lester cheung ted kong -...

COMPACT 8X8 BUTLER MATRIX FOR ISM BAND

LESTER CHEUNG TED KONG

UNIVERSITI TEKNOLOGI MALAYSIA

PM DR. MAZLINA BT. ESA 124, LORONG BULAN,,

OFF GREEN ROAD,

93150, KUCHING, SARAWAK.


2004/2005


“I hereby declare that I have read this thesis and in my

opinion it is sufficient in terms of scope and quality for the

award of the degree of Masters of

Electrical Engineering (Electronics & Telecommunications)”

Signature : ……………………………….

Supervisor’s Name : Dr. Mazlina bt. Esa

Date : MARCH 2005



A project report submitted in partial fulfillment

of the requirements for the award of the degree of

Masters of Electrical Engineering

(Electronics & Telecommunications)

Faculty of Electrical Engineering

Universiti Teknologi Malaysia

MARCH, 2005

ii

I declare that this thesis entitled “Compact 8x8 Butler Matrix for ISM Band” is the

result of my own research except as cited in the references. The thesis has not been

accepted for any degree and is not concurrently submitted in candidature for any

other degree.

Signature : …………………………………

Name : LESTER CHEUNG TED KONG

Date : MARCH, 2005

iii

To my beloved parents -

Thanks for the smiles during the darkest days,

Thanks for the warmth during the coldest nights,

Thanks for standing by my side when I’m torn at the seams,

Thanks for understanding my wildest dreams.

iv

ACKNOWLEDGEMENT

The author is deeply indebted to a number of individuals who helped make

this thesis possible. First of all, I would like to thank my supervisor, Dr. Mazlina bt.

Esa for her valuable guidance, suggestions and full support in all aspects during the

course of this dissertation. Her achievement and interest has motivated the author to

excel towards greater heights especially in this thesis.

Appreciation is also owed to Dr. Tayeb. A. Denidni of the University of

Quebec for readily supplying his papers related to the subject material which would

not have been easily available otherwise.

Appreciation is further extended to all my friends for their support and ideas

especially Abd Shukur and Choong Lai Mun. Also my deepest appreciation and

thanks to other members of my study group; Samsul Dahlan, Teddy Purnamirza,

Kesavan and Abdul Hafiizh.

My deepest appreciation, thanks and love goes to my family members who

have always been very supportive and willing to share all my joy and pain.

Finally, my heartfelt appreciation goes to all, who have directly or indirectly

helped me in completing my dissertation.

v

ABSTRACT

Mitigation of interference due to multiple signals coexisting in the same

frequency band can be achieved through the implementation of a smart antenna

system that employs scanning of multiple and simultaneously available beams. These

multiple and simultaneously available beams can be generated through beamforming

networks such as the Butler Matrix, of which its design is the focus of this thesis. By

employing passive devices in a butterfly layout configuration, a completely planar

microstrip 8x8 Butler Matrix to perform in the ISM Band of 2.4 GHz to 2.5 GHz was

designed. Recently, research has been focusing on the miniaturization of passive

microwave devices and components. Artificial Transmissions Lines is a relatively

new method that achieves miniaturization of a transmission line through periodically

loading the line capacitively in order to lower the phase velocity characteristic of a

high impedance line to an appropriate value. The higher the impedance of the line in

question, the greater the level of miniaturization obtained. 40 percent area savings

and 59 percent area savings were achieved in the design of the passive component

hybrid and crossover structures, respectively, compared to conventional design

methods. Through the implementation of 12 hybrids, 8 fixed phase shifters and 16

crossover structures, the final layout of a compact Butler Matrix can be designed.

Simulations were performed using Microwave Office electromagnetic packages.

Design calculations were verified using MatchCad.

vi

ABSTRAK

Kesan gangguan yang disebabkan kewujudan pelbagai isyarat pada jalur

frekuensi yang sama dapat dikurangkan melalui implementasi sistem antena cerdas

yang dapat mengimbas rangkaian pelbagai alur serentak. Rangkaian pelbagai alur

serentak ini dapat dihasilkan menerusi rangkaian membentuk alur seperti Matriks

Butler yang rekabentuknya menjadi fokus utama tesis ini. Dengan menggunakan

peranti pasif dalam suatu konfigurasi rama-rama, Matriks Butler 8x8 mikrojalur yang

