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METAMATERIAL ONE DIMENSION TRANSMISSION LINE

NEW PROPOSED MICROSTRIP DESIGN

A Project

Presented to the faculty of the Department of Electrical and Electronic Engineering

California State University, Sacramento

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

Electrical and Electronic Engineering

Ou Saejao

SUMMER 2018

Ou Saejao

METAMATERIAL ONE DIMENSION TRANSMISSION LINE

NEW PROPOSED MICROSTRIP DESIGN

A Project

Ou Saejao

Approved by:

______, Committee Chair Dr. Milica Markovic

______, Second Reader Dr. Preetham Kumar

______Date

iii

Student: Ou Saejao

I certify that this student has met the requirements for format contained in the University format manual, and that this project is suitable for shelving in the Library and credit is to be awarded for the project.

______, Graduate Coordinator ______Dr. Preetham Kumar Date

Department of Electrical and Electronic Engineering

Abstract

METAMATERIAL ONE DIMENSION TRANSMISSION LINE

NEW PROPOSED MICROSTRIP DESIGN

Ou Saejao

Communication is important in society and antennas are needed to communicate effectively across large distances. Currently, the demands of transferring more data at a faster rate and over vast distances require new technology. One new proposed technology is metamaterials. This project introduced and examined metamaterials previously proposed in published literature using ADS and

HFSS. Furthermore, this project proposed a new change to the published metamaterials by replacing interdigital capacitor with metal-insulator-metal capacitor and stub inductor with spiral inductor. The new proposal is then designed and simulated with Keysight ADS, Mathworks MATLAB, and Ansoft HFSS.

The results show that the new design is more symmetric with greater peak directivity and peak gain at its center frequency.

______, Committee Chair Dr. Milica Markovic

______Date

ACKNOWLEDGEMENTS

The support from various special individuals made this document possible and I would like to deeply express my gratitude for them. First and foremost, I would like to give special thanks and appreciation to Dr. Milica Markovic for being my project advisor and committee chair. With her guidance, I was able to conduct research on metamaterial and finish this document. I would also offer my special thanks to Dr. Preetham Kumar for being my graduate coordinator and my second reader. The assistance provided by Dr. Kumar allowed me to focus on this document while fulfilling the master program requirement.

I would also like to express my appreciation for California State University of Sacramento for the opportunity to continue my pursuit in education. I would also like to express my appreciation to

United States Army for the financial assistance.

I wish to thank my family, Sou Saejao and Sarn Saejao, for hosting me in their house; Nai

Saejao and Kao Saejao, for the emotional support. I wish to deeply express my gratitude to my mom for always believing in my effort. I would also like to give special acknowledge for my friend, Russell Vink, for the grammatical review of this document.

TABLE OF CONTENTS

Page

Acknowledgement ...... vi

List of Tables ...... x

List of Figures ...... xi

List of Programming Code ...... xv

Chapter

1 INTRODUCTION ...... 1

1.1 Application ...... 1

1.2 Current Researches ...... 2

1.3 Focus of Project ...... 3

1.4 Antennas...... 4

1.4.1 Microstrip ...... 4

1.5 Radiation Mechanism ...... 5

1.5.1 Traveling Wave ...... 5

1.5.2 Resonance...... 6

2 METAMATERIAL INTRODUCTION...... 7

2.1 Metamaterials Theory ...... 8

2.2 Slip Ring ...... 9

2.3 Structural Thin Wires ...... 10

2.4 Thin Wires Slip Ring Resonator ...... 10

2.5 Transmission Line Approach ...... 11

2.5.1 Purely Left Hand Unit Cell ...... 11

2.5.2 Purely Right Hand...... 12

2.5.3 Composite Right and Left Hand ...... 14

vii

2.5.4 CRLH Balanced and Unbalanced ...... 15

2.5.5 CRLH Symmetrical...... 15

2.6 Bloch Impedance ...... 15

3 NEW MTM CELL DESIGN ...... 17

3.1 Right Hand Capacitor and Inductor Extraction ...... 17

3.2 Left Hand Capacitor and Inductor Extraction ...... 20

3.2.1 Design of MMIC Left Hand Capacitor...... 21

3.2.2 Design of MMIC Left Hand Inductor ...... 22

3.3 ADS and HFSS Comparison of MMIC Capacitor and Inductor ...... 25

4 ANALYSIS OF PREVIOUS PUBLISHED MTM ...... 29

4.1 Background Information ...... 29

4.2 HFSS Results ...... 29

4.2.1 CRLH Lump Ports Single Cell ...... 30

4.2.2 CRLH Wave Ports Single Cell ...... 33

4.2.3 CRLH Wave Ports Three Cell Symmetrical Feed ...... 37

5 NEW DESIGN RESULTS...... 41

5.1 Type A 50.8 mm (2-inch) Unit Cell ...... 41

5.1.1 ADS Results ...... 41

5.1.2 HFSS Results ...... 43

5.1.3 Type A Three 50.8 mm Unit Cell ...... 47

5.2 Type A 20 mm Unit Cell ...... 49

5.2.1 ADS Results ...... 50

5.2.2 HFSS Results ...... 51

5.3 Type B 20 mm Unit Cell ...... 53

5.3.1 Optimizing ...... 53

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5.3.2 HFSS Results ...... 54

5.3.3 Type B Three 20 mm Unit Cell ...... 59

5.3.4 Simulated Electrical Field Vector ...... 62

6 CONCLUSION ...... 64

References ...... 65

Appendices ...... 68

LIST OF TABLES

Tables Page

1. Design Specification ...... 18

2. Capacitor, Inductor, and Frequency Values ...... 21

3. MIM Capacitor Parameters ...... 22

4. SS Inductor Parameters ...... 24

5. Self and Mutual Inductance Values ...... 24

6. Real and Imaginary Impedances Comparison ...... 25

7. Parameters ...... 30

8. Lumped Elements Parameters ...... 42

9. Initial Parameters ...... 44

10. 20 mm CRLH Lumped Elements ...... 49

11. CRLH LH Capacitor and Inductor Dimensions of a 20 mm Unit Cell ...... 51

12. Type B Parameters ...... 55

13. Comparison ...... 64

LIST OF FIGURES

Figures Page

1-1. (a) Isometric view of patch antenna. (b) Side view of patch antenna ...... 5

2-1. Quadrant Sections Identifying the Different Combination of Signed Permittivity and Permeability. .. 8

2-2. (a) SRRs, (b) CSRRs, (c) DS-SSRs, (d) DS-CSRRs ...... 9

2-3. SRRs and its Equivalent Circuit Model ...... 9

2-4. (a) Thin-wire Structure exhibiting negative permittivity (εr). (b) SRRs structure exhibiting negative

permeability (μr) ...... 10

2-5. Thin Wire Slip Ring Resonator ...... 11

2-6. Left Hand Circuit per Cell ...... 12

2-7. Dispersion Graph of PLH ...... 12

2-8. Right Hand Circuit per Cell ...... 13

2-9. Dispersion Graph of PRH ...... 13

2-10. Proposed 1D CRLH MTMs. (a) Cell of CRLH MTMs. (b) Wave Propagation Comparison of

CRLH to both PLH and PRH ...... 14

2-11. Unit Cell. Interdigital Capacitor and Short Circuit Stub ...... 16

3-1. (a) MIM Capacitor, (b) Squared Spiral Inductor ...... 17

3-2. RLC Equivalent Model of Microstrip Line ...... 18

3-3. Capacitor and Inductor Extraction ...... 19

3-4. Inductor (left), Capacitor (right), and Impedance (bottom) extracted at 500 MHz ...... 19

3-5. Example Drawing of MIM Capacitor ...... 22

3-6. SS Inductor and Parameters Example ...... 25

3-7. ADS Circuits with Lumped Elements ...... 26

3-8. ADS Smith Chart ...... 26

3-9. MIM Capacitor ...... 27

3-10. MIM Capacitor Smith Chart ...... 27

3-11. SS Inductor ...... 28

3-12. SS Inductor Smith Chart ...... 28

4-1. 3D Model Design Lump Ports Single Cell ...... 31

4-2. Dispersion Graph of Lump Ports Single Cell ...... 31

4-3. Bloch Impedance of Lump Ports Single Cell ...... 32

4-4. Insertion Loss of Lump Ports Single Cell ...... 32

4-5. 3D Far Field Radiation Pattern at 2.4 GHz ...... 33

4-6. 3D Model Design Wave Ports Single Cell ...... 34

4-7. Dispersion Graph of Wave Ports Single Cell ...... 34

4-8. Bloch Impedance of Wave Ports Single Cell ...... 35

4-9. Insertion Loss of Wave Ports Single Cell ...... 35

4-10. 3D Far Field Radiation Pattern of Wave Ports Single Cell at 2.4 GHz ...... 36

4-11. 2D Far Field Radiation Plot at 0º on Polar Plot of Wave Ports Single Cell at 2.4 GHz ...... 36

4-12. 3D Model Design of Symmetrical Three Cell Structure ...... 37

4-13. Dispersion Graph of Symmetrical Three Cell Structure ...... 38

4-14. Bloch Impedance of Symmetrical Three Cell Structure ...... 38

4-15. Insertion Loss of Symmetrical Three Cell Structure ...... 39

4-16. 3D Far Field Radiation Pattern of Three Cell Structure at 2.4 GHz ...... 39

4-17. 2D Far Field Radiation Plot at 0º on Polar Plot of Three Cell Structure at 2.4 GHz ...... 40

