMC-BG case over the AMC

case in the E-plane for the dipole resonance (Figure 3.12(a)) where the radiation at graz-

ing angles is reduced by 9.5 dB due to the effect of the surface-wave band gap. Maximum

gains of 10.1 dB and 8.3 dB are obtained for the AMC-BG and AMC surfaces respec-

tively. A smaller improvement of 2.6 dB is observed for the H-plane when comparing the

radiation at grazing angles in Figure 3.12(b).

It is observed from Figures 3.12(c) and 3.12(d) that the surface resonances also provide

excellent radiation characteristics. A peak gain of 10.6 dB is observed for the AMC-BG

surface compared to 8.4 dB for the AMC surface. In addition, a more directive pattern is

Table 3.3: Resonance characteristicsResonance Case Frequency [GHz] Gain [dB] FBR [dB] Efficiency [%]

DipoleAMC-BG 5.11 10.1 12.7 > 99

AMC 4.81 8.3 12.4 > 99

SurfaceAMC-BG 5.62 10.6 13.9 > 99

AMC 5.65 8.4 10.2 97.8

HigherAMC-BG 6.36 4.8 N/A 78.8

AMC 6.56 9.3 N/A 72.2


also obtained for the AMC-BG surface compared to the AMC surface, even though the

surface resonance falls outside the band gap. It is possible that the finite nature of the

surface causes a frequency shift in the location of the surface-wave band gap resulting

in the improved directivity. Figures 3.12(e) and 3.12(f) show the degradation of the

radiation patterns at the higher order resonance in both the E and H planes.

3.7 Chapter Summary

The effect of the surface-wave band gap on the radiation characteristics of a dipole

antenna spaced closely to a HIS was quantified in this chapter through the use of an

AMC-BG and AMC unit cell. It was shown that there are multiple resonances excited in

an AMC-BG and AMC surface by a closely space dipole, most of which are functions of

the structure size and band gap location. There is also a zero-order resonance excited in

the structure that is independent of the structure size for a sufficient grid size (5x5 in this

case). It was shown that by locating the PMC frequency within the band gap (AMC-BG

case), the surface-wave radiation is reduced and highly directive patterns are achieved,

however, the PMC effect of the surface is also limited. Conversely, by separating the PMC

frequency from the band gap, the PMC effect of the surface is greatly improved but a

lower gain is observed. Specific design cases will now be presented in the next chapter

along with simulated and measured results to practically demonstrate these effects.


15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(a) Dipole Resonance: E-plane

15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(b) Dipole Resonance: H-plane

15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(c) Surface Resonance: E-plane

15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(d) Surface Resonance: H-plane

15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(e) Higher Resonance: E-plane

15 dB10 dB5 dB0 dB

−5 dB−10 dB−15 dB−20 dB−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BGAMC

(f) Higher Resonance: H-plane

Figure 3.12: AMC-BG and AMC radiation patterns for the various resonances at thefrequencies listed in Table 3.3

Chapter 4

Design Cases: Simulated &

Measured Results

4.1 Design Parameters

Design cases are now presented for the AMC and AMC-BG designs with an incorporated

feed mechanism to compare more realistic simulations that can ultimately be fabricated.

The substrate type and thickness is kept consistent with the design of the unit cells

in Chapter 3. A Rogers TMM4 substrate of thickness 1.524 mm and εr = 4.5 is used

for the surfaces. The loss tangent associated with the substrate is, tan δ = 0.002. The

dipole height above the substrate is kept fixed at 3 mm for both cases. A dipole length

of 26 mm is selected since it provides the optimum cancellation of the input reactance

based on Figures 3.5(c) and 3.5(d). Rectangular sheets of width 0.5 mm are used to

create the dipole arms as done in Chapter 3. A 5x5 surface is used in for the AMC-BG

and AMC surfaces and the size of the ground plane is reduced to 50 mm by 50 mm.

69

Chapter 4. Design Cases: Simulated & Measured Results 70

Figure 4.1: Co-axial fed dipole through ground plane with grounded second arm.

4.2 Feed Mechanism

Several feed mechanisms have been used to feed antennas on top of AMC and EBG ground

planes. The simplest method to feed a dipole is by means of a co-axial cable directly

though the ground plane as done in [32]. The outer conductor of the cable is connected

to the ground plane whereas the inner conductor is extended through the substrate,

between the patches and bent into the required antenna shape. A second conductor is

then connected directly to the ground plane to create the second branch of the dipole

as shown in Figure 4.1. The drawback of this feeding mechanism is that it does not

take into account the impedance transformation that occurs due to the transmission-line

section created between the ground plane and dipole that can effect the input impedance

seen at the connector. Moreover, this method is not feasible in situations where the gap

width is relatively small.

An extended version of this feeding technique is used in [17] where an LC matching

network is implemented between the terminals of the antenna and the ground plane to

obtain a good impedance match to 50Ω as shown in Figure 4.2. However, this method

also relies on feeding the antenna through the gap between the patches which is unfeasible

for small gap widths.

Alternatively, a side fed arrangement may also be considered where the dipole antenna

is printed on its own substrate and placed above the surface as done in [23] and [19].

The need for a matching network is then eliminated since the antenna can be fed with a

CPS feed. However, either an integrated or external balun is required for the transition


between the unbalanced coaxial feed to the balanced antenna. This adds to the size

and complexity of the prototype and, therefore, a new feeding mechanism was required.

Figures 4.3(a) and 4.3(b) show an integrated and external balun respectively being used

with a CPS-fed antenna.

