Finite Element Analysis of Emi in a Multi-Conductor Connector

FINITE ELEMENT ANALYSIS OF EMI IN A MULTI-CONDUCTOR CONNECTOR

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Mohammed Zafaruddin

May, 2013

FINITE ELEMENT ANALYSIS OF EMI IN A MULTI-CONDUCTOR CONNECTOR

Mohammed Zafaruddin

Thesis

Approved: Accepted:

______Advisor Department Chair Dr. Nathan Ida Dr. Jose Alexis De Abreu Garcia

______Committee Member Dean of the College Dr. George C. Giakos Dr. George K. Haritos

______Committee Member Dean of the Graduate School Dr. Hamid Bahrami Dr. George R. Newkome

______Date

ABSTRACT

This thesis discusses the numerical analysis of electrical multi-conductor connectors intended for operation at high frequencies. The analysis is based on a Finite

Element Tool and looks at the effect of conductors on each other with a view to design for electromagnetic compatibility of connectors. When there is an electrically excited conductor in a medium, it acts as an antenna at high frequencies and radiates electromagnetic power. If there are any conductors in its vicinity they too act as antennas and receive a part of the EM power. Electromagnetic Interference (EMI) from nearby excited conductors causes induced currents, according to Faraday‘s law, which is considered as noise for other conductors. This noise is affected by factors such as distance between the conductors, strength and frequency of currents, permittivity, permeability and conductivity of the medium between the conductors.

To reduce the effects of EMI in a shielded multi-conductor connector, values of induced currents at various distances between the conductors have been calculated and analyzed. The currents have been calculated at high excitation frequencies varying from

0.2GHz to 1GHz and distances between the conductors varying from 0.2 mm to 0.8mm.

iii

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my sincere gratitude to my advisor, Dr. Nathan Ida, for providing an opportunity to work under him, his guidance, encouragement and support during my graduate studies. His sound technical knowledge, managerial skills, confidence in me have been a source of inspiration.

I thank Dr. Ida for providing me with the software Ansoft HFSS. I would like to thank Dr. Steve Bardi from Ansoft Corporation for helping me understand technical aspects of the software tool and his support.

I would like to thank Drs. Hamid Bahrami and George Giakos for their insight and valuable ideas into the research.

I would also like to thank Dr. John Heminger who taught the course Advanced

Numerical Analysis which helped me understand the numerical and computational aspects of Finite Element Analysis in an effective way. His course work, exams and projects were very relevant, organized.

I would like to offer special thanks to the chair, Dr. De Abreu Garcia for supporting me financially by assigning a Graduate Assistantship throughout my Masters.

I acknowledge Ms. Tammy A Stitz for the help with my documentation and access to library resources.

Finally, I would like to thank my family and friends for their unconditional help and encouragement all the time. iv

TABLE OF CONTENTS Page

LIST OF TABLES ...... viii

LIST OF FIGURES ...... xi

CHAPTER

I. INTRODUCTION ...... 1

1.1 Electrical Connector...... 1

1.2 Finite Element Analysis ...... 2

1.3 Problem Description ...... 3

II. ELECTROMAGNETIC INTERFERENCE IN CABLES AND CONNECTORS ...... 4

2.1 Introduction ...... 4

2.2 Accidental Antennas ...... 5

2.3 Electromagnetic Compatibility for Cables...... 9

2.4 Electromagnetic Compatibility for Connectors ...... 12

2.5 Shielding ...... 14

III. FINITE ELEMENT METHOD AND ANALYSIS ...... 17

3.1 Finite Element Method ...... 17

3.1.1 Finite Element Formulations ...... 17

3.1.2 The Variational Method ...... 23

3.1.3 Variational Finite Element Method ...... 26

3.2 Computational Aspects in Finite Element Analysis ...... 29

3.3 Matrix Storage Schemes ...... 30

3.3.1 Compact Storage Method ...... 34

3.4 Matrix Computation ...... 35

3.4.1 Application of Boundary Conditions ...... 35

3.4.2 Direct Methods ...... 36

3.4.3 Iterative Methods ...... 42

3.5 Eigenvalues and Eigenvectors in Computational Electromgnetics ...... 47

3.6 Finite Element Based Electromagnetic Computation Software...... 49

3.6.1 Pre-Processor ...... 50

3.6.2 Processor ...... 55

3.6.3 Post-Processor ...... 55

IV. ANALYSIS AND RESULTS ...... 58

4.1 Introduction ...... 58

4.2 Description ...... 58

4.3 Dimensions ...... 60

4.4 Properties ...... 60

4.5 Case 1: Two Conductors connector ...... 60

4.5.1 Excitation ...... 61

4.5.2 Mesh ...... 61

4.5.3 Electric and Magnetic Field Plots ...... 64

4.5.4 Induced Currents ...... 68

4.6 Case 2: Three Conductors connector ...... 70

4.6.1 Excitation ...... 71

4.6.2 Mesh ...... 71

4.6.3 Electric and Magnetic Field Plots ...... 72

4.6.4 Induced Currents ...... 85

4.7 Summary ...... 91

V. CONCLUSIONS AND FUTURE WORK ...... 92

5.1 Conclusions ...... 92

5.2 Future Work ...... 92

BIBLIOGRAPHY ...... 94

APPENDICES ...... 97

APPENDIX A. LIST OF ABBREVIATIONS ...... 98

APPENDIX B. TABLES OF INDUCED CURRENTS FOR TWO CONDUCTOR CONNECTOR ...... 100 APPENDIX C. TABLES OF INDUCED CURRENTS FOR THREE CONDUCTOR CONNECTOR 106

vii

LIST OF TABLES

Table Page

3.1: Common methods for solving a system of linear equations [15] ...... 36

4.1(a): Mesh statistics (Tetrahedron edge length in [mm]) ...... 63

4.1(b): Mesh statistics (element volume [mm3])...... 63

4.2: Comparison of mesh elements [mm] for 2 conductor connector...... 64

4.3: Comparison of mesh elements [mm] for 3 conductor connector...... 72

B.1 (a): Induced Currents at 0.2GHz for 2mm spacing ...... 100

B.1 (b): Induced Currents at 0.2GHz for 4mm spacing ...... 100

B.1 (c): Induced Currents at 0.2GHz for 6mm spacing ...... 101

B.1 (d): Induced Currents at 0.2GHz for 8mm spacing ...... 101

B.2 (a): Induced Currents at 0.5GHz for 2mm spacing ...... 102

B.2 (b): Induced Currents at 0.5GHz for 4mm spacing ...... 102

B.2 (c): Induced Currents at 0.5GHz for 6mm spacing ...... 103

B.2 (d): Induced Currents at 0.5GHz for 8mm spacing ...... 103

B.3 (a): Induced Currents at 1GHz for 2mm spacing ...... 104

B.3 (b): Induced Currents at 1GHz for 4mm spacing ...... 104

viii

B.3 (c): Induced Currents at 1GHz for 6mm spacing ...... 105

B.3 (d): Induced Currents at 1GHz for 8mm spacing ...... 105

C.1 (a): Induced currents at 0.2 GHz for 2mm spacing with L, C excited ...... 106

C.1 (b): Induced currents at 0.2 GHz for 2mm spacing with L excited ...... 107

C.1 (c): Induced currents at 0.2 GHz for 2mm spacing with C excited ...... 107

C.2 (a): Induced currents at 0.5 GHz for 2mm spacing with L, C excited ...... 108

C.2 (b): Induced currents at 0.5 GHz for 2mm spacing with L excited ...... 108

C.2 (c): Induced currents at 0.5 GHz for 2mm spacing with C excited ...... 109

C.3 (a): Induced currents at 1 GHz for 2mm spacing with L, C excited ...... 110

C.3 (b): Induced currents at 1 GHz for 2mm spacing with L excited ...... 110

C.3 (c): Induced currents at 1 GHz for 2mm spacing with C excited ...... 111

C.4 (a): Induced currents at 0.2 GHz for 4mm spacing with L, C excited ...... 112

C.4 (b): Induced currents at 0.2 GHz for 4mm spacing with L excited ...... 112

C.4 (c): Induced currents at 0.2 GHz for 4mm spacing with C excited ...... 113

C.5 (a): Induced currents at 0.5 GHz for 4mm spacing with L, C excited ...... 114

C.5 (b): Induced currents at 0.5 GHz for 4mm spacing with L excited ...... 114

C.5 (c): Induced currents at 0.5 GHz for 4mm spacing with C excited ...... 115

C.6 (a): Induced currents at 1 GHz for 4mm spacing with L, C excited ...... 116

C.6 (b): Induced currents at 1 GHz for 4mm spacing with L excited ...... 116 ix

C.6 (c): Induced currents at 1 GHz for 4mm spacing with C excited ...... 117

C.7 (a): Induced currents at 0.2 GHz for 6mm spacing with L, C excited ...... 118

C.7 (b): Induced currents at 0.2 GHz for 6mm spacing with L excited ...... 118

C.7 (c): Induced currents at 0.2 GHz for 6mm spacing with C excited ...... 119

C.8 (a): Induced currents at 0.5 GHz for 6mm spacing with L, C excited ...... 120

C.8 (b): Induced currents at 0.5 GHz for 6mm spacing with L excited ...... 120

C.8 (c): Induced currents at 0.5 GHz for 6mm spacing with C excited ...... 121

C.9 (a): Induced currents at 1 GHz for 6mm spacing with L, C excited ...... 122

C.9 (b): Induced currents at 1 GHz for 6mm spacing with L excited ...... 122

C.9 (c): Induced currents at 1 GHz for 6mm spacing with C excited ...... 123

C.10 (a): Induced currents at 0.2 GHz for 8mm spacing with L, C excited ...... 124

C.10 (b): Induced currents at 0.2 GHz for 8mm spacing with L excited ...... 124

C.10 (c): Induced currents at 0.2 GHz for 8mm spacing with C excited ...... 125

C.11 (a): Induced currents at 0.5 GHz for 8mm spacing with L, C excited ...... 126

C.11 (b): Induced currents at 0.5 GHz for 8mm spacing with L excited ...... 126

C.11 (c): Induced currents at 0.5 GHz for 8mm spacing with C excited ...... 127

C.12 (a): Induced currents at 1 GHz for 8mm spacing with L, C excited ...... 128

C.12 (b): Induced currents at 1 GHz for 8mm spacing with L excited ...... 128

C.12 (c): Induced currents at 1 GHz for 8mm spacing with C excited ...... 129 x

LIST OF FIGURES

Figure Page

2.1: Radiation of Electric signals from a PCB trace [5] ...... 7

2.2: Frequencies used by communication devices [6] ...... 7

2.3: Noise emitted by electric and electronic devices [6] ...... 8

2.4: Accidental antenna behaviour of conductors [6] ...... 9

2.5: Signal cables close to CM return path ...... 10

2.6: Some arrangements of wire bundles and flat cables [6] ...... 13

2.7: Connector model for arrangement of cables shown in Figure 2.6 [6] ...... 13

2.8: Ten way connector carrying signals and power [6] ...... 14

2.9: Shielded cable and connector [6] ...... 15

3.1: Elements of a Finite Element Mesh ...... 17

3.2 (a): 2-D dielectric problem...... 19

3.2 (b): FE mesh over the solution domain in Figure 3.2(a) ...... 19

3.3: Coordinates of nodes ...... 20

3.4: Domain for section 3.1.3...... 26

3.5: Node numbering for a triangular mesh ...... 30

3.6: Non- zero elements of the coefficient matrix ...... 31

3.7: Band Matrix ...... 31

3.8: Skyline and Sparse storage schemes ...... 33

3.9: Processing time comparison between Banded storage and Sparse storage [12] ...... 34

3.10: Main steps in the Gaussian Elimination method ...... 37

3.11: Finite Element Mesh [20] ...... 51

3.12: Steps involved in Automatic Mesh Generation [14] ...... 53

3.13: Mesh generation using Delaunay‘s method ...... 54

3.14: Linear interpolation to trace equipotential lines ...... 56

3.15: Illustration of EM analysis in Ansoft HFSS [26] ...... 57

4.1: Connector ...... 59

4.2: Two conductor connector showing the conductors and dielectric sheets...... 61

4.3: Mesh for two conductor model with 2mm spacing ...... 62

4.4: E - Field for 0.2 GHz ...... 65

4.5: H - Field for 0.2 GHz ...... 65

4.6: E - Field for 0.5 GHz ...... 66

4.7: H - Field for 0.5 GHz ...... 66

4.8: E - Field for 1 GHz ...... 67 xii

4.9 :H - Field for 1 GHz ...... 67

4.10: Integration paths for each conductor ...... 68

4.11: Currents at 0.2GHz ...... 69

4.12: Currents at 0.5GHz ...... 70

4.13: Currents at 1 GHz ...... 70

4.14: Three conductors connector with conductors 1 and 2 excited...... 71

4.15: E and H field plot at 0.2 GHz for 2mm spacing ...... 73

4.16: E and H field plot at 0.5 GHz for 2mm spacing ...... 74

4.17: E and H field plot at 1 GHz for 2mm spacing ...... 75

4.18: E and H field plot at 0.2 GHz for 4mm spacing ...... 76

4.19: E and H field plot at 0.5 GHz for 4mm spacing ...... 77

4.20: E and H field plot at 1 GHz for 4mm spacing ...... 78

4.21: E and H field plot at 0.2 GHz for 6mm spacing ...... 79

4.22: E and H field plot at 0.5 GHz for 6mm spacing ...... 80

4.23: E and H field plot at 1 GHz for 6mm spacing ...... 81

4.24: E and H field plot at 0.2 GHz for 8mm spacing ...... 82

4.25: E and H field plot at 0.5 GHz for 8mm spacing ...... 83

4.26: E and H field plot at 1 GHz for 8mm spacing ...... 84

4.27: Induced currents at 0.2 GHz for 2mm spacing ...... 85 xiii

4.28: Induced currents at 0.5 GHz for 2mm spacing ...... 86

4.29: Induced currents at 1 GHz for 2mm spacing ...... 86

4.30: Induced currents at 0.2 GHz for 4mm spacing ...... 87

4.31: Induced currents at 0.5 GHz for 4mm spacing ...... 87

4.32: Induced currents at 1 GHz for 4mm spacing ...... 88

4.33: Induced currents at 0.2 GHz for 6mm spacing ...... 88

4.34: Induced currents at 0.5 GHz for 6mm spacing ...... 89

4.35: Induced currents at 1 GHz for 6mm spacing ...... 89

4.36: Induced currents at 0.2 GHz for 8mm spacing ...... 90

4.37: Induced currents at 0.5 GHz for 8mm spacing ...... 90

4.38: Induced currents at 1 GHz for 8mm spacing ...... 91

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CHAPTER I

INTRODUCTION

1.1 Electrical Connector

The term connector can be used to refer to a broad range of applications related to electrical, mechanical, mathematical and many other disciplines. An electrical connector might be described as ―an electromechanical component that provides a separable interface between two subsystems of an electrical system without unacceptable effects on the performance of the system‖ [1]. Connectors usually have metal contacts (usually plug and socket) for electrical conduction and dielectrics for electric insulation. The insulation may be lossy or lossless depending on needs. The insulation is as important as the conductors because a significant percentage of all connector failures are due to poor insulation [2]. The functional requirements of a connector depend largely on whether the contacts are used to carry a signal or to distribute power. Power connectors are used for high power applications usually conducting tens of amps of current and voltages of up to about 600V [3]. These are not the subject of interest in this work and require a different study. Signal connectors generally carry less than 1A with high frequency switching data that can reach into the GHz range.

A connector for digital data transfers usually consists of multiple conductors for data transfer, power and for reference (signal return). The ratio of these conductors i.e.,

signal to reference (S/R) is determined, to a large extent, by the common-mode noise

(cross talk) created in the conductor. At frequencies above 300MHz to over 1GHz, the connector becomes a transmission line element in the path of the signal, considering the signal delay occurring in the connector. When the frequency of the signals are over

300MHz, a small number of lines are required for signal returns and S/R ratios of 8:l or higher are workable. As frequency increases from 300 MHz to 1GHz range, issues of electromagnetic interference (EMI) became significant demanding allocation of more return lines taking the S/R ratio to 4:1 or less. Frequencies over 1GHz require allocation of nearly half of the lines for return paths to eliminate cross-talk taking the S/R ratio to nearly 1:1 [4].

Depending on the application, connectors can be shielded or unshielded.

Shielding of connectors protects them from EMI from external sources in their vicinity that radiate EM energy and to maintain electromagnetic compatibility (EMC). Also to maintain EMC in the connector itself the spacing between the conductors is varied and the signal and reference conductors are strategically placed.

1.2 Finite Element Analysis

Analysis of electromagnetic structures and the fields/signals they generate involve the solution of partial differential equations (PDEs) summarized by Maxwell‘s equations.

In general it is not possible to solve these problems analytically for any reasonably complex structure. The task has been computerized and software tools have been developed for their simulations. One of the methods implemented in many commercial software tools is the Finite Element Method (FEM). Practical application of Finite

Elements is called Finite Element Analysis (FEA). Chapter III in this work discusses in detail the mathematical, computational and implementation aspects of FEA software.