5-1. ADS New CRLH Unit Cell of 50.8 mm Line Length ...... 42

5-2. ADS Insertion Loss of 50.8 mm Unit Cell ...... 42

5-3. ADS Reflection Coefficient of 50.8 mm Unit Cell ...... 43

5-4. ADS Dispersion Graph of 50.8 mm Unit Cell ...... 43

5-5. Type A Design of 50.8 mm Unit Cell ...... 44

xii

5-6. Type A Design with De-Embedded Length for 50.8 mm Unit Cell ...... 45

5-7. Type A Insertion Loss of 50.8 mm Unit Cell...... 45

5-8. Type A Reflection Coefficient of 50.8 mm Unit Cell ...... 46

5-9. Type A Dispersion Graph of 50.8 mm Unit Cell ...... 46

5-10. Type A Bloch Impedance of 50.8 mm Unit Cell ...... 47

5-11. Type A Three Cell Design of 50.8 mm Unit Cell ...... 48

5-12. Type A Three Cell Dispersion Graph of 50.8 mm Unit Cell ...... 48

5-13. 20 mm Unit Cell Design Circuit ...... 50

5-14. 20 mm Unit Cell Dispersion Graph ...... 50

5-15. Type A 20 mm Unit CellDesign ...... 52

5-16. Type A Design of 20 mm Unit Cell with De-embedded Length ...... 52

5-17. Type A Dispersion Graph of 20 mm Unit Cell ...... 53

5-18. Optimizing Capacitor and Inductor ...... 54

5-19. Type B Design of 20 mm Unit Cell...... 55

5-20. Type B Insertion Loss of 20 mm Unit Cell ...... 56

5-21. Type B Reflection Coefficient of 20 mm Unit Cell...... 56

5-22. Type B Dispersion Graph of 20 mm Unit Cell ...... 57

5-23. Type B Bloch Impedance of 20 mm Unit Cell ...... 57

5-24. Type B 3D Far Field Radiation Pattern of 20 mm Unit Cell at 0.5 GHz ...... 58

5-25. Type B 2D Far Field Radiation Pattern of 20 mm Unit Cell at 0.5 GHz ...... 58

5-26. Type B Three Cell Structure of 20 mm Unit Cell ...... 59

5-27. Type B Three Cell Insertion Loss ...... 60

5-28. Type B Three Cell Reflection Coefficient ...... 60

5-29. Type B Three Cell Dispersion Graph ...... 61

5-30. Type B Three Cell Bloch Impedance ...... 61

xiii

5-31. Type B Three Cell Simulated at 0.5 GHz ...... 62

5-32. Type B Three Cell Simulated at 0.8 GHz ...... 63

5-33. Type B Three Cell Simulated at 0.3 GHz ...... 63

xiv

LIST OF PROGRAMMING CODES

Programing Codes Page

1. Extracted Left Hand Parameters ...... 20

2.Calculation for MIM Capacitor Length ...... 21

3.Excerpt of Calculation for SS Inductor ...... 23

4. Capacitors and Inductors Calculation for 20 mm ...... 49

5. MIM Capacitor and SS Inductor Extracted Dimensions ...... 51

LIST OF ACRONYMS

1D One Dimension 2D Two Dimension 3D Three Dimension ADS Advance Digital System BPF Band Pass Filter CRLH Composite Right Left Hand CSRRs Complementary Slip-Ring Resonators DS-CSRRs Double-Slit Complementary Slip-Ring Resonators DS-SRRs Double-Slit Slip-Ring Resonators LH Left Hand LPF Low Pass Filter HFSS High-frequency Structural Simulator HPF High Pass Filter IC&SI Interdigital Capacitor and Shunt Stub Inductor MEMS Microelectromechanical System MIM Metal-Insulator-Metal MIM&SS Meta-Insulator-Metal and Squared Spiral MIMO Multiple-input Multiple-output MMIC Monolithic Microwave Integrated Circuits MTMs Metamaterials PLH Purely Left Hand PRH Purely Right Hand RH Right Hand SNR Signal-to-noise Ratio SRRs Slip-Ring Resonators SS Squared Spiral

xvi

1 INTRODUCTION

Antenna designers are faced with difficult challenges as antenna now need to be smaller, have better signal-to-noise (SNR) ratio, be capable of tunable frequency, and have increased performance in directivity. Since a traditional solution is no longer a viable option, different methods have been proposed to increase antenna performance. One of the solutions is replacing conventional materials with artificial materials, known as metamaterials (MTMs). MTMs have the potential to address these mentioned issues.

Jamil et al., researchers at Universitiy Teknologi Petronas, shows that with metamaterials, the size of the antenna decreases the overall design and the radiation pattern performance increases based on MTMs in multiple-in multiple-output (MIMO) design [10]. In addition, Turpin et al., researchers at Pennsylvania

State University, reports that they are able to manufacture a tunable antenna with MTMs as a base [15].

They are also able to tune the frequency using different methods such as varactor diodes and switches.

These studies have shown that MTMs are the next practical solution for improving antenna design and this project will investigate their use.

1.1 Application

An example of a particular area that benefits from MTMs is satellite antennas. MTMs will improve antenna performance and could be a viable solution for addressing issues for satellites’ antenna usage as discussed by Farr and Henderson [13]. In [13], they discuss some of the challenges for the new- generation of nano satellites. For example, satellite antennas need to be small and capable of tuning different frequencies. These presented challenges could be resolved by using MTMs antenna. As discussed earlier, MTMs antenna is small and tunable, due to its unique properties which will be discussed further in this paper. Thus, the application of MTMs antenna could be the next viable solution for future satellite communication.

1.2 Current Researches

Improving antenna bandwidth is just one of the many benefits MTMs have to offer. According to

A. Radwan et al., graphene (a planar layer of carbon atoms bonded in a hexagonal structure) can be used to increase operating frequency up to THz ranges. Graphene allows for tunable resonance frequency at low bias voltage; however, some issues of antenna performance are low gain and narrow bandwidth. One out of two solutions is to use a hybrid of graphene and metal. This approach will increase the gain while keeping the bandwidth is still narrow. The other solution is to use a slip ring resonator (a MTMs approach) to increase gain and widen the bandwidth. The results provided by simulation shows that MTMs graphene will help boost the gain and bandwidth size [28].

Instead of using graphene, W. Wijayanto et al. proposed the idea of using MTMs with LiNbO3, an optical crystal substrate, to improve operational bandwidth at millimeter-wave while maintaining a small size antenna. A millimeter-wave has an operational bandwidth between 30 GHz and 300 GHz. Arrays of 2 dimensional MTM circuits are fabricated on top of the LiNbO3 substrate. Measurements were taken with a

1.55 micrometer wavelength light that propagates through the fabricated device. The results show that due to MTMs and LiNbO3 properties, optical carrier power in free space is possible at 90 GHz as the obtained power is 7 dB [29].

For devices that are not operating at extremely high frequency, MTMs are used to improve performance for managing increasing capacity of users per bandwidth. These users can be either people, locations, or devices. This mean that the wireless communication systems need to be reconfigurable for operational bandwidth. Y. Luo et al. proposed the idea of using MTMs and microelectromechanical systems (MEMS) to reconfigure the beam scanning of radiation. The tested device is a microstrip, leaky waveguide of J-Shaped MTMs with integrated MEMS. Operating at around 8 GHz, three different switches are used for scanning radiation beams. The yielded results show that the device can scan from negative to positive phase constants [30].

1.3 Focus of Project

This project will examine and research new implementation of MTMs using CAD software

Keysight ADS [32] and Ansys HFSS [33]. The new proposed design is the MIM capacitor and squared spiral (SS) inductor obtained from Bahl [5] [6]. Once parameters are extracted, MATLAB [34] is used to calculate for physical dimensions. Finally, the new design is simulated and the results are evaluated.

Chapter 1, briefly introduces MTMs and present-day challenges that MTMs could resolve.

Furthermore, Chapter 1 is a brief overview of traditional antenna types, microstrip, and radiation mechanism. Chapter 2 focuses on the definition of MTMs and its properties. The introduction and design of the new proposed changes are all recorded in Chapter 3. This pertains to parameter values, circuit design, simulation using different software, graphs, equations, etc. Chapter 4 will then exclusively analyze Caloz and Itoh’s MTMs in ADS and HFSS. Meanwhile, Chapter 5 shows the results of the new design from

HFSS simulation. Finally, Chapter 6 examines the proposed changes to MTM and contains the concluding remarks regarding the obtained results.

1.4 Antennas

According to A. Balanis [8], from the 1970’s to the 1990’s six types of traditional and fundamental antennas have been introduced: wire, aperture, microstrip, array, reflector, and lens. Wire antennas are radiating wires such as monopole or early T.V. antennas. Aperture antennas are geometrically shaped antenna such as a horn antenna. Microstrip antennas are conductive metal patches on top of a grounded substrate such as patch antenna. Array antennas are arrays of other antennas set a distinct distance for maximum performance. Reflector antennas are usually curved surface metallic antenna designed for redirecting waves such as satellite antennas. Lens antennas use lenses to collimate diverged energy into plane waves. These antennas can all be improved upon by using metamaterials; however, this project will focus solely on improvement of the microstrip antenna type.

