GOOGLE EXHIBIT 1014
`
`Page1 of 13
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`Page 1 of 13
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`GOOGLE EXHIBIT 1014
`
`

`

`Microwave Engineering
`
`Second Edition
`
`David M. Pozar
`University of Massachusetts at Amherst
`
`m
`
`JOHN WILEY & SONS, INC.
`
`New York • Chichester • Weinheim
`Brisbane • Singapore • Toronto
`
`Page 2 of 13
`
`

`

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`Thi, t'<~,l "a~ ,~tin 10/ 1.::! Times Roman by ETP HA RRI S() and
`pnnuxl :rnd ~mnd b) R.R. Donnelky & Sons Company, Crawfords ilk.
`The t,,,,e, was printed by The Lt.>high Press. Inc.
`
`Rec,,gnizi ng the importance of preserving what has been written. it is a
`policy of John Wi ley & Sons. Inc. 10 have books o f enduring vnlue publis hed
`in the
`nited States printed on acid-free paper. and we exert our best
`efforts to that end.
`
`The paper on this book was manufactured by a mill whose forest management programs inc lude
`sustained yield harvesting of its timberlands. Sustained yield harvesting ptinciples e nsure that
`the number of trees c:ut each year does not exceed the amount of new growth.
`
`Copyright © 1998. by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Cmada.
`
`Reproduction or translat.ion of any part of
`this work beyond that permitted by Sections
`107 and 108 of the 1976 United States Copyright
`Act without the permission of the copyright
`owner is unlawful. Requests for permission
`or further information should be addressed to
`the Permissions Department. John Wiley & Sons. Inc.
`
`Library of Congress Cataloging in Publicarion Dara
`Pozar, David M.
`Microwave engineering/ David M. Pozar. ·- 2nd ed.
`p.
`cm.
`ISBN_ 0-47 1-17096-8 (cloth : alk. paper)
`l. Microwaves. 2. Microwave devices
`. 3. Microwave circuits.
`l. Title.
`TK7876.P69 1998
`621.381 13--dc20
`
`~
`
`97-20878
`C IP
`
`Printed in the United States o f Arnericu
`
`JO 9 8 7 6 5 4 3 2
`
`Page 3 of 13
`
`

`

`Contents
`
`1
`
`ELECTROMAGNETIC THEORY
`
`1
`
`1.1
`
`1
`
`2
`
`•
`
`A Short History of Microwave
`
`1.6
`
`1.7
`
`1.8
`
`1.9
`
`Introduction to Microwave Engineering
`Applications of Microwave Engineering
`3
`Engineering
`5
`1.2 Maxwell's Equations
`9
`1.3 Fields in Media and Boundary Conditions
`•
`12
`Fields at a General Material Interface
`Fields at a Dielectric
`Interface
`14
`•
`Fields at the Interface with a Perfect Conduc tor (Electric
`•
`14
`Wall)
`The Magnetic Wall Boundary Condition
`15
`•
`l S
`The Radiation Condition
`16
`1.4 The Wave Equation and Basic Plane Wave Solutions
`16
`•
`The Helmholtz Equation
`Plane Waves in a Lossless
`16
`•
`Medium
`Plane Waves in a General Lossy Medium
`19
`Plane Waves in a Good Conductor
`1.5 General Plane Wave Solutions
`21
`25
`Circularly Polarized Plane Waves
`Energy and Power
`26
`Power Absorbed by a Good Conductor
`29
`Plane Wave Reflection from a Media Interface
`31
`•
`General Medium
`Lossless Medium
`Conductor
`34
`•
`Perfect Conductor
`36
`36
`Impedance Concept
`39
`Oblique Incidence at a Dielectric Interface
`Parallel Polarization
`40
`•
`Perpendicular Polarization
`Total Reflection and Surface Waves
`43
`Some Useful Theorems
`45
`The Reciprocity Theorem
`45
`The Uniqueness Theorem
`49
`
`18
`
`•
`
`30
`32
`•
`
`Good
`•
`The Surface
`
`41
`
`•
`
`•
`
`Image Theory
`
`47
`
`•
`
`ix
`
`Page 4 of 13
`
`