Our proposed feed mechanism provides a simple solution to the limitations of the

methods outlined above and eliminates the need for a balun as shown in Figure 4.4. A

co-axial cable is used to feed the dipole antenna through the ground plane but instead

of terminating the outer conductor at the bottom ground, it is extended all the way

through the substrate up to the required antenna height. The centre conductor is then

bent to create one branch of the dipole whereas a second conductor can be connected to

the outer conductor to create the second branch. In this way, a 50Ω feed is provided right

up to the terminals of the dipole and only a small unbalanced section exists between the

antenna and the ground plane. More importantly, the AMC-BG and AMC surfaces can

be centered on the outer conductor of the cable which is then used as the via for the

center patch circumventing the need for feeding the antenna through the gaps between

the patches.

Figure 4.2: Matching network implemented with co-axial feed through ground. From[17] c© IEEE 2008


(a) Integrated balun. (b) External bazooka balun.

Figure 4.3: Side-fed antennas with integrated and external baluns. From [23] c© IEEE2009 and [19] c© IEEE 2011 respectively.

Figure 4.4: Co-axial feed extending through ground plane. The outer conductor servesas the via for the center patch.

4.3 Simulated Results

4.3.1 Reference Case

A 26mm dipole above a grounded substrate is simulated first to characterize the effect of

the feed and establish a reference case. The dipole arm connected to the center conductor

is located exactly 3 mm above the surface of the substrate whereas the second dipole arm

is flush with the termination of the outer conductor above the substrate at 2.5 mm as

shown in Figure 4.5(a). The 50Ω coaxial cable used to feed the antenna is extended to

the radiation boundary defining the simulation space and is excited by means of a wave

port.

Figure 4.5(b) shows a comparison of the return loss characteristics when the dipole

is excited with the coaxial feed as opposed to a lumped port. A resonance is introduced


3 mm

2.5 mm13 mm

1.524 mm

(a) Dipole on a grounded substrate.

3.5 4 4.5 5 5.5 6 6.5 7−30

−25

−20

−15

−10

−5

0

X: 4.48Y: −15.92

Frequency [GHz]

|S11

| [d

B]

X: 5.12Y: −4.321

Coaxial FeedLumped Port

(b) |S11| comparison.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(c) Patterns at 4.48 GHz.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(d) Patterns at 5.12 GHz.

Figure 4.5: Reference case characteristics

at 4.48 GHz as a result of this feed arrangment and the monopolar radiation patterns

in Figure 4.5(c) reveal that this resonance is due to the short feed section above the

ground plane. This resonance occurs in close proximity to the design frequency of 5 GHz

and hence can potentially interefere with the dipole and surface resonances. However,

it will later be seen that by introducing the AMC-BG and AMC surfaces, the monopole

mode is shifted to a lower frequency and does not interfere with the dipole and surface

resonances. The dipole resonance occurs at 5.1 GHz and is poorly matched due to the

close proximity of the ground plane with an input impedance of 12Ω. The E and H-planes


in Figures 4.5(c) and 4.5(d) are with reference to the dipole and represent the XZ and

YZ cuts respectively.

4.3.2 AMC Case

A 5x5 AMC grid is then introduced into the substrate as shown in Figure 4.6(a) and the

dipole parameters are kept consistent with those of the reference case. The resulting |S11|

is plotted in Figure 4.6(b). The input impedance response of the antenna is plotted on a

Smith Chart in Figure 4.6(c) from 4.5 GHz to 6 GHz after de-embedding the wave port

to the end of the coaxial cable to eliminate the phase delay associated with the cable.

An observation of Figure 4.6(c) reveals that the dual resonance has been detuned

due to an additional capacitance by introducing the AMC surface. By experimenting

with the relative lengths of the dipole arms, it is discovered that this capacitance can

be compensated for by offsetting the dipole arms about the feed. The length of the arm

connected to the center conductor is increased by 1.5 mm and the length of the second

arm is reduced by the same amount resulting in a total offset of loffset = 3 mm. The

resulting |S11| and input impedance responses are plotted along side the loffset = 0 mm

case in Figures 4.6(b) and 4.6(c). The dashed black circle in Figure 4.6(c) represents the

VSWR 2 : 1 circle.

From Figure 4.6(b), it is noted that the dipole resonance at 4.89 GHz is well matched

due to the PMC effect of the AMC surface. In addition, the (βd)x,y = 0 surface resonance

is matched at a higher frequency (5.54 GHz) and merged with the dipole resonance to

create a wideband response. A total bandwidth of 0.95 GHz is obtained from 4.75 −

5.70 GHz. The radiation pattern at each of the resonances is shown in Figures 4.7(a)

and 4.7(b). The E-plane and H-plane represent the XZ and YZ cuts respectively. A

maximum gain of 7.9 dB and an efficiency of 95.5% is noted for the dipole resonance

at 4.89 GHz and a gain of 8.0 dB and efficiency of 94.0% for the surface resonance at

5.54 GHz. The maximum gain is skewed off normal at the surface resonance (5.54 GHz)


50 mm

13 mm +Loffset/2

50 mm

13 mm - Loffset/2

y

x

z

(a) Dipole on a 5x5 AMC grid.

3.5 4 4.5 5 5.5 6 6.5 7−45

−40

−35

−30

−25

−20

−15

−10

−5

0

X: 4.89Y: −44.49

Frequency [GHz]

|S11

| [d

B]

X: 5.54Y: −18.35

loffset

= 0 mm

loffset

= 3 mm

0.95 GHz

(b) |S11| response.

loffset

= 0 mm

loffset

= 3 mm

(c) Zin response.

Figure 4.6: 5x5 AMC response.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(a) Patterns at 4.89 GHz.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(b) Patterns at 5.54 GHz.

Figure 4.7: 5x5 AMC radiation patterns.


and occurs at approximately 18o in the E-plane. This is attributed to a current imbalance

in the dipole arms as a result of the offset about the feed since the pattern asymmetry is

not observed when using a lumped port excitation with a symmetric dipole as in Chapter

3.

