ABSTRACT 
 
 
 
 
Title of Dissertation: ELECTROMAGNETIC INTERFERENCE 
REDUCTION USING ELECTROMAGNETIC 
BANDGAP STRUCTURES IN PACKAGES, 
ENCLOSURES, CAVITIES, AND ANTENNAS   
  
 Baharak Mohajer Iravani, Doctor of Philosophy, 2007 
Dissertation Directed By: Associate Professor  Omar M. Ramahi,  
Mechanical Engineering Department 
Electrical and Computer Engineering Department 
 
 
Electromagnetic interference (EMI) is a source of noise problems in electronic 
devices. The EMI is attributed to coupling between sources of radiation and 
components placed in the same media such as package or chassis. This coupling can 
be either through conducting currents or through radiation. The radiation of 
electromagnetic (EM) fields is supported by surface currents. Thus, minimizing these 
surface currents is considered a major and critical step to suppress EMI. 
In this work, we present novel strategies to confine surface currents in 
different applications including packages, enclosures, cavities, and antennas. The 
efficiency of present methods of EM noise suppression is limited due to different 
drawbacks. For example, the traditional use of lossy materials and absorbers suffers 
from considerable disadvantages including mechanical and thermal reliability leading 
to limited life time, cost, volume, and weight. In this work, we consider the use of 
Electromagnetic Band Gap (EBG) structures. These structures are suitable for 
 
 
suppressing surface currents within a frequency band denoted as the bandgap. Their 
design is straight forward, they are inexpensive to implement, and they do not suffer 
from the limitations of the previous methods. 
 A new method of EM noise suppression in enclosures and cavity-backed 
antennas using mushroom-type EBG structures is introduced. The effectiveness of the 
EBG as an EMI suppresser is demonstrated using numerical simulations and 
experimental measurements. 
To allow integration of EBGs in printed circuit boards and packages, novel 
miniaturized simple planar EBG structures based on use of high-k dielectric material 
(?
r 
> 100) are proposed. The design consists of meander lines and patches. The 
inductive meander lines serve to provide current continuity bridges between the 
capacitive patches. The high-k dielectric material increases the effective capacitive 
load substantially in comparison to commonly used material with much lower 
dielectric constant. Meander lines can increase the effective inductive load which 
pushes down the lower edge of bandgap, thus resulting in a wider bandgap. 
Simulation results are included to show that the proposed EBG structures provide 
very wide bandgap (~10GHz) covering the multiple harmonics of of currently 
available microprocessors and its harmonics. 
To speed up the design procedure, a model based on combination of lumped 
elements and transmission lines is proposed. The derived model predicts accurately 
the starting edge of bandgap. This result is verified with full-wave analysis.  
Finally, another novel compact wide band mushroom-type EBG structure 
using magneto-dielectric materials is designed. Numerical simulations show that the 
 
 
proposed EBG structure provides in-phase reflection bandgap which is several times 
greater than the one obtained from a conventional EBG operating at the same 
frequency while its cell size is smaller. This type of EBG structure can be used 
efficiently as a ground plane for low-profile wideband antennas. 
 
 
 
 
 
 
 
 
 
 
 
 
 
ELECTROMAGNETIC INTERFERENCE REDUCTION USING 
ELECTROMAGNETIC BANDGAP STRUCTURES IN PACKAGES, 
ENCLOSURES, CAVITIES, AND ANTENNAS   
 
 
 
By 
 
 
Baharak Mohajer Iravani 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2007 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Associate Professor Omar M. Ramahi, Co-chair/Supervisor 
Professor Victor Granatstein, Chair/Advisor 
Professor Issak D. Mayergoyz 
Associate Professor Bruce Jacob 
Professor Amr Baz 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Baharak Mohajer Iravani 
2007 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ii 
Dedication 
 
 
 
 
 
To my great family especially my dear mother Hilda and my dear father Parviz 
To my lovely and supportive relatives 
To the memories of my beloved grandparents and aunts 
 
 
 
 
 
 
 
 
 
 
 
 
iii 
Acknowledgements 
I am grateful to the Lord for his mercy and blessings. He is giving me the chance of 
progressing and understanding a little from infinite logics existent in this world. I 
would like to thank my family especially my parents, relatives, and friends for their 
sincere love, ultimate support, and continuous encouragement in every step of my 
life. I would like to offer my thanks to Prof. Omar Ramahi for supervising and 
guiding me during different stages of my Ph.D. research. He was aggressive to make 
me fruitful, modest to make me confident of my knowledge and logic, pushing to 
make me hit the top, consultant to choose the best way, advisor to suggest ideas in 
situations in which I had been stuck, and decent to give me chance to develop my 
own ideas. I would like to thank Prof. Victor Granatstein, Prof. Issak Mayergoyz, 
Prof. Bruce Jacob, and Prof. Amr Baz for serving on my defense committee. I am 
thankful to Dr. Granatstein for accepting to be my official advisor at the last stage of 
my doctorate study. I am grateful to Prof. Zaki for advising me one year, who 
provided me great opportunity to achieve theoretical and practical experience about 
microwave components. I would like to thank Dr. Mahmoud EL Sabbagh for his 
support and advice both during the early steps of my research at University of 
Maryland and the last steps at University of Waterloo, Canada. I am thankful to Dr. 
Mehdi Kalantari at University of Maryland for his helpful discussions and 
encouragements. I would like to thank Mr. Jay Pyle in Machine Shop at Institute for 
Research in Electronics and Applied Physics (IREAP), University of Maryland, for 
his advice and help in building different microwave components. I am thankful to Ms. 
Maria Hoo for her help at office of Graduate Studies, Department of Electrical and 
 
iv 
Computer Engineering, University of Maryland. I would like to thank my friends Dr. 
Shahrooz Shahparnia, Dr. Mohammad Haeri Kermani, Dr. Telesphor Kamgaing, Dr. 
Lin Li, and Dr. Xin Wu in Electromagnetic Compatibility and Propagation Lab at 
University of Maryland for their support, discussions, and cooperation. My thanks are 
due to my friends Dr. Yunchi Zhang and Mr. Mohamed Fahmi in Microwave Lab at 
University of Maryland, Dr. Jorge Alfonso Ruiz Cruz at Universidad Autonoma de 
Madrid/ Spain, and Mr. Seokjin Kim at University of Maryland for their help, 
support, discussions, and guide. Finally, I would like to thank my friends Mrs. Leila 
Yousefi, Mr. Samer Abdullah, Mr. Muhammad Boybay, and Mr. Ali Kabiri in 
Electromagnetic group at University of Waterloo for being supportive. 
 
 
 
 
 
 
 
 
 
 
 
 
 
v 
Table of Contents 
ABSTRACT.................................................................................................................. ii 
Dedication..................................................................................................................... ii 
Acknowledgements......................................................................................................iii 
Table of Contents.......................................................................................................... v 
List of Tables .............................................................................................................. vii 
List of Figures............................................................................................................viii 
Chapter 1 : Introduction................................................................................................ 1 
1.1. Background on Electromagnetic Noise ............................................................. 1 
1.2. Problems and Challenges................................................................................... 4 
Chapter 2 : Electromagnetic Bandgap Structures ......................................................... 9 
2.1. EBG Structures .................................................................................................. 9 
2.2. Characterization of EBG Structures................................................................. 11 
2.2.1. Scattering Parameters................................................................................ 12 
2.2.2. Dispersion Diagram .................................................................................. 12 
2.3. Numerical Study of EBG Structures................................................................ 18 
Chapter 3 : Miniaturized EBG Structures................................................................... 30 
3.1. Miniaturization of Embedded EBG Structures between a Single Pair of Planes 
using High-K Dielectrics ........................................................................................ 30 
3.1.1. Numerical Characterization of Embedded Mushroom-Type EBGs made of 
High-k Dielectrics............................................................................................... 31 
3.2. Miniaturization of Planar EBG Structures using High-K Dielectrics.............. 38 
3.2.1. Numerical Characterization of Miniaturized-Wideband Planar EBG 
Structures ............................................................................................................ 44 
3.3. Modeling of the Proposed Planar EBG Structures .......................................... 49 
3.3.1. Setup and Model Formulation .................................................................. 52 
3.3.2. Characterization of EBG using Full-Wave Simulator .............................. 55 
3.3.3. Model of Patch.......................................................................................... 62 
3.3.4. Model of Patch-Meander Line Step Discontinuity ................................... 64 
3.3.5. Model of Meander Line ............................................................................ 67 
3.3.5.A. Computation of Inductance............................................................... 69 
3.3.5.B. Computation of Parasitic Capacitances............................................. 76 
3.3.6. Model of Coupling Effect between Patch and Meander line.................... 83 
3.3.7. Model of Electrical Length in Planar EBG Structure ............................... 91 
3.3.8. The Model of a Unit Cell of the Proposed Planar EBG Structure............ 92 
3.3.9. Case Studies and Performance of the Developed Model........................ 101 
3.3.10. Characterization of the Proposed Planar EBG Structure ...................... 107 
3.3.11. Effects of Variation of Elements in Circuit-Transmission Line Model on 
Bandgaps........................................................................................................... 121 
Chapter 4 : Switching Noise and Electromagnetic Noise in High Speed Packages . 140 
4.1. Analytical Model of EM Noise in Packages.................................................. 141 
4.2. SSN and EMI Noise Mitigation in Packages using High-K Embedded 
Capacitance........................................................................................................... 147 
4.2.1. Analytical Relationship between Package Topology and Noise 
Suppression....................................................................................................... 147 
 
vi 
4.2.2. Case Studies............................................................................................ 161 
Chapter 5 : Application of EBG Structures .............................................................. 168 
5.1. EMI Reduction............................................................................................... 168 
5.1.1. Reduction of EMI in Enclosures and Chassis using the EBG Structures168 
5.1.1.A. Modeling ......................................................................................... 169 
5.1.1.B. Numerical Case Study..................................................................... 170 
5.1.2. Reduction of EMI Coupling between Cavities and Antennas using EBG 
Structures .......................................................................................................... 174 
5.1.2.A. Modeling ......................................................................................... 174 
5.1.2.B. Numerical Case Study..................................................................... 175 
5.1.2.C. CBS Antenna System with Presence of EBG Structures ................ 186 
5.1.2.D. Experimental Study......................................................................... 201 
5.1.3. EMI and SSN Noise Mitigation in IC Packages using Miniaturized Planar 
EBGs................................................................................................................. 209 
5.1.3.A. Modeling ......................................................................................... 209 
5.1.3.B. Numerical Case Studies .................................................................. 209 
5.2. Design of Low Profile Antenna ..................................................................... 218 
5.2.1. Limitations in Design of Low Profile wideband Antennas .................... 219 
5.2.1.A. Concept and Modeling of Mushroom-Type EBG Structures Using 
Magneto-Dielectric Substrates...................................................................... 220 
5.2.1.B. Numerical Study of EBG Structures Made of Magneto-Dielectric 
Substrates ...................................................................................................... 222 
Chapter 6 : Conclusions and Future Work................................................................ 228 
6.1. Conclusions.................................................................................................... 228 
6.2. Future Work................................................................................................... 231 
Appendix................................................................................................................... 232 
Bibliography ............................................................................................................. 235 
 
 
 
 
 
 
 
 
 
vii 
List of Tables 
Table  3.I. Specifications of the different square patch mushroom-type EBG structures 
embedded in parallel planes and related suppression bands (design parameters 
are shown in Figure  3.1). The thickness of parallel plane is equal to 116 ?m and 
?
r1
 = 4.4 for all samples....................................................................................... 38 
Table  3.II Specifications of the different planar EBG patterns and related suppression 
band (design parameters are shown in Figure  3.12). For all samples h = 114.3 
?m, ?
r 
= 140, and tan ? = 0.0015. ....................................................................... 48 
Table  3.III. Specifications of the different planar EBG patterns and related 
suppression band (design parameters are shown in Figure  3.12). For all samples 
h = 114.3 ?m and N = 2. ..................................................................................... 49 
Table  5.I. Bandgap Obtained From Numerical Simulation and Experimental 
Measurements ................................................................................................... 208 
Table  5.II. In-phase reflection band obtained from numerical simulation for different 
EBG structures. The EBG structures have common specification of a = 0.5 mm, 
g = 0.15 mm, h = 1.57 mm, d = 0.1 mm, and ?
r 
= 2.51. The value of relative 
permeability, ?
r
, is varying from 1 to 9. The magnetic loss tangent for material 
with ?
r
 > 1 is equal to 0.01 (tan ?
m 
= 0.01). ...................................................... 227 
 
 
 
 
 
 
 
 
 
 
 
viii 
List of Figures 
Figure  2.1. Square patch mushroom-type EBG structure (a) Perspective view. (b) Top 
view. Relevant design parameters are shown on the diagrams........................... 10 
Figure  2.2. A schematic showing the simulation setup used to extract the bandgap of 
an EBG structure using scattering parameters. In this setup S
21
 represents the 
power delivered at port 2 when port 1 is excited. ............................................... 12 
Figure  2.3. Schematic showing the simulation setup used to extract the bandgap of an 
EBG structure using dispersion diagrams. (a) Computational domain containing 
single cell. Periodic boundary conditions are placed at x-z and y-z planes and an 
absorbing condition (PML) is placed on the top face x-y plane. (b) The 
irreducible Brillouin zone triangle. ..................................................................... 15 
Figure  2.4. Dispersion diagram for the EBG structure having 4 mm square patches on 
a substrate with ?
r
 = 2.2....................................................................................... 16 
Figure  2.5. Dispersion diagram for the EBG structure having 4 mm square patches on 
a substrate with ?
r
 = 4.8....................................................................................... 17 
Figure  2.6 Dispersion diagram for the EBG structure having 2.6 mm square patches 
on a substrate with ?
r
 = 3..................................................................................... 17 
Figure  2.7. Numerical setup used to extract the bandgap of EBG structures. ............ 20 
Figure  2.8. Magnitude of S
21
 for the setup shown in Figure  2.7 with and without and 
EBG surface with the following parameters a = 2.6 mm, h = 1.54 mm, d = 0.8 
mm, g = 0.4 mm, and ?
r 
= 2. The design parameters are shown in Figure  2.1. .. 20 
Figure  2.9. Frequency bandgap versus the patch size. (a) 4 rows of EBG patches, (b) 
8 rows of EBG patches. EBG parameters: ?
r 
= 3, h = 1.54 mm, d = 0.8 mm, and g 
= 0.4 mm. ............................................................................................................ 21 
Figure  2.10. Frequency bandgap versus the gap between patches. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, ?
r 
= 4.4, h = 
1.54 mm, and d = 0.8 mm. .................................................................................. 22 
Figure  2.11. Frequency bandgap versus the gap between patches. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, ?
r 
= 3, h = 
1.54 mm, and d = 0.8 mm. .................................................................................. 23 
Figure  2.12. Frequency bandgap versus the height of via. (a) 4 rows of EBG patches, 
(b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g = 0.4 mm, ?
r 
= 4.4, 
and d = 0.8 mm. .................................................................................................. 24 
Figure  2.13. Frequency bandgap versus the height of via. (a) 4 rows of EBG patches, 
(b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, g = 0.4 mm, ?
r 
= 3, 
and d = 0.8 mm. .................................................................................................. 25 
 
ix 
Figure  2.14. Frequency bandgap versus the diameter of via. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g = 0.4 mm, ?
r 
= 4.4, and h = 1.54 mm. ...................................................................................... 26 
Figure  2.15. Frequency bandgap versus the diameter of via. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, g = 0.4 mm, 
?
r 
= 3, and h = 1.54 mm....................................................................................... 27 
Figure  2.16. Frequency bandgap versus the relative permittivity of substrate. (a) 4 
rows of EBG patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g 
= 0.4 mm, d = 0.8 mm, and h = 1.54 mm. .......................................................... 28 
Figure  2.17. Frequency bandgap versus the relative permittivity of substrate (a) 4 
rows of EBG patches, (b) 8 rows of EBG patches. EBG parameters: are a = 2.6 
mm, g = 0.4 mm, d = 0.8 mm, and h = 1.54 mm. ............................................... 29 
Figure  3.1. Side view of embedded mushroom-type EBG between parallel planes. 
The design parameters are included in the figure. .............................................. 31 
Figure  3.2. Physical-based model for a solid patch mushroom-type EBG embedded in 
between parallel planes. C
1
 models the capacitance between the patch and the 
top plane, C
2
 is the capacitance between the patch and the bottom plane, and L 
represents the inductance of via.......................................................................... 32 
Figure  3.3. Simulation setup used to extract the suppression band of the EBG 
structure from computed S
21
. The computational domain is ideal parallel plate 
waveguide with PMC boundary on the sides...................................................... 33 
Figure  3.4. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different patch sizes. The EBG structures have the following parameters: ?
r1
 
= 4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, g = 200 ?m, and v = 200 ?m (design 
parameters are shown in Figure  3.1)................................................................... 34 
Figure  3.5. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different gap sizes. The EBG structures have the following parameters: ?
r1
 = 
4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 1 mm, and v = 200 ?m (design 
parameters are shown in Figure  3.1)................................................................... 34 
Figure  3.6. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different via diameters. The EBG structures have the following parameters: 
?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 1 mm, and g = 40 ?m (design 
parameters are shown in Figure  3.1)................................................................... 35 
Figure  3.7. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
different values of dielectric relative permittivity for media 1. The EBG 
structures have the following parameters: h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 
0.4 mm, g = 40 ?m, and v = 40 ?m (design parameters are shown in Figure  3.1).
............................................................................................................................. 35 
 
x 
Figure  3.8. Magnitude of S
21
 for different embedded EBG structures in parallel 
planes. The samples have the following parameters: (a) ?
r1
 = 4.4, h
1
 = 100 ?m, 
?
r2
 = 50, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. (b) ?
r1
 = 4.4, h
1
 = 
100 ?m, ?
r2
 = 70, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. (c) ?
r1
 = 
4.4, h
1
 = 108 ?m, ?
r2
 = 70, h
2
 = 8 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m 
(design parameters are shown in Figure  3.1). ..................................................... 37 
Figure  3.9. Planar inductors. (a) Archimedean spiral inductor. (b) Meander line 
inductor. .............................................................................................................. 40 
Figure  3.10. Top views of a unit cell of different planar EBG structures. The meander 
line as an inductive bridge connecting the patches in EBG structures consists of 
(a) one turn, (b) two turns, and (c) three turns. ................................................... 40 
Figure  3.11. Top view of a unit cell of a planar EBG structure. The meander line as an 
inductive bridge providing current continuity between the patches in the EBG 
pattern is connected to two adjacent patches with a margin............................... 41 
Figure  3.12. Planar EBG structure with two-turn meander lines as inductive bridges. 
(a) top view of the unit cell, (b) top view of the meander line bridge, (c) 
perspective view of the total structure, (d) top view of the total structure. The 
design parameters are shown in the diagram. ..................................................... 42 
Figure  3.13. Physical-based model of the planar EBG structure. C
1
 represents the 
mutual coupling capacitance between arms of the meander line. C
2
 is the mutual 
coupling capacitance between meander line and two adjacent patches. C is the 
capacitance between the patch and the bottom plane, and L represents the 
inductance of meander line. ................................................................................ 44 
Figure  3.14. Schematic showing the simulation setup and location of the ports to 
extract the suppression band of the EBG structure from computed S
21.
............. 46 
Figure  3.15. Magnitude of S
21
 versus frequency for the EBG structure shown in 
Figure  3.12 using the setup shown in Figure  3.14. The parameters of the EBG 
structure are: P
W
 = 1.4 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 
1.34 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015 (design parameters are 
shown in Figure  3.12). The reference case which is parallel plane is included for 
the purpose of comparison. ................................................................................. 47 
Figure  3.16. A unit cell of the planar EBG structure with two-turn meander bridges. 
(a) Setup used for extracting the bandgap of EBG structure in 2-D propagation 
using dispersion diagram. The boundary setup is shown in the figure. (b) Top 
view of EBG structure. ....................................................................................... 54 
Figure  3.17. Infinite 1-D array of the proposed planar EBG...................................... 54 
Figure  3.18. A unit cell of the planar EBG structure with two-turn meander bridge. (a) 
Setup used for extracting the bandgap of EBG structure in 1-D propagation using 
dispersion diagram. The boundary setup is shown in figure. (b) Top view of 
 
xi 
EBG structure. If PMC boundaries are replaced by a pair of PBC boundaries it 
resembles 2-D propagation which is different from Figure  3.16........................ 55 
Figure  3.19. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 
140, and tan ? = 0.0015. ..................................................................................... 58 
Figure  3.20. Dispersion diagram of the planar EBG structure shown in Figure  3.16 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 
140, and tan ? = 0.0015. ..................................................................................... 59 
Figure  3.21. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
500 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, 
and tan ? = 0.0015. ............................................................................................. 60 
Figure  3.22. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 
200, and tan ? = 0.0015. ..................................................................................... 61 
Figure  3.23. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 20 mm, h = 
114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 mm, M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and 
tan ? = 0.02. ........................................................................................................ 62 
Figure  3.24. a) Patch of the planar EBG pattern which has short electrical length and 
low impedance. b) Equivalent lumped-element circuit. ..................................... 63 
Figure  3.25. (a) Asymmetrical discontinuity between patch and meander line with 
margin = 0 (refer to Figure  3.11). (b) Equivalent lumped- element circuit........ 66 
Figure  3.26. (a) Symmetrical discontinuity between microstrip lines. (b) Equivalent 
lumped- element circuit. ..................................................................................... 66 
Figure  3.27. (a) Asymmetrical discontinuity between microstrip lines with margin ? 0 
(refer to Figure  3.11). (b) and (c) Symmetrical discontinuity between microstrip 
lines. Design parameters are shown in the figure. (C
D
, L
DP
, L
DM
), (C
D1
, L
DP1
, 
L
DM1
), and (C
D2
, L
DP2
, L
DM2
) show the elements modeling the step discontinuity 
at the connections (a), (b), and (c), respectively. The equivalent circuit model is 
shown in Figure  3.25 (b)..................................................................................... 67 
Figure  3.28. Three-dimensional view of meander line inductor. Design parameters are 
included in the figure. ......................................................................................... 68 
Figure  3.29. Equivalent circuit model of meander line. ............................................. 69 
 
xii 
Figure  3.30. Magnetic flux lines showing the magnetic coupling between two 
adjacent lines for two cases: (a) positive mutual inductance and (b) negative 
mutual inductance. .............................................................................................. 71 
Figure  3.31. Magnetically coupled traces in a meander line inductor. The mutual 
coupling between pairs of arms and bridges is marked with solid lines and 
dashed lines, respectively.................................................................................... 71 
Figure  3.32. Magnetically coupled strips. Design parameters for computing mutual 
coupling are shown in the figure......................................................................... 74 
Figure  3.33. Image of meander line by ground plane................................................. 75 
Figure  3.34. Parasitic capacitances in the meander pattern. ....................................... 77 
Figure  3.35. Parasitic capacitances between the strip and the ground plane in a 
microstrip line. .................................................................................................... 78 
Figure  3.36. Constituents of (a) the even mode capacitance and (b) the odd mode 
capacitance in a pair of symmetric coupled microstrip lines.............................. 80 
Figure  3.37. Parasitic capacitances for a pair of symmetric coupled strip lines......... 81 
Figure  3.38. Parasitic capacitances for the three symmetric coupled microstrip lines, 
where the coupling effect between the non-adjacent arms is neglected. ............ 82 
Figure  3.39. (a) Side view of patch and meander-arm microstrips as a pair of 
asymmetric coupled microstrip lines. (b) Side view of a patch microstrip 
between four meander-arm mirostrips. The capacitances modeling the coupled 
lines are shown in the figure. (c) Capacitive equivalent circuit model of the 
asymmetric coupled patch-arm microstrips. ....................................................... 87 
Figure  3.40. (a) A meander line between two patches. The bridge connecting the first 
arm to patch is marked in the figure. (b) Equivalent circuit modeling: 1) the 
mutual coupling effect between the patch and the adjacent meander-arm and 2) 
the connecting bridge-trace. (c) Equivalent circuit modeling the total meander 
line (refer to Figure  3.29). In this model, the effect of mutual coupling between 
the patch and the meander is considered as interline parasitic capacitance........ 89 
Figure  3.41. Different types of fringing capacitances per unit length for the coupled 
and uncoupled traces in the meander line are marked with different colors and 
symbols. The 3-turn meander has the following parameters: M
Li
 = M
Lo
 = M
L
, 
(M
Bi
 = M
Bo
 = M
B
) ? M
G
, and P
W
 ? M
W
 (The design parameters are shown in 
Figure  3.25 and Figure  3.28)............................................................................... 91 
Figure  3.42. Simple physical-based circuit model for the proposed planar EBG 
structure............................................................................................................... 93 
Figure  3.43. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the simple physical-based circuit model shown in Figure  3.42. The 
specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 
 
xiii 
mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015.
............................................................................................................................. 94 
Figure  3.44. Advanced physical-based circuit model for the proposed planar EBG 
structure............................................................................................................... 95 
Figure  3.45. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the advanced physical-based circuit model shown in Figure  3.44. 
The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 
0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 
0.0015.................................................................................................................. 96 
Figure  3.46. Simple equivalent circuit-transmission line model for the proposed 
planar EBG structure........................................................................................... 98 
Figure  3.47. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the simple equivalent circuit-transmission line model shown in 
Figure  3.46. The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and 
tan ? = 0.0015. .................................................................................................... 98 
Figure  3.48. Advanced equivalent circuit-transmission line model for the planar 
proposed EBG structure...................................................................................... 99 
Figure  3.49. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the advanced equivalent circuit-transmission line model shown in 
Figure  3.48. The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and 
tan ? = 0.0015. .................................................................................................. 101 
Figure  3.50. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line 
model and (b) numerical full-wave analysis by Ansoft HFSS. The specifications 
of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. .................... 102 
Figure  3.51. The attenuation diagram in ?-? zone for the planar EBG structure shown 
in Figure  3.18 is extracted using advanced equivalent circuit-transmission line 
model for (a) a unit cell of EBG and (b) 4 unit cells of EBG. The 20 dB 
attenuation level is marked in the graph. The specifications of EBG are as 
follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ........................................... 103 
Figure  3.52. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line 
model and (b) numerical full-wave analysis by Ansoft HFSS. The specifications 
of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 200, and tan ? = 0.0015. .................... 104 
 
xiv 
Figure  3.53. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line 
model and (b) numerical full-wave analysis by Ansoft HFSS. The specifications 
of EBG are as follows: P
W
 = 1.6 mm, h = 500 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. .................... 106 
Figure  3.54. Dispersion diagram in ?-? zone for the planar EBG structure shown  in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line 
model and (b) numerical full-wave analysis by Ansoft HFSS. The specifications 
of EBG are as follows: P
W
 = 20 mm, h = 114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 mm, 
M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and tan ? = 0.02. .................................... 106 
Figure  3.55. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the 
dispersion relation are extracted numerically by Ansoft HFSS. The polynomial 
fit of these data points of the propagating modes show dispersion graphs. The 
specifications of the EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 
mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015.
........................................................................................................................... 108 
Figure  3.56. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the 
dispersion relation are extracted numerically by Ansoft HFSS. The polynomial 
fit of these data points of the propagating modes show dispersion graphs. The 
specifications of the EBG are defined in Figure  3.55....................................... 109 
Figure  3.57. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the 
dispersion relation are extracted numerically by Ansoft HFSS. The polynomial 
fit of these data points of the propagating modes show dispersion graphs. The 
specifications of the EBG are defined in Figure  3.55....................................... 110 
Figure  3.58. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.55. 
The Vg graphs are obtained from the polynomials fit of the propagating modes.
........................................................................................................................... 114 
Figure  3.59. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.56. 
The Vg graphs are obtained from the polynomials fit of the propagating modes.
........................................................................................................................... 115 
Figure  3.60. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.57. 
The Vg graphs are obtained from the polynomials fit of the propagating modes.
........................................................................................................................... 116 
Figure  3.61. Radiation loss (1- |S
11
|
2
 - |S
21
|
2
) versus frequency for two cases: 1) the 
EBG patterned parallel planes and 2) solid parallel planes as reference. The S-
 
xv 
parameters are extracted using the setup shown in Figure  3.14. The planar EBG 
structure and the design parameters are shown in Figure  3.12. The parameters of 
the EBG structure are: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. .................... 119 
Figure  3.62. Radiation loss (1- |S
11
|
2
 - |S
21
|
2
) versus frequency for two cases: 1) the 
EBG patterned parallel planes and 2) solid parallel planes as reference. The S-
parameters are extracted using the setup shown in Figure  3.14. The planar EBG 
structure and the design parameters are shown in Figure  3.12. The parameters of 
the EBG structure are: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 200, and tan ? = 0.0015. .................... 120 
Figure  3.63. Relationship between the dispersion/attenuation diagram and the 
inductance L modeling the meander line in the planar EBG structure shown in 
Figure  3.18. The value of L is equal to (a) 0.2921 nH, (b) 0.5921 nH, and (c) 
0.8921 nH. Figure  3.48 demonstrates the equivalent circuit-transmission line 
model of EBG. The nominal values of elements in the model are extracted for 
the EBG with the following specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 
0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 
0.0015................................................................................................................ 122 
Figure  3.64. Relationship between the dispersion/attenuation diagram and the 
capacitance C
L2 
modeling the meander line in the planar EBG structure shown in 
Figure  3.18. The value of C
L2
 is equal to (a) 1.2926 pF, (b) 1.7926 pF, and (c) 
2.2926 pF. Figure  3.48 demonstrates the equivalent circuit-transmission line 
model of EBG. The nominal values of elements in the model are extracted for 
the EBG with following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 
0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 
0.0015................................................................................................................ 124 
Figure  3.65. Relationship between the dispersion/attenuation diagram and the (C
M_PM
 
|| C
L1
 || C
M_PM
) modeling the interline coupling between the meander line and 
two adjacent patches and between the arms of meander line in the planar EBG 
structure shown in Figure  3.18. The value of the equivalent capacitance is equal 
to (a) 0.19337 pF, (b) 0.39337 pF, and (c) 0.59337 pF. Figure  3.48 demonstrates 
the equivalent circuit-transmission line model of EBG. The nominal values of 
elements in the model are extracted for the EBG with following the 
specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 
1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ..................................... 125 
Figure  3.66. Relationship between the dispersion/attenuation diagram and the 
inductance L
PH
 modeling the patch in the planar EBG structure shown in Figure 
 3.18. The value of L
PH
 is equal to (a) 0.0287 nH, (b) 0.0587 nH, and (c) 0.0887 
nH. Figure  3.48 demonstrates the equivalent circuit-transmission line model of 
EBG. The nominal values of elements in the model are extracted for the EBG 
with following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ... 126 
 
xvi 
Figure  3.67.  Relationship between the dispersion/attenuation diagram and the 
capacitance C
PH
 modeling the patch in the planar EBG structure shown in Figure 
 3.18. The value of C
PH
 is equal to (a) 17.903 pF, (b) 27.903 pF, and (c) 37.903 
pF. Figure  3.48 demonstrates the equivalent circuit-transmission line model of 
EBG. The nominal values of elements in the model are extracted for the EBG 
with following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ... 128 
Figure  3.68. Relationship between the dispersion/attenuation diagram and the length 
of patch-microstrip d
1
 modeling the phase difference along the patch in the 
planar EBG structure shown in Figure  3.18. The value of d
1
 is equal to (a) 0.75 ? 
P
W
, (b) P
W
, and (c) 1.25 ? P
W
. P
W
 is the width of patch-microstrip, the spacing 
between two consecutive patches d
2
 = 2 ? M
B
 + N ? M
W 
+ (N-1) ? M
G
, and the 
length of a unit cell of EBG d = d
1
 + d
2
. Figure  3.48 demonstrates the equivalent 
circuit-transmission line model of EBG. The nominal values of elements in the 
model are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, 
h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 
140, and tan ? = 0.0015. ................................................................................... 130 
Figure  3.69. Relationship between the dispersion/attenuation diagram and the length 
of meander-microstrip d
2
 modeling the phase difference across the meander line 
in the planar EBG structure shown in Figure  3.18. The value of d
2
 showing the 
spacing between the two consecutive patches is equal to (a) 0.75 ? D
2
, (b) D
2
, 
and (c) 1.25 ? D
2
 where D
2
 = 2 ? M
B
 + N ? M
W 
+ (N-1) ? M
G
. M
W
 is the width of 
meander-microstrip, the length of patch d
1
 = P
W
, and the length of a unit cell of 
EBG d = d
1
 + d
2
. Figure  3.48 demonstrates the equivalent circuit-transmission 
line model of EBG. The nominal values of elements in the model are extracted 
for the EBG with following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 
0.0015................................................................................................................ 131 
Figure  3.70 Relationship between the dispersion/attenuation diagram and the 
inductance L
DP
 modeling the step discontinuity between the patch and the 
meander line in the planar EBG structure shown in Figure  3.18. The value of L
DP
 
is equal to (a) 0.9511 pH, (b) 5.9511 pH, and (c) 10.9511 pH. Figure  3.48 
demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 
0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ............ 132 
Figure  3.71. Relationship between the dispersion/attenuation diagram and the 
inductance L
DM
 modeling the step discontinuity between the patch and the 
meander line in the planar EBG structure shown in Figure  3.18. The value of 
L
DM
 is equal to (a) 43.576 pH, (b) 93.576 pH, and (c) 143.576 pH. Figure  3.48 
demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with 
 
xvii 
following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 
0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ............ 133 
Figure  3.72. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the 
meander line in the planar EBG structure shown in Figure  3.18. The value of C
D
 
is equal to (a) 0.6468 pF, (b) 1.6468 pF, and (c) 2.6468 pF. Figure  3.48 
demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 
0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ............ 135 
Figure  3.73. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the 
meander line in the planar EBG structure shown in Figure  3.18. The value of C
D
 
is equal to (a) 0.7066 pF, (b) 1.7066 pF, and (c) 2.7066 pF. Figure  3.48 
demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 1.6 mm, h = 500 ?m, M
W 
= 0.02 mm, M
B 
= 
0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. ............ 138 
Figure  3.74. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the 
meander line in the planar EBG structure shown in Figure  3.18. The value of C
D
 
is equal to (a) 0.22452 pF, (b) 0.72452 pF, and (c) 1.22452 pF. Figure  3.48 
demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 20 mm, h = 114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 
mm, M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and tan ? = 0.02. ............................ 139 
Figure  4.1. Single plane pair in the package consisting of metal-dielectric-metal 
layers is modeled as parallel plate waveguide. (a) 3-D view.  (b) Top view. 
Typical dimensions of a microprocessor package are: a = b = 4 cm. Each via 
shown in the figure represents a port. Port 1 and Port 2 are located at (x
1
 = 1 cm, 
y
1
 = 1 cm), and (x
2
 = 2 cm, y
2
 = 2 cm), respectively......................................... 144 
Figure  4.2. Magnitude of (a) trans-impedance between ports 1 and 2, and (b) input 
impedance at port 2 for the package shown in Figure  4.1. The plane separation is 
h = 200 ?m. The Impedances are obtained through three different methods 
including: analytical method using Equation  4-1 plotted by the solid line, 
analytical method using Equation  4-2 plotted by the circle symbol line, and full-
wave simulation using HFSS plotted by the dashed line.................................. 146 
Figure  4.3. Normalized skin depth versus frequency for a single plane pair in the 
package shown in Figure  4.1. The dimensions of the layer are: a = b = 4 cm, h = 
114.3 ?m, and S/m. 108.5
7
?=? ..................................................................... 149 
Figure  4.4. (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 200 ?m 
 
xviii 
and dielectric material is FR-4. The Impedance from Equation  4-2 is plotted by 
the solid line. The impedance value at the resonant modes of a plane pair is 
marked by square symbol. The approximate value of impedance from Equation 
 4-11, Equation  4-13, and Equation  4-14 is marked by star symbol.................. 155 
Figure  4.5.  (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation 
is h = 200 ?m and dielectric material is FR-4. The Impedance from Equation  4-2 
is plotted by the solid line. The impedance value at the resonant modes of a 
plane pair is marked by square symbol. The approximate value of impedance 
from Equation  4-11, Equation  4-13, and Equation  4-14 is marked by star 
symbol............................................................................................................... 156 
Figure  4.6.  (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 200 ?m 
and dielectric material has permittivity of 140 and loss tangent of 0.0015. The 
Impedance from Equation  4-2 is plotted by the solid line. The impedance value 
at the resonant modes of a plane pair is marked by square symbol. The 
approximate value of impedance from Equation  4-11, Equation  4-13, and 
Equation  4-14 is marked by star symbol........................................................... 157 
Figure  4.7. (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation 
is h = 200 ?m and dielectric material has permittivity of 140 and loss tangent of 
0.0015. The Impedance from Equation  4-2 is plotted by the solid line. The 
impedance value at the resonant modes of a plane pair is marked by square 
symbol. The approximate value of impedance from Equation  4-11, Equation 
 4-13, and Equation  4-14 is marked by star symbol. ......................................... 158 
Figure  4.8. (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 100 ?m 
and dielectric material has permittivity of 140 and loss tangent of 0.0015. The 
Impedance from Equation  4-2 is plotted by the solid line. The impedance value 
at the resonant modes of a plane pair is marked by square symbol. The 
approximate value of impedance from Equation  4-11, Equation  4-13, and 
Equation  4-14 is marked by star symbol........................................................... 159 
Figure  4.9. (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation 
is h = 100 ?m and dielectric material has permittivity of 140 and loss tangent of 
0.0015. The Impedance from Equation  4-2 is plotted by the solid line. The 
impedance value at the resonant modes of a plane pair is marked by square 
symbol. The approximate value of impedance from Equation  4-11, Equation 
 4-13, and Equation  4-14 is marked by star symbol. ......................................... 160 
Figure  4.10. Magnitude of (a) input impedance at port 2, and (b) trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. Impedances are 
provided for two different configurations. The impedance for plane separation h 
= 200 ?m is plotted by the dashed line and the impedance for h = 100 ?m is 
 