`

`X
`
`Contents
`
`2
`
`TRANSMISSION LINE THEORY
`
`56
`
`59
`
`68
`
`•
`
`76
`
`Point of Interest: Decibels
`
`•
`
`The Slotted
`
`83
`•
`
`The Multiple Reflection Viewpoint
`
`85
`
`2.1 The Lumped-Element Circuit Model for a Transmission Line
`56
`Wave Propagation on a Transmission Line
`58
`•
`The Lossless Line
`2.2 Field Analysis of Transmission Lines
`59
`•
`Transmission Line Parameters
`60
`The Telegrapher Equations Derived
`63
`from Field Analysis of a Coaxial Line
`•
`Propagation Constant,
`Impedance, and Power Flow for the Lossless Coaxial Line
`64
`2.3 The Terminated Lossless Line
`65
`Special Cases of Lossless Terminated Lines
`72
`and Nepers
`73
`2.4 The Smith Chart
`The Combined Impedance-Admittance Smith Chart
`Line
`79
`2.5 The Quarter-Wave Transformer
`The Impedance Viewpoint
`83
`87
`2.6 Generator and Load Mismatches
`Generator Matched to Loaded
`•
`Load Matched to Line
`89
`Line
`89
`•
`Conjugate Matching
`89
`90
`Lossy Transmission Lines
`The Low-Loss Line
`•
`90
`Terminated Lossy Line
`Attenuation
`94
`
`The
`The Distortionless Line
`•
`92
`The Perturbation Method for Calculating
`•
`93
`The Wheeler Incremental Inductance Rule
`96
`
`•
`
`3
`
`TRANSMISSION Ll.~ES AND WAVEGUIDES
`
`104
`
`2.7
`
`3.1
`
`3.2
`
`3.3
`
`3.4
`
`3.5
`
`105
`•
`
`111
`
`))4
`
`•
`
`TE Modes
`
`117
`
`General Solutions for TEM, TE, and TM Waves
`TEM Waves
`TE Waves
`107
`•
`109
`•
`TM Waves
`l IO
`Attenuation Due to Dielectric Loss
`112
`Parallel Plate Waveguide
`•
`112
`TM Modes
`TEM Modes
`120
`Rectangular Waveguide
`•
`•
`T&no Modes of a Partially
`120
`125
`TM Modes
`TE Modes
`•
`/31
`130
`Loaded Waveguide
`Point of Interest: Waveguide Flanges
`132
`Circular Waveguide
`TM Modes
`TE Modes
`133
`141
`Coaxial Line
`•
`141
`TEM Modes
`Interest: Coaxial Connectors
`
`•
`
`137
`
`Higher-Order Modes
`146
`
`143
`
`•
`
`Point of
`
`Page 5 of 13
`
`

`

`Contents
`
`xi
`
`147
`•
`
`Point of Interest:
`
`3.6 Smface Waves on a Grounded Dielectric Slah
`TM Modes
`147
`•
`TE Modes
`150
`Root-Fi,ufo,g Algoritl11m
`152
`3.7 Sttipline
`153
`Fonnulas for Propagation Constanl. Characteristic Impedance. and
`Attenuation
`154
`•
`An Approximate Electrostatic Solution
`3.8 Microstrip
`160
`Fommlas for Effective Die lectric Constant. Characteristic lmpcdunce, a nd
`An Approximate Electrostatic Solution
`Attenuation
`162
`•
`3.9 The Transverse Resonance Technique
`167
`TM Modes for the Parallel Plate Waveguide
`168
`Partially Loaded Rectangular Waveguide
`3. 10 Wave Velocities and Dispersion
`170
`Group Velocity
`170
`
`157
`
`164
`
`•
`
`TEo,, Modes of a
`
`169
`
`173
`3.11 Summary of Transmission Lines and Waveguides
`Other Types of Lines and Guides
`174
`•
`Point of Interest.· Power Capacity
`176
`of Transmission Lines
`
`4
`
`MICROWAVE NETWORK ANAL VSIS
`
`182
`
`183
`Impedance and Equivalent Voltages and Currents
`Equivalent Voltages and Currents
`183
`•
`The Concept
`of Impedance
`187
`•
`Even and Odd Properties of Z(w) and f (<.,,•)
`Impedance and Admittance Matrices
`191
`Reciprocal Networks
`193
`•
`Lossless Networks
`4.3 The Scattering Matrix
`196
`•
`199
`Reciprocal Networks and Lossless Networks
`Planes
`202
`•
`Generalized Scattering Parameters
`205
`Interest: The Vector Network Analyzer
`The Transmission (ABCD) Matrix
`206
`Relation to Impedance Matrix
`209
`•
`Networks
`210
`213
`Signal Flow Graphs
`Decomposition of Signal Flow Graphs
`Analyzer Calibration
`217
`•
`Microwave Circuits · 222
`222
`Discontinuities and Modal Analysis
`Modal Analysis of an H-Plane Step in Rectangular Waveguide
`of Interest: Microstrip Discontinuity Compensation
`229
`Excitation of Waveguides- Electric and Magnetic Currents
`Current Sheets That Excite Only One Waveguide Mode
`230
`Excitation from an Arbitrary Electric or Magnetic Current Source
`
`4.1
`
`4.2
`
`4.4
`
`4.5
`
`4.6
`
`4.7
`
`190
`
`195
`
`A Shift in Reference
`204
`•
`Point of
`
`Equivalent Circuits for Two-Port
`
`Application to TRL Network
`•
`214
`Point of Interest: Computer-Aided Design f or
`
`225
`
`•
`
`Poi111
`
`230
`•
`232
`
`Mode
`
`Page 6 of 13
`
`