1.4.1 Microstrip

Microstrip antennas have been widely popular since around the 1970s. Some basic characteristics of microstrip are two conductive metal surfaces with a substrate between them. One side is a grounded plane, while the other side is the design fabricated by an etching machine. Thick substrate provides better efficiency and larger bandwidth and vice versa for small substrate. An example of a microstrip patch antenna is seen on Figure 1-1 provided by C. Balanis [8]. The source is provided by an identified edge and is usually attached to a feed for impedance matching purposes. Matching in microstrip describes the process of optimizing a line to obtain desired gain, bandwidth, and resonance frequency to the load without changing the load. An example of this feed can be seen in Figure 1-1(a); the feed is the line connecting the left edge to the patch.

(a) (b)

Figure 1-1: (a) Isometric view of patch antenna. (b) Side view of patch antenna [8]

1.5 Radiation Mechanism

C. Balanis [8] describes how radiation propagates due to oscillation of charges over time and when there is any curved, bent, discontinuous, terminated, or truncated conductor. During operation, microstrip generates radiation due to some of its properties. For example, some microstrip are either terminated or truncated from the ground resulting in radiation. This occurs due to an unbalanced charge between the two surfaces. These microstrip radiation properties are classified and focused on by A. Ochetan and G.

Lojewski [16] and are referred to as waveguided and resonance. These attributes are important and must be considered when designing a microstrip antenna as it affects the performance of the board.

1.5.1 Traveling Wave

Traveling wave describes the propagation of waves through a waveguide by radiating throughout the length of the structure. These are typically known as leaky-wave antenna. Waveguide structures support slow-waves and fast-waves.

Slow-wave occurs when phase velocity is less than the speed of light (vp ≤ c) or (vp/c ≤ 1) where vp is phase velocity and c is speed of light. This is generally seen in surface wave antenna. According to C.

Balanis [8] who references C. H. Walter [20], a slow wave structure is “an antenna which radiates power flow from discontinuities in the structure that interrupt a bound wave on the antenna surface”. This means that slow-wave traveling through waveguided structure frequently occurred during nonuniformities, curvatures, and discontinuities.

Fast-wave occurs when phase velocity exceeds the speed of light (vp ≥ c) or (vp/c ≥ 1). Fast-wave generally occurs in leaky-wave structure which is described by C. Balanis [8] as “an antenna that couples power in small increments per unit length, either continuously or discretely from a traveling wave structure to free-space”. Furthermore, Leaky-wave can be identified in two categories, uniform or quasi-uniform, as described by D. R. Jackson, et al. [11].

1.5.2 Resonance

In resonant antenna, there is narrow-bandwidth and higher dispersion using resonance mechanism

[2]. Resonance occurs when the antenna is terminated in an open or short circuit at the load or at the end of the feed line. The operating wave produces a standing wave where amplitudes at each point are constant.

This occurs when current traveling from the source, the incident wave, is superimposed by the current traveling from the load, the reflected wave. Resonance is also the reason why the desired operating bandwidth occurs at different frequencies called resonant frequencies. Resonance is important in metamaterial as each cell is terminated to the ground. MTMs are also used to improve on zero-order (when phase constant is zero) devices proposed by C. Lai et el [9]. The proposal in the article states that MTMs could be used to create a zero-order design allowing for more miniature antennas as the resonance is independent from physical length.

2 METAMATERIAL INTRODUCTION

Refraction describes the phenomenon of the wave characteristic when traversing from one medium to another. In nature, refraction tends to have a positive index. Counterintuitive, artificial structures could have negative refraction. Materials with this property are identified as metamaterials

(MTMs) and the broad definition of electromagnetic MTMs defined by Caloz and Itoh [1] is “artificial effectively homogeneous electromagnetic structures with unusual properties not readily available in nature”. The term effectively homogeneous refers to the structural cell size, p, and is less than the quarter of guided wave length, λ/4. The purpose of the artificial effectively homogeneous electromagnetic structures is to ensure the refractive phenomena dominates over the scattering/diffraction property in MTMs, where

MTMs behave similar to real material. In 1967, V. Veselago [21] introduced this concept by identifying a homogenous electromagnetic structure with unique electromagnetic properties which have negative permittivity (εr) and permeability (μr). Equation 3.1 shows a general case of for both positive and negative refraction. In [21], the effect of negative ε and μ creates a left-handed (LH) backward-wave propagation often abbreviated to LH wave. Similarity, the right-handed (RH) forward-wave propagation is shortened to

RH. This concept is shown in Figure 2-1 as illustrated by Caloz and Itoh [1]. Fundamentally, backward- wave propagation consists of negative refraction index such that the phase velocity is anti-parallel to group velocity.

Refraction Index 푛 = ±√ε푟μ푟 (3.1)

Figure 2-1: Quadrant Sections Identifying the Different Combination of Signed Permittivity and Permeability. [1]

2.1 Metamaterials Theory

In order to achieve negative refraction, the materials need to exert negative εr and μr. Veselago

[21] introduces backward-wave propagation or “LH” to emphasize the propagation of electromagnetic waves with the electric field. To summarize Veselago’s paper described by Caloz and Itoh [1], if a backward-wave propagation or LH material exists, negative refraction also exists such that currently defined phenomena must be redefined to incorporate this LH propagation. Examples of these phenomena are Doppler’s effect, Snell’s Law, Vavilov-Cerenkov, etc. Currently there is no natural material that is able to exhibit this characteristic; however, artificial structures can be made to mimic this behavior. The first structure to fully incorporate the negative refraction is the thin wire slip ring resonator which has the form of a lattice crystal (LC).

2.2 Slip Ring

In 1995, Pendry et al. [14] experimented with Slip-Ring Resonators (SRRs) which exhibit a positive ε and negative μ. Following Pendry’s SRRs, complementary split-ring resonators (CSRRs) were proposed by F. Falcone et al. in 2004 [22] [23]. Double-Slit Complementary Split Ring Resonators (DS-

CSRRs) and Double-Slit Split-Ring Resonators (DS-SRRs) was introduced by R. Marques et al. [24] as reported by S. Jindal [17]. The four types of SRRs are shown in Figure 2-2 as illustrated by N.

Wiwatcharagoses [3]. A SRRs equivalent model is shown in Figure 2-3 provided by Caloz and Itoh [1].

Overall, SRRs have different modifications, but consistently provide negative permeability.

Figure 2-2: (a) SRRs, (b) CSRRs, (c) DS-SSRs, (d) DS-CSRRs. [3]

Figure 2-3: SRRs and its Equivalent Circuit Model [1]

2.3 Structural Thin Wires

J.B. Pendry [25] [26] et al. experimented with Thin-Wire Structure which is parallel thin wires along the z plane and each thin wire is separated by a certain distance to the nearest thin wires in the x and y planes. This exhibits a negative εr and positive μr. An illustration is shown in Figure 2-4.

Figure 2-4: (a) Thin-wire Structure exhibiting negative permittivity (εr). (b) SRRs structure exhibiting negative permeability (μr) [24,25, 1].

2.4 Thin Wires Slip Ring Resonator

Combining structural thin wires with SRRS creates a composite which demonstrates a negative ε and negative μ with the cell size p less than the guided wavelength (λg). This composite material is then classified as metamaterials as explained by Caloz and Itoh [2]. The experiment was first setup by a team at

UCSD led by D. R. Smith. An illustration of their experiment can be seen in Figure 2-5.

Figure 2-5: Thin Wire Slip Ring Resonator [26, 1]

2.5 Transmission Line Approach

The physical application of MTMs, that is thin wires and SRR, introduces backward-wave propagation. Around June 2002, three different groups proposed a transmission line approach to achieve negative permittivity (εr) and permeability (μr) for a single cell 1 Dimensional 1D transmission line. An equivalent circuit is shown in Figure 2-6, where CL and LL represent the capacitor and inductor required to mimic backward wave propagation or Purely Left Hand (PLH). When creating PLH, there exist parasitic effects that exhibit Purely Right Hand (PRH) nature. Thus, PRH material currently does not exist and this creates a Composite Right and Left Hand (CRHL) material.

2.5.1 Purely Left Hand Unit Cell

The PLH circuit is realized as shown below. This circuit model is also identical to that of a high pass filter and is classified as PLH. Caloz and Itoh [1] [2] showed that phase velocity is negative and antiparallel to the group velocity. This means that the n is negative along with μr and εr. PLH also has a unique dispersion of its phase constant, βPLH, as shown in Figure 2-7.

Figure 2-6: Left Hand Circuit per Cell

Figure 2-7: Dispersion Graph of PLH

LH Phase Constant βPLH = 1/ω√C퐿L퐿 (3.2)

LH Angular Frequency ωL = 1/√C퐿L퐿 (3.3)

LH Characteristic Impedance ZPLH = √C퐿L퐿 (3.4)

2.5.2 Purely Right Hand

The PRH circuit is shown below and as mentioned before, PRH is related to MTMs transmission line due to its parasitic nature. The circuit model of PRH is identical to that of low pass filter. The phase velocity is positive and parallel to the group velocity. Its n, μr, and εr are all positive. The dispersion of phase constant is linear as shown in Figure 2-9.

Figure 2-8: Right Hand Circuit per Cell

Figure 2-9: Dispersion Graph of PRH

RH Phase Constant βPRH = ω√C푅L푅 (3.5)

RH Angular Frequency ωR = 1/√C푅L푅 (3.6)

RH Characteristic Impedance ZPRH = √C푅L푅 (3.7)

2.5.3 Composite Right and Left Hand

Due to the parasitic nature of transmission lines, there is no pure PLH materials; thus, a composite material exists which is known as composite right and left hand (CRLH). A circuit-equivalent model of the

CRLH unit cell is shown in Figure 2-10 along with its dispersion characteristic as shown by Caloz and Itoh

[1]. This circuit exerts a negative n with negative phase velocity that is antiparallel to the group velocity.

