Astudy on the Effects of Ground Via Fences

A Study On The Effects Of Ground Via Fences, Embedded Patterned Layer, And Metal Surface Roughness On Conductor Backed Coplanar Waveguide

Item Type text; Electronic Dissertation

Authors Sain, Arghya

Publisher The University of Arizona.

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A STUDY ON THE EFFECTS OF GROUND VIA FENCES, EMBEDDED PATTERNED LAYER, AND METAL SURFACE ROUGHNESS ON CONDUCTOR BACKED COPLANAR WAVEGUIDE

Arghya Sain

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

The University of ARIZONA

2015 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Arghya Sain, titled A Study on the Effects of Ground Via Fences, Embedded Patterned Layer, and Metal Surface Roughness on Conductor Backed Coplanar Waveguide and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy.

Date: 10/19/2015 Kathleen L. Melde, Ph.D.

Date: 10/19/2015 Hao Xin, Ph.D.

Date: 10/19/2015 Janet M. Roveda, Ph.D.

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the graduate college.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Date: 10/19/2015 Dissertation Director: Kathleen L. Melde, Ph.D.

STATEMENT BY AUTHOR

This dissertation has been submitted in the partial fulfillment of the requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the library.

Brief quotations from this dissertation are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder.

SIGNED: Arghya Sain

ACKNOWLEDGEMENTS

Althea Gibson said, “No matter what accomplishments you make, somebody helped you”. True to her saying, this dissertation would not have been accomplished without the guidance, help and support of several people. So, I would like to take this opportunity to express my deep gratitude to those individuals, who have not only helped me with this work but also have allowed me to grow and learn throughout this journey. First of all, I would like to thank Dr. Kathleen L. Melde for giving me the opportunity to work on this project, for being patient, open and honest with me, and for allowing me to work from home when needed. I really appreciate your guidance and the fact that you encouraged me to publish papers and to sit for my written and oral comprehensive exams. I feel both honored and lucky to have had you as my research advisor. I would like to express my gratitude to Dr. Hao Xin and Dr. Janet M. Roveda for being on my dissertation defense committee and for providing feedback on the dissertation. I would like to thank Dr. Janet M. Roveda, Dr. Hal S. Tharp, Dr. Hao Xin, and Dr. Harold Parks for being on my written and/or oral comprehensive exams. I would also like to thank Tami Whelan for helping me to understand and meet the degree requirements and for providing me with important documentations on short notice. In addition, I greatly appreciate Leo Enfield (and his team) for troubleshooting any computer problems in a timely manner. I have many fond memories of my time spent in Tucson because of my friends who also worked at the High Frequency Packaging and Antenna Design Lab. Thank you Chase, Ho-Shin, Ian, Marcos, Nobuki, Prabhat, Sean, and Sung not only for all the discussions and help with my research work but also for all the laughs and the good times. You guys really made this entire journey enjoyable. I am grateful to both Jennifer and Min for trusting me and giving me the opportunity to work in their group at Intel Corporation as an intern. Both of you, along with Alaa, Emily, Mo, Jason, Jongbae, Matthew, and Ray have helped me gain not only valuable industry experience but also made the internship fun and interesting. This journey would not have been possible without my mom Kaberi and dad Swapan. Thank you for your unconditional love, constant support and for always providing for me without asking for anything in return. No amount of words can sum up the sacrifices the two of you have made for my education and happiness. Finally, I must thank my lovely girlfriend Leah for all her love, patience and encouragement. Thank you for being supportive of my work and for being by my side through thick and thin. I really appreciate you for being a constant source of joy and motivation throughout this journey. Your love means a lot to me.

CONTENTS

LIST OF FIGURES………………………………………………………………………… 8

LIST OF TABLES…………………………………………………………………………. 14

ABSTRACT………………………………………………………………………………… 16

CHAPTER 1: INTRODUCTION……………………………………………………... 18 1.1 Signal in frequency and time domain……………………... 20 1.2 Transmission lines…………………………………………. 23 A. Stripline……………………………………………… 24 B. Microstrip……………………………………………. 26 C. Coplanar waveguide…………………………………. 27 D. Coupled transmission lines………………………….. 28 1.3 Embedded patterned layer (EPL)………………………….. 29 1.4 Simulation tools…………………………………………… 31 A. ANSYS High Frequency Structure Simulator (HFSSTM)……………………………………………. 32 B. ANSYS Q3D Extractor (Q3D)……………………… 33 C. ANSYS Q2D (Q2D)………………………………… 33 D. Agilent Advanced Design System (ADS)…………… 35 E. MathWorks® MATLAB® (MATLAB®)…………... 36 1.5 Thesis goals……………………...... 37 1.6 Thesis outline……………………………………………… 38

CHAPTER 2: GROUNDED COPLANAR WAVEGUIDES (GCPWs) VIA FENCE ANALYSIS…………………………………………………... 40 2.1 Introduction………………………………………………... 40 2.2 Traditional via picket fence theory………………………... 45 A. Case 1………………………………………………... 47 B. Case 2 (Shorting plates along all three edges of both coplanar side grounds)……………………………… 48 2.3 Comparison between simulation and measurement……….. 50 2.4 Impact of via fence location and dimension on GCPW performance……………………………………………….. 52 A. Varying distance between via fence and signal trace (VL)………………………………………………….. 53 B. Varying the via to via pitch in a via fence (VP)……... 55 C. Varying separation between ends of GCPW and start of via fence edge (ES)……………………………….. 56 2.5 Alternate grounding structures for improving GCPW performance……………………………………………….. 61 A. Increasing the number of ground via fences connecting the coplanar side grounds to the lower ground plane…………………………………………. 61

B. Coplanar side grounds with periodic cutouts………... 65 2.6 Summary…………………………………………………... 66

CHAPTER 3: GROUNDED COPLANAR WAVEGUIDE WITH EMBEDDED PATTERNED LAYER (EPL)…………………………………… 67 3.1 Introduction………………………………………………... 67 3.2 Lumped element model for transmission lines……………. 71 A. Resistance (R)……………………………………….. 74 B. Inductance (L)……………………………………….. 76 C. Capacitance (C) and Conductance (G)………………. 78 3.3 Lumped element model for transmission line including floating metal effects………………………………………. 81 A. Self-resistance (RSFM) of a floating metal…………… 91 B. Self-inductance (LSFM) of a floating metal…………... 92 C. Mutual inductance (MFMST) between a floating metal and a signal trace…………………………………….. 94 3.4 Parametric study of EPL with GCPW……………………... 98 A. Varying EPL thickness (TEPL)……………………….. 99 B. Varying separation between signal trace and EPL (HEPL)………………………………………………... 103 C. Varying EPL pitch along x (PXEPL) and y (PYEPL) axis…………………………………………………... 105 D. Varying EPL metal conductivity…………………….. 108 E. Varying EPL individual element geometry…………. 110 F. Varying the number and location of EPL columns….. 112 3.5 Comparison of the effect of EPL on different types of transmission lines………………………………………….. 115 3.6 Summary…………………………………………………... 117

CHAPTER 4: EMBEDDED PATTERNED LAYER (EPL) IN A THREE CONDUCTOR GROUNDED COPLANAR WAVEGUIDE SYSTEM……………………………………………...... 120 4.1 Introduction………………………………………………... 120 A. Quiet mode…………………………………………... 123 B. Even mode…………………………………………… 123 C. Odd mode……………………………………………. 124 4.2 Impact of EPL on coupled grounded coplanar waveguides.. 126 4.3 Summary…………………………………………………... 134

CHAPTER 5: BROADBAND CHARACTERIZATION OF COPLANAR WAVEGUIDE INTERCONNECTS WITH ROUGH CONDUCTOR SURFACES…………………………………………………………… 136 5.1 Introduction………………………………………………... 136 5.2 Properties of random rough surfaces……………………… 143 5.3 Generating a random rough surface……………………….. 147 5.4 Reducing computational complexity……………………… 150

5.5 Results and comparison…………………………………… 152 5.6 Computational requirements and model generation………. 162 5.7 Summary…………………………………………………... 164

CHAPTER 6: CONCLUSION AND FUTURE WORK……………………………... 165

APPENDIX I DERIVATION OF RESONANT FREQUENCY EQUATION OF RECTANGULAR CAVITIES WITH DIFFERENT BOUNDARY CONDITIONS………………………………………………………… 168

APPENDIX II REFLECTION COEFFICIENT AT DIELECTRIC AND PEC OR PMC INTERFACE……………………………………………….. 176

APPENDIX III PLOTS OF SCATTERING PARAMETERS FROM CHAPTER 3 WHEN RENORMALIZED TO STANDARD 50 Ω IMPEDANCE… 178

APPENDIX IV PLOTS OF SCATTERING PARAMETERS FROM CHAPTER 4 WHEN RENORMALIZED TO STANDARD IMPEDANCES……... 181

REFERENCES……………………………………………………………………………... 183

LIST OF FIGURES

Fig. 1-1 Comparison between (a) pulse and an ideal square wave in time domain; (b) frequency content of a pulse and an ideal square wave in frequency domain…………………………………………………….. 22 Fig. 1-2 (a) Pulse in time domain; (b) Frequency content of the pulse; (c) Pulse recreated with only h1 and first 6 odd harmonics; (d) Pulse recreated with first 3 odd harmonics and first 36 odd harmonics……. 23 Fig. 1-3 Pulse recreated from harmonics with (a) amplitude distortion, (b) phase distortion, (c) both amplitude and phase distortion, and (d) shows the impact of harmonic distortion on pulse edge rate………… 24 Fig. 1-4 (a) Image of cross-sectional view of a centered stripline; (b) Plot of characteristic impedance of a centered stripline versus ratio of signal width and substrate thickness for different dielectric constant and metallic thickness……………………………………………………. 25 Fig. 1-5 (a) Image of cross-sectional view of a microstrip; (b) Plot of characteristic impedance and effective permittivity of a microstrip versus ratio of signal width and substrate thickness for different metallic thickness………...... 26 Fig. 1-6 (a) Image of cross-sectional view of a CPW; (b) Plot of characteristic impedance and effective permittivity of a CPW versus aspect ratio for different metallic thickness……………………………………….. 28 Fig. 1-7 (a) Image of cross-sectional view of a coupled microstrip transmission line; (b) Plot of even and odd mode characteristic impedance of a coupled microstrip transmission line versus ratio of signal width and substrate thickness for different spacing between traces………………………………………………………………….. 29 Fig. 1-8 Image of (a) cross-sectional view and (b) top view of a microstrip transmission line with EPL…………………………………………... 30 Fig. 1-9 Image of (a) cross-sectional view and (b) top view of a coupled microstrip transmission line system with EPL……………………….. 31 Fig. 1-10 Accurate modeling flowchart for microstrip transmission line [30].… 36 Fig. 2-1 3-D representations of (a) CPW, (b) FG-CBCPW, and (c) GCPW….. 43 Fig. 2-2 Plot of electric field distribution in the cross-section of (a) CPW, (b) FG-CBCPW, and (c) GCPW…………………………………………. 43 Fig. 2-3 Image of (a) top layer and (b) cross-sectional view of a GCPW…….. 45 Fig. 2-4 Plot of (a) insertion loss and (b) power loss versus frequency for FG- CBCPWs with varying SGW………………………………...... 48 Fig. 2-5 Top layer view of FG-CBCPWs with coplanar side ground edges shorted to the lower ground plane……………………………………. 49 Fig. 2-6 Plot of insertion loss versus frequency for a FG-CBCPW with un- grounded and grounded CSG edges………………………………….. 49 Fig. 2-7 Plot of electric field distribution in log scale for (a) FG-CBCPW at 73.3 GHz for (fr)420, (b) Case 1 at 82.4 GHz for (fr)42.50, and (c) Case 2 at 82 GHz for (fr)42.50……………………………………..………… 50 Fig. 2-8 Image of the cross-section of a fabricated GCPW…………………… 51

Fig. 2-9 Plot of (a) insertion loss and (b) return loss versus frequency for measured and simulated data…………………………………………. 51 Fig. 2-10 Plot of electric field distribution in log scale for a fabricated GCPW structure at (a) 35.5 GHz for (fr)00.50, and (b) 39 GHz for (fr)10.50……. 52 Fig. 2-11 Plot of (a) insertion loss and (b) power loss versus frequency for GCPWs with varying VL…………………………………………….. 53 Fig. 2-12 Plot of electric field distribution in log scale for GCPW with (a) VL = 1200 μm at 38.5 GHz for (fr)00.50, (b) VL = 1200 μm at 68.6 GHz for (fr)40.50, and (c) VL = 2700 μm at 72.9 GHz for (fr)41.50………….. 54 Fig. 2-13 Plot of insertion loss versus frequency for GCPWs with varying VP.. 55 Fig. 2-14 Plot of electric field distribution in log scale for GCPW with (a) VP = 2873 μm at 32.5 GHz, (b) VP = 2873 μm at 78 GHz, and (c) VP = 1436.5 μm at 68.5 GHz………………………………………………. 56 Fig. 2-15 Plot of insertion loss versus frequency for GCPWs with varying ES... 57 Fig. 2-16 Plot of electric field distribution in log scale for GCPW with (a) ES = 127 μm, and (b) ES = 414.3 μm at 100 GHz…………………………. 58 Fig. 2-17 Plot of insertion loss versus frequency in GCPWs with varying SGW………………………………………………………………….. 59 Fig. 2-18 Image of a section of a CCPW...... 59 Fig. 2-19 Plot of insertion loss versus frequency for GCPW and CCPW with similar physical dimensions………………………………………….. 59 Fig. 2-20 Plot of electric field distribution in log scale for (a) GCPW, and (b) CCPW at 100 GHz…………………………………………………… 60 Fig. 2-21 Plot of insertion loss versus frequency in GCPWs with varying VR… 60 Fig. 2-22 Top layer view of a GCPW with three via fences on each CSG……... 62 Fig. 2-23 Plot of insertion loss versus frequency in GCPWs with one or two via fences on each CSG and varying VFP…………………………… 63 Fig. 2-24 Plot of insertion loss versus frequency in GCPWs with one or three via fences on each CSG and varying VFP…………………………… 63 Fig. 2-25 Plot of insertion loss versus frequency in GCPWs with one or two via fences and varying VFP, when VL = 400 μm and VP is less than λ/2 at 100 GHz……………………………………………………….. 64 Fig. 2-26 Plot of insertion loss versus frequency in GCPWs with one or two via fences and varying VFP, when VL = 1200 μm and VP is less than λ/2 at 100 GHz……………………………………………………….. 64 Fig. 2-27 Top layer view of a GCPW with slots cut out in each CSG…………. 65 Fig. 2-28 Plot of insertion loss versus frequency for GCPWs without and with slots of varying SW…………………………………………………... 66 Fig. 3-1 Image of (a) top, (b) cross-sectional, and (c) diametric view of a GCPW with EPL……………………………………………………... 67 Fig. 3-2 Image of a 3 x 3 EPL unit cell………………………………………... 69 Fig. 3-3 General lumped element model of a transmission line………………. 72 Fig. 3-4 Simplified lumped element model of a transmission line……………. 73

Fig. 3-5 Image of (a) a conductor of rectangular cross-section, (b) current distribution in the cross-section of a conductor at DC, (c) current distribution in the cross-section of a conductor at high frequencies; (d) Plot of skin depth versus frequency with varying resistivity...... 75 Fig. 3-6 Simplified pictorial representation of a current carrying loop interacting with an adjacent loop via magnetic fields………………... 77 Fig. 3-7 Image of a signal trace and a floating metal in close proximity of each other in (a) diametric and (b) side view………………………… 81 Fig. 3-8 Image of current distribution at high frequency in a signal trace and a floating metal when they are in close proximity of each other in (a) diametric and (b) side view...... 82 Fig. 3-9 Modified lumped element model of a transmission line in the presence of a floating metal………………………………………….. 82 Fig. 3-10 Cross-section of a transmission line (a) without floating metal, (b) with floating metal directly under the signal trace, and (c) with floating metal not directly below the signal trace……………………. 84 Fig. 3-11 Plot of p.u.l. capacitance versus floating metal width with varying floating metal thickness………………………………………………. 86 Fig. 3-12 Image of (a) floating metal, (b) current distribution in floating metal at high frequency, and (c) floating metal being represented as a rectangular current loop made of a conductor with rectangular cross- section………………………………………………………………… 90 Fig. 3-13 Plot of floating metal self-resistance versus frequency with varying floating metal thickness and width……...... 92 Fig. 3-14 Plot of floating metal self-inductance versus frequency with varying floating metal thickness and length…………………………………... 94 Fig. 3-15 Magnetic flux density at any arbitrary point due to a current sheet of finite width and infinite length……………………………………….. 95 Fig. 3-16 Setup for calculating the magnetic flux density at any arbitrary point on a surface enclosed by a floating metal due to current through signal trace……………………………………………………………. 96 Fig. 3-17 Plot of mutual inductance versus frequency with varying floating metal thickness and separation between floating metal and signal conductor……………………………………………………………... 97 Fig. 3-18 Plot of (a) insertion loss, (b) return loss, and (c) insertion loss phase versus frequency for GCPWs without and with EPL of varying TEPL, with ports renormalized to line impedance...... 100 Fig. 3-19 Eye diagram of 1 Gbps bit stream (a) at transmitter, (b) at receiver of a GCPW without EPL, (c) at receiver of a GCPW with EPL of TEPL = 9 μm, and (d) at receiver of a GCPW with EPL of TEPL = 27 μm…………………………………………………………………….. 102 Fig. 3-20 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying HEPL, with ports renormalized to line impedance……………………...... 104 Fig. 3-21 Top view of a GCPW with (a) EPL, and (b) solid metal strips embedded in the substrate……………………………………………. 105

Fig. 3-22 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PXEPL, with ports renormalized to line impedance…...... 106 Fig. 3-23 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PYEPL, with ports renormalized to line impedance……...... 108 Fig. 3-24 Bar plot showing variation in p.u.l. R, L, G, and C in GCPWs without and with EPL when the substrate dielectric constant is varied keeping the loss tangent constant…………………………………….. 110 Fig. 3-25 Top view of GCPWs with different EPL individual element shape…. 111 Fig. 3-26 Top view of GCPWs with a (a) 3 x 1, (b) offset 3 x 1, (c) 3 x 2, (d) offset 3 x 2, and (e) 3 x 3 EPL unit cell……………………………… 113 Fig. 3-27 Images of cross-section of GCPWs without and with different EPL arrangement…………………………………………………………... 116 Fig. 3-28 Images of cross-section of microstrip transmission lines without and with EPL……………………………………………………………… 116 Fig. 3-29 Images of cross-section of striplines without and with different EPL arrangement…………………………………………………………... 116 Fig. 4-1 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide………………... 121 Fig. 4-2 Lumped element model of two lossless coupled transmission line….. 122 Fig. 4-3 Even mode (a) electric, and (b) magnetic field pattern in a coupled transmission line [25]………………………………………………… 123 Fig. 4-4 Odd mode (a) electric, and (b) magnetic field pattern in a coupled transmission line [25]………………………………………………… 124 Fig. 4-5 General signature of saturated FEXT and NEXT voltage noise……... 125 Fig. 4-6 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with a 1 column EPL…………………………………………………………………… 126 Fig. 4-7 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with a 3 column EPL…………………………………………………………………… 127 Fig. 4-8 Plot of (a) insertion loss, (b) return loss, (c) NEXT, and (d) FEXT versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to line impedance…………………………………………………….. 129 Fig. 4-9 Plot of (a) differential insertion loss (SDD21), (b) differential return loss (SDD11), and (c) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to line impedance…………………………... 131 Fig. 4-10 Image of cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with an asymmetric EPL placement…... 133 Fig. 4-11 Plot of (a) differential insertion loss (SDD21), and (b) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with asymmetric EPL placement, with ports renormalized to 100 Ω differential impedance……………………….. 133

Fig. 5-1 Cross-sectional image of a rough conductor over ground plane [89]... 137 Fig. 5-2 3-D model of a 90o peel strength test setup…………………………... 137 Fig. 5-3 Diagram of 2-D corrugated copper surface for Hammerstad model [25]…………………………………………………………………… 138 Fig. 5-4 Hemispherical model shown as a single surface protrusion with (a) top and (b) side views [25]...... 139 Fig. 5-5 Magnified (5000x) SEM photograph of a copper surface at an angle of 30o [25] ………………...... 139 Fig. 5-6 Model cross-section of copper spheres which form pyramid like structures on flat copper conductors [25]…..………………………… 140 Fig. 5-7 Plot of 퐾퐻, 퐾퐻푒푚푖, and 퐾퐻푢푟푎푦 versus frequency [25]……………...... 141 Fig. 5-8 3-D representation of a conductor backed coplanar waveguide……... 142 Fig. 5-9 SEM images of copper and dielectric with (a) 1 oz. electrodeposited, (b) 0.5 oz. electrodeposited, (c) 1 oz. rolled, and (d) 0.5 oz. rolled copper foils [101] ……………………………………………………. 144 Fig. 5-10 Magnified (2500x) SEM photographs of (a) electrodeposited and (b) rolled annealed copper foils with 퐻푟푚푠 = 3.0 and 0.5 μm, respectively [91]……………………………………………………… 144 Fig. 5-11 3-D surface profile measurement of a copper foil [92]………………. 145 Fig. 5-12 1-D Gaussian rough surface with unity 퐻푟푚푠 and (a) λ = 0.1 mm, (b) λ = 0.1 mm, and (c) λ = 1.0 mm………...... 146 Fig. 5-13 1-D rough surface with (a) Gaussian and (b) exponential ACFs with unity 퐻푟푚푠 and λ = 0.1 mm…………………………………………... 146 Fig. 5-14 (a) 3-D surface plot and (b) contour plot of a randomly generated rough surface with 퐻푟푚푠 = 5.5 μm and λ = 3 μm……………………. 148 Fig. 5-15 Signal conductor with (a) smooth and (b) randomly roughened surface………………………………………………………………... 150 Fig. 5-16 Plot of attenuation coefficient versus frequency for the six simulation cases mentioned in Table 5.1………………………………………… 152 Fig. 5-17 Bar plot of memory, simulation time, matrix size and number of tetrahedrons for simulation cases mentioned in Table 5.1…………… 152 Fig. 5-18 Plot of (a) attenuation coefficient per unit length, (b) enhancement factor, and (c) insertion loss versus frequency with varying λ (1.0 and 4.0 μm) and constant 퐻푟푚푠 (1.0 μm) for a 7-in-long CB-CPW……… 153 Fig. 5-19 Plot of (a) attenuation coefficient per unit length, (b) enhancement factor, and (c) insertion loss versus frequency for constant λ (3.0 μm) and varying 퐻푟푚푠 (1.0, 2.5, and 5.5 μm) for a 7-in-long CB-CPW….. 155 Fig. 5-20 Plot of (a) attenuation coefficient per unit length and (b) enhancement factor versus frequency for varying λ and 퐻푟푚푠 (퐻푟푚푠 = λ; 0.5, 1.0 and 2.5 μm) for a 7 in-long CB-CPW…………... 157 Fig. 5-21 Plot comparing measured results from [3] with simulated results from HFSSTM…………………………………………………………. 159 Fig. 5-22 Cross-sectional view of current distribution in (a) portions of roughened signal and coplanar side ground and (b) zoomed in image of a portion of the roughened conductors in (a) at 40 GHz…………... 160

Fig. 5-23 Cross-sectional view of CB-CPW with the sides of the conductors inclined at an angle θ…………………………………………………. 160 Fig. 5-24 Cross-sectional view of the electric field distribution in CB-CPW with (a) smooth and (b) rough conductor surface at 40 GHz, in log scale…………………………………………………………………... 161 Fig. 5-25 Cross-sectional view of the mesh generated by HFSSTM; (a) Meshing in the entire cross-section of the interconnect, (b) meshing in the metal (signal and coplanar side ground) with a roughened surface, and (c) zoomed in image of the meshing in the signal conductor and substrate interface…………………………………………………….. 162 Fig. 5-26 Bar plot for memory, simulation time, matrix size and number of tetrahedrons for CB-CPWs with varying extent of conductor surface roughness……………………………………………………………... 164 Fig. I-1 (a) Top layer and (b) cross-sectional view of a rectangular patch formed by CSG and LGP of a FG-CBCPW………………………….. 168 Fig. I-2 Shows (a) the 3-D simplified model of the rectangular cavity formed by a CSG and LGP of a FG-CBCPW along with the locations of the (a) PECs and (c) PMCs with respect to the cavity…………………… 169 Fig. I-3 Location of PECs and PMCs for the rectangular cavity in Case 1…... 173 Fig. I-4 Location of PECs and PMCs for the rectangular cavity in Case 2…... 174 Fig. II-1 Perpendicularly polarized plane wave incident at an angle on an interface between a dielectric and a PEC or PMC…………………… 176 Fig. III-1 Plot of (a) insertion loss, (b) return loss, and (c) insertion loss phase versus frequency for GCPWs without and with EPL of varying TEPL, with ports renormalized to 50 Ω impedance…………………………. 178 Fig. III-2 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying HEPL, with ports renormalized to 50 Ω impedance…………………………………….. 179 Fig. III-3 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PXEPL, with ports renormalized to 50 Ω impedance…………………………………….. 179 Fig. III-4 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PYEPL, with ports renormalized to 50 Ω impedance…………………………………….. 180 Fig. IV-1 Plot of (a) insertion loss, (b) return loss, (c) NEXT, and (d) FEXT versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to 50 Ω impedance…………………………………………………… 181 Fig. IV-2 Plot of (a) differential insertion loss (SDD21), (b) differential return loss (SDD11), and (c) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to 100 Ω differential impedance……………. 182

LIST OF TABLES

Table 1.1 Technology trends associated with on-chip clock frequency, transistor count and density, chip area and chip I/O count for different product generations associated with high performance MPUs and ASICs according to ITRS [1]…………………………… 19 Table 1.2 Comparison between static, quasi-static, and full wave EM solvers [29]………………………………………………………………….. 35 Table 2.1 Comparison of resonant frequencies obtained from simulations and analytical equations…………………………………………………. 47 Table 3.1 List of variable dimensions associated with EPL…………………... 98 Table 3.2 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying TEPL at 40 GHz…………………………………….. 99 Table 3.3 α, ZO and εeff of a GCPW without and with EPL of varying TEPL at 40 GHz……………………………………………………………… 99 Table 3.4 Eye height and eye width at the receiver of a 18 cm long GCPW without and with EPL of varying TEPL……………………………... 102 Table 3.5 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying HEPL at 40 GHz…………………………………...... 104 Table 3.6 α, ZO and εeff of a GCPW without and with EPL of varying HEPL at 40 GHz……………………………………………………………… 104 Table 3.7 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying PXEPL at 40 GHz…………………………………… 106 Table 3.8 α, ZO and εeff of a GCPW without and with EPL of varying PXEPL at 40 GHz……………………………………………………………… 106 Table 3.9 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying PYEPL at 40 GHz…………………………………… 107 Table 3.10 α, ZO and εeff of a GCPW without and with EPL of varying PYEPL at 40 GHz……………………………………………………………… 108 Table 3.11 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying EPL metal conductivity at 40 GHz………………... 109 Table 3.12 α, ZO and εeff of a GCPW without and with EPL of varying EPL metal conductivity at 40 GHz………………………………………. 109 Table 3.13 Per unit length R, L, G, and C values of a GCPW without and with EPL made of different individual element shape at 40 GHz……….. 112 Table 3.14 α, ZO and εeff of a GCPW without and with EPL of varying EPL made of different individual element shape at 40 GHz…………….. 112 Table 3.15 Per unit length R, L, G, and C values of a GCPW without and with EPL with varying number of columns and column location at 40 GHz…………………………………………………………………. 114 Table 3.16 α, ZO and εeff of a GCPW without and with EPL of varying EPL with varying number of columns and column location at 40 GHz…. 114 Table 3.17 Percentage change in characteristic impedance and effective permittivity of different transmission line types due to EPLs at 40 GHz…………………………………………………………………. 117

Table 3.18 Summary of the effect of EPL parameters on GCPW p.u.l. R, L, G, C parameters, attenuation coefficient, characteristic impedance, and effective permittivity……………………………………………...... 118 Table 4.1 List of variable dimensions associated with conductor backed edge coupled coplanar waveguide………………………………………... 121 Table 4.2 List of variable dimensions associated with EPL…………………... 127 Table 4.3 Per unit length Cg, CM, LO, and LM values of a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz……………………………………………………………… 128 Table 4.4 Different modal impedances associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz……………………………………………………………… 128 Table 4.5 Different modal propagation velocities associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz………………………………………………… 128 Table 4.6 NEXT and FEXT associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz…………………………………………………………………. 128 Table 5.1 Percentage of conductor surface area roughened………………….... 151

ABSTRACT

Electrical engineers have responded to the increasing demand for circuit speed and functionality by reducing transistor feature size and increasing on-chip transistor density. Consequently, interconnect density, both on-chip and the system level is also increasing. Increasing circuit speed translates into shorter clock cycles and signals with faster edge rates, which have multi-GHz bandwidth. Densely packed parallel interconnects will cause signal integrity problems not only due to the increase in crosstalk noise but also due to the intrinsic low pass filter characteristics of the interconnects. The lossy nature of the interconnects is also going to increase due to metal surface roughness at higher frequencies, which will further degrade the signal quality at the receiver input.