4.3.3 AMC-BG Case

A 5x5 AMC-BG grid is simulated next with the remaining geometry identical to the

reference case. A similar de-tuned response is observed, as shown in Figure 4.8, and the

dipole is offset about the feed point by 2 mm as in the AMC case to compensate for the

parasitic capacitance. The tuned |S11| response and input impedance are plotted along

side the untuned case in Figures 4.8(b) and 4.8(c).

50 mm

13 mm +Loffset/2 50 mm

13 mm - Loffset/2

y

x

z

(a) Dipole on a 5x5 AMC-BG grid.

3.5 4 4.5 5 5.5 6 6.5 7−45

−40

−35

−30

−25

−20

−15

−10

−5

0

X: 5.27Y: −15.12

Frequency [GHz]

|S11

| [d

B] X: 5.49

Y: −15.22

loffset

= 0 mm

loffset

= 2 mm

0.52 GHz

(b) |S11| response.

loffset

= 0 mm

loffset

= 2 mm

(c) Zin response.

Figure 4.8: 5x5 AMC-BG response.


Once again, by merging the dipole resonance with the (βd)x,y = 0 surface resonance a

broadband response is observed from 5.13−5.65 GHz with a total bandwidth of 0.52 GHz.

However, the dipole and surface resonances appear to be sufficiently merged in this case

making it difficult to individually distinguish each resonance. As a result, the radiation

patterns are plotted at 5.27 GHz and 5.49 GHz to quantify the maximum gain, as shown

in Figure 4.8(b). The resulting patterns are shown in Figure 4.9. A maximum gain of

8.9 dB with an efficiency of 94.0% is observed for the dipole resonance at 5.27 GHz and

maximum gain of 9.2 dB with an efficiency of 93.2% is observed for the surface resonance

at 5.49 GHz. More importantly, it is seen that the maximum gain at the surface resonance

is observed at broadside and is not skewed as in the AMC case. It is possible that this

improvement is a combination of the smaller offset (2 mm vs 3 mm) for the AMC-BG

case over the AMC case and the presence of the band-gap. A detailed investigation of

this, however, is beyond the scope of this work.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(a) Patterns at 5.27 GHz.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(b) Patterns at 5.49 GHz.

Figure 4.9: 5x5 AMC radiation patterns.


4.3.4 Comparison of Simulated Results

Figure 4.10 shows a comparison of the simulated |S11| responses for a dipole of fixed

length (26 mm) and fixed height (3 mm) above a 5x5 AMC-BG and AMC surface.

Both these designs exhibit a dual resonance behaviour with the lower resonance corre-

sponding to the dipole resonance and the upper resonance corresponding to the surface

resonance. The resonances have been merged in both cases to extend the bandwidth of

the antenna/surface combination. A bandwidth of 0.52 GHz or 9.6% is observed for the

AMC-BG case whereas a bandwidth of 0.95 GHz or 18.2% is noted for the AMC case. It

is likely that this improved bandwidth is a result of the better matching characteristics

due to an improved impedance response (see Figure 3.10) of the AMC surface over the

AMC-BG surface. A further investigation of this improvement is, however, outside the

scope of this work.

The radiation patterns at each of the resonances for the two surface cases are then

compared in Figures 4.11(a) through 4.11(d). At the dipole resonance, a maximum gain

of 8.9 dB is observed for the AMC-BG case compared to a maximum gain of 7.9 dB for

the AMC case. Moreover, radiation at θ = ±90o is improved by approximately 15 dB in

the E-plane for the AMC-BG case over the AMC case. The FBR at the dipole resonance

3.5 4 4.5 5 5.5 6 6.5 7−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

AMC−BGAMC

Figure 4.10: Comparison of |S11| for the AMC-BG and AMC cases.


is 11.9 dB for the AMC-BG case compared to 11.2 dB for the AMC case. At the surface

resonance, a maximum gain of 9.2 dB is noted for the AMC-BG case versus a maximum

gain of 8.0 dB for the AMC case. Also, the maximum AMC gain occurs at 18o off

broadside and pattern distortion is observed. Moreover, the AMC-BG surface resonance

pattern is observed to be more symmetric than that of the AMC pattern. FBRs of

12.3 dB and 9.8 dB are calculated at the surface resonance for the AMC-BG and AMC

cases respectively. Hence, it is seen seen that by placing the PMC frequency within the

band-gap, higher gains and FBRs may be achieved.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

AMC−BG (f = 5.27 GHz) AMC (f = 4.89 GHz)

(a) Dipole Resonance (E-Plane).

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°


(b) Dipole Resonance (H-Plane).

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°


(c) Surface Resonance (E-Plane).

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°


(d) Surface Resonance (H-Plane).

Figure 4.11: Radiation pattern comparison at dipole resonance and surface resonance forAMC-BG and AMC cases.


4.4 Fabrication

A Rogers TMM4 substrate of thickness 60 mils (1.524 mm) and copper cladding 1 oz. was

ordered from Rogers Corporation to prototype the boards. The two options available for

prototyping were a wet etching process or milling process. In wet etching, the board is first

developed using a standard photolithographic procedure and then immersed into a ferric

chloride solution to etch away the excess copper. Fine resolutions (up to 100 µm) can be

achieved using this process. The milling process uses a variety of drilling and routing bits

to mill away the copper to create the prototype and resolutions of up to 200 µm can be

achieved. A significant disadvantage of the wet etching process compared to the milling

process is that vias have to be drilled in manually compared to the automated, more

precise drilling done by the milling machine. Therefore, it was decided not to proceed

with a wet etching process.

Gerber files of the design were generated in Agilent Momentum and provided to

Electro Circuit Inc., located in Scarborough, Ontario, for fabrication. The center hole

radius for each prototype was increased to 1.5 mm in anticipation of feeding through the

coaxial cable and allowing room for the connector to be used. The feature dimensions

were measured upon recieving the boards to ensure accuracy of the fabrication process.