planar dapat direkabentuk untuk kegunaan dalam Jalur ISM yang merangkumi 2.4

GHz hingga 2.5 GHz. Baru-baru ini, penyelidikan juga tertumpu kepada pengecilan

rekabentuk peranti gelombang mikro pasif serta komponennya. Talian Penghantaran

Buatan merupakan kaedah baru yang berjaya mengecilkan talian penghantaran

melalui pembebanan kapasitif berkala bagi mengurangkan ciri halaju fasa talian

penghantaran bergalangan tinggi kepada nilai yang dikehendaki. Galangan yang

lebih tinggi akan dapat mencapai tahap pengecilan yang lebih besar. Penjimatan

keluasan sebanyak 40 peratus untuk rekabentuk komponen hibrid pasif dan 59

peratus untuk rekabentuk komponen litar crossover dapat dicapai berbanding dengan

kaedah konvensional. Menerusi implementasi 12 hibrid, 8 penganjak fasa tetap dan

16 litar crossover, konfigurasi akhir Matrix Butler yang padat dapat direkabentuk.

Simulasi telah dilaksanakan melalui pakej perisian elektromagnet Microwave Office

manakala pengiraan rumus rekabentuk telah disahkan melalui perisian MathCad.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT (ENGLISH) v

ABSTRAK (BAHASA MELAYU) vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xiii