1.3 Problem Description

Following the fact that multi-conductor signal connectors must be designed with care to avoid EMI/EMC, the present work undertakes the calculation of the effects of signals on neighboring conductors in a connector structure. The purpose is the calculation of induced currents in conductors based on solving the field equations as a function of frequency and conductor separation, taking into account the dielectric of the connector as well as an external conducting shield. A simple connector made by sandwiching two dielectrics with two and three conductors in each case is simulated using a commercial software tool. The distance between the conductors is varied in steps as the induced currents on the conductors are measured in each case by varying the operating frequency.

The implementation and results are discussed in Chapter IV. The results of this work points to the need to either separate the conductors widely or to introduce reference conductors to separate the signal paths. It is hoped that these results will help in the design of better connectors.

The structure modeled here is also applicable to shielded and unshielded transmission lines as the model is a full-wave model, that is, it takes into account the propagation effects along conductors and in dielectrics. The size of the model has been specifically selected to be large enough to show these effects.

CHAPTER II

ELECTROMAGNETIC INTERFERENCE IN CABLES AND CONNECTORS

2.1 Introduction

Electromagnetic Interference (EMI) is the effect of electromagnetic waves on the performance of an electric circuit in its vicinity. EMI is caused by radiation of EM energy from electrical conductors at high frequencies. EMI could be intentional (for example in signal jamming or due to licensed transmitters) or unintentional as is usually found in analog and digital systems. The source of interference may not necessarily lie outside the circuit, but can be inside it when one part of the circuit interferes with another.

From the idea of EMI one defines a very important factor called Electromagnetic

Compatibility (EMC). Products and systems which use or generate signals and/or electronics are said to be Electromagnetically Compatible when their emissions do not cause unacceptable interference with other electrical and/or electronic products and systems, and when they have sufficient immunity to operate as required in their electromagnetic environment [5].

To transmit and receive high frequency EM signals one resorts to antennas of various sizes and shapes. But antennas are nothing more than conductors specifically designed to be good at converting conducted waves (electrical signals) into radiated waves in the air and vice versa [5]. The same applies to conductors used to conduct 4

signals. When high frequency signals usually above few hundred MHZ are transmitted over cables, the conductors act as accidental (unintentional) antennas and radiate the signals. These signals are picked up by nearby conductors that also act as accidental antennas and cause signal integrity (SI) problems in the circuit, that is, they degrade the signal due to introduction of unwanted signals at various levels. At best, this reduces data rates, at worst, it renders the system unusable. An example is cross talk seen in copper wire telephone cables which degrade the signal to noise ratio (SNR). Possible solutions to this problem are to increase the distance between the circuits or to shield the cables.

These may not always be possible. Shielding increases the cost, size and weight of conductors and the cost of manufacturing may be prohibitive.

2.2 Accidental Antennas

The antennas we use are made of conductors, designed to transmit and receive electromagnetic waves, composed of electric (E) and magnetic (H) fields. But the principles of working of antennas apply to any other conductors and hence any conductor carrying an AC current acts as accidental antennas. Figure 2.1 shows electric field leak produced by a PCB trace connecting pins of two IC‘s. This filed, together with the accompanying magnetic field cause propagation of energy away from the conductor.

Time varying electric fields produce changing magnetic fields in accordance with

Ampere‘s law:

(2.1)

Where H is the magnetic field intensity, J is the free (source) current density and D is the electric flux density. Faraday‘s law provides a second relation:

(2.2)

By using the constitutive relations (that is, the material relations) between the fields, relations that introduce material properties:

(2.3)

One obtains a wave equation of the following form:

̅ ̅ ̅ (2.4)

̅ ̅ ̅ (2.5)

These fields form an electromagnetic wave that propagates in space and in media and interact with any structure in their path. The properties of the media in which they propagate define their properties including speed of propagation and attenuation.

In general, the radiation from signal conductors is weak compared to antennas but may be significant enough to cause EMC problems. The following figures show, in a schematic way, the relative levels of interfering signals from various sources based on antenna lengths at different frequencies. Figure 2.2 shows the frequencies used in various communication devices.

Figure 2.1: Radiation of Electric signals from a PCB trace [5]

Figure 2.2: Frequencies used by communication devices [6]

The figure shows frequencies being used for domestic AC power, radio, TV, cables, cellphones and personal communication devices over the range 10Hz to 2.5GHz. The spectrum also contains noise added by switching devices, rectifiers, microprocessors etc. 7

Figure 2.3 shows the sources of noise over the previous frequency domain, again indicating relative levels.

Figure 2.3: Noise emitted by electric and electronic devices [6]

Figure 2.3 suggests the need to contain the noise emitted by various devices, their cables and connectors. Usually antennas are driven at one of their ends with a low impedance source and a high impedance load at the other end. If an antenna has a length of quarter wavelength (λ/4) it is said to be resonant and therefore ideal at the specific frequency at which it is resonant. Any antenna length that is not resonant will be much less efficient at transmitting signals. The same applies to signal conductors. Those consuctors that are multiples of quarter wavelengths in length are efficient antennas whereas other lengths will not. Figure 2.4 shows this schematically.

Figure 2.4: Accidental antenna behaviour of conductors [6]

In addition to these effects, electromagnetic waves along conductors can be reflected from loads if these are not matched to the impedance of the lines. Proper impedance matching at the terminals of conductors is a good approach to minimize reflections, avoiding resonance and hence making them very poor accidental antennas [7].

2.3 Electromagnetic Compatibility for Cables

High frequency noise is encountered mostly in high frequency digital data conductors and PWM power signals but AC and DC power lines can also carry fluctuating RF noise. The conductors in digital electronic devices, rectifiers, local oscillators have differential mode (DM) and common mode (CM) currents. DM mode signals are typically the wanted signals. In this signaling scheme signals are transmitted across two wires to reduce CM voltage such as the difference of grounding level between signal source and the load, and to reduce induced noise on signal transmission wires. It 9

also reduces radiated emission from signal transmission line. However DM signals are often converted to CM noise mainly due to unbalanced impedance between signal source, transmission line and load [8]. This is the main cause for EMC emission problems above

1MHz and even the performance of EMI filters gets worse in the high frequency range

(f>10MHz) [9]. The CM to DM signal conversions are main cause for immunity problems above 1MHz. Because DM noise is easy to identify and control but CM noise is hard to characterize, more concerns arise with CM noise [9].

Apart from filtering, CM noise can be eliminated by using the conductive metal body/chassis as a return path. The chassis should be connected to the 0V reference of the

DM signal circuit. Also the CM circuit current loop should have smallest possible area to reduce emissions and to improve immunity. This can be done by running the signal conductors very close to the chassis (or ground) or the CM return path. The following circuit explains this requirement schematically.

Signal cable (has both send and Device 1 return conductors) is routed very Device 2 close to CM return path

Input Output

CM return path (Chassis or 0V) Figure 2.5: Signal cables close to CM return path

In Figure 2.5, devices 1 and 2 could be either in same product or in the same system but in different products. The lines in red show cables containing both send and return paths.

They are run close to the CM return path shown in black. Instead of metal chassis, metal cable tray, metal plate, metallized surfaces or foil wrapped cardboards can be used.

Sometimes capacitors can be used in series with the CM return path to achieve galvanic isolation at power supply frequencies while allowing RF CM current to be grounded.

It may not always be possible to use chassis/ground as a CM return path, as it may not be available and may not be possible to create an artificial one as said before. This may be the case in portable electronic equipment such as mobile phones, laptops etc., aircrafts, ships where ground is not available and a sheet metal is used as ground. In such cases greater control to minimize DM to CM conversion is required and higher filtering can be used.

In Figure 2.5 if there were multiple inputs and outputs (connected with two conductors for send and return), with all the return conductors connected with a low resistance common ground (chassis) then the current path is as described next. We know that at low frequencies the impedance is dominated by resistance whereas at high frequencies inductance dominates the impedance of circuits. The inductance of a loop increases with its radius. So, at high frequencies the impedance of the circuit will be lower if its area is minimized. It can be concluded that at low frequencies (< 1KHz) the return currents will flow through the low resistance chassis and at high frequencies return currents will flow through conductors which are physically closest to the send

conductors. This is an important design aspect and is used in the connector design section.

2.4 Electromagnetic Compatibility for Connectors

There are many types of connectors available in the market generally with poor

EMC performance [6]. To improve their performance, as suggested in the above section there should be adequate number of return conductors placed and properly spaced for minimum interference. These additional conductors should be connected to chassis or 0V reference planes at both ends via low impedance RF bonds (direct connection or series capacitance). Ideally if there are a series of conductors like a ribbon cable then there should be return conductors placed beside each send conductor. Consider an arrangement of cables as shown in Figure 2.6. It shows flat cables and bundles of connectors. The red conductors represent the signal conductors and the blue represent the return conductors.

These sorts of arrangements can guarantee improvement in EMC more than 10dB, up to a minimum of 200MHz.

A model of connectors for the arrangement of conductors as in Figure 2.6 can be seen in Figure 2.7. It can be observed that for a flat cable and bundle model with fewer return conductors, the arrangement is made such that the return conductors are in the vicinity of each signal carrying conductor. When there are equal number of send and return conductors they are placed adjacent to each other. For a twisted pair conductor, the return pin in the connector should be closest to the send pin.

shown in the Figure 2.8 works best. For the ten-way connector shown, conductors are arranged such that there is at least one return conductor close to power and signal conductors so as to reduce EMI.

Figure 2.8: Ten way connector carrying signals and power [6]

2.5 Shielding

Shielding of cables and connectors is done by surrounding them with metal layers. For good EMI screening the cables and connectors need to be shielded all along them including. There should be no gaps in the shield. Shielding a cable with a low impedance metal layer screens EMI signals from reaching the conductors inside and make them less susceptible to interference. They also prevent interference from signals being transmitted by the conductor. The shield needs to be grounded or connected to the chassis. Like any other conductor the shield has a skin depth (δ) given by

√ (2.6)

Where ρ is the resistivity of the shield, ω is angular frequency of the interfering signal and μ is the magnetic permeability of the shield.

The current induced in the shield will be

(2.7)

Where Js is the current density on the surface

d is depth from surface

J is current density at any point in the shield

Figure 2.9: Shielded cable and connector [6] The current density decreases by a factor of 1/3 (approximately 0.37) for every skin depth. The shield should be many skin depths in thickness at the lowest frequency to be shielded for good shield effectiveness. The shield of the connector should be bonded to

the shield of the cable. A shielded cable and connector is shown in Figure 2.9. In EM simulation software a shield is implemented as a perfect electric conductor. That means that the electric field on the surface of a perfect conductor is orthogonal to the conductor.

CHAPTER III

FINITE ELEMENT METHOD AND ANALYSIS

3.1 Finite Element Method

The Finite Element Method (FEM) is based on discretization of the problem domain into many small regions called ―finite elements‖. The elements can be two or three dimensional with different types of geometry (ex: triangles, tetrahedra, hexahedra).

These elements form a mesh over the solution domain. Figure 3.1 shows a triangular and tetrahedral element with their nodes. Smaller, denser elements produce solutions that are more accurate.

Node

Triangular Element Tetrahedral Element

Figure 3.1: Elements of a Finite Element Mesh

3.1.1 Finite Element Formulations

Use of FEM reduces PDE‘s to a system of algebraic equations. To understand the finite element process, we use here an electrostatic problem. Although the purpose of this work is the analysis of high frequency issues in connector, the discussion here is valid. 17

Certainly, in the high frequency case, one models the full Maxwell‘s equations but the process is very similar.

Electrostatic problems often lead to Poisson‘s and Laplace equations. For example the electric potential distribution in the dielectric domain where an electrostatic field exists is given by

(3.1)

Where ρ is volume charge density, ε is the electric permittivity of the dielectric and V is scalar electric potential. If there are no free charges in the domain then equation (3.1) will become Laplace equation as shown (assuming there is only one dielectric in the domain)

(3.2)

Solving these equations by analytical methods for large problems is impossible. The finite element method is a tool is used to solve partial differential equations (PDEs). The

FEM reduces PDE‘s to a system of algebraic equations for steady state problems which can be solved using numerical analysis methods. Time-dependent problems can also be solved by coupling the finite element method with a time approximation method such as the finite difference method. FEM implementation is done generally, using either the variational approach or the Galerkin method. Application of variational FEM to physical problems is based on minimization of energy. It reduces the energy variational equations by matrix equations (discussed in next section) which can be easily solved using advanced numerical analysis methods.

We start by discussing the application of FEM to a 2-D problem as shown in

Figure 3.2 (a) by discretizing it using a triangular element mesh. The problem domain has two different dielectrics with potentials applied at two boundaries. When the domain is discretized the elements are created such that each element is in its own boundary (i.e. elements cannot cross material boundaries). One element cannot have two different materials within it. Also the mesh needs to be conforming (All nodes coincide with vertices of triangles). A non-conforming mesh yields incorrect results unless the formulation is modified specifically to include non-conforming elements.

V=Va V=Va

ε ε0 0

ε1 d ε1

V=Vb V=Vb Figure 3.2 (a): 2-D dielectric problem Figure 3.2 (b): FE mesh over the solution domain in Figure 3.2(a) This example is demonstrated with first order finite elements. The potential varies linearly with in the element as

( ) (3.3)

Higher order elements exist and yield better results compared to first order elements but they are more complex to implement and consume more processing time when implemented in software packages. Although most commercial FEM tools include high

order elements in practice, first order elements are more popular since one can always increase the number of elements to obtain the same level of accuracy as with high-order elements.

P3(x3,y3)

P1(x1,y1)

P2(x2,y2)

Figure 3.3: Coordinates of nodes

Figure 3.3 shows coordinates of an element with its nodes P1, P2, P3 locally numbered as

1, 2, and 3. When the element is seen from the complete mesh the numbering will be different. The numbering directly corresponds to storage patterns in a matrix and they are also renumbered for effective storage as discussed in section 3.3. The element in the figure should satisfy Equation (3.3), resulting in three equations

Solving the three equations for a1, a2, a3 in terms of the node potentials and their coordinates we get

∑

Where D is twice the area of the element given by

√ ( )( )( ) (3.6)

Where‗s‘ is the semi perimeter of the triangle given by and a,b,c are the sides of triangle.

Substituting the values of a1, a2, a3 in Equation (3.3) gives

( ) ∑ ( ) (3.7)

Where

And

Similarly p2, p3, q2, q3, r2, r3 are obtained by cyclic permutation of the indices i.e. 1→2,

2→3, 3→1.

These formulations are similar for any element shape (although there are other methods that can be used as well). The difference between elements manifests itself during the computation phase of the FEM.

Consider now the calculation of the electric field (E) in the element using the above formulations.

( ̂ ̂ ) (3.8)

Using Equation (3.3)

( )

and ( )

We can see that Ex and Ey are constant values in the triangle.

Writing Ex and Ey in terms of node potentials and coordinates using Equation (3.5)

∑ ( ̂ ( ) ̂ ( ))

or ∑ ( ̂ ̂ ) (3.9)

∴ ( ) (3.10)

( ) (3.9)

The electric field remains constant throughout the element (for first order triangular elements). This is a disadvantage as this may introduce discretization errors and does not allow the plot of isolines of the electric field. This can be overcome by a two-step algorithm [10]. In step one the constant electric field in an element is calculated as discussed above. In the second step the 3 E-field nodal values are calculated as the weighted mean of constant electric field of the elements surrounding the nodes and that

which have same permittivity as the finite element. The x component of electric field of a node k and element j ( ) is calculated as

∑ (3.10) ∑ for all elements connected to node k and have the same permittivity as that of element j.

The weighing function can be the area of the element i, the interior angle of the node k in element i, the distance between node k and the centroid of element i, or it may be 1.

Obviously each choice has its own properties and effect on accuracy.

3.1.2 The Variational Method

As said earlier one of the methods for formulation of the FEM is the Variational method. The calculation procedure resulting from the application of the variational method in conjunction with FEM is called variational FEM. It is a purely mathematical method used to convert the formulation into algebraic equations. The idea behind it is to minimize the energy function corresponding to the equation to be solved. Physically it corresponds to an equilibrium state of any physical system possessing potential energy.

Consider an equation ―ax+b=0‖ to be solved. One way of finding a solution to it is to minimize the function (assuming of course that a functional exists)

( )

It has a minimum at x when . This is the same as solving the original equation

―ax+b=0‖

An explanation of the variational method is given below by applying it to a functional F in the 3-D Cartesian domain. Let F, defined in volume v is function of a variable P with partial derivatives P'x, P'y, P'z. Then

( ) ∫

For F to be stationary (minimum), a small variation in P is δP and the corresponding variation in F is

∫ ( ) (3.11)

( ) ( ) ( ) Where , ,

Solving Equation (3.11) we arrive at

∫ . ⃗/ ∫ ⃗ ̂ (3.12) where S is the surface enclosing the volume v, ̂ is normal unit vector on S and

̂ ⃗ ̂ ̂ (3.13)

Where ̂ ̂ ̂ are unit vectors in the direction of x, y, z respectively.