xix 
plotted by the solid line. The first three resonances are marked with different 
symbols in the plot and table. The similar symbols in each graph show the 
corresponding resonances. ................................................................................ 164 
Figure  4.11. Magnitude of trans-impedance between ports 1 and 2 for the package 
under study shown in Figure  4.1. The impedance is illustrated for three different 
configurations: (a) (?
r 
= 4.3, tan ? = 0.02),? (b) (?
r 
= 30, tan ? = 0.02), and (c) (?
r 
= 
30, tan ? = 0.0015). In all configurations, the plane separation is h = 150 ?m. 
The first three resonances are marked with different symbols in the plot and 
table. The similar symbols on the curves show the corresponding resonances.165 
Figure  4.12. Magnitude of input impedance at port 2 for the package under study 
shown in Figure  4.1. The impedance is illustrated for three different 
configurations: (a) (?
r 
= 4.3, tan ? = 0.02), (b) (?
r 
= 30, tan ? = 0.02), and (c) (?
r 
= 
30, tan ? = 0.0015). In all configurations, the plane separation is h = 150 ?m. 
The first four resonances are marked with different symbols in the plot and table. 
The similar symbols on the curves show the corresponding resonances.......... 166 
Figure  4.13. Magnitude of (a) input impedance at port 2, and (b) trans-impedance 
between ports 1 and 2 for the package under study shown in Figure  4.1. The 
impedance is illustrated for three different cases of (?
r 
= 4.3, tan ? = 0.02, h = 
200 ?m), (?
r 
= 100, tan ? = 0.0015, h = 100 ?m), and (?
r 
= 140, tan ? = 0.0015, h 
= 50 ?m)............................................................................................................ 167 
Figure  5.1. Enclosure cavity with two EBG ribbons. ............................................... 169 
Figure  5.2. Schematic of two back-to-back perfectly conducting boxes. One of them 
contains the excitation probe and the second contains the receiver probe. Both of 
them have a small aperture in the middle of their external y-z plane with 
dimensions a= 20 mm, b= 4 mm. The EBG surface is placed as a ribbon 
completely surrounding the structure. (a) Front view. (b) Side view. .............. 172 
Figure  5.3. Magnitude of S
21
 between two probes positioned as shows in Figure  5.2 
(b). The EBG structures consists of 4 ? 4 mm patches with 0.4 mm gap and ?
r
 = 
2.2...................................................................................................................... 173 
Figure  5.4. Magnitude of S
21
 between two probes positioned as shows in Figure  5.2 
(b). The EBG structures consists of 4 ? 4 mm patches with 0.4 mm gap and ?
r
 = 
4.8...................................................................................................................... 173 
Figure  5.5. Air-filled rectangular cavity-backed antenna fed with a probe oriented in 
the y-direction. (a) 3-D view, (b) top view and (c) side view (x-y plane)......... 175 
Figure  5.6. Three dimensional schematic diagram showing two identical cavity-
backed antennas mounted on a rectangular ground plane.  The structure of each 
antenna is as described in Figure  5.5. ............................................................... 176 
Figure  5.7. (a) Side view and (b) top view of a CBS antennas system showing the 
placement of the EBG patches.......................................................................... 177 
 
xx 
Figure  5.8. Magnitude of S
11
 of two identical CBS antennas mounted on a rectangular 
ground plane (without the EBG structures). This antenna resonates at both 7.5 
and 12.6 GHz as marked on the plot................................................................. 179 
Figure  5.9. Coupling between the two antennas is represented by the magnitude of the 
S
21
 parameter. Two different EBG designs for two operating frequencies are 
implemented and simulated. Pattern 1 corresponds to a patch size of 4 mm and ?
r
 
of 4.8. Pattern 2 corresponds to a patch size of 2.6 mm and ?
r
 of 3. ................ 180 
Figure  5.10. Top view of the magnitude of the total surface current density J
tot
 (a) 
without the EBG structure when the antenna is operating at 7.5 GHz, (b) with the 
EBG Pattern 1 when the antenna is operating at 7.5 GHz, (c) without the EBG 
structure when the CBS antenna is operating at 12.6 GHz, and (d) with the EBG 
Pattern 2 when the antenna is operating at 12.6 GHz. Due to symmetry, only half 
of the structure is shown. .................................................................................. 182 
Figure  5.11. E-Field distribution in the y-z plane (intersecting the two antennas) when 
one of the antennas is excited at 7.5 GHz while the second one is receiving. (a) 
Without and (b) with the EBG Pattern 1........................................................... 183 
Figure  5.12. E-Field distribution in the y-z plane (intersecting the two antennas) when 
one of the antennas is excited at 12.6 GHz while the second one is receiving. (a) 
Without and (b) with the EBG Pattern 2........................................................... 184 
Figure  5.13. Radiated field patterns at 7.5 GHz with and without EBG Pattern 1. The 
rgain pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth.
........................................................................................................................... 185 
Figure  5.14. Radiated field patterns at 12.6 GHz with and without EBG Pattern 2. The 
gain pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth. 186 
Figure  5.15. Top view of the CBS antennas system showing the placement of the 
EBG structures all around the apertures and between the antennas on a common 
plane.................................................................................................................. 188 
Figure  5.16. Coupling between the two antennas operating at 12.6 GHz for three 
different setups is represented by the magnitude of S
21
. 1) Without EBG 
structures, 2) with EBG Pattern 2 only between antennas, and 3) with EBG 
Pattern 2 around and in between the apertures. Separation between antennas is 
40 mm. .............................................................................................................. 188 
Figure  5.17. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system: 1) without EBG structures, 2) with EBG Pattern 2 only between 
antennas, and 3) with EBG Pattern 2 around and in between the apertures. The 
pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth. 
Separation between antennas is 40 mm. ........................................................... 189 
Figure  5.18. Coupling between the two antennas operating at 12.6 GHz for 2 different 
setups is represented by the magnitude of S
21
. 1) Without EBG structures and 2) 
with EBG Pattern 2 around and in between the apertures of antennas. Separation 
between antennas is 13.84 mm. ........................................................................ 190 
 
xxi 
Figure  5.19. Radiated field patterns at 12.6 GHz for 2 different setups for the antenna 
system: 1) without EBG structures and 2) with EBG Pattern 2 around and in 
between the apertures of antennas. The pattern is in the y-z plane (E-plane) 
where 0 corresponds to the azimuth. Separation between antennas is 13.84 mm.
........................................................................................................................... 191 
Figure  5.20. Top view of the CBS antennas system showing the placement of a slab 
of the EBG structures between the antennas on a common plane. The slab is 
partially covering the separation between antennas.......................................... 193 
Figure  5.21. Coupling between the two antennas operating at 12.6 GHz for three 
different setups is represented by the magnitude of S
21
. 1) Without EBG 
structures. 2) With slab of EBG Pattern 2 between antennas. The width of slab is 
40 mm. 3) With slab of EBG Pattern 3 between antennas. The width of slab is 
22.4 mm. Separation between antennas is 40 mm. ........................................... 194 
Figure  5.22. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system. 1) Without EBG structures. 2) With slab of EBG Pattern 2 
between antennas. The width of slab is 40 mm. 3) With slab of EBG Pattern 3 
between antennas. The width of slab is 22.4 mm. Separation between antennas is 
40 mm. The pattern is in the y-z plane (E-plane) where 0 corresponds to the 
azimuth.............................................................................................................. 195 
Figure  5.23. Coupling between two antennas operating at 12.6 GHz for three different 
setups is represented by the magnitude of the S
21
. 1) Without EBG structures. 2) 
With EBG Pattern 3 only between antennas. 3) With EBG Pattern 3 around and 
between the apertures of antennas. Separation between antennas is 22.4 mm. 197 
Figure  5.24. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system: 1) without EBG structures, 2) with EBG Pattern 3 only between 
antennas, and 3) with EBG Pattern 3 around and in between the apertures of 
antennas. The pattern is in the y-z plane (E-plane) where 0 corresponds to the 
azimuth. Separation between antennas is 22.4 mm. ......................................... 198 
Figure  5.25. Top view of the CBS antenna system showing the placement of a ribbon 
of EBG structures around the aperture. The distance of this ribbon from antenna 
is determined by the two parameters: d
1
 and d
2
................................................ 199 
Figure  5.26. Radiated gain patterns at 7.5 GHz obtained for four different setups for a 
single antenna system. 1) Without EBG structures. 2) With ribbon of EBGs 
distant from antenna by d
1
 = 4.77 mm and d
2 
= 3.02 mm. 3) With ribbon of 
EBGs distant from antenna by d
1
 = 10.17 mm and d
2 
= 8.42 mm. 4) With ribbon 
of EBGs distant from antenna by d
1
 = 15.57 mm and d
2 
= 13.82 mm (refer to 
Figure  5.25). The pattern is in the y-z plane (E-plane) where 0 corresponds to the 
azimuth.............................................................................................................. 201 
Figure  5.27. (a) Setup used to assess coupling mitigation using EBG structures. (b) 
Close up view of antenna system with presence of EBG structure. EBG 
structures with different configurations (patch sizes of 2 mm to 5 mm) in the 
 
xxii 
area between two antennas separated by 16 cm are implemented. Width of EBG 
separator is 6 cm. .............................................................................................. 203 
Figure  5.28. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 2 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 
substrate. ........................................................................................................... 204 
Figure  5.29. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 3 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 
substrate ............................................................................................................ 205 
Figure  5.30. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 4 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 
substrate ............................................................................................................ 206 
Figure  5.31. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 5 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 
substrate. ........................................................................................................... 207 
Figure  5.32. Bandgap obtained using numerical simulation. EBG parameters: ?
r 
= 4.4, 
h = 1.54 mm, d = 0.8 mm, and g = 0.4 mm. ..................................................... 208 
Figure  5.33. Single plane pair in the package consisting of metal-dielectric-metal 
layers is modeled as parallel plate waveguide. (a) 3-D view.  (b) Top view. 
Typical dimensions of a microprocessor package are: a = b = 4 cm. Each via 
shown in the figure represents a port. Port 1 and Port 2 are located at (x
1
 = 1 cm, 
y
1
 = 1 cm), and (x
2
 = 2 cm, y
2
 = 2 cm), respectively......................................... 210 
Figure  5.34. Test Setup 1. The ports in the package under study are isolated with EBG 
pattern, (a) 3-D view and (b) top view. The specifications of package and ports 
are shown in Figure  5.33. In this experiment the EBG sample No. 7, specified in 
Table  3.III, is used. Each port is centered in the middle of the EBGs patch. The 
size of patch is 5 cells ? 5 cells......................................................................... 212 
Figure  5.35. Magnitude of S
21
 between ports 1 and 2 in the single plane pair under 
study for four different scenarios (design parameters and port setup are shown in 
Figure  5.33). The first and second cases show embedded capacitance. The third 
case shows test setup 1 (refer to Figure  5.34). The fourth case corresponds to test 
setup 2 (refer to Figure  5.36). In all configurations h = 114.3 ?m. .................. 213 
Figure  5.36. Test Setup 2. The second port in the package under study is isolated with 
EBG pattern, (a) perspective view and (b) top view. Specifications of package 
and ports are shown in Figure  5.33. In this experiment the EBG sample No. 7, 
specified in Table  3.III, is used. The second port is centered in a patch of EBGs. 
The size of patch is d
P
 ? d
P
, where in two different experiments d
P
 = {5 cells, 9 
cells}. ................................................................................................................ 214 
 
xxiii 
Figure  5.37. Magnitude of S
21
 between ports 1 and 2 in the single layer package under 
study for two different cases in test setup 2 (refer to Figure  5.36). .................. 215 
Figure  5.38. Radiation loss, (1- |S
11
|
2
 - |S
21
|
2
), versus frequency for a single plane pair 
under study for three different scenarios. The first case shows the test setup 1 
(refer to Figure  5.34). The second and the third cases show the embedded 
capacitance (refer to Figure  5.33) ..................................................................... 217 
Figure  5.39. Square patch mushroom-type EBG structure (a) Perspective view. (b) 
Top view. Relevant design parameters are shown on the diagrams. ................ 222 
Figure  5.40. Setup used to extract the reflection phase of an EBG structure. The 
schematic shows the boundary setup for a unit cell. The unit cell is excited 
through normal incident plane wave. The reflected waves recorded at the 
observation plane are used to compute the phase. ............................................ 225 
Figure  5.41. Reflection phase of EBG structures. The in-phase reflection phase is 
marked on each graph. The conventional EBG structure introduced in  [85] is 
specified by: a = 2.44 mm, g = 0.15 mm, h = 1.57 mm, d = 0.25 mm, ?
r 
= 1
,
 and 
?
r 
= 2.51. The novel EBG structure is specified by: a = 0.5 mm, g = 0.15 mm, h = 
1.57 mm, d = 0.1 mm, ?
r 
= 2.51, ?
r 
= 6, and tan ?
m 
= 0.01................................ 226 
Figure A. 1.  Microstrip line. Design parameters are shown in the figure................ 234 
 
 
 
 
 
 
 
 
 
 
 
 
1 
Chapter 1 : Introduction 
 
 
1.1. Background on Electromagnetic Noise 
Electromagnetic Interference (EMI) adversely affecting electronic devices is a 
critical challenge facing designers of electronic equipment operating at low threshold 
voltage levels. These challenges are intensified by increase in system speed and 
integration driven by commercial imperatives to increase functionality while reducing 
space and power consumption, resulting in inexpensive hardware, wide range of 
functionality with longer battery life. EMI can be generated from or attributed to 
different hardware stages including the device, the chip, the package, the printed 
circuit board (PCB), the interconnects, the components, the chassis, and the 
peripherals. Mitigation of noise in all system levels is necessary due to strict 
requirements in new design technologies. It is important to note that in addition to 
disrupting interference due to unintentional radiation, EMI is also of concern when 
using intentional radiators (i.e., antennas) placed in close proximity of each other, or 
that share common reference planes. Continuous advances in communication and 
electronic circuits increase the need for complicated systems with several 
electromagnetic sources that operate in the same environment such as array of 
antennas or several electronic circuits located in different compartments in chassis. 
The proximity of these radiating sources increases the probability of interference, or 
mutual coupling between them. 
 
2 
The most common type of noise in high speed packages is simultaneous 
switching noise (SSN). SSN, also referred to as voltage bounce or delta-I noise, is 
caused by the switching of transistors between two different logic levels. When 
switching takes place, a surge of current travels between power and ground planes 
causing a voltage drop between them and a voltage drop across segments of the 
power plane itself. These voltage drops are directly proportional to the current 
switching rate and the effective inductance of current paths. SSN was studied 
extensively in the past decade  [1]- [4]. The propagating wave between metallic layers 
of laminated substrate acting as parallel plate waveguide which is supported by 
traveling current on planes can interfere with other via passing through or connected 
to these planes. When the traveling wave reaches the edges, it will partially radiate 
out and the rest will reflect back inside the package. The radiated wave from package 
can interfere with the other electronics located in the same media. Another design 
artifact that exacerbates the noise induced by switching is the natural resonance of the 
package itself. If the natural resonant frequencies of the package encompass the 
switching frequency or the clock frequency and its harmonics, then the potential for 
further disturbance to the reference voltage levels increase and equally important, the 
potential for EMI increases. The resonant frequencies of the package may be pushed 
out of the band of interest if the package size is reduced. However, such a solution is 
not viable as higher integration is demanding larger size packages.  
Packages are designed to provide structural and mechanical integrity and EM 
shielding to insure proper functionality of the integrated circuit (IC). A package is an 
intermediate stage which transfers input/output (IO) signals between IC and PCBs 
 
3 
through the connections such as pins and balls. The EMI noise with the 
external/internal sources can lead to currents that flow through the interconnects all 
over the package. Therefore mitigation of noise at the early stage such as IC package 
level becomes critical considering the increasing demands to produce electronic 
devices with higher speeds, compact size, longer life-time batteries, and higher 
functionality. As a result, it is necessary to develop new designs for IC packages. In 
between packaging techniques of microprocessor which is the heart of most 
electronic devices are developing fast. In typical modern processor architecture, the 
L2 cache memory and microprocessor chip are packaged together to speed up the 
CPU execution time. The forthcoming generation of processors will have several 
parts of chipsets integrated in the same package  [5], thus resulting in complex 
integrated circuits and highly dense power dies.  
Package integration is expected to reduce the cost and the complexity of 
interconnects. However, higher density and faster processors lead to higher levels of 
electromagnetic noise. If the noise is not managed effectively such that it remains 
below acceptable levels, then the performance can be compromised and the potential 
for the device to electromagnetically interfere with other independent devices 
increases. Therefore, one of the key challenges for package designers is noise 
mitigation at every stage of the system. In fact, design for electromagnetic noise 
management becomes one of the key imperatives in the design process.  
 Enclosures and chassis which are designed to provide structural and 
mechanical integrity are also intended to provide EM shielding to insure proper 
functionality of the system. The mechanism of coupling between internal 
 
4 
electromagnetic sources and the external electromagnetic environment takes place 
either by direct penetration of electromagnetic fields through openings or through 
induced currents on conducting surfaces. Once currents exist on the internal surface 
of an enclosure, they travel to the external surface through openings such as 
apertures, seams, gaskets, etc. Once currents reach external surfaces, they travel and 
penetrate other cavities sharing the same surface, thus creating fields in these so-
called victim cavities. Therefore, designs for enclosures and chassis managing 
electromagnetic noise are necessary. 
 
1.2. Problems and Challenges 
EMI can be generated from or attributed to different hardware stages 
including devices, chips, packages, printed circuit boards (PCB), interconnects, 
components, chassis, and peripherals. To minimize EMI, which could mean either 
reducing radiation from the equipment or increasing the immunity to external 
electromagnetic stimuli; various strategies are applied that are typically specific to 
each of these level. Electromagnetic noise or interference can be mitigated using one 
or more of several strategies. These strategies include, but are not limited to, isolation 
of critical components, shielding, grounding, matching, filtering, addition of lossy 
materials and absorbers and finally the possibility of redesigning the circuitry of the 
victim device or the circuitry of the source of EMI  [6]- [17].  Eliminating the source of 
EMI is clearly desirable, but does not necessarily translate or equate to reduction in 
the susceptibility of the device to external sources. In fact, strong interference can 
occur at frequencies different from the frequencies corresponding to the device 
 
5 
switching speed (and its significant harmonics), which makes the containment 
strategies even more challenging. 
The techniques proposed to reduce coupling noise between cavities as a 
constitutive part of the cavity-backed antennas include: placement of lossy materials 
and absorbers between the radiating elements, change in antenna configurations (E-
plane or H-plane), increasing the distance between radiating elements, cutting a slit 
on the common surface or a combination of these methods  [6]. However, while lossy 
materials have desirable electromagnetic features, mechanical and thermal properties 
can severely limit their applicability. Additionally, cost can be a detrimental factor as 
these materials need to be engineered to work over specific frequency bands.  Also 
the other mentioned methods are not efficient, as they lead to increase in size and 
weight, or change in the setup and structure of system. 
The problem of mitigating noise in IC packages remains a major challenge. In 
 [18], a solution based on stitching via and retreated power planes was introduced to 
reduce package radiation. Most noise mitigation strategies presented in the literature, 
however, were applied to PCBs. In  [9],  [12] dissipative and lossy materials were used 
to damp down the noise level. Yet, the thermal and mechanical factors can limit the 
reliability and the applicability of this method. In  [13], decoupling capacitors were 
applied which are effective for suppressing noise from DC to a few hundred mega 
hertz. In  [14],  [15], embedded capacitance was studied. In this method, decoupling 
level depends on the limitations in the fabrication techniques, such as: limitation in 
using different dielectric materials or minimum isolation distance between two 
metallic layers (minimum dielectric thickness between two consecutive layers). 
 
6 
Generally, by using embedded capacitance methodology, EM waves within the entire 
frequency band are still guided. Therefore, the application of this method completely 
depends on the level of noise tolerance in electronic circuits. In  [16], power islands 
method, which isolates the source of noise from susceptible components, was used. 
Efficiency of this method depends on the level of components? density and topology. 
In  [17], via stitching was used. It is an effective method in reducing emission from 
PCB package and coupling to periphery. Efficiency of this method which increases 
the dimension of board depends on the size requirements and fabrication techniques. 
The concept of using electromagnetic bandgap (EBG) structures for suppressing SSN 
and EMI was introduced in  [3],  [19]- [25]. EBG structures consist of dielectric 
substrate with periodic metallic patterns. They are designed to function as band-stop 
filters. This method is effective in that band-stop or bandgap is located in a range 
from few hundred megahertz to above. The bandgap is a function of the dimensions 
and the materials of both metallic patterns and substrate  [26]- [28].  In  [4],  [19]- [25], 
an additional layer of metal for patches of the EBG structure is sandwiched between 
two layers of the package. Each patch is connected to one of these two metallic layers 
through a via. Recently, new planar EBGs were proposed to reduce fabrication cost. 
In these structures, the patterns were etched on one of the metallic layers instead of 
having additional third layer and vias  [29]- [31]. Although, these new planar EBGs 
reduced the fabrication cost, their size was impractical for numerous applications and 
for use in suppressing noise in IC packages. More recently in  [20],  [32], an EBG 
structure suitable for use in IC packages was proposed. While the size of the EBG 
structure introduced in  [32] was suitable for IC packages, it provided a narrow 
 
7 
bandgap which has to be fine tuned to coincide with a specific switching frequency, 
clock frequency, or higher-order harmonics.  
In enclosures, the integrity of shielding compromised with size and number of 
apertures designed for airflow and heat dissipation  [33]. To reduce EMI in enclosures, 
absorbers and lossy materials  [8],  [11] and combination of lossy materials and dual 
perforated ventilation screens  [10] have been used. Although, lossy materials have 
desirable EM features their mechanical and thermal properties can severely limit their 
applicability. In addition, cost can be a detrimental factor as these materials need to 
be engineered to work over specific frequency bands. Also, designs with dual 
aperture screens change the size and weight of enclosure. In other methods 
implemented to reduce EMI, designers change the dimensions of box or openings till 
natural resonances of box and openings do not coincide with clock frequency of 
system and its significant harmonics. Generally, methods which change the 
dimension and weight and others which apply additional materials may add cost and 
size to enclosures.   
Challenges with the present methods of noise reduction can be summarized as 
follows: 1) geometry dependence; 2) material dependence; 3) no guarantee of 
effectiveness in the microwave region; 4) presence of reliability issues. All these 
limitations and drawbacks in the methods of noise suppressions in the PCBs, the 
packages, the enclosures, the cavities, and the antennas are showing that the problems 
related to EMI noise suppression are not solved yet! In the following chapters, new 
solutions for EMI noise reduction in stages including: the packages, the enclosures, 
the cavities, and the antennas will be proposed and studied. 
 
8 
This dissertation is organized as follow. In  0, the EBG structures are 
introduced. Then, we study the characterization of EBG structures for applications 
where they are exposed to open media (they are not confined in closed media such as 
embedded EBGs in parallel plate waveguide). Also, we investigate the relationship of 
the suppression band versus different constituent materials and structures of EBG. 
In  Chapter 3, the concept of miniaturization of EBG structures using very 
high-k materials for mushroom- and planar-type structures is discussed. These novel 
EBGs are characterized. Also, we discuss the modeling of the proposed planar EBG 
structures. The developed model is providing efficient initial start in the design 
procedure. 
The SSN and EMI noise suppression in IC packages using very high-k 
material is discussed in  Chapter 4. We study analytically and numerically the 
limitations of using high-k materials in EM noise suppression in packages. 
In  Chapter 5, we discuss the new applications of EBG structures under two 
categories as follows: 1) EM noise reduction and 2) design of low profile antennas. In 
this chapter, the potential of EBG structures in suppressing EMI noise in enclosures 
and chassis, cavities and cavity-backed antennas, and packages are demonstrated. We 
assess the performance in enclosures and chassis numerically, in cavities and cavity-
backed antennas numerically and experimentally, and in IC packages numerically.  
The design of novel EBG structures using magneto-dielectric materials 
applied in the design of low profile wideband antennas is discussed in  Chapter 5. 
Finally the conclusions and the future work are presented in  Chapter 6. 
 
9 
Chapter 2 : Electromagnetic Bandgap Structures 
In this chapter, we present different methods to characterize EBG structures. 
We will study numerically the effect of different design parameters on the 
suppression bandgap. The data obtained in this chapter will be used in subsequent 
chapters in different case studies to develop the concepts and to validate the 
efficiency of the proposed methods. 
 
2.1. EBG Structures 
EBG structures operating in the high frequency and the microwave frequency 
bands are periodic or quasi-periodic patterns created by metallic inclusions in 
dielectric or magnetic material  [26]- [28]. In the literature, the EBG structures have 
been referred to as high-impedance surfaces (HIS), metallo-dielectric material, 
negative materials, and other designations. In this study, we adopt a simple structure 
of mushroom-type EBG patterns represented generically by the schematic shown in 
Figure  2.1. These structures are characterized by periodic metallic patches connected 
to a common ground (or reference) plane through shorted stubs or plated through-
holes (commonly referred to as vias). In essence, the EBG structures are comprised of 
a Frequency Selective Surface (FSS) positioned on top of a conductor-backed 
material with vias positioned between the FSS and the metallic backing, hence 
electrically shorting them together. The material between the patches and the ground 
plane can be dielectric, magnetic, or combination of the two. In this study, we 
consider the metallic patches to be square. However, this choice does not diminish the 
 
10 
generality of our study as other designed topologies of similar size can be ptimized to 
achieve higher suppression. Design parameters of the EBG structure under study are 
illustrated in Figure  2.1. These parameters include the size of individual patches, the 
gap between adjacent patches, the diameter of stubs (vias), the thickness, and the 
material of the substrate. The geometrical features of the structure and the material of 
the substrate are directly related to the frequency range of the stop band.  
 
 
Figure  2.1. Square patch mushroom-type EBG structure (a) Perspective view. (b) Top 
view. Relevant design parameters are shown on the diagrams. 
 
A major property of the EBG structure is the suppression of surface waves as 
originally postulated and experimentally demonstrated by the ground breaking work 
of Sievenpiper et al.  [26]- [28]. This fundamental property of EBG structures makes 
 
11 
them play a highly critical role in the field of EMI/EMC and signal integrity. In 
effect, the EBG structure acts as a medium with either effective negative ?
r
 or 
effective negative ?
r
 occurring within a specific frequency band. In fact, EBG 
structure could be viewed as a plasma medium resonating over a band of frequencies 
which can be produced inexpensively using PCB technology. 
 
2.2. Characterization of EBG Structures 
The frequency bandgap of EBGs is directly related to the geometrical 
parameters and material parameters of the host medium. The EBG structures are 
essentially electrically-small resonators. Within the bandgap where they provide wave 
suppression, their size is much smaller than the wavelength. Analytical formulation of 
a precise relationship between the effective bandwidth (bandgap) and the geometrical 
and material parameters is difficult. In  [34],  [35], the authors have obtained circuit 
models for EBGs embedded in power planes. However, for the cases of EBG 
structures used in open systems, as in this study, we could not find any analytical 
expressions or accurate circuit models to give a reasonably accurate prediction of the 
bandgap. Therefore, and until a highly accurate closed-form analytic expression is 
developed, the stop band of an EBG structure may be generated directly or indirectly 
using numerical simulation tools. There are different numerical techniques to extract 
the frequency stop band of a structure which vary in complexity and efficiency. Using 
the scattering parameters (S-parameters) to characterize the bandgap is the most 
direct method while the dispersion diagram  [36],  [37] is the indirect one. 
 
 
12 
2.2.1. Scattering Parameters 
In this method the bandgap can be specified by extracting the S-parameters 
numerically between two ports placed across the EBG structure as shown in Figure 
 2.2. For this purpose full-wave EM simulators such as Ansoft HFSS  [38] can be used. 
EBG structure with specific bandgap can be designed using this method by trial and 
error procedure. This method is the most direct one because not only it characterizes 
the location of the bandgap, the bandwidth, and the center frequency but also, it 
provides the attenuation level of the signal at different frequencies. 
 
 
Port 1 Port 2
EBG structures
 
Figure  2.2. A schematic showing the simulation setup used to extract the bandgap of 
an EBG structure using scattering parameters. In this setup S
21
 represents the power 
delivered at port 2 when port 1 is excited. 
 
2.2.2. Dispersion Diagram 
In this method, the bandgap is specified by extracting the dispersion diagram 
which describes the propagation characteristics of an infinitely periodic structure 
composed of EBG patches. Dispersion diagram is numerically extracted using full-
wave solver, by considering a single patch (unit cell) and applying periodic boundary 
condition (PBC) on the sides of the cell to mimic the presence of the cell in a inifinite 
 
13 
array of periodic structure, and absorbing boundary condition (ABC) or perfectly 
matched layer (PML) on the top open wall as shown in Figure  2.3 (a)  [36].   
Dispersion diagrams show the relationship between wave numbers and 
frequency. These diagrams present the propagating modes and the bandgaps that can 
potentially exist between such modes (in a periodic structure at a given frequency of 
operation, many modes propagating in different directions may be excited)  [37]. 
Brillouin, in his theory of wave propagation in periodic structures  [39], states that for 
any periodic structure there are certain vectors (i.e., directions) in the unit cell of the 
periodic structure that constitute a boundary region of propagation called irreducible 
Brillouin zone. According to that theory, deriving the propagating modes in the 
direction of these vectors suffices to cover all possible directions of wave propagation 
within the lattice. Therefore, the problem of deriving the propagating modes excited 
at a certain frequency reduces to finding such modes only in the directions of the 
vectors of the irreducible Brillouin zone. For the type of structure considered in this 
work, the border of the irreducible zone is illustrated in Figure  2.3 (b) and it consists 
of the direction pointing from ? to X, from X to M and from M back to ?.  
Therefore, in light of Brillouin theory, a dispersion diagram will consist of 
three regions. In each region, we consider the paropagation wave vector ? which is 
translated into a phase shift between the sides of the unit cell shown in Figure  2.3 (a). 
This translation allows the derivation of dispersion diagram using traditional 
eigenmode solver based on full-wave analysis. In these simulations, the structure of 
unit cell and the required phase shifts (shown as Phase 1 and Phase 2 in Figure  2.3 
(a)) are given to the simulator. The simulator calculates the frequencies of 
 
14 
propagating waves that would generate such phase shifts. For a wave propagating in 
the x-direction with no y variation, Phase 1 is varied between 0 and 180
0
 and Phase 2 
is kept constant at zero degree. This corresponds to the ? to X direction. The X to M 
direction corresponds to Phase 1 being constant and equal to 180
0 
and Phase 2 
varying from 0 and 180
o
. This represents the second region in the dispersion diagram. 
The third region is represented by the M to ? direction in which both phases are equal 
and changing from 180
o 
back to 0.
 
 For wave propagation in free space, as there is no 
dispersion, the diagram constitutes of straight lines in the first and third regions and a 
quadratic graph in the second region. The following equation shows the frequency-
phase relationship for wave propagation in free space: 
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?????
=?????+
=?????
=
dddd
d
c
ddd
d
c
ddd
d
c
df
1211
12
2
2
2
211
0:)(
2
2
0:)(
2
00:)(
2
)(
?????
?
??????
?
????
?
?  
 Equation  2-1 
where d = a + g is the size of unit cell (periodicity length), c is the speed of light in 
free space, ?
1 
= ?
x
 and ?
2 
= ?
y
 are the wave numbers in the ?-X and the X-M 
directions. 
With the presence of EBG structures, the surface becomes dispersive; 
therefore the frequency-phase relationship of propagating modes will not be the same 
as in free space. A gap between the upper limit of any propagating mode and the 
intersection of the free space propagation line with the next propagating mode 
represents a region in which the surfaces do not support any propagation. 
 
 
15 
 
 
X M 
? 
(a) 
 
(b) 
 
Figure  2.3. Schematic showing the simulation setup used to extract the bandgap of an 
EBG structure using dispersion diagrams. (a) Computational domain containing 
single cell. Periodic boundary conditions are placed at x-z and y-z planes and an 
absorbing condition (PML) is placed on the top face x-y plane. (b) The irreducible 
Brillouin zone triangle.  
 
 
As an example to show how the Brillouin theory is used, we consider a 4 mm 
patch with a substrate of ?
r
 = 2.2, g = 0.4 mm, d = 0.8 mm, and h = 1.54 mm. The 
dispersion diagram for this structure obtained using HFSS is shown in Figure  2.4. 
From the dispersion diagram, the frequency band where no propagating modes exist 
is the corresponding bandgap of the structure. From Figure  2.4, the bandgap is 
observed to be between 8.4 GHz and 9.6 GHz. Also, the dispersion diagrams of two 
 
16 
other cases are shown in Figure  2.5 and Figure  2.6. The bandgap of EBG with a = 4 
mm, ?
r
 = 4.8, g = 0.4 mm, d = 0.8 mm, and h = 1.54 mm is from 7 GHz to 8 GHz as 
shown in Figure  2.5. The bandgap of EBG with a = 2.6 mm, ?
r
 = 3, g = 0.4 mm, d = 
0.8 mm, and h = 1.54 mm is between 11.9 GHz and 16 GHz as shown in Figure  2.6. 
 
 
0
5
10
15
20
25
30
?
 First propagating mode
 Second propagating mode
 Third propagating mode
 Propagation in vacuum
Bandgap (no-propagation band)
X M?
F
r
equency
 (G
Hz)
Phase (?d)
 
Figure  2.4. Dispersion diagram for the EBG structure having 4 mm square patches on 
a substrate with ?
r
 = 2.2. 
 
17 
 
Figure  2.5. Dispersion diagram for the EBG structure having 4 mm square patches on 
a substrate with ?
r
 = 4.8. 
 
Figure  2.6 Dispersion diagram for the EBG structure having 2.6 mm square patches 
on a substrate with ?
r
 = 3. 
0
5
10
15
20
25
30
35
40
?
Phase(?d)
Bandgap (no propagation band)
 First propagating mode
 Second propagating mode
 Third propagating mode
 Propagation in vacuum
M? X
Freq
uency
 (GH
z
)
0
2
4
6
8
10
12
14
16
18
20
22
?
Bandgap (no-propagation band)
 First propagating mode
 Second propagating mode
 Third propagating mode
 Propagation in vacuum
 
? MX
Fr
e
q
ue
n
c
y
 (
G
Hz
)
Phase(?d)
 
18 
2.3. Numerical Study of EBG Structures 
As we mentioned in Section  2.1, there are no exact analytical formulas that 
relate the structural parameters of the EBG structures to the frequency stop band. 
Therefore, our goal here is to determine the effect of varying the EBG parameters on 
the width of the bandgap and its lower and upper frequency limits based on direct 
numerical simulation using HFSS.  
The numerical setup consists of two parallel plate waveguides and the EBG 
structure to be tested is placed between them as shown in Figure  2.7. The EBG 
surface in Figure  2.7 is considered to be semi-infinite (infinite in the x-direction and 
finite in the y-direction) which is modeled in HFSS using symmetric boundary 
conditions. As a measure of the performance of the EBG structure, we consider the 
transmission loss (S
21
) between the two ports shown in Figure  2.7 with and without 
the EBG surface.  For example, in Figure  2.8 we show the magnitude of S
21
 for an 
EBG structure with the parameters a = 2.6mm, h = 1.54 mm, d = 0.8 mm, g = 0.4 
mm, and ?
r 
= 2 (parameters are shown in Figure  2.1). The bandgap used throughout 
this study is defined as the difference between the two frequency points where the S
21
 
suppression exceeds 3dB. The two frequency points defining the edge of the bandgap 
will be referred to as the lower and the upper frequency points. Figure  2.9-Figure  2.17 
summarize the results of numerous simulations showing the relationship between the 
bandgap and the different parameters of the structure. Since the EBG patches are 
positioned in the direction of propagation of the waves, it is expected that the more 
EBG patches are used the suppression goes higher as shown in Figure  2.9 (a) and 
Figure  2.9 (b) where 4 rows and 8 rows of patches were used, respectively. Also, 
 
19 
from these figures we realize that increasing the patch size shifts the bandgap 
downward. In the following simulations, the distance between patches is 0.4 mm and 
via diameter is 0.8 mm.  
Figure  2.10 and Figure  2.11 show the bandgap in relation to the separation 
between patches. We observe that increasing the gap between the patches causes the 3 
dB bandgap to shift upward. Figure  2.12 and Figure  2.13 show the relationship 
between the bandgap and the height of via. Increasing the height of via shifts the 
bandgap downward. Figure  2.14 and Figure  2.15 show the relationship between the 
bandgap and the diameter of via. The trend of change of bandgap with diameter of via 
is not very clear. Figure  2.16 and Figure  2.17 show the relationship between the 
bandgap and the relative permittivity of the substrate. Increasing the permittivity 
shifts the bandgap downward. Generally, the relationship between the bandgap and 
the design parameters are not linear.  
 
 
20 
Pa
ra
llel
 p
la
te
 W
av
eg
ui
de
EBG structure
under test
R
e
p
e
a
t
i
n
g
?
?
R
e
p
e
a
t
i
n
g
Pa
ra
lle
l p
la
te 
W
av
eg
ui
de
Port 1
Port 2
 
Figure  2.7. Numerical setup used to extract the bandgap of EBG structures.  
 
Figure  2.8. Magnitude of S
21
 for the setup shown in Figure  2.7 with and without and 
EBG surface with the following parameters a = 2.6 mm, h = 1.54 mm, d = 0.8 mm, g 
= 0.4 mm, and ?
r 
= 2. The design parameters are shown in Figure  2.1.  
8 1012141618202
-40
-35
-30
-25
-20
-15
-10
 without EBG
 with EBG (?
r
=2)
M
a
gnitude of S
21
 (dB)
Frequency (GHz)
3dB Bandgap 
 
21 
 
Figure  2.9. Frequency bandgap versus the patch size. (a) 4 rows of EBG patches, (b) 
8 rows of EBG patches. EBG parameters: ?
r 
= 3, h = 1.54 mm, d = 0.8 mm, and g = 
0.4 mm. 
 
2.0 2.5 3.0 3.5 4.0 4.5 5.0
6
8
10
12
14
16
18
20
F
r
eq
uen
c
y (G
Hz)
Patch Size (mm)
(a) 
2.0 2.5 3.0 3.5 4.0 4.5 5.0
6
8
10
12
14
16
18
20
Fr
eq
uen
c
y (G
Hz)
Patch Size (mm)
(b) 
 
22 
 
Figure  2.10. Frequency bandgap versus the gap between patches. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, ?
r 
= 4.4, h = 1.54 
mm, and d = 0.8 mm. 
 