The vias had been plated to create an electrical connection between the patches and the

ground plane and a conductivity test was conducted using a digital multimeter to confirm

the connection. The center hole was not plated since the outer conductor of the coxial

cable would be used to create an electrical connection between the center patch and

ground plane. The prototyped AMC-BG and AMC boards are shown in Figures 4.12(a)

and 4.12(b) respectively.

An RG − 402 coaxial cable with Zo = 50Ω was used to create the antenna feed and

dipole arms. The cable was connectorized at one end using an SMA connector, the other

end of the cable was then stripped to reveal the inner conductor that would be used to

create one of the dipole arms. The center hole on each of the boards had to be filed to


(a) 5x5 AMC-BG prototype. (b) 5x5 AMC-BG prototype.

Figure 4.12: Fabricated AMC-BG and AMC surface protoypes.

increase its radius and make room for the connector to ensure a snug fit with the board.

The cable and connector were fed through the hole and the center conductor was bent

in to shape to create the dipole arm. The height of the dipole arm above the substrate

surface was measured using a vernier caliper and the outer conductor was further stripped

as required to ensure the height to the center conductor arm above the surface matched

the simulated design height. Once the center conductor had been adjusted, the connector

was soldered to the ground plane and the portion of outer conductor of the coaxial cable

above the substrate was soldered to the top patch. The inner conductor of a different

piece of RG−402 cable was then extracted and soldered to the outer conductor above the

substrate to create the second dipole arm. A Vernier caliper was then used to measure

the two dipole arms and a pair of pliers was used to trim the arms to the required length.

Figure 4.13 shows the final prototyped designs.

4.5 Measured Results

S-parameter measurements of the fabricated prototypes were carried out using an Agilent

E8364B Vector Network Analyzer (VNA). The VNA was calibrated using an Agilent

85033E calibration kit using a 1-port short-open-load procedure over the frequency range


Figure 4.13: Final fabricated prototypes. AMC-BG board with dipole antenna shown onleft and AMC board with dipole antenna on right.

3.5 4 4.5 5 5.5 6 6.5 7−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

MeasuredSimulated

(a) AMC-BG |S11| response.

3 3.5 4 4.5 5 5.5 6 6.5−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

MeasuredSimulated

(b) AMC |S11| response.

Figure 4.14: Comparison of measured and simulated |S11| responses.

of interest prior to conducting any |S11| measurements. The measured |S11| responses

for the AMC-BG and AMC cases is compared against the simulated results as shown in

Figures 4.14(a) and 4.14(b).

Minor discrepencies are noted in the measured and simulated results, particularly for

the AMC-BG case, and can be attributed to the coarse tolerances in prototyping the

designs. The AMC-BG case is re-simulated with the dipole length and offset adjusted to


3.5 4 4.5 5 5.5 6 6.5−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

MeasuredSimulated

(a) AMC-BG |S11| response.

3.5 4 4.5 5 5.5 6 6.5−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

MeasuredSimulated

(b) AMC |S11| response.

Figure 4.15: Comparison of measured and tuned simulated |S11| responses.

12.5 mm and 2.5 mm respectively. The AMC case in not adjusted since the measured

dipole and surface resonances are well described by the original parameters. The final

simulated and measured |S11| responses for the AMC-BG and AMC cases are shown in

Figures 4.15(a) and 4.15(b). It is noted in Figure 4.15(a) that for the AMC-BG case,

the dipole and surface resonance are completely merged and indistiguishable. It is likely

that this is a result of the tolerances associated with the fabrication procedure since the

length of the dipole and its height above the surface had to be manually adjusted. An

alternative approach could have been to fabricate the dipole on a low dielectric constant

substrate and glue it to the AMC-BG surface. This would have allowed much finer control

of the dipole features and position. One disadvantage of this alternative approach is that

the dipole substrate can influence the surface resonance and hence should be taken into

account during the design of the unit cells. Regardless, for proof of concept purposes,

the current AMC-BG design demonstrates the improvement in bandwidth over a single

resonance and the improvement in gain over the AMC case as discussed below.

Radiation patterns were then measured at the anechoic chamber at the University of

Toronto. The chamber was calibrated using two DRH-0118 broadband horn antennas.

Once the calibration process was complete, the antenna under test (AUT) was placed


on a rotating pedestal and the power from a transmitting horn antenna was sampled at

each angle increment to determine the gain of the antenna as a function of angle. Since

the setup is limited to measuring single planes rather than a complete 3D radiation

pattern, the E and H cut-planes of the antenna were measured. The co-polarization

and cross-polarization were measured individually for each cut plane by rotating the

transmitting horn with respect to the AUT. For the AMC case, since the measured

dipole and surface resonances were easily distinguished, patterns were measured at both

resonances. However, for the AMC-BG case, pattern measurements were made at the

lowest point of the |S11| response since the dipole and surface resonances were completely

merged and not distinguishable. The measured and simulated radiation patterns for the

AMC and AMC-BG cases are shown in Figure 4.16 where the total gain is plotted.

Excellent agreement is seen between the measured and simulated results for both

AMC and AMC-BG cases. A small discrepancy is noted in the radiation pattern of the

surface resonance for the AMC case where the maximum measured gain is approximately

1.4 dB lower than the maximum simulated gain. It is seen from the E-plane pattern for

this case that the measured radiation pattern is not as skewed as the simulated pattern

due to the higher gain measured in the negative half of the E-Plane. It is likely that the

coarse fabrication tolerances are responsible for this deviation, however, the qualitative

features of the pattern are fully captured. Table 4.1 summarizes the simulated and

measured resonant frequencies and gains for the AMC-BG and AMC cases. It is seen

from the results in Table 4.1 that a gain improvement of 1.2 dB is measured by designing

the unit cell so that the PMC frequency and surface-wave band gap coincide.