Embedded Patterned Layer (EPL), which is a patterned floating metal layer between a signal trace and its return path shows promise in reducing far-end crosstalk (FEXT). EPL also allows designers to modify the characteristic impedance of interconnects by varying the different physical parameters of the EPL.

This dissertation analyzes the effect of EPL on conductor backed coplanar waveguides (CB-CPW). CB-CPWs excite higher order modes at high frequencies, so work was done to understand the effect of different ground via fence parameters in suppressing the higher modes which helps increase the interconnect bandwidth. A CB-

CPW with ground via fence is called a grounded coplanar waveguide (GCPW). A very basic lumped element model transmission line model was developed to account for the effect of floating metals near a transmission line. This model was then used to explain the effect of EPL on a GCPW with large bandwidth. EPL reduces the characteristic impedance of the transmission line. Engineers can then design narrow high impedance

16 transmission lines and use EPL to reduce the impedance to a desired value. This also allows reduction in crosstalk by increasing the spacing between the transmission lines.

The EPL also reduced the differential impedance of a grounded conductor backed edge coupled coplanar waveguide, when it was used for differential signaling. Care must be taken to make sure that the EPL is symmetric to both the legs of the differential pair to avoid differential to common mode energy conversion, which can cause electromagnetic interference (EMI) problems. EPL reduced FEXT while increasing near-end crosstalk

(NEXT), when the coupled transmission line system was used for single ended signaling.

Finally, a statistical method for modeling transmission line metal surface roughness in three dimensional (3D) full wave electromagnetic solvers was developed to account for increased attenuation in transmission lines, at high frequencies, due to metal surface roughness.

CHAPTER 1: INTRODUCTION

Electronic devices have become an integral part of the human society over the duration of a few decades. These omnipresent devices are improving the quality of life via their applications in diverse markets like communication networks, computation and data processing, environment, health care, home appliances and entertainment, personal communication, and transportation. The large scale low cost production of electronic devices allows ordinary humans to buy electronic devices of their choice, which in turn has led to the rapid proliferation of these devices into human society. Wafer size increase, reduction in transistor feature size, and yield improvements can be attributed to the low manufacturing cost associated with integrated circuits, which in turn reduces the cost of the electronic devices.

The ever increasing demand for bandwidth, functionality, and speed, due to emerging markets like cloud computing, gaming, and mobile video entertainment, requires the development of high performance communication network systems, desktops, laptops, mobile devices, and servers. The falling transistor feature size results in faster transistors and improved circuit speed. The chip size does not scale down with the transistor size instead circuit designers make use of the increased circuit transistor density to increase the number of on-chip system functionalities by adding more transistors to the same (or reduced) die area. This increasing transistor count, to support the increasing number of on-chip functionalities, increases the number of signal

Input/Output (I/O) terminals (pads/pins) required to move signals from and to an integrated circuit (IC) at higher transmission rates due to increasing on-chip circuit speed.

Additional power and ground terminals are also required to maintain power and signal

18 integrity. Table 1.1 shows the projected technology trends in terms of on-chip clock frequency, chip size, transistor density and maximum number of chip I/O terminals for high performance microprocessors (MPU) and application specific integrated circuits

(ASIC), according to International Technology Roadmap for Semiconductors (ITRS) [1].

Table 1.1 Technology trends associated with on-chip clock frequency, transistor count and density, chip area, and chip I/O count for different product generations associated with high performance MPUs and ASICs according to ITRS [1] Year of production 2011 2015 2019 2023 MPU physical gate length (nm) 24.00 17.00 11.70 8.10 On-chip local clock frequency 3.744 4.380 5.124 5.994 (GHz) Number of transistors per MPU 4,424 8,848 35,391 70,782 chip at production (Millions) ASIC 14,599 46,348 116,790 294,293 Chip area at production MPU 260 164 260 206 (mm2) ASIC 858 858 858 858 Transistor density (Million 1,702 5,402 13,612 34,300 transistors/cm2) Maximum number of chip MPU 3,072 3,072 3,072 3,072 I/Os ASIC 4,800 5,600 6,200 6,840 Maximum number of total package 5,094 6,191 7,525 9,148 pins

The increase in on-chip signal I/O, power and ground terminals, in turn, increases the interconnect density on-chip, package substrate, and system printed circuit board

(PCB). Interconnects are used to distribute clock and other signals to the different functional blocks of an IC while also providing the necessary power and ground connections. This increased interconnect density results in more closely spaced parallel interconnects which results in increased crosstalk (XT) among interconnects. Increasing circuit speed is the result of shorter clock cycles and faster rise and fall times of the signal waveforms. Signals with short periods and fast switching speeds have multi-GHz bandwidth. Therefore, interconnects that transmit such high speed signals must be

19 analyzed using the transmission line theory [2]. These interconnects (or transmission lines) are lossy and act as low pass filters. As a consequence, the transmission lines attenuate the high frequency components of a signal more than the low frequency components and also change the phase relationships among the different frequency components. Also, transmission line edge roughness increases the transmission line resistivity [1]. Any signal passing through a transmission line will be distorted by the time it reaches the receiver due to the low pass, lossy nature of the transmission line, inter symbol interference (ISI), and crosstalk from neighboring transmission lines. This inability of transmission lines, both on-chip and off-chip, to transmit distortion and interference free multi-GHz bandwidth signals results in the interconnect bottleneck associated with high speed circuits.

In signaling, information is transmitted in time domain through data streams which contain pulses of varying width. So it is important to understand how the signals in time domain relate to information in frequency domain. The following section shows the frequency component of signals with different rise and fall times and how the signal is distorted when the magnitude and phase relationship among the different frequency components is changed.

1.1 Signal in frequency and time domain

A periodic waveform in the time domain can be reconstructed by the summation of a series of sine waves with proper amplitude and phase information. Discrete Fourier transform (DFT) of a time domain signal extracts the amplitude and phase information associated with the different harmonics that constitute the signal. This information about a signal’s frequency content is required to determine the bandwidth of interconnects. The

20 fundamental frequency or first harmonic is determined by the period (T) of the signal and the subsequent harmonics as separated from each other by an amount equaling the fundamental frequency. The DC value or zeroth harmonic (h0) of the signal is its mean value.

A pulse transitions from 0 to a fixed voltage after some finite rise time (Tr) and stays fixed at this voltage for a certain time (THIGH) and then falls back to 0 after some finite fall time (Tf). It then stays at 0 for the rest of the time (TOFF) in the period T. The pulse on time (TON) is the summation of TR, THIGH, and TF. The duty cycle (DC) of the pulse is the ratio of TON to T. An ideal square wave is symmetrical with zero TR and TF and has 50% DC and it has a peak voltage of 1V [3]. A comparison of a pulse and an ideal square wave is shown in Fig. 1-1 (a). The comparison of the frequency content of the two is shown in Fig. 1-1 (b). From Fig. 1-1 (b), it is clear that an ideal square wave contains only odd harmonics (h1, h3, h5,…) while a pulse consists of both odd and even harmonics. Also, the amplitude of the harmonics associated with the pulse fall to zero much faster than the ideal square wave. So, the bandwidth of a signal is dependent on the rise and fall times. Signals with sharper rise and fall times have larger bandwidths. The amplitude of each harmonic depends on the duty cycle and rise time of the trapezoidal pulse [4].

The bandwidth of a time domain signal is determined by the highest frequency component that must be included to approximate all the important features of the time domain signal. A signal starts to resemble the real time domain signal as more and more harmonics are added with proper amplitude and phase information, see Fig. 1-2 (c), (d).

However, as more and more harmonics are added the overshoot near the transition gets

21 worse due to Gibb’s phenomenon. Gibb’s phenomenon is the result of approximating a periodic signal with discontinuities with a finite series of continuous functions [5]. Fig. 1-

2 (a) shows a pulse in time domain while Fig. 1-2 (b) shows the harmonic contents of the signal in frequency domain.

(a) (b) Fig. 1-1 Comparison between (a) pulse and an ideal square wave in time domain; (b) frequency content of a pulse and an ideal square wave in frequency domain

Transmission lines are used to propagate signals from one point to another. But transmission lines act as low frequency filters since conductor and dielectric losses attenuate high frequency components more than low frequency harmonics of a signal.

This results in reduced signal bandwidth and slower rise and fall time. Transmission lines also alter the phase relationship between harmonics. Only amplitude distortion of harmonics results in a smooth waveform with overall low amplitude, see Fig. 1-3 (a) [4].

Phase distortion causes the resultant waveform to peak instead of a flat top, see Fig. 1-3

(b) [4]. Finally, when both amplitude and phase of the harmonics are distorted the resultant waveform has reduced amplitude with peaking and it is spread outside the region occupied by the original waveform, see Fig. 1-3 (c) [4]. This can result in unwanted inter-symbol interference (ISI).

(a) (b)

(c) (d) Fig. 1-2 (a) Pulse in time domain; (b) Frequency content of the pulse; (c) Pulse recreated with only h1 and first 6 odd harmonics; (d) Pulse recreated with first 3 odd harmonics and first 36 odd harmonics

1.2 Transmission lines

Transmission lines are broadly classified as homogeneous and inhomogeneous depending on the uniformity of the dielectric medium surrounding the signal traces [6]. In a homogeneous transmission line, the surrounding dielectric is uniform and the propagating modes are transverse electromagnetic (TEM) in nature. Inhomogeneous transmission lines support quasi-TEM waves because the field lines between the signal and ground trace are not completely confined in the substrate. Microwave integrated circuits (MICs) require transmission lines which are planar in configuration. Planar transmission lines allow easy circuit fabrication using photolithography and etching on

23 substrates [7]. Planar transmission lines have metal strips that lie completely in parallel planes [8]. This section compares the three most popular planar transmission line structures used in integrated circuit design.

(a) (b)

(c) (d) Fig. 1-3 Pulse recreated from harmonics with (a) amplitude distortion, (b) phase distortion, (c) both amplitude and phase distortion, and (d) shows the impact of harmonic distortion on pulse edge rate

A. Stripline

Striplines have evolved from the coaxial transmission lines and were the first planar transmission lines to be used in microwave printed circuits [9], [10]. A typical centered stripline consists of a signal conductor between two ground planes, see Fig. 1-4

(a). The signal conductor is equidistant from both ground planes and is embedded in a dielectric material separating the ground planes. Striplines are homogeneous and support

24 a pure TEM wave. These transmission lines provide excellent high frequency performance with minimized radiation and are immune to incoming spurious signals.

However, striplines are more difficult and expensive to fabricate than microstrips and coplanar waveguides. Striplines are harder to prototype and troubleshoot post fabrication compared to microstrips as well [11]. Striplines also need via connections to short the two metal ground planes and deny designers direct access to the signal trace for mounting components. Stripline applications are generally restricted to passive components like filters and directional couplers [12]. The characteristic impedance of a stripline is determined by the width of the signal trace, thickness and dielectric constant of the substrate [13], [14], [15]. For a given substrate thickness and line impedance, striplines have narrower signal conductor compared to microstrips because of the second ground plane. Fig. 1-4 (b) shows the variation in line impedance of a stripline with substrate height and signal width for given dielectric constant and metal thickness.

(a) (b) Fig. 1-4 (a) Image of cross-sectional view of a centered stripline; (b) Plot of characteristic impedance of a centered stripline versus ratio of signal width and substrate thickness for different dielectric constant and metallic thickness

B. Microstrip

Microstrips are one of the most popular planar transmission line used in microwave printed circuit board (PCB) applications. It is made of a signal conductor and ground plane separated by a dielectric medium, see Fig. 1-5 (a). Microstrips are selected over striplines due to simplicity. It is an inhomogeneous transmission line and supports quasi-TEM waves. Microstrips are dispersive because of the difference in phase velocity of the EM waves in the air and the dielectric. These are simple, easy to fabricate, and allow easy integration of active and passive components in series. However, via holes are required to connect components to the ground plane [12]. It is also easier to probe integrated components and perform post fabrication circuit adjustments and measurements. The characteristic impedance and effective dielectric constant of a microstrip circuit is dependent on the substrate thickness and width of the signal conductor [7], [15], [16]. The upper and lower limits of achievable line impedance with a microstrip are decided by the minimum line width fabricable and excitation of higher order modes, respectively [12]. Fig. 1-5 (b) shows the variation in line impedance of a microstrip with substrate height and signal width for different metal thickness.

(a) (b) Fig. 1-5 (a) Image of cross-sectional view of a microstrip; (b) Plot of characteristic impedance and effective permittivity of a microstrip versus ratio of signal width and substrate thickness for different metallic thickness

C. Coplanar waveguide

Coplanar waveguides (CPWs) are the most frequently used planar transmission lines in integrated circuits after microstrips. In CPWs, the signal and the two ground planes (running adjacent and parallel to the signal) are fabricated on the same side of the dielectric, creating a ground signal ground (GSG) trace configuration, see Fig. 1-6 (a).

Like microstrips, CPWs are also inhomogeneous and support quasi-TEM waves. CPWs can achieve lower dispersion than microstrips via tight coupling between signal and ground conductors [17]. The fields in CPWs are less confined than microstrips making

CPWs more sensitive to shields placed on top of the transmission lines [7]. CPWs allow easy series and shunt surface integration of active and passive components. CPWs also offer low inductive ground connection compared to microstrips by eliminating vias required for connecting components to ground [12]. CPWs provide better isolation than microstrips due to the GSG configuration which in turn allows for higher packaging density and smaller chip size [18]. CPWs suffer from higher losses and have lower power handling capability compared to microstrips. A conductor backing is introduced to improve the power handling capability of the CPWs creating a conductor backed CPW

(CB-CPW). The characteristic impedance of CPWs is determined by the width of the signal, substrate thickness, and the spacing between the signal and ground planes in the

GSG configuration [15], [18]. Fig. 1-6 (b) shows the variation in line impedance of a

CPW with aspect ratio for different metal thickness. Aspect ratio is defined as the ratio of signal width (W = 2a) to the separation between the two coplanar side grounds (W + 2G

= 2b).

(a) (b) Fig. 1-6 (a) Image of cross-sectional view of a CPW; (b) Plot of characteristic impedance and effective permittivity of a CPW versus aspect ratio for different metallic thickness

D. Coupled transmission lines

A simple coupled transmission line structure consists of two transmission lines, which are parallel to and in close proximity of each other. These transmission lines are coupled to each other via electromagnetic fields generated when signals propagate through them. A pair of coupled lines can support two different modes of propagation.

Even mode is excited when both the traces are excited by similar signals. Odd mode is excited when the traces are excited by signals of opposite polarity. These modes have different characteristic impedance while effective dielectric constant is different for inhomogeneous transmission lines only. Coupled transmission lines can be classified as edge coupled (lines side by side) and broadside coupled (lines on top of each other in adjacent layers). Coupled lines are used in microwave components like directional couplers, filters and phase shifters. Coupled lines are also used in differential signaling.

Differential signaling uses the odd mode of propagation in a coupled transmission line.

The odd mode excitation creates a virtual reference plane between the two conductors thus providing a continuous return path to the current. This makes differential signaling less susceptible to the harmful impacts of non-ideal reference planes. Equations to

28 calculate the mode impedances and effective dielectric constants can be found in [7] and

[18] for microstrip and CPWs, respectively. Authors in [19] provide equations for calculating frequency dependent mode impedances and effective dielectric constants for microstrip lines. Equation for even and odd mode impedances for coupled striplines can be found in [13]. Fig. 1-7 (a) shows the cross-section of a pair of coupled microstrip lines. The coupling between two lines is dependent on the separation (S) between the two lines. Fig. 1-7 (b) shows that even and odd mode impedances (ZOE and ZOO) approach the characteristic impedance of a single microstrip line with increasing S. The equations used to calculate the impedances shown in Fig. 1-7 (b) assume t = 0. According to [7], with increasing t ZOE and ZOO fall.

(a) (b) Fig. 1-7 (a) Image of cross-sectional view of a coupled microstrip transmission line; (b) Plot of even and odd mode characteristic impedance of a coupled microstrip transmission line versus ratio of signal width and substrate thickness for different spacing between the traces

1.3 Embedded patterned layer (EPL)

An embedded patterned layer (EPL) consists of a thin, planar, and patterned layer of floating metallic elements, which are situated between the signal line and ground plane

(or return path) of a transmission line. Fig. 1-8 shows the image of an EPL in a microstrip transmission line while Fig. 1-9 shows the image of an EPL in a coupled microstrip

29 transmission line system. The individual EPL elements are similar to the dummy metals, which are used in very large scale integration (VLSI) circuits for reducing layout pattern dependent dielectric thickness variation. The EPL can modify the characteristic impedance and propagation properties of a transmission line. The effect of EPL on coupled microstrip transmission lines was analyzed in [20]. EPL was used in [21] to design passive radio frequency (RF) circuits like binomial matching transformer,

Wilkinson power divider and stepped impedance filter. These three structures were designed using the impedance changing ability of the EPL. This impedance lowering effect of EPL on a microstrip line was validated using simulation and measurement in

[21]. However, the basic understanding of how the EPL interacts with the transmission lines in terms of the per unit length resistance, inductance, conductance and capacitance is lacking. One of the goals of this research work is to develop a modified lumped circuit element model of the transmission line while taking into account the effect of neighboring floating metals. This will help explain the effect of EPL on transmission line properties.

(a) (b) Fig. 1-8 Image of (a) cross-sectional view and (b) top view of a microstrip transmission line with EPL

(a) (b) Fig. 1-9 Image of (a) cross-sectional view and (b) top view of a coupled microstrip transmission line system with EPL

Closed form design equations act as a good starting point for designing transmission lines with a desired characteristic impedance. However, an electromagnetic simulators should be used to accurately predict its electromagnetic behavior in frequency or time domain. EM simulators allow characterizing any structure without building and measuring it in a laboratory. This in turn quickens the design process while lowering costs and optimizing performance. There are different types of electromagnetic field simulators and the following sections presents a brief discussion on the EM field simulators used in this work.

1.4 Simulation tools

Maxwell’s equations can be used to describe the electromagnetic (EM) behavior of a transmission line and the electromagnetic interaction among transmission lines. It is extremely difficult to solve these equations analytically and as such electromagnetic field simulators are used to model any structure and numerically solve the Maxwell’s equations to accurately analyze and determine the current and field distribution in the structure. EM field simulators can be broadly classified into time and frequency domain.

Frequency and time domain simulators solve Maxwell’s equation in frequency and time domain, respectively. There are many more ways to classify EM field solvers [22], [23]

31 and these are commercially available from different vendors. Each of these codes has its pros and cons and one must choose the EM solvers depending on the scope and type of structure being analyzed. This research work was performed using some of the frequency domain EM field solvers listed below.

A. ANSYS High Frequency Structure Simulator (HFSSTM):

HFSSTM is a finite element method (FEM) based 3D full wave electromagnetic field simulator [24]. Time varying EM fields are calculated by full wave solvers without placing any constraints on the EM field behavior in any dimension [23]. HFSSTM is capable of simulating complex 3D structures and can model effects like frequency dependent losses, dispersion and radiation [25]. HFSS divides the finite and bounded problem space into smaller tetrahedral (four sided pyramid) regions and solves for the fields within each tetrahedron. These fields are related to each other such that Maxwell’s equations are satisfied at the inter-tetrahedron boundaries [26]. This process in turn results in a field solution for the entire problem space and HFSSTM then determines the generalized scattering (S) parameters. The entire tetrahedron collection is called finite element mesh. Dividing the problem space into smaller section also has the added advantage of conforming to arbitrary shapes. Waveports and lumped ports are generally used to excite transmission line structures in HFSSTM. Waveports are applied externally to a structure and generate Gamma, S, admittance (Y), and impedance (Z) parameters while lumped ports are applied internally to the model and do not generate Gamma.

While both port types allow for impedance renormalization de-embedding is possible with waveports only. According to [23], full wave modeling techniques are most effective when the device dimension varies from 0.10λ to 10λ.

B. ANSYS Q3D Extractor (Q3D):

Q3D is a quasi-static 3D field solver. The EM fields calculated using quasi-static solvers vary slowly with time [23]. According to [27], quasi static approximation holds when the waves in a transmission line propagate in the transverse electromagnetic (TEM) mode. This requirement is met when signal wavelength greatly exceeds the separation between conductors. Accordingly, quasi-static approximation is valid when the model size is less the 0.10λ. Q3D uses a combination of method of moments (MoM) and FEM to compute resistance (R), inductance (L), conductance (G) and capacitance (C) matrices

[28]. FEM divides the problem space into smaller tetrahedral elements and solves for the fields inside each element while MoM divides up the volumes and surfaces of the modeled structures into smaller tetrahedral and triangular structures, respectively to represent the charges and currents present [26]. FEM is used to solve for DC R values and MoM is used to calculate C, G, AC L and DC L values. In Q3D, a collection of conductors separated by a non-conducting material is called a net. For any net to be included in the L, R matrix it must have a source and at least one source associated with it [28]. Source and sink define the location where current enters and exits the net.

However defining source and sink on nets is not required for calculating C, G matrices.

Q3D allows for a process called matrix reduction which lets the user to change conductor definitions and generate matrices containing fewer elements post simulation.

C. ANSYS Q2D (Q2D):

Q2D comes bundled with Q3D and is a 2D quasi-static, FEM field solver. 2D solvers work under the assumption that physical geometry and materials are uniform along the length of the transmission line and therefore can be represented by the cross

33 section of the transmission line. So, any transmission line discontinuity like vias and bends will not be accounted for by the 2D solvers. 2D solvers are easier and faster to set up compared to the 3D solvers. 2D modeling is more efficient than 3D modeling and 2D simulations optimized for a particular application can be more accurate than similar 3D simulations [23]. However, 2D field solvers do not calculate the fringing fields at the ends of the conductors [27]. Q2D calculates per unit length RLGC parameter matrices, characteristic impedance matrices, crosstalk coefficients etc. for single or multi signal conductor transmission line system [28]. Q2D is much easier to set up compared to Q3D.

Q2D uses AC conduction and Eddy current field solvers to C, G and R, L matrices, respectively [26]. Matrix reduction is also available in Q2D. Q2D calculates loop inductance and resistance while Q3D calculates partial inductance and resistance. Also, unlike Q3D, the L and R matrices are not separated into AC and DC cases in Q2D.

HFSSTM and Q3D, unlike Q2D, allow engineers to model and simulate complicated transmission line structures in 3D with non-uniform physical geometry and materials.

This provides engineers with a greater degree of flexibility and help understand the overall electrical behavior of the modeled structure. Table 1.2 from [29] shows the comparison between static, quasi-static and full wave EM simulators. Also, authors in

[30] have presented an accurate modeling flowchart for microstrip transmission lines in

GHz regime which is shown in Fig. 1-10.

In addition to these tools, two simulation tools that do not exactly fit the description of EM field solvers were also extensively used in this work. These tools are described below.

D. Agilent Advanced Design System (ADS):

ADS is an electronic design automation (EDA) software for RF, microwave and high speed digital circuit design, simulation and optimization [31]. ADS comes with a vast array of simulators like linear/nonlinear frequency domain simulator and transient time domain simulator. It is a circuit simulator like simulation program with integrated circuit emphasis (SPICE) but its device libraries are focused towards microwave devices.

It easier to incorporate the effects of linear and nonlinear circuit components into S- parameter analysis in ADS and it can be used to account for EM effects like crosstalk

(XT), simple transmission line behavior. Tools like LineCalc in ADS are used to calculate transmission line design parameters for given impedance and vice versa. This provides the designers with a starting point for their design which they can analyze further in EM field solvers. It also allows for 3D planar and full wave EM simulations through its Momentum (uses Method of Moments) and finite element method (FEM) simulators, respectively. ADS is also used for post processing and displaying data.

Table 1.2 Comparison between static, quasi-static and full wave EM solvers [29] Item Static Solver Quasi-static Solver Full wave solver

휕퐻⃗⃗ 휕퐻⃗⃗ ∇ × 퐸⃗ = 0 ∇ × 퐸⃗ = −휇 ∇ × 퐸⃗ = −휇 휕푡 휕푡

휕퐸⃗ Equations ∇ × 퐻⃗⃗ = 휎퐸⃗ ∇ × 퐻⃗⃗ = 휎퐸⃗ ∇ × 퐻⃗⃗ = −휀 + 휎퐸⃗ 휕푡 ∇. 휀퐸⃗ = 휌 ∇. 휀퐸⃗ = 휌 ∇. 휀퐸⃗ = 휌 ∇. 휇퐻⃗⃗ = 0 ∇. 휇퐻⃗⃗ = 0 ∇. 휇퐻⃗⃗ = 0 Skin Effects No Yes Yes Displacement Current No No Yes Tool Example SPICE Q3D Extractor HFSSTM

E. MathWorks® MATLAB® (MATLAB®):

MATLAB® (“MATrix LABoratory”) is a high performance coding language for data analysis, generating 2D and 3D graphics, mathematical computations, simulations etc. MATLAB® also contains application/problem space specific toolboxes. Radio frequency (RF) networks designed, modeled, and analyzed using RF ToolboxTM. This toolbox makes it easy to read and write Touchstone® file format, define RF networks, manipulate S-parameters, perform network parameter conversions, and plot data in rectangular or polar form and Smith charts [32].