0.2 0.4 0.6 0.8
5
6
7
8
9
10
F
r
equ
e
ncy (G
Hz)
gap between patches (mm)
(a)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
4
5
6
7
8
9
10
11
12
F
r
eq
uen
c
y (
G
Hz)
gap between patches (mm)
(b)
 
23 
 
Figure  2.11. Frequency bandgap versus the gap between patches. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, ?
r 
= 3, h = 1.54 
mm, and d = 0.8 mm. 
 
0.2 0.3 0.4 0.5 0.6 0.7 0.8
6
8
10
12
14
16
18
20
22
F
r
equen
c
y (GHz)
gap between patches (mm)
0.20.30.40.50.6
8
10
12
14
16
18
20
22
F
r
equen
c
y (GHz)
gap between patches (mm)
(a)
(b)
 
24 
 
Figure  2.12. Frequency bandgap versus the height of via. (a) 4 rows of EBG patches, 
(b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g = 0.4 mm, ?
r 
= 4.4, and d = 
0.8 mm. 
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
5
6
7
8
9
10
11
12
Freque
ncy (GHz)
via height (mm)
0.20.40.60.81.01.21.41.61.82.02.2
6
7
8
9
10
11
12
Fr
eq
ue
nc
y (GHz)
via height (mm)
(a)
(b)
 
25 
 
Figure  2.13. Frequency bandgap versus the height of via. (a) 4 rows of EBG patches, 
(b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, g = 0.4 mm, ?
r 
= 3, and d = 
0.8 mm. 
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
8
10
12
14
16
18
20
22
Frequency
 (
GHz)
via height (mm)
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
10
12
14
16
18
20
22
Frequency (GHz)
via height (mm)
(b)
(a)
 
26 
 
Figure  2.14. Frequency bandgap versus the diameter of via. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g = 0.4 mm, ?
r 
= 4.4, 
and h = 1.54 mm. 
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
6.5
7.0
7.5
8.0
8.5
Fre
quen
c
y (GHz
via diameter (mm)
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
6
7
8
9
10
11
12
13
14
F
r
e
que
ncy (GHz
)
via diameter (mm)
(a)
(b)
 
27 
 
Figure  2.15. Frequency bandgap versus the diameter of via. (a) 4 rows of EBG 
patches, (b) 8 rows of EBG patches. EBG parameters: a = 2.6 mm, g = 0.4 mm, ?
r 
= 3, 
and h = 1.54 mm. 
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
10
12
14
16
18
20
Frequen
c
y (GHz)
via diameter (mm)
(a)
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
10
12
14
16
18
Frequency (GHz)
via diameter (mm)
(b)
 
28 
 
Figure  2.16. Frequency bandgap versus the relative permittivity of substrate. (a) 4 
rows of EBG patches, (b) 8 rows of EBG patches. EBG parameters: a = 4 mm, g = 
0.4 mm, d = 0.8 mm, and h = 1.54 mm. 
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
6
7
8
9
10
11
Frequency
 (GHz)
relative permitivity (?
r
)
(a)
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
6
7
8
9
10
11
12
13
14
15
16
17
Frequency
 (GHz)
relative permitivity (?
r
)
(b)
 
29 
 
Figure  2.17. Frequency bandgap versus the relative permittivity of substrate (a) 4 
rows of EBG patches, (b) 8 rows of EBG patches. EBG parameters: are a = 2.6 mm, g 
= 0.4 mm, d = 0.8 mm, and h = 1.54 mm. 
 
2345
8
10
12
14
16
18
Frequen
c
y (GHz)
relative permitivity (?
r
)
2345
10
12
14
16
18
20
22
Freq
uency
 (GHz)
relative permitivity (?
r
)
(a)
(b)
 
30 
Chapter 3 : Miniaturized EBG Structures 
In this chapter, we investigate the use of very high-k dielectrics (?
r
 ? 100) in 
the design of EBG structures as strong contenders of surface wave suppressor for 
EMI/EMC applications. The basic idea behind the designs presented here is the use of 
high-k dielectrics to increase the effective capacitive load. The increased effective 
capacitance shifts the bandgap to lower frequencies, or alternatively, allows for 
maintaining the same frequency performance but with smaller size patches.  
 
3.1.  Miniaturization of Embedded EBG Structures between a Single Pair of Planes 
using High-K Dielectrics 
The concept of embedding mushroom-type EBG structures for suppressing 
SSN and EMI has been studied in  [4],  [19]- [25]. Already there are several models 
based on the physics of structures and the lumped elements and transmission lines 
which characterize embedded EBGs and predicts the bandgaps  [35],  [40] with good 
approximation. Therefore, we do not intend to speak about the modeling of 
mushroom-type EBGs embedded in a plane pairs. However, briefly and using 
numerical tools we show the possibility of shrinking the dimensions of EBGs using 
very high-k dielectrics. This type of EBGs due to addition of layers and having vias 
are costly to manufacture. Recently, planar EBGs as a solution to these drawbacks 
were introduced  [29]- [31]. We continue our studies on planar EBGs in Section  3.2. 
 
 
31 
3.1.1.  Numerical Characterization of Embedded Mushroom-Type EBGs made of High-k 
Dielectrics 
The square patch mushroom-type EBG studied in  0 is embedded between two 
metallic layers as shown in Figure  3.1. The design parameters including size of the 
square patch, diameter of the via, gap between the patches, thickness of the filling 
dielectric materials; the one surrounding the EBG structures and the one located 
between the patch and the top metallic plane; and a relative permittivity of these 
dielectrics are shown on the same figure. The physical-based model for a unit cell of 
EBG embedded between pairs of plane is shown in Figure  3.2  [41]. The resonant 
frequency (center frequency) of the unit cell is given by 
()
21
2
1
CCL
f
c
+
=
?
. 
Therefore, by increasing the capacitive load or the inductive load the bandgap can be 
moved to lower frequencies or the same frequency performance can be maintained by 
reducing the size of inclusions. The effect of design parameters on the first 
suppression band is demonstrated numerically as follows. 
 
Dielectric2
Dielectric1
h
2
h
1
d
v
EBG Structures
?
r2
?
r1
Metallic Plane
Metallic Plane
g
 
Figure  3.1. Side view of embedded mushroom-type EBG between parallel planes. 
The design parameters are included in the figure. 
 
 
32 
C
2
C
1
L
 
Figure  3.2. Physical-based model for a solid patch mushroom-type EBG embedded in 
between parallel planes. C
1
 models the capacitance between the patch and the top 
plane, C
2
 is the capacitance between the patch and the bottom plane, and L represents 
the inductance of via. 
 
The suppression bandgap of EBGs is extracted from the computed S-
parameters. For this purpose, we used the setup which is parallel plate waveguide 
loaded with 4 cells of EBG structures as shown in Figure  3.3. The EBGs are analyzed 
up to 20 GHz. The effective suppression band is defined to be the frequency band 
where suppression exceeds 20 dB. The relationship of the suppression band and the 
patch size for EBG structures, with the following parameters: ?
r1
 = 4.4, h
1
 = 100 ?m, 
?
r2
 = 50, h
2
 = 16 ?m, g = 200 ?m, and v = 200 ?m, is shown in Figure  3.4. Increasing 
capacitive loads (C
1
, C
2
) through increasing patch dimension shifts the bandgap 
downward. The suppression band in relation to gap separation is plotted in Figure  3.5. 
The EBG parameters are as follows: ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 
1 mm, and v = 200 ?m. As it is clear from graphs, in this scale the effect of gap on the 
lower edge of suppression band is not noticeable. The relationship between the 
suppression band and diameter of via is plotted in Figure  3.6. The EBG parameters 
are as follows: ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 1 mm, and g = 40 ?m. 
 
33 
Decreasing the via diameter shifts the lower edge of the suppression band downward. 
This decrement equates the increment of the capacitive loads (C
1
, C
2
) due to 
increment of the effective patch area and the increment of inductance L  [40]. The 
relationship between the suppression band and dielectric relative permittivity of 
media 1 is shown in Figure  3.7. The parameters of the EBG structure are as follows:  
h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. Using 
high-k materials in media 1 increases the capacitance C
1
. As a result, the suppression 
band shifts to lower frequencies. Decreasing h
1
 and h
2
 has similar effects, as it causes 
the capacitive loads (C
1
, C
2
) to increase correspondingly.  
 
 
Port 
1
Port 
2
 
Figure  3.3. Simulation setup used to extract the suppression band of the EBG 
structure from computed S
21
. The computational domain is ideal parallel plate 
waveguide with PMC boundary on the sides.  
 
34 
02468101214161820
-220
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 d = 400 ?m
 d = 800 ?m
 d = 1.2 mm
 d = 1.6 mm
 d = 2 mm
M
a
gni
t
ude of 
S
21
 (dB
)
Frequency (GHz)
 
Figure  3.4. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different patch sizes. The EBG structures have the following parameters: ?
r1
 = 4.4, 
h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, g = 200 ?m, and v = 200 ?m (design parameters 
are shown in Figure  3.1).  
0 2 4 6 8 101214161820
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 g = 40 ?m
 g = 80 ?m
 g = 120 ?m
 g = 160 ?m
 g = 200 ?m
M
a
g
n
i
tu
de
 of S
21
 (dB
)
Frequency (GHz)
 
Figure  3.5. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different gap sizes. The EBG structures have the following parameters: ?
r1
 = 4.4, 
h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 1 mm, and v = 200 ?m (design parameters are 
shown in Figure  3.1).  
 
35 
0 2 4 6 8 101214161820
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 v = 40 ?m
 v = 80 ?m
 v = 120 ?m
 v = 160 ?m
 v = 200 ?m
Magni
t
ud
e o
f
 S
21
 (dB
)
Frequency (GHz)
 
Figure  3.6. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
the different via diameters. The EBG structures have the following parameters: ?
r1
 = 
4.4, h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 1 mm, and g = 40 ?m (design parameters 
are shown in Figure  3.1).  
6 8 10 12 14 16 18 20 22 24 26 28 30
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 ?
1
 = 4.4
 ?
1
 = 50
M
a
gni
t
ude of S
21
 (dB
)
Frequency (GHz)
 
Figure  3.7. Magnitude of S
21
 for the embedded EBG structures in parallel planes with 
different values of dielectric relative permittivity for media 1. The EBG structures 
have the following parameters: h
1
 = 100 ?m, ?
r2
 = 50, h
2
 = 16 ?m, d = 0.4 mm, g = 40 
?m, and v = 40 ?m (design parameters are shown in Figure  3.1).  
 
36 
In Figure  3.8, the magnitude of S
21
 versus frequency is plotted for three 
different EBG structures embedded between plane pairs. The design parameters of the 
first sample referred to by the symbol (a) on the figure are: ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 
= 50, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. The parameters of sample 
(b) are: ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 = 70, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 
40 ?m. Sample (c) has the following parameters: ?
r1
 = 4.4, h
1
 = 108 ?m, ?
r2
 = 70, h
2
 = 
8 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. We can identify that increasing the 
capacitive loads (C
1
, C
2
) shifts the suppression band to lower frequencies. Additional 
type of samples are summarized in Table  3.I. Theses samples are sorted in such a way 
to show the effect of increment of the effective capacitive load on the suppression 
bandgap. The distance between two planes (the total thickness) is fixed at 116 ?m and 
the permittivity of media 1 is equal to 4.4. It is obviously clear in Table  3.I the 
advantage of applying thin layer of very high-k material (?
r
 ? 100) between patch and 
upper metallic plane to provide suppression band at lower frequencies.  
 
 
 
 
 
 
 
   
 
 
37 
 
 
 
0 2 4 6 8 10 12 14 16 18 20
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 a
 b
 c
M
a
gnitude of S
21
 (dB)
Frequency (GHz)
 
Figure  3.8. Magnitude of S
21
 for different embedded EBG structures in parallel 
planes. The samples have the following parameters: (a) ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 = 
50, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. (b) ?
r1
 = 4.4, h
1
 = 100 ?m, ?
r2
 
= 70, h
2
 = 16 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m. (c) ?
r1
 = 4.4, h
1
 = 108 ?m, 
?
r2
 = 70, h
2
 = 8 ?m, d = 0.4 mm, g = 40 ?m, and v = 40 ?m (design parameters are 
shown in Figure  3.1).  
 
 
 
 
 
 
38 
Table  3.I. Specifications of the different square patch mushroom-type EBG structures 
embedded in parallel planes and related suppression bands (design parameters are 
shown in Figure  3.1). The thickness of parallel plane is equal to 116 ?m and ?
r1
 = 4.4 
for all samples. 
0.8-20
+
111520100.44200
1.1-20
+
111520100.44150
1.2-20
+
111520100.44120
1.4-20
+
211420100.44150
1.5-20
+
211420100.44120
1.6-20
+
211420100.43120
1.9-20
+
211440100.43120
2.1-20
+
211440100.43100
2.5-20
+
311340100.43100
3.2-20
+
411240400.4100
3.8-20
+
411240400.470
5.4-20
+
810840400.470
7.7-20
+
1610040400.470
9.1-20
+
1610040400.450
Suppression 
Band (GHz)
h
2
(?m)
h
1
(?m)
v
(?m)
g
(?m)
d
(mm)
? 
r2
+
Shows suppression band extends beyond 20 GHz
 
 
3.2. Miniaturization of Planar EBG Structures using High-K Dielectrics 
A planar EBG structure consists of two metallic layers isolated by a substrate. 
It is constructed by periodically patterning one of those metallic layers. Using spiral-
inductors is the best choice to increase the net inductance of EBG structures leading 
 
39 
to a widening of the bandgap and decreasing the lower edge of bandgap. The 
innermost turn of spiral inductors are connected to other circuitries using wire-bonds, 
air-bridges or bridges through additional layers (In that case, the inductor isn?t 
planar). The value of inductance may increase because of having positive mutual 
inductance between turns. However, fabrication of this type of inductor is costly. To 
reduce manufacturing cost, planar inductors such as Archimedean spiral inductor or 
meander line inductor shown in Figure  3.9 can be used. The inductance per unit area 
provided with these types of inductors is not as high as the one obtained from spiral-
inductor with bridge. Reduction in the value of inductance is due to the negative 
mutual couplings between adjacent turns. Later, in the modeling section, we will 
discuss more about this effect. Choosing the planar inductor is a trade off between 
cost and performance. The efficacy of implementing meander lines was shown in 
 [42]. In this work, meander lines are used as inductive bridges connecting the patches. 
The top views of a unit cell of a few planar EBG structures consisting of different 
meander lines with dissimilar number of turns and unequal lengths are shown in 
Figure  3.10. The meander lines are connected to the patches at the edge without any 
margin. Figure  3.11 shows the top view of a unit cell of EBG pattern with two-turn 
meander lines connected with a margin to the patch. In this study, unless it is clearly 
stated, we are dealing with structures with no margins. A planar EBG structure 
consisting of two-turn meander lines and its critical design parameters are shown in 
Figure  3.12. The design parameters include: size of the patch, dimensions of the 
meander line, thickness and dielectric constant of the substrate. The number of line 
 
40 
turns in meander is represented by N. In this work, it is assumed that the meander line 
parameters (M
G
, M
W
, M
L
, M
B
) are constant along the line to simplify the study. 
 
(a)
(b)
 
Figure  3.9. Planar inductors. (a) Archimedean spiral inductor. (b) Meander line 
inductor.   
 
(a) (b) (c)
 
Figure  3.10. Top views of a unit cell of different planar EBG structures. The meander 
line as an inductive bridge connecting the patches in EBG structures consists of (a) 
one turn, (b) two turns, and (c) three turns.  
 
41 
Margin
 
Figure  3.11. Top view of a unit cell of a planar EBG structure. The meander line as an 
inductive bridge providing current continuity between the patches in the EBG pattern 
is connected to two adjacent patches with a margin. 
 
The physical-based model of a unit cell of planar EBG structure existing in a 
1-D array of EBG structures is shown in Figure  3.13. In this model, the capacitance 
between patterned layer and ground is given by: 
hAC ?=  
 Equation  3-1 
where ? and h are the permittivity and the thickness of the substrate, respectively. A is 
the area of the patterned planar conductive part. According to  [26],  [43], coupling 
capacitance between two adjacent arms of the meander line can be approximated as: 
)(cosh
)(
10
GW
L
cmm
MM
M
C
?
+
=
?
??
 
 
 Equation  3-2 
Similarly, the coupling capacitance between each patch and its adjacent meander arm 
can be approximated as the geometrical mean of two capacitances as: 
 
42 
(b)
(c)(a)
Periodicity (d)
(d)
P
w
M
e
a
n
de
r
 W
i
dt
h
 
(
M
W
)
M
ean
der
 L
e
ngt
h (
M
L
)
Gap between Meander Lines 
(M
G
)
Co
n
n
e
c
t
i
n
g Br
i
d
ge
 
Len
g
th
 (
M
B
)
Substrate
h
Conductive patch
Ground plane
Conductive Meander Line
?
r
, ?
M
e
a
n
de
r
 W
i
dt
h
 
(
M
W
)
M
ean
der
 L
e
ngt
h (
M
L
)
Co
n
n
e
c
t
i
n
g Br
i
d
ge
 
Len
g
th
 (
M
B
)
M
e
a
n
de
r
 W
i
dt
h
 
(
M
W
)
M
ean
der
 L
e
ngt
h (
M
L
)
Co
n
n
e
c
t
i
n
g Br
i
d
ge
 
Len
g
th
 (
M
B
)
 
Figure  3.12. Planar EBG structure with two-turn meander lines as inductive bridges. 
(a) top view of the unit cell, (b) top view of the meander line bridge, (c) perspective 
view of the total structure, (d) top view of the total structure. The design parameters 
are shown in the diagram. 
 
cppcmmcpm
CCC ??=  
 Equation  3-3 
where C'
cmm
 represents the coupling between two adjacent arms which are spaced 
from each other by a gap equal to M
B
. It is given similar to Equation  3-2 by: 
)(cosh
)(
10
BW
L
cmm
MM
M
C
?
+
=?
?
??
 
 Equation  3-4 
 
43 
Similarly, C'
cpp
 is the coupling between two adjacent patches which are spaced from 
each other by a gap equal to M
B
 and given by: 
)(cosh
)(
10
BW
L
cpp
MP
M
C
?
+
=?
?
??
 
 Equation  3-5 
The total mutual coupling between adjacent arms of meander line can be 
approximated by: ( )12
1
?= NCC
cmm
. Similarly, C
2
 showing the total mutual coupling 
between meander line and two adjacent patches is defined by: 2
2 cpm
CC = . The total 
inductance of meander line is represented by L. The center frequency (resonant 
frequency) of the unit cell of EBG is moved to lower values by increasing the 
effective capacitance (C
eff
) or the effective inductance (L
eff
) according to the 
relationship ( )
effeffc
CLf ?2/1= . L
eff
 is equal to L and ( ) ()
21
||
2
||
2
CC
CC
C
eff
+= . 
Therefore, to have a bandgap at lower frequencies, one needs to increase C
eff
 or L
eff
. 
By inspection of formulas for C, C
cmm
, and C
cpm
, the capacitance is increased by 
increasing the metallic pattern dimensions or the dielectric constant. However, using 
high dielectric constant have the distinct advantage of allowing miniaturization of 
EBG patterns in addition to shifting the bandgap to lower frequencies. In following 
sections, we will discuss about the subject of accurate modeling. It will be shown that 
the effective inductance can be increased by increasing the length of meander line and 
the gap between adjacent arms of meander line. Increasing the gap will reduce the 
magnetic coupling. 
   The progress in material science has provided composite dielectric materials 
with very high dielectric constants (?
r
 ? 100) known as high-k dielectrics. Currently, 
 
44 
several companies are producing these types of materials with low losses. Following, 
we will show that the miniaturization of EBG structures is possible due to availability 
of these commercial high-k materials. Also, we will show advantages of combining 
these high-k materials with high effective inductive loads to have wide suppression 
bands at low frequencies. 
 
(a)
(b)
C
1
C
2
L
2
C
2
C
 
Figure  3.13. Physical-based model of the planar EBG structure. C
1
 represents the 
mutual coupling capacitance between arms of the meander line. C
2
 is the mutual 
coupling capacitance between meander line and two adjacent patches. C is the 
capacitance between the patch and the bottom plane, and L represents the inductance 
of meander line. 
 
3.2.1. Numerical Characterization of Miniaturized-Wideband Planar EBG Structures 
We used commercial software, Ansoft?s HFSS, to validate the improvement 
obtained from the combination of high-k dielectric substrate and meander line bridge 
 
45 
in the proposed planar EBG structure. The EBGs are analyzed up to 14 GHz. If the 
suppression band goes beyond 14 GHz, we have indicated the upper bound of band 
by 14
+
 GHz. The setup used to extract the effective suppression band is illustrated in 
Figure  3.14. The ports defined for this structure are separated by N cells (in our case 
N = 4). Each port is located between the patterned layer and ground. The magnitude 
of the S
21
 parameter (defined between two ports) is indicative of the suppression 
band. Figure  3.15 shows the magnitude of S
21
 for the EBG pattern consisting of two-
turn meanders as shown in Figure  3.12  with the following parameters: P
W
 = 1.4 mm, 
h = 114.3 ?m (4.5 mil), M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.34 mm, M
G 
= 0.02 
mm, ?
r 
= 140, and tan ? = 0.0015. The reference case which is a similar setup without 
EBG pattern is included in the same figure. The effective suppression band 
considered throughout this study is defined to be the frequency band where the 
suppression exceeds 20 dB. For the proposed structures, a very wide suppression 
band (3.5 ? 13.9 GHz) is achieved.  
Few samples are summarized in the Table  3.II to show the effect of design 
parameters on the suppression bandgap. Samples No. 1 to 6 are tabulated in such a 
way that only one design parameter in each sample is different from the previous one. 
From these results, we conclude the following: 1) increasing the dimensions of patch 
shifts the bandgap to lower frequencies. This increment is equivalent to increasing the 
effective capacitive load. 2) The longer meander line which provides larger inductor 
not only shifts the bandgap to lower frequency but also widens the effective 
suppression band. In sample No. 6 compared to sample No. 5, a longer meander line 
with three turns is used. However, the patch size is decreased to keep approximately 
 
46 
the same unit cell size. As can be concluded from results of the suppression band, a 
comparable or even better performance is obtained. Generally, results in Table  3.II 
show the efficiency of using larger inductive load. Comparing samples No. 5 and No. 
7 in this table emphasizes that not only the length of meander line is an important 
factor to clarify the suppression band but also the connection point between the patch 
and the meander line is important too. Sample No. 6 with a margin equal to 0 mm is 
providing larger bandgap than sample No. 7 with a margin equal to 0.1 mm (refer to 
Figure  3.11). All other design parameters are the same for both samples. However, 
the length of meander line in sample No. 5 is slightly longer than the length of 
meander line in sample No. 7. The step discontinuity in microstrip line can be 
modeled by a T-equivalent circuit. The values of lumped elements modeling the 
discontinuities with different margins are different. Later, in modeling section, we 
will include the discontinuity effect in our modeling. 
 
 
Port 1
Port 2
 
Figure  3.14. Schematic showing the simulation setup and location of the ports to 
extract the suppression band of the EBG structure from computed S
21. 
 
47 
0 2 4 6 8 10 12 14
-160
-140
-120
-100
-80
-60
-40
-20
0
20
 Without EBG Pattern (Reference)
 With EBG Pattern
Ma
gnitu
de of 
S
21
 (dB)
Frequency (GHz)
20 dB Suppression Band
Ma
gnitu
de of 
S
21
 (dB)
 
Figure  3.15. Magnitude of S
21
 versus frequency for the EBG structure shown in 
Figure  3.12 using the setup shown in Figure  3.14. The parameters of the EBG 
structure are: P
W
 = 1.4 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.34 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015 (design parameters are shown in 
Figure  3.12). The reference case which is parallel plane is included for the purpose of 
comparison. 
 
 Another set of samples consisting of two-turn meanders are listed in Table 
 3.III. These samples have been chosen in such a way that their suppression band 
covers the operating frequency of current microprocessors and resonant frequencies 
of the package. The data in this table clearly shows the advantage of using high 
permittivity dielectrics compared to conventional low ones (here FR-4) in reducing 
size of unit cell of EBGs. By comparing these cases, it is concluded that a size 
 
48 
reduction of more than 10 times is achieved. Also, we could approximately keep a 
similar wide suppression band. The effective wide suppression band in proposed 
planar EBGs can be easily tuned to filter EMI and electromagnetic noises in EMC 
applications. The sample designs provided here shows efficacy of combining very 
high-permittivity materials with inductive bridges in design of miniaturized wideband 
planar EBG structures. 
Generally, the design of samples provided in this section has not been 
optimized. Optimizing inductive part of the design to provide more inductive load in 
small area can lead to a wider band in addition to smaller unit cell size. In the 
following section, we will continue our study on the modeling and the 
characterization of the proposed EBG structure proposed in this section. 
 
 
Table  3.II Specifications of the different planar EBG patterns and related suppression 
band (design parameters are shown in Figure  3.12). For all samples h = 114.3 ?m, ?
r 
= 140, and tan ? = 0.0015. 
3.85-14.11.580.021.240.020.021.42
(margin = 0.1 mm)
7
3.5-14
+
1.560.021.240.020.021.336
3.45-141.580.021.240.020.021.425
3.5-13.91.580.021.340.020.021.424
3.7-11.31.50.021.340.020.021.413
4.9-9.71.50.020.440.020.021.412
7.9-14
+
1.10.020.440.020.02111
Suppression 
Band (GHz)
d
(mm)
M
G
(mm)
M
L
(mm)
M
W
(mm)
M
B
(mm)
P
W
(mm)
# Turns
N 
Sample 
N0.
+
shows suppression band extends beyond 14 GHz.
 
 
49 
Table  3.III. Specifications of the different planar EBG patterns and related 
suppression band (design parameters are shown in Figure  3.12). For all samples h = 
114.3 ?m and N = 2. 
1.9-10.81.780.021.60.020.021.60.00152007
3-11.31.380.021.20.020.021.20.00153006
3-11.31.580.021.40.020.021.40.00152005
3.4-11.51.380.021.20.020.021.20.00152004
2.9-12.51.780.021.60.020.021.60.00151403
3.5-13.91.580.021.340.020.021.40.00151402
1.3-13.821.80.2200.20.2200.024.31
Suppression 
Band (GHz)
T
(mm)
M
G
(mm)
M
L
(mm)
M
W
(mm)
M
B
(mm)
P
W
(mm)
tan ??
r
Sample 
N0.
 
 
3.3. Modeling of the Proposed Planar EBG Structures 
In the last few years, different methods have been used to characterize the 
EBG structures. The characterization is satisfied by relating the bandgap and center 
frequency to constituent materials and dimensions of inclusions. These methods 
include: 
? Numerical analysis using various methods such as the Finite Elements Method 
(FEM), The Finite-Difference Time-domain (FDTD) method, ?etc.  
? Measurements 
? Equivalent circuit modeling  [20],  [26],  [33],  [34],  [41] 
? Equivalent circuit-transmission line models  [35],  [40],  [44] 
Among the characterization methods, experimental methods consisting of fabrication 
of structures followed by measurements is favored due to its accuracy, but it is costly 
and time consuming. The full-wave numerical analysis such as commercial software 
 
50 
Ansoft?s HFSS based on FEM, provides accurate results, however, at the cost of 
expensive computations in terms of time and memory. More often one encounters 
convergence or memory problem due to large number of meshes in the computational 
domain. The analysis using methods which are not full-wave based may compromise 
between accuracy, time, and cost. To use time and resources efficiently, several 
methods based on analyzing infinite periodic structures have been developed. These 
methods approximately characterize the required structures in a defined application 
based on analyzing a unit cell of the structure and implementing proper periodic 
boundary condition based on Floquet theorem  [45]. Simplified setups decrease 
simulation time significantly. However, they are not still efficient to characterize a 
large number of structures to prepare data tables which may be used as guide for 
designing new EBG structures for the required operating regimes. Model-based 
methods were developed as a solution to these drawbacks. Generally, model-based 
methods give a qualitative perspective about the relationship between the bandgap 
and the topology and the composition of EBG structures. However, deriving a model 
which may predict accurately the dispersive effects through the entire range of 
frequencies is not viable, because of several EM effects that become strong at the 
higher frequencies and are typically not included in the models. Therefore, for 
accurate designs, further tunings and adjustments of the specifications of EBG 
structures are necessary using full-wave numerical analysis methods. As a result, the 
combination of two or more of the above mentioned methods are used to design new 
EBGs using trial and error procedure. A good initial guess using models, may 
decrease the load of work needed.  
 
51 
In  [26], a simple lumped element model for the mushroom EBG structures 
was proposed. This model approximately works for normal wave incidence at low 
frequencies where the dimensions of the EBG are much smaller than the wavelength 
in media. Later, a model based on both transmission line theory and circuit elements 
was developed  [44]. This model partially overcoming previous limitation can predict 
the bandgap with higher accuracy. Authors in  [20],  [34] developed a method for 
extracting the parameters of the model for a mushroom-type EBG embedded in 
parallel plate waveguide. This method uses simple formulas to derive initially the 
value of the lumped elements in the model and later uses a numerical algorithm for 
curve fitting on the S-parameters graph. This method usually is not efficient for 
characterization and designing of new EBG structures. Mostly, this method could be 
used for post analyzing, such as computing the delay time for the logic change in 
digital circuits connected to power planes or computing reservoir capacitors needed in 
different locations in power planes to improve the functioning of electronic circuits. 
In  [41], a physical-based equivalent circuit model for mushroom EBGs embedded in 
parallel plate waveguide was derived. This model predicts only the center frequency 
of bandgap. Recently, authors in  [35],  [40] used the transmission line method to 
improve the modeling of embedded EBGs between parallel plate waveguide. Those 
models can predict the bandgap as well as the center frequency. Following, we are 
going to derive a model for planar EBG proposed in this study to get idea about the 
relation of bandgap to the constitutive materials and their dimensions. In Section  3.2, 
a physical-based equivalent circuit model was introduced to justify the advantage of 
using high-k materials in design of miniaturized planar EBG structure operating at 
 
52 
low frequencies. In this section, we derive an equivalent circuit-transmission line 
model which is more accurate in providing information about bandgaps.  
 
3.3.1. Setup and Model Formulation  
Detailed analysis of periodic structures based on transmission line theory and 
Floquet theorem is given in  [45],  [46]. A unit cell of planar EBG structure is shown in  
Figure  3.12. The computational domain and the boundary setup for extracting 
dispersion diagram for 2-D propagation is shown in Figure  3.16. A pairs of PBC 
setup for any two side walls in front of each other resembles the periodicity in that 
direction.  To extract a 1-D equivalent circuit-transmission line model for a wave 
propagating through EBG patterns in x-direction with xkk
x
?= , the infinite 2-D array 
of EBG structures shown in Figure  3.12 may be reduced to a 1-D infinite array by 
applying perfect magnetic conducting (PMC) boundaries on both sides of the array as 
shown in Figure  3.17. Therefore, H-field is equal to zero on the sides of array. With 
this assumption, we have ignored the side coupling effects. A unit cell of the 
proposed planar EBG structure is shown in Figure  3.18. The setup shows the 
computational domain and the boundary setup which can be used to extract the 
bandgap using the dispersion diagram. This is the basic setup which will be used in 
later subsections for both the numerical full-wave analysis and the modeling for 1-D 
propagation (?-? region in the Brillouin zone).  To prevent confusion, it should be 
mentioned that if we consider periodic boundary instead of PMC boundary in Figure 
 3.18 then dispersion in ?-? and ?-? regions in the Brillouin zone can be extrated. 
Results characterize new 2-D structure where in one direction its patches are 
 
53 
connected to each other through a meander line and in the other direction there is a 
gap between the two adjacent patches. 
The frequency independent lumped elements used in the model represent the 
meander line, the patch, the mutual coupling between the meander line and the patch, 
and the step discontinuity between the patch and the meander line. The transmission 
lines connecting lumped elements are known to transfer impedance and to model the 
phase shift increasing with frequency. The transfer ABCD matrix presenting a unit 
cell of the EBG modeled with lumped elements and transmission lines is: 
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
mm
mm
DC
BA
DC
BA
 
 Equation  3-6 
Considering the infinite array of EBG structure as a transmission line made of 
engineered materials with the effective propagation constant k
x
 and the effective 
characteristic impedance Z
0,eff
, the ABCD parameters for this transmission line 
corresponding to a unit cell of EBG are presented by: 
?
?
?
?
?
?
?
=
?
?
?
?
?
?
)cos()sin(
)sin()cos(
,0
,0
dkdkjY
dkjZdk
DC
BA
xxeff
xeffx
 
 Equation  3-7 
where d is the size of a unit cell (periodicity length). By equating Equation  3-6 and 
Equation  3-7 the dispersive behavior of structure is predicted. In the following 
subsections, we introduce the relationship between physical parameters of the 
structure and the lumped elements, and compute the elements of the transfer matrix 
A
m
, B
m
, C
m
, and D
m
. Without loss of generalization, the metallization thickness t and 
the metallic and dielectric losses are neglected. 
 
54 
 
(a) (b)
PBC boundary
PBC boundary
PML boundary
 
Figure  3.16. A unit cell of the planar EBG structure with two-turn meander bridges. 
(a) Setup used for extracting the bandgap of EBG structure in 2-D propagation using 
dispersion diagram. The boundary setup is shown in the figure. (b) Top view of EBG 
structure. 
 
 
x
z
y
k
x
E
H
?
???
???
Structure is repeating 
to infinity
Structure is repeating 
to infinity
PMC Boundary
 
Figure  3.17. Infinite 1-D array of the proposed planar EBG. 
 
 
55 
 
(a) (b)
PBC boundary
PBC boundary
PMC boundary
PML boundary
 
Figure  3.18. A unit cell of the planar EBG structure with two-turn meander bridge. (a) 
Setup used for extracting the bandgap of EBG structure in 1-D propagation using 
dispersion diagram. The boundary setup is shown in figure. (b) Top view of EBG 
structure. If PMC boundaries are replaced by a pair of PBC boundaries it resembles 
2-D propagation which is different from Figure  3.16. 
 
 
3.3.2. Characterization of EBG using Full-Wave Simulator 
In this subsection to assess the performance of the model that will be 
developed in the following subsections, we are extracting the dispersion diagram of a 
few selective planar EBG structures using the setup shown in Figure  3.18. We use 
 
56 
full-wave numerical simulator Ansoft?s HFSS and the irreducible Brillouin zone for 
this purpose. The selected samples are covering the combination of different 
specifications for the dielectric substrate such as high or low relative permittivity and 
thin or thick dielectric thickness. The first sample which is made of thin slab of high-
k material has the following parameters: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. The 
dispersion diagram for this sample is shown in Figure  3.19.  The first three modes and 
the bandgaps in between are clarified in the figure. Also, the dispersion diagram of 
this sample for the unit cell shown in Figure  3.16 is plotted in Figure  3.20. 
Comparison of results in Figure  3.19 and Figure  3.20 shows that the first and second 
modes are similar in both structures. However, there is a degenerate or very close 
mode to the second propagating mode in the structure of Figure  3.16. Again the third 
mode in Figure  3.18 is compatible with the fourth mode of Figure  3.16. (This way of 
mapping between modes of two unit cells is seen in the other chosen samples. The 
samples will be introduced in the following paragraphs).  
The second sample which is made of thicker slab with the same permittivity is 
specified with the following design parameters: P
W
 = 1.6 mm, h = 500 ?m, M
W 
= 0.02 
mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
Figure  3.21 shows the dispersion diagram for the second sample. The first three 
modes and the bandgaps in between are clarified in the figure. The third case study is 
made of a thin slab with higher permittivity. This sample is specified by: P
W
 = 1.6 
mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 
200, and tan ? = 0.0015. The dispersion diagram including the first three propagating 
 
57 
modes is shown in Figure  3.22. The bandgaps in between of those modes are marked 
in the figure. The fourth sample is made of thin slab of low permittivity material. The 
design parameters for this sample are as: P
W
 = 20 mm, h = 114.3 ?m, M
W 
= 0.2 mm, 
M
B 
= 0.2 mm, M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and tan ? = 0.02. The dispersion 
diagram for this case study is shown in Figure  3.23. The first three propagating modes 
and the bandgaps in between of these modes are clarified in Figure  3.23. 
The small sudden variations seen in the graphs of propagating modes such as 
zigzag changes are numerical errors. Later, we will return back to these graphs and 
continue our studies with curve fitting on the f-?d points in the dispersion diagrams to 
characterize performance of EBG structures in detail. In the following subsections, 
we continue the modeling discussion. 
 
 
 
 
 
 
 
 
 
 
 
 
 
58 
 
 
 
  
Figure  3.19. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and 
tan ? = 0.0015. 
 
 
 
 
 
 
0
1
2
3
4
5
6
MX? ?
Bandgap
 First propagating mode
 Second propagating mode
 Third propagating mode
Frequency
 (
G
H
z
)
phase (?d)
 
59 
 
 
 
 
Figure  3.20. Dispersion diagram of the planar EBG structure shown in Figure  3.16 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and 
tan ? = 0.0015. 
 
 
 
 
 
 
0
1
2
3
4
5
6
X ?? M
Bandgap
 First propagating mode
 Second propagating mode
 Third propagating mode
 Fourth propagating mode
Fr
equ
e
ncy
 (
G
Hz)
phase (?d)
 
60 
 
 
 
 
 
Figure  3.21. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 500 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? 
= 0.0015. 
 
 
 
 
 
0
1
2
3
4
5
? ?MX
Bandgap
 First propagating mode
 Second propagating mode
 Third propagating mode
Fr
e
q
ue
nc
y (
GHz)
phase (?d)
 
61 
 
 
 
 
 
Figure  3.22. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 200, and 
tan ? = 0.0015. 
 
 
 
0
1
2
3
4
5
?? MX
Bandgap
 First propagating mode
 Second propagating mode
 Third propagating mode
Freq
uency
 (
G
Hz)
phase (?d)
 
62 
 
Figure  3.23. Dispersion diagram of the planar EBG structure shown in Figure  3.18 
extracted using HFSS.  The design parameters of the EBG are: P
W
 = 20 mm, h = 
114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 mm, M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and tan 
? = 0.02. 
 