Since the two integrals in Equation (3.12) are independent, each has to be zero to satisfy the equation:

∫ . ⃗/ (3.14)

⃗ ̂ (3.15) ∫

δP is a variable, therefore Equation (3.14) can be rewritten as

. ⃗/

Substituting ⃗ from Equation (3.12), the above equation reduces to Euler‘s equation given by

( ) ( ) ( )=0 (3.16)

Now looking at Equation (3.15), either or ⃗ ̂

The term is a Dirichlet boundary condition. Since , P is a constant on the boundary of surface S.

The term ⃗ ⃗ ̂ represents a Neumann boundary condition, which is

(3.17)

̂ Where ̂ ̂ ̂

Summarizing the variational method we can say that to find the value of P, by minimizing the functional F the following conditions must be satisfied:

 For functional F to be valid it must have an integrand that satisfies Euler‘s

equation (Eq. (3.16))

 Either of the boundary conditions i.e., Dirichlet (P constant at the boundary), or

Neumann (Eq. (3.17)) should be satisfied.

3.1.3 Variational Finite Element Method

Finite Element Method associated with the variational method results in calculation technique called as variational finite element method. This method is primarily used in computer software programs to solve the EM equations via numerical techniques. This section discusses the formulation of the matrix equations starting from the PDEs.

To do so, we apply this method to an electrostatic field function for a dielectric medium over the domain shown in the Figure 3.4

b d a c k h i f g

e m

j l

Figure 3.4: Domain for section 3.1.3 The field function (energy) over the 2-D domain is given as

∫ ( ) ∫ ( ) (3.18)

The functional can be split in the domain as

∑ (3.19)

Assuming that the mesh has K nodes, we minimize the functional for the node potential

∑ (3.20)

All the terms in the equation are zero except for the elements sharing the node

∑ (3.21)

Applying this discretization to Equation (3.18), we get (considering element i and nodes k=1,2 or 3 belonging to the element)

∫ ( ) (3.22)

and ∫ ( ) (3.23)

The first part (Part1) of Equation (3.23) can be written as (note that E is constant in an element):

∫ . / ( ) (3.24)

Using Equation (3.9), the above equation reduces to

0 ∑ ( ̂ ̂ ) 1

Using the vector identity ̅ ̅ ̅ and solving, the equation reduces to

,( ) ( ) ( ) -

Writing the above equation in matrix form for nodes 1, 2 and 3 of element i

[ ][ ] (3.25)

It should be observed that the coefficient matrix is symmetric and is called ―elemental matrix‖ or ―stiffness matrix‖. For a (n×k) matrix its general term is given by

( ) ( ) (3.26)

Calculating the second part (Part2) of Equation (3.23)

∫ ( )

Solving the above integral and representing the result in matrix form leads to

[ ] (3.27)

Substituting Equations (3.25) and (3.27) in Equation (3.23):

[ ][ ] [ ] (3.28)

[ ]

This is the basic building block of a finite element solution. It is only for one element i of the mesh derived for electrostatic potential for a dielectric medium. In general this equation looks like

[ ][ ] [ ] (3.29)

[ ]

The vector Q is called source vector of the problem which could be charges, currents or permanent magnets depending on the problem.

Applying Equation (3.29) to Equation (3.20) results in an assembly of all elemental matrices:

(3.30)

[ ] [ ] [ ]

As said earlier the node numbers 1, 2 and 3 were local to an element and correspond to some global numbering as seen from the matrix. Equation (3.30) can be represented in matrix equation as

, -* + * + (3.31)

This system is often called a global system of equations. Before the equation is solved the

Dirichlet boundary condition (constant value at a node) has to be applied on the system, which simplifies the coefficient matrix. The Neumann boundary condition is implicit in the variational method and need not be applied (i.e. a Neumann boundary is not enforced explicitly).

3.2 Computational Aspects in Finite Element Analysis

There are a few factors which need to be considered for practical implementation of the FEM in computational software. Memory size and computational speed are two of these. These are critical issues because the large coefficient matrices generally associated 29

with electromagnetic problems may result in hours and even days of simulation time [11].

The coefficient matrix [D] in Equation (3.31) is always symmetric except for the case of moving bodies. The program needs to store only half of the matrix with the principal diagonal when [D] is symmetric and store the complete matrix when [D] is not symmetric. This amounts for considerable saving in terms of storage. There are multiple ways of storing the coefficient matrix to be solved on a parallel machine. The performance of each storage scheme depends on the matrix pattern for a given solution method. A few of them are briefly described below.

3.3 Matrix Storage Schemes

The matrix for the finite element mesh is a band matrix. The following mesh diagram shows how it happens

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80 Figure 3.5: Node numbering for a triangular mesh

The above figure shows a mesh with nodes numbered from 1 to 80. As an example consider node 26. The non-zero terms involved for node 26 are 26, 16, 17, 27,

36, 35 and 25. Similarly for node 27 they are 27, 17, 18, 28, 37, 36 and 26. This means that the non-zero terms range from 16 to 36 for node 26 and 17 to 37 for node 27. The following figure shows the matrix structure for the nodes 26, 27 and 28.

For node 26 16 17 25 26 27 35 36

For node 27 17 18 26 27 28 36 37

For node 28 18 19 27 28 29 37 38

Figure 3.6: Non- zero elements of the coefficient matrix

As can be seen in the above figure the non-zero elements of the matrix occur in diagonals. Extending the matrix structure to the complete mesh gives a band matrix as shown in figure below.

Principal diagonal Super-diagonals (ν) Semi Bandwidth (ν + 1)

0 Sub-diagonals (µ)

0 N× N×(ν+1) Band width = µ + ν + 1 N Figure 3.7: Band Matrix

The diagonal connecting the first element of row 1, second element of row 2 and so on is called the principal diagonal. The diagonals above it are called super diagonals

(ν) and the ones below it are called sub diagonals (µ). Mathematically a band matrix can be defined as a matrix A=An×n. If an element ai,k=0 whenever i>k+ µ for k=1,2,….,n- µ -

1 and ak,j=0 whenever j>k+ν for k=1,2,….,n-ν-1, then A is a band matrix with µ sub- diagonals, ν super-diagonals and a bandwidth of µ + ν + 1. It is an inefficient use of computer memory to save the coefficient matrix in an N×N array. All that is needed is an array big enough to hold the band structures in Figure 3.7 [12]. For symmetric matrices super and sub diagonals are identical. It is therefore sufficient to save either the super or sub diagonals along with the principal diagonal as shown in the figure.

This method of saving the node potentials of a mesh depends on uniform numbering of the nodes over the domain. The density of the band can be varied by renumbering the nodes in the mesh. When the nodes are randomly numbered the resulting matrix would be very sparse and thus has a large bandwidth. It can be made dense if the matrix is processed by minimum profile scheme based renumbering schemes [13]. This method may not be efficient when the band matrix is sparsely filled. For the example explained above, for node 26 the elements 18 to 24 and 28 to 34 are zero. When denoted by matrix it leads to zeros between the super and sub diagonals, thus resulting in loosely filled bands. These kinds of matrices can be stored in form of skyline (envelope) storage or sparse storage method, as shown in figure below. The skyline matrix is a special case of band matrix which stores all the entries from the first non-zero to the last non-zero element of each column, resulting in more compact data. Sparse matrix stores only the

non-zero entries of the skyline matrix. Thus it is more efficient than the other two schemes. However this argument holds good only in case of sequential computation [12].

Actual results of parallel computation (multicore processors are also a classification of parallel computers) show that banded matrices perform faster when the matrices are dense (Figure 3.9) and become slower when the matrices are very sparse.

x x x x x x x = non - zero element x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

Skyline Sparse Figure 3.8: Skyline and Sparse storage schemes As shown in the Figure 3.9, when a matrix is stored in banded form with half bandwidth of 11 and a density of 87% in sparse storage compared to banded storage, the performance of banded storage is much better for fewer processors and it converges to that of skyline and sparse storage methods (but still better) when the number of parallel processors is increased to 16. The performance is reversed when the half bandwidth is 52 for banded storage and the matrix density is 25% for sparse storage.

Matrix: 240 degrees of freedom, 87% Banded storage: Half bandwidth = 78 matrix density (with renumbering) Skyline storage: Avg. half bandwidth = 52 Banded storage: Half bandwidth = 11 Std. deviation of column heights = 4.7 Sparse storage: Matrix density = 87% Sparse storage: Matrix density = 25% (w.r.t. size of banded storage) (w.r.t. size of banded storage)

Figure 3.9: Processing time comparison between Banded storage and Sparse storage [12] .

3.3.1 Compact Storage Method

This is one more method to overcome the matrix sparsity problem. This storage method is particularly employed when iterative solution techniques are used and is not compatible with direct solution methods as discussed in the Matrix Computation, Section

3.4. The non-zero locations in the matrix can be saved using a pointer array. The 2D pointer array saves all the locations of randomly placed non-zero elements of the matrix.

For each node it stores the location of non-zero neighboring nodes. The actual values of the indexed terms are saved in another 2D array called coefficient array. The method does not require the nodes to be numbered in any order to obtain a compact matrix, as the

bandwidth has no effect in this method. This storage method is particularly advantageous in 3D applications because the matrices are large but very sparse

3.4 Matrix Computation

3.4.1 Application of Boundary Conditions

Before the solution process begins, the potentials at the boundary nodes for which

Dirichlet conditions are specified needs to be imposed. If for example the application of boundary conditions leads to the matrix equation AX=B, with a matrix element say x1=k

[ ][ ] [ ] (3.32)

The modified boundary condition can be represented as in the above form with the diagonal element as one and off diagonal elements as zero. Upon matrix multiplication we get x1=k. The matrix looks simple and processing is fast (fewer variables) but has a disadvantage. As discussed above the coefficient matrix for EM problems is symmetric and the storage only needs to accommodate half the bandwidth (either upper or lower band). If say suppose the upper band is stored then we lose the coefficients a21 and a31 as their corresponding terms have been replaced by zero. To eliminate this issue, the diagonal element is replaced with a very large number (C) compared to the off diagonal elements with typical values of 1014 [14].

[ ][ ] [ ] (3.33)

The matrix equation for the first row reduce to

(3.34) or

This procedure yields the same solution with no noticeable error and maintains the symmetry of the matrix.

The solution of the system of equation can be performed by multiple methods.

Some of the commonly used ones in finite element programs are listed below

Table 3.1: Common methods for solving a system of linear equations [15] Method Type Condition Gaussian elimination Elimination Nonzero pivot elements and nonsingular coefficient matrix. Gaussian elimination Elimination Nonsingular rearranged matrix. with partial pivoting Gauss-Jordan Elimination Nonzero pivot elements and nonsingular coefficient matrix. Crout LU factorization Nonzero diagonal elements in the L matrix. Doolittle LU factorization Nonzero diagonal elements in the U matrix. Cholesky LU factorization Positive-definite coefficient matrix. Thomas LU factorization Tridiagonal coefficient matrix. Jacobi Iterative Diagonally dominant coefficient matrix. Gauss-Seidel Iterative Diagonally dominant coefficient matrix. Only the Gaussian Elimination and Cholesky methods are discussed below as they are most widely employed in Finite Element programs. Many schemes have been developed based on them and are being implemented. Those are not discussed here as they are beyond the scope of this work.

3.4.2 Direct Methods

These methods in the absence of round-off errors produce the exact answer after finitely many arithmetic operations.

a) Gaussian Elimination Method

This method has two parts. First, is the forward elimination and the second is called backward substitution. Before applying the method, the coefficient matrix is appended to get the augmented matrix (A|b). The forward elimination aims at reducing the augumented matrix to upper triangular matrix U using the following equations.

⁄ (3.35)

Ax=b

Row Operations Ux=v

Backward Substitutions

Solution

Figure 3.10: Main steps in the Gaussian Elimination method The following example demonstrates the working procedure.

Below is the banded augmented matrix that is to be converted to upper triangular matrix through the elimination process.

(3.36)

[ ]

The operations are performed column wise. To eliminate elements below the first diagonal elements the row operation defined in the above equation is performed with i=2, k=1 and j=1 to n. The row operation applies throughout the row including b‘s until all the rows below the principal diagonal are processed. The elements of the matrix after the row operation are replaced by their new value denoted here by ‘

(3.37)

[ ]

A similar operation on the second column eliminates the elements below the second element of the principal diagonal.

(3.38)

[ ]

After all the columns are operated upon we obtain an upper triangular matrix as shown below

(3.39)

[ ] 38

The next step of the method is backward substitution which yields the final solution of the augmented matrix according to the following steps

∑ (3.40)

Variations in Gauss elimination methods such as Gaussian elimination with maximum column pivoting (partial pivoting) exist. This scheme is the same as the discussed above, except that after the operations in each column, the rows are rearranged such that the first element of the next row to be operated upon has the maximum magnitude in the column below the .principal diagonal element.

The effectiveness of a method used in computational programs is determined by storage and computational cost. Computational cost of a particular method is measured by the number of algebraic operations the computer must perform to obtain a solution.

Multiplication and division are generally more expensive compared to addition and subtraction as they are more time consuming for a computer. While counting the number of operations, addition and subtraction are counted the same as they take the same amount of time. Similarly multiplication and division are counted as identical operations although division takes more time than multiplication. The computation cost of solving a

n x n linear system using Gaussian Elimination is approximately multiplications and additions.

Related to Gaussian elimination are LU factorization methods. These involve reducing the coefficient matrix to the product of a lower triangular matrix L and a upper triangular matrix U. There are different types of LU factorization methods with some variations like Crout, Doolittle, Choleski and Thomas methods as listed in Table 3.1. Out 39

of these Choleski is the most widely employed in FE programs and will be discussed below. For example Doolittle‘s method has the same computation cost as that of

Gaussian Elimination i.e., approximately multiplications and additions; hence no significant advantage.

b) Choleski Method

This method is applicable to the coefficient matrices only when they are positive definite. A matrix is said to be positive definite when it is symmetric and there exists a vector such that

. (3.41)

Let A be positive definite, then there exists a unique lower triangular matrix L with positive diagonal entries such that [A]=[L]∙[L]T. Therefore Choleski decomposition is the product of lower triangular matrix and its conjugate transpose. The decomposition is the first part of the solution towards finding the unknowns of the equation Ax=b. Next is the solution for the vector x. These steps can be demonstrated as follows

[ ] [ ] [ ]

Upon solving the above equation we obtain the algorithm for the elements of the triangular matrix, which is the following

i) For k=1 to n

√ ∑ (3.42)

ii) For i=k+1 to n

∑ (3.43)

For Hermitian matrices

i) For k=1 to n

√ ∑ (3.44)

ii) For i=k+1 to n

∑ (3.45)

(Where indicates complex conjugate)

T The above steps yield the triangular matrices L and L . To solve the system Ax=b the following procedure is employed

, -, -x (3.46)

The above equation can be simplified by writing

, - (3.47)

, -x (3.48)

The above two equations are solved first for and then is used to solve for x using backward substitution as discussed above in the Gaussian elimination method.

This method is faster when than Gaussian elimination because the number of steps involved in Choleski factorization is lower, but it cannot be applied to non-symmetric matrices.

3.4.3 Iterative Methods

An iterative method is a repetition of a set of steps. When an iteration starts the variables have some initial values and are updated at the end of each iteration. At some point they converge to a predefined error value and the iteration ends. These produce a

(k) ( ) sequence, say {x } such that where A . The difference between and ( ) is the error which needs to be reduced to a predefined value.

The need for iterative methods arises when the number of nodes is large and the storage of banded matrices becomes impractical. Once the dimension of coefficient matrix becomes very large (several thousand) and it becomes very sparse (say 99.9% of its entries are zeros) as is common in FEM solutions, iterative methods become more efficient [16]. As discussed in the previous section, compact storage method is used for iterative solution methods. Some of the iterative solution methods are discussed below beginning with easy to understand methods, before discussing the more efficient but more complex conjugate gradient method

a) Jacobi‘s Method

In this method the values of a variable are updated based on values of other variables in the system from their previous iteration. This method requires that the coefficient matrix be positive definite for the solution to converge. Consider the rth equation in the system of linear equation A

∑ (3.49)

(3.50)

From the above equation can be written as 42

∑ (3.51)

th k For the k iteration the value of xr is updated based on the current values of x to produce

th (k+1) the (k+1) iteration of xr i.e., xr

( ) ( ) ∑ (3.52)

( ) is not the exact solution of A but an approximate solution based on the predefined error as follows:

‖ ( ) ( )‖ (3.53)

(The infinity norm is defined as‖ ‖ x ∑ | |)

The iteration stops when the error criterion is met, else it repeats.

b) Gauss-Seidel Method

In this iterative method the values of a variable is updated based on current values of other variables in the system. Because of this dependence on current values, convergence of the solution is faster than the Jacobi method.

The following steps demonstrate the Gauss-Seidel method.