Fig. 1-10 Accurate modeling flowchart for microstrip transmission lines [30]

1.5 Thesis goals

From the previous sections, it is clear that transmission lines must have multi-

GHz bandwidth to support high speed signaling and transmission line conductor surface roughness increases its resistivity and therefore the losses associated with it. The increasing circuit speeds along with densely routed parallel transmission lines increases crosstalk between neighboring transmission lines. EPL provides a unique way of reducing far-end crosstalk while reducing the characteristic impedance of the transmission lines. However, a basic circuit model to explain the effects of EPL on transmission lines is required. Conductor backed coplanar waveguides are selected for this work since these are easy to measure using probe stations. The goals of this thesis are as follows:

. Understand the reason behind higher order mode excitation in conductor backed

coplanar waveguides and analyze the effect of different ground via fence parameters

on the bandwidth of a conductor backed coplanar waveguide.

. Develop a modified lumped element model for transmission lines by taking into

account the effect of neighboring floating metals and then use this model to explain

the effect of EPL on grounded coplanar waveguides.

. Analyze the effect of EPLs on grounded conductor backed edge coupled coplanar

waveguide in terms of per unit length self-capacitance, self-inductance, mutual

capacitance, and mutual inductance, when this three conductor system is used for

either single ended or differential signaling.

. Develop a statistical way of modeling metal surface roughness using 3D full wave

electromagnetic solvers for transmission lines.

1.6 Thesis outline

This thesis is divided into six chapters including the introductory and concluding ones. Chapter 2 analyzes the impact of ground via fences on grounded coplanar waveguide bandwidth. Chapter 3 and Chapter 4 are devoted to examining the effect of

EPLs on a grounded coplanar waveguide and a grounded conductor backed edge coupled coplanar waveguide, respectively. Chapter 5 studies the effects of conductor surface roughness on conductor backed coplanar waveguide.

In Chapter 2, presents a brief discussion on the development, evolution and properties of the coplanar waveguide and then develops the basic theory behind picket via fence and resonances associated with conductor backed coplanar waveguide. The effect of ground via fence location and dimension on grounded coplanar waveguide bandwidth is analyzed and a comparison between measured and simulated data is presented. Two new alternative structures that can be used to improve grounded coplanar waveguide bandwidth are also suggested.

In Chapter 3, a standard transmission line is analyzed in terms of the lumped element model. The standard transmission line lumped element model is modified to include the effect of floating metals present near a transmission line. This modified transmission line model is used to explain the effect of EPL on grounded coplanar waveguides. The effect of EPL on grounded coplanar waveguide, microstrip and stripline is also compared.

In Chapter 4, briefly describes single ended and differential signaling schemes, and the effect of odd and even mode of signal propagation on transmission line impedance and propagation speed. The effect of EPL on a grounded conductor backed

38 edge coupled coplanar waveguide is analyzed when this multi conductor transmission line system is used for single ended and differential signaling.

In Chapter 5, the effect of conductor surface roughness on conductor backed coplanar waveguides is analyzed. The properties of random rough surface is described and a statistical method of modeling rough surface in a 3D full wave solver is developed.

The results obtained from this new technique is compared to the data available in the literature for a microstrip line for validation.

Finally, Chapter 6 summarizes the main results obtained from this work and presents possible future work.

CHAPTER 2: GROUNDED COPLANAR WAVEGUIDE (GCPW) VIA FENCE ANALYSIS

2.1 Introduction

A conventional coplanar waveguide (CPW) is a planar transmission line and consists of a center conductor with semi-infinite ground planes on either side fabricated on top of a dielectric substrate. The CPW structure was first proposed by C. P. Wen in

[33]. The CPW supports quasi transverse electromagnetic (TEM) mode of propagation.

The effective dielectric constant (εeff), characteristic impedance (ZO), and attenuation (α) of the lines are determined by the dielectric constant of the substrate (εr), substrate thickness (h), center conductor (signal) width (S), and the gap (G) between signal and coplanar side grounds (CSGs) on either side [15], [18]. Frequency dependency of εeff and

α of CPW with finite metal thickness (T) and conductivity (σ) has been analyzed in [34].

The uniplanar construction of a CPW simplifies and accelerates the manufacturing and the testing process by allowing the use of automatic assembly equipment and computer controlled on wafer measurement, respectively [18]. Unlike the conventional CPWs, in a practical circuit the coplanar side grounds are of finite width. A CPW structure supports two fundamental modes called the even and odd modes [35], [36]. The even and odd modes are also referred to as coplanar waveguide and slot-line modes, respectively. The dominant CPW mode becomes leaky above a critical frequency and travels away from the CPW at an angle in the form of surface waves in the substrate [37]. This energy leakage can cause unwanted crosstalk (XT) and package effects [37]. The lowest order surface wave that can be supported by CPW with either semi-infinite or finite width coplanar side ground is TM0 and TE0, respectively. Authors in [38] and [39], report a

40 highly dispersive dominant mode called CPW surface wave like mode in CPWs with either semi-infinite or finite width coplanar side grounds. Air bridges are used to suppress slot-line and other higher order modes by equalizing ground potentials on both coplanar side grounds of a CPW [40]. CPWs with air bridges have been analyzed using a 3-D finite difference method (FDM) and full wave frequency domain transmission line matrix

(FDTLM) method in [41] and [42], respectively.

CPWs have several advantages over microstrip lines, which include ease of fabrication, reduced radiation losses and ease of series and parallel insertion of active and passive components for microwave integrated circuits (MICs) and monolithic MICs

(MMICs) [18]. However, CPWs suffer from higher losses and are more sensitive to interaction effects of nearby structures [43]. To allow easy implementation of mixed coplanar and microstrip circuits an additional lower ground plane (LGP) is used [44]. A

CPW with a lower ground plane is called conductor backed coplanar waveguide (CB-

CPW). The lower ground plane can be used for heat sinking purposes [45] while providing mechanical support [46] and preventing the electric fields from coupling to interconnects on lower circuit layers. Authors in [44] have derived the quasi-static equations for calculating εeff and ZO of a CB-CPW. The characteristics of a CB-CPW resembles that of a microstrip transmission line when G increases for a given h while the

CB-CPW behaves like a CPW when h is increased while keeping G constant [46].

Authors in [47] have characterized CB-CPW using on wafer measurements and measured

εeff, ZO, and α, which are in good agreement with the theory provided in [44] and [46].

The theory behind the resonant behavior of CB-CPWs was developed and validated through full wave analysis and measurements in [48]. Finite ground conductor backed

41 coplanar waveguide (FG-CBCPW) was characterized in [49] to determine the optimum width of the coplanar side grounds using conformal mapping and finite difference time domain (FDTD) method. The performance of a FG-CBCPW is independent of the coplanar side ground width when the coplanar side ground width is twice the width of the center conductor but is less than one eighth of the wavelength in the dielectric substrate.

[50] presents closed form equations for FG-CBCPW in the frequency domain while accounting for metal loss and dispersion.

A FG-CBCPW with finite width substrate supports the dominant CPW mode. The

FG-CBCPW can also be treated as a system of three coupled microstrip transmission lines. Thus a microstrip like (MSL) mode inherently exists in a FG-CBCPW [51]. The lowest order MSL mode resembles the parallel plate transmission line mode. Higher order MSL modes are excited as the coplanar side ground width is increased. Authors in

[51] suggest grounding the coplanar side grounds to get rid of the MSL modes. The moding problem in FG-CBCPWs can be eliminated by shorting the coplanar side grounds to the lower ground plane using via holes [52]. FG-CBCPWs with vias shorting the coplanar side grounds to the lower ground plane are called grounded coplanar waveguides (GCPWs) in this work. Via holes when placed at the location of maximum electric field are effective in suppressing higher order modes [53], [54]. According to

[53], resonant frequencies can be predicted via patch antenna theory. [55] presents four different ways of avoiding or suppressing higher order modes one of which is using shorting pins to connect the coplanar side grounds to the lower ground plane. However, there is no discussion regarding the effects of different via fence design parameters on the bandwidth of a GCPW. The bandwidth of a GCPW is defined as the highest frequency

42 after which the insertion loss falls below -3dB in this work. Fig. 2-1 (a), (b), and (c) show the 3-D schematic representation of a CPW, FG-CBCPW, and GCPW, respectively. Fig.

2-2 (a), (b), and (c) show the electric field distribution in the cross-section of a CPW, FG-

CBCPW, and GCPW, respectively.

(a) (b)

Vias are metal filled holes etched, drilled or punched in a substrate either to electrically connect different layers or to provide low impedance grounding for

43 transistors, transmission lines etc. [56], [57] and [58]. Via placement impacts routing density on a printed circuit board (PCB) by controlling the number of traces that can be routed through the space between adjacent vias [59]. The cost of a PCB can be reduced by reducing the number of layers, which can be accomplished by increasing the routing density. Through strata and through silicon vias have been modeled and analyzed in [60] and [61], respectively. Electrical and mechanical properties of vias have been discussed in details in [27], [59]. In this chapter, the effect of different via fence parameters on

GCPW bandwidth is studied and possible ways of improving GCPW performance while meeting fabrication requirements are suggested.

The results presented here are based on GCPWs on copper (Cu) clad Megtron6 dielectric material (referred to as core or substrate) in this work. Megtron6 has the relative permittivity (εr) of 3.5 and loss tangent (tan δ) of 0.002. Substrate thickness (h) is

101.6 μm. The thickness (T) and conductivity (σ) of Cu are 17.78 μm (0.5 oz.) and

5.8E+07 S/m, respectively. The signal conductor width (S) is 177.8 μm and the spacing between the signal conductor and a coplanar side ground is G = 101.6 μm. The interconnect length (L) is 6000 μm. Fig. 2-3 (a) and (b) show the top layer and cross- sectional view of a GCPW, respectively. The three dimensional (3D) modeling and full wave simulation were performed using HFSSTM [62]. HFSSTM is a full wave electromagnetic simulator which solves Maxwell’s equations directly using the finite element method (FEM). The Touchstone® files are extracted from HFSSTM and plotted using ADS [63]. HFSSTM was also used to plot the electric field distribution in the interconnects.

(a)

(b) Fig. 2-3 Image of (a) Top layer and (b) cross-sectional view of a GCPW

In Fig. 2-3, the parameter SGW denotes the coplanar side ground width. The parameters VP and VR represent via to via pitch and via radius, respectively. The variable VL is the center to center distance between the signal trace and a via fence. The via fence consists of a line of vias located along the length of the GCPW on a particular coplanar side ground. The parameter ES represents the separation between GCPW edge and center of the via closest to the edge in a via fence. GCPW edge is defined as the location where probes (or feeds) come in contact with the GCPW for performing measurements.

2.2 Traditional via picket fence theory

In a FG-CBCPW, a coplanar side ground along with the lower ground plane act as a rectangular patch antenna. The rectangular patch acts like a resonant cavity with open circuit at the edges [64]. The length, width, height of this cavity is equal to L, SGW, and h of the FG-CBCPW, respectively. The energy from the signal conductor is capacitively

45 coupled to the rectangular patch [65] exciting higher order modes in the cavity. Most of the electromagnetic energy gets coupled to the cavity at its resonant frequencies and very little signal propagates down the signal. The fields between the coplanar side ground and lower ground plane are cosinusoidal in nature. The fringing fields along the sides are neglected because h is very small. The resonant frequencies (fr) for such a cavity are given by the following equation, see Appendix I.

1 푚휋 2 푛휋 2 푝휋 2 (푓푟)푚푛푝 = √( ) + ( ) + ( ) (2.1a) 2휋√휇휀 퐿 푆퐺푊 ℎ

푚휋 훽 = , 푚 = 0,1,2, … 푥 푙 푛휋 훽 = , 푛 = 0,1,2, … 푚 = 푛 = 푝 ≠ 0 (2.1b) 푦 푤 푝휋 훽 = , 푝 = 0,1,2, … 푧 ℎ }

In Eq. 2.1a, ε and μ represent the permittivity and permeability of the dielectric material.

Since the substrate is non-magnetic (휇푟 = 1) and electrically h is very small (h ≪ 휆) Eq.

2.1a can be modified and represented as

퐶 푚 2 푛 2 (푓푟)푚푛0 = √( ) + ( ) , (2.1c) 2√휀푟 퐿 푆퐺푊 where C is the speed of light in free space. Table 2.1 shows the comparison between calculated and simulated resolvable resonant frequencies for a FG-CBCPW with SGW,

L, h, and εr of 3200 μm, 6000 μm, 101.6 μm and 3.5, respectively. It is possible to get rid of the higher order modes by reducing the coplanar side ground width but this is not a practical approach for complex coplanar ICs designed on substrates of higher εr [54]. Fig.

2-4 (a) shows that reducing SGW result in higher resonant frequencies while Fig. 2-4 (b) show the power loss associated FG-CBCPW with varying coplanar side ground width.

The power loss can be calculated from the scattering parameters using the following equation.

2 2 푃표푤푒푟 퐿표푠푠 = 1 − |푆11| − |푆21| (2.2)

Table 2.1 Comparison of resonant frequencies obtained from simulations and analytical equations (f ) (GHz) (f ) (GHz) (f ) r mn0 r mn0 r mn0 from 2.1c from simulation (fr)110 28.4 27.5 (fr)210 36.6 35.5 (fr)310 47.3 46.5 (fr)120 51.9 50.5 (fr)220 56.8 56.0 (fr)410 59.0 58.0 (fr)320 64.2 63.5 (fr)420 73.3 71.5 (fr)130 76.3 77.5 (fr)230 79.8 81.0 (fr)620 94.6 93.0

The boundary conditions are going to change when vias are used to connect the coplanar side grounds to the lower ground plane. This change in boundary conditions is only going to shift the resonant frequencies of the different modes but will not get rid of them. To make the analysis simple assume that instead of vias solid shorting plates are used to connect the coplanar side grounds to the lower ground plane. These solid shorting plates are placed at the coplanar side ground edges as described in the two cases below, see Fig. 2-5.

A. Case 1

In this case, the solid shorting plates are located only at the coplanar side ground edges furthest away from the signal trace. As a consequence, each coplanar side ground has a single shorting plate that runs the entire length (x-axis) of the coplanar side ground.

The waves along the length and width (y-axis) inside the cavity will be cosinusoidal and sinusoidal in nature, respectively. Eq. 2.1c can still be used to predict the resonant

47 frequencies in the cavity but the mode number n = 0.5, 1.5, 2.5,… instead of whole numbers (0, 1, 2, 3,…).

(a)

(b) Fig. 2-4 Plot of (a) insertion loss and (b) power loss versus frequency for FG-CBCPWs with varying SGW

B. Case 2: (Shorting plates along all three edges of both coplanar side grounds)

In this case, the solid shorting plates are located at all the coplanar side ground edges except for the edge closest to the signal trace. So, each coplanar side ground has a shorting plate running the length of the coplanar side ground and two additional shorting plates running the width of the coplanar side ground. As such the waves along the length

48 and width inside the cavity will be sinusoidal in nature. Eq. 2.1c can still be used to predict the resonant frequencies in this modified cavity but the mode numbers m and n will be natural numbers (1, 2 ,3,…) and odd multiples of half (0.5, 1.5, 2.5,…) instead of whole numbers. Fig. 2-6 shows the insertion loss for FG-CBCPW, Case 1 and Case 2.

Fig. 2-7 shows the electric field distribution in FG-CBCPW, Case 1 and Case 2 at different frequencies.

Fig. 2-5 Top layer view of FG-CBCPWs with coplanar side ground edges shorted to the lower ground plane

Fig. 2-6 Plot of insertion loss versus frequency for a FG-CBCPW with un-grounded and grounded CSG edges

(a) (b)

(c) Fig. 2-7 Plot of electric field distribution in log scale for (a) FG-CBCPW at 73.3 GHz for (fr)420, (b) Case 1 at 82.4 GHz for (fr)42.50, and (c) Case 2 at 82 GHz for (fr)42.50

From the above discussion it is clear that adding vias to the FG-CBCPW structure is not going to get rid of the higher order modes instead it only results in excite higher order modes that satisfy the boundary conditions at the via fence location.

2.3 Comparison between simulation and measurement

In order to validate the simulated results, a GCPW of L = 6000 μm was fabricated and measured using Agilent Technologies E8361A performance network analyzer (PNA)

[66] and Cascade Microtech ® air coplanar probes (ACP) 250 ground signal ground

(GSG) probes [67]. The fabricated GCPW has h = 101.6 μm, S = 177.8 μm, G = 101.6

μm, SGW = 1200 μm, VL = 1212.7 μm, VR = 76.2 μm, VP = 508 μm, and ES = 460 μm.

Megtron6 (εr = 3.5 and tanδ = 0.002) is the substrate (core) and cladded by Cu of T =

17.78 μm. For structural stability, the GCPW has a 508 μm thick (including prepreg layer) Megtron6 dielectric layer beneath the lower ground plane as supporting layer. This supporting layer is isolated from the GCPW due to the lower ground plane and thus does not have any impact on the GCPW performance. Fig. 2-8 shows the schematic cross- section of the fabricated GCPW. This fabricated structure was simulated in HFSSTM. A good correlation between measured and simulated data is obtained within the limits of experimental errors, see Fig. 2-9. Fig. 2-10 shows the electric field distribution in the fabricated GCPW structure.

Fig. 2-8 Image of the cross-section of a fabricated GCPW

(a)

(b) Fig. 2-9 Plot of (a) insertion loss and (b) return loss versus frequency for measured and simulated data

(a) (b) Fig. 2-10 Plot of electric field distribution in log scale for a fabricated GCPW structure at (a) 35.5 GHz for (fr)00.50, and (b) for 39 GHz for (fr)10.50

2.4 Impact of via fence location and dimension on GCPW performance

In this section, the influence of different via fence parameters on the GCPW bandwidth will be analyzed. The via fence parameters that will be analyzed include VL,

VP, and ES. The ground via fence helps reduce the electrical coplanar side ground width.

The effective side ground width (eSGW) can be calculated from the following equation.

푆 푒푆퐺푊 = 푉퐿 − 푉푅 − 퐺 − ( ⁄2) (2.3)

A. Varying distance between via fence and signal trace (VL)

In this section VL is varied from 400 to 2700 μm with SGW = 3200 μm, VP =

287.3 μm, ES = 127 μm, and VR = 76.2 μm. GCPW performance shows tremendous improvement when VL is reduced. Fig. 2-11 (a) and (b) show the insertion and power loss of a GCPW with varying VL, respectively. With reducing VL, eSGW also falls preventing the coplanar side grounds from exciting higher order modes. The higher order modes start getting excited at 38.5 GHz instead of 16 GHz when VL is reduced from

2700 to 1200 μm. The eSGW for these VLs are approximately quarter wavelength (λ/4) at these frequencies. However, there are no resonances for VL = 400 μm up to 100 GHz because this is less than λ/4 at 100 GHz. Excitation of higher order modes can be avoided in a GCPW by placing the via fence as close to the signal conductor as possible. Fig. 2-12 shows the electric field distribution in a GCPW when VL = 1200 and 2700 μm.

(a)

(b) Fig. 2-11 Plot of (a) insertion loss and (b) power loss versus frequency for GCPWs with varying VL

(a) (b)

(c) Fig. 2-12 Plot of electric field distribution in log scale for GCPW with (a) VL = 1200 μm at 38.5 GHz for (fr)00.50, (b) VL = 1200 μm at 68.6 GHz for (fr)40.50, and (c) VL = 2700 μm at 72.9 GHz for (fr)41.50

B. Varying the via to via pitch in a via fence (VP)

In this section VP is varied from 2873 to 359.125 μm with SGW = 3200 μm, VL

= 400 μm, ES = 127 μm, and VR = 76.2 μm. Simulations show that reducing VP improves the insertion loss of a GCPW, see Fig. 2-13. The first resonance occurs at 32.5

GHz when VP is 2873 μm, which is approximately λ/2 at this frequency. The resonances are pushed out to higher frequencies as VP is reduced. No resonances are present for VP

= 359.1 μm up to 100 GHz since this length is less than λ/4 at 100 GHz. Energy coupled to the coplanar side grounds will leak into the region beyond the via fence when VP is large. This leaked energy will generate higher order modes in the structure deteriorating

GCPW performance. The excitation of higher order modes in a GCPW can be suppressed by keeping the via pitch less than λ/4 at the highest frequency of operation or by making

VP as small as possible while meeting fabrication guide lines. Fig. 2-14 shows the electric field distribution in GCPWs with different via pitch at different frequencies.

Fig. 2-13 Plot of insertion loss versus frequency for GCPWs with varying VP

Fig. 2-14 Plot of electric field distribution in log scale for GCPW with (a) VP = 2873 μm at 32.5 GHz, (b) VP = 2873 μm at 78 GHz, and (c) VP = 1436.5 μm at 68.5 GHz

From Fig. 2-14, it is clear that the coupled electromagnetic energy excites higher order modes between adjacent vias in a via fence. The resonant cavity formed by the coplanar side ground and lower ground plane can be described as made of smaller cavities of length VP and width SGW. This energy then flows into the region of coplanar side ground beyond the via fence exciting other higher order modes in the cavity of dimension L and (SGW-VL). This process occurs for each adjacent via pair in the via fence and as a consequence these leaked fields either interact constructively or destructively with one another to generate the final higher order modes in the coplanar side ground segment beyond the via fence.

C. Varying separation between ends of GCPW and start of via fence edge (ES)

This section shows the impact of increasing ES on GCPW behavior. The GCPW dimensions are as follows: SGW = 1200 μm, VL = 376.2 μm, VP = 287.3 μm, and VR

76.2 μm. Fig. 2-15 shows the insertion loss of the GCPW when ES is varied from 127 μm to 701.6 μm.

Fig. 2-15 Plot of insertion loss versus frequency for GCPWs with varying ES

The GCPW bandwidth decreases with increasing ES as higher order modes get excited at lower frequencies. No higher order modes are excited for the GCPW with ES =

127 μm up to 100 GHz, higher order modes appear at 46 and 34 GHz for ES = 414.3 and

701.6 μm, respectively. Fig. 2-16 shows the electric field distribution in a GCPW at 100

GHz when ES is varied. From the figure it is clear that energy is leaked out into the coplanar side ground region beyond the via fence when ES is large enough thereby generating higher order modes.

The performance of a GCPW does not change with varying SGW when ES, VL and VP are kept at a minimum. Fig. 2-17 shows the variation in insertion loss in GCPWs with varying SGW when ES = 127 μm, VL = 400 μm, and VR = 76.2 μm. As ES and VP approach 0, a GCPW starts to resemble a channelized coplanar waveguide (CCPW) [3].

A CCPW structure can be defined as a CPW with lower ground plane and with solid conducting walls connecting the coplanar side grounds to the lower ground plane, see

Fig. 2-18. The solid conducting walls along with the lower ground plane constitute the metal channel in the interconnect. Since all the grounds are connected together in a

(a) (b) Fig. 2-16 Plot of electric field distribution in log scale for GCPW with (a) ES = 127 μm, and (b) ES = 414.3 μm at 100 GHz.

CCPW, the higher order parallel plate modes get eliminated. Fig. 2-19 shows very little difference in the insertion loss between a CCPW and a GCPW. The VL for the CCPW is

400 μm, while VP = ES = 0 μm. The grounding walls are 76.2 μm thick. Fig. 2-20 shows the electric field distribution in the two interconnects. From this discussion, it is predictable that for a given VP, ES, and VL the GCPW performance should improve with increasing VR. These expected trends are observed in Fig. 2-21 when VR is varied in

GCPWs with SGW = 3200 μm, ES = 127 μm, VL = 400 μm and VP = 287.3 μm.

Fig. 2-17 Plot of insertion loss versus frequency in GCPWs with varying SGW

Fig. 2-18 Image of a section of a CCPW

Fig. 2-19 Plot of insertion loss versus frequency for GCPW and CCPW with similar physical dimensions

(a) (b) Fig. 2-20 Plot of electric field distribution in log scale for (a) GCPW, and (b) CCPW at 100 GHz

Fig. 2-21 Plot of insertion loss versus frequency in GCPWs with varying VR

2.5 Alternate grounding structures for improving GCPW performance

The mechanical performance of a PCB will be adversely affected due to warpage, which is the result of mismatch in coefficient of thermal expansion (CTE) between the metal in high density via and the surrounding dielectric. Therefore, it may not be possible to place vias in close proximity of one another due to fabrication and routing limitations.

This can result in degradation of GCPW performance as energy gets coupled to the coplanar side grounds exciting higher order modes. The bandwidth of a GCPW can be increased by suppressing these higher order modes. This section provides possible approaches to improve GCPW bandwidth. The basic GCPW dimensions common to the following section are as follows. The GCPW has SGW = 3200 μm, VR = 76.2 μm, VP =

1436.5 μm, VL = 400 μm and ES = 127 μm.

A. Increasing the number of ground via fences connecting the coplanar side grounds to the lower ground plane

Here adding extra ground via fences to each coplanar side ground of a GCPW with moding issues is suggested to improve the GCPW performance. The via fence pitch

(VFP) is varied from 200 μm to 600 μm. The first via fence (VF1) on each coplanar side ground is closest to the signal conductor. This fence was there on the original GCPW with higher order mode excitation problem. The third via fence (VF3) on each coplanar side ground is furthest away from the signal. The second via fence (VF2) is positioned between VF1 and VF3. VF2 is offset from VF1 and VF3 along x-axis by VO = 718.25

μm. Either VF2 or both VF2 and VF3 are added to the original GCPW to improve bandwidth. ES for VF1 and VF3 is 127 μm while for VF2 is 845.25 μm. Fig. 2-22 shows the top layer view of a GCPW with 3 via fences on each coplanar side ground.

Fig. 2-22 Top layer view of a GCPW with three via fences on each CSG

Increasing the number of via fences improves the bandwidth of the GCPW by pushing the resonances out to higher frequencies. For a GCPW with two via fences (VF1 and VF2) and VFP = 200 μm, the higher order modes are excited at 88.5 GHz compared to 45 GHz for the original GCPW with VF1, see Fig. 2-23. As VFP is increased from 200 to 600 μm, higher order modes get excited at 73 GHz instead of 88.5 GHz. This trend is expected from previous sections. A GCPW with VF1 and VP = 718.25 μm can also be described as a GCPW with two fences (VF1 and VF2) and VFP = 0 μm. Similar trends are observed in GCPWs with three via fences, see Fig. 2-24. However, the improvement in performance over the GCPW with two via fences is not that significant. The performance improvement is also less when VFP is increased. With three via fences, the modes get excited at 91.5 GHz compared to 88.5 GHz for the two via fence case with

VFP = 200 μm. The improvement in performance of a GCPW with three via fences is only 0.5 GHz compared to the GCPW with two via fences when VFP is 600 μm.

Fig. 2-23 Plot of insertion loss versus frequency in GCPWs with one or two via fences on each CSG and varying VFP

Fig. 2-24 Plot of insertion loss versus frequency in GCPWs with one or three via fences on each CSG and varying VFP

These trends are consistent when VL is increased to 1200 μm. But the improvement in GCPW bandwidth is not that drastic. The improvement in bandwidth for the GCPW with two via fences is 2.5 GHz and 0.5 GHz for VFP = 200 μm and 600 μm, respectively. Another 0.5 GHz improvement in bandwidth is observed in a GCPW with three via fences. However, when VP is less than λ/2 at the highest frequency of operation adding extra via fences do not result in a huge improvement of GCPW performance. Fig.