Table 4.1: Summary of Measured and Simulated ResultsSurface

CaseResonant Freq. [GHz] Gain [dB]

Type Simulated Measured Simulated Measured

AMCDipole 4.89 4.84 7.9 7.8Surface 5.54 5.57 8.0 6.6

AMC-BG Merged 5.36 5.36 9.1 9.0


10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(a) AMC Dipole Resonance: E-plane.

10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(b) AMC Dipole Resonance: H-plane.

10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(c) AMC Surface Resonance: E-plane.

10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(d) AMC Dipole Resonance: H-plane.

10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(e) AMC-BG Resonance: E-plane.

10 dB

0 dB

−10 dB

−20 dB

−30 dB

0°

45°

90°

135°

180°

225°

270°

315°

(f) AMC-BG Resonance: H-plane.

Figure 4.16: Comparison of measured and simulated radiation patterns for AMC andAMC-BG cases. — Simulated - - - Measured. Patterns are plotted at frequencies listedin Table 4.1

Chapter 5

Conclusion

5.1 Summary

The mushroom-like structure was examined in this thesis with the goal of establishing

a relationship between the scattering and dispersion characteristics of the unit cell and

determining the effect of the bound surface-wave band gap on the performance of a closely

spaced dipole antenna.

It was shown in Chapter 2 that the resonances that occur in the reflection phase profile

of a unit cell for on-axis TE and TM polarized incident waves are equivalent to the TE

and TM eigenmodes of the unit cell corresponding to a zero degree phase shift along

the direction of propagation. The approach was then extended to off-axis incidence for

TE and TM polarized waves and a complete mapping of the reflection phase resonances

to the dispersion diagram eigenmodes was presented. An NRI-TL model was then used

to investigate the dispersion characteristics of the mushroom-like structure and it was

shown that the unit cell resonances with a zero degree phase shift correspond to the series

and shunt resonances off the NRI-TL unit cell. The boundary conditions at each of these

resonances was then investigated analytically. It was shown that for an eigenmode to

manifest as a resonance in the reflection phase profile, the field profile of the excitation

86

Chapter 5. Conclusion 87

must be consistent with that of the eigenmode.

Two unit cell designs were presented in Chapter 3 to examine the effect of the surface-

wave band gap on the performance of a dipole antenna. The unit cells were designed

to have identical reflection phase profiles but varying band-gap locations. The first unit

cell, referred to as the AMC-BG case, was designed to have the zero-degree reflection

phase frequency coincide with the location of the band gap whereas in the second unit

cell, referred to as the AMC case, the two properties were seperated. A parametric study

of the surface size and dipole length was carried out for each unit cell case and it was

shown that in each of the unit cell cases, a surface resonance is excited in close proximity

to the zero degree reflection phase frequency. In addition to the surface resonance, which

was shown to be independent of the structure size, a dipole resonance is also observed

which is naturally a function of the dipole length. It was demonstrated that, to maintain

an optimum impedance match to 50Ω, the dipole resonance should be tuned to lie in

the vicinity of the surface resonance. The improvement in input impedance at resonance

for the dipole antenna was quantified for the AMC-BG and AMC cases and simulated

results demonstrated enhanced performance for the AMC case over the AMC-BG case.

An examination of the radiation patterns, however, described an increased gain and FBR

for the AMC-BG case over the AMC case.

A novel antenna feed mechanism was proposed in Chapter 4 and practical design

cases were presented for the AMC-BG and AMC cases. Simulated results confirmed an

improvement in antenna gain and symmetric radiation patterns for the AMC-BG case

over the AMC case. The simulated prototypes were then fabricated and tested to confirm

the performance improvements. A maximum measured gain of 9.0 dB at broadside and

symmetric radiation patterns were observed for the AMC-BG case. A maximum gain

of 7.8 dB and 6.6 dB was measured at the dipole and surface resonances for the AMC

case and pattern distortion was noted for the AMC case at the surface resonance with

maximum gain tilted off broadside by 20o.


5.2 Future work

The work presented in this thesis has raised some interesting questions about the design

of radiating elements over AMCs. Specifically, it was demonstrated in Chapter 3 that

an additional resonance is observed for the AMC-BG case that does not appear in the

AMC case. This resonance was shown to be a function of the surface size, however,

its properties were not fully investigated. Figure 5.1 shows the |S11| response, input

impedance and radiation patterns for a dipole antenna of length 33.5 mm at a height of

3 mm above a 5x5 AMC-BG grid. A narrow band resonance is observed at 3.82 GHz

which is well outside the ±90o in-phase reflection bandwidth of the AMC-BG unit cell

(4.66GHz to 5.38GHz). Good radiation patterns are observed at this resonance as well

and a maximum gain of 7.3 dB is noted at broadside as shown in Figure 5.1(c). The E

and H-planes are with reference to the dipole antenna.

It is likely that this additional resonance is related to the fvia eigenmode discussed in

Chapter 2. This conjecture could be further examined to study the true nature of the

resonance and explain how it is excited by dipole source. A dual-band dipole could then

potentially be designed to be matched at both the resonances occuring in the AMC-BG

case to create a dual-band, highly directive antenna.