3.3.3. Model of Patch 
The characteristic impedances of the patch and the meander line are denoted 
by Z
P
 and Z
M
, respectively (Formulas of the characteristic impedance and the 
effective relative dielectric permittivity for microstrip line are provided in Appendix). 
Z
P
 is much smaller than Z
M
 as the width of patch microstrip is much wider than the 
width of the meander line microstrip. At low frequencies, patch microstrip line 
behaves as an electrically short length line which has low impedance compare to 
adjacent meanders. For that scenario, the patch is modeled by a T-equivalent circuit 
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
? ?MX
Bandgap
 First propagating mode
 Second propagating mode
 Third propagating mode
Frequency (
G
Hz)
phase (?d)
 
63 
as shown in Figure  3.24  [47]. The series reactance and shunt susceptance are given 
by: 
cZ
P
C
P
Z
B
P
rePW
PH
g
W
P
?
?
?
??
?
?
?
?
?
?
?
?
?
21
 
 Equation  3-8 
c
ZP
L
P
Z
X PrePW
PH
g
W
P
22
?
?
?
??
?
?
?
?
?
?
?
?
?  
 Equation  3-9 
where P
W
 is the length of square patch, ?
g
 is the guided wavelength in dielectric 
substrate, c is the speed of light in free space and ?
reP
 is the effective relative 
permittivity of dielectric substrate. If Z
P
 << Z
M
 then the patch may be approximated 
only by capacitance C
PH
 at low frequencies and the series inductance L
PH
 could be 
neglected. 
 
(b)(a)
Z
P
Z
M
Z
M
P
W
TT'
C
PH
L
PH
L
PH
TT'
j
X
2
j?
j
X
2
 
Figure  3.24. a) Patch of the planar EBG pattern which has short electrical length and 
low impedance. b) Equivalent lumped-element circuit. 
 
64 
3.3.4. Model of Patch-Meander Line Step Discontinuity  
The discontinuity between the patch and the meander line in our planar EBG 
design is asymmetrical as shown in Figure  3.25 (a). Here, the meander line is 
connected to the patch at the edge (margin = 0). P
W
 is the width of patch and M
W
 is 
the width of the meander line. The T-equivalent circuit modeling the discontinuity is 
shown in Figure  3.25 (b). This type of discontinuity can be modeled approximately as 
follows. If we ignore the fringing field-effect along the side where the patch is 
uniformly connected to the meander then we can place PMC boundary (or H-
symmetry boundary) along that side. With this assumption, we can replace the 
asymmetrical discontinuity with a symmetrical one as shown in Figure  3.26. The 
width of microstrip lines in the new step discontinuity is double the widths in original 
structure: W
1
 = 2P
W
 and W
2
 = 2M
W
. The elements modeling the symmetrical 
discontinuity are given by  [47],  [48]: 
L
LL
L
L
WW
W
21
1
1
+
=  
 Equation  3-10 
L
LL
L
L
WW
W
21
2
2
+
=  
 Equation  3-11 
where the inductance per unit length in the microstrip line L
wi
 (i = 1, 2) is given by: 
c
Z
L
reiCi
Wi
?
=  
 Equation  3-12 
 
65 
2
2
1
2
1
1000987.0
?
?
?
?
?
?
?
?
?=
re
re
C
C
Z
Z
hL
?
?
 
 Equation  3-13 
Z
Ci
 and ?
rei
 are the characteristic impedance and the effective relative permittivity of 
microstrip line with strip-width W
i
, respectively. h is the dielectric thickness in micro 
meter. The capacitance is given by: 
?
?
?
?
?
?
?
?
?
?
+
+
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?=
8.0
264.0
258.0
3.0
100137.0
1
1
1
1
1
2
1
1
h
W
h
W
W
W
Z
hC
re
re
C
re
?
?
?
 
 Equation  3-14 
C is in pF, L and L
i
 are in nH. The lumped elements of the model in Figure  3.25 and 
Figure  3.26 are mapped as follows: C
D
 = C, L
DP
 = L
1
, and L
DM
 = L
2
. In this 
approximation, the value of capacitance C
D 
is almost twice the value of capacitance 
for symmetrical step discontinuity with the same dimensions as the original 
asymmetrical one  [49],  [50]. 
It is to be noted that in this study we have not considered any case study with 
the general form of asymmetric step discontinuity where microstrips are connected 
with some margin from the sides. However, we propose to approximate the value of 
the capacitance and the inductance as the geometric mean of related values for two 
symmetric cases as shown in Figure  3.27. The accuracy of this proposal has not been 
verified and it remains as a point of investigation in future work. 
 
 
 
 
66 
 
(a) (b)
C
D
L
DP
L
DM
TTT
P
W
M
W
 
Figure  3.25. (a) Asymmetrical discontinuity between patch and meander line with 
margin = 0 (refer to Figure  3.11). (b) Equivalent lumped- element circuit. 
 
 
(a) (b)
C
L
1
L
2
TTT
W
1 W
2
 
Figure  3.26. (a) Symmetrical discontinuity between microstrip lines. (b) Equivalent 
lumped- element circuit. 
 
 
67 
(a)
(b)
T
P
W
M
W
W'
1
W'
2
M
W
W'
1
W'
1
T'
M
W
W'
2
W'
2
T''
D2D1D
CCC =
(c)
DP2DP1DP
LLL =
DM2DM1DM
LLL =
 
Figure  3.27. (a) Asymmetrical discontinuity between microstrip lines with margin ? 0 
(refer to Figure  3.11). (b) and (c) Symmetrical discontinuity between microstrip lines. 
Design parameters are shown in the figure. (C
D
, L
DP
, L
DM
), (C
D1
, L
DP1
, L
DM1
), and 
(C
D2
, L
DP2
, L
DM2
) show the elements modeling the step discontinuity at the 
connections (a), (b), and (c), respectively. The equivalent circuit model is shown in 
Figure  3.25 (b). 
 
3.3.5. Model of Meander Line  
A 3-D view of a planar meander line and its design parameters is shown in 
Figure  3.28. The meander line which is placed on the top of metal backed dielectric 
substrate consists of N metallic arms and (N + 1) metallic bridges connecting arms. 
The meander line in Figure  3.28 has 6 arms and 7 bridges. For simplicity of study, it 
 
68 
is assumed that all traces in the meander have same width, M
W
; separation gap 
between two adjacent arms, M
G
; arm length, M
L
; input and output connecting bridges? 
length, M
Bi 
= M
Bo 
= M
B
; and input and output arms? length, M
Li
 = M
Lo
 = M
L
. However, 
the introduced models are general and easily can be applied to any configuration. The 
thickness and relative permittivity of dielectric substrate are h and ?
r
, respectively. 
The ?-equivalent circuit model of meander line is shown in Figure  3.29.  The inductor 
is the dominant element in this model up to the first self resonant frequency. The 
capacitors model the parasitic effects. It should be noted, in this approximate 
modeling, we have not taken into account the effect of right angle bends in meander 
line which becomes important when a rigorous model is required at higher 
frequencies. 
 
M
L
i
M
L
M
G
M
W
M
Bi
?
r
h
M
L
o
M
Bo
Arms
Bridges
 
Figure  3.28. Three-dimensional view of meander line inductor. Design parameters are 
included in the figure.   
 
 
69 
C
L2
C
L2
C
L1
L
 
Figure  3.29. Equivalent circuit model of meander line.   
 
3.3.5.A. Computation of Inductance  
In this subsection, we provide a formula for meander line inductance based on 
the Greenhouse formulas  [51]. The expression for inductance includes the self 
inductance of all sections in the inductor pattern (total of N arms and (N+1) bridges), 
the mutual inductance between all magnetically coupled arms and bridges, and the 
effect of ground. It is assumed that the meander line is electrically short (the total 
length of meander line is much less than the wavelength at the maximum operating 
frequency in the system). The root of this assumption is in the original formulation 
which was done assuming the magnitude and phase of current are constant across the 
line. Later, we will discuss about the effects related to phase changes along the 
electrically long lines. In the meander line, the mutual inductance between adjacent 
arms is negative due to opposite directions of current. The mutual inductance between 
arms separated by another arm is positive as the direction of current-flow is similar in 
both. Figure  3.30 (a) shows magnetic flux lines for two lines carrying current in the 
same direction providing positive mutual inductance. Figure  3.30 (b) shows magnetic 
flux lines for two lines carrying current in opposite directions providing negative 
 
70 
mutual inductance. Therefore, in the meander line the mutual inductance between one 
arm and all the magnetically coupled arms are sequentially alternating between 
negative and positive. The coupled arms are marked by solid line in Figure  3.31. The 
mutual inductance between one bridge and all the magnetically coupled bridges are 
positive. In this study, as a first-order approximation, we have considered only the 
coupling between adjacent bridges indicated by dashed marker in Figure  3.31. 
Coupling between other pairs of bridges are negligible as there is not any overlap 
along the traces as well as the increased gap between them. The total inductance of 
meander line pattern on top of thick dielectric or suspended in air (we have not 
discussed the effect of presence of ground yet) is given by: 
()
?? ??
?
=
?
==
+?+
+
=
+?+=
1
11 1
12,12)(2,2
12
1
212
N
j
jN
i
N
i
iijii
j
N
i
iP
MMLL  
 Equation  3-15 
where the first sum represents the total self inductance, the second sum gives the total 
mutual inductance between coupled arms, and the third sum gives the total mutual 
inductance between adjacent bridges. We intend to use the developed model in the 
design of EBG structures with miniaturized dimensions, where the small separations 
between the arms and the long overlap length between arms can produce considerable 
increase in the total mutual coupling. Therefore, to develop a highly accurate model, 
it is better not to approximate the term of mutual coupling between arms by a first-
order representation.  
 
 
 
 
71 
 
+?
(a) (b)
??
 
Figure  3.30. Magnetic flux lines showing the magnetic coupling between two 
adjacent lines for two cases: (a) positive mutual inductance and (b) negative mutual 
inductance. 
 
 
i=1
i=
2
i=5
i=
6
i=
4
i=3
i=2N+1
i=
2N
i=2N-1
???
i=
2N
-
2M
1,3
M
5,6
M
3,
5
M
2N
-
1
,2N
+
1
i=6
M
2,2N
M
2,2N-2
M
2,6
M
2,4
M
4,2N
M
4,2N-2
M
4,6
M
6,2N
M
6,2N-2
i=
2
i=
6
i=
4
i=
2N
i=
2N
-
2M
1,3
M
5,6
M
3,
5
M
2N
-
1
,2N
+
1
 
Figure  3.31. Magnetically coupled traces in a meander line inductor. The mutual 
coupling between pairs of arms and bridges is marked with solid lines and dashed 
lines, respectively. 
 
 
72 
From topology and composition of the structure, the inductances can be 
defined. The self inductance of ith section is given by  [52],  [53]: 
?
?
?
?
?
?
?
?
+
++
?
?
?
?
?
?
?
?
+
?=
?
i
W
W
i
ii
l
tM
tM
l
lL 2235.0193.1ln102
4
 
 Equation  3-16 
where it is assumed t = 0 and l
i
 is the length of line. All dimensions are in ?m and L
i
 
is in nH. The mutual inductance between two parallel filaments which are completely 
overlapping, as shown in Figure  3.32 (a), with length m and geometric mean distance 
d
GMD
 between them is computed by  [51],  [53],  [54]: 
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++?=
?
m
d
m
d
d
m
d
m
mM
GMDGMD
GMD
GMD
dm
GMD
2/1
2
2
2/1
2
2
4,
11ln102  
 Equation  3-17 
where all dimensions are in ?m and M
 
is in nH. The geometric mean distance 
between parallel strips in the meander which are distant from center to center by d
ave
 
is calculated as  [51],  [54]: 
()()
()()()()()?
?
?
?
?
?
?
?
+++++
?=
L
108642
660
1
360
1
168
1
60
1
12
1
lnln
WaveWaveWaveWaveWave
aveGMD
MdMdMdMdMd
dd
 
 Equation  3-18 
 
To compute the mutual coupling between parallel strips, we need to consider different 
configurations  [54] including: 1) the case without any overlap shown in Figure  3.32 
(b). For that case, the mutual inductance, M, is given by: 
 ( ) ( )
GMDGMDGMDGMD
dpndpmdpdpnm
MMMMM
,,,,
2
++++
+?+=  
 
73 
 Equation  3-19 
If the length p is equal to zero as shown in Figure  3.32 (c) then M is obtained from the 
relation: 
( )
GMDGMDGMD
dndmdnm
MMMM
,,,
2 +?=
+
 
 Equation  3-20 
2) The case with partial overlap shown in Figure  3.32 (d) where M is given by: 
( ) ( )
GMDGMDGMDGMD
dpndpmdpdpnm
MMMMM
,,,,
2
???+
+?+=  
 Equation  3-21 
3) The case with overlap but unequal strip-lengths as shown in Figure  3.32 (e). M is 
computed as: 
( ) ( )
GMDGMDGMDGMD
dqdpdqmdpm
MMMMM
,,,,
2 +?+=
++
 
 Equation  3-22 
If the length p is equal to zero as shown in Figure  3.32 (f) then M is given by: 
( )
GMDGMDGMD
dqdndm
MMMM
,,,
2 ?+=  
 Equation  3-23 
In those examples we will study later to asses our modeling due to choice of equal 
arms along the meander, only cases (a) and (c) in Figure  3.32 will be used. However, 
for general modeling of a meander line with arbitrary shape all the cases mentioned 
above are needed. 
 
 
74 
n
m
d
ave
p
n
m
d
ave
p
(b)
(d)
n
m
d
avepq
(c)
m
n
d
ave
(e)
n
m
d
ave
q
(f)
m
d
ave
(a)
 
Figure  3.32. Magnetically coupled strips. Design parameters for computing mutual 
coupling are shown in the figure. 
 
In the following, we apply the image theory to include the effect of ground 
plane on the total inductance of the meander line  [55]. It is very important to consider 
this effect because of the trends of size shrinking as followed in the design of 
electronics which are affecting design architectures. Figure  3.33 shows a meander 
line and its image modeling the effect of ground plane. The image is located at a 
distance equal to twice the substrate-thickness from the actual meander location. The 
current in the image meander is in the reverse direction leading to a total negative 
mutual inductance represented by L
MIP
. The mutual inductance between the 
magnetically coupled sections in the actual and the image meanders are computed 
through the formulas stated earlier. In those formulas, the geometric mean distance is 
 
75 
approximated by the distance between centers of strips. The decrement of the 
dielectric thickness leading to closer location of the ground plane to the meander 
pattern causes the total value of inductance to decrease. Therefore, the total net 
inductance of meander line is given by: 
MIPP
LLL +=  
 Equation  3-24 
where the mutual inductance of image pattern is computed by: 
()
()
?
?
?+
=
=
++
=
+
?
?
?
?
?
?
?
?
?
=
12
1
1
2,
)22(5.0
0
12,
even is  f1
odd is  f1
N
j
N
i
ij
ij
N
i
ij
j
MIP
jiM
jiM
L  
 Equation  3-25 
It is noted that the mutual inductance between image and actual meanders does not 
need to be considered twice as the image pattern is not part of the actual one.  
 
h
h
I
I
Actual Meander Line Pattern
Image Meander Line Pattern
L
o
c
a
t
i
on
 
o
f
 G
r
o
u
nd
 
P
la
n
e
i=1
i
=
2
i=3
j=1
j
=
2
j=3
i
=
4
j
=
4
 
Figure  3.33. Image of meander line by ground plane. 
 
76 
3.3.5.B. Computation of Parasitic Capacitances  
There are two types of parasitic capacitances in meander line model, C
L1
 and 
C
L2
 as shown in Figure  3.29. C
L1
 representing the parasitic capacitance in the meander 
pattern consists of different interline capacitances such as C
in
, C
top
, and C
bottom
 
between adjacent arms and C
top
, C
bottom
, and C
side
 between adjacent bridges as shown 
in Figure  3.34. In this modeling, due to our assumption of zero metallization 
thickness, t = 0, C
in
 and C
side
 are negligible. Also, C
top
 and C
bottom
 between arms are 
much greater than C
top
 and C
bottom
 between bridges in our meander designs where we 
intentionally want large value of inductance per area. Therefore, the parasitic interline 
capacitance between two coupled adjacent arms j and j+1 is given by: C
g_j
 = C
top_arm_j
 
+ C
bottom_arm_j
 or ( )
jgdjgajgjg
CClC
____
5.0 += . l
g_j
 representing the effective coupled 
length between two arms is approximated by: l
g_j
 = Minimum(l
j
, l
j+1
). 0.5 ? C
ga_j
 and 
0.5 ? C
gd_j
 are the total fringing capacitance per unit length in gap between two arms 
through air and dielectric, respectively. Regarding our assumption that the strip width 
is constant along the meander line, two adjacent arms in meander pattern are 
symmetric coupled lines. Also, in this modeling we are approximating the parasitic 
capacitance by considering only the coupling effect between adjacent arms. In 
another word, the coupling between the non-adjacent arms as well as the coupling 
between the patch and the non-adjacent arms is neglected. Therefore, due to the 
assumption of equal gap between arms M
G
, C
ga
 = C
ga_j
 and C
gd
 = C
gd_j 
for 1 ? j ? N-1.  
Later in this subsubsection, we will provide formulas to compute C
ga
 and C
gd
. Now, 
C
L1
 which is the equivalent capacitance resulting from series of gap capacitances 
C
g_j
?s  in (N-1) consecutive inter-arm gaps is computed by: 
 
77 
?
?
=
=
1
1
_
1
1
1
N
j
jg
L
C
C  
 Equation  3-26 
As it is assumed that all arms are equal in length, C
L1 
is given by: 
1
_
1
?
=
N
C
C
jg
L
 
 Equation  3-27 
 
C
br
idge
_bot
tom
C
arm
_b
ott
om
C
arm_in
C
bridge_side
C
arm_top
C
b
ri
d
g
e
_
to
p
 
Figure  3.34. Parasitic capacitances in the meander pattern. 
 
 
C
L2 
represents half of the parasitic capacitance between the meander line strip 
and the ground plane. To compute C
L2
, we have to consider bridge-sections and arm-
sections separately due to the different coupling effects. The capacitance per unit 
length between the strip and the ground in bridge-sections which are uncoupled 
microstrip lines is given by  [53]: 
 
78 
M
reM
GD
cZ
C
?
=  
 Equation  3-28 
where ?
reM
 is the effective relative permittivity of the meander microstrip. C
GD
 is the 
result of parallel combination of three capacitances: 
FPGD
CCC 2+=  as shown in 
Figure  3.35. C
p
, the per unit length parallel plate capacitance between the strip and 
the ground plane is given by: 
h
M
C
WrM
P
??
0
=  
 Equation  3-29 
Where ?
rM
 is the relative permittivity of the meander microstrip. The fringing 
capacitance per unit length between the uncoupled strip and the ground is given by: 
?
?
?
?
?
?
?
?
?=
P
M
reM
F
C
cZ
C
?
5.0  
 Equation  3-30 
For two symmetrically coupled microstrip lines, the capacitance per unit 
length between the strip and the ground is defined by  [47],  [48]: 
 
 
C
P
C
F
C
F
 
Figure  3.35. Parasitic capacitances between the strip and the ground plane in a 
microstrip line. 
 
79 
eGD
CC =  
 Equation  3-31 
The even mode capacitance per unit length, C
e
, is given by 
FFPe
CCCC ?++=  as 
shown in Figure  3.36 (a). It is worth to mention that the interline mutual capacitance 
between symmetric lines discussed earlier in this subsubsection can also be defined in 
terms of the odd and the even mode capacitances by ()
eojgjg
CClC ?=
__
5.0 . C
o
, the 
odd mode capacitance per unit length is given by 
gdgaFFPo
CCCCCC ++?++=   [47], 
 [48], as shown in Figure  3.36 (b). C'
F
, the fringing capacitance between the meander-
strip and the ground in presence of another similar strip is given by  [48]: 
4
1
10
tanh1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
=?
reM
rM
G
G
F
F
h
M
M
h
A
C
C
?
?
 
 Equation  3-32 
where 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??=
h
M
A
W
5.133.2exp1.0exp .  C
gd
, the fringing capacitance per unit length 
between two symmetrical coupled microstrip lines through the dielectric gap is given 
by  [48]: 
()
esosgd
CCC ?= 5.0  
 Equation  3-33 
where C
es
 and C
os
 are representing the even and the odd mode capacitances per unit 
length in the symmetrically coupled strip lines, respectively. A pair of symmetrically 
coupled strip lines and the representing parasitic capacitances in the configuration are 
shown in Figure  3.37. The even and the odd capacitances for the configuration of two 
symmetrically coupled arm-strips are given by  [48]: 
 
80 
()
()
oex
kK
kK
C
x
x
rMxs
 ,       4
0
=
?
= ??  
 Equation  3-34 
K(k
x
) and K(k'
x
) are the elliptic function and its complement, respectively. The 
functional ratio K(k
x
)/K(k'
x
) is approximated by  [48],  [56]: 
()
()
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
??
?+
??
?
?
?
?
?
?
?
?
?
+
=
?
5.00 f
1
1
2ln
15.0 f
1
1
2ln
1
2
2
x
x
x
x
x
x
x
x
k i
k
k
k i
k
k
kK
kK
?
?
 
 Equation  3-35 
The argument k
x
 is given by  [48]: 
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
? +
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
? +
?
?
?
?
?
?
?
?
=
ex
h
MM
h
M
ox
h
MM
h
M
k
WGW
WGW
x
 if
4
tanh
4
tanh
 if
4
coth
4
tanh
??
??
 
 Equation  3-36 
 
H-Symmetry Wall
(a)
C
PC
F
C'
F
C
P
C
F
C'
F
+++ +++
E-Symmetry Wall
(b)
C
ga
C
PC
F
C'
F
C
P
C
F
C'
F
C
gd
C
ga
C
gd
+++ - - -
 
Figure  3.36. Constituents of (a) the even mode capacitance and (b) the odd mode 
capacitance in a pair of symmetric coupled microstrip lines. 
 
 
81 
C
PsC
Fs
C'
Fs
C
Ps
C
Fs
C'
Fs
C
gd
C
gd
C
Ps
C
Fs
C'
Fs
C
Ps
C
Fs
C'
Fs
C
gd
C
gd
 
Figure  3.37. Parasitic capacitances for a pair of symmetric coupled strip lines. 
 
where 
2
1
xx
kk ?=? . C
ga
, the fringing capacitance per unit length between two 
symmetrical coupled microstrip lines through the air gap is given by  [48],  [57]: 
F
a
cpsga
CCC ??= 5.0  
 Equation  3-37 
where C?
cps
 represents the fringing capacitance per unit length between the symmetric 
coupled coplanar lines when the dielectric material is air. The width of strips and the 
separation between them are similar to the arm-strips in the original meander 
structure and they are equal to M
W
 and M
G
, respectively. C?
cps
 is given by  [48]: 
()
()kK
kK
C
a
cps
?
=
0
2?  
 Equation  3-38 
where 
WG
G
MM
M
k
2+
=  and 
2
1 kk ?=? . ?C
F 
is the fringing capacitance per unit length 
in the air not in the air gap between the two symmetric coupled arm-strips. It is 
approximated by  [57]: 
( ) ( )
a
Fs
a
F
a
Fs
a
FF
CCCCC ???+?=?  
 Equation  3-39 
 
82 
a
F
C , 
a
F
C? , 
a
Fs
C , and 
a
Fs
C?  are the fringing capacitances per unit length in the symmetric 
coupled microstrip and strip lines with air as a dielectric material. The value of those 
capacitances are obtained using the formulas discussed earlier in this subsubsection 
when substituting ?
r
 = 1. It should be considered that in the symmetric coupled strip 
lines the fringing capacitances are defined as 
a
Ps
a
es
a
gd
a
Ps
a
os
a
Fs
a
Fs
CCCCCCC ?=??=?+ 5.05.0 , where 
a
P
a
Ps
CC = . 
To model the parasitic capacitance of a meander line, we deal with other 
configurations such as having three or more symmetric coupled microstrip lines. 
However, as mentioned earlier, we are approximating our modeling by neglecting the 
coupling between non-adjacent lines. Figure  3.38 shows the approximated parasitic 
capacitances of three symmetric coupled lines. For the middle strip, the parasitic 
apacitance per length between the strip and the ground is given by 
FPGD
CCC ?+= 2 . 
 
 
C
P
C
F
C'
F
C
P
C'
F
C'
F
C
P
C
F
C'
F
C
ga
C
ga
C
gd
C
gd
C
ga
C
ga
C
gd
C
gd
 
Figure  3.38. Parasitic capacitances for the three symmetric coupled microstrip lines, 
where the coupling effect between the non-adjacent arms is neglected. 
 
 
83 
In our EBG structure, to compute both the parasitic capacitances between the 
strip and the ground for the first and the last arms, and the coupling effect between 
meander line and the adjacent patches, we need to model the asymmetric coupled 
microstrip lines. In the next subsection, the available model and expressions are 
discussed. 
 
3.3.6. Model of Coupling Effect between Patch and Meander line  
In the proposed planar EBG structures, the configuration of a patch and an 
adjacent arm is asymmetric coupled microstrip lines as shown in Figure  3.39 (a). All 
capacitances per unit length characterizing that configuration including the parallel 
plate capacitances and the fringing capacitances are shown in that figure  [58]. The 
unit cell of EBG structure under study may include several coupled microstrip lines 
parallel to each other such as a patch sandwiched between two arms from each side as 
shown in Figure  3.39 (b). The capacitances per unit length modeling those microstrips 
are shown in that figure. The capacitive equivalent circuit model of asymmetric 
coupled patch-arm configuration is shown in Figure  3.39 (c), where C
GD_M 
is 
representing the capacitance between the first arm and the ground plane. C
GD_P 
is the 
capacitance between the patch and the ground and C
M_PM
 is the mutual coupling 
between the patch and the arm. To complete the total value of the parasitic 
capacitance between the meander line and the ground discussed in Subsubsection 
 3.3.5.B, the capacitance C
GD_M
 is included in the value of C
L2
. We can replace C
PH
 in 
Subsection  3.3.3 with C
GD_P 
which is approximating the capacitive load with higher 
accuracy due to the consideration of the fringing capacitances in presence of coupled 
 
84 
lines. The lumped elements in Figure  3.39 (c) for the configuration shown in Figure 
 3.39 (a) are approximated by: 
( ) lCCCC
PHFPHFPHPPGD
???++=
____
 
 Equation  3-40 
( ) lCCCC
MRFMRFMRPMGD
???++=
____
 
 Equation  3-41 
( ) lCCC
PMgdPMgaPMM
?+=
___
 
 Equation  3-42 
where ?l is the effective overlapping length between the patch and the meander arm. 
For the EBG structures under study ?l is equal to M
L 
- M
W
. C
P_PH
 and C
P_MR
 represent 
the parallel plate capacitance per unit length for the patch and the meander-strip, 
respectively. C
F_PH
 and C?
F_PH
 demonstrate the fringing capacitances per unit length 
for the two coupled symmetric patch-microstrip lines. Similarly, C
F_MR
 and C?
F_MR
 
represent the fringing capacitances per unit length for the two coupled symmetric 
meander-microstrip lines. The separation between the two symmetric strips is the 
same as the one in the original setup, M
B
. Referring to the expressions provided for 
the meander line model in Subsubsection  3.3.5.B, C
P_MR
 = C
P
 and C
F_MR
 = C
F
. C''
F_MR
 
is obtained from the formula C'
F 
by replacing a proper separation between coupled 
lines which corresponds to: (M
B
 ? M
G
). If in the provided expressions, we replace 
the specifications of meander with the specifications of patch including: the width 
(P
W
 ? M
W
), the characteristic impedance (Z
P
 ? Z
M
), and the relative effective 
permittivity (?
reP
 ? ?
reM
) then the values of C
P_PH
, C
F_PH
, and C''
F_PH
 are easily 
 
85 
obtained. C
gd_PM
, the capacitance per unit length through the dielectric gap is given by 
 [58]: 
MRgdPHgdPMgd
CCC
___
5.0=  
 Equation  3-43 
where 0.5 ? C
gd_MR
 represents the total fringing capacitance per unit length between 
the two symmetric coupled meander-strips through the dielectric gap. That 
capacitance is computed through the formula provided for C
gd
 in Subsubsection 
 3.3.5.B by replacing the proper separation gap (M
B
 ? M
G
). Similarly, 0.5 ? C
gd_PH
, 
the total fringing capacitance per unit length between the two symmetric coupled 
patch-strips through the dielectric gap is computed. The fringing capacitance per unit 
length through air gap is given by  [58]: 
PMF
a
PMcpsPMga
CCC
___
5.0 ??=  
 Equation  3-44 
where 
a
PMcps
C
_
 is the total fringing capacitance per unit length between the 
asymmetric coplanar strips that have same dimensional configuration as the patch-
arm microstrips under study. It means that width of strips is the same as width of the 
patch and the meander-arm (P
W
, M
W
) and the separation gap is also the same (M
B
). 
The dielectric material is air. That capacitance is given by  [58],  [59]: 
()
()kK
kK
C
a
PMcps
?
=
0_
2?  
 Equation  3-45 
 
 
86 
where 
?
?
?
?
?
?
+
?
?
?
?
?
?
+
++
=
B
W
B
W
B
W
B
W
M
M
M
P
M
M
M
P
k
11
1
 and 
2
1 kk ?=? . ?C
F_PM 
shows the fringing 
capacitance per unit length in the air not in the air gap between the asymmetric 
coupled patch-arm microstrips. It can be given by
1
: 
( )( )
()
a
MRFs
a
MRF
a
MRFs
a
MRF
a
PHFs
a
PHF
a
PHFs
a
PHF
a
MRFs
a
MRF
a
MRFs
a
MRF
a
PHFs
a
PHF
a
PHFs
a
PHF
PMF
CCCCCCCC
CCCCCCCC
C
________
________
_
?????+?+?????+?
?????+??????+?
=?  
 Equation  3-46 
where 
a
PHF
C
_
, 
a
PHF
C
_
?? , 
a
MRF
C
_
, 
a
MRF
C
_
?? , 
a
PHFs
C
_
, 
a
PHFs
C
_
?? , 
a
MRFs
C
_
, and 
a
MRFs
C
_
??  represent 
the fringing capacitances per unit length in the symmetric coupled microstrip and 
strip lines between the strips and the ground where the dielectric material is air. The 
configurations of the two coupled lines are meander-meander and patch-patch which 
are labeled by the subscripts MR and PH. The expressions for these capacitances are 
identical to those provided earlier in Subsubsection  3.3.5.B, except the relative 
permittivity which should be considered as ?
r
 = 1. When the structure under study is 
consisting of several coupled lines similar to Figure  3.39 (b), the expressions defining 
the lumped elements in the equivalent circuit model should be modified to include the 
                                                 
 
 
 
1
 Here, we are suggesting a more inclusive formula for the fringing field effects in air 
than the formula provided in  [58]. Therefore, our suggested formula may provide 
better approximation. 
 
87 
coupling effects (the same issue was discussed previously while modeling the 
meander line in Subsubsection  3.3.5.B).  
 
(a)
(b)
C
ga_PM
C
P_PH
C
F_PH
C''
F_PH
C
P_
M
R
C
F_
MR
C''
F_
MR
C
gd_PM
(c)
C
M_PM
C
GD_P
C
GD_M
C
ga_PM
C
ga_PM
C
P_PH
C''
F_
P
H
C
P_MR
C''
F_
MR
C
gd_PM
C
P_MR
C''
F_
MR
C''
F_
P
H
C
gd_PM
C
P_MR
C
F_
MR
C'
F_
MR
C
P_MR
C
F_
MR
C'
F_
MR
C'
F_
MR
C'
F_
MR
C
ga_MM
C
gd_MM
C
ga_MM
C
gd_MM
C
P_
M
R
C
F_
MR
C''
F_
MR
C
P_
M
R
C
F_
MR
C''
F_
MR
C''
F_
P
H
C''
F_
MR
C''
F_
MR
C''
F_
P
H
C
F_
MR
C'
F_
MR
C
F_
MR
C'
F_
MR
C'
F_
MR
C'
F_
MR
C''
F_
P
H
C''
F_
MR
C''
F_
MR
C''
F_
P
H
C
F_
MR
C'
F_
MR
C
F_
MR
C'
F_
MR
C'
F_
MR
C'
F_
MR
 
Figure  3.39. (a) Side view of patch and meander-arm microstrips as a pair of 
asymmetric coupled microstrip lines. (b) Side view of a patch microstrip between 
four meander-arm mirostrips. The capacitances modeling the coupled lines are shown 
in the figure. (c) Capacitive equivalent circuit model of the asymmetric coupled 
patch-arm microstrips. 
 
To complete the modeling in this part, we should consider the bridge 
connecting the patch to adjacent arm as shown in Figure  3.40 (a). If the bridge-length, 
M
B
, is a considerable portion of the total distance separating the two consecutive 
patches, then we can complete the modeling by considering the bridge-trace 
 
88 
inductance, L
B
, the trace to ground capacitance, C
B
, and the mutual coupling 
capacitance between the patch and meander-arm, C
M_PM
, as one circuit block The 
complete ?-equivalent circuit model of this case is shown in Figure  3.40 (b). 
However, if the bridge-length, M
B
, is very small then L
B
 and C
B
 may be included in 
the total inductance and capacitance modeling the meander line (L and C
L2
) discussed 
in Subsection  3.3.5. In this case, C
M_PM
 is modeled in series with C
L1
 and the total 
combination is parallel to L as shown in Figure  3.40 (c). In another word, we can 
model the coupling capacitor between the patch and the meander as part of the 
interline parasitic capacitances in the meander pattern. That is acceptable because we 
are developing an approximate model for initial guess in EBG design procedure. 
 
 
 
 
89 
T T' T''
???
(a)
(c)
C
L1
C
L2
L
C
L2
C
M_PM
C
M_PM
TT'
(b)
TT'
C
B
C
B
C
M_PM
L
B
 
Figure  3.40. (a) A meander line between two patches. The bridge connecting the first 
arm to patch is marked in the figure. (b) Equivalent circuit modeling: 1) the mutual 
coupling effect between the patch and the adjacent meander-arm and 2) the 
connecting bridge-trace. (c) Equivalent circuit modeling the total meander line (refer 
to Figure  3.29). In this model, the effect of mutual coupling between the patch and the 
meander is considered as interline parasitic capacitance. 
 
Example?We are computing the parasitic capacitance between pattern and 
ground for a meander line with six arms, N = 6, as shown in Figure  3.41. The 
meander line is connected to adjacent patches at the edge (margin = 0). Specifications 
of the meander are as follows: 1) identical line-width along the pattern M
W
; 2) equal 
separating gap between the arms M
G
; and 3) equal arm-length M
L
. All the 
coupled/uncoupled traces are partially modeled with the different fringing 
 
90 
capacitances per unit length which are marked with different symbols and colors in 
that figure. The meander line consists of 7 (N + 1) bridge-sections which are 
uncoupled lines. The fringing capacitances per unit length in both sides of those 
traces are identical and they are equal to C
F
. The total length of those bridges in the 
pattern is equal to ()( )( )
WGWB
MMNMM +?++ 15.02  and the parasitic capacitance to 
ground per unit length is given by 
FPGD
CCC 2+= . In the meander pattern, four (N ? 
2) arms are the symmetric coupled lines from both sides (one arm is sandwiched 
between two similar arms). Therefore, the fringing capacitances per unit length for 
both sides are identical and they are equal to C'
F
. The total length of that type of arm 
is equal to ()( )
WL
MMN ??2  and the parasitic capacitance to ground per unit length is 
given by 
FPGD
CCC ?+= 2 .The last two remaining arms in the meander pattern (the first 
and the last ones) are the coupled lines sandwiched between a similar arm and a 
patch. Therefore, the fringing capacitance in the side corresponding to arm is C'
F
 and 
in the side corresponding to patch is C''
F
. The total length of these two arms is equal 
to ()
WL
MM ?2  and the parasitic capacitance to ground per unit length is equal to 
FFPGD
CCCC ?+??+= . Therefore, for that 3-turn meander line the total parasitic 
capacitance to ground is given by: 
()()( )[]( ) ( )( )( )
[]
()()()()()
[]
FFPWL
FPWLFPWGWB
FFPWL
FPWLFPWGWBL
CCCMM
CCMMCCMMMM
CCCMM
CCMMNCCMMNMMC
?+??+?+
?+?+++++=
?+??+?+
?+??+++?++=
2           
24255.02        
2           
22 215.022
2
 
 Equation  3-47 
 
91 
C
F 
Uncoupled
C'
F
Symmetric Coupled
C''
F
Asymmetric Coupled
 
Figure  3.41. Different types of fringing capacitances per unit length for the coupled 
and uncoupled traces in the meander line are marked with different colors and 
symbols. The 3-turn meander has the following parameters: M
Li
 = M
Lo
 = M
L
, (M
Bi
 = 
M
Bo
 = M
B
) ? M
G
, and P
W
 ? M
W
 (The design parameters are shown in Figure  3.25 and 
Figure  3.28). 
 
3.3.7. Model of Electrical Length in Planar EBG Structure 
As reported by other researchers and mentioned earlier, the equivalent circuit-
transmission line model can capture higher frequency effects more properly than the 
equivalent circuit model. The main reason for this superiority is that the transmission 
lines models the phase shift which is a function of frequency. The phase shift is the 
delay in the propagating field across the structure and it is increasing with frequency. 
To consider the phase delay across the unit cell of EBG structure, we are including 
two different types of microstrip transmission lines in our model consisting of patch-
microstrip and meander-microstrip with total length of d
1
 and d
2
, respectively. Where 
d = d
1
 + d
2
 represents the periodicity-length (or the length of a unit cell of EBG 
structure), d
1
 represents the length of a patch, and d
2
 represents the length of 
 
92 
separation between the two consecutive patches. Therefore, the patch-microstrip 
models the phase difference across the patch and the meander-microstrip models the 
phase difference across the meander line (not along the meander line). The basic idea 
of modeling the phase delay in a meander line by the proposed method which is 
working properly as will be shown in later subsections is coming from the concept of 
modeling of meander-antennas used by Endo et al.  [60]. In  [60], the meander-antenna 
was modeled as a linear dipole antenna with inductive loading. It is possible to 
include the effect of phase difference between adjacent arms in meander line with 
higher accuracy by considering the coupling effects vectorially as  [55]. However, in 
our applications the proposed level of accuracy is working well enough. The ABCD 
parameters for the microstrip line specified by the characteristic impedance, the 
relative effective permittivity, and the length (Z
0
, ?
re
, d
0
) are available in microwave 
fundamental books such as  [46].  
 