The equations listed below are discussed and derived by Caloz and Itoh [1].

Figure 2-10: Proposed 1D CRLH MTMs. (a) Cell of CRLH MTMs. (b) Wave Propagation Comparison of CRLH to both PLH and PRH. [1]

−1 푖푓 ω < min(휔 , 휔 ) 퐿퐻 푟푎푛푔푒 Sign Function 푠(ω) = { 푠푒 푠ℎ (3.8) +1 푖푓 ω > min(휔푠푒, 휔푠ℎ) 푅퐻 푟푎푛푔푒

2 2 2 CRLH Phase Constant βCRLH = √(ω/ω푅) + (ω퐿/ω) − κω퐿 (3.9)

Guided Wavelength λg = 2π/|βCRLH|

2 2 2 λg = 2π/√(ω/ω푅) + (ω퐿/ω) − κω퐿 (3.10)

CRLH Characteristic Impedance ZCRLH = ZPRH = ZPLH (3.11)

1 Per-Unit-Length Impedance 푍′ = j(ω퐿푅 − ) (3.12) 휔퐶퐿

1 Per-Unit-Length Admittance 푌′ = j(ω퐶푅 − ) (3.13) 휔퐿퐿

Complex Propagation 훾 = 훼 + 푗훽 = √푍′푌′

2 2 2 훾 = 푗푠(휔)√(ω/ω푅) + (ω퐿/ω) − κω퐿 (3.14)

Resonance Frequency Series ωse = 1/√L푅C퐿 (3.15)

Resonance Frequency Shunt ωsh = 1/√L퐿C푅 (3.16)

Constant κ = 퐿푅퐶퐿 + 퐿퐿퐶푅 (3.17)

4 Center Resonance Frequency ω0 = 1/√√L푅퐶푅퐿퐿C퐿 (3.18)

2.5.4 CRLH Balanced and Unbalanced

CRLH is considered balanced when its resonances ωse and ωsh are equal to each other and CRLH is unbalanced when they are not equal to each other. For the unbalanced case, the group velocities are 0 and, as a result, a stop band or gap emerges. For the balanced case, the resonance is suppressed as ωse = ωsh and this allows “matching over an infinite bandwidth” according to Caloz and Itoh [1]. In this project, both the unbalanced and balanced case is evaluated; however, the design only focuses on the balanced case.

2.5.5 CRLH Symmetrical

For repeated cell with external Ports, a matching design is required. This is accomplished simply by adding a feed at both ends to make the circuit symmetrical. If the circuit does not have symmetrical feed, then Zin does not equal Zout or S11 does not equal S22. If there is symmetrical feed, then Zin is equal to Zout or S11 is equal to S22.

2.6 Bloch Impedance

Previously, the characteristic impedance has been defined as Equation 3.11 for a LC network. Itoh and Caloz [1] redefine this new characteristic impedance as Bloch Impedance or ZB for MTMs. For a unit cell, ZB is defined in Equation 3.18. For a more general symmetrical case, ZB is defined as Equation 3.19.

√(푍푌/2)2+푍푌 ZB Unit Cell 푍 = (3.18) 퐵 푌

2 2 2 √((ω/ω푠푒) −1) ω퐿 ω ZB General 푍퐵 = 2 − { [( ) − 1]} (3.19) ((ω/ω푆ℎ) −1) 2ω ω푠푒

Caloz and Itoh proposed the MTM Unit Cell 1D transmission line as shown in Figure 2-11 [1].

The interdigital capacitor represents the series capacitor, CL, with N fingers. The length of the finger is determined by the expected parasitic inductor, LR. The short circuit stub represents the shunt inductor, LL.

Using parameter extraction, the width and length of the stub is dependent on LL and CR. P is the length per cell where each cell is composed of an interdigital capacitor and a short circuit stub. Wc is the width of the interdigital capacitor and within it, the spacing, width, and length, lc, of the fingers needs to be solved. Ws is the width of the stub and ls is length of the stub.

Figure 2-11: Unit Cell. Interdigital Capacitor and Short Circuit Stub [1]

3 NEW MTM CELL DESIGN

The proposed new design is replacing the interdigital capacitor to a monolithic microwave integrated circuits (MMIC) metal-insulator-metal (MIM) capacitor and the shunt stub inductor to a MMIC squared spiral inductor. The MIM capacitor is a capacitor that has two microstrip ribbons separated by an insulator. The two metal strips will be stacked horizontally as shown in Figure 3-1. Figure 3-1 shows the squared spiral that will be used in this proposal. The unit cell is formed by adding a MIM capacitor and a shunt squared spiral inductor.

(a) (b) Figure 3-1: (a) MIM Capacitor, (b) Squared Spiral Inductor [6]

The purpose of this proposal is to provide additional methods in the creation of microstrip MTMs.

In addition, the design and the simulated results will be analyzed for comparison between this thesis proposal and Caloz and Itoh’s proposal – interdigital capacitor and short stub or shunt stub inductor.

3.1 Right Hand Capacitor and Inductor Extraction

The first process for designing is to define the requirements. The design specification for the proposed MTM is shown in Table 1. The next step is to extract the lumped elements from a microstrip line with a certain length. For this project, the line length is 50.8 mm or 2 inches and the equivalent model of a microstrip line is seen in Figure 3-2. The schematic that is necessary to extract the parameters is shown in

Figure 3-3. First the microstrip line is grounded to calculate for the inductor because when the microstrip line is grounded then the inductance can be observed in the reflection coefficient. The capacitance is evaluated in a similar fashion using an open microstrip line. The simulated results from ADS,

ParameterExtract, are shown in Figure 3-4 using Equation 3.19 and 3.20 for lumped elements and 3.20 for

18 characteristic impedance from Dr. Markovic’s lecture [31]. The capacitor value is 5.693 pF and the inductor value is14.23 nH with characteristic impedance of 50 Ω. The conductivity of metal is set to the

ADS default setting for simplicity. To summarize, the capacitor and inductor values are extracted at 50.8 mm microstrip line.

Right Hand Inductor Value L = imag(Z11)/width (3.19)

Right Hand Capacitor Value L = imag(Y11)/width (3.20)

Characteristic Impedance 푍표 = √Z11/Y11 (3.21)

Table 1: Design Specification Design Specification Values Center Frequency, W0 500 MHz System Impedance 50 Ohm Substrate Height 1.57 mm Substrate Material RT Duroid 5880 Dielectric Constant 2.2 Metal Thickness 35 μm Roughness 2.1 mil

Figure 3-2: RLC Equivalent Model of Microstrip Line

Figure 3-3: Capacitor and Inductor Extraction

Figure 3-4: Inductor (left), Capacitor (right), and Impedance (bottom) extracted at 500 MHz

3.2 Left Hand Capacitor and Inductor Extraction

The right-hand values are then used to obtain the left-hand capacitor and inductor values. This is accomplished by using Equations provided by Caloz and Itoh [1]; however, some values are rearranged to solve for the unknown values. This is accomplished by equating the resonant frequencies of the left and right hand components as Equation 3.15 and 3.16 is equal to each other for a balanced case. Rearranging this equality, a ratio is obtained as shown in Equation 3.22, where R is the ratio. To solve for the left hand capacitor and inductor, the right hand (RH) and left hand (LH) is multiplied by that ratio. Afterwards, the ratio is tuned around 1 until ω0 is equal to 3.14 GHz from Equation 3.18 and is shown Programming Code

1 in MATLAB file TwoInchLine. The ratio is found to be 1.252, the calculated LH capacitor is 7.124 pF, the LH inductor is 17.816 nH, and the center angular frequency is 3.141 Giga (1*109) radiance per second or approximately 500 MHz. Programming Code 1 is used for these calculated values and Table 2 shows the summarized values.

Programming Code 1: Extracted Left Hand Parameters

퐶 퐿 Ratio R = 퐿 = 퐿 (3.22) 퐶푅 퐿푅

Table 2: Capacitor, Inductor, and Frequency Values Nomenclature Values Right Hand Capacitor, CR 5.693 pF Right Hand Inductor, LR 14.23 nH Left Hand Capacitor, CL 7.124 pF Left Hand Inductor, LL 17.816 nH Center Frequency, W0 500 MHz Center Angular Frequency 3.14G rps

3.2.1 Design of MMIC Left Hand Capacitor

Previously, the LH capacitor was evaluated to be 7.124 pF. From Bahl’s book [5], Equation 3.23 and 3.24 is utilized to build a physical MIM capacitor. There are three parameters that can affect the physical build of the capacitor and they are: DMC (the insulated thickness), ZoMC (width of the capacitor line), and ErMC (the dielectric constant of insulator). The values of these parameters are defined and shown in Table 3. DMC and ErMC is chosen to be a thin piece of paper. ErMC is specified as 4.8 mm, the same physical width for a 50 Ohm line and matches Table 1 specifications. From Equation 3.23, LP (parasitic inductance of the MIM capacitor) and CL (LH capacitor) is considered extremely small and this equates to

2 W0 being divided by a large value. Therefore, the effective capacitance is multiplied by 1 plus a small value or for simplicity, the effective capacitance is approximately CL. The line length of the effective capacitance is calculated to be 14.775 mm, roughly 15 mm. Programming Code 2 from TwoInchLine

MATLAB file is used for calculation and Figure 3-5 is an approximate drawing of the physical build of

MIM capacitor provided from Bahl [5].