LIST OF NOTATION xix

LIST OF APPENDICES xxi

CHAPTER I INTRODUCTION 1

1.1 Project Background 1

1.2 Objective 4

1.3 Scope Of Project 5

1.4 Summary Of Chapters 7

viii

CHAPTER II MICROSTRIP TRANSMISSION LINE THEORY 8

2.1 Transmission Line Basics 8

2.2 General Impedance Definition 10

2.3 Microstrip Transmission Line s 12

2.4 Propagation Constant and Phase Velocity 15

2.5 Return Loss and Insertion Loss 17

2.6 General Description of Microstrip Lines 18

2.7 Waves on Microstrip 19

2.8 Microstrip Circuit Advantages 22

2.9 Microstrip Circuit Disadvantages 22

CHAPTER III ANTENNA ARRAY AND BEAMFORMING 24

3.1 Fundamental Array Theory 24

3.2 Scanning and Collimation of Linear and Planar Arrays 27

3.3 Beamforming Networks 28

3.4 Butler Matrix 29

CHAPTER IV BUTLER MATRIX PASSIVE COMPONENTS 34

4.1 Branch Line Hybrids 35

4.2 Broadbanded Branch Line Hybrid Coupler 39

4.3 Crossover 40

4.4 Phase Shifters 42

CHAPTER V ARTIFICIAL TRANSMISSION LINES 43

5.1 Analysis of Periodic Capacitive Loaded Transmission

Lines 44

5.2 Guidelines for ATL Design 48

ix

5.3 ATL Quadrature Hybrid 50

5.4 ATL Broadbanded Branch Line Hybrid Coupler 50

5.5 ATL Crossover 51

CHAPTER VI METHODOLOGY AND DESIGN 52

6.1 Design Material Specification and Preliminary Design

Calculation 54

6.2 Passive Component Design 56

6.2.1 Quadrature Hybrid 58

6.2.2 ATL Quadrature Hybrid 59

6.2.3 Perpendicular Arm Conventional Quadrature

Hybrid 61

6.2.4 ATL Perpendicular Arm Quadrature Hybrid 62

6.2.5 Broadbanded Quadrature Hybrid 63

6.2.6 ATL Broadbanded Quadrature Hybrid 64

6.2.7 Crossover 65

6.2.8 ATL Crossover 66

6.2.9 Broadbanded Crossover 67

6.2.10 ATL Broadbanded Crossover 68

6.2.11 Phase Shifters 69

6.3 Butler Matrix Layout 69

6.4 Simulation Tool 74

CHAPTER VII SIMULATION RESULTS AND DISCUSSION 76

7.1 Quadrature Hybrid Simulation Results 76

7.2 ATL Quadrature Hybrid Simulation Results 78

7.3 Perpendicular Arm Quadrature Hybrid Simulation Results 80

7.4 ATL Perpendicular Arm Quadrature Hybrid Simulation

Results 81

7.5 Broadbanded Quadrature Hybrid Simulation Results 83

x

7.6 ATL Broadbanded Quadrature Hybrid Simulation Results 85

7.7 Crossover Simulation Results 87

7.8 ATL Crossover Simulation Results 89

7.9 Broadbanded Crossover Simulation Results 91

7.10 Broadbanded ATL Crossover Simulation Results 92

7.11 Butler Matrix Simulation Results 94

7.12 Additional discussion on ATL hybrid design 97

CHAPTER VIII CONCLUSION 99

8.1 Conclusion 99

8.2 Recommendations and Future Work 101

REFERENCES 102

APPENDIX 106

Appendix A INITIAL MATHCAD CALCULATIONS 106

Appendix B MATERIAL SPECIFICATION AND

GENERAL PARAMETERS AND

CALCULATIONS 108

Appendix C PHASE SHIFT CALCULATIONS 112

Appendix D ATL CALCULATIONS 113

Appendix E MITERED BEND CALCULATION 119

Appendix F BUTLER MATRIX GRAPHS 120

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Circuit and Antenna Substrate Requirements 21