3.3.8. The Model of a Unit Cell of the Proposed Planar EBG Structure 
In this subsection, different models with different levels of accuracy in 
predicting bandgaps of an EBG structure are studied. These models are showing the 
effect of added elements in the configuration. We realize that modeling is not easy 
when the frequency is increasing. However, we could predict lower edge of bandgap 
with good accuracy which may decrease the load of full-wave simulations during the 
design procedure of EBG structure. 
In Figure  3.42 a very simple circuit model for a unit cell of the proposed 
planar EBG structure including two capacitances each modeling half of the patch and 
 
93 
an inductance modeling the meander line is shown in Figure  3.18. The dispersion 
diagram of a unit cell of the first case study (defined previously in Subsection  3.3.2) 
extracted using this simple circuit model is shown in Figure  3.43. The design 
parameters of the EBG structure are repeated here for convenience:  P
W
 = 1.6 mm, h 
= 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, 
and tan ? = 0.0015. The ABCD transfer matrix defined in Equation  3-6 is given by: 
?
?
?
?
?
?
?
?
?
?
?
?
++
+
=
?
?
?
?
?
?
?
?
2
1
2
2
1
1
22
1
12
1
1
y
y
y
y
y
yy
y
DC
BA
mm
mm
 
 Equation  3-48 
where ()2
1 PH
Cjy ?=  and ( )Ljy ?1
2
= . The dispersion extracted from numerical 
analysis was demonstrated in Figure  3.19. Comparison in the ?-? zone shows that 
this model which is describing the EBG structure as a low pass filter could not predict 
accurately the lower edge of the bandgap.  
 
C
PH
C
PH
L
2
1
2
1
 
Figure  3.42. Simple physical-based circuit model for the proposed planar EBG 
structure. 
 
94 
 
Figure  3.43. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the simple physical-based circuit model shown in Figure  3.42. The 
specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
An advanced physical-based model which includes physical specifications at 
low frequencies is shown in Figure  3.44. The model includes: 1) the inductances and 
the capacitances modeling the patch. 2) The inductances and the capacitances 
modeling the step discontinuities between the patch and the meander line. 3) The 
inductances and the parasitic capacitances modeling the meander line. 4) The 
capacitances modeling the coupling effect between the meander and the two adjacent 
patches. The dispersion diagram for the same EBG structure mentioned above is 
0 500 1000 1500 2000 2500
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eque
ncy
 (
G
H
z
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone Boundary
(K
x
 = ? / d)
 
95 
extracted using this advanced circuit model shown in Figure  3.45. The ABCD matrix 
for this model is given by: 
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
++
+
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
+
=
?
?
?
?
?
?
?
?
1
1
1
1
1
1
                         
12
1
1
1
1
1
1
1
1
1
2
1
2
5
3
5
5
43
43
5
4
7
6
7
2
6
6
77
6
5
4
5
5
43
43
5
3
1
2
1
2
z
z
z
z
z
z
z
z
zz
zz
z
z
y
y
y
y
y
yy
y
z
z
z
z
zz
zz
z
z
z
z
z
z
DC
BA
mm
mm
 
 Equation  3-49 
where 
()
PH
Cj
z
?
2
1
= , 
PH
Ljz ?=
2
, 
DP
Ljz ?=
3
, 
DM
Ljz ?=
4
, 
()
D
Cj
z
?
1
5
= , 
26 L
Cjy ?= , 
and 
()Lj
CCCjy
PMMLPMM
?
?
1
||||
_1_7
+= . Comparing this result with those in Figure 
 3.19 and Figure  3.43 shows the improvement in the prediction of the lower edge of 
bandgap. Also, the next propagating mode has appeared. However, the model is not 
still acceptable and the error is high.  
 
 
2
1
C
L1
C
L2
L
C
L2
C
M_PM
C
M_PM
L
DM
L
DP
C
D
L
DM
L
DP
C
D
L
PH
C
PH
2
1
L
PH
C
PH
 
Figure  3.44. Advanced physical-based circuit model for the proposed planar EBG 
structure. 
 
 
96 
 
Figure  3.45. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the advanced physical-based circuit model shown in Figure  3.44. The 
specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
To improve our model, we consider the phase delay by including the 
transmission lines in the advanced circuit model. The simple equivalent circuit-
transmission line model under study is shown in Figure  3.46. The different 
transmission lines included in the model consider the phase shift across the patch and 
the meander line separately. The dispersion diagram for this configuration is shown in 
Figure  3.47. The ABCD matrix for this model is given by: 
0 500 1000 1500 2000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eque
ncy
 (
G
H
z
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone Boundary
(K
x
 = ? / d)
 
97 
() ()
()()
() ()
()()
() ()
()()
() ()
()()
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
=
?
?
?
?
?
?
?
?
1
1
1
2cos2sin
2sin2cos
1
1
1
                        
2cos2sin
2sin2cos
12
1
1
2cos2sin
2sin2cos
1
1
1
2cos2sin
2sin2cos
1
1
1
1
2
1
2
11
11
5
3
5
5
43
43
5
4
22
22
7
6
7
2
6
6
77
6
22
22
5
4
5
5
43
43
5
3
11
11
1
2
1
2
z
z
z
z
ddjY
djZd
z
z
z
z
zz
zz
z
z
ddjY
djZd
y
y
y
y
y
yy
y
ddjY
djZd
z
z
z
z
zz
zz
z
z
ddjY
djZd
z
z
z
z
DC
BA
PPP
PPP
MMM
MMM
MMM
MMM
PPP
PPP
mm
mm
??
??
??
??
??
??
??
??
 
 Equation  3-50 
where 
()
PH
Cj
z
?
2
1
= , 
PH
Ljz ?=
2
, 
DP
Ljz ?=
3
, 
DM
Ljz ?=
4
, 
()
D
Cj
z
?
1
5
= , 
26 L
Cjy ?= , 
and 
()Lj
CCCjy
PMMLPMM
?
?
1
||||
_1_7
+= . The characteristic impedance and the 
propagation constant of patch-microstrip are shown by pairs of (Z
P
, ?
P
), respectively. 
d
1
 is the length of a patch. The characteristic impedance and propagation constant of 
meander-microstrip are shown by pairs of (Z
M
, ?
M
), respectively. d
2
 is the distance 
between two consecutive patches. Comparison between the dispersion diagram in this 
figure and the previous one shown in Figure  3.19 shows that although the second 
propagating mode is well predicted and the third propagating mode has inaccurately 
located, the model could not correctly predict the lower edge of the first bandgap 
which is our main target in this modeling. 
 
98 
C
L1
C
L2
L
C
L2
C
M_PM
C
M_PM
2
1
L
PH
C
PH
Z
M
, ?
M
2
d
2
2
1
L
PH
C
PH
Z
P
, ?
P
2
d
1
L
DM
L
DP
C
D
Z
M
, ?
M
2
d
2
L
DM
L
DP
C
D
Z
P
, ?
P
2
d
1
 
Figure  3.46. Simple equivalent circuit-transmission line model for the proposed 
planar EBG structure. 
 
 
Figure  3.47. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the simple equivalent circuit-transmission line model shown in Figure 
 3.46. The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 
0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
quenc
y (
G
H
z
)
?
x
?
x
Light Line in
Dielectric
Brillouin Zone
Boundary
 
99 
To improve our modeling we need to consider an important fact. We are 
modeling the wave propagation in the x-direction in a 1-D infinite array of EBG 
structures (refer to Figure  3.17) but when the frequency increases the propagation in 
the y-direction starts to be effective. In the ?-? zone, the propagation constant in the 
y-direction is equal to zero (k
y
 = 0). Therefore, to include the propagation in y-
direction in our model, it is enough to consider the following elements: 1) the open 
end (or the capacitance modeling the open end microstrip), 2) the patch-microstrips 
modeling the phase delay for wave propagating in y-direction across the patch, and 3) 
the inductances and capacitances modeling the patch in y-direction. The advanced 
equivalent circuit-transmission line model for the proposed planar EBG structure is 
shown in Figure  3.48. The dispersion diagram for this model is shown in Figure  3.49. 
The ABCD matrix for this model is given by: 
 
2
L
P
H
2
Z
P
,
 
?
P
2
Z
P
,
 
?
P
2
L
P
H
d
1
2
d
1
2
L
PH
2
1
C
PH
Z
M
, ?
M
2
d
2
L
PH
2
1
C
PH
Z
P
, ?
P
2
d
1
L
DM
L
DP
C
D
C
L1
C
L2
L
C
L2
C
M_PM
C
M_PM
Z
M
, ?
M
2
d
2
L
DM
L
DP
C
D
Z
P
, ?
P
2
d
1
2
L
P
H
2
Z
P
,
 
?
P
2
Z
P
,
 
?
P
2
L
P
H
d
1
2
d
1
2
 
Figure  3.48. Advanced equivalent circuit-transmission line model for the planar 
proposed EBG structure. 
 
100 
() ()
()()
() ()
()()
() ()
()()
() ()
()()
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
=
?
?
?
?
?
?
?
?
1
1
1
2cos2sin
2sin2cos
1
1
1
                        
2cos2sin
2sin2cos
12
1
1
2cos2sin
2sin2cos
1
1
1
2cos2sin
2sin2cos
1
1
1
1
2
1
2
11
11
5
3
5
5
43
43
5
4
22
22
7
6
7
2
6
6
77
6
22
22
5
4
5
5
43
43
5
3
11
11
1
2
1
2
z
z
z
z
ddjY
djZd
z
z
z
z
zz
zz
z
z
ddjY
djZd
y
y
y
y
y
yy
y
ddjY
djZd
z
z
z
z
zz
zz
z
z
ddjY
djZd
z
z
z
z
DC
BA
PPP
PPP
MMM
MMM
MMM
MMM
PPP
PPP
mm
mm
??
??
??
??
??
??
??
??
 
 Equation  3-51 
where 
()
z
Cj
z
PH
+=
?
2
1
, 
()
PH
P
P
Lj
dj
Z
z ?
?
+=
2tan
1
, 
PH
Ljz ?=
2
, 
DP
Ljz ?=
3
, 
DM
Ljz ?=
4
, 
()
D
Cj
z
?
1
5
= , 
26 L
Cjy ?= , and ( )
()Lj
CCCjy
PMMLPMM
?
?
1
||||
_1_7
+= . 
The characteristic impedance and the propagation constant of patch-microstrip are 
shown by pairs of (Z
P
, ?
P
), respectively. d
1
 is the length of a patch. The characteristic 
impedance and the propagation constant of meander-microstrip are shown by pairs of 
(Z
M
, ?
M
), respectively. d
2
 is the distance between two consecutive patches. The 
comparison of results in this diagram and the numerical ones in Figure  3.19 shows a 
prediction with high accuracy for the first three propagating modes indicated in the 
figures. The developed advanced equivalent circuit-transmission line model will be 
used in the rest of this study.  
 
 
 
101 
 
Figure  3.49. Dispersion diagram of the planar EBG structure shown in Figure  3.18 is 
modeled by the advanced equivalent circuit-transmission line model shown in Figure 
 3.48. The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 
0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
3.3.9. Case Studies and Performance of the Developed Model 
In this subsection, we are going to asses the performance of the developed 
circuit-transmission line model in the ?-? region of the Brillouin zone for the 
samples defined in Subsection  3.3.2. In the Figure  3.50 for the first case study where 
?
r
 = 140 and h = 114.3 ?m the comparison between the dispersion diagrams extracted 
by (a) the developed model and (b) the full-wave numerical simulator is shown. 
Clearly, the agreement between graphs for the first three propagating modes is 
0 500 1000 1500 2000 2500 3000 3500 4000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eque
ncy
 (
G
H
z
)
?
x
?
x
Light Line
in Dielectric
Brillouin Zone
Boundary (K
x
 = ? / d)
 
102 
excellent. The attenuation diagram extracted from the equivalent circuit-transmission 
line model for a unit cell of the first sample is shown in Figure  3.51 (a). The 
attenuation diagram for the four samples is plotted in Figure  3.51 (b) which shows the 
suppression level that could be reached in the bandgaps if the source and the 
destination ports are distant from each other by four EBG unit cells. By increasing the 
number of EBG unit cells between the two ports, we can increase the attenuation 
level. The 20 dB suppression level is marked in Figure  3.51 (b). 
 
 
0 200 400 600 800 1000 1200 1400 1600 1800
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
 First propagating mode 
 Second propagating mode 
 Third propagating mode 
 Light line in dielectric
Fr
e
q
ue
n
c
y (
G
Hz
)
phase constant (?
x
)
0 1000 2000 3000 4000 5000 6000 7000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eq
ue
nc
y
 (
G
Hz
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone Boundary (K
x
 = ? / d)
First Propagating Mode
Second Propagating Mode
Third Propagating Mode
(a) (b)
Fr
e
q
ue
n
c
y (
G
Hz
)
Fr
eq
ue
nc
y
 (
G
Hz
)
Fr
eq
ue
nc
y
 (
G
Hz
)
 
Figure  3.50. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line model 
and (b) numerical full-wave analysis by Ansoft HFSS. The specifications of EBG are 
as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
103 
0 1 2 3 4 5 6 7 8 9 10
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
X: 1.509
Y: -20.11
X: 3.932
Y: -20.11
A
t
t
e
nua
t
i
on
 f
o
r
 
4
 
U
n
i
t
 C
e
l
l
 of
 
E
B
G
 
S
21
(d
B
)
Frequency (GHz)
0 2 4 6 8 10
-120
-100
-80
-60
-40
-20
0
A
t
t
e
nua
t
i
on pe
r
 Un
i
t
 
C
e
l
l
 of
 E
B
G
 
(
d
B
)
Frequency (GHz)
(a) (b)
A
t
t
e
nua
t
i
on
 f
o
r
 
4
 
U
n
i
t
 C
e
l
l
 of
 
E
B
G
 
S
21
(d
B
)
A
t
t
e
nua
t
i
on pe
r
 Un
i
t
 
C
e
l
l
 of
 E
B
G
 
(
d
B
)
A
t
t
e
nua
t
i
on
 f
o
r
 
4
 
U
n
i
t
 C
e
l
l
 of
 
E
B
G
 
S
21
(d
B
)
A
t
t
e
nua
t
i
on pe
r
 Un
i
t
 
C
e
l
l
 of
 E
B
G
 
(
d
B
)
 
Figure  3.51. The attenuation diagram in ?-? zone for the planar EBG structure shown 
in Figure  3.18 is extracted using advanced equivalent circuit-transmission line model 
for (a) a unit cell of EBG and (b) 4 unit cells of EBG. The 20 dB attenuation level is 
marked in the graph. The specifications of EBG are as follows: P
W
 = 1.6 mm, h = 
114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and 
tan ? = 0.0015. 
 
 
For the second case study where ?
r
 = 200 and h = 114.3 ?m, the comparison 
between the dispersion diagrams extracted by (a) the developed model and (b) the 
full-wave numerical simulator is shown in Figure  3.52. Here, also the agreement is 
very good between the first three propagating modes. The comparison for the third 
case study where ?
r
 = 140 and h = 500 ?m between the dispersion diagrams extracted 
by (a) the developed model and (b) the full-wave numerical analysis is shown in the 
Figure  3.53. The first and the third propagating modes are in excellent agreement. 
However, the second propagating mode extracted by the developed model covers 
larger frequency band around the frequency region predicted by the full-wave 
 
104 
analysis. This difference comes from approximate models implemented in our 
developed model for the different configurations of microstrips such as the step 
discontinuity discussed earlier in the modeling subsections. In Figure  3.54, for the 
fourth case study where ?
r
 = 4.3 and h = 114.3 ?m the comparison between the 
dispersion diagrams extracted by (a) the developed model and (b) the full-wave 
numerical simulator is shown. Diagrams show that the first propagating mode and as 
a result the lower edge of the bandgap are predicted accurately. However, the higher 
order propagating modes are not accurately predicted.  
 
 
0 1000 2000 3000 4000 5000 6000 7000 8000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
uen
c
y (
G
H
z
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone Boundary (K
x
 = ? / d)
First Propagating Mode
Second Propagating Mode
Third Propagating Mode
0 200 400 600 800 1000 1200 1400 1600 1800
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
 First propagating mode 
 Second propagating mode 
 Third propagating mode 
 Light line in dielectric
Fr
eque
nc
y
 (GH
z
)
phase constant (?
x
)
(a) (b)
Fr
e
q
uen
c
y (
G
H
z
)
Fr
e
q
uen
c
y (
G
H
z
)
Fr
eque
nc
y
 (GH
z
)
 
Figure  3.52. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line model 
and (b) numerical full-wave analysis by Ansoft HFSS. The specifications of EBG are 
as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 200, and tan ? = 0.0015. 
 
 
105 
It can be concluded that 1) the developed model based on the available low 
frequency approximated circuit models for the different microstrip configurations is 
working excellent for the thin slab of high-k dielectric materials in the range of 
couples of GHz. 2) The model is well behaving for the increasing thickness of slab of 
high-k materials in the range of couples of GHz. But, 3) the developed model is 
inaccurate for thin low permittivity dielectrics in the range of couples of GHz. 
However, 4) for all the combination of specifications in the dielectric the performance 
of model is excellent for extracting the first propagating mode. It means the 
developed model can predict the lower edge of the bandgap with high accuracy. 
Therefore, the model is excellent for low GHz frequency range. The main reason for 
the mentioned level of performance for the developed model is that: we have 
approximated a considerable portion of transmission lines in the structure as 
uncoupled microstrips. Also, we have neglected non-adjacent couplings. In the 
uncoupled microstrips the fields produced around the structures do not affect each 
other. This assumption in the structures under study is much more valid in the 
microstrips made of high-k materials than the ones made of conventional dielectrics 
such as FR-4. In the structures made of high-k, the fringing fields outside the strip are 
bounded to strip and concentrated at the edge of strip. Therefore, the approximation 
used in the developed model about the couplings between the constituting parts is 
working with higher accuracy in high-k materials. In another word, to improve the 
performance in the model we may need to consider non-adjacent coupling. 
 
106 
0 1000 2000 3000 4000 5000 6000 7000 8000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
n
c
y
 (
G
Hz
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone Boundary (K
x
 = ? / d)
First Propagating Mode
Second Propagating Mode
Third Propagating Mode
(a) (b)
0 200 400 600 800 1000 1200 1400 1600 1800
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
 First propagating mode 
 Second propagating mode 
 Third propagating mode 
 Light line in dielectric
F
r
e
quency
 (
G
Hz)
phase constant (?
x
)
Fr
e
q
ue
n
c
y
 (
G
Hz
)
Fr
e
q
ue
n
c
y
 (
G
Hz
)
F
r
e
quency
 (
G
Hz)
 
Figure  3.53. Dispersion diagram in ?-? zone for the planar EBG structure shown in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line model 
and (b) numerical full-wave analysis by Ansoft HFSS. The specifications of EBG are 
as follows: P
W
 = 1.6 mm, h = 500 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, 
M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
0 20 40 60 80 100 120 140
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
 First propagating mode 
 Second propagating mode 
 Third propagating mode 
 Light line in dielectric
F
r
e
que
nc
y
 (
G
Hz
)
phase constant (?
x
)
0 100 200 300 400 500 600 700
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Wave Number, K
x
Fr
equ
e
ncy (
G
H
z
)
?
x
?
x
Light Line in Dielectric
Brillouin Zone
Boundary (K
x
 = ? / d)
First Propagating Mode
Second Propagating Mode
Third Propagating Mode
(a) (b)
F
r
e
que
nc
y
 (
G
Hz
)
Fr
equ
e
ncy (
G
H
z
)
Fr
equ
e
ncy (
G
H
z
)
 
Figure  3.54. Dispersion diagram in ?-? zone for the planar EBG structure shown  in 
Figure  3.18 is extracted using (a) advanced equivalent circuit-transmission line model 
and (b) numerical full-wave analysis by Ansoft HFSS. The specifications of EBG are 
as follows: P
W
 = 20 mm, h = 114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 mm, M
L 
= 20 mm, 
M
G 
= 0.2 mm, ?
r 
= 4.3, and tan ? = 0.02. 
 
107 
We continue our studies in later subsections to investigate about the 
relationship between the bandgap and the elements included in the model. We will 
present how the values of elements are affecting the bandgap and controlling the 
location of the bandgap in different frequency regimes.  
 
3.3.10. Characterization of the Proposed Planar EBG Structure 
In this subsection, we investigate about characteristics of the planar EBG 
structures in more details. For this purpose, we continue our study on the first sample 
discussed in Subsection  3.3.2. We fit the best and the lowest order polynomials on the 
data points of propagating modes in dispersion diagrams extracted by the full-wave 
numerical analysis using Ansoft HFSS. We do not intend to fit a high order 
polynomial which exactly follows the variations of data points in dispersion diagram. 
We want a simple approximate graph to study and extract information about our 
suggested EBGs. The fitted polynomials are plotted for each region of irreducible 
Brillouin zone: ?-?, ?-?, and ?-? in Figure  3.55 to Figure  3.57, respectively. The 
light lines in dielectric plotted in those regions are defined by: 
()
{
?
?
?
?
?
?
?
?
?
?
?
==???
?
?
?
=
=
???+
?
?
?
=
=
???
=
],0[  Zone   2
],0[
  Zone   
 
0
],0[
  Zone   
22
?
?
?
?
?
?
dkdkdk
dk
dk
dk
dk
dk
dk
d
yxx
y
x
y
y
x
x
 
 Equation  3-52 
 
 
 
108 
 
0.0000 1.5708 3.1416
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
?0
? X
 First propagating mode         Polynomial fit of first mode data
 Second propagating mode     Polynomial fit of Second mode data
 Third propagating mode        Polynomial fit of third mode data
                                                          Light line in dielectric
F
r
e
que
ncy
 (
G
Hz)
phase (?d)
 
Figure  3.55. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the dispersion 
relation are extracted numerically by Ansoft HFSS. The polynomial fit of these data 
points of the propagating modes show dispersion graphs. The specifications of the 
EBG are as follows: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
 
 
109 
 
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
0
1
2
3
4
5
6
??
MX
 First propagating mode           Polynomial fit of first mode data
 Second propagating mode      Polynomial fit of second mode data
 Third propagating mode         Polynomial fit of third mode data
                                                           Light line in dielectric
Frequ
e
ncy
 (
G
Hz)
phase (?d)
?2
Frequ
e
ncy
 (
G
Hz)
 
Figure  3.56. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the dispersion 
relation are extracted numerically by Ansoft HFSS. The polynomial fit of these data 
points of the propagating modes show dispersion graphs. The specifications of the 
EBG are defined in Figure  3.55. 
 
 
 
 
 
 
 
110 
 
 
012345
0
1
2
3
4
5
6
?0
M?
 First propagating mode         Polynomial fit of first mode data
 Second propagating mode     Polynomial fit of second mode data
 Third propagating mode        Polynomial fit of third mode data
                                                          Light line in dielectric
Frequ
e
ncy
 (G
Hz)
phase (?d)
?2
Frequ
e
ncy
 (G
Hz)
 
Figure  3.57. Dispersion diagram of the planar EBG structure shown in Figure  3.18 in 
the ?-? region. The data points of the propagating modes representing the dispersion 
relation are extracted numerically by Ansoft HFSS. The polynomial fit of these data 
points of the propagating modes show dispersion graphs. The specifications of the 
EBG are defined in Figure  3.55. 
 
 
 
 
111 
At the edge of the bandgap, the lower and the upper propagating modes are 
slow modes. For slow modes compared to light line, the group velocity defined by 
?
?
?
?
=
?1
Vg  is much lower. Therefore, the slow modes are very dispersive. As it is 
clear from the dispersion diagrams of the sample under study in three regions, the 
variation of the frequency versus the phase for the second and the third propagating 
modes is very small. The dispersion graphs are approximately straight lines with zero 
degree tilt which show that the group velocities of these modes are very low. For the 
miniaturized planar EBG sample, the group velocity of the first three propagating 
modes and the light line in dielectric for the ?-?, ?-?, and ?-? regions are plotted 
in Figure  3.58 to Figure  3.60, respectively. The group velocity of light line is defined 
by: 
()
()
{
?
?
?
?
?
?
?
?
?
?
?
?
?
==???
?
?
?
=
=
???
?
?
?
?
?
?
?
?
+
?
?
?
=
=
???
=
?
],0[  Zone   
2
],0[
  Zone   
2
2
 
0
],0[
  Zone   
2
2
1
?
?
?
?
??
?
??
?
?
?
dkdk
c
dk
dk
c
df
c
df
c
dk
dk
c
Vg
yx
r
y
x
r
r
r
y
x
r
 
 Equation  3-53 
It is clear in the diagrams that the group velocity of the first propagating mode at the 
lower edge of the first bandgap abruptly and very sharply changes to zero. In the three 
regions of the Brillouin zone, the group velocity of the second and the third 
propagating modes are more than 10 times slower than the group velocity of the light 
in the dielectric. This means that for the existing modes, the media is completely 
dispersive.  
 
112 
From the dispersion diagrams and the group velocity diagrams in Brillouin 
zone we conclude that the lower edge of the first bandgap is located at 1.9 GHz. 
However, if we consider the 20 dB suppression band for this sample which is labeled 
in Table  3.III as sample No. 3, we notice that this band extends from 2.9 GHz to 12.5 
GHz. Therefore, the lower edge of the suppression band is not the same as the lower 
edge of the bandgap for this sample. In addition, it is expected although there is not 
such a rule that the suppression band is wider than the bandgap and it encompass the 
bandgap. The difference at the lower edge of the two extracted bands is attributed to 
the following reasons: 1) the dispersion diagram is extracted for an idealistic case 
where the periodic array of EBG structures is infinite. However, the suppression band 
is extracted for a finite number of EBG structures where in our case; we set up 4 units 
of EBG structure between the two measuring ports. 2) In a bandgap which is defined 
with upper cut off frequency (f
CU
) of lower propagating mode and lower cut off 
frequency (f
CH
) of upper propagating mode, evanescent modes are propagating. The 
power of these evanescent modes is decreasing at f
CU
 and increasing at f
CH
. In another 
word, the attenuation diagram of bandgap is increasing at f
CU
 and decreasing at f
CH
. 
Therefore, if there is not infinite number of EBG structures between the source and 
the receiver ports then the 20 dB suppression band limits might not locate precisely at 
the edge of the bandgap. This fact is shown in Figure  3.51. 
The 20 dB suppression band of this sample is very wide. It is consisting of 
upper part of the first bandgap, several higher order propagating modes, and several 
higher order bandgaps as partially shown in dispersion diagrams. A bandgap is a gap 
of propagating modes, however, a suppression band is not. A suppression band is a 
 
113 
gap of power. In another words: in the suppression band, the level of transferred 
power between two ports (S
21
), where EBG structures are located between them, 
either through evanescent modes or propagating modes is reduced by more than 20 
dB. It is noted that the bandgap is the inherent feature of periodic bandgap structure 
which directly relates to topology and composition of structure and not to losses of 
constructive materials. However, in measuring the suppression band we consider 
losses including metallic loss and dielectric loss. Lossess in the structure contribute to 
the attenuation level, however, considering the range of frequencies where we are 
studying these structures and the presence of limited numbers of structures in our 
setup reveals that the value of loss is not in the order of 20 dB.  So we need to find 
out what is happening to the energy of these slow propagating modes. Our guess is: 
these modes instead of propagating through structures may radiate out. Therefore, the 
EBG sample, similar to a bandstop filter, blocks through-propagation in the 
suppression band. 
 
 
 
 
 
 
 
 
 
 
114 
 
 
 
0123456
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Region:  ? ???> X
 First propagating mode
 Second propagating mode
 Third propagating mode
 Light line in dielectric
Vg (
G
r
oup Vel
o
ci
ty
)
   
(Mm
/
s)
Frequency (GHz)
 
Figure  3.58. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.55. The Vg 
graphs are obtained from the polynomials fit of the propagating modes. 
 
 
 
 
 
 
 
 
115 
 
 
0123456
0
50
100
150
200
250
300
350
400
450
Region:  X ???> M
 First propagating mode  
 Second propagating mode
 Third propagating mode 
 Light line in dielectric
Vg (
G
r
oup Vel
o
cit
y
) 
  
(Mm
/
s)
Frequency (GHz)
123456
-5
0
5
10
15
20
25
30
35
40
45
50
Vg  
 (M
m/
s
)
Frequency (GHz)
Vg (
G
r
oup Vel
o
cit
y
) 
  
(Mm
/
s)
Vg  
 (M
m/
s
)
Vg  
 (M
m/
s
)
 
Figure  3.59. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.56. The Vg 
graphs are obtained from the polynomials fit of the propagating modes. 
 
 
 
 
 
 
 
 
116 
 
 
 
 
0123456
0
2
4
6
8
10
12
14
16
18
Region:  M ???> ?
 First propagating mode
 Second propagating mode
 Third propagating mode
 Light line in dielectric
V
g
 (G
ro
u
p
 V
e
lo
city
)  
 (M
m
/
s)
Frequency (GHz)
 
Figure  3.60. Group velocity of the propagating modes of the planar EBG structure 
and the light line in the dielectric in the ?-? region as defined in Figure  3.57. The Vg 
graphs are obtained from the polynomials fit of the propagating modes. 
 
 
 
 
 
 
 
117 
Following, we investigate the hypothesis of energy radiation from opening of 
the patterned structure to surrounding media nearby the slow mode frequency 
regimes. The solid parallel plate radiates out energy at the edges to surrounding 
environment. The radiated power at the excited resonant frequency of the parallel 
plate is stronger. For example, if we consider a setup consisting of a finite parallel 
plate with two ports inside, the total sum of the normalized transferred energy 
between two ports and the normalized reflected energy to source at the considered 
frequency is not equal to unity. In a lossy-opened system, the radiation loss can be 
characterized by the quantity of radiation loss parameter which is defined as: 1- |S
11
|
2
 
- |S
21
|
2
. In our study, we calculate the radiation loss parameter for the EBG structures 
by using the S-parameters extracted from the setup shown in Figure  3.14. For 
comparison purpose, the radiation loss parameter for the solid parallel planes setup 
with the same dimensions and port locations as the EBG patterned structure is 
extracted.  
The radiation loss for the first sample is shown in Figure  3.61. The 20 dB 
suppression band of this sample which is extended from 2.9 GHz to 12.5 GHz is 
marked in the figure. The figure clearly shows that the local peaks of radiation loss 
from solid parallel planes are excessively reduced by the EBG patterning of one of 
the planes. Therefore, patterning not only does not increase the radiation to 
surrounding environment but also it decreases EMI in the band where EBG is 
effective. In the suppression band, there are few local peaks in the radiation loss 
diagram of the EBG patterned planes. Comparing the frequency band of these peaks 
with the frequency band of the slow propagating modes in the dispersion diagram of 
 
118 
Figure  3.20 shows that the frequency regions could be mapped with acceptable 
accuracy. The second and third local peaks which are compatible with the third and 
the fourth propagating modes are marked in Figure  3.61. Therefore, the results for the 
first sample verify our proposal about the radiating emissions from openings of EBG 
patterns to surrounding environment at the frequency bands where dispersive modes 
exist. We could say that these slow modes radiate out instead of propagating through 
structures. 
To investigate more about the slow modes which exists in the suppression 
band of planar EBG structures, we look to the second EBG sample. The radiation loss 
diagram for this sample and the reference case are plotted in Figure  3.62. The 20 dB 
suppression band which extends from 1.9 GHz to 10.8 GHz is marked in the figure. 
The reduction of radiation loss in the frequency band where the EBG pattern is 
effective is obvious in the figure. Also, comparing the propagating modes from 
dispersion diagram with the local peaks of radiation loss diagram for the EBG 
patterned planes proves the validity of our idea. Therefore, the dispersive slow modes 
radiate out instead of propagating through structure. These radiating emissions may 
interfere with the performance of other parts of a system which should be considered 
in design of a system with high functionality.  
 
 
 
 
 
 
119 
 
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
 EBG Patterned Plane (First Case Study)
 Solid Plane
Rad
i
ation L
o
ss
Frequency (GHz)
20 dB Suppression Band
Dispersive Modes
Rad
i
ation L
o
ss
 
Figure  3.61. Radiation loss (1- |S
11
|
2
 - |S
21
|
2
) versus frequency for two cases: 1) the 
EBG patterned parallel planes and 2) solid parallel planes as reference. The S-
parameters are extracted using the setup shown in Figure  3.14. The planar EBG 
structure and the design parameters are shown in Figure  3.12. The parameters of the 
EBG structure are: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 
1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
 
 
120 
 
012345678910112131415
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
 EBG Patterned Plane (Second Case Study)
 Solid Plane
Radiation Los
s
Frequency (GHz)
20 dB Suppression Band
Dispersive Modes
Radiation Los
s
 
Figure  3.62. Radiation loss (1- |S
11
|
2
 - |S
21
|
2
) versus frequency for two cases: 1) the 
EBG patterned parallel planes and 2) solid parallel planes as reference. The S-
parameters are extracted using the setup shown in Figure  3.14. The planar EBG 
structure and the design parameters are shown in Figure  3.12. The parameters of the 
EBG structure are: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 
1.6 mm, M
G 
= 0.02 mm, ?
r 
= 200, and tan ? = 0.0015. 
 
 
 
 
121 
3.3.11. Effects of Variation of Elements in Circuit-Transmission Line Model on Bandgaps 
In this subsection, we study the effects of variation of the value of elements in 
the equivalent circuit?transmission line model of the planar EBG structure on its 
bandgaps. This study will help us to control the location of bandgaps by changing the 
physical parameters in the structure. Primarily, our focus is on the miniaturized EBG 
structures made of thin slab of high-k materials. At the end of this subsection, we will 
briefly speak about the possible existing differences in EBG structures made of 
dielectrics with low permittivity or higher thickness. Figure  3.63 to Figure  3.72 
present the relationships of the location of bandgaps and the attenuation levels versus 
the different values of elements modeling a unit cell of the first sample of planar EBG 
structure discussed in Subsection  3.3.2. We varied the value of elements once at a 
time in these figures where the values of other elements are equal to their nominal 
values. The nominal values are extracted using our developed model. For the first 
study case, the nominal values of elements in Figure  3.48 are as follows: C
PH
 = 27.9 
pF, L
PH
 = 0.0587 nH, L = 0.5921 nH, C
L1
 = 0.3039 pF, C
L2
 = 1.7926 pF, C
M_PM
 = 
1.0633 pF, L
DP
 = 5.9511 pH, L
DM
 = 0.0936 nH, C
D
 = 1.6468 pF, d = 1.78 mm, d
1
 = 
P
W
 = 1.6 mm, and d
2
 = d - d
1
 = 0.18 mm. Figure  3.63 shows the variations of the 
dispersion/attenuation diagram for the different values of L = {0.2921, 0.5921, 
0.8921} nH. This change affects the lower edge of the first bandgap. Increasing L 
pushes the bandgap to lower frequencies. Also, this increment increases widths of the 
first three bandgaps which is clear in the plot. Also the attenuation level in the 
bandgaps increases. Generally, it is concluded that increasing L can provide a wider 
suppression bandgap with higher attenuation level. 
 
122 
 
 
0 500 1000 1500 2000 2500
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
F
r
equ
e
ncy (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
F
r
equ
e
ncy (
G
H
z
)
 
Figure  3.63. Relationship between the dispersion/attenuation diagram and the 
inductance L modeling the meander line in the planar EBG structure shown in Figure 
 3.18. The value of L is equal to (a) 0.2921 nH, (b) 0.5921 nH, and (c) 0.8921 nH. 
Figure  3.48 demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with the following 
specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
 
123 
The variations of the dispersion/attenuation diagram for the different values of 
C
L2
 = {1.2926, 1.7926, 2.2926} pF are presented in Figure  3.64. There is not any 
noticeable effect at the lower edge of the first bandgap by this change. However, 
increasing C
L2
 pushes the second and the third propagating modes to lower 
frequencies. Yet, these variations compared to the variations resulting from change of 
inductance are small. Increasing C
L2
 decreases the attenuation level in the first and the 
second bandgaps. Also, the second and the third propagating modes with the lower 
value of C
L2
 are more dispersive and slower modes which may widen the suppression 
band and increase the filtering effect. The change of C
L2
 obviously affects the higher 
order modes. The smaller C
L2
 provides the wider third bandgap. Therefore, we 
conclude that decreasing C
L2
 may provide a better effective suppression band. 
The variations of the dispersion/attenuation diagram for the different values of 
the equivalent capacitance which is resulting from the following combination (C
M_PM
 
|| C
L1
 || C
M_PM
) = {0.19337, 0.39337, 0.59337} pF are plotted in Figure  3.65. Change 
of the value of interline parasitic capacitances on the lower edge of the first bandgap 
is negligible. Increasing these capacitances slightly shifts the second and the third 
propagating modes to lower frequencies and at the same time the attenuation level in 
the second and the third bandgaps increases. It is concluded that increasing interline 
coupling capacitances provides a suppression band with higher efficiency. 
 
 
 
 
 
124 
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eque
ncy
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
eque
ncy
 (
G
H
z
)
 
Figure  3.64. Relationship between the dispersion/attenuation diagram and the 
capacitance C
L2 
modeling the meander line in the planar EBG structure shown in 
Figure  3.18. The value of C
L2
 is equal to (a) 1.2926 pF, (b) 1.7926 pF, and (c) 2.2926 
pF. Figure  3.48 demonstrates the equivalent circuit-transmission line model of EBG. 
The nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
 
 
125 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
eque
ncy
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
eque
ncy
 (
G
H
z
)
 
Figure  3.65. Relationship between the dispersion/attenuation diagram and the (C
M_PM
 
|| C
L1
 || C
M_PM
) modeling the interline coupling between the meander line and two 
adjacent patches and between the arms of meander line in the planar EBG structure 
shown in Figure  3.18. The value of the equivalent capacitance is equal to (a) 0.19337 
pF, (b) 0.39337 pF, and (c) 0.59337 pF. Figure  3.48 demonstrates the equivalent 
circuit-transmission line model of EBG. The nominal values of elements in the model 
are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, h = 114.3 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? 
= 0.0015. 
 