Programming Code 2: Calculation for MIM Capacitor Length

2 푊0 Effective Capacitance 퐶푒푓푓 = 퐶퐿 ∗ (1 + ( 2)) (3.23) (1/(퐶퐿퐿푝))

퐶푒푓푓∗36∗푝푖∗퐷푀퐶 Capacitance Line Length 퐿퐶푙 = −15 (3.24) 퐸푟푀퐶∗10 ∗푍표푀퐶

Table 3: MIM Capacitor Parameters Defined Parameters Values DMC 88 μm ZoMC 4.8 mm ErMC 2 W0 3.141 G rads/sec Results Ceff 7.124 pF Line Length, cL 7.3 mm

Figure 3-5: Example Drawing of MIM Capacitor [5]

3.2.2 Design of MMIC Left Hand Inductor

Equation 3.25 is the given effective inductance from Bahl [6] where WP is the parallel resonant frequency and Cp is the associated parasitic capacitance that exists in the squared spiral (SS) from Equation

3.26. Since the parallel resonant frequency is much greater than W0, the effective inductance is approximately LL (LH inductor). In order to design a simple spiral, 2 turns (n) and 8 sections (m) is set as static variables. The parameter lL (length of line or initial squared length), WL (width of line), and SL

(spacing between strip) are tuned until a calculated inductance value is approximately equal to the effective inductance. This is accomplished by using mutual inductance approach. The first step is calculating the SS

self inductances (Equation 3.27) where tL is height of the substrate and Kg is a constant variable. The next step is calculating the positive mutual inductances and the negative mutual inductances (Equation 3.28).

Negative mutual inductance does not refer to negative values but it represents two currents flowing in opposite directions. The final step is summing all inductances as shown in Equation 3.29 and the calculated squared spiral inductor is approximately 17.44 nH. Programming Code 3 shows an excerpt of the programing code that is used for finding the SS physical dimension. Table 4 lists the dimension of the SS and Table 5 lists the self and mutual inductance values.

Programming Code 3: Excerpt of Calculation for SS Inductor

2 푊0 Effective Inductance 퐿푒푓푓 = 퐿푙/(1 − ( ) ) (3.25) 푊푃

Parallel Resonant Frequency 푊푃 = 1/푠푞푟푡(퐿퐿퐶푃) (3.26)

Microstrip Inductance

−4 푙퐿 푊퐿+푡퐿 퐿푖(푛퐻) = 2 ∗ 10 ∗ 푙퐿 (ln ( ) + 1.193 + ( )) ∗ 퐾푔 (3.27) 푊퐿+푡퐿 3푙퐿

Mutual Inductance

2 2 −4 푙퐿 푙퐿 1/2 푙퐿 1/2 푑 푀푎,푏 = 2 ∗ 10 ∗ 푙퐿 (ln ( + (1 + 2 ) ) − (1 + 2 ) + ) (3.28) 푑 푑 푑 푙퐿

푚 푛 푚−4 푚−2 Total Inductance 퐿 = ∑푖=1 퐿푖 + 2 ∗ [∑푗=1(∑푖=1 푀푖,푖+4푗 − ∑푖=1 푀푖,푖+2푗)] (3.29)

푊 푊 Constant Variable 퐾푔 = 0.57 − 0.145 푙푛( 퐿) , 퐿 > 0.05 (3.30) 푡퐿 푡퐿

Table 4: SS Inductor Parameters Defined Parameters Values lL, Squared Length 8500 μm WL, Width of line 1100 μm SL, Spacing 600 μm n 2 m 8 tL 35 μm Kg 621.58*10-3 Results Leff 17.816 nH Calculated SS Inductance 17.44 nH

Table 5: Self and Mutual Inductance Values Self Values Mpos Values Mneg Values Inductance nH Inductance nH Inductance nH L1 3.435 Mpos1,5 1.787 Mneg1,3 0.736 L2 2.869 Mpos2,6 1.344 Mneg2,4 0.592 L3 2.569 Mpos3,7 1.119 Mneg3,5 0.577 L4 2.036 Mpos4,8 0.743 Mneg4,6 0.433 L5 1.756 - - Mneg5,7 0.418 L6 1.267 - - Mneg6,8 0.274 L7 1.015 - - Mneg1,7 0.577 L8 0.59 - - Mneg2,8 0.433 Ltotal 15.54 Mpostotal 4.995 Mnegtotal 4.043

Figure 3-6: SS Inductor and Parameters Example [6]

3.3 ADS and HFSS Comparison of MMIC Capacitor and Inductor

Once the dimensions are found for the capacitor and inductor, both elements are built and simulated in HFSS and compared with an ideal case from ADS results. The HFSS results are simulated from FinalCapItself and FinalIndItself with previously obtained dimensions from earlier sections. All

HFSS simulations are appropriately de-embedded. The ADS result is from LengthExtraction with the effective capacitance and inductance values. Comparisons are made using Smith Chart and measuring the reflection coefficient as shown in the following figures below under this section. The real and imaginary part impedances (Imp) of the devices from the smith charts are recorded in Table 6 showing the differences and the errors. Errors exist due to parasitic effect of the real device simulated in HFSS where ADS assume an ideal case for both the capacitor and inductor.

Table 6: Real and Imaginary Impedances Comparison Nomenclature ADS HFSS Error % Real Part Imp of Series Capacitor 1 0.8829 11.7 Imaginary Part Imp of Series Capacitor 0.9 0.6491 27.8 Real Part Imp of Shunt Inductor 0 0.0084 0 Imaginary Part Imp of Shunt Inductor 1.122 1.338 19.3

Figure 3-7: ADS Circuits with Lumped Elements

Figure 3-8: ADS Smith Chart

Figure 3-9: MIM Capacitor

Figure 3-10: MIM Capacitor Smith Chart

Figure 3-11: SS Inductor

Figure 3-12: SS Inductor Smith Chart

4 ANALYSIS OF PREVIOUS PUBLISHED MTM

4.1 Background Information

A. Cloninger [4] examined, designed, and simulated the proposed 1D CRLH MTMs. Cloninger’s results show that PLH lump elements is similar to a High Pass Filter (HPF). The dispersion graph of this

PLH simulation looks similar to Figure 2-7. Cloninger’s results also show that PRH lump elements is identical to a Low Pass Filter (LPF). The dispersion graph from her simulated PRH result appears similar to

Figure 2-9. The simulated result of the unit-cell CRLH shows a Bandpass Filter (BPF) and the dispersion graph is similar to a balanced case from Figure 2-10 (b) where 휔0 is equal to 휔se and 휔sh. Furthermore, she analyzed insertion loss, wave propagation, Bloch impedance, dispersion graph, etc. for a balanced, symmetrical CRLH MTMs with 3 cells. Cloninger’s results displayed identical characteristics from the

Caloz and Itoh [1] simulation and Cloninger is able to validate the interdigital capacitor and short circuit stub as a 1D MTMs transmission line, microstrip.

4.2 HFSS Results

This section investigates the microstrip approach of a MTMs transmission line. This is accomplished through extracting parameters and converting them to an interdigital capacitor and short circuit stub. The procedure on converting can be found in references from Bahl, Caloz, Itoh, etc. This section also investigates Ansoft HFSS software to further understand creating a design that could replicate their experiments. Ansoft [7] defined complex propagation, 훽p, as shown in Equation 4.1 and the Bloch

Impedance, ZB, as Equation 4.2 using obtained measurements from S parameters. Zo is system impedance and using LineCalc from ADS with the given material from Table 1, the width of the line is determined to be approximately 4.8 mm.

−1 1−푆11푆22+푆12푆21 훽p β = cos ( ) (4.1) 2푆21

2jZ표S21sin(β푝) ZB Measured 푍퐵 = (4.2) (1−S11)(1−S22)−S21S12

4.2.1 CRLH Lump Ports Single Cell

Lumped port for a single cell is examined in this section. The parameters that are used are from

Ansoft [7] initial parameters as seen in Table 7. The dispersion graph should show that approximately at

2.4 GHz, the traveling wave should be a standing wave when 훽p is at 0 degrees. The wave should be forward wave for frequency that is operating greater than 2.4 GHz. Similarly, the wave should be backward wave for frequency that is operating below 2.4 GHz. As for ZB, the Bloch impedance imaginary and real should both be equal to approximately 0 at 2.4 GHz. A lumped port single cell unit is simulated in HFSS

LumpPortSingleCell file at a frequency of 2.4 GHz and a sweep from 1GHz to 6 GHz. From Figure 4-2, when 훽p is equal to 0, the center frequency is approximately 1.4 GHz. Furthermore, the Bloch impedance shows that approximately at 1.4 GHz, the real and imaginary is equal to 0. In addition, a radiation pattern is shown in Figure 4-5 simulated at 2.4 GHz and the pattern shows non-uniformity. The results from

LumpPortSingleCell do not agree with the above predictions because of unwanted series inductance that is between the unit cell and ports. The lump ports simulation is slightly faster than wave ports simulation, but there is no option for de-embedding the system which causes inaccurate measurements.