`

`Xii
`
`Contents
`
`4.8
`
`.237
`Excitation of Waveguides- Aperture Coupling
`Coupling Through an Aperture in a T ransverse Waveguide W~II
`Coupling Through an Aperture in the Broad Wall of a W aveguide
`
`240
`243
`
`•
`
`5
`
`IMPEDANCE MATCHING AND TUNING
`
`251
`
`•
`
`Point of
`
`254
`257
`
`Series Stubs
`
`262
`
`•
`
`270
`
`. 252
`5. 1 Matching with Lumped Elements (L Networks)
`Analytic Solutions
`253
`•
`Smith Chart Solutions
`Interest: Lumped Elements for Microwave Integrated Circuits
`5.2 Single-Stub Tuning
`258
`Shunt Stubs
`259
`•
`5.3 Double-Stub Tuning
`266
`Analytic Solution
`Smith Chart Solution
`266
`271
`5.4 The Quarter-Wave Transformer
`275
`5.5 The Theory of Small Reflections
`Multisection Transformer
`•
`Single-Section Transformer
`276
`5.6 Binomial Multisection Matching Transformers
`278
`5.7 Chebyshev Multisection Matching Transformers
`282
`Chebyshev Polynomials
`283
`•
`Design of Chebyshev Transformers
`5.8 Tapered Lines
`288
`Exponential Taper
`290
`Taper
`291
`5.9 The Bode-Pano Criterion
`
`277
`
`285
`
`•
`
`Triangular Taper
`
`29 1
`
`•
`
`Klopfenstein
`
`295
`
`6
`
`MICROWAVE RESONATORS
`
`300
`
`6.1 Series and Parallel Resonant Circuits
`Series Resonant Circuit
`300
`•
`Loaded and Unloaded Q
`306
`6.2 Transmission Line Resonators
`Short-Circuited >../ 2 Line
`306
`Open-Circuited >../ 2 Line
`3 11
`6.3 Rectangular Waveguide Cavities
`Resonant Frequencies
`313
`•
`Circular Waveguide Cavities
`318
`•
`Resonant Frequencies
`3 18
`323
`6.5 Dielectric Resonators
`Resonant Frequencies of TEo16 Mode
`6.6 Fabry-Perot Resonators
`328
`Stability of Open Resonators
`330
`
`300
`Parallel Resonant Circuit
`
`303
`
`•
`
`Short-Circuited A/ 4 Line
`
`3 10
`
`•
`
`306
`•
`
`313
`Q of the TE,oe Mode
`
`3 15
`
`Q of the TE,unt Mode
`
`320
`
`324
`
`6.4
`
`Page 7 of 13
`
`