Figure  3.66 presents the variations of the dispersion/attenuation diagram for 
the different values of L
PH
 = {0.0287, 0.0587, 0.0887} nH. The effects of these 
variations are negligible on the lower edge of the first bandgap. However, increasing 
 
126 
L
PH
 pushes the second, the third, and the fourth propagating modes shown in the plot 
to lower frequencies. Also, this increment decreases the attenuation level in the first 
and second bandgap. Therefore, designing for low values of L
PH
 may be more 
efficient in providing a suppression band which will be starting at low frequencies. 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
uency
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
uency
 (
G
H
z
)
 
Figure  3.66. Relationship between the dispersion/attenuation diagram and the 
inductance L
PH
 modeling the patch in the planar EBG structure shown in Figure  3.18. 
The value of L
PH
 is equal to (a) 0.0287 nH, (b) 0.0587 nH, and (c) 0.0887 nH. Figure 
 3.48 demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with following the 
specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
127 
The variations of the dispersion/attenuation diagram for the different values of 
C
PH
 = {17.903, 27.903, 37.903} pF are shown in Figure  3.67. Increasing C
PH
 pushes 
the lower edge of the first bandgap to lower frequencies. The changes of the second 
and the fourth propagating modes are negligible and the third propagating mode shifts 
to lower frequencies. Therefore, by increasing C
PH
, widths of the first and the third 
bandgaps and their attenuation levels increase. However, the width of the second 
bandgap and its attenuation level decrease. The third propagating mode, by increasing 
C
PH
, becomes more dispersive or slower mode which may result in widening of the 
suppression band and increasing of the filtering effect. As a result, increasing the 
value of C
PH
 increases the efficiency of suppression bandgap. Generally, in the 
designed EBG structures the value of inductance L is considerably high; therefore, 
variation in the value of L shows its effects more obviously than the variations in 
other elements. It should be considered that in the designs, the inductance is in the 
order of 10
-9
 but the capacitance is in the order of 10
-12
.  
Figure  3.68 shows the variations of the dispersion/attenuation diagram for the 
different values of d
1
 = {0.75 ? P
W
, P
W
, 1.25 ? P
W
}. Increasing the length of patch-
microstrip pushes all the propagating modes to lower frequencies. However, the 
changes in location of the second propagating mode are more than the changes in the 
locations of other propagating modes in these diagrams. Therefore, the first bandgap 
narrows down and reversely, the second bandgap widens. The level of attenuation in 
the first bandgap decreases and in the second bandgap increases. The visible higher 
order propagating modes in the diagrams are slow modes. So it is possible by 
increasing d
1 
we widen the suppression band and increase the filtering effect. 
 
128 
Therefore, we conclude that increasing d
1
 may increase efficiency and width of the 
suppression bandgap.  
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.67.  Relationship between the dispersion/attenuation diagram and the 
capacitance C
PH
 modeling the patch in the planar EBG structure shown in Figure 
 3.18. The value of C
PH
 is equal to (a) 17.903 pF, (b) 27.903 pF, and (c) 37.903 pF. 
Figure  3.48 demonstrates the equivalent circuit-transmission line model of EBG. The 
nominal values of elements in the model are extracted for the EBG with following the 
specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 
mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
129 
The variations of the dispersion/attenuation diagram for the different values of 
d
2
 = {0.75 ? D
2
, D
2
, 1.25 ? D
2
} where
 
D
2
 = 2 ? M
B
 + N ? M
W 
+ (N-1) ? M
G
 are shown 
in Figure  3.69. Increasing d
2
 negligibly pushes the first, the second, and the third 
propagating modes down in frequencies. Therefore, the change of the first and the 
second bandgaps are negligent. Increasing d
2
 shifts the fourth propagating mode to 
lower frequencies so the third bandgap narrows down. We conclude from these 
results that the decrease of d
2
 in the planar EBG structures made of thin slab of high-k 
dielectrics may increase width of the suppression band. 
The variations of the dispersion/attenuation diagram for the different values of 
L
DP
 = {0.9511, 5.9511, 10.9511} pH are demonstrated in Figure  3.70. The change of 
value of this element affects the first, the second, and the third propagating modes and 
as a result the first and the second bandgaps are negligibly affected. Increasing L
DP
 
pushes the fourth propagating mode down in frequencies. Therefore, the third 
bandgap narrows down. It can be concluded that we can get a more efficient 
suppression bandgap by decreasing L
DP
. 
The variations of the dispersion/attenuation diagram for the different values of 
L
DM
 = {43.576, 93.576, 143.576} pH are shown in Figure  3.71. Increasing L
DM
 
pushes the first propagating mode to lower frequencies. The effect of this change on 
the second and the third propagating modes is negligible. Therefore, the first and the 
second bandgaps slightly change. Increasing L
DM
 shifts the fourth propagating mode 
down in frequencies; this will narrow the third bandgap. As a conclusion, decreasing 
L
DM
 may widen the suppression bandgap but increasing L
DM
 may push the bandgap to 
lower frequencies. Designing L
DM
 with low value can be more efficient. 
 
130 
 
0 500 1000 1500 2000 2500 3000 3500
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
que
ncy
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
Fr
e
que
ncy
 (
G
H
z
)
 
Figure  3.68. Relationship between the dispersion/attenuation diagram and the length 
of patch-microstrip d
1
 modeling the phase difference along the patch in the planar 
EBG structure shown in Figure  3.18. The value of d
1
 is equal to (a) 0.75 ? P
W
, (b) P
W
, 
and (c) 1.25 ? P
W
. P
W
 is the width of patch-microstrip, the spacing between two 
consecutive patches d
2
 = 2 ? M
B
 + N ? M
W 
+ (N-1) ? M
G
, and the length of a unit cell 
of EBG d = d
1
 + d
2
. Figure  3.48 demonstrates the equivalent circuit-transmission line 
model of EBG. The nominal values of elements in the model are extracted for the 
EBG with following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, 
M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
131 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
equency (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
Fr
equency (
G
H
z
)
 
Figure  3.69. Relationship between the dispersion/attenuation diagram and the length 
of meander-microstrip d
2
 modeling the phase difference across the meander line in 
the planar EBG structure shown in Figure  3.18. The value of d
2
 showing the spacing 
between the two consecutive patches is equal to (a) 0.75 ? D
2
, (b) D
2
, and (c) 1.25 ? 
D
2
 where D
2
 = 2 ? M
B
 + N ? M
W 
+ (N-1) ? M
G
. M
W
 is the width of meander-
microstrip, the length of patch d
1
 = P
W
, and the length of a unit cell of EBG d = d
1
 + 
d
2
. Figure  3.48 demonstrates the equivalent circuit-transmission line model of EBG. 
The nominal values of elements in the model are extracted for the EBG with 
following the specifications: P
W
 = 1.6 mm, h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? = 0.0015. 
 
 
 
132 
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.70 Relationship between the dispersion/attenuation diagram and the 
inductance L
DP
 modeling the step discontinuity between the patch and the meander 
line in the planar EBG structure shown in Figure  3.18. The value of L
DP
 is equal to (a) 
0.9511 pH, (b) 5.9511 pH, and (c) 10.9511 pH. Figure  3.48 demonstrates the 
equivalent circuit-transmission line model of EBG. The nominal values of elements in 
the model are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, 
h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, 
and tan ? = 0.0015. 
 
 
 
 
133 
 
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.71. Relationship between the dispersion/attenuation diagram and the 
inductance L
DM
 modeling the step discontinuity between the patch and the meander 
line in the planar EBG structure shown in Figure  3.18. The value of L
DM
 is equal to 
(a) 43.576 pH, (b) 93.576 pH, and (c) 143.576 pH. Figure  3.48 demonstrates the 
equivalent circuit-transmission line model of EBG. The nominal values of elements in 
the model are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, 
h = 114.3 ?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, 
and tan ? = 0.0015. 
 
 
 
 
134 
Figure  3.72 shows the variations of the dispersion/attenuation diagram for the 
different values of C
D
 = {0.6468, 1.6468, 2.6468} pF. Increasing C
D
 negligibly 
pushes the first propagating mode down in frequencies. However, the second and the 
third propagating modes obviously shift to lower frequencies. As a result, the first 
bandgap narrows down but change the in width of the second bandgap is very small. 
The change of the fourth propagation mode is negligible. Therefore, width of the third 
bandgap increases. We conclude that decreasing C
D
 may widen the first bandgap but 
effectsw on the suppression bandgap may be negligible. 
The conclusions up to this end are related to planar EBG structures made of 
thin slab of high-k materials. We can not generalize these results to materials with 
lower permittivity or higher thickness. For example, in the fourth sample discussed in 
Subsection  3.3.2 where ?
r
 = 4.3 and h = 114.3 ?m the second and the third 
propagating modes are more sensitive to variation of L
PH
 and C
PH
. The variation of 
elements of the step discontinuity does not affect these modes noticeably. However, 
in the third sample discussed in Subsection  3.3.2 where ?
r
 = 140 and h = 500 ?m the 
variation of elements of the step discontinuity affects the second and the third 
propagating modes clearly. 
 
 
 
 
 
 
 
135 
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.72. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the meander 
line in the planar EBG structure shown in Figure  3.18. The value of C
D
 is equal to (a) 
0.6468 pF, (b) 1.6468 pF, and (c) 2.6468 pF. Figure  3.48 demonstrates the equivalent 
circuit-transmission line model of EBG. The nominal values of elements in the model 
are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, h = 114.3 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? 
= 0.0015. 
 
 
 
136 
In continue of our discussion, the relationship between the 
dispersion/attenuation diagram and the capacitance of step discontinuity for the third 
and the fourth case studies are plotted in Figure  3.73 and Figure  3.74, respectively. 
The nominal values of the elements of the model shown in Figure  3.48 for the third 
sample are as follows: C
PH
 = 6.5507 pF, L
PH
 = 0.17897 nH, L = 0.62373 nH, C
L1
 = 
0.38232 pF, C
L2
 = 0.83432 pF, C
M_PM
 = 1.5698 pF, L
DP
 = 0.046319 nH, L
DM
 = 
0.31122 nH, C
D
 = 1.7066 pF, d = 1.78 mm, d
1
 = P
W
 = 1.6 mm, and d
2
 = d - d
1
 = 0.18 
mm. The variations of the dispersion/attenuation diagram for the different values of 
C
D
 = {0.7066, 1.7066, 2.7066} pF are shown in Figure  3.73. This figure shows that 
the increasing C
D
 pushes the propagating modes and the bandgaps to lower 
frequencies. This increment narrows the first bandgap and widens the second and the 
third bandgaps. The third propagating mode becomes a slower mode by this 
increment. Therefore, if this increment combines the second and the third bandgaps 
then the suppression band is wide. We may conclude that decreasing C
D
 widens the 
first bandgap and increasing C
D
 pushes the first narrow bandgap to lower frequencies. 
Totally, the effect of this change on the suppression bandgap is not very clear.  
The nominal values of the elements of the model shown in Figure  3.48 for the 
fourth sample are as follows: C
PH
 = 136.73 pF, L
PH
 = 0.0703 nH, L = 22.847 nH, C
L1
 
= 0.074304 pF, C
L2
 = 4.074 pF, C
M_PM
 = 0.69867 pF, L
DP
 = 0.018134 nH, L
DM
 = 
0.10791 nH, C
D
 = 0.72452 pF, d = 21.8 mm, d
1
 = P
W
 = 20 mm, and d
2
 = d - d
1
 = 1.8 
mm. Figure  3.74 shows the variations of the dispersion/attenuation diagram for the 
different values of C
D
 = {0.22452, 0.72452, 1.22452} pF. Figure  3.74 demonstrates 
that this change has negligible effect on the propagating modes and the bandgaps. So 
 
137 
we can not clearly change the efficiency of the suppression band by varying this 
element. 
After studying the relationship between the dispersion diagram and the 
elements modeling the EBG, by considering 1) the frequency region or the 
suppression band where the EBG is effective and 2) the available spacing for a unit 
cell design we can change the proper elements in the model or basically the physical 
design parameters to get the required suppression band. Therefore, the procedures of 
design might be done by using recursive optimization code. We can manually define 
the priority to change the elements in model by defining weight coefficients for these 
elements. Similar studies to what we did in this subsection could introduce the weight 
coefficients. 
 
 
 
 
 
 
 
 
 
 
 
 
 
138 
 
 
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
5
6
7
8
9
10
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.73. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the meander 
line in the planar EBG structure shown in Figure  3.18. The value of C
D
 is equal to (a) 
0.7066 pF, (b) 1.7066 pF, and (c) 2.7066 pF. Figure  3.48 demonstrates the equivalent 
circuit-transmission line model of EBG. The nominal values of elements in the model 
are extracted for the EBG with following the specifications: P
W
 = 1.6 mm, h = 500 
?m, M
W 
= 0.02 mm, M
B 
= 0.02 mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, ?
r 
= 140, and tan ? 
= 0.0015. 
 
 
 
 
139 
 
 
0 100 200 300 400 500
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
4.4
4.8
Wave Number, K
x
Fr
e
q
ue
nc
y
 (
G
H
z
)
(a) Atte.
(a) Disp.
(b) Atte.
(b) Disp.
(c) Atte.
(c) Disp.
K
x
 = ? / d
Fr
e
q
ue
nc
y
 (
G
H
z
)
 
Figure  3.74. Relationship between the dispersion/attenuation diagram and the 
capacitance C
D
 modeling the step discontinuity between the patch and the meander 
line in the planar EBG structure shown in Figure  3.18. The value of C
D
 is equal to (a) 
0.22452 pF, (b) 0.72452 pF, and (c) 1.22452 pF. Figure  3.48 demonstrates the 
equivalent circuit-transmission line model of EBG. The nominal values of elements in 
the model are extracted for the EBG with following the specifications: P
W
 = 20 mm, h 
= 114.3 ?m, M
W 
= 0.2 mm, M
B 
= 0.2 mm, M
L 
= 20 mm, M
G 
= 0.2 mm, ?
r 
= 4.3, and 
tan ? = 0.02. 
 
 
 
140 
Chapter 4 : Switching Noise and Electromagnetic Noise in High 
Speed Packages 
The most common type of noise in high speed packages is the simultaneous 
switching noise (SSN). SSN, also referred to as voltage bounce or delta-I noise, is 
caused by the switching of digital circuit between two different logic levels. When 
switching takes place, a surge of current travels between power and ground planes 
causing a voltage drop between them and a voltage drop across segments of the 
power plane itself. These voltage drops are directly proportional to the current 
switching rate and the effective inductance of current paths. SSN was studied 
extensively in the past decade  [1]- [4]. There are waves propagating between metallic 
layers of laminated substrate acting as parallel plate waveguide. These waves 
supported by traveling current on planes can interfere with other vias passing through 
these planes or connected to these planes. When the traveling waves reach the edges, 
part of their power radiate out and the remainder reflect back inside the package. The 
waves radiated out of package may interfere with other surrounding electronics. 
Another design artifact that exacerbates the noise induced by switching is the natural 
resonance of the package itself. If the natural resonant frequencies of the package 
encompass the switching frequency or the clock frequency and its harmonics, then the 
potential for further disturbance to the reference voltage levels increase and equally 
important the potential for EMI increases.  
In the following, we explain briefly the range of frequencies where we have to 
protect vias and interconnects from EM interference. There are vias in the package 
which are routing the signal lines through layers from one trace to another. The power 
 
141 
spectral density of the electrical signal on signal lines (also known as data bus and 
address bus) is concentrated around the operating frequencies and their harmonics. 
Therefore, to provide clean data we have to mitigate the electromagnetic noise around 
the operating frequency and its related harmonics. Suppressing noise from clock-
transmission lines is equally important. The bus and clock operate at the same 
frequency; however, the rate of external clock in a system discussed extensively in 
the subject of noise suppressions in PCB boards is a fraction of the internal clock or 
core clock of the state-of-the-art processors which is the subject of our discussion in 
this chapter. The internal bus or the backside bus in processor package is operating at 
the same speed of the processor. The external bus or front side bus (FSB) connecting 
the processor to external memory and peripherals is operating at the rate of external 
clock on the PCB.  
Other types of vias in the package connect the power planes to electronic 
circuits to provide power and ground. Switching or change of logic in those circuits 
leads to a surge of current from power planes to circuits through vias which induces 
EMI noise in whole band of frequencies. Therefore, for the suppression of switching 
noise, the interest is in broad band suppression. 
 
4.1. Analytical Model of EM Noise in Packages 
Coupling of EMI noise in a multilayer package occurs both transversally and 
vertically. The field propagating in transverse direction can couple to vias and 
interconnects passing through layers  [2],  [3] or can couple vertically, through cut-outs 
and apertures in the planes, to another layers  [63],  [64] and traps the vias in those 
 
142 
layers. Also, the field can penetrate through conductor plane; however this 
penetration is negligible beyond tens of MHz where the thickness of metallic planes 
is much larger than the skin depth. In this study, we mainly focus on the transverse 
coupling between vias and interconnects located in the same layer over the range of 
frequencies above hundreds of megahertz. Therefore, we analyze only a single plane 
pair. 
Characterization of scattering parameters between interconnects in boards or 
packages is important for understanding the potential of the board or the package to 
transfer electromagnetic energy or noise between two specific points or ports. 
Impedance parameters which could be extracted from S-parameters are alternatively 
used for characterization of interconnects  [43]. Analyzing S-parameters or Z-
parameters help the package designers to estimate the frequency range where the EMI 
noise and switching noise are strong. In this section, and without loss of generality, 
we consider impedance parameters. Our analysis is based on the principle of 
resonators modeled with lumped elements  [61],  [62]. Following, we introduce this 
model and verify its performance. 
A 3-D view of the single plane pair is shown in Figure  4.1 (a). The structure 
considered here is basically a parallel plate waveguide. The top and bottom metallic 
plates are rectangular with dimensions (a ? b) and conductivity ?. A dielectric with 
thickness h, permittivity ?, loss angle ?, and permeability ? is sandwiched between 
the metallic plates. The dimensions of the microprocessor package satisfy the 
following conditions: a >> h, b >> h, and h << ?, where ? is the wavelength in the 
dielectric material. With such conditions, the complex impedance between two 
 
143 
different ports, as shown in Figure  4.1 (b), using planar resonator model  [61],  [62] is 
given by : 
 
()
??
==
?
=
N
n
M
m mn
mnjmni
ij
kkab
NN
hjZ
00
22
)( ???  
 Equation  4-1 
Equation  4-1 may be approximated by an equivalent circuit model  [62] as: 
??
==
++
=
N
n
M
m
mnmn
mn
mnjmni
ij
GCj
Lj
NN
Z
00
1
)(
?
?
?  
 Equation  4-2 
where k = k' - jk? is the wave number with ???=?k  and ()2tan hk
s
????? +=?? . 
)(
2
?? abhL
mnmn
= , habC
mn
?= , ?
?
?
?
?
?
+=
hh
ab
G
smn
mnmn
?
??
?
tan , and                                                             
)cos()cos( bynaxmN
iimnmni
????= .                                                                                                      
Here, m and n are the indices of the (m,n) mode corresponding to the resonant 
frequency f
mn
. ???? /2/1
mnmnmnmnmn
kfCL ===  and 
()()
22
// bnamk
mn
?? += . )/(2 ????
mnsmn
=  is the skin depth for the (m,n) mode 
and the ratio h
s
/?  represents the normalized skin depth with respect to the separation 
between the metallic layers. ?
m
, ?
n
 = 1 for m, n = 0 and ?2 otherwise. (x
i
, y
i
) and (x
j
, 
y
j
) are the coordinates of the via (port) at the ith and jth locations, respectively. The 
dimension of each port is neglected. Z
ii
 is the input impedance at the ith port. Z
ij
 is the 
mutual impedance (trans-impedance) between the ith and jth ports. Input and mutual 
impedances introduce noise voltage and/or bounce in the voltage level in power 
planes.  
 
144 
(a)
a
h
y
z
x
b
?
r
, ?
(b)
Port 1
Port 2
y
x
(x
1
, y
1
)
(x
2
, y
2
)
 
Figure  4.1. Single plane pair in the package consisting of metal-dielectric-metal 
layers is modeled as parallel plate waveguide. (a) 3-D view.  (b) Top view. Typical 
dimensions of a microprocessor package are: a = b = 4 cm. Each via shown in the 
figure represents a port. Port 1 and Port 2 are located at (x
1
 = 1 cm, y
1
 = 1 cm), and (x
2
 
= 2 cm, y
2
 = 2 cm), respectively. 
 
 
In the following, we discuss a numerical experiment to validate the analytical 
expression for the impedance Z
ij
 extracted from the resonator model based on the 
planar structure. Ansoft?s HFSS is used to perform full-wave simulation. The case 
study is a 4 cm ? 4 cm package. Conductor is copper and dielectric is FR-4 with 
 
145 
thickness of 200 ?m. The first port is located at (1 cm, 1 cm) and the second one at (2 
cm, 2 cm). Figure  4.2 (a) and (b) show a comparison between the magnitudes of the 
impedances Z
21
 and Z
22
 calculated using full-wave simulation and the calculations 
obtained from two different analytical formulas including Equation  4-1 incorporating 
the resonator model and Equation  4-2 showing the resonator model approximated 
with the equivalent circuit lumped elements. These results confirm that Equation  4-2 
is a good approximation for Equation  4-1.   
Furthermore, the magnitude of Z
21
 from the planar analysis compares very 
favorably to that obtained using numerical full-wave analysis. However, the 
magnitude of the input impedance, Z
22
, is in close agreement at low frequencies and 
the discrepancy increases at higher frequencies. This difference is expected as in full-
wave simulation at the port discontinuity several modes including propagating and 
evanescent waves with fields in x, y, and z directions are excited. However, in planar 
analysis only E-fields in z-direction are considered. At the port junction, and at higher 
frequencies, the variation of the field in all directions becomes more complex causing 
more variation from planar analysis. Evanescent fields, on the other hand, diminish in 
strength as the distance from the excited port increases, therefore, we expect a good 
match between the different calculation methods for the trans-impedance Z
21
 as 
shown in Figure  4.2 (b). 
 
 
 
 
 
146 
 
2 4 6 8 10 12 14
2
4
6
8
10
12
Frequency (GHz)
M
a
gni
t
ude
 of
 Tr
a
n
s
-
I
m
pe
da
nc
e
 Z
21
 
(
ohm
)
2 4 6 8 10 12 14
5
10
15
20
Frequency (GHz)
M
a
gni
t
ude
 of
 I
nput
 I
m
pe
da
nc
e
 Z
22
 
(
ohm
)
Analytical from Equation 4-1
Analytical from Equation 4-2
HFSS Simmulation
M
a
gni
t
ude
 of
 Tr
a
n
s
-
I
m
pe
da
nc
e
 Z
21
 
(
ohm
)
M
a
gni
t
ude
 of
 I
nput
 I
m
pe
da
nc
e
 Z
22
 
(
ohm
)
 
Figure  4.2. Magnitude of (a) trans-impedance between ports 1 and 2, and (b) input 
impedance at port 2 for the package shown in Figure  4.1. The plane separation is h = 
200 ?m. The Impedances are obtained through three different methods including: 
analytical method using Equation  4-1 plotted by the solid line, analytical method 
using Equation  4-2 plotted by the circle symbol line, and full-wave simulation using 
HFSS plotted by the dashed line. 
 
 
 
147 
4.2. SSN and EMI Noise Mitigation in Packages using High-K Embedded 
Capacitance 
Following we will discuss about the concept of EM noise mitigation in IC 
packages using very high value embedded capacitance. The embedded capacitance 
technique as a strong EM noise suppressor has been used in the microwave regimes in 
the PCBs. We investigate about this technique in IC packages using very high-k 
materials which are newly commercially available. We need this study to assess the 
performance of one of our proposals related to the application of EBG structures in IC 
packages.  
In the processor packaging compare to PCBs, we need to suppress EM noise 
at higher frequencies where the loss tangent becomes comparable or even higher than 
normalized skin depth. Therefore, in our analysis we include the dielectric losses as 
well as the skin depth effect for accurate modeling. Our Objective is to study 
analytically the effect of embedded capacitance and the limitations of using high-k 
materials. 
 
4.2.1. Analytical Relationship between Package Topology and Noise Suppression 
The complex resonant frequencies for each mode of the package layer are 
extracted from Equation  4-2. For typical dimensions of IC packages such as 
microprocessors, the ratio 
mnmnmn
CLG 4
2
is much smaller than 1. For such case, the 
complex resonant frequencies of the package layer in S-domain (Laplace domain) 
reduce to:  
 
148 
?
?
?
?
?
?
?
?
???=?=
mn
mnmn
mn
mn
mn
smnsmnmn
C
LG
j
C
G
jS
8
1
2
2
???  
 Equation  4-3 
where ?
smn
 and ?
smn
 represent the wave attenuation and phase constant respectively.                       
Substituting L
mn
, C
mn
, and G
mn
 in Equation  4-3, the second term of the imaginary part 
is given by:  
2
2
tan
88
?
?
?
?
?
?
?
?
+==
hh
ab
C
L
C
G
I
smnmn
mn
mn
mn
mn
?
?
??
 
 Equation  4-4 
Equation  4-4 shows that both the dielectric losses and the skin depth are 
affecting the imaginary part of the resonant frequency. The normalized skin depth 
versus frequency is plotted in Figure  4.3 for a plane pair with the following 
parameters: a = b = 4 cm, h = 114.3 ?m, and copper conductors. This figure indicates 
the range of frequencies where each type of loss term is dominant and where both of 
them need to be considered. If one of the terms is dominant over the other then 
Equation  4-4 can be rewritten as: 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
>>
<<
?
h
h
ab
h
h
ab
?
smn
mn
smn
???
??
??
??
?
tan)(tan
8
tan
4
2
3
 
 Equation  4-5 
For the typical dimensions and materials used in microprocessor packages, the 
computed value of I is always close to few Hertz (
mn
? ?<< ). Therefore, the complex 
resonant frequencies are approximated as: 
mn
mn
mn
mn
j
C
G
S ???=
2
 
 
149 
 Equation  4-6 
Substituting C
mn
 and G
mn
 in Equation  4-6, the wave attenuation constant ?
smn
 is then 
expressed as: 
?
?
?
?
?
?
+?=
h
smnmn
smn
?
?
?
? tan
2
 
 Equation  4-7 
 
0 5 10 15 20 25 30 35 40 45 50
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Frequency (GHz)
?
s
 / h
tan ? = 0.02
tan ? = 0.0015
 
Figure  4.3. Normalized skin depth versus frequency for a single plane pair in the 
package shown in Figure  4.1. The dimensions of the layer are: a = b = 4 cm, h = 
114.3 ?m, and S/m. 108.5
7
?=?  
 
Equation  4-7 shows that increasing the dielectric losses or decreasing the dielectric 
thickness h, will lead to increase of the wave attenuation constant approximately at 
 
150 
the resonant frequency ?
mn
. If any of the terms in Equation  4-7 is dominant over the 
other, we can rewrite Equation  4-7 as: 
?
?
?
?
?
>>?
<<?
=
h
hh
smn
mn
smnsmnmn
smn
?
???
?
?
??
?
tantan5.0
tan
2
 
 Equation  4-8 
Thus, for dielectrics where the loss tangent is much smaller than the normalized skin 
depth, the wave attenuation constant becomes inversely proportional to the layer 
separation h. Conversely, if the loss tangent is much larger than the normalized skin 
depth, the wave attenuation constant becomes directly proportional to the loss 
tangent. Shrinking the thickness of dielectric between the metallic layers and/or 
increasing dielectric loss lead to suppress of the EMI noise at the resonant 
frequencies. 
Including both, the dielectric losses and skin depth effect, and rearranging 
Equation  4-2, the mutual impedance can be expressed as: 
()
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
+
+
+?
?
?
?
?
?
+
?
?
?
?
?
?
+
?=
??
==
??
?
??
??
??
?
??
?
??
?
?
22
2
222
2
200
tantan
tan
)(
cmn
smn
mn
cmn
cmn
smn
mn
smn
mnN
n
M
m
ijmnij
F
h
F
j
F
h
hh
Z
 
 Equation  4-9 
where  
)cos()cos()cos()cos(
22
bynaxmbynaxm
ab
jjii
mn
ijmn
????
??
=?
 and 
() 1/
22
?= ???
mncmn
F . 
 
151 
The real part of mutual impedance at the resonant frequency of the (m
k
, n
k
) mode is: 
{ }
()
?
??
?
??
?
??
?
??
?
h
F
h
h
h
ZReal
N
n
M
m
cmnnm
smn
mn
smn
mnijmn
nsm
nm
nijm
nmij
k
n
k
mkk
k
n
k
mnm
kk
kk
kk
kk
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
+?
?
?
?
?
?
+
?
?
?
?
?
?
+?
+
?
?
?
?
?
?
?
?
+
?
=
==
?
00
22
2
2
tan
tan
tan
)( 
),(),(
 
 Equation  4-10 
If we assume that the resonant frequencies of the different modes are distant enough, 
then we can neglect effects of mutual coupling between different modes.  So, at each 
resonant frequency, the effect of the self resonant frequency on the real part of 
impedance is much larger than the effect of the resonant frequencies of other modes. 
The second term in Equation  4-10 which represents all the mutual coupling terms is 
neglected compared to the first term. The real part of the mutual impedance of the 
resonant frequency 
kk
nm
?  is approximated as: 
{ }
?
?
?
?
?
?
?
?
+
?
?
h
h
ZReal
kk
kk
kk
kk
nsm
nm
nijm
nmij
?
??
?
?
tan
)(  
 Equation  4-11 
Similarly, the imaginary part of the mutual impedance at the resonant frequency of 
the (m
k
,n
k
) mode can be written as: 
{ ()
( )
()
??
==
+?
?
?
?
?
?
+
?
=
?
N
n
M
m
nmcmn
smn
nm
mn
ijmnnmcmn
nm
nmij
kk
kk
kk
k
n
k
mnm
kk
kk
F
h
F
h
ZI
00 2
2
2
2
tan
)( mag
),(),(
?
?
?
?
?
?
??
?
 
 Equation  4-12 
At each resonant frequency of the (m
k
,n
k
) mode, the self resonant frequency does not 
contribute to the imaginary part of the  mutual impedance in Equation  4-12. The 
contribution of the other resonant frequencies to the double summation can be 
 
152 
estimated as follows. Let us define the ratio 
kk
nmmn
?? to be equal x. Then the 
argument of the sum in Equation  4-12 is expressed as: 
()
222
21
2
)1()(
1
)(
?++
?
=
xxx
x
xf
???
?
.  
Where 
22
1
tan
mn
h
ab
??
?
?? =
,
kk
nmmn
h
ab
?????
?
?
223
2
2
= , and
22
mn
h
ab
??
?
? = .  
The practical dimensions and materials of the microprocessor packages imply the 
following conditions: 
1
22
<<
?
=
mnnm
ijmn
kk
ab
???
? , ?
1 
 << 1, ?
2
  << 1, and ? << 1.  
If the condition () xxx
21
2
1 ??? +>>?  is satisfied then we have two different 
approximations for the function f. The first approximation is: 
1
)1(
)(
2
<<
?
?
x
xf
?
?
 if ( ) ?? >>?1
2
x . Therefore, the effect of high 
order resonant frequency modes on the (m
k
,n
k
)  mode can be ignored. The second 
approximation is: 
??
??
??
ab
h
xf
kk
nm
mn
22
)( ??  for the lower-order resonant 
frequencies (x << 1). This value can be very small using high-k thin dielectric. Thus 
only around x = 1, f(x) ? 0. Therefore, with the assumption: 1) the resonant modes are 
not coupled to each other, and 2) the filling between layers is high permittivity, thin 
dielectric, it can be concluded that: 
( ) 0)(  ?
kk
nmij
ZImag ?  
 Equation  4-13 
 
153 
From Equation  4-11 and Equation  4-13, the mutual impedance at the frequency 
kk
nm
? is represented as: 
?
?
?
?
?
?
?
?
+
?
?
h
h
Z
kk
kk
kk
kk
nsm
nm
nijm
nmij
?
???
?
tan
 
)(  
 Equation  4-14 
Validity of above mentioned assumptions through selection of dielectric 
material and its thickness in a package layer leading to approximate formula Equation 
 4-14 are studied through numerical examples. The magnitude, the real part, and the 
imaginary part of input and trans-impedances shown respectively by (a), (b), and (c) 
for a single plane pair defined in Figure  4.1 are illustrated in Figure  4.4 - Figure  4.9 
for the three different configurations. The solid line shows the actual value of 
impedance obtained by Equation  4-2 in the total frequency range under consideration. 
The square symbol shows the actual value of impedance at the resonant modes of a 
plane pair. The star symbol shows the approximate value of impedance at resonant 
modes obtained from Equation  4-11, Equation  4-13, and Equation  4-14. In the first 
configuration, FR-4 with thickness equal to 200 ?m is used. Figure  4.4 shows the 
input impedance at port 2, Z
22
. Figure  4.5 shows the trans-impedance between port 1 
and 2, Z
21
. In the second configuration, a dielectric with permittivity equal to 140, 
loss tangent equal to 0.0015, and thickness equal to 200 ?m is used. Figure  4.6 shows 
Z
22 
and Figure  4.7 shows Z
21
. In the third configuration, a dielectric with permittivity 
equal to 140, loss tangent equal to 0.0015, and thickness equal to 100 ?m is used. 
Figure  4.8 shows Z
22
 and Figure  4.9 shows the Z
21
. These results show that the 
approximation for the real part of the impedance, (Equation  4-11), works well. 
 
154 
Neglecting the imaginary part at resonant modes (Equation  4-13) leads to an error in 
the actual value of the magnitude of the impedance. However, this error at excited 
resonant modes is negligibly small (the excited modes are defined by the location of 
the port in the package layer). In other words, at the excited resonant frequencies 
which are shown by local peaks at the magnitude of impedance graph (or more 
correctly the frequency at which the imaginary part of impedance is equal to zero), 
the approximate expression for the impedance (Equation  4-14) estimates the 
acceptable impedance. At other resonant modes (the ones that are not excited), the 
value of the impedance is expressed by the coupling effects. It means other adjacent 
modes which are not vanished at the frequency of interest contribute to the value of 
impedance. 
Equation  4-14 indicates that the mutual impedance at each resonant frequency 
mode is decreased by reducing the separation between the planes, h, increasing the 
permittivity of the dielectric, ?, or increasing the loss tangent of the dielectric. Again 
two cases for approximating Equation  4-14 occur: If h
kk
nsm
/tan ?? << , then Equation 
 4-14 simplifies to:  ( ) ( )
kkkkkkkk
nsmnmnijmnmij
hZ ???? ??
2
)( . This means that 
2
)( hZ
kk
nmij
??  and 
75.0
)(
?
???
kk
nmij
Z . If h
kk
nsm
?? >>tan  then ( ) ( )
kkkkkk
nmnijmnmij
hZ ??? ??)( . Thus, 
hZ
kk
nmij
?)(?  and 
5.0
)(
?
???
kk
nmij
Z .  
 
 
 
 
 
 
155 
 
(a)
(b)
(c)
Excited Resonances
 
 
Figure  4.4. (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 200 ?m and 
dielectric material is FR-4. The Impedance from Equation  4-2 is plotted by the solid 
line. The impedance value at the resonant modes of a plane pair is marked by square 
symbol. The approximate value of impedance from Equation  4-11, Equation  4-13, 
and Equation  4-14 is marked by star symbol. 
 
 
 
 
 
156 
 
(a)
(b)
(c)
Excited Resonance
 
 
Figure  4.5.  (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation is h 
= 200 ?m and dielectric material is FR-4. The Impedance from Equation  4-2 is 
plotted by the solid line. The impedance value at the resonant modes of a plane pair is 
marked by square symbol. The approximate value of impedance from Equation  4-11, 
Equation  4-13, and Equation  4-14 is marked by star symbol. 
 
 
 
 
157 
 
(a)
(b)
(c)
Excited Resonances
 
 
Figure  4.6.  (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 200 ?m and 
dielectric material has permittivity of 140 and loss tangent of 0.0015. The Impedance 
from Equation  4-2 is plotted by the solid line. The impedance value at the resonant 
modes of a plane pair is marked by square symbol. The approximate value of 
impedance from Equation  4-11, Equation  4-13, and Equation  4-14 is marked by star 
symbol. 
 
 
 
 
158 
 
(a)
(b)
(c)
Excited Resonances
 
 
Figure  4.7. (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation is h 
= 200 ?m and dielectric material has permittivity of 140 and loss tangent of 0.0015. 
The Impedance from Equation  4-2 is plotted by the solid line. The impedance value at 
the resonant modes of a plane pair is marked by square symbol. The approximate 
value of impedance from Equation  4-11, Equation  4-13, and Equation  4-14 is marked 
by star symbol. 
 
 
 
 
159 
 
(a)
(b)
(c)
Excited Resonances
 
 
Figure  4.8. (a) Magnitude, (b) real part, and (c) imaginary part of input impedance at 
port 2 for the package shown in Figure  4.1. The plane separation is h = 100 ?m and 
dielectric material has permittivity of 140 and loss tangent of 0.0015. The Impedance 
from Equation  4-2 is plotted by the solid line. The impedance value at the resonant 
modes of a plane pair is marked by square symbol. The approximate value of 
impedance from Equation  4-11, Equation  4-13, and Equation  4-14 is marked by star 
symbol. 
 
 
 
 
160 
 
Excited Resonances
 
Figure  4.9. (a) Magnitude, (b) real part, and (c) imaginary part of trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. The plane separation is h 
= 100 ?m and dielectric material has permittivity of 140 and loss tangent of 0.0015. 
The Impedance from Equation  4-2 is plotted by the solid line. The impedance value at 
the resonant modes of a plane pair is marked by square symbol. The approximate 
value of impedance from Equation  4-11, Equation  4-13, and Equation  4-14 is marked 
by star symbol. 
 