Table 7: Parameters Variables Values Center Frequency 2.4 GHz System Impedance 50 Ohm Substrate Height 1.57 mm Substrate Material RT Duroid 5880 Unit-Cell 11.4 mm Stub Length 10.9 mm Stub width 1.00 mm Finger Length 10.2 mm Finger Width 0.30 mm Spacing between finger 0.20 mm Via radius 0.12 mm

Figure 4-1: 3D Model Design Lump Ports Single Cell

Figure 4-2: Dispersion Graph of Lump Ports Single Cell

Figure 4-3: Bloch Impedance of Lump Ports Single Cell

Figure 4-4: Insertion Loss of Lump Ports Single Cell

Figure 4-5: 3D Far Field Radiation Pattern at 2.4 GHz

4.2.2 CRLH Wave Ports Single Cell

This section focuses on examining wave ports with the same parameters used in Table 1. With wave ports, the option for de-embedding is possible and the length of the de-embedded line is 5 mm. From

Figure 4-7 and Figure 4-8 provided by WavePort simulation, the dispersion graph and block impedance graph agree with the previous prediction. At approximately 2.4 GHz, 훽p is equal to 0 and the Bloch impedance real and imaginary values are also equal to 0 Ohm at 2.4 GHz. The dispersion graph also looks identical to Ansoft’s [7] MTM example. The LH wave seems to start below approximately 2.4 GHz and the

RH wave starts above approximately 2.4 GHz. The insertion loss is identical to a bandpass filter, except with losses at certain frequencies. The radiation pattern looks more uniform and more symmetrical than the lumped port design at 2.4 GHz. The peak directivity is 0.738 and peak gain is 0.554 at 2.4 GHz

Figure 4-6: 3D Model Design Wave Ports Single Cell

Figure 4-7: Dispersion Graph of Wave Ports Single Cell

Figure 4-8: Bloch Impedance of Wave Ports Single Cell

Figure 4-9: Insertion Loss of Wave Ports Single Cell

Figure 4-10: 3D Far Field Radiation Pattern of Wave Ports Single Cell at 2.4 GHz

Figure 4-11: 2D Far Field Radiation Plot at 0º on Polar Plot of Wave Ports Single Cell at 2.4 GHz

4.2.3 CRLH Wave Ports Three Cell Symmetrical Feed

This section examines a wave port symmetrical, balanced 3-cell structure. The 3D model is simulated in the HFSS ThreeCellWavePort file and the results are seen below in this section. The balanced

3-cell structure adds a smaller interdigital capacitor at both ends and a shunt stub near the first port. Due to these new additions, the dispersion graph and Bloch impedance do not match the previous prediction. The complex propagation, 훽p is equal to zero at approximately 2GHz. Also, the real and imaginary Bloch impedance of the structure are both almost equal to 0 Ohm at approximately 2.05 GHz. This indicates that the center frequency is roughly 2 GHz. LH wave is below approximately 2 GHz and RH wave is above about 2 GHz. Additional tuning is required for both interdigital capacitor and shunt stub inductor in order to achieve a center frequency of 2.4 GHz. The insertion loss also indicated a bandpass filter effect, but the loss looks irregular as the loss fluctuates throughout the frequency sweep compared to the single cell insertion loss. The radiation pattern also shows sign of uniformity and symmetry at 2.4 GHz. The peak directivity is 0.954 and peak gain is 0.805 at 2.4 GHz

Figure 4-12: 3D Model Design of Symmetrical Three Cell Structure

Figure 4-13: Dispersion Graph of Symmetrical Three Cell Structure

Figure 4-14: Bloch Impedance of Symmetrical Three Cell Structure

Figure 4-15: Insertion Loss of Symmetrical Three Cell Structure

Figure 4-16: 3D Far Field Radiation Pattern of Three Cell Structure at 2.4 GHz

Figure 4-17: 2D Far Field Radiation Plot at 0º on Polar Plot of Three Cell Structure at 2.4 GHz

5 NEW DESIGN RESULTS

Chapter 5 discusses the new proposed design. First design is Type A and it refers to the SS inductor where the spiral is outward away from the capacitor as shown in Figure 5-5. Three HFSS simulation is performed for Type A and they are 50.8 mm unit cell, three 50.8 mm unit cell, and 20 mm unit cell. The second type is Type B and the spiral for Type B is inward towards the capacitor as shown in

Figure 5-15. Type B has two different models and they are 20 mm unit cell and three 20 mm unit cell.

5.1 Type A 50.8 mm (2-inch) Unit Cell

Previously in Chapter 4, the values for left-hand and right-hand elements were obtained. Table 8 summarizes the extracted parameters from the earlier section. This section focuses on combining and simulating the MIM capacitor and SS insulator in ADS for a 50.8 mm or 2-inch microstrip unit cell. The first step is to use the capacitor and inductor values to build a circuit identical to Figure 2-10.The next step is to analyze the insertion loss, reflection coefficient, and the dispersion graph. Afterward, the physical design is built on HFSS and the results are further analyzed.

5.1.1 ADS Results

Figure 5-1 shows the schematic for a CRLH unit cell with the obtained lumped elements values and the schematic is built in ADS, 508mm folder. The CRLH unit cell is simulated between 1MHz and 1

GHz with a step of 1MHz. The insertion loss from Figure 5-2 shows a bandpass filter with center frequency at 500 MHz and the reflection coefficient also exhibit that, at 500 MHz, the return loss is below -70 dB.

The dispersion graph shows an ideal MTMs property with center angular frequency at 3.142 Giga radiance per second (Grad/s) or 500 MHz. Above 500 MHz, forward wave is expected and below 500 MHz, a backward wave is expected. Furthermore, the dispersion graph shows a balanced CRLH with an expected standing wave at 500 MHz.

Table 8: Lumped Elements Parameters Elements Values Right Hand Capacitor 5.693 pF Right Hand Inductor 14.23 nH Left Hand Capacitor 7.124 pF Left Hand Inductor 17.816 nH

Figure 5-1: ADS New CRLH Unit Cell of 50.8 mm Line Length

Figure 5-2: ADS Insertion Loss of 50.8 mm Unit Cell

Figure 5-3: ADS Reflection Coefficient of 50.8 mm Unit Cell

Figure 5-4: ADS Dispersion Graph of 50.8 mm Unit Cell

5.1.2 HFSS Results

From Section 3.3, the MIM capacitor and SS inductor are combined into a 50.8 mm or 2-inch unit cell structure as shown in Figure 5-5. The design dimension of the LH capacitor and inductor parameters are listed in Table 9. This design is then built and simulated at 500 MHz in HFSS FinalFirstDesign file with a frequency sweep from 0.1 GHz to 3 GHz and is labeled as Type A. The length is de-embedded to 5 mm from ports 1 and 2 respectively as shown in Figure 5-6. From the dispersion graph, the center frequency is approximately at 0.5 GHz and the bandgap is not narrow. This means that this configuration is slightly not balanced. Between 0.45 GHz and 0.55 GHz, the wave should look similar to a standing wave.

Forward wave is expected above approximately 0.55 GHz and backward wave is expected below roughly

045 GHz. The Bloch impedance graph shows that the real and imaginary impedance is equal to 0 Ohm approximately at 0.53 GHz. The insertion loss shows a bandpass filter as expected. From Figure 5-8, the reflection coefficient is approximately -15.5 dB at 500 MHz. The peak directivity is 3.0266 and peak gain is 1.3609 at 500 MHz.

Table 9: Initial Parameters Defined Parameters Values Zo, Width of Capacitor 4.8 mm lC, Length of Capacitor 7.3 mm lL, Squared Spiral Length 8.5 mm WL, Width of Inductor Line 1.1 mm SL, Spacing Between Spiral 0.6 mm p, Cell Y Length 50.8 mm H, Substrate Height 1.57 mm th, Copper Thickness 35 μm

Figure 5-5: Type A Design of 50.8 mm Unit Cell

Figure 5-6: Type A Design with De-Embedded Length for 50.8 mm Unit Cell

Figure 5-7: Type A Insertion Loss of 50.8 mm Unit Cell

Figure 5-8: Type A Reflection Coefficient of 50.8 mm Unit Cell

Figure 5-9: Type A Dispersion Graph of 50.8 mm Unit Cell

Figure 5-10: Type A Bloch Impedance of 50.8 mm Unit Cell

5.1.3 Type A Three 50.8 mm Unit Cell

The next goal is to evaluate the three-cell structure for this 50.8 mm unit cell. The unit cell is repeated three times on the y-axis and is built in HFSS ThreeCell2Inches file. The three-cell structure is simulated at 0.5 GHz with a frequency sweep from 0.1GHz to 2GHz. The three-unit cell structure is shown

Figure 5-11 and the dispersion graph is shown in Figure 5-12. The dispersion graph shows that around 500

MHz, the 훽p is narrower than the unit cell indicating a more balanced structure. Above approximately 0.5

GHz, forward wave is expected and backward wave is expected below 0.5 GHz. The peak directivity and gain are lower than that of the single cell. The three-unit cell peak directivity is 0.221 and the peak gain is

0.12 simulated at 500 MHz.

Figure 5-11: Type A Three Cell Design of 50.8 mm Unit Cell

Figure 5-12: Type A Three Cell Dispersion Graph of 50.8 mm Unit Cell

5.2 Type A 20 mm Unit Cell

The next step is to investigate miniaturizing the original 50.8 mm design into a 20 mm unit cell.