Consider rth equation in the system of linear equation A

∑ (3.54)

( ) ( ) This equation is used to compute , and so on. By the time we want to

( ) ( ) ( ) compute we have terms up to but not the terms starting from .

( ) Therefore, to calculate we use all the current available values.

( ) ( ) ∑ (3.55)

( ) Where denotes the most current value of known. With these, we can rewrite the equation as

( ) ( ) ( ) ∑ ∑ (3.56)

The convergence criterion remains the same as that of Jacobi‘s method.

A significant advantage of this method over Jacobi‘s is that it requires less memory as there is no need to save previous values. The currently calculated variable values can replace the previous ones. But a disadvantage is that, this method is not as easily parallelizable as Jacobi‘s method.

c) Successive Over-Relaxation Method

The process of correcting an equation by modifying one unknown is called relaxation

[16]. The procedure of Gauss-Seidel was successive relaxation. The convergence of the solution can be accelerated by over-relaxing, i.e., making a bigger correction. When the relaxation factor and one over-corrects by that factor in each step the method is called Successive Over-Relaxation (SOR) Method. If , the method is called under- relaxation and delays the convergence of the solution. If then the method is the classical Gauss-Seidel. The following equations demonstrate the SOR method

Consider the rth equation in the system of linear equations A

∑ (3.57)

For the (k+1)th iteration we can write the change in solution as

( ) ( ) (3.58)

( ) To obtain SOR is multiplied by ( ) and added to the original value of as

( ) ( ) ( ) (3.59)

The performance of this method as any other iterative method depends strongly on boundary (Dirichlet) conditions. If there are many nodes at which electric potentials are imposed then the convergence of solution is usually very fast.

d) Incomplete Choleski Conjugate Gradient Method

This method uses the Conjugate Gradient (CG) method with preconditioning of the matrix [A]. This method is applicable for symmetric matrices only. The preconditioner

[T], of a matrix [A] is such that [T][A] has a smaller condition number compared to [A], where the condition number determines the worst case of how much the solution can change in proportion to small changes in the coefficient matrix [A]. A matrix is preconditioned to accelerate the solution process and hence reduce the number of iterations. Ideally a preconditioner of [A-1] would converge and produce the solution in one step. But this is not practical as computing [A-1] is in itself as complex as finding the solution without the preconditioning.

In this method [, -, - ] is used as a preconditioner where , - is obtained from

Choleski factorization of existent non-zero elements of A. Therefore [, -, - ] is an incomplete Choleski factorization of [A]. Using Equations (3.5) to (3.8) the terms of

[, -, - ] occupying non-zero positions of the matrix A are calculated. This method requires 50% more storage compared to SOR methods, since the addresses of non-zero

elements are held by the pointer array. The next step is the application of CG method with preconditioning. The following is the algorithm for the method.

Initialization:

(3.60)

( ) (3.61)

Iteration:

( ) (3.62)

(3.63)

(3.64)

( ) (3.65) ( )

( ) (3.66)

This method (and its variations) is considered to be the most effective iterative method for FEM programs. This method is two to three orders of magnitude faster than point

Gauss-Seidel method and about 30 times faster than block SOR method with optimum relaxation factor [17].

These methods are most commonly employed in FE programs because of good performance. However for specific problems of EM computations other types of methods are used. For example to model EM scattering from infinitely periodic grating structures a hybrid finite element/ rigorous coupled wave analysis (FE/RCWA) method has been developed and claims better performance than regular methods. The computational cost

for matrix factorization and storage are calculated as N2/125 and N4/3/25 respectively which is more efficient than a fully 3D FEM approach which has computational costs and storage proportional to N2 and N1.5, respectively [18].

3.5 Eigenvalues and Eigenvectors in Computational Electromgnetics

Mathematically an Eigenvalue and Eigenvector are defined as

If and with ; then is an eigenvalue of A with eigenvector

The eigenvector of a matrix [A] is a vector belonging to the complex space that, when multiplied by [A] gives the same vector times a scalar. The scalar is the eigenvalue and the vector is the eigenvector. An eigenvalue paired with its corresponding eigenvector is called an eigensystem of the matrix.

Eigensystems are important in the engineering fields of computational physics, mechanics, image processing, electromagnetics and many others. In electromagnetics problems eigensystems define many important concepts. For example in a waveguide or cavity resonator the propagating TE or TM waves have distinct modes of propagation.

The cutoff frequencies of the modes are the eigenvalues of the system of equations shown below. TM and TE modes are calculated from the following (these are Helmholtz equations and may be formulated in FEM)

( ) (3.67)

( ) (3.68)

One more example of significance of eigenvalues is in elimination of resonance in a high frequency integrated system. Integrating different circuits on the same chip often leads to undesired coupling and sometimes to system failure. An example is high speed switching signals in digital circuits causing ringing in power supply lines and in the output circuit. It in turn couples through the common substrate and adds noise to other analog signals on the same chip. Although the manufactured package meets the design targets, it sometimes fails because of the resonance caused by coupling between the die and package. The resonance frequency can be calculated by solving the quadratic eigenvalue problem which can be converted to generalized eigenvalue problem by linearization [19].

A generalized eigenvalue problem is of the form

, - , - (3.69)

Where [A] and [M] are usually symmetric and positive definite but are in general complex-valued matrices.

Eq (3.32) can be converted to standard eigenvalue problem by rewriting the equation as

, - , - (3.70) or , - (3.71) where , - , - , -

To solve the standard eigenvalue problem there are many methods available, but these will not be discussed here as the current research does not involve finding resonant

frequencies of the structure and the methods are rather specialized. Some of the standard methods used in FE programs are the Jacobi transformation, QR and QZ methods and the

Givens transform. But with these traditional methods the order of complexity remains high (it is in order of N3 for QR method, for example). With methods such as Arnoldi based eigenvalue solution the complexity is reduced to O(N) for some particular cases

[19].

3.6 Finite Element Based Electromagnetic Computation Software

Finite element software tools implement the finite element methods to solve for partial differential equations which are complex and take considerable time when applied to practical problems. These tools were initially developed in 1970‘s for mechanical in civil engineering for structural analysis of complex structures, later developed for vibrational analysis and heat transfer analysis. Soon after FEM software tools were developed for EM simulations to find electric and magnetic field intensities, fluxes, to calculate currents, potentials and scattering parameters for high frequency analysis of EM related structures. There are also software tools particularly developed specifically to analyze high power electrical devices.

These software packages are part of the more general domain of computer aided design (CAD). A CAD software for EM field analysis must have three modules. These may be available in a single package or in individual packages. Integration of the modules into one package saves time and reduces complexity for the user. These are a)

Pre-processor b) Processor and c) Post-processor. These are described below

3.6.1 Pre-Processor

The pre-processor module‘s function is to take input from the user and discretize the problem domain by generating a finite element mesh. Based on its function this section is divided into two parts: the design setup section and the mesh generation section. a) Design Setup

The design setup part of the software is the initial part with which the user interacts to begin a project. It is usually a 3D interface provided so that the user inputs the structure to be analyzed. The user can either draw the structure manually using the graphical interface or can input a file created using other CAD tools (eg. AutoCAD). The supported file formats vary by the software provider. While setup is provided user should be careful to choose the right design properties such as material and electrical parameters of the device (electrical permittivity, magnetic permeability, electrical conductivity), boundary conditions and the like. The user should provide an excitation source (wave port, current or voltage excited conductor, transmission lines, etc.), frequency of operation, mesh design parameters, error limits and more. Mesh characterization is the most important part of the software as the accuracy of the solution depends on the mesh. The user specifies mesh element size and other parameters such as how much the mesh needs to be refined. Once the input is setup it is validated for design errors and analyzed. b) Mesh Generation

Generation of a mesh is a core part of FE software, as it discretizes the problem domain and affects accuracy. FEA uses arrays of points called nodes that form a grid

called mesh. Figure 3.11 shows a triangular mesh on a surface. FE mesh generation techniques can be broadly classified as semi-automatic and automatic.

Figure 3.11: Finite Element Mesh [20]

In semi-automatic method nodes are manually specified on the boundary and the program generates the mesh inside the boundaries based on some predefined principles.

An example of semi-automatic mesh generation scheme is described by using digital image processing (DIP). Such a method has been developed to generate meshes for complex topologies such as a human body where contours of regions cannot be mathematically defined and hence the process cannot be automated. DIP techniques help to extract topological information from images (CT scans) and to differentiate boundaries, so that the nodes can be manually placed along the boundaries [21].

Automatic mesh generation can be done by many methods [22][23][24]. One of the methods used in EFCAD (Electromagnetic Finite element Computer Aided Design) will be discussed here as an example of automatic mesh generation. It is based on three principal operations:

i) Closing of acute angles ii) Cuts to cancel concavities (interior angle greater than 180 ) iii) Closing a region using elemental centroids

To better understand the working of this method we apply it to a 2-D polygon as shown in Figure 3.12. Depending on the density of the mesh defined by user the ―standard segment‖ is defined with a particular length. The principal operations are applied on this object. As shown in Figures 3.13(a) and 3.13(b), operation (i) divides the polygon edges into standard segments and the acute angles in the region are closed by the elements T1,

T2 and T3. Operation (ii) searches for concavities in the remaining region, and if one exists, it is cut into two regions R1 and R2 as can be seen from Figure 3.12(c). The line which cuts the region is divided into standard segments with a new node defined.

Operations (i) and (ii) are repeated in R1 to close acute angles and concavities in the region. In R2 since there are no acute angles, centroid B is calculated and divided into standard segment AC, if AB is larger than it. Figure 3.12(e) shows closure of acute angles by T8, T9, and T10 formed due to creation of the new segment. Again a new centroid B1 is calculated as there are no acute angles. The remainder region is closed forming triangles from T11 to T16.

(b) (a)

R R1 T1 1 T1 R2

R2 T T C B T3 T2 3 2

T 11 T16 T8 B1 T9 T 10 T14 A (e) A (f) Figure 3.12: Steps involved in Automatic Mesh Generation [14]

Ideally the mesh should be made of equilateral triangles. But this is difficult to achieve for an irregular region. To obtain better aspect ratio of element lengths a smoothing procedures are applied. Smoothing procedures include diagonal changes and re-centering of centroids of domains. If a triangle of a mesh has sides a, b and c then its quality factor is defined as

( )( )( )

(3.72) 53

where ( )⁄ (3.73)

For the complete mesh an average q is determined. It is 1 for equilateral triangle. A mesh with quality factor of 0.85 or over is considered to be good.

One of the widely used methods in mesh generation programs is the Delaunay method. It defines a mesh for a set of existing nodes and does not create new nodes. Its principle is that the circumcircle defined by any triangle in the mesh should not contain other nodes except those three nodes forming the triangle. Figure 3.13 shows the mesh formation

Delaunay‘s method can be extended to 3D objects. When the element is a tetrahedron formed by four nodes the sphere formed by the nodes should not contain any other node.

Figure 3.13: Mesh generation using Delaunay‘s method These automatic mesh generation methods do not guarantee limited error. Instead, a method called adaptive mesh generation has been developed and currently available in almost all commercially available software packages. In the setup stage the user defines the error percentage and the maximum number of iterations to be carried out to optimize the solution. In this method the FE model is created simultaneously with the solution. It is

generated iteratively, starting with a rough approximation. The approximation is refined successively to minimize the error in the solution. The mesh generation, solution, and post-processing phases are in constant interaction to optimize the solution. The method is automatic and does not require user intervention [25]. Methods are also being developed to generate meshes using artificial neural networks [24] in an attempt to optimize the solution.

3.6.2 Processor

After the discretization process the FE program transfers the command to the processing section. As described earlier in the section from 3.1 through 3.4, matrix storage and computation routines are performed after the given boundary conditions, source excitations and other setup data have been transferred into matrix forms. This stage primarily provides the results as potentials at the nodes of the mesh or, in some

FEM methods, along the edges of elements. The electric field, magnetic field and any other EM parameters are calculated from the post-processing module of the package.

3.6.3 Post-Processor

The task of the post-processor module is to analyze the results. It calculates the

EM parameters from the data obtained from the processing stage. It can be either in the same software or separate tools such as MATLAB may be used to analyze and display the data. It is consuming considerable amount of time depending on the desired parameters. It can be used to visualize results such as equipotential lines on the structure and also to calculate exact numerical values.

The user can plot or view equipotential lines on the structure as well as other quantities such as intensity, energy or gradients. This helps in analyzing the problem in qualitative ways and gives an idea to the user if the obtained solution is as desired or needs to be improved before final calculations. Sometimes logical errors that occurred in the setup section are rectified after the qualitative analysis. The problem can be re- processed with a higher mesh density for more accurate solution.

If the equipotential lines are desired to be drawn at points where nodes of the mesh do not exist then the values at the desired position are calculated by interpolation.

Figure 3.14 shows an example how the interpolation is done.

4 3.5 3.5 a P 3 1 b

Figure 3.14: Linear interpolation to trace equipotential lines

The values shown at the nodes in Figure 3.14 are the potentials calculated in the processing step. If a line is to be traced with value 3.5, point P is placed between 2 and 4 such that

(3.74)

The value (a+b) is known so the value of ‗a‘ can be calculated from the equation and the position of P is found.

If the plots are found to be true, user can proceed with numerical calculations. The user can find potentials at any point in the solution domain and the fields E, D, H, B obtained from derivatives of potentials, scattering parameters, currents, voltages, forces, impedances and others. In this research the post-processing module was used to plot E and H fields in the problem domain and to calculate induced currents on the conductors in an electrical connector.

A flow chart shows the working of Ansoft HFSS [26] (commercially available

EM simulator) with the three modules described above.

Design Desig n Boundaries Design Solution Desig Type n Mesh Desig Operationsn 1. Pre-Processor Excitations Design Geometry/Materials Design

Analyze Desig n Mesh Desig Refinementn Solve Desig n 2. Processor Solution Desig nSetup No Converged 3. Post-Processor 2D ReDesigports/Fieldsn Yes

UpdateDesig n Finished Desig n Figure 3.15: Illustration of EM analysis in Ansoft HFSS [26] 57

CHAPTER IV

ANALYSIS AND RESULTS

4.1 Introduction

A frequent EMC problem is predicting and controlling radiation from an interconnected system, such as a personal computer with peripherals. Connectors are one of the reasons for the EMC issues. Due to the lack of convenient measuring techniques exact data is not readily available [27]. Two of the ways to address the problem are to increase the gap between the conductors and shield the connector. Miniature devices require the connector‘s size to be minimum and shielding connectors increases the overall manufacturing cost of the device. This work calculates the distance between the conductors in a partially shielded multi-conductor connector, to reduce the effects of induced currents arising within the connector.

4.2 Description

To discuss the behavior of conductors we use here a very simple model made by sandwiching two dielectric sheets with the conductors constituting the contacts passing between them. The dimensions of the dielectric sheets selected here are larger than for real conductors, primarily to contain the fields and to indicate that large structures can be analyzed equally well. The conductors themselves are also long but their thickness and

separation are realistic. The upper and lower surfaces of the connector are perfect electrical boundaries (metal ground planes) for electromagnetic immunity purposes and to avoid emissions from the connector. We will analyze this model with two conductors in the first case and three in the second, as an example of multi-conductor analysis. It should be noted again that the dimensions selected are arbitrary and in fact, smaller overall dimensions result in smaller models and faster solution times. From the point of view of analysis the ―difficult‖ issues are connection to sources and the modeling of edges and these are fully modeled in the present analysis. The model employed here can equally well apply to multi-conductor transmission lines. Figure 4.1 shows an overall view of the model with two conductors.

Figure 4.1: Connector

4.3 Dimensions

The dielectrics are 101.6x101.6x3.175 mm each. The conductors are 101.6mm long and have a radius of 0.5mm each. The conductors are initially separated 2mm apart

(center to center) but later are moved to 4mm, 6mm and 8mm to observe the changes in induced fields due to the separation. This means that the gap between the conductors surface is 1mm, 3mm, 5mm, and 7mm respectively for the above cases. The overall dimensions of the model are 101.6x101.6x6.35 mm

4.4 Properties

Any electrical material is characterized by three properties, its electrical permittivity (ε), magnetic permeability (μ) and electric conductivity (ζ). Here the

-4 dielectrics have a relative permittivity εr=3, relative permeability μr=1 and ζ=10

Siemens/m, qualifying them as lossy dielectrics. The conductors are made of copper with

7 εr=1, μr=0.999991, ζ=5.8x10 S/m

4.5 Case 1: Two Conductors connector

In this case the connector has two conductors along the y axis, each placed symmetrically away from the edge center along the x axis. The conductor closer to the origin is labeled conductor 1 or L (left as seen from the xz plane with positive x-axis pointing to the right side of the page) and the other as conductor is labeled as 2 or R as shown in Figure 4.2. Also the dielectric on the top is Dielectric 1 and the bottom one is

Dielectric 2.

Conductor 1 (L)

Conductor 2 (R)

Figure 4.2: Two conductor connector showing the conductors and dielectric sheets. 4.5.1 Excitation

Conductor 1 is excited with a sinusoidal current of amplitude 1A and phase 0° as shown in Figure 4.2. The direction of current is along the y-axis. The frequency of the current is varied from 0.2 GHz to 0.5 and 1 GHz. The other conductor is unexcited and we calculate induced currents on both of them.