An alternative design methodology would consider using two sources to excite both

the fgap and fvia resonance in an attempt to create a MIMO antenna. It has been shown

in [44,45], that the fvia eigenmode can be made to radiate by feeding one of the mushroom-

like unit cells from the via as shown in Figure 5.2(a) to create a monopolar radiation

pattern. A dipole source, such as the one presented in this thesis, can then be used to

excite the fgap eigenmode and create an orthogonal, patch-type radiation pattern. In fact,

a first attempt at this has been conducted in [19] as shown in Figure 5.2(b), although, the

authors claim that it is not possible to overlap the two bands shown in Figure 5.2(c) to

create a true MIMO antenna with pattern diversity. However, it has been shown in this

thesis that, by properly designing the unit cell, the fgap and fvia resonances can occur at


3 3.5 4 4.5 5 5.5 6−40

−35

−30

−25

−20

−15

−10

−5

0

X: 3.82Y: −39.36

Frequency [GHz]

|S11

| [d

B]

(a) |S11| response.

3 3.5 4 4.5 5 5.5 6−1000

−750

−500

−250

0

250

500

750

1000

Frequency [GHz]

(Zin

) [Ω

]

RealImaginary

(b) Input Impedance.

10 dB

5 dB

0 dB

−5 dB−10 dB

−15 dB

−20 dB

−25 dB

0°

45°

90°

135°

180°

225°

270°

315°

E−PlaneH−Plane

(c) Radiation Patterns at 3.82 GHz.

Figure 5.1: Performance of 33.5 mm dipole on a 5x5 AMC-BG grid at a height of 3 mm.

the same frequency and the bands in Figure 5.2(c) can indeed be made to overlap.

These research directions could further the theoretical understanding of EBG struc-

tures and possibly culminate in new applications of the mushroom-like and associated

structures to address the growing challenges related to antenna design.


(a) Via-fed Folded Monopole. (b) Multiple port antenna radiating orthogonalmodes.

(c) S-parameter response of antenna from (b).

Figure 5.2: Folded NRI-TL Monopole shown in (a) from [44] c© IEEE 2008. Antennaprototype from [19] shown in (b) with S-parameter response shown in (c) c© IEEE 2011.

Appendix A

Parameter Extraction

This appendix details the method used to extract the host TL parameters and the loading

element values used to compare the dispersion curves generated by eigenmode simulations

to the ones predicted by NRI-TL theory. The transverse boundaries of the unit cell act

as H-walls for on-axis propagation where the transverse phase shift, (βd)y, is set to zero

[34]. This can be justified by considering on-axis propagation on a 2D transmission line

grid. A zero transverse phase shift can be considered analogous an even-mode excitation

which corresponds to H-walls between adjacent transmission line columns as shown in

Figure A.1(a). Therefore, when extracting the TL parameters or the loading element

values for a specific geometry, the transverse boundaries can be set to H-walls as shown

in Figure A.1(b).

A.1 Transmission Line Parameters

The transmission line parameters of interest are the characteristic impedance, Zo and

phase velocity, vφ. The characteristic impedance is easily obtained in Ansoft HFSS by

using a waveport excitation for the transmission line at either end of the simulation space

and plotting the port impedance over the frequency range as shown in Figure A.2(b).

Equation A.1 can be used to calculate the phase velocity where dTL is the length of the

91

Appendix A. Parameter Extraction 92

zx

Ground

H-wall H-wallH-wall

+ +

(βd)y=0

(a) (βd)y = 0 resulting in even-modeexcitation

Ground

H-wallH-wall

d

w

z

y

Radiation Boundary

t

(b) Front view of simulation setup

Figure A.1: Transverse H-walls used for on-axis propagation

TL section and φTL can be read off at a specific frequency point by plotting the phase

of the transmission coefficient, S21 versus frequency as shown in Figure A.2.

vφ =ω

β=ωdTLφTL

(A.1)

A.2 Loading Element Parameters

Loading element values for the gap capacitances and via inductances can be obtained by

treating the loading as a lumped element. To achieve this, the waveports at either end

of the simulation space are de-embedded to the center of the loading element as shown

in Figure A.3(a). Then, by treating the elements as lossless lumped elements, the 4x4 Z

or Y matrices obtained from the simulation results can be used to determine the series

capacitance or shunt inductance values [37].

The lumped capacitance can be extracted from the ABCD matrix for a series load,


Wa

vep

ort

1

Wa

vep

ort

2

Radiation Boundary

TL section

dTLx

z

(a) Simulation setup for host TL parame-ters

2 4 6 8 10 12 140

25

50

75

Frequency [GHz]

Zo

[Ω]

Port 1Port 2

(b) Characteristic Impedance for Case 1

2 4 6 8 10 12 14−200

−160

−120

−80

−40

0

40

80

120

160

200

X: 5Y: −151.1

Frequency [GHz]

Ph

ase

Sh

ift

[deg

]

(c) TL phase shift for Case 1

Figure A.2: Extraction of host TL parameters

ZL, as shown below:

A B

C D

=

1 ZL

0 1

, then

Cgap = − j

ωIm(ZL)where ZL = B =

Z11Z22 − Z12Z21

Z21

(A.2)

Similarly, for a shunt load, YL, the inductance is calculated as:

A B

C D

=

1 0

YL 1

, then

Lvia = − j

ωIm(YL)where YL = C = −Y11Y22 − Y12Y21

Y21(A.3)


The simulation setup and extracted capacitance for the geometry of Case 1 is shown

in Figure A.3

Cgap

Wa

vep

ort

1

Wa

vep

ort

2

Radiation Boundary

x

z

(a) Simulation setup for lumped elementparameters

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

Frequency [GHz]

Cap

acit

ance

[p

F]

(b) Gap capacitance for Case 1

Figure A.3: Extraction of lumped capacitance

Appendix B

Notes on the HFSS Eigenmode

Solver

The Ansoft HFSS eigenmode solver has been heavily used in this work to generate the

dispersion curves of various unit cell geometries in Chapters 2 and 3. When performing

those eigenmode simulations, it was noted that it is possible for the solver to return an

unrealistic eigenmode of the geometry in question. This appendix details the methodol-

ogy used to distinguish between the physical and unphysical resonances of the geometry

being simulated.