2-25 shows the insertion loss in GCPWs with two via fences when VP = 718.25 μm and

VO = 359.125 μm with VL = 400 μm. This trend is consistent when VL is increased to

1200 μm, see Fig. 2-26

Fig. 2-25 Plot of insertion loss versus frequency in GCPWs with one or two via fences and varying VFP, when VL = 400 μm and VP is less than λ/2 at 100 GHz

Fig. 2-26 Plot of insertion loss versus frequency in GCPWs with one or two via fences and varying VFP, when VL = 1200 μm and VP is less than λ/2 at 100 GHz

B. Coplanar side grounds with periodic cutouts

The bandwidth of a GCPW can also be increased by making slots into the coplanar side grounds. These slots are located at a distance SLo = 400 μm from the center of the signal conductor. The slot length (SL) is 2990.5 μm and slot width is SW = 762

μm. Slot to via pitch (SVP) is 718.25 μm while slot to slot pitch (SP) is 1436.5 μm. Fig.

2-27 shows the top layer view of a GCPW with slots in the coplanar side grounds.

Fig. 2-27 Top layer view of a GCPW with slots cut out in each CSG

The bandwidth of a GCPW is improved with slots in the coplanar side grounds.

As the SW is varied from 152.4 to 762 μm, the higher order modes start getting excited at higher frequencies. The bandwidth of the GCPW increases with SW. The resonances are observed at 57.5 and 62 GHz with slot width of 152.4 μm and 762 μm, respectively. Fig.

2-28 shows the insertion loss of GCPWs with and without slots in the coplanar side grounds. So, the GCPW bandwidth has increased by 17 GHz with slots in the coplanar side grounds.

Fig. 2-28 Plot of insertion loss versus frequency for GCPWs without and with slots of varying SW

2.6 Summary

In this chapter a brief theory behind the excitation of higher order resonant modes in FG-CBCPWs has been discussed. Reducing the coplanar side ground width improves

GCPW performance in the absence of via fences but this is not a practical approach in case of MICs and MMICs. A GCPW is fabricated and measured. The measurement and simulation data show good correlation within the limits of experimental errors. The ground via fences should be placed as close to the signal conductor as possible and the pitch between adjacent vias in a via fence should be less than quarter wavelength at the highest frequency of operation of the GCPW to suppress exciting higher order modes.

The width of the coplanar side ground has no impact on GCPW behavior when via fences are present. Two methods to improve GCPW bandwidth are also presented when it is not possible to have small via to via pitch due to fabrication or routing limitations.

CHAPTER 3: GROUNDED COPLANAR WAVEGUIDE WITH EMBEDDED PATTERNED LAYER (EPL)

3.1 Introduction

An embedded patterned layer (EPL) consists of a thin, planar, and patterned layer of floating metallic elements between the signal trace and return plane of a transmission line. The transmission line used in this work is a grounded coplanar waveguide (GCPW).

The GCPW is designed to have a large bandwidth by suppressing higher order modes with appropriately positioned via fences. The GCPW behavior was analyzed in the previous chapter. Fig. 3-1 (a) shows the top view of a GCPW with EPL. The lower ground plane and the substrate are not shown in the picture for simplicity and clarity. Fig.

3-1 (b) shows the cross-section of a GCPW with EPL. It can be clearly seen that the EPL layer is embedded in the substrate, which separates the signal trace and coplanar side grounds from the lower ground plane. Fig. 3-1 (c) shows the diametric view of a GCPW with EPL. The substrate is not shown for clarity. Fig. 3-2 shows the top view of an EPL unit cell. The effect of EPL on the GCPW performance is analyzed using HFSSTM [62] and Q3D [68].

(a)

(b)

The GCPW has Megtron6 and copper (Cu) as substrate and conductor, respectively. The relative dielectric constant (휀푟) and loss tangent (tan δ) of Megtron6 are

3.5 and 0.002, respectively. The conductor thickness (T), and conductivity (σ) are 17.78

μm and 5.8E+07 S/m, respectively. The transmission line length (L) is 6000 μm. Signal trace width (S) is 177.8 μm and the gap (G) between the signal trace and one of the coplanar side grounds is 101.6 μm. The substrate thickness (h) is 101.6 μm. The coplanar side ground width (SGW) is 1200 μm. The center to center distance between the signal trace and a via fence (VL) is 400 μm. The separation between GCPW end and the

68 beginning of a via fence (ES) is 127 μm. Via radius (VR) and via to via pitch (VP) are

76.2 μm and 287.3 μm, respectively. The variables LEPL and WEPL denote the length and width of individual metallic elements that constitute the EPL. The pitch between two of these metallic elements along the x and y direction are denoted by PXEPL and PYEPL, respectively. The parameter TEPL is the thickness of the EPL while the variable HEPL is the separation between the top surface of the EPL and the bottom surface of the signal trace.

Fig. 3-2 Image of a 3x3 EPL unit cell

Metallic structures that are not connected to any voltage reference or voltage source are called floating metals. The potential of a floating metal is neither fixed nor zero but instead it varies with its environment [69]. Floating or grounded metallic structures, called dummy metals, are used in very large scale integration (VLSI) circuits to reduce layout pattern dependent dielectric thickness variation [70]. Dummy metals increase interconnect capacitance, signal delay, crosstalk noise, and power consumption

[71]. The electrical effects of dummy metals on interconnect capacitance has been studied in [72], [73], [74] and [75]. The increase in capacitance due to dummy metals can be explained by thinking of a floating metal placed between the two plates of a parallel

69 plate capacitor. Assume that the parallel plate capacitor has a fixed charge density of opposing polarities on each plate. If a floating metal is inserted between the two plates of the capacitor, the charges in the floating metal will be redistributed on the floating metal surface such that the electric field inside the metal is zero. The electric field intensity remains unchanged because the charge density is constant however the voltage between the two plates decreases. This results in increased capacitance. The dummy metals increase interconnect loss [76], [77] and [78], and reduce interconnect inductance [77] due to eddy currents. The reduction in the inductance can be explained by thinking of a transmission line near a floating metal plane. The net inductance for the transmission line is calculated by integrating the flux density between the signal and return path of the transmission line and then dividing the total flux by the current [69]. The time varying current in the signal and return path of the transmission line will induce image currents in the floating ground plane. The image currents will produce magnetic fields in the opposite direction to the fields produced by the two conductors of the transmission line thereby reducing the net flux and thus the inductance between the conductors of the transmission line.

The impact of floating metals on interconnect capacitance and inductance is explained at a very basic level in the previous paragraph. However, it is pertinent that the impact of floating metals on transmission lines be explained in terms of lumped element model (also called RLCG model) for transmission lines to gain better understanding. In the following section the basic lumped element model for transmission lines is discussed as a warm up before developing a modified lumped element model for transmission lines with floating metals in section 3.3. In section 3.4, a parametric study of the effects of EPL

70 on GCPW is performed and finally in section 3.5 the effect of EPL on different types of transmission line is presented.

3.2 Lumped element model for transmission lines

The electrical length of a transmission lines can be either a significant fraction or many times a wavelength. As a consequence, voltages and currents vary in magnitude and phase over the length of the transmission line, making it a distributed network parameter [79]. Lumped element components used in circuit theory on the other hand are electrically small such that voltages and currents do not vary in magnitude or phase between the terminals of the components. However, a transmission line can be broken down into small segments of length (Δz) such that each of these segments can be represented using lumped elements. The length Δz should be chosen such that the delay of the line should be less than one tenth of the signal rise time in time domain [25]. In case of frequency domain, Δz should be less than one tenth of the wavelength of the highest frequency of interest [25]. The minimum number of segments (Ns) of length Δz required to model the transmission line in time and frequency domain can be calculated using following equations [25], [80].

푇푖푚푒퐷표푚푎푖푛 퐿 10퐿√휀푟 푁푠 = ⁄ = ⁄ (3.1) ∆푧 (푡푟퐶)

퐹푟푒푞푢푒푛푐푦퐷표푚푎푖푛 퐿 10퐿 푁푠 = ⁄ = ⁄ (3.2) ∆푧 휆푓푚푎푥

The variables L, εr, tr, C, and λfmax represent length of the transmission line, transmission line substrate relative permittivity, rise time of signal in time domain, speed to electromagnetic waves in free space and wavelength at the maximum frequency of interest, respectively.

Transmission lines must have at least two conductors (a signal line and a return path) for transverse electromagnetic (TEM) wave propagation. A voltage difference exists between the signal trace and the return path when a signal is propagating down the transmission line. Also, the current flowing through the signal trace and return path are equal and flow in opposite directions completing the current loop. The signal trace and return path will have resistance and inductance associated with them. A capacitance will appear between the two conductors because they are separated from each other by a dielectric material. Since the dielectric material is not a perfect there will be some leakage current through the dielectric leading to dielectric loss. Using these concepts, a small transmission line section of length Δz can be modeled as shown in Fig. 3-3.

Fig. 3-3 General lumped element model of a transmission line

In Fig. 3-3, RST and RRP represent the per unit length (p.u.l.) resistance of the signal trace and return path. The variables LST and LRP denote the p.u.l. self-partial inductance of the signal trace and return path, respectively. The parameter MSR represents the p.u.l. mutual partial inductance between the signal trace and return path. The variables CTXL and GTXL are per unit length shunt capacitance and conductance of the transmission line. CTXL and GTXL represent the electric field and dielectric loss associated

72 with a transmission line. The resistance and inductance values are combined to obtain a simpler lumped element transmission line model shown in Fig. 3-4.

Fig. 3-4 Simplified lumped element model of a transmission line

In Fig. 3-4, RTXL represents the per unit length combined resistance of the signal trace and the return path while LTXL represents the per unit length loop self-inductance of the signal trace and return path. RTXL and LTXL represent the conductor loss and magnetic field associated with a transmission line. The parameters, RTXL and LTXL are calculated using the following equations.

푅푇푋퐿 = 푅푆푇 + 푅푅푃 (3.3)

퐿푇푋퐿 = 퐿푆푇 + 퐿푅푃 − 2푀푆푅 (3.4)

The characteristic impedance (ZO) and complex propagation coefficient (γ) associated with the transmission line can be calculated using the equations

{푅푇푋퐿 + 푗(휔퐿푇푋퐿)} 푍푂 = √ ⁄ , (3.5a) {퐺푇푋퐿 + 푗(휔퐶푇푋퐿)}

훾 = 훼 + 푗훽 = √{푅푇푋퐿 + 푗(휔퐿푇푋퐿)}{퐺푇푋퐿 + 푗(휔퐶푇푋퐿)}, (3.6a) where α, β, and ω represent the attenuation coefficient, propagation coefficient, and radian frequency, respectively. Attenuation coefficient is the sum of the conductor loss

73 and dielectric loss associated with the transmission line. The radian frequency is related to frequency (f) through the following equation.

휔 = 2휋푓 (3.7)

The following subsections briefly describe the resistance (R), inductance (L), capacitance (C) and conductance (G) parameters used to model a transmission line.

A. Resistance (R)

Resistance is the opposition a current encounters while flowing through the cross- section of a conductor. Resistance can be classified as AC and DC resistance. DC resistance (RDC) of a conductor, see Fig. 3-5(a), of length (l), width (S), thickness (T), and resistivity (ρ) for current flowing along its length is given by the following equation.

휌푙 푅퐷퐶 = ⁄(푆푇) (3.8)

The current flows through the entire cross-section of a conductor at DC, see Fig.

3-5 (b), however it is confined to the surface of the conductor due to skin effect at high frequencies, see Fig. 3-5 (c). 63.6% of current density is confined within a skin depth (δ) from the conductor surface due to skin effect [25]. The skin depth and AC resistance

(RAC) are calculated using the following equations [81].

휌 훿 = √ ⁄ (3.9) (휋푓휇표)

휌푙 푅퐴퐶 = ⁄{2훿(푆 + 푇)} (3.10)

(a) (b)

(c) (d) Fig. 3-5 Image of (a) a conductor of rectangular cross-section, (b) current distribution in the cross-section of a conductor at DC, (c) current distribution in the cross-section of a conductor at high frequencies; (d) Plot of skin depth versus frequency with varying resistivity

From the above discussion, it is clear that the resistance of a conductor increases as the square root of frequency due to skin effect. Fig. 3-5 (d) shows the variation in skin depth with frequency for different conductivities. The current distribution in the signal trace of a transmission line is concentrated to the surface that is closest to the return path at high frequencies. The fields between the signal trace and return path pull the charges in the signal trace to the surface closest to the return path [80]. This concentration of current on adjacent conductor surfaces is called proximity effect [4]. Due to proximity effect, the current in the return path starts to concentrate under the signal trace instead of flowing through the entire cross-section of the return plane thereby increasing the resistance of

75 the return path [82]. This increase in resistance with increasing frequency also increases resistive losses resulting in reduced signal magnitude.

B. Inductance (L)

Current flowing through any conductor will generate concentric magnetic field lines surrounding the conductor. The direction of the magnetic field lines can be determined by the right hand rule. Inductance is the ratio of the total magnetic flux through a surface to the current generating the magnetic flux. Similarly, the self- inductance a current carrying loop (Loop1) is defined as the ratio of the total magnetic flux (ψ1) passing through the surface (S1) surrounded by Loop1 to the current (I1) through

Loop1, see Fig. 3-6. Mutual loop inductance between Loop1 and an adjacent loop

(Loop2) the ratio of total magnetic flux (ψ2) (generated due to current I1 in Loop1) passing through the surface (S2) of Loop2 to the current (I1) in Loop1, see Fig. 3-6.

Inductance depends on the permeability of the conductor, shape and dimension of the current loop and the properties of the surrounding medium [3], [81]. Fig. 3-6 shows a very simplistic diagram of magnetic field lines generated by the current carrying loop

(Loop1) interacting with a neighboring loop (Loop2). The open surfaces surrounded by

Loop1 and Loop2 are marked as S1 and S2, respectively. The variables B1 and B12 represent the magnetic flux density passing through surface S1 and S2, respectively.

Surfaces S1 and S2 are represented using colors pink and green, respectively, in Fig. 3-6.

Both B1 and B12 are generated due to current (I1) flowing through Loop1. The self- inductance (L11) and mutual inductance (M12) can be calculated using the following equations [81].

(∫ 퐵1. 푑푠) 휓1 푆1 ⁄ 퐿11 = ⁄ = (3.11) 퐼1 퐼1

(∫ 퐵12. 푑푠) 휓2 푆2 ⁄ 푀12 = ⁄ = (3.12) 퐼1 퐼1

Fig. 3-6 Simplified pictorial representation of a current carrying loop interacting with an adjacent loop via magnetic fields

A time varying current (I1(t)) through Loop1 will result in time varying magnetic flux densities B1(t) and B12(t) passing through surface S1 and S2, respectively.

From Faraday’s law, it can be stated that B1(t) and B12(t) passing through S1 and S2, respectively will induce voltages V1 and V12 around the periphery of Loop1 and Loop2, respectively. According to Lenz’s law, the polarity of V1 and V12 will always oppose the change in magnetic flux through Loop1 and Loop2, respectively. The induced voltages can be calculated using the following equations [81].

푑휓1⁄ 푑퐼1(푡)⁄ 푉11 = 푑푡 = 퐿11 { 푑푡} (3.13)

푑휓2⁄ 푑퐼1(푡)⁄ 푉12 = 푑푡 = 푀12 { 푑푡} (3.14)

The total loop inductance of a current loop is the sum of internal inductance (Lint) and external inductance (Lext) [25]. The current flowing through the cross-section of a conductor, when skin depth is comparable to the conductor thickness, generates magnetic field lines inside the conductor. These magnetic field lines inside a conductor gives rise to internal inductance. The current flowing through a conductor is pushed towards the

77 conductor periphery due to skin effect which results in fewer magnetic field lines inside the conductor lowering internal inductance [82]. External inductance is due to the loop geometry and is independent of frequency. So with increasing frequency internal inductance falls and external inductance starts to dominate. Eventually at high frequencies the total loop inductance is solely due to the external inductance [4].

The definition of self and mutual loop inductance presented above is only valid when one can accurately calculate the number of magnetic field lines passing through the current loop but it is difficult to determine complete current loops for complicated PCB circuits [4]. The concept of partial self and mutual inductance is used to calculate inductances when understanding of the complete current loop is lacking. A detailed analysis of loop and partial inductance is presented in [81].

C. Capacitance (C) and Conductance (G)

Capacitance (C) between two conductors separated by a dielectric is the measure of the charge storing capability of the two conductor system when a potential difference exists between the two conductors. However, the capacitance between the two conductors is dependent on the shape and dimension of the conductors and the properties of the dielectric medium surrounding the conductors.

A capacitor with an ideal lossless dielectric has no losses associated with it and

0 the displacement current (Idisp) flowing through the capacitor is exactly 90 out of phase with the time varying voltage applied between the plates. Also, there is no current flowing through the capacitor at DC because an ideal dielectric does not have conduction electrons. The current (Idisp) through such a capacitor with capacitance (C) when time

78 varying voltage (V(t)) is applied across the capacitor is calculated using the following equation [3].

푑푉(푡)⁄ 퐼푑푖푠푝 = 퐶 { 푑푡} (3.15)

However, in practice capacitors have non-ideal dielectric materials with finite resistivity and therefore have conduction and displacement current flowing between the two plates. The conduction current is due to the movement of ions in the dielectric, which will result in DC dielectric power loss. The voltage applied across the two plates of a capacitor will re-align the electric dipoles in the dielectric in a direction opposite to the applied electric field. This movement of the positive and negative end of dipoles towards a capacitor’s negatively and positively charged plates, respectively, is equivalent to a transient current flowing through the capacitor [3]. If a sinusoidal voltage is applied across the capacitor plates the dipoles will also be rotating back and forth sinusoidally generating the AC displacement current. However, energy will be used to polarize the material and for rotating the dipoles with the sinusoidally changing electric field between the capacitor plates. This energy loss associated with the displacement current results in

AC dielectric power loss. Since dielectric conductivity is very small, AC dielectric power loss is the dominant loss mechanism in a dielectric material. This dielectric loss associated with the dielectric in a capacitor can be modeled as a resistor (RDL) and is related to the capacitance (C) of the capacitor using the following equation [3].

1 푅퐷퐿 = ⁄(휔퐶푡푎푛 훿) (3.16)

In Equation 3.13, f and tan δ represent the frequency of operation and loss tangent of the dielectric, respectively. Loss tangent of a dielectric is the product of dipole density, dipole moment and the maximum angle of rotation of dipoles under applied electric field

79 in a dielectric [3]. The loss tangent is slightly frequency dependent but for most practical purposes these small variations are neglected and it is treated as constant with frequency.

The signal trace and return path of a transmission line act as the two conductors of a capacitor and therefore have a finite capacitance and dielectric loss associated with them.

The dielectric loss in a transmission line is represented by a shunt resistor with per unit length conductance (GTXL) in parallel with the per unit length capacitance (CTXL) of the transmission line and these are related by the following equation [25], [3].

퐺푇푋퐿 = 휔퐶푇푋퐿푡푎푛 훿 (3.17)

From the above discussion it is clear that RTXL increases with the square root of frequency while the term (ωLTXL) in 3.5a and 3.6a increases linearly with frequency. So at high frequencies RTXL ≪ ωLTXL. Also GTXL ≪ ωCTXL, when tan δ ≪ 1. These conditions result in the low-loss approximation, which simplifies 3.5a and 3.6a as shown below.

퐿푇푋퐿 푍푂 = √ ⁄ (3.5b) 퐶푇푋퐿

푅푇푋퐿 훾 = 훼 + 푗훽 = [0.5 {( ⁄ ) + (퐺푇푋퐿푍푂)}] + 푗(휔√퐿푇푋퐿퐶푇푋퐿) (3.6b) 푍푂

The effective dielectric constant (휀푒푓푓) associated with the transmission line can be calculated using the equation

2 훽퐶 2 휀푒푓푓 = ( ⁄휔) = 퐿푇푋퐿퐶푇푋퐿퐶 , (3.18) where C is the speed of electromagnetic waves in free space.

In the following section the lumped element transmission line model will be modified to account for the impact of floating metals near transmission lines.

3.3 Lumped element model for transmission line including floating metal effects

A changing current (iSignal) through a signal trace will cause eddy currents (iEddy) to flow in the floating metal in a direction opposite to the current in the signal trace according to Lenz’s law, see Fig. 3-8. These eddy currents in the floating metal will result in Joule losses that increase with frequency due to skin effects [83]. Fig. 3-7 (a) and (b) show the diametric and side view, respectively, of a signal trace along with a floating metal. Fig. 3-8 (a) and (b) show the current distribution in the signal trace and floating metal in diametric and side view, respectively. The current in the signal trace is flowing from the left to right while the eddy current in the floating metal is circulating in an anti- clockwise direction. The variables S, T, and L represent the width, metallic thickness and length, respectively, of the signal trace. The parameters FW, FT, and FL denote the width, thickness and length, respectively, of the floating metal. The skin depth is denoted by δ. The variable SepSF represents the separation between the signal trace and the floating metal.

(a) (b) Fig. 3-7 Image of a signal trace and a floating metal in close proximity of each other in (a) diametric and (b) side view

(a) (b) Fig. 3-8 Image of current distribution at high frequency in a signal trace and a floating metal when they are in close proximity of each other in (a) diametric and (b) side view

Resistance RFM is used to account for the per unit length loss due to Joule heating in the floating metal. The floating metal has a per unit length self-inductance (LFM) associated with it and the per unit length mutual inductance between the signal trace and the floating metal is represented by the variable MTXF. The capacitance per unit length of the transmission line associated with the signal trace also changes due to the presence of the floating metal and this in turn also changes the conductance of the transmission line according to 3.17. The modified per unit length capacitance and conductance of a transmission line in presence of floating metals can be represented by CTXF and GTXF, respectively. Fig. 3-9 shows the modified lumped element model of a transmission line in the presence of a floating metal.

Fig. 3-9 Modified lumped element model of a transmission line in the presence of a floating metal

Applying Kirchhoff’s current law to the transmission line model depicted in Fig.

3-9 one can write the following equation.

푖푆푖푔푛푎푙(푧 + ∆푧, 푡) − 푖푆푖푔푛푎푙(푧, 푡) =

−퐺푇푋퐹∆푧푣푠푖푔푛푎푙(푧 + ∆푧, 푡) − 퐶푇푋퐹∆푧{휕푣푠푖푔푛푎푙(푧 + ∆푧, 푡)⁄휕푡} (3.19a)

Dividing both sides of 3.19(a) by Δz and taking the limit Δz→0 gives

휕푖푆푖푔푛푎푙(푧, 푡)⁄휕푧 = −퐺푇푋퐹푣푠푖푔푛푎푙(푧, 푡) − 퐶푇푋퐹{휕푣푆푖푔푛푎푙(푧, 푡)⁄휕푡}. (3.19b)

In sinusoidal steady state (푒푗휔푡), 3.19(b) simplifies to

휕퐼푆푖푔푛푎푙(푧)⁄휕푧 = −(퐺푇푋퐹 + 푗휔퐶푇푋퐹)푉푠푖푔푛푎푙(푧). (3.19c)

The capacitances between the signal trace and the return path of a transmission line are shown in Fig. 3-10 (a). The variables CoSR and CfSR represent the per unit length

(p.u.l) overlap and fringe capacitance, respectively. Overlap capacitance (Co) is the capacitance between the overlapping surfaces of two conductors located in different metal layers and is calculated using the parallel plate capacitor approximation [84].

Fringe capacitance (Cf) is the capacitance between the sidewall of one conductor and the top or bottom of another conductor situated in a different metal layer [85]. The per unit length capacitance (CTXL) of the transmission line can be calculated using the equation

퐶푇푋퐿 = 푝푎푟푎푙푙푒푙(퐶표푆푅, 퐶푓푆푅1, 퐶푓푆푅2) = 퐶표푆푅 + 퐶푓푆푅1 + 퐶푓푆푅2. (3.20a)

The fringe capacitance in Fig. 3-10 (a) are symmetric and the per unit length overlap capacitance CoSR can be calculated using the equation

퐶표푆푅 = (휀표휀푟푆)⁄ℎ. (3.21)

Therefore, 3.20a can be rewritten as

퐶푇푋퐿 = {(휀표휀푟푆)⁄ℎ} + 2퐶푓푆푅. (3.20b)

(a)

(b)

(c) Fig. 3-10 Cross-section of a transmission line (a) without floating metal, (b) with floating metal directly under the signal trace, and (c) with floating metal not directly below the signal trace

Consider the case where a floating metal of width FW and thickness FT is placed at a vertical distance SepSF directly below the signal trace (Fig. 3-10 (b)), the per unit length capacitance (CTXF) of the structure will increase. The fringe capacitances will remain unaffected but there will be p.u.l overlap capacitances CoSF and CoFR between the signal trace and the floating metal and between the floating metal and the return path, respectively. CTXF is calculated using the following equation.

퐶푇푋퐹 = 푝푎푟푎푙푙푒푙{퐶표푆푅1, 퐶표푆푅2, 퐶푓푆푅1, 퐶푓푆푅2, 푠푒푟푖푒푠(퐶표푆퐹, 퐶표퐹푅)}, (3.22a)

∴ 퐶푇푋퐹 = 퐶표푆푅1 + 퐶표푆푅2 + 2퐶푓푆푅 + {(퐶표푆퐹퐶표퐹푅)⁄(퐶표푆퐹 + 퐶표퐹푅)} (3.22b)

The overlap capacitances in 3.22b can be calculated using the following equations

퐶표푆푅1 = (휀표휀푟푆1)⁄ℎ, (3.22c)

퐶표푆푅2 = (휀표휀푟푆2)⁄ℎ, (3.22d)

퐶표푆퐹 = (휀표휀푟퐹푊)⁄푆푒푝푆퐹, (3.22e)

퐶표푆푅 = (휀표휀푟퐹푊)⁄(ℎ − 퐹푇 − 푆푒푝푆퐹). (3.22f)

Substituting these in 3.22b, 3.22b can be rewritten as

퐶푇푋퐹 = {(휀표휀푟(푆 − 퐹푊))⁄ℎ} + (2퐶푓푆푅) + {(휀표휀푟퐹푊)⁄(ℎ − 퐹푇)}. (3.22g)

From 3.22g, it is clear that the capacitance of the transmission line will increase due to floating metal. Increase in width and/or thickness of the floating metal will increase CTXF.

It is completely independent of the separation (SepSF) between the signal trace and the floating metal. Fig. 3-11 shows the variation in capacitance with varying FW and FT.

Fig. 3-11 Plot of p.u.l. capacitance versus floating metal width with varying floating metal thickness

Consider the case where the floating metal is not located directly under the signal trace but is displaced by a horizontal distance Dy from the signal trace edge in Fig. 3-10

(c). In this case, the floating metal is interacting with the fringing fields of the signal trace. So, there will be a per unit length fringing capacitance (CfSF) and overlap capacitance (CoFR) between the signal trace and the floating metal and between the floating metal and the return path, respectively. To keep Fig. 3-10 (c) simple, the total fringing capacitance between the signal trace and return path is lumped into a single fringing capacitance CfSR12. Since the floating metal only interacts with a fraction of the fringing fields, the range of CfSR12 is 퐶푓푆푅1 ≤ 퐶푓푆푅12 ≤ (퐶푓푆푅1 + 퐶푓푆푅2). The p.u.l overlap capacitance (CoSR) between the signal trace and return path stays unaffected.