4.5.2 Mesh

In Ansoft HFSS, the basic element of the geometry is a four sided tetrahedron

(pyramid). The geometry is divided into a large number of tetrahedra which is referred to as the finite element mesh. The mesh here is adaptive; this means the size of the mesh varies in the geometry until the solution converges to a predefined criterion. Here, the maximum length of the tetrahedron edge is 5.08mm and the criterion for adaptive mesh is maximum increment in energy of 0.1J. 61

(a)

(b) Figure 4.3: Mesh for two conductor model with 2mm spacing

Figure 4.3 and Table 4.1 show the mesh plot and mesh statistics for a two conductor connector with 2mm spacing between the conductors. The 3 - D plot in Figure

4.3(a) shows the effect of adaptivity with high density of tetrahedral elements closer to the conductors. This is because of the much higher electric field near the conductors. The plot in Figure 4.3 (b) shows only the mesh on the surface of the connector. Table 4.1 shows the number of elements in each component of the connector for the mesh of Figure

4.3.

Table 4.1(a): Mesh statistics (Tetrahedron edge length in [mm]) for two conductor connector with 2mm spacing # of Min. edge Max. edge RMS edge Tetrahedrons length length length Dielectric 1 15828 0.740156 6.432219 3.590975 Dielectric 2 15771 0.644423 5.692851 3.615385 Conductor 1 2184 0.392861 4.838649 2.456383 Conductor 2 1892 0.382676 5.057902 2.687853

Table 4.1(b): Mesh statistics (element volume [mm3]) for two conductor connector with 2mm spacing Min. Max. Mean # of Tetrahedro Tetrahedron Tetrahedron Std. Dev. Tetrahedrons n. volume . volume volume (volume) Dielectric 1 15828 8.0x10-6 0.001457 0.003200 0.002717 Dielectric 2 15771 5.9x10-6 0.001480 0.003200 0.002717 Conductor 1 2184 8.8x10-7 0.000508 0.000050 0.000050 Conductor 2 1892 5.0x10-8 0.000609 0.000050 0.000050 When the distance between the conductors is increased in steps of 2mm up to

8mm, the mesh changes accordingly and the number of elements in the mesh also changes slightly as can be observed from Table 4.2. 63

Table 4.2: Comparison of mesh elements [mm] for 2 conductor connector Spacing # of Tetrahedrons Min. edge length Max. edge length 2 35675 0.382684 6.432219 4 35952 0.385937 7.709154 6 34861 0.5562447 5.899099 8 36433 0.3826840 6.965619

4.5.3 Electric and Magnetic Field Plots

The electric and magnetic field in the connector are generated by the applied current and hence they concentrate around the conductors. Their intensity decreases as we move away from the conductors. The electric, magnetic fields, current and voltage can be calculated in the postprocessor part of the software tool. Figures 4.4 to 4.9 show the magnitude of complex E and H fields plotted in the cross-sectional view (xz-plane) of the connector. The E and H fields are plotted by varying distances between the conductors for different values of excitation currents. The view is magnified and only shows the vicinity of the conductors.

The H-field plots are less accurate compared to E-field plots because in Ansoft

HFSS, H is calculated from E as

Spacing 4mm Spacing 2mm

Scale for E Field [V/m] Spacing 6mm Spacing 8mm Figure 4.4: E - Field for 0.2 GHz

Spacing 4mm Spacing 2mm

Scale for H Field [A/m] Spacing 6mm Spacing 8mm Figure 4.5: H - Field for 0.2 GHz

Spacing 2mm Spacing 4mm

Scale for E Field Spacing 6mm Spacing 8mm [V/m] Figure 4.6: E - Field for 0.5 GHz

Spacing 4mm Spacing 2mm

Scale for H Field Spacing 6mm Spacing 8mm [A/m] Figure 4.7: H - Field for 0.5 GHz

Spacing 4mm Spacing 2mm

Spacing 8mm Scale for E Field Spacing 6mm [V/m] Figure 4.8: E - Field for 1 GHz

Spacing 4mm Spacing 2mm

Scale for H Field Spacing 6mm Spacing 8mm [A/m] Figure 4.9: H - Field for 1 GHz

4.5.4 Induced Currents

The current in the conductor is calculated as ∮H.dl where H is the complex vector

H field data everywhere in the modeled geometry. To calculate the currents a circular integration path is defined across the conductor perpendicular to the direction of flow of current. The radius of the integration path is 0.6mm (Figure 4.10) so that it includes the conductor in which the current is calculated. For conductors that have an applied current, this calculation gives the total current (applied plus induced) whereas in conductors that have no applied current, the calculation results in the induced current alone.

Figure 4.10: Integration paths for each conductor

The currents are calculated by running VB scripts in HFSS. The values are plotted in

Figures 4.11 to 4.13. The actual currents are listed in Appendix B.

As can be observed from the figures, the direction of induced currents in the unexcited conductor (R) opposes the direction of current in excited conductor (L) and its magnitude decreases as the conductor R is moved away from L as expected.

The figures are divided into two subplots, one for the conductors that are excited and the other for conductors that are not excited. The colors in the figure represent the 68

four cases of spacing between the conductors; that is, red for 2mm, green for 4mm, blue for 6mm and black for 8mm. The solid lines represent positive magnitudes indicating excited conductor and dotted lines represent negative magnitudes indicating induced currents which are opposite in direction to the supplied current. The left ‗<‘ and right ‗>‘ pointing markers indicate the left and right conductors respectively. The currents are calculated at various points along the conductor.

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.2 G H z

0.95

0.9

C o l o r f o r 2 m m 0.85

C o l o r f o r 4 m m 0.8

C o l o r f o r 6 m m 0.75

C o l o r f o r 8 m m 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 p o s i t i v e m a g n i t u d e 0.35 n e g a t i v e m a g n i t u d e 0.3 C o n d u c t o r 1 ( L ) 0.25

0.2 C o n d u c t o r 2 ( R )

M a g n i t u d e o f i n d u c e d c u r r ent n dr u c ci e du r uodf e t ag Mn i 0.15

0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.11: Currents at 0.2GHz

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.5 G H z

0.95

0.9

C o l o r f o r 2 m m 0.85

C o l o r f o r 4 m m 0.8

C o l o r f o r 6 m m 0.75

C o l o r f o r 8 m m 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 p o s i t i v e m a g n i t u d e 0.35 n e g a t i v e m a g n i t u d e 0.3 C o n d u c t o r 1 ( L ) 0.25

0.2 C o n d u c t o r 2 ( R )

M a g n i t u d e o f i n d u c e d c u r r ent n dr u c ci e du r uodf e t ag Mn i 0.15

0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.12: Currents at 0.5GHz

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 1 G H z

0.95

0.9

C o l o r f o r 2 m m 0.85

C o l o r f o r 4 m m 0.8

C o l o r f o r 6 m m 0.75

C o l o r f o r 8 m m 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 p o s i t i v e m a g n i t u d e 0.35 n e g a t i v e m a g n i t u d e 0.3 C o n d u c t o r 1 ( L ) 0.25

0.2 C o n d u c t o r 2 ( R )

M a g n i t u d e o f i n d u c e d c u r r ent n dr u c ci e du r uodf e t ag Mn i 0.15

0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.13: Currents at 1 GHz 4.6 Case 2: Three Conductors connector

In this case three conductors are modeled keeping the overall dimensions the same as for the two conductor model. Conductors 1 (L), 2 (C) and 3 (R) are placed equidistant at 2, 4, 6, and 8mm from their centers. Conductor 2 is kept the center so that the geometry is symmetrical. To find induced currents, conductors 1 and 2 are excited at first and induced currents are calculated on all the three conductors. Later conductor 1 70

and 2 are excited in turn, one conductor at a time and the induced currents are calculated on each conductor in each case. This procedure is repeated for the four different conductors spacing. All of the properties remain the same as the Case 1 except that the number of conductors changes. The geometry is shown in Figure 4.14 (with conductors 1 and 2 excited with currents in the same direction).

Conductor 2 (C)

Conductor 1 (L)

Conductor 3

Figure 4.14: Three conductors connector with conductors 1 and 2 excited. 4.6.1 Excitation

The conductors are excited with a sinusoidal current of amplitude 1A and phase

0° as shown in Figure 4.14. The direction of current is along the y axis. The frequency of the current is varied from 0.2 GHz to 0.5 and 1 GHz.

4.6.2 Mesh

The mesh properties remain the same as described in section 4.2 although, necessarily, the number of elements and the element distribution change to accommodate 71

the changed geometry. The maximum length of the tetrahedron edge is 5.08mm and the criterion for adaptive mesh is maximum increment in energy of 0.1J.

When distance between the conductors is increased in steps of 2mm (up to 8mm), the mesh changes and the number of elements in the mesh also changes slightly as can be seen in Table 4.3.

Table 4.3: Comparison of mesh elements [mm] for 3 conductor connector Spacing # of Tetrahedrons Min. edge length Max. edge length 2 50625 0.374266 5.798693 4 49964 0.382684 6.049365 6 47277 0.378919 6.038265 8 46995 0.382684 5.954852

4.6.3 Electric and Magnetic Field Plots

Similar to section 4.4.3 the E and H fields are plotted by varying the distances between the conductors for different values of excitation currents. Figures 4.15 to 4.26 show the magnitude of complex E and H fields plotted in the cross-sectional view (xz- plane) of the connector.

E E Field with L C excited Field with L excited

Scale for E Field [V/m]

E Field with C excited

H Field with L C excited H Field with L excited

Scale for H Field [A/m]

H Field with C excited Figure 4.15: E and H field plot at 0.2 GHz for 2mm spacing

E Field with L C excited E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited H Field with L excited

Scale for H Field [A/m] H Field with C excited Figure 4.16: E and H field plot at 0.5 GHz for 2mm spacing

E Field with L C excited E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited H Field with L excited

Scale for H Field [A/m]

H Field with C excited Figure 4.17: E and H field plot at 1 GHz for 2mm spacing

E Field with L C excited E Field with L excited

Scale for E Field E Field with C excited [V/m]

H Field with L C excited H Field with L excited

H Field with C excited Scale for H Field [A/m] Figure 4.18: E and H field plot at 0.2 GHz for 4mm spacing

E Field with L C excited E Field with L excited

Scale for E Field E Field with C excited [V/m]

H Field with L C excited H Field with L excited

Scale for H Field [A/m] H Field with C excited

Figure 4.19: E and H field plot at 0.5 GHz for 4mm spacing

E Field with L C excited E Field with L excited

E Field with C excited Scale for E Field [V/m]

H Field with L C excited H Field with L excited

Scale for H Field H Field with C excited [A/m] Figure 4.20: E and H field plot at 1 GHz for 4mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m]

H Field with C excited Figure 4.21: E and H field plot at 0.2 GHz for 6mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m] H Field with C excited Figure 4.22: E and H field plot at 0.5 GHz for 6mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m]

E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m] H Field with C excited Figure 4.23: E and H field plot at 1 GHz for 6mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m] H Field with C excited Figure 4.24: E and H field plot at 0.2 GHz for 8mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m]

H Field with C excited Figure 4.25: E and H field plot at 0.5 GHz for 8mm spacing

E Field with L C excited

E Field with L excited

Scale for E Field [V/m] E Field with C excited

H Field with L C excited

H Field with L excited

Scale for H Field [A/m] H Field with C excited Figure 4.26: E and H field plot at 1 GHz for 8mm spacing

4.6.4 Induced Currents

Similar to Section 4.4.4, the plots for induced currents have been plotted. Figures

4.27 to 4.37 are divided into two subplots, one for the conductors that are excited and the other for conductors that are not excited. The colors in the figures 4.27 to 4.37 represent three cases of conductors being excited according to the three groups of columns in Table

C.1 of Appendix C. The solid lines represent positive magnitudes indicating excited conductors and dotted lines represent negative magnitudes indicating induced currents that are opposite in direction to the supplied current. The left ‗<‘ and right ‗>‘ pointing markers indicate the left (L) and right (R) conductors respectively. The marker ‗O‘ represents middle conductor (C). The actual current values are listed in Appendix B.

As shown in Figure 4.27 when L (red solid line with ‗<‘ marker) and C (red solid line with ‗O‘ marker) are excited the induced current on R is around 0.4A (red dotted line with ‗>‘ marker) throughout its length. The green and blue color groups are similar.

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.2 G H z f o r 2 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.5 C o n d u c t o r 1 ( L )

0.4 C o n d u c t o r 2 ( C )

0.3 C o n d u c t o r 3 ( R )

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.27: Induced currents at 0.2 GHz for 2mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t 85

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.5 G H z f o r 2 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.5 C o n d u c t o r 1 ( L )

0.4 C o n d u c t o r 2 ( C )

0.3 C o n d u c t o r 3 ( R )

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.28: Induced currents at 0.5 GHz for 2mm spacing

M a g n i t u d e o f i n d u c e d c u e r r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 1 G H z f o r 2 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.5 C o n d u c t o r 1 ( L )

0.4 C o n d u c t o r 2 ( C )

0.3 C o n d u c t o r 3 ( R )

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.29: Induced currents at 1 GHz for 2mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.2 G H z f o r 4 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.14 C o n d u c t o r 1 ( L ) 0.12 C o n d u c t o r 2 ( C ) 0.1

0.08 C o n d u c t o r 3 ( R )

0.06

0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.30: Induced currents at 0.2 GHz for 4mm spacing

M a g n i t u d e o f i n d u c e d c u e r r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.5 G H z f o r 4 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.14 C o n d u c t o r 1 ( L ) 0.12 C o n d u c t o r 2 ( C ) 0.1

0.08 C o n d u c t o r 3 ( R )

0.06

0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.31: Induced currents at 0.5 GHz for 4mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 1 G H z f o r 4 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.14 C o n d u c t o r 1 ( L ) 0.12 C o n d u c t o r 2 ( C ) 0.1

0.08 C o n d u c t o r 3 ( R )

0.06

0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.32: Induced currents at 1 GHz for 4mm spacing

M a g n i t u d e o f i n d u c e d c u e r r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.2 G H z f o r 6 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.05 C o n d u c t o r 1 ( L )

0.04 C o n d u c t o r 2 ( C )

0.03 C o n d u c t o r 3 ( R )

0.02

0.01

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.33: Induced currents at 0.2 GHz for 6mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t 88

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.5 G H z f o r 6 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.05 C o n d u c t o r 1 ( L )

0.04 C o n d u c t o r 2 ( C )

0.03 C o n d u c t o r 3 ( R )

0.02

0.01

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.34: Induced currents at 0.5 GHz for 6mm spacing

M a g n i t u d e o f i n d u c e d c u e r r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 1 G H z f o r 6 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.05 C o n d u c t o r 1 ( L )

0.04 C o n d u c t o r 2 ( C )

0.03 C o n d u c t o r 3 ( R )

0.02

0.01

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.35: Induced currents at 1 GHz for 6mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.2 G H z f o r 8 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.016 C o n d u c t o r 1 ( L ) 0.014

0.012 C o n d u c t o r 2 ( C )

0.01 C o n d u c t o r 3 ( R )

0.008

0.006

0.004

0.002

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.36: Induced currents at 0.2 GHz for 8mm spacing

M a g n i t u d e o f i n d u c e d c u e r n t

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 0.5 G H z f o r 8 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.016 C o n d u c t o r 1 ( L ) 0.014

0.012 C o n d u c t o r 2 ( C )

0.01 C o n d u c t o r 3 ( R )

0.008

0.006

0.004

0.002

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.37: Induced currents at 0.5 GHz for 8mm spacing M a g n i t u d e o f i n d u c e d c u e r n t 90

D i s t a n c e f r o m x z p l a n e V s C u r r e n t o n c o n d u c t o r s a t 1 G H z f o r 8 m m s p a c i n g

0.95

0.9

L , C e x c i t e d c a s e 0.85

L e x c i t e d c a s e 0.8

C e x c i t e d c a s e 0.75

p o s i t i v e m a g n i t u d e 0.7 0 0.5 1 1.5 2 2.5 3 3.5 4 n e g a t i v e m a g n i t u d e 0.016 C o n d u c t o r 1 ( L ) 0.014

0.012 C o n d u c t o r 2 ( C )

0.01 C o n d u c t o r 3 ( R )

0.008

0.006

0.004

0.002

0 0 0.5 1 1.5 2 2.5 3 3.5 4 D i s t a n c e f r o m x z p l a n e [ i n ] Figure 4.38: Induced currents at 1 GHz for 8mm spacing 4.7 Summary M a g n i t u d e o f i n d u c e d c u e r r n t

From the above finite element analysis the approximate values of induced currents for two and three conductors with various distances at different frequencies have been calculated. It is observed that the value of strongest induced current was 0.44A on conductor R when conductors L and C were excited at 1A and the three conductors were closest, with 2mm gap between their centers. The weakest induced current was 29.2µA on conductor R when conductor L was excited at 1A, the three conductors were farthest with 8mm gap between their centers. From an EMC or Signal Integrity point of view these results indicate that the induced currents are relatively high and likely to cause interference or degrade signals unless properly separated or, more likely, by interleaving signal conductors with return conductors so that different signals are widely separated.