B.1 Simulation Procedure

B.1.1 Fundamental Modes

The parameters of Case 1 as described in Section 2.1.1 are used to generate the geometry

of the unit cell with linked PBCs. The first stage of the simulation involves determining

the eigenmodes for the fundamental backward and forward wave modes (referred to as

the TM0 mode). The size of the air region between the unit cell and PML is set to six

times the substrate thickness. The minimum solution and PML frequencies are set to

95

Appendix B. Notes on the HFSS Eigenmode Solver 96

0.5 GHz and the number of eigenmodes to be solved is set to 1. The remaining PML

parameters are left at their default values. A parametric sweep is then performed for

the phase shift across the x-directed PBCs, 0o ≤ (βd)x ≤ 180o, in 5o increments with

(βd)y = 0. The resulting dispersion curve is plotted in Figure B.1.

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

βd [Deg.]

f[G

Hz]

TM0Light Line

Figure B.1: TM0 mode for Case 1 showing erroneous eigenmode solution at (βd)x = 0

B.1.2 Higher Order Modes

The next stage involves determining the eigenmode solutions for the higher order modes.

The PML is regenerated and the minimum solution and PML frequencies are set to

5 GHz, which is at the peak of the fundamental mode and the number of eigenmode

solutions is set to 6 to account for various higher order modes that may occur. The

parametric sweep, 0o ≤ (βd)x ≤ 180o with (βd)y = 0, is then repeated to generate the

dispersion diagram. The resulting modes are plotted in Figure B.2.

B.2 Mode Selection

The fundamental mode of the dispersion diagram is examined first and a discrepency is

noted at the (βd)x = 0 point with an eigenmode solution at 3.6 GHz. This result occurs


0 30 60 90 120 150 1800

2

4

6

8

10

12

14

16

18

20

βd [Deg.]f

[GH

z]

Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6Light Line

Figure B.2: Case 1 dispersion profile showing 6 higher order modes

due to the minimum frequency setting on the solution setup frequency and is discarded

since an eigenmode solution of 0 GHz is expected at the (βd)x = 0 point for this lowest

order mode.

Determining the physical higher order modes is slightly more challenging since there

is no pre-determined expectation for where the modes should occur. If the analysis is

limited to the first three higher order modes then the resulting dispersion curve is given

by Figure B.3. The resulting curves appear haphazard and incomplete, therefore, this

approach is incorrect.

0 30 60 90 120 150 1800

2

4

6

8

10

12

14

16

18

20

βd [Deg.]

f[G

Hz]

Mode 1Mode 2Mode 3Light Line

Figure B.3: Case 1 dispersion profile showing first 3 higher order modes


The correct approach is to examine the field distribution of each eigenmode and to

ensure that the dispersion curve is for each mode is continuous. For example, consider

the six resonances occuring on the (βd)x = 0 axis. By examining the field distributions,

the resonances of the unit cell can be seperated from potential unphysical resonances.

Figure B.4 shows a perspective view of the Poynting vector plotted at different elevations

above the unit cell and the electric field on the top surface of the unit cell. All fields

are plotted using the same scale and the PBCs have been hidden from view for clarity.

It is observed that the field distributions of the first three modes reflect resonances of

the unit cell and correspond to either TE or TM modes. However, the last three modes

exhibit extremely large magnitudes of the Poynting vector in region bounded between the

PML and the top surface of the unit cell. These modes could potentially be resonances

of the entire simulation setup and therefore need to be discarded when examining the

eigenmodes of the unit cell. The same analysis is then conducted at each phase shift

increment to determine the nature of the mode (TE vs TM) as well as whether it is

physical or unphysical. This analysis need only be conducted to identify modes lying

within the light cone. From Figure B.2 it can be seen that the modes lying outside

the light cone are much easier to distinguish and can be determined by requiring the

dispersion curve to be continuous. Field profiles outside the light cone still need to be

examined to distinguish between the curves corresponding to the TM and TE modes.

Once this exercise is completed, the correct dispersion curves can be plotted for each

geometry.

It is observed that for this particular case, the first three modes observed on the

(βd)x = 0 axis are valid eigenmodes of the geometry. It is important to note that this

is not always the case and unphysical modes may appear prior the the physical ones.

Table B.1 lists the eigenmode solutions returned by the solver for all three geometry

cases discussed in Table 2.1.1. It is seen that for Cases 1 and 2, the first three modes

are the correct modes whereas for Case 3, the correct modes are given by modes 1,4


x y

z

(a) Mode 1

x y

z

(b) Mode 2

x y

z

(c) Mode 3

x y

z

(d) Mode 4

x y

z

(e) Mode 5

x y

z

(f) Mode 6

Figure B.4: E-field and Poynting vector distributions at eigenmode solutions

and 5. The determination of whether a mode is correct in this specific case is based

upon the relative magnitudes of the quality factors. Since the unphysical modes show

disproportionately large fields in the air region, their quality factors are much lower than

those of the physical modes. This shortcut is valid only for modes within the light cone.

Outside the light cone, the continuity of the dispersion curve must be enforced.

Table B.1: Eigenmode solutions for Cases 1 through 3

Mode#Case 1 Case 2 Case 3

Freq. [GHz] Q Freq. [GHz] Q Freq. [GHz] Q1 5.71 + j0.56 4.78 6.46 + j0.06 50.07 6.61 + j0.07 50.472 5.72 + j0.60 4.77 6.77 + j0.81 4.24 7.29 + j4.76 0.913 6.45 + j0.06 50.01 6.78 + j0.81 4.23 7.29 + j4.76 0.914 7.26 + j4.99 0.88 7.30 + j4.92 0.89 8.45 + j1.22 3.505 7.26 + j4.99 0.88 7.30 + j4.92 0.89 8.45 + j1.22 3.506 10.82 + j7.58 0.87 10.91 + j7.57 0.88 10.88 + j7.47 0.88


In addition to the above procedure, one additional test can be introduced to distin-

guish between the physical and unphysical eigenmodes returned by the solver. Since the

unphysical modes are resonances of the simulation setup including the PML as opposed

to resonances of the unit cell, changing the size of the simulation space should affect the

position of the unphysical resonance without affecting the physical resonance. Table B.2

show the returned eigenmodes for two different sized air regions between the PML and

the unit cell for Case 1. The air region in the second simulation was increased to twice its

original value and the frequency shift in the unphysical resonances is observed whereas

the physical resonances remain constant.