6.1 RO4003 Material Specification taken from Rogers 4003 brochure 54

6.2 Input Phase Shift designed into the RF ports 70

7.1 Performance of the Standard Quadrature Hybrid 77

7.2 Performance of the ATL Quadrature Hybrid 78

7.3 Performance of the Perpendicular Arm Quadrature Hybrid 80

7.4 Performance of the ATL Perpendicular Arm Quadrature Hybrid 82

7.5 Performance of the Broadbanded Quadrature Hybrid 84

7.6 Performance of the ATL Broadbanded Quadrature Hybrid 86

7.7 Performance of the Narrowband Crossover 88

7.8 Performance of the Narrowband ATL Crossover 90

xii

7.9 Performance of the Broadband Crossover 91

7.10 Performance of the Broadband ATL Crossover 93

xiii

LIST OF FIGURES

FIGURE TITLE PAGE

1.1 Block Diagram of a conventional 8X8 Butler Matrix 5

1.2 Quadrature hybrid 6

1.3 Crossover circuit 6

2.1 The electromagnetic field around a microstrip 19

2.2 Space waves propagated through the substrate 20

2.3 Surface wave transmission in a substrate 20

3.1 Block diagram of a conventional 8x8 Butler Matrix 30

4.1 Microstrip realization of quadrature hybrid 36

4.2 Wideband or broadbanded 90° hybrid 39

4.3 Crossover structure 41

4.4 A broadbanded crossover structure made up of cascaded

xiv

wideband quadrature hybrids 41

5.1 ATLs created via periodic loading of capacitive stubs and its

equivalent electrical circuit model 45

5.2 The optimized 2.45 GHz central frequency ATL quadrature hybrid 50

5.3 The optimized 2.45 GHz central frequency wideband ATL

quadrature hybrid 51

5.4 ATL Crossover implemented through the cascade of two ATL

quadrature hybrids seen in Figure 5.2 51

5.5 Broadbanded version of the ATL crossover 52

6.1 Design Methodology 53

6.2 Material specification is entered into the MSUB component of

the Microwave Office Schematic Design 56

6.3 An equivalent substrate model is entered for Electromagnetic

Simulation in Microwave Office 56

6.4 Sample microstrip circuit schematic elements used for the design 57

6.5 Schematic dagram for a conventional quadrature branch line

coupler 58

6.6 Optimized conventional quadrature branch line coupler layout 58

6.7a Schematic diagram for an ATL quadrature branch line coupler 59

6.7b Schematic diagram blow-up for an ATL quadrature branch line

coupler 60

xv

6.8 Optimized ATL quadrature branch line coupler layout 60

6.9 Schematic diagram for a perpendicular arm quadrature hybrid

coupler 61

6.10 Optimized perpendicular arm quadrature branch line coupler

layout 61

6.11 Schematic diagram for an ATL perpendicular arm quadrature

hybrid coupler 62

6.12 Optimized ATL perpendicular arm quadrature branch line

coupler layout 62

6.13 Schematic diagram for a broadbanded quadrature hybrid 63

6.14 Optimized broadbanded quadrature branch line layout 63

6.15 Schematic diagram for an ATL broadbanded quadrature hybrid 64

6.16 Optimized ATL broadbanded quadrature branch line coupler

layout 64

6.17 Schematic diagram for a conventional narrowband crossover 65

6.18 Optimized ATL conventional narrowband crossover layout 65

6.19 Schematic diagram for an ATL narrowband crossover 66

6.20 Optimized ATL narrowband crossover layout 66

6.21 Schematic diagram for a conventional broadband crossover 67

xvi

6.22 Optimized broadband crossover layout 67

6.23 Schematic diagram for an ATL broadband crossover 68

6.24 Optimized ATL broadband crossover layout 68

6.25 Butler Matrix layout mind map 71

6.26 Conceptual layout of the 8x8 Butler Matrix 72

6.27 Layout of the antenna stage 72

6.28 Layout of the penultimate stage 73

6.29 Layout of the second stage 73

6.30 Layout of the RF stage 73

6.31 Final layout of the 8x8 Butler Matrix 74

7.1 Curves showing the performance of the standard quadrature hybrid 76

7.2 Phase response of the standard quadrature hybrid 77

7.3 Curves showing the performance of the ATL quadrature hybrid 78

7.4 Phase response of the ATL quadrature hybrid 79

7.5 Curves showing the performance of the perpendicular arm


7.6 Phase response of the perpendicular arm quadrature hybrid 81

7.7 Curves showing the performance of the ATL perpendicular

xvii

arm quadrature hybrid 82

7.8 Phase response of the ATL perpendicular arm quadrature hybrid 83

7.9 Curves showing the performance of the broadbanded quadrature

hybrid 84

7.10 Phase response of the broadbanded quadrature hybrid 85

7.11 Curves showing the performance of the ATL broadbanded


7.12 Phase response of the ATL broadbanded quadrature hybrid 86

7.13 Curves showing the performance of the narrowband crossover 87

7.14 Phase response of the narrowband crossover 88

7.15 Curves showing the performance of the narrorband ATL crossover 89

7.16 Phase response of the narrowband ATL crossover 90

7.17 Curves showing the performance of the broadband crossover 91

7.18 Phase response of the broadband crossover 92

7.19 Curves showing the performance of the broadband ATL crossover 93

7.20 Curves showing the coupling at port 1R 95

7.21 Phase response at port 1R 96

7.22 An arbitrary ATL hybrid 97

xviii

7.23 Conceptual design of the ATL hybrid 98

xix

LIST OF NOTATION

α Attenuation constant

β Propagation constant

εeff Effective dielectric constant

εo Dielectric constant of free space

εr Dielectric constant / permittivity

φ Electrical Length

Γin Input reflection coefficient

Γo Reflection coefficient

λ Wavelength

λg Guided wavelength

λo Free space wavelength

µo Permeability of free space

ω Angular frequency

c Velocity of light

C Capacitance

Cp Periodic Capacitance

f Frequency

G Conductance

h Substrate height

I Current

j Complex number

k Complex propagation constant

L Inductance

Pi Incident power

xx

Pmax Peak handling capability

Pr Reflected power

Pt Transmitted power

R Resistance

t Conductor thickness

V Voltage

vp Phase velocity

w Conductor width

weff Effective conductor width

Zf Wave impedance in free space

Z0 Characteristic impedance

xxi

LIST OF APPENDICES

APPENDIX TITLE PAGE

A INITIAL MATHCAD CALCULATION 103

B MATERIAL SPECIFICATION AND GENERAL

PARAMETERS AND CALCULATIONS 105

C PHASE SHIFT CALCULATIONS 109

D ATL CALCULATIONS 110

E MITERED BEND CALCULATION 116

F BUTLER MATRIX GRAPHS

CHAPTER 1

INTRODUCTION

1.1 Project Background

Increasing importance is placed upon the mobility of portable devices for

embedded information and telecommunication, including PDAs, pagers, cellular

phones and active badges. Advances in sensor integration and electronic

miniaturization make feasible the production of sensing devices equipped with

significant processing memory and wireless communication capabilities to create

smart environments where scattered sensors can coordinate to establish networks.

Increasing wireless technologies are developed, e.g. IEEE 802.11, Bluetooth,

and Home Radio Frequency (HomeRF) that promises to outfit portable and

embedded devices with high bandwidth, localized wireless communication

capabilities that may reach the globally wired Internet.

The 2.4 GHz Industry, Scientific and Medical (ISM) unlicensed band

constitutes a popular frequency band suitable to low cost radio solutions such as

those proposed for Wireless Personal Area Networks (WPANs) and Wireless Local

Area Networks (WLANs), more so due to its almost global availability. Sharing of

2

the spectrum among various devices in the same environment, however, may lead to

severe interference and significant performance degradation [1].

Primary and secondary users are allowed in the 2.4 GHz ISM band.

Secondary use are unlicensed, although may have to follow rules of respective

national communications commissions relating to total radiated power and use of the

spread spectrum modulation schemes. As long as these rules are adhered to,

interference among the various uses is not addressed. Therefore, the major downside

of the unlicensed ISM band is that the frequencies must be shared and potential

interference tolerated. Although spread spectrum and power rules are quite effective

when dealing with multiple users in the band of radios that are physically separated,

the same cannot be said for close proximity radios.