CTXF for this structure is calculated using the following equation

퐶푇푋퐹 = 푝푎푟푎푙푙푒푙{퐶표푆푅, 퐶푓푆푅12, 푠푒푟푖푒푠(퐶푓푆퐹, 퐶표퐹푅)}, (3.23a)

∴ 퐶푇푋퐹 = 퐶표푆푅 + 퐶푓푆푅12 + {(퐶푓푆퐹퐶표퐹푅)⁄(퐶푓푆퐹 + 퐶표퐹푅)}. (3.23b)

Unlike 3.22g, 3.23b is dependent on both the vertical (SepSF) and horizontal (Dy) distance between the signal trace and the floating metal. Any increase in SepSF and/or

Dy will reduce the fringe capacitance between the signal trace and the floating metal thereby reducing the value of series (CfSF, CoFR). This in turn will reduce CTXF in 3.23b.

The fringing capacitance (CfSF) between the floating metal and the signal trace can be calculated using the following equation when S = FW and T = FT [85].

2 2 4휀표휀푟 푆푒푝푆퐹 + (휂푇) + √퐷푦 + (휂푇) + (2휂푇푆푒푝푆퐹) 퐶푓푆퐹 = 푙푛 ( ) 휋 퐷푦 + 푆푒푝푆퐹

2푆 퐷푦 + 푇 2휀표휀푟훼푆 (푙푛 (1 + ) + 푒푥푝 (− )) 퐷푦 3퐷푦 + 2푆 퐷푦 + 푇 (휋훼푆) + (푆푒푝푆퐹 + 푇) (푙푛 (1 + ) + 푒푥푝 (− )) 퐷푦 3퐷푦

2푇 푆푒푝푆퐹 + 푆 2휀표휀푟훽푇 (푙푛 (1 + ) + 푒푥푝 (− )) 푆푒푝푆퐹 3푆푒푝푆퐹 + 2푇 푆푒푝푆퐹 + 푆 (휋훽푇) + (퐷푦 + 푆) (푙푛 (1 + ) + 푒푥푝 (− )) 푆푒푝푆퐹 3푆푒푝푆퐹

2 2 + [(휀표휀푟⁄휋)√(퐷푦푆푒푝푆퐹)⁄(퐷푦 + 푆푒푝푆퐹)]. (3.24a)

The variables η, α, and β are calculated using

2 2 휂 = 푒푥푝[(푆 + 퐷푦 − √퐷푦 + 푇 + (2푆푒푝푆퐹푇))⁄(휏푆)], (3.24b)

훼 = 푒푥푝[− (푆푒푝푆퐹 + 푇)⁄(퐷푦 + 푆)], (3.24c)

훽 = 푒푥푝[− (퐷푦 + 푆)⁄(푆푒푝푆퐹 + 푇)], (3.24d) where τ (= 3.7) is a geometry independent fitting parameter and η,α, and β are valid over a certain range specified in [85]. A very general equation for calculating the p.u.l total capacitance (CTXF) of a signal trace to in presence of a floating metal is given by the equation [86]

1 1 퐶푇푋퐹 = 퐶푠푔 + {1⁄{( ⁄ ) + ( ⁄ )}}, (3.25) 퐶푠푓 퐶푓푔

87 where 퐶푠푔, 퐶푠푓 and 퐶푓푔 are the capacitances between signal to ground, signal to floating metal and floating metal to ground, respectively.

Now applying Kirchhoff’s voltage law to the modified transmission line model shown in Fig. 3-9, one can write

푣푆푖푔푛푎푙(푧 + ∆푧, 푡) − 푣푆푖푔푛푎푙(푧, 푡) = −[푅푇푋퐿∆푧푖푠푖푔푛푎푙(푧, 푡) +

퐿푇푋퐿∆푧{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡} − 푀푇푋퐹∆푧{휕푖퐸푑푑푦(푧, 푡)⁄휕푡}]. (3.26a)

Dividing both sides of 3.26(a) by Δz and taking the limit Δz→0 gives

휕푣푆푖푔푛푎푙(푧, 푡)⁄휕푧 = −[푅푇푋퐿푖푠푖푔푛푎푙(푧, 푡) + 퐿푇푋퐿{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡} −

푀푇푋퐹{휕푖퐸푑푑푦(푧, 푡)⁄휕푡}]. (3.26b)

The eddy current flowing through the floating metal is the result of the voltage generated across it due to changing current in the signal conductor. This induced voltage (vinduced) can be calculated using 3.14.

푣푖푛푑푢푐푒푑(푧, 푡) = 푀푇푋퐹∆푧{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡} (3.27)

The floating metal has a complex impedance (ZFM) associated with it due to its per unit length resistance (RFM) and self-inductance (LFM). The per unit length floating metal impedance is calculated using the following equation.

푍퐹푀 = 푅퐹푀 + 푗휔퐿퐹푀 (3.28)

The eddy current generated due to the induced voltage can be calculated using Ohm’s law.

푖퐸푑푑푦(푧, 푡) = 푣푖푛푑푢푐푒푑(푧, 푡)⁄(푍퐹푀∆푧) (3.29a)

푖퐸푑푑푦(푧, 푡) = [푀푇푋퐹{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡}]⁄(푅퐹푀 + 푗휔퐿퐹푀) (3.29b)

Substituting 3.29b in 3.26b gives the following equation.

휕푣푆푖푔푛푎푙(푧, 푡)⁄휕푧 = − [푅푇푋퐿푖푠푖푔푛푎푙(푧, 푡) + 퐿푇푋퐿{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡} −

푀푇푋퐹 {휕 {{푀푇푋퐹{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡}}⁄(푅퐹푀 + 푗휔퐿퐹푀)}⁄휕푡}] (3.26c)

Simplifying 3.26c gives

휕푣푆푖푔푛푎푙(푧, 푡)⁄휕푧 = −[푅푇푋퐿푖푠푖푔푛푎푙(푧, 푡) + 퐿푇푋퐿{휕푖푠푖푔푛푎푙(푧, 푡)⁄휕푡} −

2 2 2 {푀푇푋퐹⁄(푅퐹푀 + 푗휔퐿퐹푀)}{휕 푖푠푖푔푛푎푙(푧, 푡)⁄휕푡 }]. (3.26d)

In sinusoidal steady state (푒푗휔푡), 3.26d simplifies to

휕푉푆푖푔푛푎푙(푧)⁄휕푧 = −[푅푇푋퐿 + 푗휔퐿푇푋퐿 +

2 2 {휔 푀푇푋퐹⁄(푅퐹푀 + 푗휔퐿퐹푀)}]퐼푆푖푔푛푎푙(푧). (3.26e)

3.26e is further simplified by multiplying the numerator and denominator of the third term in 3.26e with the complex conjugate of its denominator and then separating out the real and imaginary terms. The simplified equation is shown below.

2 2 2 2 2 휕푉푆푖푔푛푎푙(푧)⁄휕푧 = − [{푅푇푋퐿 + {(휔 푀푇푋퐹푅퐹푀)⁄(푅퐹푀 + (휔 퐿퐹푀))}} +

2 2 2 2 2 {푗휔 {퐿푇푋퐿 − {(휔 푀푇푋퐹퐿퐹푀)⁄(푅퐹푀 + (휔 퐿퐹푀))}}}] 퐼푆푖푔푛푎푙(푧). (3.26f)

From 3.26f it is clear that resistive losses and inductance of a transmission line will increase and decrease, respectively, when a floating metal is in close proximity of the transmission line. This change in resistance and inductance is dependent on the self- resistance and self-inductance of the floating metal and the mutual inductance between the floating metal and the transmission line. Also, the net capacitance of a transmission line will increase due to floating metals which in turn will increase the conductance of the transmission line. The following subsections are devoted to relating the floating metal physical dimensions to its self-resistance and to briefly discuss the self-inductance of the floating metal and the mutual inductance between the signal trace and the floating metal.

The skin effect confines most of the current distribution in the floating metal to one skin depth below the conductor surface as depicted in Fig. 3-5. The analysis presented below for a floating metal can be approximated by treating the floating metal as a rectangular current loop made out of a conductor of rectangular cross-section as shown in Fig. 3-12. The circulating eddy current in the floating metal forms the rectangular loop.

The rectangular loop in Fig. 3-12 (c) has four segments. Segments Seg1 and Seg3 are parallel to the z-axis and have FL, FW, and δ as length, width, and thickness, respectively. While segments Seg2 and Seg4 are parallel to the x-axis and have FT, FW, and δ as length, width, and thickness, respectively.

(a) (b)

(c) Fig. 3-12 Image of (a) floating metal, (b) current distribution in floating metal at high frequency, and (c) floating metal being represented as a rectangular current loop made of a conductor with rectangular cross-section

A. Self-resistance (RSFM) of a floating metal

Resistance of a current carrying conductor is directly proportional to the length of the conductor and is inversely proportional to the area of cross-section of the conductor.

In case of the circulating current in the floating metal, the length of the floating metal

(lFM) is calculated using the following equation.

푙퐹푀 = (2퐹퐿) + {2(퐹푇 − (2훿))} = 2{퐹퐿 + 퐹푇 − (2훿)} (3.30)

The area of cross-section (AFM) through which the current flows is given by

퐴퐹푀 = 훿 ∗ 퐹푊. (3.31)

The self-resistance (RSFM) of a floating metal can be calculated using the following equation [77]

푅푆퐹푀 = 휌(푙퐹푀⁄퐴퐹푀) = 휌{{2{퐹퐿 + 퐹푇 − (2훿)}}⁄(훿 ∗ 퐹푊)}, (3.32) where ρ is the resistivity of the metal. From 3.32 it is clear that self-resistance of a floating metal is going to increase when either the length (FL) and/or thickness (FT) of

91 the floating metal is increased (Fig.3-13). On the other hand, floating metal resistance is going to decrease with increasing floating metal width (FW), see Fig. 3-13.

Fig. 3-13 Plot of floating metal self-resistance versus frequency with varying floating metal thickness and width

B. Self-inductance (LSFM) of a floating metal

Non uniform current distribution in conductors due to proximity effect and skin effect makes it difficult to derive closed form equations for calculating self-inductance of conductors. So authors in [77] and [81] suggest using numerical computing or 3D field solvers for calculating self-inductances associated with conductors. Self-inductance is shown to increase with frequency, FT and FL in [77]. An analytical equation to calculate the self-impedance of a floating metal is developed by treating the floating metal as a rectangular current loop and compared with the data presented in [77] below.

The self-inductance (LSFM) of the floating metal can be calculated using the following equation.

퐿푆퐹푀 = {퐿푃푆푒푔1 + 퐿푃푆푒푔3 − 푀푃푆푒푔1푆푒푔3 − 푀푃푆푒푔3푆푒푔1}

+{퐿푃푆푒푔2 + 퐿푃푆푒푔4 − 푀푃푆푒푔2푆푒푔4 − 푀푃푆푒푔4푆푒푔2} (3.33a)

In 3.33a, the variables LPSeg1, LPSeg2, LPSeg3, and LPSeg4 represent the self-partial inductance of Seg1, Seg2, Seg3, and Seg4, respectively, which constitute the rectangular current loop in Fig. 3-12 (c). The parameters MPSeg1Seg3, MPSeg3Seg1, MPSeg2Seg4, and

MPSeg4Seg2 represent the partial mutual inductance between Seg1 and Seg3 for current in

Seg1, Seg3 and Seg1 for current in Seg3, Seg2 and Seg4 for current in Seg2, and Seg4 and Seg2 for current in Seg4, respectively. Mutual inductance between odd and even numbered segments are neglected because they are perpendicular to each other. The partial mutual inductances are subtracted from the partial-self inductances because the current in the two segments is travelling in opposite direction. The odd numbered sections have the same physical dimensions therefore their self-partial inductances are identical (LPSeg1 = LPSeg3). The same principle also holds true for the even numbered section (LPSeg2 = LPSeg4). Also, mutual inductance between two conductors are symmetric so MPSeg1Seg3 = MPSeg3Seg1 and MPSeg2Seg4 = MPSeg4Seg2. Due to these simplifications, 3.30a can be rewritten as follows.

퐿푆퐹푀 = (2퐿푃푆푒푔1 − 2푀푃푆푒푔1푆푒푔3) + (2퐿푃푆푒푔2 − 2푀푃푆푒푔2푆푒푔4) (3.33b)

∴ 퐿푆퐹푀 = 2(퐿푃푆푒푔1 + 퐿푃푆푒푔2 − 푀푃푆푒푔1푆푒푔3 − 푀푃푆푒푔2푆푒푔4) (3.33c)

The equations for calculating the self-partial inductance of a segment and mutual partial inductance between two segments are provided below [87].

퐿푃푆푒푔1 = {휇표⁄(2휋)}[{퐹퐿{푙푛((2퐹퐿)⁄(퐹푊 + 훿))}} + (0.5퐹퐿) +

{0.2235(퐹푊 + 훿)}] (3.34a)

퐿푃푆푒푔2 = {휇표⁄(2휋)}[{퐹푇{푙푛((2퐹푇)⁄(퐹푊 + 훿))}} + (0.5퐹푇) +

{0.2235(퐹푊 + 훿)}] (3.34b)

2 2 푀푃푆푒푔1푆푒푔3 = {휇표⁄(2휋)} [{퐹퐿{푙푛((퐹퐿 + √퐹퐿 + 퐹푇 )⁄퐹푇)}} −

√퐹퐿2 + 퐹푇2 + 퐹푇] (3.34c)

2 2 푀푃푆푒푔2푆푒푔4 = {휇표⁄(2휋)} [{퐹푇{푙푛((퐹푇 + √퐹퐿 + 퐹푇 )⁄퐹퐿)}} −

√퐹퐿2 + 퐹푇2 + 퐹퐿] (3.34d)

The self-inductance of the floating metal obtained with 3.33c increases with frequency,

FL and FT as predicted in [77]. However, 3.33c over predicts the self-inductance of the floating metal but it is the same order of magnitude with the data presented in [77], see

Fig. 3-14.

Fig. 3-14 Plot of floating metal self-inductance versus frequency with varying floating metal thickness and length

C. Mutual inductance (MFMST) between a floating metal and a signal trace

The mutual inductance (MFMST) between the floating metal and the signal trace can be calculated by determining the magnetic flux generated due to current in the signal conductor passing through the surface enclosed by the rectangular current loop shown in

Fig. 3-12 (c). In order to determine the magnetic flux passing through the rectangular current loop the magnetic flux density at the enclosed surface should be calculated. To

94 simplify this process, the signal trace will be treated as current sheet of infinite length and finite width S. According to [81], the magnetic flux density (B) at any point (Y, Z) due a current sheet of infinite length along the x-axis centered at the origin (Fig. 3-15) is given by the following equation.

−1 2 2 2 퐵⃗ = [{(휇표퐼)⁄(2휋)}푡푎푛 {(푆푍)⁄{푌 + 푍 − (0.5푆) }}]푦̂ −

2 2 2 2 [{(휇표퐼)⁄(4휋)}푙푛 {{(푌 + (0.5푆)) + 푍 }⁄{(푌 − (0.5푆)) + 푍 }}] 푧̂ (3.35a)

Fig. 3-15 Magnetic flux density at any arbitrary point due to a current sheet of finite width and infinite length

The variable I in 3.35a is the surface current through the current sheet and is flowing in the negative x direction. Since the floating metal in the analysis is directly under the signal trace, Y = 0 in 3.35a and the magnetic field B is only along the positive y direction. Under this situation, 3.35a simplifies into

−1 퐵⃗ = [{(휇표퐼)⁄(휋)}푡푎푛 {푆⁄(2푍)}]푦̂. (3.35b)

For calculating the mutual inductance between the signal trace and the floating metal, assume that the signal trace and the floating metal of length FL are centered at the

95 origin and are separated from each other by SepSF as shown in Fig. 3-16. The current in the signal trace is confined to the surface of the metal and is flowing in the negative x direction.

Fig. 3-16 Setup for calculating the magnetic flux density at any arbitrary point on a surface enclosed by a floating metal due to current through signal trace

The total magnetic flux (휓푆퐹푀) passing through the surface SFM (represented using the color green in Fig. 3-16) enclosed by the rectangular loop formed by the floating metal, due to the current in the signal trace can be calculated using

푥2 푍 (휇표퐼) 2 −1 푆 휓푆퐹푀 = { ⁄휋} ∫ ∫ [푡푎푛 { ⁄ }] 푑푍푑푥, (3.36a) 푍1 2푍 푥1 where

푍1 = 푆푒푝푆퐹 + 훿, (3.36b)

푍2 = 푆푒푝푆퐹 + 퐹푇 − 훿, (3.36c)

푥1 = −(0.5퐹퐿) + 훿, (3.36d)

푥2 = (0.5퐹퐿) − 훿. (3.36e)

The following equation is obtained after solving 3.36a within the limits of integration.

휇표퐼 −1 0.5푆 휓푆퐹푀 = ( ⁄ ) {퐹퐿 − (2훿)} [{{푍2푡푎푛 ( ⁄ )} − 휋 푍2

2 2 −1 0.5푆 푆 (푆 + (4푍2 )) {푍1푡푎푛 ( ⁄푍 )}} + {{( ⁄4)푙푛 ( ⁄ 2 2 )}}] (3.36f) 1 (푆 + (4푍1 ))

The mutual inductance (MFMST) between the floating metal and the signal trace can be calculated using the following equation.

휇표 −1 0.5푆 푀퐹푀푆푇 = 휓푆퐹푀⁄퐼 = ( ⁄ ){퐹퐿 − (2훿)} [{{푍2푡푎푛 ( ⁄ )} − 휋 푍2

2 2 −1 0.5푆 푆 (푆 + (4푍2 )) {푍1푡푎푛 ( ⁄푍 )}} + {{( ⁄4)푙푛 ( ⁄ 2 2 )}}] (3.37) 1 (푆 + (4푍1 ))

Fig. 3-17 Plot of mutual inductance versus frequency with varying floating metal thickness and separation between floating metal and signal conductor

From Fig. 3-17 it is clear that the mutual inductance increases with increasing FT and with reducing SepSF. The magnetic flux density at a point (Y, Z) in 3.35a is also going to fall as the point moves away from the current sheet. Therefore, the mutual inductance between the signal trace and the floating metal will also fall if the floating metal is moved away from the signal trace in vertical and/or horizontal direction.

From 3.5b and 3.6b, it is clear that changing p.u.l resistance, inductance, conductance and capacitance of a transmission line will also change its characteristic impedance, attenuation coefficient and propagation coefficient. A floating metal increases the p.u.l resistance and conductance of a transmission line which in turn will increase its attenuation coefficient. Floating metals also increase capacitance and reduce inductance in a transmission line which will change the propagation coefficient of the transmission line thereby changing the effective permittivity of the system. This reduced inductance and increased capacitance will also result in reduced characteristic impedance of a transmission line. So, the following section analyzes the impact of EPL on GCPWs in terms of per unit length resistance, inductance, conductance, and capacitance via parametric analysis. The changes in α, ZO, and εeff are also analyzed.

3.4 Parametric study of EPL with GCPW

In this section the effect of different EPL parameters on GCPW attenuation coefficient, characteristic impedance and effective permittivity will be analyzed. The dimensions associated with the GCPW are provided in Section 3.1. The different dimensions associated with the EPL, see Fig. 3-1 and Fig. 3-2, are provided in Table 3.1.

These values do not change in this section unless specified.

Table 3.1 List of variable dimensions associated with EPL Variable Size (μm) Variable Size (μm) Variable Size (μm) LEPL 150.00 HEPL 25.40 PXEPL 195.00 WEPL 150.00 TEPL 9.00 PYEPL 195.00

The percentage change in different GCPW parameters due to EPL is calculated using the equation

(푃푁 − 푃푁 ) % ∆푃푁 = { 푤푖푡ℎ 퐸푃퐿 푤푖푡ℎ표푢푡 퐸푃퐿 ⁄ } × 100, (3.38) 푃푁푤푖푡ℎ표푢푡 퐸푃퐿

98 where PN represents the parameter name.

A. Varying EPL thickness (TEPL)

In this section, the thickness of the EPL layer is varied from 9 μm to 27 μm. The self-inductance and resistance of floating metals increase with increasing thickness and so does the mutual inductance between the floating metal and the signal trace. According to 3.26f, the net p.u.l. resistance (R) and inductance (L) of the transmission line will increase and decrease, respectively, when TEPL is increased. Increasing TEPL will also increase the net p.u.l. capacitance (C) of the transmission line thereby increasing the associated p.u.l. conductance (G). Table 3.2 shows the p.u.l. resistance, inductance, conductance and capacitance of a GCPW with and without EPL at 40 GHz. Table 3.3 shows the attenuation coefficient, characteristic impedance and effective permittivity associated with the cases mentioned in Table 3.2. Increasing TEPL can reduce the characteristic impedance of the GCPW by 15.50% while the effective permittivity of the transmission line in increased by 9.90%.

Table 3.2 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying TEPL at 40 GHz

Table 3.3 α, ZO, and εeff of a GCPW without and with EPL of varying TEPL at 40 GHz

Setup α (Np/m) ZO (Ω) εeff % Δα % ΔZO % Δεeff No EPL 3.76 50.47 2.65 TEPL = 9 μm 5.22 46.58 2.82 38.76 -7.70 6.50 TEPL = 27 μm 5.80 42.62 2.91 54.32 -15.56 9.90

The increased attenuation and impedance mismatch due to EPL will deteriorate the insertion loss and return loss associated with GCPWs with EPL compared to GCPW

99 without EPL, see Fig. 3-18 (a), (b). GCPWs with EPL look electrically longer than

GCPWs without EPL since the effective dielectric permittivity is increased due to EPL, see Fig. 3-18 (c).

(a)

(b)

100

(c) Fig. 3-18 Plot of (a) insertion loss, (b) return loss, and (c) insertion loss phase versus frequency for GCPWs without and with EPL of varying TEPL, with ports renormalized to line impedance

From Table 3.2 it is evident that the increase in attenuation coefficient of the GCPWs with EPL is more than 25%, however from Fig. 3.18 (a) the increase in insertion loss does not look that drastic. This is due to the fact that the simulated GCPW insertion loss is for a 6000 μm long line while the attenuation coefficient presented in Table 3.2 is in terms of m-1. Once the attenuation coefficients in Table 3.2 are converted to dB/m and the loss for a 6000 μm long GCPW is calculated the numbers so obtained are very close to the ones obtained from Fig. 3-18 (a). Addition of the EPL does not change the insertion loss of GCPWs that much.

An eye diagram is used to analyze signal quality in the time domain after it is transmitted through GCPWs with and without EPL. Lossy transmission lines reduce eye height (EH) and eye width (EW). It is desired that the eye opening be as large as possible because it implies lower bit error rate (BER) and good signal quality. For this section, a

1-gigabit per second (Gbps) bit stream is transmitted over 18 cm long GCPW with and without EPL, without applying any equalization scheme either at the transmitter or at the

101 receiver. The pulse rise and fall time is 250 ps and the associated voltage swing is ±1 V.

Table 3.4 shows the eye height and eye width at the transmitter (row 1) and at the receiver (rows 2 - 4). The eye height and width is reduced as TEPL is increased. Increasing

TEPL increases attenuation coefficient of the GCPW which results in eye height reduction.

The capacitance of the GCPW with EPL increases with increasing TEPL which increases the charging and discharging time thus affecting eye width. Fig. 3-19 (a) shows the eye diagram at the transmitter (50 Ω) while Fig. 3-19 (b) - (d) show the eye diagram at the receiver (50 Ω). GCPWs with EPL have reduced eye height and eye width. The reduction in EH and EW is 34% and 12%, respectively, when the EPL thickness is increased from 9

μm to 27 μm.

Table 3.4 Eye height and eye width at the receiver of 18 cm long GCPW without and with EPL of varying TEPL Setup EH (mV) EW (ps) % ΔEH % ΔEW At transmitter 988 1000

No EPL 746 995 TEPL = 9 μm 620 950 -16.89 -4.52 TEPL = 27 μm 490 875 -34.32 -12.06

(a) (b)

102

(c) (d) Fig. 3-19 Eye diagram of 1 Gbps bit stream (a) at transmitter, (b) at receiver of a GCPW without EPL, (c) at receiver of a GCPW with EPL of TEPL = 9 μm, and (d) at receiver of a GCPW with EPL of TEPL = 27 μm

B. Varying separation between signal trace and EPL (HEPL)

In this section, HEPL is varied from 25.4 μm to 50.8 μm. The self-inductance and self-resistance of the floating metals stay unaffected when HEPL is varied according to

3.32 and 3.33c, respectively. The mutual inductance between the signal trace and the floating metal will decrease with increasing HEPL according to 3.37. So, as HEPL is increased the GCPW p.u.l. resistance and inductance will start to decrease and increase, respectively, according to 3.26f. The capacitance is expected to fall as HEPL is increased because the capacitance between signal trace and floating metal falls as separation is increased. This in turn will reduce conductance. So, as HEPL is increased the different transmission line parameters start approaching the values of the transmission line without

EPL (Table 3.5 and 3.6). This same behavior is visible in the insertion loss and return loss plots in Fig. 3-20.

103

Table 3.5 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying HEPL at 40 GHz

Table 3.6 α, ZO, and εeff of a GCPW without and with EPL of varying HEPL at 40 GHz

Setup α (Np/m) ZO (Ω) εeff % Δα % ΔZO % Δεeff No EPL 3.76 50.47 2.65 HEPL = 25.4 μm 5.22 46.58 2.82 38.76 -7.70 6.50 HEPL = 50.8 μm 4.72 48.21 2.72 25.66 -4.48 2.59

(a)

(b) Fig. 3-20 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying HEPL, with ports renormalized to line impedance

104

C. Varying EPL pitch along x (PXEPL) and y (PYEPL) axis

In this section, the EPL pitch along x (PXEPL) and y (PYEPL) axis are varied separately. PXEPL is varied from 195 μm to 292.5 μm while keeping LEPL = WEPL = 150

μm and PYEPL = 195 μm. It is seen that with increasing PXEPL p.u.l. R, L, G, and C values fall, see Table 3.7. The mutual capacitance between the individual elements (floating metals) also increase when PXEPL is decreased resulting in increased capacitance. This increased capacitance allows the individual elements constituting the EPL to electrically look like a solid metallic strip running along the length of the GCPW, see Fig. 3-21. This in turn increases the overall impact of the EPL on the transmission line characteristics.

According to [20], solid floating metallic strips generate unwanted resonances in microstrip transmission lines. The variation in attenuation coefficient, characteristic impedance and effective permittivity is shown in Table 3.8. Fig. 3.22 (a) and (b) show the insertion loss and return loss associated with varying PXEPL, respectively.

(a) (b) Fig. 3-21 Top view of a GCPW with (a) EPL, and (b) solid metal strips embedded in the substrate

105

Table 3.7 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying PXEPL at 40 GHz

Table 3.8 α, ZO, and εeff of a GCPW without and with EPL of varying PXEPL at 40 GHz

(a)

106

(b) Fig. 3-22 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PXEPL, with ports renormalized to line impedance

PYEPL is varied from 175 μm to 215 μm while keeping LEPL = WEPL = 110 μm and PXEPL

= 195 μm. Increasing PYEPL results in reduced field interaction between the signal trace and the EPL columns that do not lie directly under the signal trace. This results in reduced mutual capacitance between individual EPL elements, between signal trace and the EPL. Also the mutual inductance between the signal trace and individual elements is reduced. Therefore, increasing PYEPL results in reduction of resistance, capacitance and conductance of a GCPW and increase in its inductance, see Table 3.9. Accordingly, attenuation coefficient, characteristic impedance and effective permittivity falls with increasing PYEPL (Table 3.10). Fig. 3.23 (a) and (b) show the insertion loss and return loss associated with varying PYEPL, respectively.