 
 
 
161 
Therefore, in IC packages similar to PCB packages  [14],  [15], three factors 
contribute to noise suppression: Decreasing the thickness of the package, increasing 
the permittivity of the dielectric material of the package, or increasing the loss in the 
dielectrics. However, increasing loss is not an optimum solution for EMI and SSN 
noise suppression. The heat generated due to losses needs to be transferred out of the 
package, otherwise the performance of electronic circuit degrades and other types of 
noise related to heating of the circuit increases. Another drawback of using high-loss 
substrates is the decrease in the bandwidth of signal traces which is a concern for high 
data transfers rates in the future generation of microprocessors  [65]. Overall, the 
embedded capacitance method does not eliminate EM wave in any frequency range. 
Decoupling level depends to the limitations in fabrication techniques, such as: 
limitation in using different dielectric materials and minimum isolation distance 
between two metallic layers (dielectric thickness). 
 
4.2.2. Case Studies 
We follow our study on microprocessor package. In present microprocessor 
packaging technology, the typical dimension of a package is approximately 4 cm ? 4 
cm. The height of layers is varying from tens of ?m to hundreds of ?m. Conductors 
are made of copper. The permittivity of the filling dielectric is 4.3. The speed of the 
processor is less than 4 GHz. Without any loss of generality, these practical 
dimensions are used in the numerical experiments provided here. The goal here is to 
provide a solution to minimize the EMI within a broad-band of frequencies including 
the operating frequency of the microprocessor and the package resonant frequencies. 
 
162 
To this end, we focus on the impedance parameters of two interconnects located 
within the package (see the ports in Figure  4.1). The first interconnect (port) is 
located at (x
1
 = 1 cm, y
1
 = 1 cm). The second one is located at (x
2
 = 2 cm, y
2
 = 2 cm). 
The magnitude of the input impedance at port 2 and the trans-impedance 
between the two ports, for two different cases h =100, 200 ?m are shown in Figure 
 4.10. The first three resonances are marked with different symbols in the plot and 
table. The similar symbols in each graph show corresponding resonances. As can be 
seen from the tables, the corresponding resonance frequencies have changed slightly 
which is expected from 
smn
?  in Equation  4-3. Here, 
mn
?  is not function of h, and I 
(defined in Equation  4-4) is function of h but it is negligibly small. Figure  4.10 shows 
for the second case (case with larger h), the magnitude of impedance has decreased at 
resonances. Figure  4.11 and Figure  4.12 show the magnitude of trans-impedance 
between the ports and input impedance at port 2, respectively, for three different 
cases: (?
r
 = 4.3, ?tan ? = 0.02),? ?
r
 = 30, tan ? = 0.02), and (?
r
 = 30 and tan ? = 0.0015). 
For these three cases, the dielectric thickness h = 150 ?m. In Figure  4.11 and Figure 
 4.12, a couple of lower frequencies are marked with different symbols in the plot and 
table. In each plot, similar symbols on the curves show the corresponding resonances. 
For the second and third cases the resonant frequencies are shifted to lower values by 
the approximate ratio ? {?
r1 
/ (?
r2
 or ?
r3
)} = 0.3786 using 
smn
?  in Equation  4-3 where 
smn
?  is inversely proportional to square root of permittivity. I is function of ? and loss 
tangent but it is negligibly small. Those slight differences in resonance frequencies 
between the second and third case are due to the differences in the loss tangent.  At 
the corresponding resonant frequencies in the second case which has same loss 
 
163 
tangent as the first one, the decrease of the magnitude of the impedance by increasing 
?
r
 is obvious in both figures. In the third case, we have increased the permittivity but 
decreased loss tangent in comparison to the first case which is inefficient in 
suppressing the magnitude of impedance at resonances. 
The progress in material science provides composite dielectrics with very high 
permittivity (?
r
 ? 100) known as high-k dielectrics. In fact, several companies have 
started commercially producing such material with low loss. Those dielectrics are the 
one referred to during our analysis henceforth. They provide effective embedded 
capacitance for interference noise filtering in the packaging technologies. This fact is 
shown in Figure  4.13 (a) and (b) which respectively presents the input impedance at 
port 2 and the trans-impedance between ports in the package under study for three 
different cases of (?
r 
= 4.3, tan ? = 0.02, h = 200 ?m), (?
r 
= 100, tan ? = 0.0015, h = 
100 ?m), and (?
r 
= 140, tan ? = 0.0015, h = 50 ?m). The specifications of the high-k 
dielectrics used here are obtained from  [66]. As shown in Figure  4.13, the 
combination of parameters for the third case has provided the lowest maximum of 
magnitude of impedance in the entire band which leads to decrease in interference. 
Generally by using this method, the resonances with the decreased magnitude shift to 
lower frequencies.  
 
 
 
 
 
 
164 
 
 
2 4 6 8 10 12 14
5
10
15
20
Frequency (GHz)
M
agn
i
t
ud
e of
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
200 ?m
100 ?m
2 4 6 8 10 12 14
2
4
6
8
10
12
Frequency (GHz)
M
agn
i
t
ude
 of
 T
r
ans
-
I
m
ped
anc
e Z
21
 (o
hm
)
200 ?m
100 ?m
(a)
(b)
M
agn
i
t
ud
e of
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
M
agn
i
t
ude
 of
 T
r
ans
-
I
m
ped
anc
e Z
21
 (o
hm
)
M
agn
i
t
ud
e of
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
M
agn
i
t
ude
 of
 T
r
ans
-
I
m
ped
anc
e Z
21
 (o
hm
)
M
agn
i
t
ud
e of
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
M
agn
i
t
ude
 of
 T
r
ans
-
I
m
ped
anc
e Z
21
 (o
hm
)
6.7337.19114.97.201
7.3925.09717.35.101
9.6033.60723.143.609
|Z
22
| (ohm)
at 
h = 100 ?m
F (GHz)Z
22
(ohm)
at 
h = 200 ?m
F (GHz)
2.89514.56.46214.49
3.85410.228.7910.22
5.2047.23212.17.232
|Z
21
| (ohm)
at 
h = 100 ?m
F (GHz)Z
21
(ohm)
at 
h = 200 ?m
F (GHz)
 
Figure  4.10. Magnitude of (a) input impedance at port 2, and (b) trans-impedance 
between ports 1 and 2 for the package shown in Figure  4.1. Impedances are provided 
for two different configurations. The impedance for plane separation h = 200 ?m is 
plotted by the dashed line and the impedance for h = 100 ?m is plotted by the solid 
line. The first three resonances are marked with different symbols in the plot and 
table. The similar symbols in each graph show the corresponding resonances.  
 
165 
5.5185.4761.6165.4924.66614.49
6.7673.8712.1443.8696.29510.22
8.2772.7382.8872.7388.6097.232
|Z
21
| 
(ohm)
F (GHz)|Z
21
| 
(ohm)
F (GHz)|Z
21
| 
(ohm)
F (GHz)
2 4 6 8 10 12 14
2
4
6
8
?
r
 = 4.3,  tan? = 0.02
Frequency (GHz)
2 4 6 8 10 12 14
0.5
1
1.5
2
2.5
?
r
 = 30,  tan
?
 = 0.02
Frequency (GHz)
M
agni
t
u
de of
 T
r
ans
-
I
m
p
edanc
e
 Z
21
 (o
h
m
)
2 4 6 8 10 12 14
2
4
6
8
10
?
r
 = 30,  tan? = 0.0015
Frequency (GHz)
(b)
(a)
(c)
M
agni
t
u
de of
 T
r
ans
-
I
m
p
edanc
e
 Z
21
 (o
h
m
)
(a) (b) (c)
 
Figure  4.11. Magnitude of trans-impedance between ports 1 and 2 for the package 
under study shown in Figure  4.1. The impedance is illustrated for three different 
configurations: (a) (?
r 
= 4.3, tan ? = 0.02),? (b) (?
r 
= 30, tan ? = 0.02), and (c) (?
r 
= 30, 
tan ? = 0.0015). In all configurations, the plane separation is h = 150 ?m. The first 
three resonances are marked with different symbols in the plot and table. The similar 
symbols on the curves show the corresponding resonances.  
 
 
166 
15.633.0595.5593.05216.318.066
8.6872.7343.7822.72110.827.193
10.181.9354.1071.92912.285.099
12.321.3685.3091.36516.243.608
|Z
22
| 
(ohm)
F (GHz)|Z
22
| 
(ohm)
F (GHz)|Z
22
|
(ohm)
F (GHz)
2 4 6 8 10 12 14
5
10
15
?
r
 = 4.3, tan? = 0.02
Frequency (GHz)
2 4 6 8 10 12 14
2
4
6
?
r
 = 30,  tan? = 0.02
Frequency (GHz)
M
a
g
n
i
t
u
de o
f
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
2 4 6 8 10 12 14
5
10
15
?
r
 = 30,  tan? = 0.0015
Frequency (GHz)
(b)
(a)
(c)
(a) (b) (c)
M
a
g
n
i
t
u
de o
f
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
M
a
g
n
i
t
u
de o
f
 I
n
put
 I
m
ped
anc
e Z
22
 (o
hm
)
 
Figure  4.12. Magnitude of input impedance at port 2 for the package under study 
shown in Figure  4.1. The impedance is illustrated for three different configurations: 
(a) (?
r 
= 4.3, tan ? = 0.02), (b) (?
r 
= 30, tan ? = 0.02), and (c) (?
r 
= 30, tan ? = 0.0015). 
In all configurations, the plane separation is h = 150 ?m. The first four resonances are 
marked with different symbols in the plot and table. The similar symbols on the 
curves show the corresponding resonances. 
 
 
 
 
167 
 
 
 
(a) (b)
2 4 6 8 10 12 14
2
4
6
8
10
12
Frequency (GHz)
?
r
 = 4.3, tan? = 0.02, h = 200 ?m
2 4 6 8 10 12 14
0.5
1
1.5
2
Frequency (GHz)
M
agni
t
ude
 of
 T
r
ans
-I
m
pedanc
e 
Z
21
 (
ohm
)
?
r
 = 100, tan? = 0.0015, h = 100 ?m
2 4 6 8 10 12 14
0.1
0.2
0.3
0.4
Frequency (GHz)
?
r
 = 140, tan? = 0.0015, h = 50 ?m
2 4 6 8 10 12 14
5
10
15
20
Frequency (GHz)
?
r
 = 4.3, tan? = 0.02, h = 200 ?m
2 4 6 8 10 12 14
2
4
Frequency (GHz)
M
a
gni
t
u
d
e
 of
 I
n
pu
t
 I
m
pe
da
nc
e Z
22
 (o
hm
)
?
r
 = 100, tan? = 0.0015, h = 100 ?m
2 4 6 8 10 12 14
0.5
1
1.5
Frequency (GHz)
?
r
 = 140, tan? = 0.0015, h = 50 ?m
M
agni
t
ude
 of
 T
r
ans
-I
m
pedanc
e 
Z
21
 (
ohm
)
M
a
gni
t
u
d
e
 of
 I
n
pu
t
 I
m
pe
da
nc
e Z
22
 (o
hm
)
 
Figure  4.13. Magnitude of (a) input impedance at port 2, and (b) trans-impedance 
between ports 1 and 2 for the package under study shown in Figure  4.1. The 
impedance is illustrated for three different cases of (?
r 
= 4.3, tan ? = 0.02, h = 200 
?m), (?
r 
= 100, tan ? = 0.0015, h = 100 ?m), and (?
r 
= 140, tan ? = 0.0015, h = 50 
?m). 
 
 
 
 
 
 
 
168 
Chapter 5 : Application of EBG Structures 
In the following sections we will demonstrate effectiveness of implementing 
EBG structures in few applications categorized under ?EMI Reduction? and ?Design 
of Low Profile Antenna?. 
 
5.1. EMI Reduction 
Under this category, we will study the advantages of using EBG structures for 
the suppression of EM noise. Suggested applications are: 
? Reduction of EMI in Enclosures and Chassis using the EBG Structures 
? Reduction of EMI Coupling between Cavities and Antennas using EBG 
Structures 
? EMI and SSN Noise Mitigation in IC Packages using Miniaturized Planar EBGs 
 
5.1.1. Reduction of EMI in Enclosures and Chassis using the EBG Structures 
In enclosures and chassis,  the mechanism of coupling between an internal 
electromagnetic source and the external electromagnetic environment takes place 
either by direct penetration of electromagnetic fields through openings or through 
induced currents on conducting surfaces. Currents on the internal walls supporting the 
fields inside the enclosure can travel out through openings and apertures to outside 
walls. Once these currents reach external surfaces or the currents induced by external 
fields on outer walls, they travel and penetrate other cavities sharing the same surface 
and create fields in these so-called victim cavities. A highly effective way to prevent 
 
169 
the field coupling as described is proposed here. This consists of employing EBG 
patches, which play a critical role in an important class of EMI/EMC applications  [4],  
 [19]- [28],  [67]- [70],  [71], on the surface of the structure thus blocking the flow of 
surface currents. The proposed technique consists of surrounding the perimeter of 
critical openings and apertures of an enclosure or package with a ribbon of EBG 
surface designed around the frequency of interest as shown in Figure  5.1. The EBG 
structure acts as a band stop filter which eliminates the interfering surface currents 
arising from radiators. 
 
 
Figure  5.1. Enclosure cavity with two EBG ribbons. 
 
5.1.1.A. Modeling 
From a purely electromagnetic perspective, this set of problems can be 
described and modeled as coupling between antennas or radiators. Therefore, it is 
necessary primarily to determine the frequency ranges that are important for accurate 
 
170 
functioning of the system. Then, we need to design EBG structure that covers the 
frequency range of interest. This frequency band can be either narrowbnad, wideband 
or even consisting of multiple bands  [20],  [23],  [24]. In our study, we use a simple 
form of EBG patterns known as mushroom structures which were discussed in  0.   
 
5.1.1.B. Numerical Case Study   
 To demonstrate the effectiveness of this technique, two identical metallic 
boxes of sizes 80 mm ? 160 mm ? 60 mm are joined together as shown in Figure  5.2. 
Both boxes are completely closed except for one side of each box which has a 20 mm 
? 4 mm aperture in the middle as shown in Figure  5.2. A 10 mm probe is located 
inside of each box normal to the aperture. The probe is distant by 20 mm from corner 
in both x and y directions. The direction of the probe is parallel to the shorter side of 
the aperture (in order to maximize its coupling to the aperture). This setup ensures 
that any possible coupling between the two boxes is only through surface currents. 
(Perfectly conducting boxes are used in order to eliminate any field penetration 
through the skin. However, for the frequency range of interest, the skin depth is much 
smaller than the thickness of metals used in practical enclosures.)   
This particular example represents a wide class of problems where coupling 
takes place between two separate compartments associated with a single chassis 
(common reference plane). In this chosen example, the emphasis is not on the type 
and geometry of excitation, but rather on the potential of EBG structures to reduce 
coupling through the suppression of surface waves. The frequency of significant 
coupling depends on several factors including the probe location and size, enclosure 
 
171 
and aperture dimensions.  Here, we consider coupling in the 7-10 GHz frequency 
range. This range includes the frequency at which the apertures and probes are 
resonant (7.5 GHz).  
The EBG structure chosen to suppress coupling in this example is picked up 
from those data obtained previously in Section  2.3, Figure  2.16, with ?
r
 of 2.2 and 
patch size of 4 mm. The effectiveness of the performance of the EBG ribbon can be 
gauged by considering the power coupling between the two different probes shown in 
Figure  5.2 (b). The power coupling between the two ports is proportional to the 
transmission coefficient S
21
. Figure  5.3 shows simulation results for the entire 
structure in Figure  5.2 with and without the presence of the EBG ribbon. The width of 
the ribbon is 88 mm. The application of the EBG ribbon resulted in coupling 
reduction between the two probes within the operating frequency range of interest. 
Figure  5.4 shows the magnitude of S
21
 between two ports with and without the 
presence of EBG for a second studied case. The specifications of this structure are 
chosen from Section  2.3, Figure  2.16, with ?
r
 of 4.8 and patch size of 4 mm. This 
pattern is effective in reducing the EMI coupling in the frequency range 5.8 GHz to 9 
GHz. 
 
 
 
 
 
 
 
172 
 
 
Figure  5.2. Schematic of two back-to-back perfectly conducting boxes. One of them 
contains the excitation probe and the second contains the receiver probe. Both of 
them have a small aperture in the middle of their external y-z plane with dimensions 
a= 20 mm, b= 4 mm. The EBG surface is placed as a ribbon completely surrounding 
the structure. (a) Front view. (b) Side view.  
 
 
 
 
x z
y
 
173 
 
Figure  5.3. Magnitude of S
21
 between two probes positioned as shows in Figure  5.2 
(b). The EBG structures consists of 4 ? 4 mm patches with 0.4 mm gap and ?
r
 = 2.2. 
 
Figure  5.4. Magnitude of S
21
 between two probes positioned as shows in Figure  5.2 
(b). The EBG structures consists of 4 ? 4 mm patches with 0.4 mm gap and ?
r
 = 4.8. 
6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
-140
-120
-100
-80
-60
 Without EBG
 With EBG
M
agnitud
e of S21 (dB
)
Frequency (GHz)
5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
-120
-100
-80
-60
 Without EBG
 With EBG
M
agn
itu
d
e
 of S
21 (dB
)
Frequency (GHz)
 
174 
5.1.2. Reduction of EMI Coupling between Cavities and Antennas using EBG Structures 
Continuous advances in communication and electronic circuits increase the 
need for complicated systems with several electromagnetic sources that operate in the 
same environment such as array of antennas and different electronics in 
compartments of a chassis. The proximity of these radiation sources increases the 
probability of interference, or mutual coupling between them. To this end, we 
consider the case of two apertures or cavities sharing a common reference plane.  
More specifically, we consider the cavities as a constitutive part of Cavity-Backed 
Slot (CBS) antennas. CBS antennas have been used in space vehicles, satellites, radar 
and mobile telephony as they are easy to manufacture, small in size, and light 
weighted.  In this study, we propose to implement EBG structures designed around 
the frequency of interest in the common surface between two cavities to reduce the 
coupling between them. 
 
5.1.2.A. Modeling 
In order to suppress surface current traveling between cavities, EBG structure 
is designed to cover the operating frequency of the cavity-backed antennas or the 
electronic systems located in cavities that are the compartments of chassis. EBG 
structure is designed using trial and error procedure educated by the data obtained 
from HFSS simulator mentioned in  0.   
 
 
175 
5.1.2.B. Numerical Case Study 
The case study chosen here will be that of a two CBS antennas system 
discussed in detail in  [7]. This two-antenna system is reproduced in Figure  5.5 and 
Figure  5.6. The common reference plane (Ground plane) is finite with dimension of 
10 cm ? 6 cm and antennas are separated from each other by 4 cm. The EBG surface 
is placed between the two antennas as shown in Figure  5.7. 
 
 
z
x
y
(a)
1.143 cm
2.286 cm
5.85089 cm 1.905 cm
(b)
x
z
0.127 cm
1.016 cm
0.6985 cm
(c)
x
y
 
Figure  5.5. Air-filled rectangular cavity-backed antenna fed with a probe oriented in 
the y-direction. (a) 3-D view, (b) top view and (c) side view (x-y plane). 
 
176 
 
 
x
y
z
 
Figure  5.6. Three dimensional schematic diagram showing two identical cavity-
backed antennas mounted on a rectangular ground plane.  The structure of each 
antenna is as described in Figure  5.5. 
 
 
 
 
 
 
 
 
 
177 
 
 
(a)
(b)
y
4cm
6cm
10cm
x
a
g
y
z
 
Figure  5.7. (a) Side view and (b) top view of a CBS antennas system showing the 
placement of the EBG patches.   
 
 
 
178 
In order to design an EBG structure having the desired characteristics, the 
CBS antenna system (the two antennas and ground plane) without the EBG structure 
is first simulated and its operational frequencies are derived. These frequencies 
correspond to the frequencies at which each antenna has its maximum power 
emission, in another words, they are the frequencies of minimum reflection, S
11
. 
Figure  5.8 shows the result of this simulation and the resonant frequencies of 
antennas: 7.5 GHz, 10.7 GHz, and 12 GHz. It should be noted that since there is a 
strong reflection around 10.4 GHz (S
11
 is close to 0dB), the resonance at 10.7 GHz is 
not considered a working frequency both in  [7] and in this work. For the other two 
operating frequencies, two designs are considered: an EBG structure with an effective 
operating frequency range from 7 GHz to 8 GHz, and another with an effective 
operating frequency range from 11.9 to 16 GHz. For the 7 GHz to 8 GHz range, we 
choose an EBG structure, henceforth referred to as Pattern 1, with 4 mm patch size 
and substrate permittivity ?
r
 = 4.8. For the 11.9 GHz to 16 GHz, we choose an EBG 
structure, henceforth referred to as Pattern 2, with 2.6 mm patch size and permittivity 
?
r
 = 3.0. These designs were obtained using the method described in  0, where the data 
were shown in Figure  2.16 and Figure  2.17.  
Figure  5.9 shows the transmission parameter, S
21
 (where one antenna is acting 
as a source and the other as a receiver) between the CBS antennas with and without 
EBG structures. These simulations show that significant coupling reduction is 
achieved, reaching up to 35 dB at certain frequencies. As a comparative reference, in 
 [7] the largest achieved coupling reduction obtained using lossy material was 7 dB.  
We note that over very narrow bands, the coupling reduction due to the 
 
179 
implementation of the EBG surface was slight, and in fact, an enhancement in the 
coupling was observed between 6.5 GHz and 7 GHz. However, we need to remember 
that the frequency band over which the coupling enhancement takes place is outside 
the operating frequency band of each antenna. 
 
 
 
Figure  5.8. Magnitude of S
11
 of two identical CBS antennas mounted on a rectangular 
ground plane (without the EBG structures). This antenna resonates at both 7.5 and 
12.6 GHz as marked on the plot. 
 
   
5 6 7 8 9 10 11 12 13 14 15
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
 
 
7.5 GHz
12.6 GHz
Ma
gnitude of 
S
11
 (dB)
Frequency (GHz)
 
180 
 
Figure  5.9. Coupling between the two antennas is represented by the magnitude of the 
S
21
 parameter. Two different EBG designs for two operating frequencies are 
implemented and simulated. Pattern 1 corresponds to a patch size of 4 mm and ?
r
 of 
4.8. Pattern 2 corresponds to a patch size of 2.6 mm and ?
r
 of 3. 
 
Figure  5.10 shows the reduction of surface waves and currents due to the 
presence of EBG structures. As it is evident from these simulations, when one of the 
antennas is excited, surface currents are generated and travel to the second antenna, 
therefore creating strong coupling. In Figure  5.10 (b), the EBG structure with an 
operating frequency band from 7 GHz to 8 GHz was applied while the first antenna is 
excited at 7.5 GHz. The decrease in coupling is obvious compared to Figure  5.10 (a) 
which shows the surface currents without the EBG surface. Figure  5.10 (d) shows 
coupling reduction compared to Figure  5.10 (c) where an EBG structure with an 
effective bandgap from 11.9 GHz to 16 GHz is applied while the antenna is excited at 
5678910112131415
-70
-60
-50
-40
-30
-20
 
 
 (without EBG)
 (with EBG, Pattern 1)
 (with EBG, Pattern 2)
Ma
gn
i
t
u
de of
 
S
21
 (dB
)
Frequency (GHz)
 
181 
12.6 GHz. Figure  5.11 and Figure  5.12 show the E-field distribution in the y-z plane 
cross section intersecting the two antennas.   
A disadvantage of using EBG surface between the two CBS antennas is a 
change in the radiation pattern and directivity of each antenna. The amount of this 
change depends on the proximity of the EBG structures to the antennas. It is easy to 
infer such consequence from the surface current plots shows in Figure  5.10. The 
presence of EBG surface results an asymmetric distribution of the surface currents, 
consequently an asymmetric radiated field pattern.  Since the EBG structure does not 
support any wave propagation at a frequency that falls within its bandgap, it is 
expected that the directive pattern of the antennas would rotate in the direction 
opposite to the side in which the EBG structures are located. In other words, since the 
flow of energy in directions close to the EBG structure is prohibited, the antenna gain 
in such directions is expected to drop. In order to investigate this prediction, we 
extracted the radiation patterns of one of the antennas for three different cases: First, 
without the EBG structure; second, with the EBG structure of Pattern 1 (4 mm square 
patches with ?
r
 = 4.8), and third, with the EBG structure of Pattern 2 (2.6 mm square 
patches with ?
r
 = 3).  Figure  5.13 shows a comparison between the first and second 
case while Figure  5.14 shows a comparison between the first and third case. From 
these results, we conclude that the EBG structures lead to reduced coupling between 
the two antennas. However, this reduction takes place at the expense of symmetry in 
the antenna pattern, and more specifically, at the expense of radiated gain in the 
direction of the adjacent antenna. We note that the decrease of gain by implementing 
lossy material was previously reported in  [7]. 
 
182 
 
(a)
(b)
(c)
(d)
y
x
  
Figure  5.10. Top view of the magnitude of the total surface current density J
tot
 (a) 
without the EBG structure when the antenna is operating at 7.5 GHz, (b) with the 
EBG Pattern 1 when the antenna is operating at 7.5 GHz, (c) without the EBG 
structure when the CBS antenna is operating at 12.6 GHz, and (d) with the EBG 
Pattern 2 when the antenna is operating at 12.6 GHz. Due to symmetry, only half of 
the structure is shown. 
 
 
 
183 
 
 
 
 
y
z
(a) (b)
EBG 
structures
 
Figure  5.11. E-Field distribution in the y-z plane (intersecting the two antennas) when 
one of the antennas is excited at 7.5 GHz while the second one is receiving. (a) 
Without and (b) with the EBG Pattern 1. 
 
 
 
 
 
 
 
 
184 
 
 
 
 
y
z
EBG 
structures
(a)
(b)
 
Figure  5.12. E-Field distribution in the y-z plane (intersecting the two antennas) when 
one of the antennas is excited at 12.6 GHz while the second one is receiving. (a) 
Without and (b) with the EBG Pattern 2. 
 
 
 
 
 
 
 
 
185 
 
 
 
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-30
-20
-10
0
10
 
 
 without EBG
 with EBG
 
Figure  5.13. Radiated field patterns at 7.5 GHz with and without EBG Pattern 1. The 
rgain pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth.  
 
186 
-60
-50
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-60
-50
-40
-30
-20
-10
0
10
 
 
 
 without EBG
 with EBG
 
Figure  5.14. Radiated field patterns at 12.6 GHz with and without EBG Pattern 2. The 
gain pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth.  
 
5.1.2.C. CBS Antenna System with Presence of EBG Structures 
In this subsubsection, we continue our studies on the effects of presence of 
EBG structures in the CBS antennas system. We investigate about EMI coupling 
reduction between antennas as well as gain pattern. In the previous subsubsection, we 
concluded that the gain pattern is changing while reducing interference between two 
antennas. We investigate about EMI coupling reduction between antennas as well as 
gain pattern. For this purpose, we are covering not only the spacing between antennas 
but also all around the opening of CBS antennas by the EBG Pattern 2 which is 2.6 
mm square patches with ?
r
 = 3 as shown in Figure  5.15. The size of this patch of EBG 
 
187 
structures implemented on common surface is 12 cells ? 28 cells which are covering 
around the apertures and the separation between antennas (intersections of apertures 
and EBG structures are removed from the slab). The size of slab of EBG structures 
implemented only between antennas was 12 cells ? 14 cells. Figure  5.16 shows the 
magnitude of S
21
 for the setup covering around and between apertures. For 
comparison purpose, the insertion loss curves for setups without EBG and with EBG 
located only in between antennas are included in the same figure. The system with 
EBG structures located in between antennas is performing more efficient than the 
system with EBG structures placed all around the antennas reducing coupling effect. 
However, both systems have similar coupling reduction performance of 18 dB at the 
operating frequency of antennas 12.6 GHz. The radiation patterns of these setups are 
illustrated in Figure  5.17. As it can be seen from the radiation pattern, the broadside 
gain of the antenna with added EBG structures decreases. This reduction for the case 
with EBG in between is 3 dB and for the case with EBG around and between 
antennas is 2 dB. The presence of EBG structures all around the apertures of antennas 
prevent the flowing of surface current on the common surface in all directions at the 
operating frequency of EBG pattern. Therefore, the directivity of antenna in those 
directions is decreasing and the radiation bandwidth becomes narrower. The radiation 
pattern obtained for the case where the EBGs are in all sides is almost symmetric 
similar to the case without EBGs. 
 
 
188 
y
x
 
Figure  5.15. Top view of the CBS antennas system showing the placement of the 
EBG structures all around the apertures and between the antennas on a common 
plane.  
11 12 13 14 15
-70
-60
-50
-40
-30
-20
18 dB
Distance Between Antennas = 40 mm
 Without EBG
 With EBG Only Between Antennas (a = 2.6 mm, ?
r
 = 3 )
 With EBG Around and Between Antennas (a = 2.6 mm, ?
r
 = 3 )
M
agnitude of 
S
21
 (dB
)
Frequency (GHz)
 
Figure  5.16. Coupling between the two antennas operating at 12.6 GHz for three 
different setups is represented by the magnitude of S
21
. 1) Without EBG structures, 2) 
with EBG Pattern 2 only between antennas, and 3) with EBG Pattern 2 around and in 
between the apertures. Separation between antennas is 40 mm. 
 
189 
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-40
-30
-20
-10
0
10
at ? = 0  Gain = {10.2051, 7.1826, 8.09527} dB
Distance Between Antennas = 40 mm
 Without EBG
 With EBG Only Between Antennas (a = 2.6 mm, ?
r
 = 3 )
 With EBG Around and Between Antennas (a = 2.6 mm, ?
r
 = 3 )
 
Figure  5.17. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system: 1) without EBG structures, 2) with EBG Pattern 2 only between 
antennas, and 3) with EBG Pattern 2 around and in between the apertures. The pattern 
is in the y-z plane (E-plane) where 0 corresponds to the azimuth. Separation between 
antennas is 40 mm. 
 
We have studied previously in  0 that the periodicity of EBGs is an important 
factor which affects the effective suppression bandwidth. As the number of EBGs 
between the two sources increases, we get wider and better suppression. However, the 
need for integrity and presence of several electromagnetic sources close to each other 
is another concern. Therefore, we are going to study the effect of placement of CBS 
antennas in proximity to each other. The separation between two antennas is reduced 
to 13.84 mm. A patch of 16 cells ? 20 cells of EBG Pattern 2 is implemented on 
common surface, around and between the apertures of antennas. Between the two 
 
190 
antennas, we have only 4 rows of EBGs. The magnitude of S
21
 for this setup with and 
without EBGs is shown in Figure  5.18. Comparing the data in Figure  5.18 to Figure 
 5.16 shows that the interference coupling increases as we decrease the distance 
between these cavities. Also, we realize that the level of coupling suppression as well 
as the effective bandwidth for EBGs decreases. At the operating frequency of 
antennas 12.6 GHz, the EMI suppression is equal to 12 dB. Totally, the EMI coupling 
increases by more than 10 dB if we decrease the distance between antennas in 
presence of EBGs.  
 
11 12 13 14 15
-44
-42
-40
-38
-36
-34
-32
-30
-28
-26
-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
12 dB
Distance Between Antennas = 13.84 mm
 Without EBG
 With EBG Around and Between Antennas (a = 2.6 mm, ?
r
 = 3 )
Magnitude of 
S
21
 (dB)
Frequency (GHz)
 
Figure  5.18. Coupling between the two antennas operating at 12.6 GHz for 2 different 
setups is represented by the magnitude of S
21
. 1) Without EBG structures and 2) with 
EBG Pattern 2 around and in between the apertures of antennas. Separation between 
antennas is 13.84 mm. 
 
191 
The radiation patterns of these setups, which are distant by 13.84 mm, are illustrated 
in Figure  5.19. Comparison between this figure and Figure  5.17 shows that the 
broadside gain for the setups without EBG is decreased from 10.2 dB to 8.5 dB. This 
is expected as we have more interference coupling between two cavities. 
Implementing EBGs is improving the broadside gain by almost 0.7 dB. This effect is 
different from the results of earlier setups.  
 
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-30
-20
-10
0
10
at ? = 0  Gain = {8.52608, 9.24395} dB
Distance Between Antennas = 13.84 mm
 Without EBG
 With EBG Around and Between Antennas (a = 2.6 mm, ?
r
 = 3 )
 
Figure  5.19. Radiated field patterns at 12.6 GHz for 2 different setups for the antenna 
system: 1) without EBG structures and 2) with EBG Pattern 2 around and in between 
the apertures of antennas. The pattern is in the y-z plane (E-plane) where 0 
corresponds to the azimuth. Separation between antennas is 13.84 mm. 
 
 
 
192 
From the obtained results we conclude the following: when we are working 
with cavity-backed antennas we have trade off between different parameters such as: 
the separating space between radiators, the tolerating limit for EMI coupling, and the 
gain of antennas. For example, if we compare two antennas distant by 40 mm without 
presence of any EBGs with the case where two antennas distant by 13.84 mm and 
surrounded by EBG patterns then we see that in the second case we have almost 5 dB 
less EMI coupling but the first case has almost 1 dB better broadside gain. In the 
second case, the space between two antennas is decreased by more than 26 mm but at 
the same time we have added the engineered material to our common ground. 
Considering that the EBG structures used in this study are not optimized, we conclude 
that we could design cavity-backed antennas system where antennas are closer to 
each other and having acceptable radiation pattern and at the same time with less EMI 
coupling. 
In  0, it is mentioned that the relationship between different design parameters 
of EBG structure and the suppression band is not linear. Also, the results of this 
subsubsection, up to here, confirm that to study an antenna system in presence of 
EBGs we have to consider the entire system. The effect of implementation of 
different EBG structures on the specifications and performance of the antenna system 
is not consistent. Therefore, these types of antenna systems for efficient performance 
should be optimized in the presence of EBGs. For further investigation, we study the 
same antenna system separated by 40 mm at the operating frequency of 12.6 GHz. 
New EBG structure named Pattern 3 specified by 1.2 mm square patches with ?
r
 = 6 
and via diameter 0.5 mm is implemented between the two antennas. The size of this 
 
193 
slab of EBGs is 18 cells ? 14 cells located in the middle of two antennas and its width 
is 22.4 mm as shown in Figure  5.20. The coupling between the two antennas is shown 
in Figure  5.21. The insertion loss curves for setups without and with EBG Pattern 2 
placed only in the gap between antennas are included in the same figure. The results 
show that at the operating frequency of antenna, Pattern 2 suppresses interference 
coupling by 8 dB better than Pattern 3. The radiation patterns of these three cases 
included in Figure  5.22 show that at the broadside, the gain is degraded by almost 0.6 
dB when using Pattern 3 while the degradation was 3 dB when using Pattern 2. 
Therefore, in this setup again we are experiencing a tradeoff between the EMI 
suppression and the gain pattern degradation.  
 
y
x
 
Figure  5.20. Top view of the CBS antennas system showing the placement of a slab 
of the EBG structures between the antennas on a common plane. The slab is partially 
covering the separation between antennas.  
 
 
 
 
194 
 
 
  
11 12 13 14 15
-70
-60
-50
-40
-30
-20
Distance Between Antennas = 40 mm
10 dB
18 dB
 Without EBG
 With EBG (a = 1.2 mm, ?
r
 = 6 ), EBG Slab = 22.4 mm
 With EBG (a = 2.6 mm, ?
r
 = 3 ), EBG Slab = 40 mm
M
agnitude of S
21
 (dB)
Frequency (GHz)
 
Figure  5.21. Coupling between the two antennas operating at 12.6 GHz for three 
different setups is represented by the magnitude of S
21
. 1) Without EBG structures. 2) 
With slab of EBG Pattern 2 between antennas. The width of slab is 40 mm. 3) With 
slab of EBG Pattern 3 between antennas. The width of slab is 22.4 mm. Separation 
between antennas is 40 mm. 
 
 
195 
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-40
-30
-20
-10
0
10
at ? = 0  Gain = {10.2051, 8.57823, 7.1826} dB
Distance Between Antennas = 40 mm
 Without EBG
 With EBG (a = 1.2 mm, ?
r
 = 6 ), EBG Slab = 22.4 mm
 With EBG (a = 2.6 mm, ?
r
 = 3 ), EBG Slab = 40 mm
 
Figure  5.22. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system. 1) Without EBG structures. 2) With slab of EBG Pattern 2 between 
antennas. The width of slab is 40 mm. 3) With slab of EBG Pattern 3 between 
antennas. The width of slab is 22.4 mm. Separation between antennas is 40 mm. The 
pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth. 
 
Now we look at new configurations for the CBS antenna system in which the 
antennas are separated by 22.4 mm. The antennas are operating at 12.6 GHz. Three 
different cases are considered: 1) without EBG structures, 2) with EBG structures 
between antennas, and 3) with EBG structures around and between apertures of 
antennas. The EBG Pattern 3 is used in the simulations. For the second setup, the size 
of the slab of EBGs is 18 cells ? 14 cells which fully cover the separation between 
 
196 
antennas. For the third setup, the size of the slab of EBGs is 22 cells ? 36 cells which 
cover the area around the apertures and the seperation between antennas (intersection 
of apertures and the EBG structures are removed from the slab). The level of EMI 
coupling for these configurations is shown in Figure  5.23. For the second case, at 12.6 
GHz the EM suppression is equal to 13 dB. It means that in this configuration there is 
almost 8 dB less coupling between the two antennas compared to the case where 
antennas were separated by 40 mm without any EBG structures (refer to the original 
study case). The gain patterns of those three configurations are shown in Figure  5.24. 
The results indicate that by implanting EBG pattern between the antennas, the 
broadside gain is reduced by 2.4 dB. The setup with EBGs around the apertures and 
between antennas reduces the broadside gain by 1.2 dB. Applying EBGs all around 
the apertures did not improve the symmetry of radiation pattern. 
We present in Figure  5.17, Figure  5.19, and Figure  5.24 the gain patterns of 
three different antenna systems without EBG structures. These configurations have 
different separation distances between antennas. The results show that the broadside 
gain is 10.2 dB, 7.2 dB, and 8.5 dB for the separations 40 mm, 22.4 mm, and 13.84 
mm, respectively. Therefore, the change of gain versus the gap distance is not 
monotonic. However, the results obtaind from the figures presenting the magnitude of 
S
21
 for these configurations show that the coupling interference increases by 
decreasing the seperation between antennas. As a result, we do conclude that changes 
in the gain pattern are not certainly predictable.  
 