Smart Antenna Systems are defined when multiple antenna elements are

combined with a signal-processing capability to optimize the radiation and/or

reception pattern automatically in response to the signal environment. An antenna is

defined as the structure associated with the region of transition between a guided

wave and a free-space wave, or vice versa. Antennas convert electrons to photons or

vice versa. An antenna is the port through which radio frequency (RF) energy is

coupled from the transmitter to the outside world and, in reverse, to the receiver from

the outside world.

Antennas have typically been the most neglected of all personal

communication system components. However, the way energy is distributed and

collected from the ambient space has a profound influence on the spectrum

efficiency, cost in setting up new networks and the network service quality.

Utilizing adaptive beamforming techniques allows an antenna array to reject

interfering signals having a direction of arrival different from that of a desired signal.

Additionally, interfering signals having different polarization states from the desired

3

signal can also be rejected should a multi-polarized array be used, even if the signals

have the same direction of arrival. Improvements in capacity of wireless

communication systems can be improved by exploiting these capabilities.

Two or more antenna elements that are spatially arranged and electrically

interconnected to produce a directional radiation pattern is termed as an array. The

feed network which forms the interconnection between elements can provide fixed

phase to each element or can form a phased array. In optimum and adaptive

beamforming, optimal received signal is obtained by adjusting the phases (and

usually the amplitudes) of the feed network. Performance of an array is influenced by

the geometry of an array and the patterns, orientations, and polarizations of the

elements.

Multiple simultaneously available beams can be generated using array beam

forming techniques [2, 3, 4]. Furthermore, the formed beams can be shaped to have

high gain and low sidelobes, or controlled beamwidth. Adaptive beamforming

techniques can adjust the array pattern dynamically to optimize certain

characteristics of the signal received. In beam scanning, a single main lobe of an

array is steered and the direction varied either continuously or in discrete steps.

Antenna arrays using adaptive beamforming techniques can reject interfering signals

having a direction of arrival different from that of a desired signal. Multi-polarized

arrays can also reject interfering signals having different polarization states from the

desired signal, even if the signals have the same direction of arrival.

Beamforming is defined as focusing the energy radiated by an aperture

antenna along a specific direction. The beamformer is therefore the device or

apparatus that performs this function [4]. To form multiple beams, an array of N

antenna elements in connected to a beamformer of N beam ports. This architecture

acts as though the antennas are directional, forming beams in orthogonal directions

with increased directivity and described as a beamforming network.

4

The Butler Matrix (BM) is an example of a multiple-beamed antenna system.

The system measures angle of arrival on multiple input signals when multiple

receivers are used in conjunction with the Butler Matrix. Linear arrays and a beam-

forming network are utilized and it characteristically has binary numbers of input

antennas, 2n, where n is an integer.

One characteristic of NXN Butler Matrices is that the overall circuit grows

exponentially as N increases. Although increased N will mean more multiple beams

in an 8X8 design compared to a 4X4, this advantage comes with it the expense of

increased circuit size.

This dissertation utilizes capacitive loading of transmission lines to reduce

the size of the passive components of the Butler Matrix, specifically the branch line

hybrid and crossover circuits. By implementing these components in a final 8x8

Butler Matrix design, an overall decreased area can be obtained. A planar design was

simulated for microstrip implementation.

A further contribution of this dissertation is the reusability of the hybrid and

crossover structures in other microwave circuit designs.

1.2 Objectives

To design a compact planar 8x8 Butler Matrix for use in an ISM band Smart

Antenna System.

5

1.3 Scope Of Project

The scope of this dissertation relates to the design of the 8X8 Butler Matrix to

function over the ISM band constituted by a frequency band of 2.4 GHz to 2.5 GHz.

A Butler Matrix comprises of 12 hybrids, 8 fixed phase shifters and 16 crossovers as

shown in Figure 1.1.

Figure 1.1 Block Diagram of a Conventional 8X8 Butler Matrix

The individual passive components were designed in microstrip. The

quadrature hybrid, as shown in Figure 1.2 is a power divider that generates a 90

degree phase shift between its two outputs. In Figure 1.1, it is evident that the

crossing of lines between the various components is unavoidable. This crossing of

lines poses a design problem issue to the layout of a planar design. By implementing

6

a crossover circuit that will have isolation between series ports but allow the good

coupling between diagonal input and output ports, the crossing of lines issue can be

resolved. This circuit is shown in Figure 1.3.