These issues are of particular concern in small connectors in which case, some form of mitigation such as shielding may become necessary.

CHAPTER V

CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

This thesis presented issues of EMI in multi-conductor connectors and has discussed in detail their implementation in FEA software tools. In Chapter 4, values of induced currents were calculated in a two conductor connector as well as a three conductor connector for high frequency excitation currents.

It should be noted that the currents used here for excitation are very large (1A).

Signals are likely to be much lower except, of course, connectors that lead to power devices such as antennas. In most cases the currents are much smaller than that and will produce proportionally lower induced currents. Nevertheless the conclusions above are valid since the ratio between induced and applied currents remains the same and the likelihood of interference and signal degradation depends on that ratio not on the absolute magnitude of the induced currents.

5.2 Future Work

The work of this thesis can be extended with smaller size connectors. The behavior of multi-conductor connectors can be analyzed for lossy dielectrics by introducing frequency and temperature dependency of dielectrics. Connectors being used

for specific purposes such as in low power communication devices can be analyzed.

Another way of reducing the EMI, not implemented in this thesis is by terminating the conductors with impedances such that the reflections are reduced at the terminations and hence reducing EMI. These models can be integrated with the use of more return conductors in the design which reduces EMI in the connector.

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APPENDICES

APPENDIX A

ABBREVIATIONS

A.1 List of Abbreviations

2-D Two Dimensional

3-D Three Dimensional

CAD Computer Aided Design

CM Common Mode

CRT Cathode Ray Tube

DM Differential mode

E Electric Field Intensity

EM Electromagnetic

EMC Electromagnetic Compatibility

EMI Electromagnetic Interference

FE Finite Element

FEA Finite Element Analysis

FEM Finite Element Method

H Magnetic Field Intensity

PCB Printed Circuit Board

PDE Partial Differential Equation

SNR Signal to Noise Ratio

SOR Successive Over-Relaxation

SI Signal Integrity

TE Transverse Electric

TM Transverse Magnetic

APPENDIX B

TABLES OF INDUCED CURRENTS FOR TWO CONDUCTOR CONNECTOR

Table B.1 (a): Induced Currents at 0.2GHz for 2mm spacing Distance

[mm] Current in L [A] Current in R [A] 0 0.9158-j0.000175 -0.3434-j0.000749 6.35 0.91-j2.515E-5 -0.3295-j0.0006699 12.70 0.9435-j5.913E-5 -0.3176-j0.0006728 19.05 0.9424-j2.693E-6 -0.3173-j0.0005812 25.40 0.9156-j1.374E-5 -0.3261-j0.0006973 31.75 0.946-j1.052E-5 -0.3274-j0.0007085 38.10 0.9157-j1.163E-5 -0.3189-j0.0006673 44.45 0.9351-j1.901E-5 -0.3182-j0.0007096 50.80 0.895-j0.00013 -0.3325-j0.0005441

Table B.1 (b): Induced Currents at 0.2GHz for 4mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8316-j4.593E-5 -0.1015-j0.0002001 6.35 0.8548-j5.323E-7 -0.1052-j0.0001774 12.70 0.8801+j6.242E-7 -0.1075-j0.0002332 19.05 0.8156-j3.096E-6 -0.1093-j0.000188 25.40 0.8199-j2.176E-6 -0.1034-j0.0002355 31.75 0.8648-j2.554E-5 -0.09365-j0.0001207 38.10 0.8709-j1.582E-6 -0.1133-j0.0002379 44.45 0.8536+j2.324E-6 -0.1037-j0.0001883 50.80 0.8443+j3.465E-5 -0.112-j0.0001315

100

Table B.1 (c): Induced Currents at 0.2GHz for 6mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8394 - j0.000159 -0.04677 - j0.0001089 6.35 0.8748 + j5.1E-8 -0.03679 - j6.101E-5 12.70 0.8556 - j1.979E-7 -0.03475 - j8.184E-5 19.05 0.8707 + j1.394E-7 -0.03747 - j7.188E-5 25.40 0.8072 + j1.789E-7 -0.03473 - j5.99E-5 31.75 0.8326 + j7.018E-8 -0.03979 - j5.951E-5 38.10 0.8158 + j8.953E-8 -0.03681 - j8.939E-5 44.45 0.8337 + j1.77E-7 -0.03626 - j8.503E-5 50.80 0.7726 - j0.0004621 -0.04306 - j1.248E-6

Table B.1 (d): Induced Currents at 0.2GHz for 8mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8086 - j0.0005556 -0.01283 + j6.506E-6 6.35 0.8353 - j6.335E-8 -0.01354 - j2.153E-5 12.70 0.79 - j8.974E-9 -0.01367 - j2.585E-5 19.05 0.8293 - j1.912E-8 -0.01417 - j3.325E-5 25.40 0.7914 - j5.449E-8 -0.01266 - j2.966E-5 31.75 0.8319 - j2.668E-8 -0.01244 - j1.372E-5 38.10 0.8058 - j5.992E-8 -0.01272 - j2.831E-5 44.45 0.8716 - j2.447E-8 -0.01312 - j2.376E-5 50.80 0.8052 - j6.571E-5 -0.01277 - j8.892E-6

101

Table B.2 (a): Induced Currents at 0.5GHz for 2mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.9157 - j0.00012 -0.3435 - j0.0004654 6.35 0.91 - j1.754E-5 -0.3296 - j0.0004204 12.70 0.9435 - j3.939E-5 -0.3178 - j0.0004234 19.05 0.9424 - j7.459E-7 -0.3176 - j0.0003535 25.40 0.9156 - j8.793E-6 -0.3266 - j0.0004372 31.75 0.946 - j6.234E-6 -0.3279 - j0.0004432 38.10 0.9157 - j7.071E-6 -0.3195 - j0.0004128 44.45 0.9351 - j1.185E-5 -0.3189 - j0.0004481 50.80 0.895 - j8.859E-5 -0.3331 - j0.0002987

Table B.2 (b): Induced Currents at 0.5GHz for 4mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8317 - j3.419E-5 -0.1016 - j0.0001191 6.35 0.8548 - j4.075E-7 -0.1053 - j0.000109 12.70 0.8802 + j3.4E-7 -0.1077 - j0.0001485 19.05 0.8156 - j2.042E-6 -0.1095 - j0.0001153 25.40 0.82 - j1.458E-6 -0.1036 - j0.0001495 31.75 0.8648 - j1.668E-5 -0.09381 - j7.231E-5 38.10 0.8709 - j1.081E-6 -0.1135 - j0.0001483 44.45 0.8537 + j1.415E-6 -0.1039 - j0.0001146 50.80 0.8443 - j5.452E-6 -0.1122 - j7.475E-5

102

Table B.2 (c): Induced Currents at 0.5GHz for 6mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8394 - j0.0001087 -0.04693 - j6.861E-5 6.35 0.8748 - j2.587E-8 -0.0369 - j3.692E-5 12.70 0.8556 - j1.929E-7 -0.03486 - j5.28E-5 19.05 0.8708 + j2.864E-8 -0.03757 - j4.481E-5 25.40 0.8072 + j4.092E-8 -0.03482 - j3.854E-5 31.75 0.8327 - j2.607E-8 -0.03987 - j3.643E-5 38.10 0.8159 - j1.577E-8 -0.03689 - j5.66E-5 44.45 0.8338 + j4.835E-8 -0.03632 - j5.426E-5 50.80 0.7725 - j0.00037 -0.04309 + j6.032E-6

Table B.2 (d): Induced Currents at 0.5GHz for 8mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8083 - j0.0004292 -0.01286 + j8.229E-6 6.35 0.8354 - j1.074E-7 -0.01358 - j1.284E-5 12.70 0.7901 - j8.284E-8 -0.01371 - j1.576E-5 19.05 0.8293 - j8.177E-8 -0.01421 - j2.101E-5 25.40 0.7915 - j1.129E-7 -0.0127 - j1.91E-5 31.75 0.832 - j8.384E-8 -0.01247 - j7.451E-6 38.10 0.8058 - j1.146E-7 -0.01276 - j1.841E-5 44.45 0.8716 - j7.701E-8 -0.01315 - j1.462E-5 50.80 0.8052 - j5.041E-5 -0.01279 - j3.842E-6

103

Table B.3 (a): Induced Currents at 1GHz for 2mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.9158 - j8.865E-5 -0.3471 - j0.0003329 6.35 0.9101 - j1.328E-5 -0.3329 - j0.0003046 12.70 0.9436 - j2.879E-5 -0.3203 - j0.000305 19.05 0.9425 - j2.418E-7 -0.3191 - j0.0002473 25.40 0.9158 - j6.341E-6 -0.3272 - j0.0003073 31.75 0.9462 - j4.286E-6 -0.3276 - j0.0003073 38.10 0.9159 - j4.968E-6 -0.3182 - j0.0002818 44.45 0.9352 - j8.341E-6 -0.317 - j0.0003082 50.80 0.8951 - j6.564E-5 -0.3309 - j0.0001827

Table B.3 (b): Induced Currents at 1GHz for 4mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8319 - j2.664E-5 -0.1025 - j8.229E-5 6.35 0.8551 - j4.089E-7 -0.1063 - j7.74E-5 12.70 0.8804 + j1.416E-7 -0.1085 - j0.0001069 19.05 0.8159 - j1.589E-6 -0.11 - j8.092E-5 25.40 0.8202 - j1.167E-6 -0.104 - j0.0001062 31.75 0.865 - j1.208E-5 -0.09401 - j4.9E-5 38.10 0.8711 - j8.751E-7 -0.1136 - j0.0001026 44.45 0.8539 + j8.852E-7 -0.1039 - j7.719E-5 50.80 0.8445 - j1.848E-5 -0.1122 - j4.672E-5

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Table B.3 (c): Induced Currents at 1GHz for 6mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8396 - j8.017E-5 -0.04697 - j4.805E-5 6.35 0.875 - j1.209E-7 -0.03692 - j2.519E-5 12.70 0.8559 - j2.524E-7 -0.03492 - j3.777E-5 19.05 0.871 - j8.495E-8 -0.03771 - j3.145E-5 25.40 0.8075 - j9.641E-8 -0.03504 - j2.763E-5 31.75 0.8329 - j1.415E-7 -0.04023 - j2.545E-5 38.10 0.8161 - j1.395E-7 -0.03728 - j4.024E-5 44.45 0.834 - j8.285E-8 -0.03675 - j3.875E-5 50.80 0.7726 - j0.0002983 -0.04358 + j7.565E-6

Table B.3 (d): Induced Currents at 1GHz for 8mm spacing

Distance [mm] Current in L [A] Current in R [A] 0 0.8084 - j0.0003406 -0.01286 + j7.972E-6 6.35 0.8356 - j1.928E-7 -0.01361 - j8.676E-6 12.70 0.7904 - j1.93E-7 -0.01377 - j1.085E-5 19.05 0.8296 - j1.789E-7 -0.0143 - j1.486E-5 25.40 0.7917 - j2.16E-7 -0.0128 - j1.371E-5 31.75 0.8322 - j1.758E-7 -0.01261 - j4.729E-6 38.10 0.8061 - j2.146E-7 -0.01292 - j1.334E-5 44.45 0.8718 - j1.614E-7 -0.01333 - j1.027E-5 50.80 0.8054 - j3.966E-5 -0.01298 - j1.836E-6

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APPENDX C

TABLES OF INDUCED CURRENTS FOR THREE CONDUCTOR CONNECTOR

Table C.1 (a): Induced currents at 0.2 GHz for 2mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.815 - j2.544E-5 0.867 - j0.0002569 -0.3868 - j0.00103 6.35 0.9269 - j3.592E-6 0.9494 - j1.176E-5 -0.4068 - j0.0007732 12.70 0.9125 - j4.342E-6 0.9467 - j8.502E-6 -0.4281 - j0.001019 19.05 0.9287 + j3.226E-6 0.9642 - j1.838E-5 -0.4392 - j0.0008829 25.40 0.8394 + j6.514E-7 0.8841 - j2.659E-5 -0.4036 - j0.001021 31.75 0.906 - j4.962E-6 0.9302 - j2.568E-5 -0.4311 - j0.0009847 38.10 0.9106 - j1.39E-6 0.9515 - j2.4E-5 -0.4292 - j0.0007329 44.45 0.9431 - j1.277E-6 0.9341 - j3.678E-5 -0.4201 - j0.0008183 50.80 0.8642 - j0.00014 0.9413 + j4.863E-5 -0.4018 - j0.0007751

106

Table C.1 (b): Induced currents at 0.2 GHz for 2mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7984 + j6.354E-5 -0.2978 - j0.0004004 -0.01482 + j5.615E-5 6.35 0.9068 - j2.784E-5 -0.3178 - j0.0006654 -0.01544 + j6.021E-5 12.70 0.9018 - j1.989E-7 -0.3306 - j0.000611 -0.01578 + j7.093E-5 19.05 0.9444 + j1.754E-5 -0.3345 - j0.0007682 -0.0166 + j8.067E-5 25.40 0.8019 - j5.905E-5 -0.3121 - j0.0007752 -0.01572 + j7.571E-5 31.75 0.8985 - j1.094E-6 -0.3281 - j0.0006409 -0.01537 + j6.548E-5 38.10 0.913 - j3.296E-5 -0.3265 - j0.0007112 -0.01659 + j6.567E-5 44.45 0.9275 + j6.476E-7 -0.3415 - j0.0006558 -0.01588 + j7.258E-5 50.80 0.8658 - j0.0001452 -0.3318 - j0.0008024 -0.01454 + j6.151E-5

Table C.1 (c): Induced currents at 0.2 GHz for 2mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.2664 - j0.0005793 0.8316 - j0.0003458 -0.2664 - j0.0005793 6.35 -0.2795 - j0.0004868 0.9199 - j7.989E-5 -0.2795 - j0.0004868 12.70 -0.284 - j0.0005257 0.9387 - j9.267E-6 -0.284 - j0.0005257 19.05 -0.3027 - j0.0004999 0.9597 - j2.899E-5 -0.3027 - j0.0004999 25.40 -0.273 - j0.00041 0.87 - j4.117E-5 -0.273 - j0.00041 31.75 -0.2913 - j0.0005537 0.9272 - j9.363E-6 -0.2913 - j0.0005537 38.10 -0.2943 - j0.0004974 0.9455 - j5.938E-5 -0.2943 - j0.0004974 44.45 -0.2827 - j0.000303 0.9491 - j3.766E-5 -0.2827 - j0.000303 50.80 -0.2658 - j0.0004881 0.9136 - j5.989E-6 -0.2658 - j0.0004881

107

Table C.2 (a): Induced currents at 0.5 GHz for 2mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.815 - j2.388E-5 0.867 - j0.0001885 -0.3873 - j0.000627 6.35 0.9269 - j2.361E-6 0.9494 - j5.033E-6 -0.4072 - j0.0004829 12.70 0.9126 - j2.837E-6 0.9467 - j4.524E-6 -0.4287 - j0.0006481 19.05 0.9288 + j1.986E-6 0.9642 - j1.17E-5 -0.4398 - j0.0005483 25.40 0.8395 + j3.18E-7 0.8842 - j1.683E-5 -0.4042 - j0.0006443 31.75 0.9061 - j3.238E-6 0.9302 - j1.64E-5 -0.4317 - j0.0006247 38.10 0.9106 - j9.614E-7 0.9516 - j1.431E-5 -0.4297 - j0.0004518 44.45 0.9432 - j8.844E-7 0.9341 - j2.266E-5 -0.4207 - j0.0005101 50.80 0.8643 - j9.893E-5 0.9414 + j2.842E-5 -0.4024 - j0.0004753

Table C.2 (b): Induced currents at 0.5 GHz for 2mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7985 + j4.117E-5 -0.298 - j0.0002207 -0.01481 + j3.693E-5 6.35 0.9068 - j1.768E-5 -0.3182 - j0.0004202 -0.01544 + j3.855E-5 12.70 0.9018 - j7.158E-8 -0.3309 - j0.0003801 -0.01577 + j4.48E-5 19.05 0.9444 + j1.208E-5 -0.3349 - j0.0004852 -0.01658 + j5.193E-5 25.40 0.8019 - j3.781E-5 -0.3125 - j0.0004919 -0.0157 + j4.927E-5 31.75 0.8986 + j1.134E-6 -0.3285 - j0.0004013 -0.01536 + j4.111E-5 38.10 0.913 - j2.071E-5 -0.3269 - j0.0004501 -0.01658 + j4.14E-5 44.45 0.9276 + j1.462E-6 -0.3419 - j0.0004082 -0.01587 + j4.597E-5 50.80 0.8658 - j0.0001004 -0.3323 - j0.0005068 -0.01455 + j3.935E-5