Mode#Airbox 1 Airbox 2

Freq. [GHz] Q Freq. [GHz] Q1 5.71 + j0.56 4.78 5.71 + j0.57 5.042 5.72 + j0.60 4.77 5.72 + j0.57 5.033 6.45 + j0.06 50.01 6.38 + j3.17 1.124 7.26 + j4.99 0.88 6.38 + j3.17 1.135 7.26 + j4.99 0.88 6.45 + j0.06 50.016 10.82 + j7.58 0.87 9.44 + j4.92 1.08

Table B.2: Eigenmode solutions for Case 1 with varying airbox sizes

Appendix C

Parametric study of dipole antenna

on EBG ground plane

The study conducted in [15] concluded that the optimum operating point for a dipole

antenna on a mushroom-like surface occurs within the 90±45o reflection phase bandwidth

of the unit cell comprising the surface. That is to say, the dipole antenna experiences its

best impedance match bandwidth within this reflection phase criterion. It was shown in

Sections 3.3 and 3.5 that multiple resonances are excited by a dipole in close proximity to

a mushroom-like surface. In particular, depending on the size of the surface, a dipole may

exhibit improvement in its input resistance closer to its PMC frequency (as in the 5x5

case in Figure 3.10) or in a lower frequency band (as in the 7x7 case in Figure 3.10) due

to the presence of a resonance in the surface that is a strong function of the surface size.

tt was also shown that this size dependent resonance does not appear in the event that

the surface-wave band gap is not present and hence cannot be attributed to the reflection

phase characteristics of the surface. The study conducted in [15] will be re-visited in this

appendix and it will be shown that if the size of the structure is reduced, the 90 ± 45o

criterion does not hold as strongly and the optimum impedance match is obtained if the

dipole is tuned to the PMC frequency of the unit.

101

Appendix C. Parametric study of dipole antenna on EBG ground plane102

C.1 Unit Cell Properties

C.1.1 Geometry

The AMC-BG unit cell of Chapter 3 is used to perform this study since it was shown

in [15] that the results scale with frequency. Table C.1.1 lists the dimensions of the unit

cell for reference. A Rogers TMM4 substrate is used which has a relative permittivity,

εr = 4.5 and a loss tangent of tanδ = 0.002.

Table C.1: Unit cell geometry.Parameter Value [mm]

Substrate Thickness, t 1.524Patch Width, w 8.4Gap Width, g 0.2Via Radius, r 0.5

C.1.2 Scattering & Dispersion Characteristics

The reflection-phase profile of the unit cell is repeated in Figure C.1(a) with the 90±45o

shaded in grey. A PMC frequency of 5 GHz is noted with the 90 ± 45o bandwidth

spanning from 4.18 to 4.87 GHz. The dispersion profile is repeated in Figure C.1(b).

C.2 Dipole performance

The study carried out in [15] is now repeated to confirm their results. A dipole is now

placed at a height of 3 mm which corresponds to 0.05λ5GHz above a 1λ5GHz x 1λ5GHz

surface. The surface size corresponds to a 7x7 grid of the unit cell in question. The

dipole length is then varied in 0.5 mm increments from 23.5 mm to 30 mm. The grid size

is then reduced to 5x5 and the parametric sweep is repeated Figures C.2(a) and C.2(b)

show the simulation setup for the 5x5 and 7x7 cases respectively.


3.5 4 4.5 5 5.5 6 6.5 7−180

−135

−90

−45

0

45

90

135

180

f [GHz]

Ref

lect

ion

Ph

ase

[Deg

.]


90±45o:4.18 − 4.87 GHz

5 GHz

(a)

0 30 60 90 120 150 1800

1

2

3

4

5

6

7

8

9

βd [Deg.]

f[G

Hz]


Band Gap: 4 − 5.25 GHz

(b)

Figure C.1: (a)Reflection-phase and (b)dispersion profile of AMC-BG unit cell.

Figure C.3 shows the |S11| response of the dipole over 5x5 and 7x7 grids. The 90±45o

bandwidth is shaded grey in both plots. It is seen that for the 7x7 case, excellent matching

is seen for the dipole within the shaded region, however, when the size of the surface is

reduced, the optimum response is occurs outside the shaded region. Hence, the 90± 45o

reflection phase criteria applies only to surface sizes that are electrically large. Once the

size of the surface is reduced, the behaviour is akin to that of a resonator coupled to a

radiating source.

(a) Dipole over 5x5 Grid (b) Dipole over 7x7 Grid

Figure C.2: Simulation setup


3.5 4 4.5 5 5.5 6 6.5 7−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

23.5mm24.5mm25.5mm26.527.5mm28.5mm29mm29.5mm30mm

(a) Dipole |S11| response over a 5x5 Grid

3.5 4 4.5 5 5.5 6 6.5 7−30

−25

−20

−15

−10

−5

0

Frequency [GHz]

|S11

| [d

B]

23.5mm24.5mm25.5mm26.527.5mm28.5mm29mm29.5mm30mm

(b) Dipole |S11| response over a 7x7 Grid

Figure C.3: Dipole responses over 5x5 and 7x7 grids

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