Table 3.9 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying PYEPL at 40 GHz

107

Table 3.10 α, ZO, and εeff of a GCPW without and with EPL of varying PYEPL at 40 GHz

(a)

(b) Fig. 3-23 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PYEPL, with ports renormalized to line impedance

D. Varying EPL metal conductivity

In this section, the EPL metal conductivity is varied from 5.8E+07 S/m to

1.79E+07 S/m and the EPL thickness is fixed at 27 μm. There is no change in the GCPW

108 p.u.l. capacitance and conductance because capacitance between two metals is independent of the conductivity of the metal. The p.u.l. resistance and inductance associated with the GCPW increases as conductivity is reduced, see Table 3.11.

However, the change in resistance is much more drastic than inductance because resistance is more strongly dependent on skin depth than inductance. Accordingly, the attenuation coefficient associated with the GCPW increases as conductivity is reduced while there is no appreciable change in characteristic impedance and effective permittivity, see Table 3.12.

Table 3.11 Per unit length R, L, G, and C values of a GCPW without and with EPL of varying EPL metal conductivity at 40 GHz

Table 3.12 α, ZO, and εeff of a GCPW without and with EPL of varying EPL metal

conductivity at 40 GHz

The substrate relative permittivity is varied from 3.5 to 10 while keeping the loss tangent fixed at 0.002 and EPL thickness fixed at 9 μm. The p.u.l. resistance and inductance do not change as relative permittivity is increased because resistance and inductance are independent of dielectric permittivity, see Fig. 3-24. However, p.u.l. capacitance and conductance increase with increasing dielectric constant as expected.

109

Fig. 3-24 Bar plot showing variation in p.u.l. R, L, G, and C in GCPWs without and with EPL when the substrate dielectric constant is varied keeping the loss tangent constant at 40 GHz

E. Varying EPL individual element geometry

In this section, the shape of the individual EPL element is varied with TEPL = 27

μm. The shapes compared are a square patch of edge length 150 μm, a disk with diameter

169.2 μm (same area as the square patch), a disk with diameter 150 μm (same dimension as the square patch), and a regular hexagon of edge length 93.06 μm (same area as the square patch). The shapes are shown in Fig. 3-25. The GCPWs with three different EPL element shapes of same area have nearly identical p.u.l. R, L, G, and C values, see Table

3.13. However, the GCPW with disks of equal dimension as the square patch have lower p.u.l. resistance, conductance and capacitance and higher inductance compared to the other shapes. Consequently, the % changes in attenuation coefficient, characteristic impedance and effective permittivity is less for the GCPW with EPL made of disks of diameter 150 μm, see Table 3.14. The mutual capacitance between the individual elements of the EPL with square patch and with disk of same area is approximately equal.

However, the mutual capacitance is less for the EPL with disks of diameter 150 μm when compared to the EPL made of square patches. The lower surface area of the disk with 150

110

μm diameter and reduced mutual capacitance can explain reduction in capacitance observed in this case. Further analysis is required to explain the effects of EPL individual element shape on p.u.l. resistance and inductance.

Fig. 3-25 Top view of GCPWs with different EPL individual element shape

111

Table 3.13 Per unit length R, L, G, and C values of a GCPW without and with EPL made of different individual element shape at 40 GHz

Table 3.14 α, ZO, and εeff of a GCPW without and with EPL made of different individual

element shape at 40 GHz

F. Varying the number and location of EPL columns

In this section, the number of columns in EPL varied from 1 to 3. The location of the EPL columns is also shifted with respect to the signal trace as well. The EPL thickness is fixed at 27 μm. Fig. 3-26 (a) and (b) show GCPWs with a 3 x 1 EPL unit cell. The EPL is located directly under the signal trace in Fig. 3-26 (a) while there is no overlap between the signal trace and EPL in Fig. 3-26 (b). Fig. 3-26 (c) and (d) show

GCPWs with a 3 x 2 EPL unit cell. In Fig. 3-26 (c), one of the EPL columns lies directly underneath the signal trace while the other one is offset from the signal trace along y axis.

In Fig. 3-26 (d), both the EPL columns are not located directly under the signal trace. Fig.

3-26 (e) shows a GCPW with a 3 x 3 EPL unit cell. It is essentially a combination of the

3 x 1 and 3 x 2 EPL unit cell shown in Fig. 3-26 (a) and Fig. 3-26 (d), respectively.

112

(a) (b)

(e) Fig. 3-26 Top view of GCPWs with a (a) 3 x 1, (b) offset 3 x 1, (c) 3 x 2, (d) offset 3 x 2, and (e) 3 x 3 EPL unit cell

From rows 2 and 3 of Table 3.15, it is clear that the effect of EPL is more pronounced when it is located directly under the signal. This is due to the better coupling between the

113 signal trace and the EPL column directly beneath it. The effect of EPL is increased as more columns are added, see rows 2, 4, and 6 of Table 3.15. From rows 2 and 5 of Table

3.15, it is seen that a single column of EPL placed under the signal trace can have a larger effect on the transmission line properties compared to a GCPW with 2 EPL columns, where the columns are not located directly under the signal trace. So, the correct placement of the EPL is more important than the number of columns for the EPL to have a bigger impact on transmission line characteristics. Table 3.16 shows the attenuation coefficient, characteristic impedance and effective permittivity associated with these cases. In the following section, the effect of EPLs on different planar transmission lines is briefly analyzed.

Table 3.15 Per unit length R, L, G, and C values of a GCPW without and with EPL with varying number of columns and column location at 40 GHz

Table 3.16 α, ZO, and εeff of a GCPW without and with EPL with varying number of columns and column location at 40 GHz

114

3.5 Comparison of the effect of EPL on different types of transmission lines

EPLs in a GCPW reduce the characteristic impedance and increase the effective dielectric constant of the GCPW. This section will compare the percent change in characteristic impedance and effective dielectric constant for different type of transmission lines due to EPLs. The three transmission line types selected are GCPW, microstrip and stripline. There are four cases associated with the GCPW as shown in Fig.

3-27. A GCPW without EPL is the nominal case and is represented by GC-nominal. GC- w-EPL involves a GCPW with EPL between the signal trace and the lower ground plane.

GC-w-FM has floating metals in the gaps between the signal trace and the coplanar side grounds. GC-w-EPLnFM is a combination of GC-w-EPL and GC-w-FM, where EPL is present not only between the signal trace and the lower ground plane but also between the signal trace and the coplanar side grounds. There are two cases associated with the microstrip line, where the nominal case is represented by MS-nominal. A microstrip line with EPL is represented through MS-w-EPL. The cross-section and the associated dimensions are shown in Fig. 3-28. A stripline without EPL is the nominal case and is represented by SL-nominal. A stripline with EPL only below the signal trace is represented by SL-w-1EPL. In SL-w-2EPLs, EPL is present both above and below the signal trace in the stripline. The cross-section for these three cases are shown in Fig. 3-29.

In all the 9 cases mentioned so far, LEPL and PXEPL are 150 μm and 195 μm, respectively.

115

GC-nominal GC-w-EPL

GC-w-FM GC-w-EPLnFM Fig. 3-27 Images of cross-section of GCPWs without and with different EPL arrangement

MS-nominal MS-w-EPL Fig. 3-28 Images of cross-section of microstrip transmission lines without and with EPL

SL-nominal SL-w-1EPL

SL-w-2EPLs Fig. 3-29 Images of cross-section of striplines without and with different EPL arrangement

Table 3.17 below shows the percentage change in impedance and effective dielectric constant due to EPLs in different transmission line cases described above. The changes in the impedance and dielectric constant is relative to the nominal case. The EPL changes impedance and effective permittivity in all three transmission lines as expected.

116

The largest and smallest increment in effective dielectric constant among all the listed cases is observed in microstrip line (MS-w-EPL) and stripline (SL-w-1EPL), respectively. GC-w-EPLnFM and GC-w-FM represent the largest and smallest decrement in characteristic impedance, respectively, among all the listed cases.

Table 3.17 Percentage change in characteristic impedance and effective permittivity of different transmission line types due to EPLs at 40 GHz

3.6 Summary

In this chapter, the transmission line RLGC model was briefly discussed. A modified transmission line RLGC model was also developed to account for the effect of floating metals on transmission line properties. The per unit length (p.u.l.) resistance, capacitance and conductance of a transmission line increases due to floating metals while the p.u.l. inductance is reduced. The change in resistance and inductance is dependent on the self-resistance and inductance of the floating metal and the mutual inductance between the signal trace and the floating metal. A brief discussion on how floating metal physical dimension impacts its self-resistance and inductance is also provided. Next a parametric study of embedded patterned layer (EPL), which is a floating patterned metal layer located between the signal trace of a transmission line and it’s return path, was

117 performed to see how it effects GCPW characteristics. The observed effects are listed below in Table 3.18. According to the parametric study, the effect of EPL on GCPW is most pronounced when thickness of the EPL (TEPL) is high, the separation between the signal trace and EPL (HEPL) is low and the pitch between the individual elements that constitute the EPL is small. The shape of the individual elements do not have significant impact on EPL performance as long as they have the same area. Increasing the number of

EPL columns increases the effect of EPL on GCPWs but placement of EPL with respect to the signal trace is very important in order to have the biggest impact on the transmission line. Three different planar transmission lines were analyzed with EPL and all of them showed similar trends with varying degree of influence.

Table 3.18 Summary of the effect of EPL parameters on GCPW p.u.l. R, L, G, C parameters, attenuation coefficient, characteristic impedance, and effective permittivity

For short transmission lines, the effect of increased attenuation coefficient due to EPL is not that drastic. The EPL has the ability to lower the characteristic impedance of the transmission line. So, a high impedance transmission line can be designed and its actual impedance can be brought down to the desired level using EPL. High impedance transmission lines have narrower signal trace width which provides the benefit of either routing more lines on a fixed board area or routing the same number of lines with reduced crosstalk (since separation between two adjacent signal traces is increased) on the same

118 board. The EPL also increases the effective dielectric constant of the transmission line which must be taken into account when designing transmission line circuits.

119

CHAPTER 4: EMBEDDED PATTERNED LAYER (EPL) IN A THREE CONDUCTOR GROUNDED COPLANAR WAVEGUIDE SYSTEM

4.1 Introduction

In the previous chapter, the effect of EPL on a two conductor (a signal trace and a return path) grounded coplanar waveguide was analyzed. However the ever increasing need for high speed data and signal transmission requires the use of multi-conductor transmission lines. Transmission lines with more than two conductors fall under the multi-conductor transmission line category [88]. In this chapter, the effect of EPL on a three conductor grounded coplanar waveguide (GCPW) will be analyzed. The three conductors include the return path and two signal traces as shown in Fig. 4-1. The structure shown in Fig. 4-1 is called grounded conductor backed edge coupled coplanar waveguide. The variable SP is the separation between the two signal traces. The definition for rest of the variables in Fig. 4-1 can be found in the previous chapter. The values of the different parameters shown in Fig. 4-1 are tabulated in Table 4.1. The dielectric and the metal are Megtron6 and copper, respectively. This chapter presents some basic concepts regarding lossless coupled transmission lines like crosstalk and different modes of signal propagation. The effect of EPL on per unit length self- capacitance, self-inductance, mutual capacitance and mutual inductance, modal impedances, modal velocities, and crosstalk of a grounded conductor backed edge coupled coplanar waveguide is presented. Also, the effect of EPL on single ended and differential signaling is shown via frequency domain simulations. Finally, the effect of asymmetrically positioned EPL on a differential transmission line is presented.

120

(a)

(b) Fig. 4.1 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide

Table 4.1 List of variable dimensions associated with conductor backed edge coupled coplanar waveguide Variable Size (μm) Variable Size (μm) Variable Size (μm) G 101.60 S 101.60 SP 101.60 SGW 800.00 L 6197.60 h 101.60 VR 76.20 VP 254.00 VL 330.20 T 17.78 ES 177.80

Any two lossless coupled transmission line can be represented using per unit length (p.u.l.) inductance and capacitance as shown in Fig. 4-2. The parameters i1 and i2 represent the current flowing through Line 1 and Line 2, respectively. The variables v1 and v2 represent the voltage between Line 1 and ground and between Line 2 and ground, respectively. Cg represents the per unit length capacitance between Line 1/Line 2 and ground while LO represents the p.u.l. self-inductance of Line 1 and Line 2. Variables CM

121 and LM represent the p.u.l. mutual capacitance and inductance, respectively between Line

1 and Line 2.

Fig. 4-2 Lumped element model of two lossless coupled transmission line

The two transmission lines interact with each other via their mutual inductance and capacitance. This interaction in turn couples energy from one transmission line to another, when any time varying signal is transmitted, resulting in crosstalk (XT).

Transmission line characteristic impedance and signal propagation speed change due to crosstalk. These in turn affect signal integrity and timing. Noise margin is also reduced because of noise coupling to adjacent transmission lines due to crosstalk. Any time varying current (i1) in Line 1 will generate voltage noise voltage (ΔvL) in Line 2 via mutual inductance according to 4.1. Similarly any time varying voltage (v1) in Line 1 will generate current noise (ΔiL) in Line 2 via mutual capacitance according to 4.2.

푑푖1 ∆푣퐿 = 퐿푀 ⁄푑푡 (4.1)

푑푣1 ∆푖퐿 = 퐶푀 ⁄푑푡 (4.2)

The characteristic impedance and propagation speed is dependent on the signal switching patterns in the two lines because the effective capacitance and inductance change with the signal switching pattern. This can be divided into three modes as described below.

122

A. Quiet mode

In this case, one of the lines is switching from either low to high or from high to low while the signal in the second line is holding its state (i.e., not switching). In this situation the characteristic impedance (ZOQ) and propagation speed (VPQ) are calculated using the following equations [25].

푍푂푄 = √퐿푂⁄(퐶푔 + 퐶푀) (4.3)

푉푃푄 = √1⁄{퐿푂(퐶푔 + 퐶푀)} (4.4)

B. Even mode

In this case, the signals in both the lines are switching from either low to high or from high to low at the same time (i.e., excited by similar signals). The electric and magnetic field distribution in case of a coupled transmission line during even mode signaling is shown in Fig. 4-3. In this situation the characteristic impedance (ZOE) and propagation speed (VPE) are calculated using the following equations [25], [3], [80].

푍푂퐸 = √(퐿푂 + 퐿푀)⁄퐶푔 (4.5)

푉푃퐸 = √1⁄{(퐿푂 + 퐿푀)퐶푔} (4.6)

(a) (b) Fig. 4-3 Even mode (a) electric, and (b) magnetic field pattern in a coupled transmission line [25]

123

C. Odd mode

In this case, the signals in both the lines are out of phase by 180o (i.e., excited by signals of opposing polarity). The electric and magnetic field distribution in case of a coupled transmission line during odd mode signaling is shown in Fig. 4-4. In this situation the characteristic impedance (ZOO) and propagation speed (VPO) are calculated using the following equations [25], [3], [80].

푍푂푂 = √(퐿푂 − 퐿푀)⁄{퐶푔 + (2퐶푀)} (4.7)

푉푃푂 = √1⁄[(퐿푂 + 퐿푀){퐶푔 + (2퐶푀)}] (4.8)

(a) (b) Fig. 4-4 Odd mode (a) electric, and (b) magnetic field pattern in a coupled transmission line [25]

The unwanted transfer of signal from one transmission line to another results in crosstalk. The transmission line that acts as the source of the noise is called the aggressor while the line in which the noise is generated is called the victim. The noise generated in the victim trace can be divided into near-end crosstalk (NEXT) and far-end crosstalk

(FEXT), respectively. Fig. 4-5 shows the general NEXT and FEXT waveform signature.

Far-end is the direction of forward propagation of the signal in the aggressor trace while near-end is the backward direction to the signal propagation in the aggressor line. In case the far-end of the aggressor and victim line and the near-end of the victim line are

124 properly terminated so that there is no reflection, the NEXT and FEXT can be calculated using the following equations [3], [80].

푉푏 퐶푀 퐿푀 푁퐸푋푇 = ⁄ = 푘푏 = 0.25 [{ ⁄ } + ( ⁄ )] (4.9) 푉푎 (퐶푂 + 퐶푀) 퐿푂

푉푓 푙푘푓 푙 퐶 퐿 퐹퐸푋푇 = ⁄ = ⁄ = [{ 푀⁄ } − ( 푀⁄ )] (4.10) 푉푎 푅푇 2푉푃푄푅푇 (퐶푂 + 퐶푀) 퐿푂

In 4.9, Vb and Va represent the voltage at the near-end of the victim trace and voltage on the aggressor, respectively. In 4.10, the variables Vf, RT, and l represent the voltage at the far-end of the victim trace, rise time of the signal in the aggressor line, and the length of the transmission lines, respectively. 4.9 and 4.10 are valid when the rise time is at least twice the line delay.

Fig. 4-5 General signature of saturated FEXT and NEXT voltage noise

The topics discussed so far are related to single ended signaling scheme. In a single ended signaling scheme, a single transmission line is dedicated to transmitting each data bit. However, single ended signaling is good only up to 2 Gbps and for higher data rates differential signaling is used. Differential signaling uses a pair of transmission lines for each data bit. However, these two transmission lines are driven 180o out of

125 phase, which helps removing the common mode noise. This is similar to the odd mode of propagation. The differential impedance (ZDiff) can be calculated from the odd mode impedance using the following equation [25], [3], [80].

푍퐷푖푓푓. = 2푍푂푂 (4.11)

From the previous chapter it is known that EPL will impact the p.u.l. capacitance and inductance of the coupled transmission lines, which in turn will affect the even and odd mode impedances, differential impedance, near-end crosstalk, and far-end crosstalk.

The following section discusses the impact of EPL on grounded conductor backed edge coupled coplanar waveguide. The simulations are performed using Q3D and HFSS [62].

4.2 Impact of EPL on coupled grounded coplanar waveguides

In this section, the effect of EPL on grounded conductor backed edge coupled coplanar waveguide modal impedances, modal velocities and crosstalk will be discussed.

The dimensions associated with the coupled GCPW are provided in Section 4.1. The different dimensions associated with the EPL, see Fig. 3-2, Fig. 4-6, and Fig. 4-7, are provided in Table 4.2. These values do not change in this section unless specified.

(a) (b) Fig. 4-6 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with a 1 column EPL

126

(a) (b) Fig. 4-7 Image of (a) top, and (b) cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with a 3 column EPL

Table 4.2 List of variable dimensions associated with EPL Variable Size (μm) Variable Size (μm) Variable Size (μm) LEPL 128.00 HEPL 25.40 PXEPL 168.60 WEPL 127.00 TEPL 27.00 PYEPL 203.20

The addition of the EPL increases the capacitance and reduces the self-inductance associated with the individual signal traces. This trend is expected from results in Chapter

3. Increasing the number of columns in the EPL increases the effect as well. The mutual capacitance and mutual inductance between the two signal traces increase and decrease, respectively, with EPL. Table 4.3 shows the p.u.l. signal trace capacitance, mutual capacitance between the signal traces, self-inductance of a signal trace, and mutual inductance between the signal traces of the grounded conductor backed edge coupled coplanar waveguide with and without EPL at 40 GHz. Since the transmission line under consideration is symmetric, the values of p.u.l. self-inductance and capacitance for a single trace are provided. This increasing capacitance and decreasing inductance values, due to EPL, in turn reduces the line impedances and propagation speeds associated with coupled transmission line system. The impedance and propagation speeds associated with the different modes are listed in Table 4.4 and Table 4.5, respectively. EPL increases and decreases NEXT and FEXT, respectively (Table 4.6). However, the percentage change in

NEXT and FEXT falls as the number of columns in EPL is increased. This can be

127 attributed to the reduction in mutual capacitance between the two signal traces as the number of columns in EPL is increased. This effect of EPL on the mutual capacitance should be investigated in future work. The percentage changes shown in Table 4.3 – 4.6 are calculated using 3.38.

Table 4.3 Per unit length Cg, CM, LO, and LM values of a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz

Table 4.4 Different modal impedances associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz

Table 4.5 Different modal propagation velocities associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz

Table 4.6 NEXT and FEXT associated with a grounded conductor backed edge coupled coplanar waveguide without and with EPL at 40 GHz

Fig. 4-8 shows the effect of EPLs on the S-parameters associated with the signal traces when they are used for single ended signaling. The effect of EPL on FEXT and NEXT are consistent with the findings in Table 4.6 when the coupled transmission line structure

128 is analyzed in frequency domain. The S-parameter data associated with ports renormalized to 50 Ω can be found in Appendix IV.

(a)

(b)

129

(c)

(d) Fig. 4-8 Plot of (a) insertion loss, (b) return loss, (c) NEXT, and (d) FEXT versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to line impedance

The differential insertion loss increases with EPL when the coupled GCPW is used for differential signaling, see Fig. 4-9. The use of EPL reduces the differential to

130 common mode signal conversion (SCD21) and this effect should be analyzed further in future work. The mixed mode S-parameter data associated with ports renormalized to 100

Ω differential impedance can be found in Appendix IV.

(a)

(b)

131

(c) Fig. 4-9 Plot of (a) differential insertion loss (SDD21), (b) differential return loss (SDD11), and (c) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to line impedance

In differential signaling, all the information is carried by two signals of opposite polarity propagating through two signal traces. Anything that makes one of the signal traces electrically different from the second signal trace of the differential pair will result in differential to common mode conversion. SCD21 represents the amount of common mode signal received at port 2 when a differential signal is injected into port 1. This differential to common mode conversion results in reduced differential signal amplitude and causes electromagnetic interference (EMI) problems. While EPL reduces SCD21 as shown in Fig.

4-9 (c), asymmetric EPL placement can also result in increasing SCD21. The asymmetric

EPL placement, see Fig. 4-10, will make Line 1 look electrically longer than Line 2 and will also result in the two lines having different impedances. In Fig. 4-10, an EPL with 2 columns is placed under Line 1 while there is no EPL under Line 2. Fig. 4-11 shows the impact of asymmetric EPL placement on grounded conductor backed edge coupled

132 coplanar waveguide differential insertion loss and SCD21. The EPL is located 25.4 μm below the signal trace. Individual EPL element are rectangular in shape and are 128 μm in length (x-axis) and 81.6 μm in width (y-axis). The EPL thickness is 27 μm. The pitch between the individual elements is 168.6 μm and 101.6 μm along x-axis and y-axis, respectively. In the following section, the findings of this chapter are summarized.

Fig. 4.10 Image of cross-sectional view of a grounded conductor backed edge coupled coplanar waveguide with an asymmetric EPL placement

(a)

133

(b) Fig. 4-11 Plot of (a) differential insertion loss (SDD21), and (b) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with asymmetric EPL placement, with ports renormalized to 100 Ω differential impedance

4.3 Summary

This chapter provides a very brief overview of the lossless coupled transmission lines involving two signal traces and a return plane. The lumped element model of a lossless coupled line was discussed and the different modes of signal propagation in a coupled line was reviewed. The effects of EPL on a grounded conductor backed edge coupled coplanar waveguide was also presented. EPL increases the signal trace capacitance and mutual capacitance between the two signal traces. The self-inductance of the signal trace and the mutual inductance between the coupled traces are reduced due to

EPL. These changes in the capacitance and inductance results in reduction of different modal impedances and signal propagation speed. EPL increases the NEXT and reduces the FEXT between the two signal traces when the coupled lines are used for single ended signaling. In case the coupled transmission lines are used for differential signaling, the

134 differential insertion loss increases when EPL is used however the EPL reduces the differential to common mode signal conversion when the EPL is symmetrically placed under the coupled transmission lines. It was also shown that asymmetrically located EPL will increase the differential to common mode signal conversion which in turn increases differential insertion loss and can cause EMI issues. A 110 Ω grounded conductor backed edge coupled coplanar waveguide uses 19% less board area when compared to a 89 Ω grounded conductor backed edge coupled coplanar waveguide. Also, a grounded conductor backed edge coupled coplanar waveguide with 110 Ω differential impedance with EPL can be used to reduce the impedance to 89 Ω. This means for this case, EPL has the potential to reduce the coupled line footprint by approximately 19%. The work done so far involved transmission lines with smooth conductors. However in practice, the conductor surface is roughened to improve the adhesion between the conductor and the dielectric. The following chapter develops a statistical method for modeling conductor surface roughness using 3D electromagnetic full wave simulator like HFSSTM. For the sake of simplicity, only CB-CPW without via and EPL will be analyzed.

135

CHAPTER 5: BROADBAND CHARACTERIZATION OF COPLANAR WAVEGUIDE INTERCONNECTS WITH ROUGH CONDUCTOR SURFACES

5.1 Introduction

The geometric dimensions, of transmission lines (interconnects), are shrinking as engineers design smaller and faster systems by moving into the high frequency regime.

Transmission lines operating at high frequencies see an increase in resistive losses that can adversely affect the performance of the electrical system. For smooth conductors, the series resistance increase with frequency is attributed to the skin depth, which is inversely proportional to the square root of frequency. In practice, the conductor surfaces are roughened by the manufacturers to promote adhesion between the dielectric and conductors in the PCBs. Fig. 5-1 shows the cross-sectional image of a rough copper trace over a rough ground plane [89]. The 90o peel strength (Fig. 5-2) is a measure of how well a conductor adheres to the dielectric material and is directly proportional to the fourth root of the thickness of deformed resin (yo) and other parameters like thickness, etc. [90].

Intentional roughening of the surface between the copper foil and the dielectric improves peel strength and adhesion between the metal and the dielectric. Conductor surface roughness increases resistive losses at higher frequencies, affecting signal integrity. The skin depth decreases with increasing signal frequency and becomes comparable to the localized peaks of the roughened conductor surface. The current follows the surface undulations of the rough conductor, which increases the effective path length and reduces the cross-sectional area of current flow, resulting in increased resistance. Experimental results show that rough conductors exhibit 10%-50% higher losses compared to that of

136 the smooth conductors [80]. High conductor surface roughness also increases dispersion and effective dielectric constant by up to 15% [91].

Fig. 5-1 Cross-sectional image of a rough conductor over ground plane [89]

Fig. 5-2 3-D model of a 90o peel strength test setup

This chapter focuses on creating a statistical way for designers to create frequency domain models and to quantify these effects in their designs. A traditional way to account for the rough conductor losses in transmission lines is through the use of the Hammerstad equation

137

푅 = 퐾퐻푅푆√푓 (5.1) where 푅푆√푓 is the skin depth resistance for a smooth conductor, and 퐾퐻is called the

Hammerstad coefficient given by

2 퐻 2 퐾 = 1 + 푎푟푐푡푎푛 [1.4 ( 푟푚푠) ] (5.2) 퐻 휋 훿 where 훿 is the skin depth, and 퐻푟푚푠 is the root mean square value of the surface roughness height [92], [25]. The Hammerstad model is not applicable when operating frequencies are greater than 5 GHz because it is based on a 2-D corrugated surface (Fig.

5-3) for the copper foil and should be used for conductor foils with 퐻푟푚푠 ≤ 2 μm [25]. To fill this void, a hemispherical model (Fig. 5-4) was introduced with a new correction factor 퐾퐻푒푚푖 that replaces 퐾퐻 in (5.1). The red dotted lines in Fig. 5-3 and 5-4 (a) show the current path in the conductor. The hemispherical model levels off at a much higher value compared to the Hammerstad model, which always levels off at a value of two. The hemispherical model is an improvement over the Hammerstad model, but it still over and under predicts loss at middle and high frequencies, respectively, and it is valid for very specific conductor profiles [89], [25]. Scanning electron microscope (SEM) images (Fig.