108

Table C.2 (c): Induced currents at 0.5 GHz for 2mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.2766 - j0.0003933 0.8314 - j0.0002601 -0.2666 - j0.0003497 6.35 -0.2996 - j0.0002821 0.9199 - j5.083E-5 -0.2797 - j0.0003039 12.70 -0.2866 - j0.0003041 0.9387 - j5.575E-6 -0.2843 - j0.0003286 19.05 -0.2955 - j0.0002883 0.9598 - j1.847E-5 -0.3029 - j0.0003067 25.40 -0.2724 - j0.0002601 0.87 - j2.705E-5 -0.2732 - j0.0002489 31.75 -0.2873 - j0.0002853 0.9273 - j5.743E-6 -0.2917 - j0.0003522 38.10 -0.2965 - j0.000364 0.9455 - j3.861E-5 -0.2946 - j0.0003122 44.45 -0.2882 - j0.0002721 0.9492 - j2.384E-5 -0.2829 - j0.0001723 50.80 -0.2626 - j0.0002812 0.9137 - j7.582E-6 -0.2661 - j0.0003003

109

Table C.3 (a): Induced currents at 1 GHz for 2mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.8153 - j2.067E-5 0.8671 - j0.0001448 -0.3886 - j0.0004383 6.35 0.9271 - j1.831E-6 0.9496 - j2.74E-6 -0.4087 - j0.0003464 12.70 0.9128 - j2.168E-6 0.9469 - j2.98E-6 -0.4305 - j0.0004651 19.05 0.929 + j1.297E-6 0.9644 - j8.415E-6 -0.4415 - j0.0003855 25.40 0.8398 + j5.395E-8 0.8844 - j1.202E-5 -0.4054 - j0.000454 31.75 0.9063 - j2.436E-6 0.9304 - j1.174E-5 -0.4326 - j0.0004389 38.10 0.9109 - j8.161E-7 0.9517 - j9.845E-6 -0.4301 - j0.000309 44.45 0.9434 - j7.52E-7 0.9343 - j1.58E-5 -0.4209 - j0.0003506 50.80 0.8645 - j7.469E-5 0.9416 + j1.872E-5 -0.4025 - j0.0003234

Table C.3 (b): Induced currents at 1 GHz for 2mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7987 + j2.912E-5 -0.2987 - j0.000135 -0.01487 + j2.396E-5 6.35 0.9069 - j1.254E-5 -0.3189 - j0.0002899 -0.01551 + j2.471E-5 12.70 0.9019 - j9.366E-8 -0.3317 - j0.0002604 -0.01584 + j2.944E-5 19.05 0.9446 + j8.865E-6 -0.3357 - j0.0003395 -0.01661 + j3.549E-5 25.40 0.802 - j2.704E-5 -0.3132 - j0.0003479 -0.0157 + j3.547E-5 31.75 0.8987 + j1.434E-6 -0.3291 - j0.0002848 -0.01535 + j3.019E-5 38.10 0.9131 - j1.472E-5 -0.3275 - j0.0003239 -0.01659 + j3.185E-5 44.45 0.9277 + j1.359E-6 -0.3424 - j0.0002936 -0.01591 + j3.615E-5 50.80 0.8659 - j7.488E-5 -0.3327 - j0.0003657 -0.01462 + j3.204E-5

110

Table C.3 (c): Induced currents at 1 GHz for 2mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.2767 - j0.0002683 0.8315 - j0.000202 -0.2671 - j0.0002451 6.35 -0.2997 - j0.0001886 0.92 - j3.615E-5 -0.2803 - j0.0002194 12.70 -0.2868 - j0.000208 0.9388 - j3.904E-6 -0.285 - j0.0002348 19.05 -0.2957 - j0.0001982 0.9598 - j1.318E-5 -0.3037 - j0.0002148 25.40 -0.2727 - j0.0001791 0.8701 - j1.964E-5 -0.2737 - j0.0001708 31.75 -0.2878 - j0.0002016 0.9274 - j4.094E-6 -0.292 - j0.0002467 38.10 -0.2973 - j0.0002637 0.9456 - j2.784E-5 -0.2948 - j0.0002148 44.45 -0.2891 - j0.0001976 0.9493 - j1.691E-5 -0.2829 - j0.0001078 50.80 -0.2634 - j0.0002051 0.9138 - j7.288E-6 -0.2661 - j0.0002027

111

Table C.4 (a): Induced currents at 0.2 GHz for 4mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7805 - j3.509E-6 0.7899 - j0.0002097 -0.104 - j0.0002631 6.35 0.841 - j9.331E-9 0.9007 - j5.572E-7 -0.1253 - j0.0002661 12.70 0.844 + j3.936E-8 0.8797 + j8.894E-7 -0.12 - j0.0001536 19.05 0.8639 + j1.108E-7 0.8879 + j3.616E-7 -0.1366 - j0.0003104 25.40 0.8181 + j3.605E-8 0.8498 + j1.842E-6 -0.1121 - j0.000225 31.75 0.8312 + j9.788E-8 0.9064 - j1.415E-6 -0.127 - j0.0002505 38.10 0.8737 - j7.383E-8 0.8828 - j1.192E-6 -0.12 - j0.0002754 44.45 0.8767 + j1.348E-7 0.8734 - j4.962E-7 -0.1269 - j0.0002652 50.80 0.8445 - j0.0006286 0.7895 - j0.0001259 -0.1257 - j0.0002839

Table C.4 (b): Induced currents at 0.2 GHz for 4mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7896 - j0.0002565 -0.09878 - j0.000122 -0.001952 + j7.982E-6 6.35 0.867 - j2.529E-7 -0.1086 - j0.0002111 -0.002147 + j7.709E-6 12.70 0.8626 + j1.045E-6 -0.1118 - j0.0001814 -0.002561 + j8.046E-6 19.05 0.8851 + j4.989E-7 -0.1134 - j0.00025 -0.002137 + j1.445E-5 25.40 0.8456 + j6.999E-7 -0.1007 - j0.0001751 -0.00226 + j1.216E-5 31.75 0.8649 + j1.563E-6 -0.1189 - j0.0001997 -0.002423 + j9.443E-6 38.10 0.9077 - j2.449E-8 -0.1192 - j0.0002266 -0.002492 + j1.04E-5 44.45 0.8907 + j1.09E-6 -0.1096 - j0.0002235 -0.002505 + j1.136E-5 50.80 0.8604 - j2.058E-5 -0.1018 - j0.0001741 -0.002266 + j9.952E-6

112

Table C.4 (c): Induced currents at 0.2 GHz for 4mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.1003 - j8.061E-5 0.7787 + j8.773E-5 -0.09235 - j0.0002002 6.35 -0.1088 - j1.098E-5 0.9045 - j1.526E-6 -0.1097 - j0.000193 12.70 -0.1065 - j0.0001791 0.8833 - j6.416E-7 -0.1018 - j0.0001898 19.05 -0.1099 - j0.0001254 0.8947 + j1.106E-7 -0.1198 - j0.0002689 25.40 -0.09902 - j0.000161 0.8216 - j1.613E-6 -0.1013 - j0.0001507 31.75 -0.1021 - j0.0001488 0.9306 - j1.184E-7 -0.1186 - j0.0002311 38.10 -0.1129 - j0.0002506 0.8995 - j9.19E-7 -0.105 - j0.0002291 44.45 -0.112 - j0.0002059 0.8638 - j9.984E-7 -0.1131 - j0.0002324 50.80 -0.1079 - j0.0001098 0.8006 - j4.057E-5 -0.1051 - j0.0001044

113

Table C.5 (a): Induced currents at 0.5 GHz for 4mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7805 - j6.396E-6 0.7899 - j0.0001426 -0.1042 - j0.0001669 6.35 0.8411 - j7.7E-8 0.9007 - j4.204E-7 -0.1255 - j0.000165 12.70 0.8441 - j4.269E-8 0.8797 + j5.043E-7 -0.1201 - j8.684E-5 19.05 0.864 + j4.395E-9 0.888 + j1.72E-7 -0.1368 - j0.0001996 25.40 0.8182 - j4.609E-8 0.8499 + j1.119E-6 -0.1124 - j0.0001394 31.75 0.8313 - j9.871E-9 0.9064 - j9.568E-7 -0.1273 - j0.0001559 38.10 0.8738 - j1.122E-7 0.8829 - j8.209E-7 -0.1202 - j0.0001716 44.45 0.8768 + j2.585E-8 0.8735 - j3.836E-7 -0.1272 - j0.0001683 50.80 0.8442 - j0.0004388 0.7895 - j9.463E-5 -0.126 - j0.0001808

Table C.5 (b): Induced currents at 0.5 GHz for 4mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7896 - j0.0001891 -0.09889 - j6.408E-5 -0.00195 + j5.112E-6 6.35 0.867 - j2.194E-7 -0.1087 - j0.0001299 -0.002146 + j4.912E-6 12.70 0.8627 + j6.073E-7 -0.1119 - j0.0001067 -0.00256 + j4.832E-6 19.05 0.8851 + j2.606E-7 -0.1136 - j0.0001537 -0.002132 + j9.49E-6 25.40 0.8457 + j3.829E-7 -0.1009 - j0.0001072 -0.002258 + j7.96E-6 31.75 0.8649 + j9.421E-7 -0.1191 - j0.0001211 -0.002425 + j5.982E-6 38.10 0.9078 - j6.423E-8 -0.1194 - j0.0001405 -0.002495 + j6.838E-6 44.45 0.8907 + j6.385E-7 -0.1098 - j0.000139 -0.002509 + j7.375E-6 50.80 0.8605 - j1.422E-5 -0.102 - j9.701E-5 -0.002272 + j6.534E-6

114

Table C.5 (c): Induced currents at 0.5 GHz for 4mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.1003 - j2.968E-5 0.7788 + j5.688E-5 -0.09246 - j0.000124 6.35 -0.1088 + j1.507E-5 0.9045 - j1.044E-6 -0.1099 - j0.0001175 12.70 -0.1067 - j0.0001122 0.8834 - j4.712E-7 -0.1019 - j0.0001172 19.05 -0.11 - j7.004E-5 0.8947 + j2.617E-8 -0.12 - j0.0001752 25.40 -0.09918 - j9.593E-5 0.8217 - j1.103E-6 -0.1015 - j8.696E-5 31.75 -0.1023 - j8.932E-5 0.9307 - j1.158E-7 -0.1189 - j0.0001456 38.10 -0.1131 - j0.0001595 0.8996 - j6.406E-7 -0.1052 - j0.0001447 44.45 -0.1122 - j0.0001261 0.8639 - j7.06E-7 -0.1133 - j0.0001468 50.80 -0.1081 - j4.704E-5 0.8006 - j3.653E-5 -0.1052 - j4.543E-5

115

Table C.6 (a): Induced currents at 1 GHz for 4mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7808 - j6.799E-6 0.7901 - j0.0001051 -0.1049 - j0.0001169 6.35 0.8413 - j1.787E-7 0.9009 - j4.046E-7 -0.1263 - j0.0001142 12.70 0.8443 - j1.49E-7 0.88 + j2.379E-7 -0.1208 - j5.59E-5 19.05 0.8642 - j1.169E-7 0.8882 + j2.77E-9 -0.1376 - j0.0001417 25.40 0.8184 - j1.536E-7 0.8501 + j6.704E-7 -0.1129 - j9.712E-5 31.75 0.8316 - j1.326E-7 0.9066 - j7.777E-7 -0.1278 - j0.0001096 38.10 0.874 - j1.943E-7 0.8831 - j7.002E-7 -0.1206 - j0.0001216 44.45 0.877 - j8.574E-8 0.8737 - j3.885E-7 -0.1275 - j0.0001208 50.80 0.8443 - j0.0003274 0.7897 - j7.362E-5 -0.1262 - j0.0001313

Table C.6 (b): Induced currents at 1 GHz for 4mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7897 - j0.0001453 -0.09941 - j4.066E-5 -0.002003 + j3.584E- 6 6.35 0.8672 - j2.643E-7 -0.1093 - j9.24E-5 -0.002194 + j3.429E- 6 12.70 0.8629 + j3.32E-7 -0.1125 - j7.351E-5 -0.0026 + j3.212E-6 19.05 0.8853 + j8.831E-8 -0.1142 - j0.0001075 -0.00215 + j6.81E-6 25.40 0.8459 + j1.663E-7 -0.1014 - j7.43E-5 -0.002264 + j5.734E- 6 31.75 0.8651 + j5.671E-7 -0.1195 - j8.236E-5 -0.002421 + j4.311E- 6 38.10 0.9079 - j1.37E-7 -0.1198 - j9.651E-5 -0.002481 + j5.123E- 6 44.45 0.8909 + j3.526E-7 -0.1101 - j9.563E-5 -0.002493 + j5.497E- 50.80 0.8607 - j1.08E-5 -0.1021 - j6.072E-5 -0.0022566 + j4.934E- 6

116

Table C.6 (c): Induced currents at 1 GHz for 4mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.1009 - j1.368E-5 0.779 + j4.029E-5 -0.09324 - j8.864E-5 6.35 -0.1094 + j1.701E-5 0.9047 - j8.517E-7 -0.1108 - j8.371E-5 12.70 -0.1073 - j8.148E-5 0.8836 - j4.453E-7 -0.1027 - j8.327E-5 19.05 -0.1106 - j4.693E-5 0.8949 - j7.548E-8 -0.1208 - j0.0001268 25.40 -0.09959 - j6.477E-5 0.8219 - j9.069E-7 -0.1019 - j5.741E-5 31.75 -0.1026 - j5.972E-5 0.9308 - j1.647E-7 -0.1192 - j0.0001013 38.10 -0.1134 - j0.0001106 0.8997 - j5.512E-7 -0.1052 - j0.0001005 44.45 -0.1124 - j8.409E-5 0.8641 - j6.117E-7 -0.1132 - j0.0001014 50.80 -0.1082 - j1.861E-5 0.8008 - j3.115E-5 -0.1051 - j2.166E-5

117

Table C.7 (a): Induced currents at 0.2 GHz for 6mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7709 - j9.333E-5 0.8069 - j0.000234 -0.0382 - j3.902E-5 6.35 0.8576 - j3.758E-8 0.8787 - j1.223E-7 -0.04262 - j7.747E-5 12.70 0.8363 - j3.869E-8 0.8366 - j1.065E-7 -0.03915 - j6.919E-5 19.05 0.8573 - j4.198E-8 0.8644 - j1.157E-7 -0.04405 - j8.892E-5 25.40 0.8166 - j3.619E-8 0.7745 - j1.692E-7 -0.03687 - j8.404E-5 31.75 0.8953 - j4.122E-8 0.8398 - j4.408E-7 -0.03618 - j3.27E-5 38.10 0.871 - j4.222E-8 0.8809 - j1.186E-7 -0.04353 - j6.607E-5 44.45 0.8649 - j3.818E-8 0.8421 - j2.906E-7 -0.04395 - j8.065E-5 50.80 0.7965 - j7.381E-5 0.7287 - j3.148E-5 -0.04613 - j8.863E-5

Table C.7 (b): Induced currents at 0.2 GHz for 6mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7685 + j0.0001065 -0.03274 + j7.419E-6 -0.0002964 + j1.35E-6 6.35 0.9207 + j1.314E-8 -0.04075 - j8.187E-5 -0.000319 + j1.278E-6 12.70 0.8567 - j6.243E-8 -0.0377 - j7.534E-5 -0.000269 + j1.27E-6 19.05 0.8938 - j6.548E-8 -0.0411 - j4.742E-5 -0.0003313 + j1.267E-6 25.40 0.7792 + j2.308E-9 -0.03545 - j3.192E-5 -0.0002818 + j9.276E-7 31.75 0.9002 - j1.56E-8 -0.03897 - j6.435E-5 -0.0002961 + j1.362E-6 38.10 0.8866 - j2.338E-7 -0.03886 - j8.278E-5 -0.0003232 + j1.401E-6 44.45 0.8694 - j2.179E-7 -0.03835 - j6.504E-5 -0.0003277 + j1.396E-6 50.80 0.8181 + j7.515E-5 -0.03475 - j6.497E-5 -0.000343 + j1.367E-6

118

Table C.7 (c): Induced currents at 0.2 GHz for 6mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.04041 - j5.348E-5 0.7936 - j0.000249 -0.03639 - j3.713E-5 6.35 -0.03768 - j6.006E-5 0.9056 - j8.306E-8 -0.03991 - j7.304E-5 12.70 -0.03623 - j7.469E-5 0.867 - j1.376E-7 -0.03848 - j8.115E-5 19.05 -0.03682 - j5.891E-5 0.8993 + j5.065E-8 -0.04189 - j9.587E-5 25.40 -0.03675 + j2.16E-5 0.8396 + j1.762E-7 -0.03491 - j6.675E-5 31.75 -0.04429 - j9.018E-5 0.8606 + j6.187E-9 -0.03452 - j3.472E-5 38.10 -0.03933 - j0.000105 0.8845 - j9.18E-8 -0.04125 - j5.796E-5 44.45 -0.0416 - j7.785E-5 0.8707 - j1.15E-7 -0.04185 - j7.319E-5 50.80 -0.04152 - j9.587E-5 0.7401 + j0.0001486 -0.04236 - j6.021E-5