 
 
197 
 
 
11 12 13 14 15
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
8 dB
13 dB
Distance Between Antennas = 22.4 mm
 Without EBG
 With EBG Only Between Antennas (a = 1.2 mm, ?
r
 = 6 )
 With EBG Around and Between Antennas (a = 1.2 mm, ?
r
 = 6 )
Magnitude of S
21
 (d
B)
Frequency (GHz)
 
Figure  5.23. Coupling between two antennas operating at 12.6 GHz for three different 
setups is represented by the magnitude of the S
21
. 1) Without EBG structures. 2) With 
EBG Pattern 3 only between antennas. 3) With EBG Pattern 3 around and between 
the apertures of antennas. Separation between antennas is 22.4 mm. 
 
 
 
 
 
198 
-40
-30
-20
-10
0
10
0
30
60
90
120
150
180
210
240
270
300
330
-30
-20
-10
0
10
at ? = 0  Gain = {7.19312, 4.78105, 6.01831} dB
Distance Between Antennas = 22.4 mm
 Without EBG
 With EBG Only Between Antennas(a = 1.2 mm, ?
r
 = 6 )
 With EBG Around and Between Antennas (a = 1.2 mm, ?
r
 = 6 )
 
Figure  5.24. Radiated Gain patterns at 12.6 GHz for three different setups for the 
antenna system: 1) without EBG structures, 2) with EBG Pattern 3 only between 
antennas, and 3) with EBG Pattern 3 around and in between the apertures of antennas. 
The pattern is in the y-z plane (E-plane) where 0 corresponds to the azimuth. 
Separation between antennas is 22.4 mm. 
 
To complete our study, in what follows, we continue our investigation on the 
effect of proximity of EBG structures to the aperture of the CBS antenna on the 
directivity pattern when the perimeter of the opening of antenna is covered with EBG 
structures. Figure  5.25 shows the top view of a single antenna system in which a 
ribbon of EBG structures is surrounding the aperture of CBS antenna. The antenna 
has the same specifications as in Figure  5.5. The width of the ribbon is 4 unit cells. 
The distance of ribbon of EBGs from aperture is determined by the two parameters: 
 
199 
d
1
 and d
2
. In our numerical studies, we assume that the ground plane is infinite. 
Therefore, the radiation pattern occupies the upper half space. The EBG pattern, with 
square patch size of 5 mm and ?
r
 = 4.4 (FR-4) which is effective at the operating 
frequency of antenna 7.5 GHz, is used in the following numerical simulations.  
 
y
x
d
1
d
1
d
2
d
2
 
Figure  5.25. Top view of the CBS antenna system showing the placement of a ribbon 
of EBG structures around the aperture. The distance of this ribbon from antenna is 
determined by the two parameters: d
1
 and d
2
.  
 
The case studies include the following antenna systems: 1) without EBG, 2) 
with EBG where d
1
 = 4.77 mm and d
2
 = 3.02 mm, 3) with EBG where d
1
 = 10.17 mm 
and d
2
 = 8.42 mm, and 4) with EBG where d
1
 = 15.57 mm and d
2
 = 13.82 mm. In the 
setups with presence of ribbon of EBGs we have changed the distance of ribbon from 
antenna aperture. Figure  5.26 shows the gain patterns of those four configurations. 
The results show that we can control the directivity pattern by changing the location 
 
200 
of EBG structures with respect to aperture of antenna. The setup without EBG has 
approximately a uniform gain pattern in the upper half space. By implementing EBG 
ribbon around the aperture in the second case study we have a broadside directive 
antenna with almost 3 dB increment in the gain. The third case study is a broadside 
directive antenna. It has almost 0.5 dB less gain at broadside compared to second case 
study. In addition, this configuration has side lobes. The fourth case study is a 
directive antenna with maximum gain at ? 30? and minimum gain at the broadside 
direction. These results confirm that we can use EBGs in designing directive 
antennas. 
 
 
 
 
201 
2
3
4
5
6
7
8
9
0
30
60
90
120
150
180
210
240
270
300
330
2
3
4
5
6
7
8
9
 Without EBG
 With EBG (d
1
 = 4.77 mm, d
2
 = 3.02 mm)
 With EBG (d
1
 = 10.17 mm, d
2
 = 8.42 mm)
 With EBG (d
1
 = 15.57 mm, d
2
 = 13.82 mm)
 
Figure  5.26. Radiated gain patterns at 7.5 GHz obtained for four different setups for a 
single antenna system. 1) Without EBG structures. 2) With ribbon of EBGs distant 
from antenna by d
1
 = 4.77 mm and d
2 
= 3.02 mm. 3) With ribbon of EBGs distant 
from antenna by d
1
 = 10.17 mm and d
2 
= 8.42 mm. 4) With ribbon of EBGs distant 
from antenna by d
1
 = 15.57 mm and d
2 
= 13.82 mm (refer to Figure  5.25). The pattern 
is in the y-z plane (E-plane) where 0 corresponds to the azimuth. 
 
5.1.2.D. Experimental Study 
In this subsubsection, we present the experimental results showing the 
performance of EBG surfaces of various patch sizes. The setup used in this 
experiment is shown in Figure  5.27 where two waveguides lead to two CBS antennas 
16 cm apart and sharing a common ground plane in a manner similar to the numerical 
 
202 
case study discussed in Subsubsection  5.1.2.B. The two ports of the waveguides are 
fed through SMA connectors and then connected to an Agilent PNA series vector 
network analyzer (VNA). Figure  5.27 also shows the placement of the EBG surface 
under study between the two antennas. The EBG structures used in this experiment 
are made of FR-4 substrate with via diameter and height of 0.8 mm and 1.54 mm, 
respectively. The distance between the patches is 0.4 mm and the width of EBG 
structures used is 6cm. EBG of different patch sizes ranging from 2 mm to 5 mm are 
tested and comparison is made to the reference case where EBG surface is removed.   
Figure  5.28-Figure  5.31 show the results of measurements. In each plot, the 
magnitude of S
21
 is presented for the cases with and without the EBG surface. The 3 
dB bandgap is marked in each graph. In Figure  5.32, we present the numerical 
simulation results obtained by the numerical simulation procedure depicted in Figure 
 2.7 with 8 infinite rows of EBG patches. Table  5.I summarizes the bandgap obtained 
from measurements and numerical simulations. 
While we have good agreement between the bandgaps calculated numerically 
and measured experimentally, we attribute the discrepancies observed between the 
bandgap edges primarily to: first, the non-ideal environment inherent in the 
experimental setup such as the finiteness of the EBG surface in contrast to the infinite 
rows which were numerically modeled using field symmetry boundary conditions, 
second, the assumption that the permittivity of the FR-4 is 4.4 whereas in reality the 
permittivity varies with frequency.  
 
 
 
203 
 
 
Figure  5.27. (a) Setup used to assess coupling mitigation using EBG structures. (b) 
Close up view of antenna system with presence of EBG structure. EBG structures 
with different configurations (patch sizes of 2 mm to 5 mm) in the area between two 
antennas separated by 16 cm are implemented. Width of EBG separator is 6 cm.  
 
 
CBS antenna 
CBS antenna 
EBG structure 
(b)
(a)
 
204 
 
 
 
8 1012141618
-70
-60
-50
-40
-30
 without EBG
 with EBG (2mm patch)
Magnitude of 
S
21
 (dB)
Frequency (GHz)
3dB Band Gap
Magnitude of 
S
21
 (dB)
 
Figure  5.28. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 2 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 substrate. 
 
 
 
 
 
 
 
 
205 
 
 
 
6 8 10 12 14 16
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
 without EBG
 with EBG (3mm patch)
Magnitude of S
21
 (dB)
Frequency (GHz)
3dB Band Gap
Magnitude of S
21
 (dB)
 
Figure  5.29. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 3 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 substrate 
 
 
 
 
 
 
 
 
206 
 
 
 
5 6 7 8 9 10 11 12 13 14 15
-70
-65
-60
-55
-50
-45
-40
-35
-30
 without EBG
 with EBG (4mm patch)
Magnitude of S
21
 (dB)
Frequency (GHz)
3dB Band Gap
Magnitude of S
21
 (dB)
 
Figure  5.30. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 4 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 substrate 
 
 
 
 
 
 
 
 
207 
 
 
 
45678910112
-80
-70
-60
-50
-40
-30
 without EBG
 with EBG (5mm patch)
Magnitude of S
21
 (dB)
Frequency (GHz)
3dB Band Gap
Magnitude of S
21
 (dB)
 
Figure  5.31. Experimental results show the coupling between the two antennas with 
and without the EBG surface as represented by the magnitude of S
21
. EBG 
parameters: 5 mm patch, h = 1.54 mm, g = 0.4 mm, d = 0.8 mm, and FR-4 substrate. 
 
 
 
 
 
 
 
 
208 
 
Figure  5.32. Bandgap obtained using numerical simulation. EBG parameters: ?
r 
= 4.4, 
h = 1.54 mm, d = 0.8 mm, and g = 0.4 mm. 
 
 
Table  5.I. Bandgap Obtained From Numerical Simulation and Experimental 
Measurements 
EBG Patch Size 
(in mm) 
Numerically Derived Bandgap 
(in GHz) 
Experimentally Derived Bandgap 
(in GHz) 
2 11.8 - 16.4 12.7 - 16.3  
3 8 - 12.8 9.7 - 14.6 
4 6.65 -11.85 7.8 - 12.1 
5 5.6 - 7.9 6.6 - 8.5 
 
 
 
2345
6
8
10
12
14
16
Frequency (GHz)
patch size (mm)
 
209 
5.1.3. EMI and SSN Noise Mitigation in IC Packages using Miniaturized Planar EBGs 
By analyzing the S-parameters between different interconnects in parallel 
metallic layers of package, it is possible to estimate the range of frequencies where 
the EMI noise and power/ground bounce are strong. The EM noise with internal and 
external sources can interfere with the operation of electronic circuits in chip. 
Therefore, this analysis is necessary for accurate operation of IC. In this subsection, 
we investigate about EM noise suppression in IC packages using EBG structures. 
This novel method of noise suppression in IC packages as strong contender of surface 
wave suppression has become possible due to design of the miniaturized EBGs 
proposed in our research and discussed earlier in  Chapter 3.  
 
5.1.3.A. Modeling 
After analyzing the S-parameters to extract the range of frequency where the 
EM noise is strong enough to degrade the operation of electronics, we need to design 
EBG structure at the requested frequency band. This frequency band can be either 
narrowband, wideband or even consisting of multiple bands. In packages, the 
engineered structures are implemented surround the source or the receiver of noise to 
block the propagation of wave. This is an efficient method to suppress EM noise in 
the packages. 
 
5.1.3.B. Numerical Case Studies 
In this subsubsection, by mean of numerical studies we demonstrate the 
efficiency of this proposed method. We continue our studies on the typical 
 
210 
microprocessor package as our case of study as considered earlier in  Chapter 4. For 
convenience, the structure of this single layer package and the setup of ports are 
redrawn in Figure  5.33. The dimensions of the package are 4 cm ? 4 cm. The 
dielectric thickness h = 114.3 ?m. Following, two different EBG setups are studied. 
We use EBG sample No. 7 from Table  3.III, which has a suppression band from 1.9 
GHz to10.8 GHz, in both setups. The design parameters of this EBG structure shown 
in Figure  3.12 are:  P
W
 = 1.6 mm, h = 114.3 ?m (4.5 mil), M
W 
= 0.02 mm, M
B 
= 0.02 
mm, M
L 
= 1.6 mm, M
G 
= 0.02 mm, and ?
r 
= 200. 
 
(a)
a
h
y
z
x
b
?
r
, ?
(b)
Port 1
Port 2
y
x
(x
1
, y
1
)
(x
2
, y
2
)
 
Figure  5.33. Single plane pair in the package consisting of metal-dielectric-metal 
layers is modeled as parallel plate waveguide. (a) 3-D view.  (b) Top view. Typical 
dimensions of a microprocessor package are: a = b = 4 cm. Each via shown in the 
figure represents a port. Port 1 and Port 2 are located at (x
1
 = 1 cm, y
1
 = 1 cm), and (x
2
 
= 2 cm, y
2
 = 2 cm), respectively. 
 
In the first setup, the two ports are isolated with the EBG patterns as shown in 
Figure  5.34. Each port is centered in a patch of EBG cells. Each patch is made of 5 
 
211 
cells ? 5 cells in both x- and y-direction. The magnitude of S
21 
for this configuration is 
illustrated in Figure  5.35. In the second setup, only the second port is isolated with the 
EBG patterns as shown in Figure  5.36. The second port is centered in the patch of 
EBG cells. The size of patch in this experiment is 5 cells ? 5 cells.  The magnitude of 
S
21 
for this configuration is also shown in Figure  5.35. Comparison is made with two 
different embedded capacitance cases studied in  Chapter 4. Embedded capacitance 
method is implemented to packages as an effective method of EMI suppression in the 
entire frequency band. These cases are applied to a single plane pair with two 
different materials of ?
r 
= 4.3 and tan ? = 0.02 (low value embedded capacitance), and 
?
r 
= 200 and tan ? = 0.0015 (high value embedded capacitance). The magnitude of S
21 
for both cases is included in Figure  5.35. The results show that to protect the ports 
from EMI noise in the package, it is enough to isolate either the source or the 
receiver. The level of EMI noise mitigation we can get from EBG patterned power 
planes compared to high embedded capacitance case is very high.  
In another study, we use the second test setup in two different cases. In the 
first case, the EBG patch is 5 cells ? 5 cells. In the second case, it is 9 cells ? 9 cells. 
The magnitude of S
21 
for each case is shown in Figure  5.37. The simulation results 
show that the level of electromagnetic noise suppression depends on the number of 
EBG unit cells. Therefore, if we increase the EBG cell population between source and 
receiver, more suppression is achieved. However, simulating a fully EBG-populated 
board is prohibitive due to the large number of meshes required to model meander 
bridges between patches.  
 
 
212 
 
 
Port 1
Port 2
Planar EBG Pattern
y
x
z
(a)
(b)
y
x
5d
5d
5d
5d
 
Figure  5.34. Test Setup 1. The ports in the package under study are isolated with EBG 
pattern, (a) 3-D view and (b) top view. The specifications of package and ports are 
shown in Figure  5.33. In this experiment the EBG sample No. 7, specified in Table 
 3.III, is used. Each port is centered in the middle of the EBGs patch. The size of patch 
is 5 cells ? 5 cells. 
 
 
 
 
213 
 
 
02468101214
-180
-160
-140
-120
-100
-80
-60
-40
-20
 without EBG ?
r
 = 4.3, tan? = 0.02
 without EBG ?
r
 = 200, tan? = 0.0015
 with EBG pattern (Setup 1) ?
r
 = 200, tan? = 0.0015
 with EBG pattern (Setup 2, d
p
 = 5d) ?
r
 = 200, tan? = 0.0015
Ma
gni
t
u
de
 of S
21
 (dB))
Frequency (GHz)
 
Figure  5.35. Magnitude of S
21
 between ports 1 and 2 in the single plane pair under 
study for four different scenarios (design parameters and port setup are shown in 
Figure  5.33). The first and second cases show embedded capacitance. The third case 
shows test setup 1 (refer to Figure  5.34). The fourth case corresponds to test setup 2 
(refer to Figure  5.36). In all configurations h = 114.3 ?m. 
 
 
 
 
 
 
214 
 
 
(a)
Port 1 Port 2
Planar EBG Pattern
y
x
z
(b)
dp
dp
y
x
 
Figure  5.36. Test Setup 2. The second port in the package under study is isolated with 
EBG pattern, (a) perspective view and (b) top view. Specifications of package and 
ports are shown in Figure  5.33. In this experiment the EBG sample No. 7, specified in 
Table  3.III, is used. The second port is centered in a patch of EBGs. The size of patch 
is d
P
 ? d
P
, where in two different experiments d
P
 = {5 cells, 9 cells}. 
 
 
 
215 
2468101214
-140
-120
-100
-80
-60
-40
 d
p
 = 5d
 d
p
 = 9d
Magnitude of S
21
 (dB))
Frequency (GHz)
 
Figure  5.37. Magnitude of S
21
 between ports 1 and 2 in the single layer package under 
study for two different cases in test setup 2 (refer to Figure  5.36).  
 
The efficacy of EBG patterning of the plane pairs in EMI and SSN noise 
suppression for through propagation was shown. In the next part of the study, we look 
at the radiation loss parameter (1- |S
11
|
2
 - |S
21
|
2
) to assess the possibility of 
interference due to outward radiation from plane pairs to surrounding media. Testing 
the performance of the other parts of a system with patterned layers is necessary to 
keep the functionality of a whole system. The radiation loss for the microprocessor 
package with the test setup 1 shown in Figure  5.34 is calculated and presented in 
Figure  5.38. For comparison purpose, the radiation loss from a single plane pair with 
two different materials of ?
r 
= 4.3 and tan ? = 0.02 providing low value embedded 
 
216 
capacitance and ?
r 
= 200 and tan ? = 0.0015 providing high value embedded 
capacitance are included in the same figure. The 20 dB suppression band of the EBG 
structure, from 1.9 GHz to 10.8 GHz, showing the effective operating frequency 
region is marked in that figure. 
It is seen that for the low value embedded capacitance case the radition loss at 
low frequency regions is negligible. However, by increasing the frequency strong 
local peaks are appearing that may interfere with other levels or parts in the package. 
The high value embedded capacitance case has reduced peaks of interference in the 
whole frequency band compared to the low embedded capacitance case. However, we 
can obviously see that the local peaks are distributed in the entire frequency band. In 
the effective band of EBG patterned case, the level of radition loss compared to two 
other cases is negligible. Out of this band, strong local peaks are seen in the plot. 
Therefore, EBG structures are effective in suppressing noise in their band not only for 
through propagation but also for radiation to surrounding environment. The effect of 
those strong local peaks, in the patterened case, on the other parts should be assessed. 
We need to evaluate the performance of system for purposes of signal integrity 
especially in low frequency regimes. 
Studies in Subsection  3.3.10 have shown that there are local peaks in the 
radiation loss diagram in the frequency bands compatible with the slow propagating 
modes of EBG pattern as shown in Figure  3.62. In those frequency regimes, shown in 
Figure  5.38, radiation exists. However, it is not strong enough to be clearly seen.  
 
217 
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
 Two Island EBG Patterned Plane
 Solid Plane, ?
r
 = 200
 Solid Plane, ?
r
 = 4.3
Radiat
ion Loss
Frequency (GHz)
20 dB Suppression Band
Radiat
ion Loss
 
Figure  5.38. Radiation loss, (1- |S
11
|
2
 - |S
21
|
2
), versus frequency for a single plane pair 
under study for three different scenarios. The first case shows the test setup 1 (refer to 
Figure  5.34). The second and the third cases show the embedded capacitance (refer to 
Figure  5.33) 
 
This discrepancy may be attributed to different structural setups. In the 
package, the larger dimensions of the parallel planes are not populated with EBG 
patterns as reverse to those results in Figure  3.62 which are coming from fully EBG 
populated parallel planes with small dimensions. The EBG patterns are radiating 
waves in those bands, however, the solid parallel planes support propagating waves. 
Therefore, in the package case the power radiated out per unit area to surrounding 
environment (computational domain) from the EBG patterns is less than the second 
 
218 
case. As a general result, even by considering those peaks from slow modes, EBG 
patterning reduces radiating emissions to surrounding environment in the suppression 
bandgap. 
 
5.2. Design of Low Profile Antenna 
EBG structures show novel features which may not occur in nature such as in-
phase reflection phase and surface wave suppression. These features have led EBGs 
to a wide range of applications including antennas. Due to their  properties, it has 
been shown that EBG structures emulate good magnetic conductors, or high-
impedance surfaces, in their bandgap when used as ground plane in the design of low 
profile antennas  [72],  [73]. This could be explained as follows. When an antenna 
placed closely on top of an EBG surface is radiating, the reflected waves from this 
surface will add up constructively to waves radiated from the antenna. Therefore, 
unlikely to an antenna system with perfect electric conductor (PEC) reflector, there is 
no need to have that distance of approximately ?/4 between antenna and reflector. 
These results can be justified by reflection coefficient of plane wave normally 
incident on surface defined by: 
0
0
?
?
+
?
=?
s
s
Z
Z
 
 Equation  5-1 
where Z
s
 is the surface impedance of the reflector and ?
0
 is the wave impedance in the 
media filling the space between reflector and antenna.  
 
219 
Previous EBG structures made of dielectric materials had desirable properties 
over a narrow frequency range, thus, their application was limited in design of narrow 
band low profile antenna. In this study, we have proposed novel EBG structures 
which make the design of low profile antennas viable. 
 
5.2.1. Limitations in Design of Low Profile wideband Antennas 
Previous EBG structures made of dielectric materials, however, had desirable 
properties over a narrow frequency range thus limiting their application in the design 
of low profile wideband antennas such as spirals. In  [74]- [76], EBG structures were 
used as a ground plane to implement low profile spiral antennas, but in those works 
the frequency independent behavior of the spiral antenna was limited by the relatively 
narrow bandgap of the EBG surface. In  [77], negative impedance converters which 
are active loads (capacitors and inductors having negative values) were inserted 
within the EBG structure leading to a significant increase in the bandgap. However, 
the implementation of active circuits adds complications and cost to the design. 
Recently, magneto-dielectric substrates were used in several applications  [78], 
 [79]. In  [78], magneto-dielectric substrates were used for miniaturization of antennas. 
In  [79] it was demonstrated that using such substrates in woodpile EBG structures 
leads to an increase in its bandgap. Furthermore,  [80] showed a method of 
synthesizing meta-ferrites with desired permeability using novel EBG structures 
which makes these materials available. In this study, our objective is to show that the 
effective bandgap in mushroom EBGs can be increased significantly due to the use of 
magneto-dielectric materials. This achievement opens the doors for designing low 
 
220 
profile wideband antennas. However, this proposed EBG structure is based on 
development in design and fabrication of low loss magneto-dielectric materials in the 
microwave region.  
  
5.2.1.A. Concept and Modeling of Mushroom-Type EBG Structures Using Magneto-
Dielectric Substrates 
The square patch mushroom-type EBG structure and its design parameters 
shown in Figure  2.1 is redrawn in Figure  5.39 for convenience. The EBG structure 
has a HIS in its bandgap at normal incidence. This surface impedance can be modeled 
as an inductor in parallel with a capacitor close to the region that it is emulating a 
PMC surface [26]. The capacitance part is due to the gap between the adjacent patches 
and is equal to  [81]: 
?
?
?
?
?
?
?
?+
=
g
DD
C
r
 
 2
ln
)1(
0
??
??
 
 Equation  5-2 
where D = a + g  is the period of unit cell. The inductance part is due to the adjacent 
vias and the metal backed substrate which can be calculated using transmission lines 
theory. The impedance of shorted transmission line while ignoring inductance of vias 
is calculated from: 
)tan( hjZ
s
??=  
 Equation  5-3 
where ? and ? are the wave impedance and the phase constant in the substrate slab, 
respectively. When h << ?, and the substrate is lossless the above impedance 
 
221 
becomes purely inductive. ? is the wavelength in substrate slab. Therefore, L is given 
by: 
hL
r
??
0
=  
 Equation  5-4 
L is a monotonic function of h and ?
r
 up to the range governed by the tangent 
function in the impedance formula. 
To use EBG surface as a ground plane in the design of low profile antennas, 
the operational frequency band is defined as the overlap of the input-match frequency 
band and surface wave frequency bandgap  [72]. The in-phase reflection bandwidth is 
the range of frequencies over which the phase of the broadside reflected wave is 
bounded by ?45?. In this band, the antenna shows good return loss. In another words, 
the reflected wave from surface is in-phase with incident wave from the antenna. 
Therefore, the interaction of EBG surface with antenna is constructive. The surface 
wave frequency bandgap is the range of frequencies over which the surface wave is 
suppressed significantly. This band improves the radiation pattern of the antenna. 
These two bands for the square patch mushroom structure overlap completely 
providing an efficient operating frequency band.  
It is well known, that bandgap for a parallel LC circuit is proportional to 
?(L/C), while the resonance frequency is proportional to 1/?(LC). It can be seen that 
using magneto-dielectric materials with relative permeability ?
r 
> 1 leading to 
increase of L has distinct advantage of increasing the bandgap besides the shifting 
down the lower edge of the bandgap. The increase in L, for a specific resonance 
frequency, has the added advantage of reducing the size of the patch as a smaller 
 
222 
capacitance would be needed to achieve the same resonant frequency. Thus, the 
bandgap increases more than before due to reduction of capacitance, C. 
 
 
Figure  5.39. Square patch mushroom-type EBG structure (a) Perspective view. (b) 
Top view. Relevant design parameters are shown on the diagrams. 
 
5.2.1.B. Numerical Study of EBG Structures Made of Magneto-Dielectric Substrates 
To verify the concept, several square patch mushroom-type EBG structures 
with magneto-dielectric substrate of permeability higher than one are designed and 
compared with a conventional EBG structure (EBG with dielectric substrate). The 
antenna under study is a wideband spiral antenna operating between 8 GHz and18 
GHz. Therefore, for low profiling this antenna our ultimate goal is design of an EBG 
structure which can cover this band of operation. According to the physical model 
 
223 
mentioned in Subsubsection  5.2.1.A and the study of effect of the design parameters 
in Section  2.3, we consider new structures with the following parameters: a = 0.5 
mm, g = 0.15 mm (D = 0.65mm), h = 1.57 mm, d = 0.1 mm, ?
r 
= 2.51, and different 
values of relative permeability 1 < ?
r 
< 9. It is assumed that all the magneto-dielectric 
substrates have tan ?
m 
= 0.01. This is the lowest value of magnetic loss which has 
been achieved and reported in design of engineered magnetic materials  [82]- [84]. 
Ansoft HFSS is used to extract in-phase reflection band for EBG structures. The setup 
is shown in Figure  5.40. For this purpose a unit cell of these structures is simulated 
using plane wave excitation with normal incidence. Periodic boundary condition is 
assigned around the cell to model an infinite EBG structure. The phase shifts between 
corresponding boundaries is equal to zero. Absorbing boundary condition is placed on 
the top of the cell. The reflected waves are recorded at an observation plane which is 
far enough from the EBG surface to make sure that evanescent modes are vanished. 
Using recorded field, the reflection phase is computed on the observation plane which 
is linearly transferred to the top surface of EBG structure  [36]. The in-phase reflection 
band shows the range of frequencies in where the phase of broadside reflections on 
top of the EBG structure is bounded by ?45?. The center frequency shows the point at 
which the reflected phase on top of the EBG structure is equal to zero. The percentage 
bandwidth is defined as the ratio of the in-phase reflection bandwidth to the center 
frequency in percentage. 
Figure  5.41 demonstrates the result of the simulation for structure with ?
r 
= 6 
where the reflection phase is presented as a function of frequency. The in-phase 
reflection phase for this structure is 7.1 GHz to 14.6 GHz as marked on the figure. To 
 
224 
show advantages of using magneto-dielectric substrate as compared to dielectric 
substrate which is conventionally used to design EBG structures, one conventional 
sample is studied as a reference. The selected EBG which is designed and tested in 
 [85] has the following specifications:  a = 2.44 mm, g = 0.15 mm (D = 2.59 mm), h = 
1.57 mm, d = 0.25 mm, ?
r 
= 1
,
 and ?
r 
= 2.51. The reflection phase of this structure is 
also plotted in Figure  5.41. The in-phase reflection phase for this structure is from 
12.1 GHz to 14.5 GHz as marked on the figure. Our EBG structure has same ?, g, and 
h as the conventional EBG structure, but its cell size is 4 times smaller. In addition 
the in-phase reflection band in the proposed EBG is 72.4% while this band is 18% for 
the conventional one. The result of simulations including in-phase reflection band, 
center frequency, and percentage bandwidth for EBG structures introduced earlier in 
this subsubsection are summarized in Table  5.II. It can be seen that the percentage 
bandwidth increases as the relative permeability increases. 
Interested readers can follow the subject of design of low profile wideband 
antennas using magneto-dielectric EBG structures in  [86]. This subject and design of 
magneto-dielectric materials in microwave frequency region is already under progress 
by members of other research groups. 
 
 
 
 
 
 
 
225 
 
 
 
PBC PBC
Ground Plane
EBG Unit Cell
Incident 
Plane Wave
ABC
Observation Plane
 
Figure  5.40. Setup used to extract the reflection phase of an EBG structure. The 
schematic shows the boundary setup for a unit cell. The unit cell is excited through 
normal incident plane wave. The reflected waves recorded at the observation plane 
are used to compute the phase. 
 
 
 
 
 
 
226 
 
 
 
6 8 10 12 14 16 18
-150
-100
-50
0
50
100
150
 The Conventional EBG
 The Novel EBG with ?
r
 = 6
R
e
flec
tion
 Ph
a
s
e (D
eg)
Frequency (GHz)
BW
2
BW
1
R
e
flec
tion
 Ph
a
s
e (D
eg)
 
Figure  5.41. Reflection phase of EBG structures. The in-phase reflection phase is 
marked on each graph. The conventional EBG structure introduced in  [85] is 
specified by: a = 2.44 mm, g = 0.15 mm, h = 1.57 mm, d = 0.25 mm, ?
r 
= 1
,
 and ?
r 
= 
2.51. The novel EBG structure is specified by: a = 0.5 mm, g = 0.15 mm, h = 1.57 
mm, d = 0.1 mm, ?
r 
= 2.51, ?
r 
= 6, and tan ?
m 
= 0.01. 
 
 
 
 
 
 
 
227 
 
 
 
 
Table  5.II. In-phase reflection band obtained from numerical simulation for different 
EBG structures. The EBG structures have common specification of a = 0.5 mm, g = 
0.15 mm, h = 1.57 mm, d = 0.1 mm, and ?
r 
= 2.51. The value of relative permeability, 
?
r
, is varying from 1 to 9. The magnetic loss tangent for material with ?
r
 > 1 is equal 
to 0.01 (tan ?
m 
= 0.01). 
8.7
10.65
15
25.9
Center 
Frequency 
(GHz)
% 83.917.35.4 ? 12.79
% 72.427.57.1 ? 14.66
% 51.337.711.3 - 193
% 30.89822 - 301
Percentage 
Bandwidth
In-phase 
Reflection 
Bandwidth (GHz)
In-phase 
Reflection Band 
(GHz)
?
 
 
 
 
 
 
 
 
 
 
228 
Chapter 6 : Conclusions and Future Work 
 
6.1. Conclusions 
The designs and analytical studies carried out through this work proved that 
EBG structures are strong contenders of EMI suppression in wide class of important 
problems where the EM wave interferes with the victim through surface wave or 
surface current, e.g., in applications where the source and victim share a common 
reference plane. 
In this research, the mushroom-type EBG structures for open media 
applications were numerically characterized. The nonlinear relationship of the 
suppression bandgap versus the constituent materials and dimensions of structure was 
studied. Through the use of full-wave modeling commercial software based on finite 
element method, High Frequency Structure Simulator (HFSS), the efficacy of the 
mushroom-type EBG structures in two generic problems involving radiating structure 
sharing a common ground plane were demonstrated. In the first problem, we 
demonstrated that EBG structures can suppress EMI in the enclosures and chassis. In 
the second application, we showed the performance of those structures in the cavities 
as a part of cavity-backed antennas system. Also, we verified the efficiency of those 
structures through laboratory measurements. From the numerical and experimental 
results presented, we conclude that the numerical simulation setup discussed in this 
work provides a reliable methodology for predicting the suppression band of the 
generic class of simple EBG structures compromising square patches on the metal 
 
229 
backed dielectric substrates with a via connection in between. We anticipate that EBG 
structures with more complex topology can be treated in identical manner. 
The EBG structures presented in this work were not optimized but rather 
arrived at using simple trial and error procedure coupled with educated guess. The 
optimization of EBG structures plays a strong role in more complex and sensitive 
applications where the radiators and victims are very close to each other and the 
higher EM noise suppression is required. In the antenna applications, it was shown 
that the design and implementation of EBG structures can amount to a balancing act 
between maintaining functionality and reducing coupling. Also, in the applications 
where several antennas are present in the same media such as array of cavity-backed 
antennas the spacing between antennas add up to mentioned trade off. Therefore, the 
entire multi antennas system with presence of EBG structure should be designed and 
optimized to satisfy the required performance.   
Through study of the effect of presence of EBG structure on the performance 
of cavity-backed antenna system by assessing the gain pattern, it was demonstrated 
that the design of directive CBS antennas using EBG structure is possible. Also, the 
gain of antenna at the direction of maximum directivity can be increased.  
In another study, we developed the concept of miniaturization of EBG 
structures using very high-k materials for mushroom- and planar-type structures. 
Also, we presented using very high value inductive bridges besides the high value 
effective capacitance made of very high permittivity dielectrics we could design 
miniaturized planar EBG structures providing wide suppression bands at the low 
frequency regions. Those novel EBG structures were numerically characterized. The 
 
230 
modeling of the proposed planar EBG structures was discussed. The effect of 
parameters in the model presenting the physical structure on the suppression band 
was studied. The developed model could provide a good initial guess for the EBG 
dimension and the structure as an initial starting point in the EBG design procedure. 
The miniaturized EBG structures presented in this study work were not optimized. 
Optimizing the structure may provide more effective ones.  
Availability of the miniaturized EBG structures promotes another set of 
EMI/EMC applications which were impractical due to sizing issues. We studied the 
efficiency of proposed EBG structures in EMI and SSN noise suppression in IC 
packages. Modern electronic designs are requiring high level of isolation in the 
integrated high power density circuits which demands more efficient methods of 
noise suppression. To asses the performance of the EBG structures we studied the 
embedded capacitance method which is well known as an efficient method of EM 
noise suppression in packages operating in microwave region. We studied 
analytically and numerically the limitation of using the embedded capacitance method 
made of very high-k dielectric materials. It was shown how a thin slab of very high-k 
material decreases the peak of EMI; however, EM waves are still guided in the entire 
frequency band. The comparison results between the two methods show that in IC 
packages the method of EM noise suppression using EBG structures is very 
promising. The applicability of the embedded capacitance method depends on the 
defined noise tolerance level in a system.  
The concept of design of novel wideband EBG structures using magneto-
dielectric materials is developed. These novel EBG structures could be applied in the 
 
231 
design of low profile wideband antennas. It was shown that the designed mushroom-
type EBG structures made of magneto-dielectric material are providing very wide 
bandgap compared to conventional designs made of only dielectric materials with 
smaller size patterns. This wide bandgap may extend from low frequencies. 
 
6.2. Future Work 
The development of the equivalent circuit-transmission line model of the 
proposed EBG structures in the two remainder regions of the irreducible Brillouin 
zone, ?-? and ?-? regions need to be completed. Completing that model can 
provide an initial start in design procedure with higher accuracy. 
The miniaturized planar EBG structures made of very high-k material should 
be fabricated and the performance has to be measured. However, the fabrication 
process with high-k materials is in very early stages and it is hardly possible to find 
laboratories dealing with the fabrication as a research. 
Also, time domain analysis of the proposed miniaturized EBG structures is 
necessary. The signal integrity with presence of these structures in packages should 
be assessed. 
Finally, experiments and measurements of packages with presence of the 
proposed EBG structures in the real scenarios have to be assessed. 
 
232 
Appendix 
The quasi-static value of the characteristic impedance Z
C
 for microstrip line 
shown in Figure A. 1 is given by  [53]: 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++
?
?
?
?
?
?
?
?
?
+
=
?
1444.1ln667.0393.1
125.0
8
ln
2
1
h
w
h
w
h
w
h
w
h
w
w
h
Z
ee
re
e
e
re
C
?
?
??
?
 
 Equation A- 1 
where ? is the wave impedance in free space and ? = 120? ohms. The effective width 
of microstrip w
e
 is obtained from  [53]: 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
=
2
2
ln1
25.1
2
4
ln1
25.1
?
?
??
?
h
w
t
h
h
t
h
w
h
w
t
w
h
t
h
w
h
w
e
 
 Equation A- 2 
To simplify the modeling it is assumed that the conductor thickness is negligible (t = 
0), therefore, w
e
 = w. The effective relative dielectric constant ?
re
 is given by  [53]: 
C
h
w
F
rr
re
??
?
?
?
?
?
?
+
+
=
2
1
2
1 ??
?  
 Equation A- 3 
where 
h
w
h
t
C
r
6.4
1?
=
?
. Therefore, in our modeling C = 0. Function F is given by  [53]: 
?
?
?
?
?
?
?
??
?
?
?
?
?
+
??
?
?
?
?
?
?+?
?
?
?
?
?
+
=?
?
?
?
?
?
?
?
1
12
1
11041.0
12
1
2/1
22/1
h
w
w
h
h
w
h
w
w
h
h
w
F  
 
233 
 Equation A- 4 
A few points to improve modeling in Section  3.3: 
? For small values of 
h
t
, the effect of strip thickness on Z
C
 and ?
re
 is insignificant 
where we have good conductor. However, in applications such as IC packaging in 
which the dielectric thickness h, is shrinking by development in design and 
fabrication technology, for accurate modeling we need to take to account the 
metalic thickness (assuming t = 0 will introduce large error in modeling). 
? If 1
min
<<
?
h
, the effect of dispersion on Z
C
 and ?
re
 is negligible. Yet, this is not the 
case when we are dealing with very high-k materials at high frequencies 
imperated by growing operating-speed in systems. Therefore, to get better 
compatibility with experimental results we need to consider the effect of 
dispersion on Z
C
 and ?
re
. 
? We need to remember that all the above mentioned formulas and the ones in the 
Section  3.3 are approximate. They are valid in certain ranges. For example, the 
accuracy of those expressions has not been confirmed for very high-k materials. 
However, those formulas are working accurate enough for quick analysis purpose 
as it is shown in our study. 
 
 
 
 
 
 
234 
 
w
? 
r
h
t
 
Figure A. 1.  Microstrip line. Design parameters are shown in the figure. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
235 
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