Figure 1.2 Quadrature hybrid

Figure 1.3 Crossover circuit

Electromagnetic simulations were performed on the individual passive

components and the final 8X8 design. Design calculations were verified using

MathCad. Analysis of the S-parameters of the component designs and final matrix

was performed to give an indication of performance.

7

1.4 Summary of Chapters

The first chapter gives an introduction to the dissertation, providing relevant

information concerning the project background, objective and the scope of the

project.

The second chapter deals with transmission line theory related to microstrip

implementation. A general description of the microstrip material is also given,

together with an overview of circuit and antenna requirements.

In chapter three, antenna array and beamforming theory are examined in

detail. Equations related to array factor derivations and beam space representation

are presented here.

An overview of the relevant theory related to the passive components

required in the final design in considered in chapter four. In this chapter, a

conceptualization of passive component theory is shown and the corresponding

number of components required for the design is also presented.

In chapter five, the theory related to the method of miniaturization, artificial

transmission lines is examined. Details of the methodology and design procedure are

expounded in the sixth chapter. The entire design process is discussed in detail in this

chapter.

Simulation results are analyzed in the seventh chapter. Finally, the whole

project is concluded in the final chapter. Suggestions for future work were given.

CHAPTER II

MICROSTRIP TRANSMISSION LINE THEORY

Higher frequencies imply decreasing wavelengths. When dealing with RF

design, the consequent spatially non-uniform voltages and currents when compared

to the size of discrete circuit elements result in a wave propagation treatment of the

problem. Conventional lumped circuit analysis is no longer applicable under these

conditions as Kirchhoff’s voltage and current laws do not account for spatial

variations. Distributed circuit representation must be considered. Spatially dependent

impedance representation of generic transmission line configurations and its

application to the analysis and design of high frequency circuits using microstrip is

considered in this chapter.

2.1 Transmission Line Basics

Transmission lines are defined as multiconductor structures that support

transverse electromagnetic (TEM) and non-TEM mode of propagation [5, 6]. Four

basic parameters describe the TEM transmission line characteristic. These are the

characteristic impedance, Zo the phase velocity, vp (or guided wavelength λg = vp/f),

the attenuation constant α and the peak handling capability Pmax in terms of the

9

physical parameters (e.g. geometrical cross section), and the properties of the

dielectric and conductor materials used.

When considering perfectly terminated TEM lines, the voltage to current ratio

of any point along the line is constant, having the units of resistance for a lossless

medium and defined as the characteristic impedance. The propagation constant of a

lossy transmission structure is a complex quantity that comprises of an attenuation

constant real portion that contains information concerning the conductor dissipation

and dielectric loss and the phase constant imaginary portion that relates to the phase

velocity. Attenuation constant is defined as

α =average power lost per unit length

2 power transmitted( ) (2.1)

The dielectric breakdown and attenuation heating will limit the power

handling capability of a transmission line. By considering both the breakdown

electric field of dry air (2.9 x 106 V/m) and by calculating the maximum field

strength, the peak power handling capability of a transmission line can be

determined. The temperature rise and of the line in an air environment will determine

the average power-handling capability of a transmission line. Parameters crucial to

average power capacity calculations are the attenuation constant, the surface area of

the line, the maximum tolerable temperature rise and the ambient temperature.

Selecting a higher temperature limit and applying forced cooling or cooling fins can

raise the average power rating.

10

2.2 General Impedance Definition

General solutions for transmission lines aligned along the z-axis result in two

exponential functions for the voltage and for the current.

V (z) = V +e−kz + V −e+kz (2.2)

I(z) = I+e−kz + I−e+kz (2.3)

The convention is that the first term represents waveforms propagating in the

+z-direction, whereas the second term denotes wave propagation in the –z-direction.

The factor k is known as a complex propagation constant.

k = kr + jki = R + jωL( ) G + jωC( ) (2.4)

The complex propagation constant equation reflects the nature of a

distributed circuit system whereby infinitesimal length is distributed across a

transmission line and can be characterized by distributed parameters resistance, R,

inductance, L, capacitance, C and conductance, G, and where all circuit parameters

are given in terms of unit length.

Rearranging the equations accordingly provides a current expression of the

following form:

I(z) =k

R + jωL( )V +e−kz −V −e+kz( ) (2.5)

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