119

Table C.8 (a): Induced currents at 0.5 GHz for 6mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7709 - j7.595E-5 0.8068 - j0.0001719 -0.03823 - j1.832E-5 6.35 0.8576 - j9.088E-8 0.8787 - j1.354E-7 -0.04268 - j4.89E-5 12.70 0.8363 - j9.429E-8 0.8367 - j1.424E-7 -0.03921 - j4.135E-5 19.05 0.8574 - j9.391E-8 0.8645 - j1.385E-7 -0.04414 - j5.538E-5 25.40 0.8166 - j9.611E-8 0.7746 - j1.84E-7 -0.03696 - j5.423E-5 31.75 0.8953 - j8.277E-8 0.8399 - j3.521E-7 -0.03627 - j1.69E-5 38.10 0.8711 - j9.078E-8 0.8809 - j1.359E-7 -0.04365 - j3.871E-5 44.45 0.865 - j8.673E-8 0.8421 - j2.52E-7 -0.04409 - j5.039E-5 50.80 0.7966 - j5.442E-5 0.7288 - j2.42E-5 -0.04629 - j5.525E-5

Table C.8 (b): Induced currents at 0.5 GHz for 6mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7686 + j7.27E-5 -0.03282 + j1.455E-5 -0.0002955 + j8.642E-7 6.35 0.9207 - j4.225E-8 -0.04089 - j5.099E-5 -0.0003183 + j7.818E-7 12.70 0.8568 - j1.022E-7 -0.03781 - j4.58E-5 -0.0002686 + j8.115E-7 19.05 0.8938 - j9.834E-8 -0.04119 - j2.723E-5 -0.0003312 + j8.048E-7 25.40 0.7793 - j7.306E-8 -0.03551 - j1.482E-5 -0.0002822 + j5.883E-7 31.75 0.9002 - j6.15E-8 -0.03904 - j3.9E-5 -0.0002967 + j8.993E-7 38.10 0.8867 - j2.075E-7 -0.03893 - j5.255E-5 -0.0003245 + j9.31E-7 44.45 0.8694 - j1.967E-7 -0.0384 - j3.942E-5 -0.0003295 + j9.189E-7 50.80 0.8182 + j5.374E-5 -0.0348 - j4.064E-5 -0.000345 + j9.012E-7

120

Table C.8 (c): Induced currents at 0.5 GHz for 6mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.04044 - j2.931E-5 0.7936 - j0.0001841 -0.03642 - j1.72E-5 6.35 -0.03772 - j3.653E-5 0.9056 - j1.039E-7 -0.03997 - j4.617E-5 12.70 -0.03628 - j4.682E-5 0.867 - j1.459E-7 -0.03854 - j5.049E-5 19.05 -0.03687 - j3.619E-5 0.8993 - j2.142E-8 -0.04197 - j6.032E-5 25.40 -0.03678 + j2.506E-5 0.8397 + j5.061E-8 -0.03498 - j3.804E-5 31.75 -0.04442 - j5.705E-5 0.8606 - j5.919E-8 -0.0346 - j1.894E-5 38.10 -0.03947 - j6.777E-5 0.8846 - j1.149E-7 -0.04136 - j3.369E-5 44.45 -0.04175 - j4.82E-5 0.8707 - j1.316E-7 -0.04199 - j4.552E-5 50.80 -0.04168 - j6.018E-5 0.7402 + j9.992E-5 -0.04249 - j3.531E-5

121

Table C.9 (a): Induced currents at 1 GHz for 6mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7712 - j6.142E-5 0.807 - j0.0001318 -0.03879 - j1.174E-5 6.35 0.8579 - j1.809E-7 0.8789 - j1.989E-7 -0.04329 - j3.66E-5 12.70 0.8366 - j1.882E-7 0.837 - j2.314E-7 -0.0397 - j2.96E-5 19.05 0.8576 - j1.837E-7 0.8647 - j2.115E-7 -0.04456 - j3.966E-5 25.40 0.8169 - j1.955E-7 0.7748 - j2.65E-7 -0.03718 - j3.882E-5 31.75 0.8955 - j1.57E-7 0.8401 - j3.704E-7 -0.03637 - j9.535E-6 38.10 0.8713 - j1.758E-7 0.8811 - j2.004E-7 -0.04367 - j2.46E-5 44.45 0.8652 - j1.703E-7 0.8423 - j2.918E-7 -0.04405 - j3.35E-5 50.80 0.7968 - j4.209E-5 0.7291 - j1.929E-5 -0.04624 - j3.672E-5

Table C.9 (b): Induced currents at 1 GHz for 6mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7689 + j5.309E-5 -0.03275 + j1.511E-5 -0.0003092 + j6.546E-7 6.35 0.9209 - j1.187E-7 -0.04085 - j3.506E-5 -0.0003323 + j5.882E-7 12.70 0.857 - j1.818E-7 -0.03782 - j3.108E-5 -0.0002776 + j6.224E-7 19.05 0.894 - j1.687E-7 -0.0413 - j1.777E-5 -0.0003386 + j6.064E-7 25.40 0.7796 - j1.817E-7 -0.03571 - j7.933E-6 -0.0002844 + j4.329E-7 31.75 0.9004 - j1.335E-7 -0.03939 - j2.711E-5 -0.0002949 + j6.467E-7 38.10 0.8869 - j2.496E-7 -0.03941 - j3.79E-5 -0.0003197 + j6.45E-7 44.45 0.8696 - j2.415E-7 -0.03894 - j2.802E-5 -0.0003225 + j6.116E-7 50.80 0.8184 + j4.021E-5 -0.03533 - j2.912E-5 -0.0003365 + j6.072E-7

122

Table C.9 (c): Induced currents at 1 GHz for 6mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.04119 - j2.108E-5 0.7937 - j0.0001415 -0.03693 - j1.122E-5 6.35 -0.03841 - j2.773E-5 0.9058 - j1.619E-7 -0.04053 - j3.49E-5 12.70 -0.03682 - j3.482E-5 0.8672 - j2.093E-7 -0.03901 - j3.684E-5 19.05 -0.03725 - j2.611E-5 0.8995 - j1.09E-7 -0.04236 - j4.339E-5 25.40 -0.03697 + j2.279E-5 0.8399 - j7.126E-8 -0.03518 - j2.505E-5 31.75 -0.04449 - j3.927E-5 0.8608 - j1.509E-7 -0.03469 - j1.126E-5 38.10 -0.03939 - j4.65E-5 0.8848 - j1.819E-7 -0.04136 - j2.094E-5 44.45 -0.04156 - j3.112E-5 0.8709 - j1.941E-7 -0.04194 - j2.982E-5 50.80 -0.04146 - j3.972E-5 0.7405 + j7.234E-5 -0.04245 - j2.163E-5

123

Table C.10 (a): Induced currents at 0.2 GHz for 8mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7499 - j0.0001439 0.8008 - j0.0005043 -0.01247 + j2.895E-5 6.35 0.8322 - j3.628E-8 0.8129 - j7.472E-8 -0.01485 - j3.502E-5 12.70 0.837 - j3.652E-8 0.85 - j4.324E-8 -0.0145 - j2.572E-5 19.05 0.7948 - j4.266E-8 0.8485 - j5.195E-8 -0.0141 - j1.178E-5 25.40 0.7763 - j3.903E-8 0.702 - j7.195E-8 -0.01126 - j1.666E-5 31.75 0.8955 - j2.915E-8 0.85 - j6.353E-8 -0.01459 - j1.05E-5 38.10 0.8587 - j3.393E-8 0.8501 - j3.326E-8 -0.01351 - j2.516E-5 44.45 0.881 - j3.273E-8 0.8519 - j4.925E-8 -0.01428 - j1.597E-5 50.80 0.8027 - j0.0001107 0.7502 - j0.0005144 -0.01111 + j7.567E-6

Table C.10 (b): Induced currents at 0.2 GHz for 8mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7764 - j0.0003319 -0.01286 + j1.225E-5 -3.49E-5 + j3.04E-7 6.35 0.8798 - j1.71E-8 -0.01465 - j2.963E-5 -4.137E-5 + j1.47E-7 12.70 0.8971 - j1.961E-8 -0.01396 - j2.964E-5 -4.078E-5 + j1.622E-7 19.05 0.8865 - j4.416E-8 -0.0138 - j1.532E-5 -3.951E-5 + j2.016E-7 25.40 0.8293 - j1.635E-8 -0.0126 - j1.82E-6 -2.927E-5 + j1.354E-7 31.75 0.8892 - j4.31E-8 -0.01438 - j2.35E-5 -3.941E-5 + j2.143E-7 38.10 0.8735 - j4.444E-8 -0.01344 - j7.189E-6 -3.508E-5 + j1.344E-7 44.45 0.8792 - j8.469E-8 -0.01374 - j3.514E-5 -3.993E-5 + j1.747E-7 50.80 0.8063 - j1.488E-5 -0.01161 + j2.258E-6 -3.564E-5 + j2.381E-7

124

Table C.10 (c): Induced currents at 0.2 GHz for 8mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.01294 - j2.367E-5 0.8064 - j2.715E-5 -0.01237 + j2.754E-5 6.35 -0.01401 - j1.388E-5 0.8649 - j3.851E-8 -0.01492 - j3.447E-5 12.70 -0.01378 - j1.967E-5 0.856 - j4.842E-8 -0.01427 - j2.57E-5 19.05 -0.01328 - j1.052E-5 0.8763 - j4.059E-8 -0.01368 - j1.44E-5 25.40 -0.0113 - j1.782E-5 0.7143 - j6.824E-8 -0.01349 - j1.629E-5 31.75 -0.01433 - j2.879E-5 0.9018 - j1.636E-8 -0.01536 - j2.064E-5 38.10 -0.01358 - j3.964E-6 0.8849 - j1.948E-8 -0.01392 - j2.902E-5 44.45 -0.01502 - j3.207E-5 0.8753 - j4.153E-8 -0.01393 - j1.679E-5 50.80 -0.01505 - j3.755E-5 0.7996 - j3.886E-7 -0.01283 - j3.477E-6

125

Table C.11 (a): Induced currents at 0.5 GHz for 8mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7499 - j0.0001035 0.8006 - j0.0003393 -0.0125 + j2.722E-5 6.35 0.8322 - j9.129E-8 0.813 - j1.205E-7 -0.01492 - j2.261E-5 12.70 0.8371 - j9.135E-8 0.8501 - j9.306E-8 -0.01455 - j1.56E-5 19.05 0.7949 - j1.058E-7 0.8486 - j1.005E-7 -0.01414 - j6.088E-6 25.40 0.7763 - j9.763E-8 0.7021 - j1.409E-7 -0.01129 - j1.037E-5 31.75 0.8955 - j7.282E-8 0.8501 - j1.079E-7 -0.01462 - j4.704E-6 38.10 0.8587 - j8.418E-8 0.8502 - j8.726E-8 -0.01354 - j1.579E-5 44.45 0.881 - j8.142E-8 0.8519 - j9.521E-8 -0.0143 - j8.492E-6 50.80 0.8027 - j8.612E-5 0.75 - j0.0003819 -0.01111 + j8.305E-6

Table C.11 (b): Induced currents at 0.5 GHz for 8mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7763 - j0.0002333 -0.0129 + j1.459E-5 -3.512E-5 + j2.241E-7 6.35 0.8799 - j6.711E-8 -0.01472 - j1.921E-5 -4.156E-5 + j9.174E-8 12.70 0.8972 - j6.636E-8 -0.01402 - j1.85E-5 -4.096E-5 + j1.035E-7 19.05 0.8866 - j8.46E-8 -0.01385 - j8.977E-6 -3.966E-5 + j1.318E-7 25.40 0.8293 - j7.169E-8 -0.01263 + j8.629E-7 -2.938E-5 + j8.779E-8 31.75 0.8893 - j8.129E-8 -0.01441 - j1.485E-5 -3.942E-5 + j1.421E-7 38.10 0.8735 - j8.646E-8 -0.01345 - j3.593E-6 -3.504E-5 + j8.577E-8 44.45 0.8793 - j1.111E-7 -0.01377 - j2.323E-5 -3.987E-5 + j1.146E-7 50.80 0.8063 - j1.811E-5 -0.01161 + j3.246E-6 -3.556E-5 + j1.646E-7

126

Table C.11 (c): Induced currents at 0.5 GHz for 8mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.01296 - j1.499E-5 0.8065 - j2.219E-5 -0.01238 + j2.615E-5 6.35 -0.01402 - j7.505E-6 0.865 - j8.494E-8 -0.01496 - j2.191E-5 12.70 -0.01381 - j1.174E-5 0.856 - j9.455E-8 -0.01431 - j1.56E-5 19.05 -0.01331 - j4.705E-6 0.8763 - j8.442E-8 -0.01371 - j7.794E-6 25.40 -0.01134 - j1.094E-5 0.7144 - j1.297E-7 -0.01353 - j9.88E-6 31.75 -0.01438 - j1.784E-5 0.9018 - j6.235E-8 -0.0154 - j1.194E-5 38.10 -0.01362 - j1.087E-7 0.8849 - j6.815E-8 -0.01396 - j1.838E-5 44.45 -0.01508 - j1.992E-5 0.8753 - j8.373E-8 -0.01397 - j9.161E-6 50.80 -0.01511 - j2.347E-5 0.7996 - j1.787E-6 -0.01285 + j7.916E-7

127

Table C.12 (a): Induced currents at 1 GHz for 8mm spacing with L, C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7501 - j7.87E-5 0.8008 - j0.000248 -0.01245 + j2.479E-5 6.35 0.8325 - j1.829E-7 0.8133 - j2.077E-7 -0.01491 - j1.517E-5 12.70 0.8373 - j1.842E-7 0.8503 - j1.785E-7 -0.01457 - j9.992E-6 19.05 0.7952 - j2.128E-7 0.8488 - j1.898E-7 -0.0142 - j3.273E-6 25.40 0.7766 - j1.959E-7 0.7024 - j2.652E-7 -0.01139 - j7.252E-6 31.75 0.8957 - j1.466E-7 0.8503 - j1.941E-7 -0.0148 - j2.942E-6 38.10 0.8589 - j1.714E-7 0.8504 - j1.742E-7 -0.01375 - j1.194E-5 44.45 0.8812 - j1.614E-7 0.8522 - j1.793E-7 -0.01455 - j6.288E-6 50.80 0.8029 - j6.794E-5 0.7501 - j0.0002942 -0.01132 + j6.702E-6

Table C.12 (b): Induced currents at 1 GHz for 8mm spacing with L excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 0.7765 - j0.0001748 -0.01278 + j1.508E-5 -3.328E-5 + j1.823E-7 6.35 0.8801 - j1.46E-7 -0.01463 - j1.269E-5 -4.053E-5 + j6.403E-8 12.70 0.8974 - j1.409E-7 -0.01398 - j1.211E-5 -4.073E-5 + j7.121E-8 19.05 0.8868 - j1.581E-7 -0.01388 - j5.576E-6 -3.99E-5 + j9.61E-8 25.40 0.8295 - j1.575E-7 -0.01274 + j1.518E-6 -2.996E-5 + j6.546E-8 31.75 0.8895 - j1.515E-7 -0.01461 - j1.094E-5 -4.062E-5 + j1.031E-7 38.10 0.8737 - j1.629E-7 -0.01372 - j2.969E-6 -3.661E-5 + j6.024E-8 44.45 0.8795 - j1.792E-7 -0.01408 - j1.809E-5 -4.196E-5 + j8.175E-8 50.80 0.8065 - j1.677E-5 -0.01191 + j2.125E-6 -3.695E-5 + j1.231E-7

128

Table C.12 (c): Induced currents at 1 GHz for 8mm spacing with C excited

Distance [mm] Current in L [A] Current in C [A] Current in R [A] 0 -0.01322 - j1.127E-5 0.8067 - j1.808E-5 -0.01246 + j2.309E-5 6.35 -0.0143 - j5.435E-6 0.8652 - j1.654E-7 -0.01507 - j1.544E-5 12.70 -0.01404 - j8.583E-6 0.8562 - j1.779E-7 -0.01442 - j1.067E-5 19.05 -0.01348 - j2.719E-6 0.8765 - j1.619E-7 -0.01382 - j4.853E-6 25.40 -0.01144 - j7.572E-6 0.7147 - j2.421E-7 -0.01364 - j6.793E-6 31.75 -0.01444 - j1.215E-5 0.902 - j1.345E-7 -0.01554 - j8.058E-6 38.10 -0.01361 + j1.514E-6 0.8851 - j1.452E-7 -0.01409 - j1.314E-5 44.45 -0.01504 - j1.324E-5 0.8755 - j1.589E-7 -0.01408 - j5.923E-6 50.80 -0.01506 - j1.579E-5 0.7999 - j2.197E-6 -0.01294 + j1.898E-6

129