5-5) show pyramid like structures formed by electrodeposited (ED) copper spheres

(snowballs) [89], [92] and [25].

Fig. 5-3 Diagram of 2-D corrugated copper surface for Hammerstad model [25]

138

(a) (b) Fig. 5-4 Hemispherical model shown as a single surface protrusion with (a) top and (b) side views [25]

Fig. 5-5 Magnified (5000x) SEM photograph of a copper surface at an angle of 30o [25]

The Huray model (Fig. 5-6) was introduced after the hemispherical model. This model includes the Huray surface roughness correction factor (퐾퐻푢푟푎푦), which replaces

퐾퐻 in (5.1). Such a structure is extremely difficult to model for simulation in 3-D electromagnetic full wave solvers because the SEM images need to be analyzed very closely to determine snowball dimensions and the number of spheres in each protrusion.

Such data is not readily available from the manufacturers. Fig. 5-7 shows the variation

139 of 퐾퐻, 퐾퐻푒푚푖 and 퐾퐻푢푟푎푦 with frequency. The effects of conductor random rough surface on attenuation have been studied for 2-D and 3-D cases using the small perturbation method second order (SPM2) and the method of moments (MoM) [93], [94]. These methods rely on a 2-D surface model that has limited accuracy since the power absorption ratio increases significantly in 3-D configurations. The SPM2 applies to surfaces with small levels of roughness [95]. The authors in [95] have developed a scalar- wave model (SWM) to simulate the effect of 3-D surface roughness over diverse roughness profiles over a wide frequency band. Much of the prior work is focused on microstrip transmission lines with specific rough surface conductor profiles and relies on simulation software that in not broadly available in the industry. There is no published technique that discusses simulating the effect of conductor surface roughness on different types of transmission lines using 3-D full wave electromagnetic solvers.

Fig. 5-6 Model cross-section of copper spheres which form the pyramid like structures on flat copper conductors [25]

This chapter presents a general way of simulating the effects of conductor surface roughness for CB-CPW using HFSSTM [96]. A practical CPW on a dielectric substrate consists of a central conductor (signal) with finite width ground planes on either side.

CB-CPW has an additional ground plane at the bottom surface of the substrate as shown in Fig. 5-8. The CB-CPW supports a low dispersion quasi-TEM mode of propagation

140

[18]. The conductor backing below the CPW adds additional shielding for interconnects that may be routed below the CB-CPW in the case of multilayer modules. CB-CPWs simplify fabrication by facilitating easy surface mounting or flip-chip attachment of active and passive devices. CB-CPW allows the use of automatic on wafer measurements for interconnection of components and characterization. CB-CPW circuits have a ground plane between two adjacent lines reducing crosstalk. The method presented in this chapter allows the analysis of conductor roughness effects on CB-CPWs for different roughness parameters and validates the simulation results with published results in literature. This methodology is applicable to other rough surface profiles and different types of transmission lines. The advantage of this approach is that designers are often familiar with interconnect modeling of smooth lines in HFSSTM, and thus existing design cases may be utilized to include the effects of surface roughness.

Fig. 5-7 Plot of 퐾퐻, 퐾퐻푒푚푖 and 퐾퐻푢푟푎푦 versus frequency [25]

The results presented here are based on CB-CPW on copper clad Megtron6. The dielectric thickness is 254 μm, the width of the signal trace is 254 μm, and the width of the gap between signal and the coplanar ground traces is 58.42 μm. The thickness and conductivity of copper is 17.78 μm and 5.8E+07 S/m, respectively. The relative

141 permittivity of the substrate is 3.5. In this work, no vias between the top and lower ground conductors were used. Simulations were run to verify that only the CB-CPW mode propagates all the way up to 40 GHz. No unusual mode behavior, such as even or odd propagating modes or signal loss, was observed.

Fig. 5-8 3-D representation of a conductor backed coplanar waveguide

The surface roughness modeling approach is based upon implementing the characteristics of the rough surface model directly into full wave electromagnetic simulators, such as HFSSTM. HFSSTM solves Maxwell’s equations directly using the finite element method (FEM). HFSSTM has been used to compute the electric field scattering off rough dielectric surfaces for calculating the scattering and emissivity of such dielectric mediums [97]. The HFSSTM results are in close agreement with the MoM, predicted and published measured results to 40 GHz.

A method of generating rough conductor surfaces with desired root mean square height, correlation length, and surface correlation function is discussed and implemented.

This technique allows for the simulation of the effects of surface roughness on different types of transmission lines. Previous studies provide insight into the variation of either insertion loss or attenuation coefficient (or enhancement factor) with frequency. This new method provides access to both parameters simultaneously for up to 40 GHz.

Interconnect performance can be better understood via field plots in the interconnects. All

142 these simulations were performed on an Intel Xeon system with 12 cores distributed across two physical processors and 256 GB of random access memory (RAM). The operating system installed is Windows Server 2008 R2 enterprise. The following section deals with the different properties of random rough surfaces.

5.2 Properties of random rough surfaces

A random rough surface is modeled as a stationary random process with zero mean and characterized by the root mean square height (퐻푟푚푠), correlation length (λ), and autocorrelation function (ACF) [94], [98]. While two rough surfaces can have the same statistical parameters, the physical profiles are not necessarily the same. The height probability distribution function (PDF) of a rough surface is assumed to be a Gaussian distribution with zero mean and 퐻푟푚푠 as the standard deviation. Additional details are described in [98] and [99].

Copper foils used as conductors in transmission lines can be broadly classified into three types: rolled annealed (RA), reverse treated (RT), and ED. 퐻푟푚푠 ranges from

0.4 to 0.5 μm, 0.5 to 0.7 μm, and 1 to 4 μm for RA, RT, and ED foils, respectively. The

ED foils can be hyper very low profile (1.5 μm ≤ 퐻푟푚푠 ≤ 2 μm) and very low profile (3

μm ≤ 퐻푟푚푠 ≤ 4 μm) [91], [100]. This is consistent with the findings in [92], which says,

퐻푟푚푠 varies from 0.3 to 5.8 μm. The correlation length is defined as the distance over which ACF falls by 1⁄푒. λ varies between 0.3 and 3.5 μm [93], [95]. Fig. 5-9 shows SEM images of copper and dielectric with different types of copper foils [91], [101]. Fig. 5-10 shows a 2500 time magnified SEM photograph of ED and RA copper foils with 퐻푟푚푠 3 and 0.5 μm, respectively [91]. Fig. 5-11 shows the 3-D surface profile measurement of a rough copper foil [92].

143

(a) (b)

(c) (d) Fig. 5-9 SEM images of copper and dielectric with (a) 1 oz. electrodeposited, (b) 0.5 oz. electrodeposited, (c) 1 oz. rolled, and (d) 0.5 oz. rolled copper foils [101].

(a) (b) Fig. 5-10 Magnified (2500x) SEM photographs of (a) electrodeposited and (b) rolled

annealed copper foils with 퐻푟푚푠 3 and 0.5 μm, respectively [91]

144

Fig. 5-11 3-D surface profile measurement of a copper foil [92]

Different random surfaces are distinguished by their ACFs, which determine the extent of roughness of the surface. For conductor surfaces, the ACFs are either Gaussian or exponential in nature. Fig. 5-12 (a), (b), and (c) show 1-D Gaussian rough surfaces with unity 퐻푟푚푠 and varying λ. The exponential surface correlation function is rougher than the Gaussian ACF for a given 퐻푟푚푠 and λ [93], see Fig. 5-13. The random rough surfaces are assumed to isotropic, stationary (translational invariance), and ergodic. A rough surface is defined as isotropic if the statistical nature of the surface is independent of the direction along the surface [98]. The isotropic nature of the surface results in λ being equal in two perpendicular directions. Theoretically, given 퐻푟푚푠, λ, and ACF, a large number of unique rough surfaces can be created. Consider that the direction of propagation of signal is along the x-axis. A normalized Gaussian correlation function is represented as in [93]

푥2+푦2 퐶(푥, 푦) = 푒푥푝 (− ). (5.3) 휆2

The Fourier transform of an ACF results in the power spectral density (PSD) function, which is another way of describing the rough surface in the spatial domain. In the case of the Gaussian ACF, the PSD is also Gaussian, given by [93]

145

퐻2 휆2 (푘2+푘2)휆2 푃(푘 , 푘 ) = 푟푚푠 exp (− 푥 푦 ). (5.4) 푥 푦 4휋 4

The exponential ACF and PSD are represented as follows [93]:

√푥2+푦2 퐶(푥, 푦) = exp (− ) (5.5) 휆

퐻2 휆2 1 푃(푘 , 푘 ) = 푟푚푠 (5.6) 푥 푦 2휋 2 2 2 3⁄2 {1+(푘푥+푘푦)휆 }

This chapter considers the Gaussian surface correlation function. To capture the behavior of the exponentially correlated rough surface, the sampling interval should be at least one-tenth of, which is too intensive a simulation to handle for the time being [98].

(a)

(b)

(c)

Fig. 5-12 1-D Gaussian rough surface with unity 퐻푟푚푠 and (a) λ = 0.1 mm, (b) λ = 0.3 mm, and (c) λ = 1 mm

(a)

146

(b)

Fig. 5-13 1-D rough surface with (a) Gaussian and (b) exponential ACFs with unity 퐻푟푚푠 and λ = 0.1 mm

5.3 Generating a random rough surface

A randomly roughened 3-D surface with Gaussian height distribution and any given surface correlation function can be generated via the linear transformation of matrices, whose components are Gaussian random numbers with zero mean and a standard deviation of one [102]. To use this, one needs to numerically solve a set of simultaneous nonlinear equations for determining the transform coefficients. A much more efficient method of generating 3-D rough surfaces with various ACFs is using the techniques of digital filter design [103]. This method uses an independent Gaussian distribution of random numbers of zero mean and unity standard deviation, that is generated using the random number generator functions built in MATLAB [104], for a mesh of points on the X-Y surfaces. This vector is convolved with the Gaussian ACF and then normalized and multiplied with 퐻푟푚푠 to obtain the random surface heights with desired surface characteristics [103], [105]. This process is given by

∞ 푧(푟) = 퐻 [푛표푟푚푎푙푖푧푒푑{ 퐶(푟 − 푟′)푍(푟′)푑푟′}] (5.7) 푟푚푠 ∬−∞ where 푧(푟) represents the random surface heights, 푟 = (푥, 푦) and 푍(푟) is the set of random numbers. Equation (5.7) can also be solved using fast Fourier transform and the

Gaussian PSD is used instead of the Gaussian ACF. A random rough surface with

147

퐻푟푚푠 = 5.5 μm and λ = 3 μm is generated via the above process and is shown in Fig. 5-

14.

(a)

(b) Fig. 5-14 (a) 3-D surface plot and (b) contour plot of a randomly generated rough surface

with 퐻푟푚푠 = 5.5 and λ = 3 μm.

The shape on the rough surface on the copper foil obtained from surface profile measurements and SEM (from [89] and [92]) have a conical shape. Thus, the model in

148 this chapter uses varying heights of the rough surface as cones of randomly varying heights and diameters. 푧(푟)obtained from (5.7) sets the cone heights, while the radii are set using another set of random numbers generated in MATLAB. The random number data are stored in an Excel sheet along with the coordinates for the cones. A visual basic

(VB) script is used to create a 50 Ω CB-CPW geometry with smooth conductor surfaces that can be imported into HFSSTM. The VB script reads the Excel file to modify the geometry to include surface roughness. The file includes the data of the randomly generated cones for the conductors on the signal and grounds. The script unites the cones to the smooth conductor surfaces. The script also ensures that in the overall geometry file a portion of the dielectric is subtracted (or added) to create a continuous interface that mimics the roughness profile between the substrate and the conductor. Special care was taken to appropriately set the zero mean to the mountains and valleys in the roughened profile. The interface between the conductor and substrate is roughened, while the interface between the conductor and air is kept smooth. A small segment of CB-CPW with smooth conductors is added to each end in order to prevent non-uniform, non-planar surfaces from touching the waveports, which is necessary for a waveport excitation in

HFSSTM. A comparison of the smooth to rough conductor surface generated by the above process is shown in Fig. 5-15. Only the surface of the conductor making contact with the substrate is roughened. Since the roughened conductor geometry presents a complex tetrahedral meshing challenge, a study was conducted to investigate the impact of how limiting the area of roughened conductors in the model affects the accuracy and computational load, which is discussed in the following section.

149

(a)

(b) Fig. 5-15 Signal conductor with (a) smooth and (b) randomly roughened surface

5.4 Reducing computational complexity

An important aspect of this work is to reduce the computational complexity and simulation time. For this purpose, different test cases are simulated with varying extent of roughness on the conductors, see Table 5.1. The rough conductor surfaces have a

Gaussian surface correlation function. The attenuation coefficients of these simulations were used to decide on the structure that will provide the desired result in a short time.

The CB-CPW has four conductors namely signal, left coplanar side ground (LCSG), right coplanar side ground (RCSG) and lower ground plane (LGP). The signal, LCSG, and

RCSG are coplanar. The area closest to the signal is roughened first in all the cases 1-5.

The case called Smooth under the Simulations column indicates that all the metal surfaces are assumed to be smooth.

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Table 5.1 Percentage of conductor surface area roughened

Simulations Percentage of conductor surface area roughened Signal LCSG RCSG LGP Smooth 0.00 0.00 0.00 0.00 Case 1 100.00 0.00 0.00 12.93 Case 2 100.00 0.00 0.00 38.79 Case 3 100.00 31.89 0.00 0.00 Case 4 100.00 31.89 31.89 0.00 Case 5 100.00 31.89 31.89 12.93

Fig. 5-16 shows the attenuation coefficient for the six cases. The results show a huge jump in losses due to the conductor surface roughness over the smooth results. The attenuation remains roughly the same for Case 1 and Case 2. This means that the roughness of the ground conductor closest to the signal has the maximum impact on attenuation. The electric field near the signal are stronger and interact more strongly with the section of conductors lying close to signal compared to areas lying further away. Case

3 and Case 4 show that losses increase with increase in number of rough conductor surfaces. This leads to the practical situation Case 5, where all the conductors are roughened but the extent of roughness is limited to save simulation time and complexity.

The interconnect with the smooth conductor surfaces encounter lowest loss while Case 5 the highest.

Next the computational effort required to perform the simulations for Case 1 to

Case 5 is analyzed. Each case is further subdivided into four sub-categories namely memory (peak amount of physical memory used), simulation time, matrix size (number of unknowns) and number of tetrahedrons. Simulation complexity and time increases with the number of conductors with rough surface see Fig. 5-17. However, Case 2 results in huge increase in simulation time and complexity even though it does not represent a practical situation.

151

Fig. 5-16 Plot of attenuation coefficient versus frequency for the six simulation cases mentioned in Table 5.1

Fig. 5-17 Bar plot of memory, simulation time, matrix size and number of tetrahedrons for simulations mentioned in Table 5.1

5.5 Results and comparison

This section shows how 퐻푟푚푠 and λ influence the losses in CB-CPWs. The enhancement factor (k) is defined as the ratio of the attenuation coefficient of a rough surface to that of a smooth surface. The trends observed in the variation of the

152 enhancement factor and insertion loss match the results presented in [91], [92] and [93].

The enhancement factor neither saturates nor shows any trends of saturation at higher frequencies. The attenuation coefficient and insertion loss increases with increased conductor roughness.

In the first case, two interconnects with constant 퐻푟푚푠 (1 μm) with λ = 1 and 4

μm, respectively were simulated. The results are shown in Fig. 5-18. The results show that the attenuation coefficient falls with increasing correlation length. Note that increasing correlation length results in a smoother conductor surface profile. The enhancement factor for λ = 1 μm is higher than λ = 4 μm, indicating higher attenuation.

This suggests an increase in the roughness of the surface with decreasing λ. The insertion loss for both the cases is essentially the same, and suggests that λ alone may not be a dominating factor in determining the extent of attenuation.

(a)

153

(b)

(c) Fig. 5-18 Plot of (a) attenuation coefficient per unit length, (b) enhancement factor, and (c) insertion loss versus frequency with varying λ (1.0 and 4.0 μm) and constant

퐻푟푚푠 (1.0 μm) for a 7-in-long CB-CPW

Fig. 5-19 shows the results when 퐻푟푚푠 is varied from 1 to 5.5 μm with λ being held constant at 3 μm. The results show that for this case, the attenuation coefficient increases with increasing root mean square height. Increasing root mean square height

154 results in a rougher conductor profile. These figures show that increasing 퐻푟푚푠 creates a rougher surface and thus increases the resistive losses. While there does not seem to be a huge difference in the insertion loss, the proliferation of low-power electrical circuits is increasing, and this means even a slight variation in power can cause a shift in the logic state of the devices, resulting in unpredictable errors.

(a)

(b)

155

(c) insertion loss versus frequency for constant λ (3.0 μm) and varying 퐻푟푚푠 (1.0, 2.5, and 5.5 μm) for a 7-in-long CB-CPW

Fig. 5-20 shows the simulated results for the case where both 퐻푟푚푠 and λ are varied. The results suggest that 퐻푟푚푠 affects the results more than λ, thus deciding the extent of the resistive losses suffered by a transmission line. Increasing 퐻푟푚푠 and decreasing λ indicate an increase in the roughness profile of a surface. In this case, both the root mean square height and correlation length are varied at the same time by the same amount (퐻푟푚푠 = 휆). Attenuation coefficient and enhancement factor both increase with increasing 퐻푟푚푠 and λ as in Fig. 5-20 (a) and (b).

156

(a)

(b) Fig. 5-20 Plot of (a) attenuation coefficient per unit length and (b) enhancement factor versus frequency for varying λ and 퐻푟푚푠 (퐻푟푚푠 = λ; 0.5, 1.0, and 2.5 μm) for a 7-in-long CB-CPW

A review of these results in the context of the law of conservation of energy provides some additional insight. A current passing through a conductor results in a magnetic field (creating inductance) that surrounds the conductor, according to the right

157 hand rule. According to [92], the total inductance in a current-carrying conductor is the sum of the external and internal inductances denoted by 퐿푒푥푡 and 퐿푖푛푡, respectively. 퐿푖푛푡 and 퐿푒푥푡 are caused by current flowing inside and on the surface of a conductor, respectively. Total inductance is reduced by 퐿푖푛푡 at high frequencies as current in the conductor is confined to the surface. Consequently, the associated magnetic field has reduced energy storage capacity. Therefore, the excess energy that can no longer be stored in the magnetic field must be dissipated by resistive losses (skin effect), thus increasing attenuation at high frequencies.

In order to validate simulated results, a comparison with published and measured results was conducted. No studies that provided sufficient details on the material samples were found for CB-CPW. A detailed study that includes both measured and simulated insertion loss with and without surface roughness for microstrip transmission lines was used [91]. HFSSTM simulation models for 50 Ω microstrip lines on 100 μm thick Rogers

ULTRALAM 3850 LCP substrate using 18 μm thick copper lines with and without surface roughness were created. The models with rough conductor surfaces use the same methodology discussed in section 5.3. The substrate has a loss tangent and dielectric constant of 0.0025 and 2.9, respectively, at 10 GHz. Two cases were simulated in

HFSSTM. The first one is a microstrip line without any conductor roughness, and the second one is a microstrip line with conductor surface roughness (퐻푟푚푠= 3 μm and λ = 1

μm). The simulated and measured results are shown in Fig. 5-21. The plots with name

“HFSS-smooth Cu” and “HFSS-Simulated 퐻푟푚푠 = 3 μm foil” are the two test cases simulated while the plots “Sonnet-smooth Cu” and “Measured 퐻푟푚푠 = 3 μm foil” are from [91]. A good correlation between measured and simulated data within the limits of

158 experimental errors is observed. There is an average difference of 0.6% and 1.2% between the results computed by the methods discussed here and the comparison data, for the smooth and rough copper foils, respectively. Note, measured data on smooth foils are not available.

Fig. 5-21 Plot comparing of measured results from [91] with simulated results from HFSSTM

The next step is to look at the current distribution on the conductor surfaces. Fig.

5-22 shows how the current on the conductors follow the surface contours of the rough surfaces. Fig. 5-22 (a) shows the current distribution in the cross-sectional slice of a portion of the signal conductor and one coplanar side ground for the roughened conductor surfaces. Fig. 5-22 (b) shows the same view, but zoomed in. The current is flowing into the plane of the page (i.e., along the length of the transmission line). The figures clearly show that the interface between the conductor and the substrate is roughened. Both plots use the same linear scale. The figures even show the appropriate current density behavior consistent with the edge effect [106]. In practice, the lines are trapezoidal in cross section with the line width at top less than that at the bottom (metal substrate interface), since in

159 wet etching the etchant has more time to interact with metal at top, see Fig. 5-23. The edges of the transmission lines also exhibit roughness that is dependent upon the circuit fabrication process. These factors will also create higher insertion loss, which could account for some measurement and simulation differences in Fig. 5-21.

(a)

(b) Fig. 5-22 Cross-sectional view of current distribution in (a) portions of roughened signal and coplanar side ground and (b) zoomed in image of a portion of the roughened conductor in (a) at 40 GHz

Fig. 5-23 Cross-sectional view of CB-CPW with the sides of the conductors inclined at an angle θ

Fig. 5-24 shows how the electric field inside the conductor falls off due to the presence of rough structure when compared to the conductor with a smooth surface. The

160 variation in depth penetration of the electric field can be attributed to the randomness of the thickness of the conductors. Fig. 5-24 shows the electric field distribution in the cross section of the CB-CPW with smooth and rough conductor surfaces at 40 GHz. The electric field is plotted in the log scale. The electric field pattern in the dielectric and air; however, remains unperturbed even in the presence of the rough conductor surface.

(a)

(b) Fig. 5-24 Cross-sectional view of the electric field distribution in CB-CPW with (a) smooth and (b) rough conductor surfaces at 40GHz, in log scale

161

5.6 Computational requirements and model generation

This section discusses some of the factors involved in creating the HFSSTM models and a comparison of approaches used to reduce the computational complexity of the rather detailed rough surface geometries. First, a study of the FEM mesh created by

HFSSTM was conducted. Fig. 5-25 shows the FEM mesh at the cross section of the line.

Fig. 5-25 (a) shows the air box region surrounding the line, and the lower region is the

Megtron6 substrate sandwiched between the copper surfaces. Fig. 5-25 (a) shows the mesh generated at one cross section of the entire structure. The mesh is very dense in the areas where the conductors have been roughened. This region is shown with a white dotted circle in Fig. 5-25 (a). Fig. 5-25 (b) shows a zoomed in image of the mesh in the region around one of the coplanar side grounds and the signal, and in Fig. 5-25 (c) shows a further zoomed in image of the meshing in the region underneath the roughened signal.

(a)

162

(b)

(c) Fig. 5-25 Cross-sectional view of the mesh generated by HFSSTM; (a) Meshing in the entire cross-section of the interconnect, (b) meshing in the metal (signal and coplanar side ground) with a roughened surface, and (c) zoomed in image of the meshing at the signal conductor and substrate interface

Another interesting aspect of this chapter involves the computational power required to perform such simulations. All these simulations were performed on an Intel

Xeon system with 12 cores distributed across two physical processors and 256 GB of random access memory (RAM). The operating system installed is Windows Server 2008

R2 enterprise. Four cases were considered: Case 1 (퐻푟푚푠 = 휆 = 0.5 휇푚), Case 2

(퐻푟푚푠 = 휆 = 1.0 휇푚), Case 3 (퐻푟푚푠 = 휆 = 2.5 휇푚), and Case 4 (퐻푟푚푠 = 5.5 휇푚; 휆 =

3.0 휇푚). Each case is further sub divided into four categories, namely memory, total simulation time, matrix size, and number of tetrahedrons. It becomes clear from Fig. 5-26 that computational complexity and time increase with the increase in roughness of the surface.

163

Fig. 5-26 Bar plot for memory, simulation time, matrix size and number of tetrahedrons for CB-CPWs with varying extent of conductor surface roughness

5.7 Summary

A technique to design conductor surface roughness for CB-CPWs using 3-D full- wave electromagnetic solvers like HFSSTM was presented. The simulation results suggest an increase in attenuation and thus enhancement factor and insertion loss as the roughness of the conductors is increased. Unlike the Hammerstad model, the enhancement factor did not saturate at two; instead it continued to increase with increasing frequency for any given rough surface. 퐻푟푚푠 played a dominant role when compared to λ. The roughness of a surface increases as 퐻푟푚푠 increases and λ decreases.

The trends seen in the enhancement factor and insertion loss for the different cases followed the trends presented in the references. The simulations also helped provide a clear picture of the current distribution in the signal conductor. The increase in computational complexity with the increase in surface roughness of the conductors in

CB-CPWs was also documented.

164

CHAPTER 6: CONCLUSION AND FUTURE WORK

This dissertation analyzed the effect of embedded patterned layer (EPL) on conductor backed coplanar waveguide (CB-CPW) and developed a circuit model to understand the impact of floating metals on transmission lines. This chapter wraps up all the finding from this research endeavor and presents ideas for future research work.

The theory behind excitation of higher order modes in conductor backed coplanar waveguide was analyzed in Chapter 2. The higher order modes restrict the bandwidth of the conductor backed coplanar waveguide. The higher order modes can be suppressed using ground via fence thereby improving the conductor backed coplanar waveguide bandwidth. The ground via fence should be as close as possible to the signal trace and via to via pitch should be less than quarter wavelength at the highest frequency of interest for best results. The knowledge acquired from this chapter was used to design a grounded coplanar waveguide (GCPW) with large bandwidth which was used to analyze the effect of EPL on GCPWs. Two new techniques (using either multiple via fences or slotted ground plane) for improving grounded coplanar waveguide bandwidth were also developed. A better understanding of how the slotted ground plane structure suppresses the higher order modes should be developed and an analysis of how this structure impacts the signal integrity of neighboring transmission lines must be performed in any future work.

In Chapter 3 and Chapter 4, the effect of EPL on a GCPW and a grounded conductor backed edge coupled waveguide was analyzed, respectively. Floating metals increase the per unit length (p.u.l.) resistance, capacitance and conductance of a transmission line but reduce the p.u.l. inductance. The change in inductance and

165 resistance depends on the mutual inductance between the floating metal and the transmission line and the self-resistance and self-inductance of the floating metal. EPL reduces the characteristic impedance and increases the effective permittivity of a transmission line. The increased resistance of the transmission line increases the insertion loss. The increased capacitance of a transmission line with EPL adversely affects signal speed by slowing the signal edge rate. The effect of EPL is more pronounced when it is closest to the signal conductor, the EPL thickness is high, and pitch between individual

EPL elements is small.

In Chapter 4, the EPL increases the p.u.l. self-capacitance of the individual signal traces and the mutual capacitance between the signal traces. The p.u.l. self-inductance of the signal traces and mutual inductance between the signal traces fall due to EPL. The reason behind reduction in the mutual inductance due to EPL needs further analysis.

Near-end crosstalk increases while far-end crosstalk decreases due to EPL. EPL increases the differential insertion loss when the coupled transmission line is used for differential signaling. The differential to common mode energy conversion is suppressed when EPL is symmetric to both the signal traces of a differential pair. This effect needs to be analyzed and understood in future work. Any asymmetric EPL placement in a differential pair will increase differential to common mode energy conversion causing EMI problems.