The first step is to find the values of CRLH lump elements from the original design. This process is accomplished by applying ratio of 50.8 mm and 20 mm to the CRLH as shown Equation 6-1 and 6-2 as x is the capacitor or inductor and y is the unknown variable. The reason why the RH is multiplied by 0.4

(20mm/50.8mm) is because the physical line is shorter. To accommodate for balancing this line reduction, the LH is multiplied by 2.5 (50mm.8/20mm). The results are shown in Table 10. The calculation is shown in the MATLAB file, TwentymmLine. Programming Code 4 shows this ratio being applied to obtain the new capacitors and inductors values. Furthermore, Programming Code 4 shows that f0, center frequency, is

500 MHz or 휔0, the center angular frequency, is 3.14 Grad/s.

50.8 푚푚 푥 Right Hand Ratio = (6-1) 20 푚푚 푦

20 푚푚 푥 Left Hand Ratio = (6-1) 50.8 푚푚 푦

Programming Code 4: Capacitors and Inductors Calculation for 20 mm

Table 10: 20 mm CRLH Lumped Elements Defined Parameters Values Right Hand Inductor 5.692 nH Right Hand Capacitor 2.276 pF Left Hand Inductor 44.539 nH Left Hand Capacitor 17.809 pF

5.2.1 ADS Results

Similar to Section 5.1.1, the new capacitors and inductors values are simulated in ADS 20mm folder. The schematic and result are shown below in this section. The dispersion graph shows that 훽p is narrower than the 50.8 mm unit cell around f0. Similar to Section 5.1.1, forward wave is expected above approximately 5GHz and reverse wave is expected below approximately 5GHz. This simulation is proof that these lumped elements values could be a CRLH MTM.

Figure 5-13: 20 mm Unit Cell Design Circuit

Figure 5-14: 20 mm Unit Cell Dispersion Graph

5.2.2 HFSS Results

Similar to Section 3.2, MATLAB is used to calculate for MIM capacitor length and SS inductor parameters. Programming Code 5 from TwentymmLine MATLAB file is an excerpt of the code that is used for calculating the MIM capacitor and SS inductor dimensions. Table 11 summarizes the new 20 mm design dimensions for a 20 mm unit cell. The 20 mm unit cell is then simulated in HFSS TypeA20mm file.

The design with de-embed consideration and the dispersion graph are shown in the following figures under this section. The dispersion result from HFSS graph is narrower than 50.8 mm unit cell around 500 MHz indicating a more balanced cell structure. At 500 MHz, the peak directivity is 2.988 and peak gain is 1.627.

Programming Code 5: MIM Capacitor and SS Inductor Extracted Dimensions

Table 11: CRLH LH Capacitor and Inductor Dimensions of a 20 mm Unit Cell Defined Parameters Values MIM Capacitor Length 18.4 mm SS Inductor Length 12 mm SS Inductor Width 1 mm SS Inductor Spacing 0.5 mm p, Cell Y Length 19.9 mm

Figure 5-15: Type A 20 mm Unit CellDesign

Figure 5-16: Type A Design of 20 mm Unit Cell with De-embedded Length

Figure 5-17: Type A Dispersion Graph of 20 mm Unit Cell

5.3 Type B 20 mm Unit Cell

In order to decrease the cell size, the SS inductor is flipped inward parallel to the MIM capacitor.

This new design will be labeled as Type B as shown in Figure 5-19. This causes positive mutual inductance as the current in the inductor spiral and the capacitor flow in the same direction. This means that there is more parasitic series inductance. To account for this unwanted effect, an optimizing process is needed to have the dispersion graph centered at 500 MHz.

5.3.1 Optimizing

During optimization, the capacitor length and inductor length are tuned, simulated, and observed.

The tuning function is accomplished using HFSS optimetrics tool and the dispersion graph result is displayed in Figure 5-18. If the capacitor length decreases or the capacitance becomes smaller, the center frequency increases. If the inductor length decreases resulting in smaller inductance, the center frequency increases. An example of this is shown in Figure 5-18 and is simulated in HFSS Optimizing file.

Figure 5-18: Optimizing Capacitor and Inductor

5.3.2 HFSS Results

After optimizing the capacitor and inductor parameters, the best narrow dispersion line center at

500 MHz is shown in Table 12. The following figures in this section shows the HFSS results simulated from TypeB20mm file. The insertion loss shows a bandpass filter as expected. The reflection coefficient is approximately -27 dB which is smaller than Type A 50.8 mm unit cell. The dispersion graph shows a more balanced 훽p than both Type A 50.8 mm unit cell and Type A 20 mm unit cell. The Bloch impedance imaginary and real values are both equal to 0 approximately at 500 MHz. The 3D and 2D radiation pattern simulated at 500 MHz shows that this new design is also uniform and symmetrical similar to Caloz and

Itoh’s design. The peak directivity and gain measured at 500 MHz is 2.99 and 1.5.

Table 12: Type B Parameters Defined Parameters Values Zo, Width of Capacitor 4.8 mm lC, Length of Capacitor 19 mm lL, Squared Spiral Length 12 mm WL, Width of Inductor Line 1 mm SL, Spacing Between Spiral 0.5 mm px, Cell X Length 16.8 mm p, Cell Y Length 20.1 mm H, Substrate Height 1.57 mm th, Copper Thickness 35 μm

Figure 5-19: Type B Design of 20 mm Unit Cell

Figure 5-20: Type B Insertion Loss of 20 mm Unit Cell

Figure 5-21: Type B Reflection Coefficient of 20 mm Unit Cell

Figure 5-22: Type B Dispersion Graph of 20 mm Unit Cell

Figure 5-23: Type B Bloch Impedance of 20 mm Unit Cell

Figure 5-24: Type B 3D Far Field Radiation Pattern of 20 mm Unit Cell at 0.5 GHz

Figure 5-25: Type B 2D Far Field Radiation Pattern of 20 mm Unit Cell at 0.5 GHz

5.3.3 Type B Three 20 mm Unit Cell

Type B 20 mm unit cell design is used to create a Type B three-cell structure by repeating the single cell three times on the y-axis. This three-cell structure is simulated in HFSS and the file name is

TypeB20mm_500. The peak directivity of this three-cell structure at 0.5 GHz is 3.07 and the peak gain is

1.815. The Insertion loss still has a bandpass filter characteristic. The reflection coefficient at 0.5 GHz is approximately -20 dB which is greater than the single cell Type B. Both the dispersion graph and Bloch impedance shows that center frequency has shifted to approximately 0.48 GHz due to series parasitic capacitance. From the dispersion graph, near 0.5 GHz, the propagating wave should be a standing wave.

Above 0.5 GHz forward wave is expected and backward wave is expected below 0.5 GHz.

Figure 5-26: Type B Three Cell Structure of 20 mm Unit Cell

Figure 5-27: Type B Three Cell Insertion Loss

Figure 5-28: Type B Three Cell Reflection Coefficient

Figure 5-29: Type B Three Cell Dispersion Graph

Figure 5-30: Type B Three Cell Bloch Impedance

5.3.4 Simulated Electrical Field Vector

Figure 5-31 is simulated form TypeB20mm_500 file at 500 MHz and it shows that the electric field vector has a characteristic similar to a standing wave. For a standing wave, the phase velocity and group velocity are approximately equal and antiparallel to each other. The Forward wave occurs when both phase velocity and group velocity are parallel. This forward wave can be seen in Figure 5-32 simulated in

TypeB20mm_800 file at 800 MHz indicating positive refraction. Finally, backward wave determines if the design is capable of negative refraction which is one of the properties of MTMs. Error! Reference source ot found. shows a backward wave from the electric field and this simulation is performed in

TypeB20mm_300 simulated at 300 MHz. These simulated results show that the new design is capable of

CRLH MTM since forward, standing, and reverse waves exist. The animation of the wave propagation can be seen at http://athena.ecs.csus.edu/~saejaoo/MTM/Animation/.

Figure 5-31: Type B Three Cell Simulated at 0.5 GHz

Figure 5-32: Type B Three Cell Simulated at 0.8 GHz

Figure 5-33: Type B Three Cell Simulated at 0.3 GHz

6 CONCLUSION

Recently, MTMs have been heavily researched due to their intrinsic characteristic of negative refraction from having negative (εr) and (μr). This also indicates that the phase and group velocity are antiparallel where they are not equal to each other. This negative refraction creates a backward wave propagation and is demonstrated in Caloz and Itoh’s [1] proposed design which is an interdigital capacitor and shunt stub inductor unit cell. This project thesis introduced a new design replacing the interdigital capacitor and shunt stub inductor with MIM capacitor and SS inductor. Both Caloz and Itoh’s design and the new proposed design are analyzed using Keysight ADS [32] and Ansoft HFSS [33]. The new design,

MIM capacitor and SS inductor, radiation pattern is symmetrical and uniform similar to Caloz and Itoh’s design. Both designs are capable of backward-wave propagation as shown by their dispersion graph and

Bloch impedance. Due to the different operating frequencies, both designs could not be accurately compared. The new design can be smaller if it is to operate at the higher frequency such as 2.4 GHz. The cell size, peak directivity, and peak gain is compared in Table 13 when simulated at their respected center frequency. The proposed change of MIM and SS is an effective MTM design. For future research, additional SS inductor could be added on the left side of the MIM capacitor from Type B 20 mm unit cell design to create a symmetrical cell structure similar to Caloz and Itoh’s design of adding additional shunt stub near port 1 and two smaller interdigital capacitors at both ends.