EPL reduces the characteristic impedance and the differential impedance of a

GCPW and a grounded conductor backed edge coupled coplanar waveguide, respectively. This can allow designers to fabricate narrow high impedance transmission lines and use EPL to reduce the impedance to a desired value. The narrow high

166 impedance transmission lines increase separation between neighboring transmission lines which can result in reduced crosstalk. This reduced transmission line footprint can also allow engineers to route more lines over a fixed area. However, EPL requires an extra board layer which will increase the board thickness and fabrication cost. So, an analysis comparing improvement in crosstalk and/or routing density versus fabrication cost must be performed in future work. This work involved inhomogeneous transmission line but future work should also analyze the impact of EPL on homogeneous transmission lines like striplines. For example, striplines have very little far-end crosstalk due to their homogeneous nature. So it will be interesting to see how the EPL affects NEXT and

FEXT in a stripline. In future it will be great to perform a comparison between the simulated and measured data.

Finally in Chapter 5, a statistical method of modeling conductor surface roughness in 3D full wave electromagnetic simulators was developed. The attenuation coefficient and insertion loss of a CB-CPW increases with increasing conductor surface roughness and frequency. The roughness of a surface increases with increasing root mean square height and with reducing correlation length. Computational complexity and requirement increases with increasing surface roughness of the conductors. The work in this chapter involved conductors of rectangular cross-section and only the conductor surface in contact with the dielectric was roughened, however, in actuality the conductor cross-sections are trapezoidal in shape and these slanted conductor edges also exhibit surface roughness. These effects should be considered in future modeling and simulations to get a better understanding of the effect of conductor cross-section and surface roughness on transmission lines at high frequencies.

167

APPENDIX I: DERIVATION OF RESONANT FREQUENCY EQUATION OF RECTANGULAR CAVITIES WITH DIFFERENT BOUNDARY CONDITIONS

In a finite ground conductor backed coplanar waveguide (FG-CBCPW), the coplanar side ground along with the lower ground plane (LGP) act as a rectangular patch antenna which acts like resonant cavity with open circuit on the sides. The length (l), width (w), and height (h) of the cavity are along x, y, and z axis, respectively (Fig. I-1). The coplanar side ground and lower ground plane in a FG-CBCPW will be made out of finite conductivity metals. But for the purpose of simplification, perfect electric conductors (PEC) will be used in place of metals. The open circuit on the sides will be modeled by placing perfect magnetic conductors (PMC). This results in a rectangular cavity with two PEC and four PMC walls.

(a) (b) Fig. I-1 (a) Top layer and (b) cross-sectional view of a rectangular patch formed by CSG and LGP of a FG-CBCPW

The two PECs have length and width of l and w, respectively. These are located on the x-y plane at z = 0 and z = h. Two PMCs are located on the x-z plane with y = 0 and y = w. These PMCs have length and height of l and h, respectively. Remainder of the PMCs are located on the y-z plane at x = 0 and x = l. These PMCs have width and height of w and h, respectively. Fig. I-2 shows the PECs and PMCs location with respect to the rectangular cavity.

168

(a) (b)

(c) Fig. I-2 Shows (a) the 3-D simplified model of the rectangular cavity formed by a CSG and LGP of a FG-CBCPW along with the locations of the (b) PECs and (c) PMCs with respect to the cavity

For this work, TMZ mode is considered and the concept of vector potentials will be used for the derivations. In a source free and lossless region, vector potential A and F have solutions of the form [107]

A(푥, 푦, 푧) = 푥̂퐴푥(푥, 푦, 푧) + 푦̂퐴푦(푥, 푦, 푧) + 푧̂퐴푧(푥, 푦, 푧) (I.1)

F(푥, 푦, 푧) = 푥̂퐹푥(푥, 푦, 푧) + 푦̂퐹푦(푥, 푦, 푧) + 푧̂퐹푧(푥, 푦, 푧) (I.2) Electric field intensity (E) and magnetic field intensity (H) can be calculated from A and F using (I.3) and (1.4), respectively [107]. Solving (I.3) and (I.4) using (I.1) and (I.2) E and H is given by (I.5) and (I.6), respectively. 1 1 E = −푗휔A − j ∇(∇. A) − ∇ × F (I.3) 휔휇휀 휀 1 1 H = ∇ × A − 푗휔F − 푗 ∇(∇. F) (I.4) 휇 휔휇휀 1 휕2퐴 휕2퐴 휕2퐴 1 휕퐹 휕퐹 E = 푥̂ [−푗휔퐴 − 푗 ( 푥 + 푦 + 푧) − ( 푧 − 푦)] 푥 휔휇휀 휕푥2 휕푥휕푦 휕푥휕푧 휀 휕푦 휕푧 1 휕2퐴 휕2퐴 휕2퐴 1 휕퐹 휕퐹 +푦̂ [−푗휔퐴 − 푗 ( 푥 + 푦 + 푧) − ( 푥 − 푧)] 푦 휔휇휀 휕푥휕푦 휕푦2 휕푦휕푧 휀 휕푧 휕푥 1 휕2퐴 휕2퐴 휕2퐴 1 휕퐹 휕퐹 +푧̂ [−푗휔퐴 − 푗 ( 푥 + 푦 + 푧) − ( 푦 − 푥)] (I.5) 푧 휔휇휀 휕푥휕푧 휕푦휕푧 휕푧2 휀 휕푥 휕푦 1 휕2퐹 휕2퐹 휕2퐹 1 휕퐴 휕퐴 H = 푥̂ [−푗휔퐹 − 푗 ( 푥 + 푦 + 푧 ) + ( 푧 − 푦)] 푥 휔휇휀 휕푥2 휕푥휕푦 휕푥휕푧 휇 휕푦 휕푧 1 휕2퐹 휕2퐹 휕2퐹 1 휕퐴 휕퐴 +푦̂ [−푗휔퐹 − 푗 ( 푥 + 푦 + 푧 ) + ( 푥 − 푧)] 푦 휔휇휀 휕푥휕푦 휕푦2 휕푦휕푧 휇 휕푧 휕푥

169

1 휕2퐹 휕2퐹 휕2퐹 1 휕퐴 휕퐴 +푧̂ [−푗휔퐹 − 푗 ( 푥 + 푦 + 푧) + ( 푦 − 푥)] (I.6) 푧 휔휇휀 휕푥휕푧 휕푦휕푧 휕푧2 휇 휕푥 휕푦 Z TM mode means transverse magnetic to z, which means 퐻푧 = 0. From (I.6) this requirement can be satisfied by setting 퐴푥 = 퐴푦 = 퐹푥 = 퐹푦 = 퐹푧 = 0, 퐴푧 ≠ 0, 휕⁄휕푥 ≠ 0 and 휕⁄휕푦 ≠ 0. With these conditions, (I.1) and (I.2) gets reduced to (I.7) and (I.8), respectively Equations (I.5) and (I.6) are reduced to (I.9) and (I.10) using the above conditions, respectively.

A(푥, 푦, 푧) = 푧̂퐴푧(푥, 푦, 푧) (I.7) F(푥, 푦, 푧) = 0 (I.8) 1 휕2퐴 1 휕2퐴 E = −푥̂ [푗 ( 푧)] − 푦̂ [푗 ( 푧)] 휔휇휀 휕푥휕푧 휔휇휀 휕푦휕푧 1 휕2퐴 −푧̂ [푗휔퐴 + 푗 ( 푧)] (I.9) 푧 휔휇휀 휕푧2 1 휕퐴 1 휕퐴 H = 푥̂ [ ( 푧)] − 푦̂ [ ( 푧)] + 푧̂[0] (I.10) 휇 휕푦 휇 휕푥 The vector potential A must satisfy the following condition ∇2A + β2A = −μJ (I.11) where β and J are propagation coefficient and electric current density, respectively. Since a source free region is being considered and 퐴푥 = 퐴푦 = 0, equation (I.11) gets reduced to 2 2 ∇ 퐴푧 + 훽 퐴푧 = 0 (I.12) Using the separation of variables method, assume that 퐴푧(푥, 푦, 푧) can be represented 퐴푧(푥, 푦, 푧) = 푑(푥)푒(푦)푓(푧). (I.13) The solutions for 푑(푥), 푒(푦) and 푓(푧) can take the forms shown in equation I.14 (a)-(c). The solutions represent standing waves because standing waves are formed inside a rectangular cavity.

푑(푥) = 퐴1 cos(훽푥푥) + 퐵1sin (훽푥푥) (I.14a)

푒(푦) = 퐴2 cos(훽푦푦) + 퐵2sin (훽푦푦) (I.14b)

푓(푧) = 퐴3 cos(훽푧푧) + 퐵3sin (훽푧푧) (I.14c) where 2 2 2 2 2 훽 = 훽푥 + 훽푦 + 훽푧 = 휔 휇휀 (I.14d)

Equation I.14d is called the constraint (dispersion) equation and 훽푥, 훽푦, and 훽푧 are propagation constants in the x, y, and z directions, respectively. Functions 푑(푥), 푒(푦) and

푓(푧) corresponding to specific eigenvalues (훽푥, 훽푦, and 훽푧) are the eigenfunctions. The unknowns 퐴1, 퐴2, 퐴3, 퐵1, 퐵2, 퐵3, 훽푥, 훽푦, and 훽푧 can be determined by substituting 퐴푧 in I.5 and I.6 and then applying the appropriate boundary conditions on the E and H field components. In this case, the two PEC walls are located at z =0 and z = h and the four PMC walls are located at x = 0, x = l, y =0 and y = w. It is known that tangential electric fields and tangential magnetic fields go to zero on PECs and PMCs, respectively. Thus applying the appropriate boundary conditions

퐸푥(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = 0) = 퐸푥(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = ℎ) =

170

퐸푦(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = 0) = 퐸푦(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = ℎ) = 0 (I.15a)

퐻푥(0 ≤ 푥 ≤ 푙, 푦 = 0,0 ≤ 푧 ≤ ℎ) = 퐻푥(0 ≤ 푥 ≤ 푙, 푦 = 푤, 0 ≤ 푧 ≤ ℎ) = 퐻푦(푥 = 0,0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 퐻푦(푥 = 푙, 0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 0 (I.15b) From Eq. I.9, the x component of E is −푗 휕2퐴 퐸 = ( 푧) (I.16a) 푥 휔휇휀 휕푥휕푧 −푗훽 훽 ∴ 퐸 = 푥 푧 푑′(푥)푒(푦)[−퐴 sin(훽 푧) + 퐵 cos (훽 푧)] (I.16b) 푥 휔휇휀 3 푧 3 푧 At z = 0, −푗훽 훽 퐸 = 0 = 푥 푧 푑′(푥)푒(푦)[−퐴 sin(0) + 퐵 cos (0)] (I.16c) 푥 휔휇휀 3 3 Only way to satisfy I.16c without leading to a trivial solution will be when

퐵3 = 0 (I.16d) At z = h, −푗훽 훽 퐸 = 0 = 푥 푧 푑′(푥)푒(푦)[−퐴 sin(훽 ℎ)] (I.16e) 푥 휔휇휀 3 푧 To satisfy I.16e without leading to a trivial solution will be when

sin(훽푧ℎ) = 0 (I.16f) 푝휋 ∴ 훽 = , 푝 = 0, 1,2,3, … (I.16g) 푧 ℎ Now, from Eq. I.10, the x component of H is 1 휕퐴 퐻 = ( 푧). (I.17a) 푥 휇 휕푦 훽 ∴ 퐻 = 푦 푑(푥)[−퐴 sin(훽 푦) + 퐵 cos (훽 푦)]푓(푧) (I.17b) 푥 휇 2 푦 2 푦 At y = 0, 훽 퐻 = 0 = 푦 푑(푥)[−퐴 sin(0) + 퐵 cos (0)]푓(푧) (I.17c) 푥 휇 2 2 Eq. I.17c is satisfied when

퐵2 = 0 (I.17d) At y = w, 훽 퐻 = 0 = 푦 푑(푥)[−퐴 sin(훽 푤)]푓(푧) (I.17e) 푥 휇 2 푦 Eq. I.17e is satisfied when

sin(훽푦푤) = 0 (I.17f) 푛휋 ∴ 훽 = , 푛 = 0, 1,2,3, … (I.17g) 푦 푤 Also from Eq. I.10, the y component of H is 1 휕퐴 퐻 = − ( 푧). (I.18a) 푦 휇 휕푥 훽 ∴ 퐻 = − 푥 [−퐴 sin(훽 푥) + 퐵 cos (훽 푥)]푒(푦)푓(푧) (I.18b) 푦 휇 1 푥 1 푥 At x = 0,

171

훽 퐻 = 0 = − 푥 [−퐴 sin(0) + 퐵 cos(0)]푒(푦)푓(푧) (I.18c) 푦 휇 1 1 Eq. I.18c is satisfied when

퐵1 = 0 (I.18d) At y = l, 훽 퐻 = 0 = − 푥 [−퐴 sin(훽 푙)]푒(푦)푓(푧) (I.18e) 푦 휇 1 푥 Eq. I.17e is satisfied when

sin(훽푥푙) = 0 (I.18f) 푚휋 ∴ 훽 = , 푚 = 0, 1,2,3, … (I.18g) 푥 푙 Now, Eq. I.13 gets reduced to 푚휋 푛휋 푝휋 퐴 (푥, 푦, 푧) = 퐴 퐴 퐴 cos ( 푥) cos ( 푦) cos ( 푧) (I.19) 푧 1 2 3 푙 푤 ℎ where 푚휋 훽 = , 푚 = 0,1,2, … 푥 푙 푛휋 훽 = , 푛 = 0,1,2, … 푚 = 푛 = 푝 ≠ 0 (I.20) 푦 푤 푝휋 훽 = , 푝 = 0,1,2, … 푧 ℎ } Resonant frequency ((푓푟)푚푛푝) for the cavity can be derived from Eq. I.14d using Eq. I.20 and is given by

1 푚휋 2 푛휋 2 푝휋 2 (푓푟)푚푛푝 = √( ) + ( ) + ( ) (I.21) 2휋√휇휀 푙 푤 ℎ

Case 1. PMC at y = w replaced by PEC

The PMC at y = w on the x-z plane is replaced with a PEC. In that case, only the boundary conditions along the y- axis will change. Eq. I.19 can be modified as shown below 푚휋 푝휋 퐴 (푥, 푦, 푧) = 퐴 퐴 cos ( 푥) [퐴 cos(훽 푦) + 퐵 sin (훽 푦)] cos ( 푧) (I.22) 푧 1 3 푙 2 푦 2 푦 ℎ The position of the PECs and PMCs are shown in Fig. I-3. Also, the boundary conditions are as follows

퐸푥(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = 0) = 퐸푥(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = ℎ) = 퐸푦(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = 0) = 퐸푦(0 ≤ 푥 ≤ 푙, 0 ≤ 푦 ≤ 푤, 푧 = ℎ) =

퐸푥(0 ≤ 푥 ≤ 푙, 푦 = 푤, 0 ≤ 푧 ≤ ℎ) = 퐸푧(0 ≤ 푥 ≤ 푙, 푦 = 푤, 0 ≤ 푧 ≤ ℎ) = 0. (I.23a)

퐻푥(0 ≤ 푥 ≤ 푙, 푦 = 0,0 ≤ 푧 ≤ ℎ) = 퐻푦(푥 = 0,0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) =.

퐻푦(푥 = 푙, 0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 0 (I.23b)

172

Fig. I-3 Location of PECs and PMCs for the rectangular cavity in Case 1

From Eq. I.9, the z component of E is 1 휕2퐴 퐸 = − [푗휔퐴 + 푗 ( 푧)] (I.24) 푧 푧 휔휇휀 휕푧2 푗 푝휋 2 ∴ 퐸 = 퐴 [−푗휔 + ( ) ] (I.25) 푧 푧 휔휇휀 ℎ Eq. I.25 is zero at y = w 푗 푝휋 2 푚휋 푝휋 퐸 = 0 = [−푗휔 + ( ) ] 퐴 cos ( 푧) 퐴 cos ( 푧) 푧 휔휇휀 ℎ 1 푙 3 ℎ

[퐴2 cos(훽푦푤) + 퐵2sin (훽푦푤)] (I.26) Eq. I.26 can be satisfied when

[퐴2 cos(훽푦푤) + 퐵2sin (훽푦푤)] = 0 (I.27)

Now, 퐻푥 can be calculated from 퐸푧 using the following equation [79], [108] 푗 휕퐸푧 휕퐻푧 퐻푥 = 2 (휔휀 − 훽 ) (I.28) 푘푐 휕푦 휕푥 2 푗휔휀훽푦 푗 푝휋 푚휋 푝휋 ∴ 퐻푥 = 2 [−푗휔 + ( ) ] 퐴1 cos ( 푧) 퐴3 cos ( 푧) 푘푐 휔휇휀 ℎ 푙 ℎ

[−퐴2 sin(훽푦푦) + 퐵2cos (훽푦푦)] (I.29) where 푘푐 is the cut off wavenumber. 퐻푥 is zero at y = 0. 2 푗휔휀훽푦 푗 푝휋 푚휋 푝휋 퐻푥 = 0 = 2 [−푗휔 + ( ) ] 퐴1 cos ( 푧) 퐴3 cos ( 푧) 푘푐 휔휇휀 ℎ 푙 ℎ [−퐴2 sin(0) + 퐵2cos (0)] (I.30) Eq. I.30 is satisfied when

퐵2 = 0 (I.31) Now, Eq. I.27 is satisfied when

cos(훽푦푤) = 0 (I.32a) 푛휋 ∴ 훽 = , 푛 = 0.5, 1.5, 2.5, … (I.32b) 푦 푤 Now, Eq. I.22 gets reduced to 푚휋 푛휋 푝휋 퐴 (푥, 푦, 푧) = 퐴 퐴 퐴 cos ( 푥) cos ( 푦) cos ( 푧) (I.33a) 푧 1 2 3 푙 푤 ℎ where

173

푚휋 훽 = , 푚 = 0,1,2, … 푥 푙 푛휋 훽 = , 푛 = 0.5,1.5,2.5, … (I.33b) 푦 푤 푝휋 훽 = , 푝 = 0,1,2, … 푧 ℎ }

Case 2. PMCs at x = 0, x = l, and y = w replaced by PECs

Finally, the PMCs at x = 0 and x = l on y-z plane and at y = w on x-z plane are replaced with PECs. Since the boundary conditions at x = 0 and x = l have changed, Eq. I.33a can be modified as follows 푛휋 푝휋 퐴 (푥, 푦, 푧) = [퐴 cos(훽 푥) + 퐵 sin (훽 푥)]퐴 퐴 cos ( 푦) cos ( 푧) (I.34) 푧 1 푥 1 푥 2 3 푤 ℎ The position of the PECs and PMCs are shown in Fig. I-4. Also, the boundary conditions are updated as follows

퐸푥(0 ≤ 푥 ≤ 푙, 푦 = 푤, 0 ≤ 푧 ≤ ℎ) = 퐸푧(0 ≤ 푥 ≤ 푙, 푦 = 푤, 0 ≤ 푧 ≤ ℎ) =. 퐸푦(푥 = 0,0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 퐸푦(푥 = 푙, 0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) =.

퐸푧(푥 = 0,0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 퐸푧(푥 = 푙, 0 ≤ 푦 ≤ 푤, 0 ≤ 푧 ≤ ℎ) = 0 (I.35a)

퐻푥(0 ≤ 푥 ≤ 푙, 푦 = 0,0 ≤ 푧 ≤ ℎ) = 0 (I.35b)

Fig. I-4 Location of PECs and PMCs for the rectangular cavity in Case 2

From Eq. I.9, the y component of E is 1 휕2퐴 퐸 = 푗 ( 푧). (I.36) 푦 휔휇휀 휕푦휕푧 −푗푛푝휋2퐴 퐴 푛휋 푝휋 ∴ 퐸 = 2 3 sin ( 푦) sin ( 푧) [퐴 cos(훽 푥) + 퐵 sin (훽 푥)] (I.37) 푦 휔휇휀푤푙 푤 ℎ 1 푥 1 푥

퐸푦 is 0 at x = 0 −푗푛푝휋2퐴 퐴 푛휋 푝휋 ∴ 퐸 = 0 = 2 3 sin ( 푦) sin ( 푧) [퐴 cos(0) + 퐵 sin (0)] (I.38) 푦 휔휇휀푤푙 푤 ℎ 1 1 Eq. I.38 is satisfied when

퐴1 = 0 (I.39) 퐸푦 is also 0 at x = l

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−푗푛푝휋2퐴 퐴 푛휋 푝휋 ∴ 퐸 = 0 = 2 3 sin ( 푦) sin ( 푧) [퐵 sin (훽 푙)] (I.40) 푦 휔휇휀푤푙 푤 ℎ 1 푥 Eq. I.40 is satisfied when

sin(훽푥푙) = 0 (I.41a) 푚휋 ∴ 훽 = , 푚 = 0, 1, 2, … (I.41b) 푥 푙 Now, Eq. I.34 gets reduced to 푚휋 푛휋 푝휋 퐴 (푥, 푦, 푧) = 퐵 퐴 퐴 sin ( 푥) cos ( 푦) cos ( 푧) (I.42a) 푧 1 2 3 푙 푤 ℎ where 푚휋 훽 = , 푚 = 1,2, … 푥 푙 푛휋 훽 = , 푛 = 0.5,1.5,2.5, … (I.42b) 푦 푤 푝휋 훽 = , 푝 = 0,1,2, … 푧 ℎ }

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APPENDIX II: REFLECTION COEFFICIENT AT DIELECTRIC AND PEC OR PMC INTERFACE

In this section the reflection coefficient at the interface between a dielectric (Medium 1) and a PEC or PMC (Medium 2) has been calculated. The dielectric is assumed to be lossless and interface is located at z =0. The interface is along the x-y plane. The incident plane wave is travelling through the dielectric and is incident on the interface at an angle 휃푖 and reflected at an angle 휃푟. The wave is perpendicularly polarized. Fig. II-1 shows the wave incident on the interface.

Fig. II-1 Perpendicularly polarized plane wave incident at an angle on an interface between a dielectric and a PEC or PMC

The incident electric and magnetic fields can be written as −푗훽(푥 sin 휃푖+푧 cos 휃푖) Ei = 푦̂퐸푂푒 (II.1) 퐸 H = (−푥̂ cos 휃 + 푧̂ sin 휃 ) 푂 푒−푗훽(푥 sin 휃푖+푧 cos 휃푖) (II.2) i 푖 푖 휂 Similarly, the reflected fields and be written as −푗훽(푥 sin 휃푟−푧 cos 휃푟) Er = 푦̂훤퐸푂푒 (II.1) 훤퐸 H = (푥̂ cos 휃 + 푧̂ sin 휃 ) 푂 푒−푗훽(푥 sin 휃푟+푧 cos 휃푟) (II.2) i 푟 푟 휂 where 훤, 휂, and 퐸푂 are the reflection coefficient at the interface, impedance of the dielectric medium, and amplitude of the incident electric field, respectively. From Snell’s law

휃푖 = 휃푟 = 휃 (II.3)

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Case 1. Medium 2 = PEC

In case of a dielectric PEC interface, the tangential electric fields at the interface should be equal to zero. 푡푎푛푔푒푛푡푖푎푙 푡푎푛푔푒푛푡푖푎푙 퐸푖 + 퐸푟 = 0 (II.4a) −푗훽푥 sin 휃 −푗훽푥 sin 휃 ∴ 퐸푂푒 + 훤퐸푂푒 = 0 (II.4b) ∴ 훤 = −1 (II.4c)

The reflection coefficient (훤) is also defined in terms of load impedance (푍퐿) and characteristic impedance (푍푂) by the following equation: 푍 −푍 훤 = 퐿 푂 (II.5) 푍퐿+푍푂 훤 is -1 in Eq. II.5, when 푍퐿 = 0, which means a short circuit. So, PEC is equivalent to a short circuit in terms of circuit theory.

Case 2. Medium 2 = PMC

In case of a dielectric PMC interface, the tangential magnetic fields at the interface should be equal to zero. 푡푎푛푔푒푛푡푖푎푙 푡푎푛푔푒푛푡푖푎푙 퐻푖 + 퐻푟 = 0 (II.6a) 퐸 퐸 ∴ − cos 휃 푂 푒−푗훽푥 sin 휃 + 훤cos 휃 푂 푒−푗훽푥 sin 휃 = 0 (II.6b) 휂 휂 ∴ 훤 = 1 (II.6c)

훤 is 1 in Eq. II.5, when 푍퐿 = ∞, which means an open circuit. So, PMC is equivalent to an open circuit in terms of circuit theory.

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APPENDIX III: PLOTS OF SCATTERING PARAMETERS FROM CHAPTER 3 WHEN RENORMALIZED TO STANDARD 50 Ω IMPEDANCE

The S-parameter plots presented in Chapter 3 (Fig. 3-18, 3-20, 3-22, and 3-23) were renormalized to the actual transmission line impedances to show the true effect of EPLs on the transmission lines. Here the S-parameters are shown with the ports renormalized to 50 Ω. This impedance mismatch between the port and the transmission line results in ripples seen in the insertion loss plots while increasing return loss.

Case 1. Varying EPL thickness (TEPL)

The figures presented here correspond to Fig. 3-18 in Chapter 3.

(a) (b)

(c) Fig. III-1 Plot of (a) insertion loss, (b) return loss, and (c) insertion loss phase versus frequency for GCPWs without and with EPL of varying TEPL, with ports renormalized to 50 Ω impedance

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Case 2. Varying separation between signal trace and EPL (HEPL)

The figures presented here correspond to Fig. 3-20 in Chapter 3.

(a) (b) Fig. III-2 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying HEPL, with ports renormalized to 50 Ω impedance

Case 3. Varying EPL pitch along x (PXEPL) and y (PYEPL) axis

Fig. III-3 corresponds to Fig. 3-22 in Chapter 3.

(a) (b) Fig. III-3 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PXEPL, with ports renormalized to 50 Ω impedance

179

Fig. III-4 corresponds to Fig. 3-23 in Chapter 3.

(a) (b) Fig. III-4 Plot of (a) insertion loss, and (b) return loss versus frequency for GCPWs without and with EPL of varying PYEPL, with ports renormalized to 50 Ω impedance

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APPENDIX IV: PLOTS OF SCATTERING PARAMETERS FROM CHAPTER 4 WHEN RENORMALIZED TO STANDARD IMPEDANCES

The S-parameter plots presented in Chapter 4 (Fig. 4-8 and 4-9) were renormalized to the actual transmission line impedances to show the true effect of EPLs on the transmission lines. Here the S-parameters are shown with the ports renormalized to 50 Ω and 100 Ω for single ended and differential signaling, respectively. This impedance mismatch between the port and the transmission line results in ripples seen in the insertion loss plots while increasing return loss.

Case 1. Single ended signaling

The figures presented here correspond to Fig. 4-8 in Chapter 4.

(a) (b)

(c) (d) Fig. IV-1 Plot of (a) insertion loss, (b) return loss, (c) NEXT, and (d) FEXT versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to 50 Ω impedance

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Case 2. Differential signaling

The figures presented here correspond to Fig. 4-9 in Chapter 4.

(a) (b)

(c) Fig. IV-2 Plot of (a) differential insertion loss (SDD21), (b) differential return loss (SDD11), and (c) SCD21 versus frequency for grounded conductor backed edge coupled coplanar waveguides without and with EPL, with ports renormalized to 100 Ω differential impedance

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