Table 13: Comparison

Design f0 Cell Size Directivity Gain Caloz and Itoh Single Cell 2.4 GHz 11.4 mm 0.738 0.554 Caloz and Itoh Three Cell 2.4 GHz 11.4 mm 0.954 0.805 Type A 50.8 mm Unit Cell 0.5 GHz 28.1 mm 3 1.36 Type A Three 50.8 mm Unit Cell 0.5 GHz 28.1 mm 0.221 0.12 Type A 20 mm Unit Cell 0.5 GHz 30.9 mm 2.988 1.627 Type B 20 mm Unit Cell 0.5 GHz 20.1 mm 2.99 1.5 Type B Three 20 mm Unit Cell 0.5 GHz 20.1 mm 3.07 1.7

References

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Appendix A

%Two Inch Line or 50.8 mm Line clc;clear;format('longeng') %Select center frequency f0 = 500e6; %Hz W0 = f0*2*pi %rad/s

%Unit Cell length converted inch to m Length = 0.0254; %m c = 3e8; %Speed of light m/s lamda = c/f0; B = 2*pi/lamda; thetaRad = B*Length; thetaDeg = thetaRad*180/pi;

%Cell length needs to be less than lamda/4 AllowedLength = lamda/4; %m %Design of lumped elements at RF, microwave consider 1/10 lamda DesLength = lamda/10; %m

%Select either capacitor or inductor. For balance case Wse=Wsh %subbing in, leads to a ratio Cl/Cr = Ll/Lr

%Given from Parameter Extraction %----Programming Code 1------%2 inch line Cr = 5.69e-12; %RH capacitor Lr = 14.23e-9; %RH inductor

%Solving for ratio based on Wse=Wsh and Z0 %2 inch line ratio = 1.252 Ll = Lr*ratio %LH inducotr Cl = Cr*ratio %LH capacitor

%Cutoff frequency at low and high freq. Wl = 1/sqrt(Ll*Cl); Wr = 1/sqrt(Lr*Cr);

%Test Wsh = 1/sqrt(Ll*Cr); %Resonance Frequency series Wse = 1/sqrt(Lr*Cl); %Resonance Frequency Shunt TestZ0 = sqrt(Lr/Cr); %RH Characteristic Imp TestZo2 = sqrt(Ll/Cl); %LH Characteristic Imp TestWo = 1/nthroot(Lr*Cr*Ll*Cl,4) %Center Resonance Frequency

%------Programming Code 2 MIM Capacitor------d = 88; %um Given gap measurement between two metal 88 um paper ZoW = 4800; %um from width of line w/r to Z0 from line calc

Er = 2.2; %Dielectric constant of substrate Erd = 2; %Dielectric constant of film between two conductors.

%Wo = (1/(Lr*Cl)); %temporary variable Ceff = Cl%*(1+W0^2/Wo^2) Effective Capacitance

%Capacitance Line Length (Lcl) Lengthcl = Ceff*36*pi*d/(Erd*10^-15*ZoW) %um

%------Progamming Code 3 Square Spiral Inductor------%Page 21 of Bahl Lumped %wp = 1/sqrt((Ll*Cp)); %Temporary Variable Le = Ll %/(1-(W0/wp)^2) Le effective inductance.

W = 1100; %um WL width of the line t = 35; %um tL thickness of line metal h = 1570; %um Substrate Hieght S = 600; %um SL spacing between strip

Kg = 0.57-0.145*log(W/h); %constant variable lmax = 8500; %um first length of the strip (leg) m = 8; n = 2; %m section of leg and n = number of minimum turn

%physical length for each line l(1) = lmax; l(2) = l(1)-W; l(3) = l(1)-W-S; l(4) = l(1)-2*W-S; l(5) = l(1)-2*W-2*S; l(6) = l(1)-3*W-2*S; l(7) = l(1)-3*W-3*S; l(8) = l(1)-4*W-3*S;

%Self Inductance ------for i = 1:m L(i) = 2*10^-4*l(i)*(log(l(i)/(W+t))+1.193+(W+t)/(3*l(i)))*Kg; %nH; %l(i+1) = l(i)-W-S; end

%Positive Inductance ------for j = 1:n for i = 1:m-4*j le = (l(i)+l(i+4*j))/2; %um Page 35 d = j*(S+W); %um Page 35 Mpos(j,i) = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))- sqrt(1+d^2/le^2)+d/le); end end

%Negative Inductance ------

le = (l(1)+l(7))/2; W2 = 2; S2 = 1; d = lmax-S2*S-W2*W; MN1_7 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(1)+l(3))/2; W2 = 1; S2 = 0; d = lmax-S2*S-W2*W; MN1_3 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(2)+l(8))/2; W2 = 2; S2 = 2; d = lmax-S2*S-W2*W; MN2_8 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(2)+l(4))/2; W2 = 1; S2 = 1; d = lmax-S2*S-W2*W; MN2_4 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(3)+l(5))/2; W2 = 2; S2 = 1; d = lmax-S2*S-W2*W; MN3_5 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(4)+l(6))/2; W2 = 2; S2 = 2; d = lmax-S2*S-W2*W; MN4_6 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(5)+l(7))/2; W2 = 3; S2 = 2; d = lmax-S2*S-W2*W; MN5_7 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(6)+l(8))/2; W2 = 3; S2 = 3; d = lmax-S2*S-W2*W; MN6_8 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le);

SumMN = MN1_7+MN1_3+MN2_8+MN2_4+MN3_5+MN4_6+MN5_7+MN6_8;

TotalL = (sum(L)+2*(sum(sum(Mpos))-sum(sum(SumMN))))*(1*10^-9)

Appendix B

%20 mm Line clc;clear;format('longeng') %Select center frequency f0 = 500e6; %Hz W0 = f0*2*pi %rad/s

%Unit Cell length converted inch to m Length = 0.0254; %m c = 3e8; %Speed of light m/s lamda = c/f0; B = 2*pi/lamda; thetaRad = B*Length; thetaDeg = thetaRad*180/pi;

%Given from Parameter Extraction %2 inch line Cr = 5.69e-12; Lr = 14.23e-9;

%sqrt(Lr/Cr) = 50 TestZ0 = sqrt(Lr/Cr);

%Solving for ratio based on Wse=Wsh and Z0 %2 inch line ratio = 1.252 Ll = Lr*ratio Cl = Cr*ratio

%------Programming Code 4 Converting to 20 mm ------Cl2 = Cl*2.5; %LH Cap Ll2 = Ll*2.5; %LH Ind Cr2 = Cr*0.4; %RH Cap Lr2 = Lr*0.4; %RH Ind

%------Programming Code 5------%------MMIC Capacitor------d = 88; %um Given gap measurement between two metal 88 um paper ZoW = 4800; %um from width of line w/r to Z0 from line calc Er = 2.2; %Dielectric constant of substrate Erd = 2; %Dielectric constant of film between two conductors.

Ceff = Cl2 %Effective Capacitance

%Capacitance Line Length (Lcl) LengthCl = Ceff*36*pi*d/(Erd*10^-15*ZoW) %um %------Square Spiral Inductor------Le = Ll2 %Effective inductance.

W = 1000; %um specify by designer t = 35; %um thickness of metal h = 1570; %um Substrate Hieght S = 500; %um for spacing. Use to adjust length lmax = 12000; %um first length of the strip (leg) Kg = 0.57-0.145*log(W/h); %constant variable m = 8; n = 2; %m section of leg and n = number of minimum turn

%Initial conditions l(1) = lmax; l(2) = l(1)-W; l(3) = l(1)-W-S; l(4) = l(1)-2*W-S; l(5) = l(1)-2*W-2*S; l(6) = l(1)-3*W-2*S; l(7) = l(1)-3*W-3*S; l(8) = l(1)-4*W-3*S;

%Self Inductance ------for i = 1:m L(i) = 2*10^-4*l(i)*(log(l(i)/(W+t))+1.193+(W+t)/(3*l(i)))*Kg; %nH; %l(i+1) = l(i)-W-S; end

%Positive Inductance ------for j = 1:n for i = 1:m-4*j le = (l(i)+l(i+4*j))/2; %um Page 35 d = j*(S+W); %um Page 35 Mpos(j,i) = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))- sqrt(1+d^2/le^2)+d/le); end end

%Negative Inductance ------le = (l(1)+l(7))/2; W2 = 2; S2 = 1; d = lmax-S2*S-W2*W; MN1_7 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(1)+l(3))/2; W2 = 1; S2 = 0;

73 d = lmax-S2*S-W2*W; MN1_3 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(2)+l(8))/2; W2 = 2; S2 = 2; d = lmax-S2*S-W2*W; MN2_8 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(2)+l(4))/2; W2 = 1; S2 = 1; d = lmax-S2*S-W2*W; MN2_4 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(3)+l(5))/2; W2 = 2; S2 = 1; d = lmax-S2*S-W2*W; MN3_5 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(4)+l(6))/2; W2 = 2; S2 = 2; d = lmax-S2*S-W2*W; MN4_6 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(5)+l(7))/2; W2 = 3; S2 = 2; d = lmax-S2*S-W2*W; MN5_7 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le); le = (l(6)+l(8))/2; W2 = 3; S2 = 3; d = lmax-S2*S-W2*W; MN6_8 = 2*10^-4*le*(log(le/d+sqrt(1+le^2/d^2))-sqrt(1+d^2/le^2)+d/le);

SumMN = MN1_7+MN1_3+MN2_8+MN2_4+MN3_5+MN4_6+MN5_7+MN6_8;

TotalL = (sum(L)+2*(sum(sum(Mpos))-sum(sum(SumMN))))*(1*10^-9)