`

`..
`
`C'lllll' .11 C1111plt1I}!
`lk!\1111a111r
`.1.14
`Cavi1y P~rturha1 i1111~
`Material Pcrt11rhati1111,
`
`1.12
`•
`
`Contents
`
`xiii
`
`/\ (i11p-C'11upk<l M1umtnp
`•
`:l.17
`/\ 11 /\ pc1111rc-(..'11upk d (':iv,ry
`340
`J40
`
`Shupe l'crturhal11Jn~
`
`143
`
`•
`
`7
`
`POWER DIVIDERS AND DIRECTIONAL
`COUPLERS
`351
`
`7. 1
`
`7.2
`
`7.3
`
`7.4
`
`7.5
`
`7.6
`
`7.7
`7.8
`
`7.9
`
`36 1
`
`Unequal Power Di vision and ./\·.way
`
`368
`Design of Multihole Couplers
`379
`
`374
`
`35 1
`Basic Properties of Dividers and Couplers
`Four-Port Networks (Directional
`•
`Three-Port Networks (T-junctions)
`35 1
`Couplers)
`354
`•
`Poinr of lmeresr: Measuring Coupler Direcrii:ity
`357
`The T-Junc.;tion Power Divider
`359
`Lossless Divider
`360
`•
`Resistive Divider
`The Wilkinson Power Divider
`363
`Even-Odd Mode Analysis
`363
`•
`Wilkinson Dividers
`367
`Waveguide Directi0naJ Couplers
`Bethe Hole Coupler
`369
`•
`The Quadrature (90°) Hybrid
`Even-Odd Mode Analysis
`380
`383
`Coupled Li ne Directional Couplers
`Design of Coupled Line
`Coupled Line Theory
`384
`•
`Couplers
`389
`Design of Multisection Coupled Line Couplers
`The Lange Coupler
`398
`401
`The 180° Hybrid
`Even-Odd Mode Analysis of the Ring Hybrid
`Analysis of the Tapered Coupled Line Hybrid
`411
`Magic-T
`411
`Other Couplers
`Point of Inte rest: The Rejlecrometer
`
`•
`
`394
`
`403
`407
`
`•
`•
`
`Even-Odd Mode
`Waveguide
`
`414
`
`8
`
`MICROWAVE FIL TEAS
`
`422
`
`423
`8.1 Periodic Structures
`•
`424
`Analysis of Infinite Periodic Structures
`Terminated Periodic
`Structures
`427
`•
`k -{3 Diagrams and Wave Velocities
`428
`8.2 Filter Design by the Image Parameter Method
`431
`43 1
`Image Impedances and Transfer Functions for Two-Port Networks
`433
`•
`m-Derived Filter Sections
`Constant-k Filter Sections
`Composite Filters
`440
`
`•
`436
`
`•
`
`Page 8 of 13
`
`

`

`xiv
`
`Contents
`
`454
`
`•
`
`Bandpass and Bandstop
`
`•
`
`Kuroda' s Ide ntities
`
`464
`
`•
`
`.
`443
`8.3 Filter Design by the Insertion Loss Method
`444
`. Maximally Flat Low-Pass Filter
`•
`Characterization by Power Loss Ratio
`Prototype
`447
`•
`Equal-Ripple Low-Pass Filter Prototype
`450
`•
`Linear Phase Low-Pass Filter Prototypes
`451
`452
`8.4 Filter Transformations
`Impedance and Frequency Scaling
`Transfonnations
`457
`462
`8.5 Filter Implementation
`462
`Richard's Transformation
`468
`Impedance and Admittance Inverters
`470
`8.6 Stepped-Impedance Low-Pass Filters
`Approximate Equivalent Circuits for Short Transmission Line Sections
`8.7 Coupled Line Filters
`474
`Filter Properties of a Coupled Line Section
`Bandpass Filters
`477
`486
`8.8 Filters Using Coupled Resonators
`Bandstop and Bandpass Filters Using Quarter-Wave Resonators
`Bandpass Filters Using Capacitively Coupled Resonators
`490
`Direct-Coupled Waveguide Cavity Filters
`493
`
`470
`
`474
`
`•
`
`Design of Coupled Line
`
`486
`•
`
`•
`
`9
`
`THEORY AND DESIGN OF FERRIMAGNETIC
`COMPONENTS
`497
`
`498
`9. 1 Basic Properties of Ferrimagnetic Materials
`The Permeability Tensor
`498
`•
`Circularly Polarized Fields
`Effect of Loss
`506
`•
`Demagnetization Factors
`508
`Interest: Permanent Magnets
`510
`9.2 Plane Wave Propagation in a Ferrite Medium
`Propagation in Direction of Bias (Faraday Rotation)
`Transverse to Bias (Birefringence)
`515
`9.3 Propagation in a Ferrite-Loaded Rectangular Waveguide
`TEm.o Modes of Waveguide with a Single Ferrite Slab
`518
`of Waveguide with Two Symmetrical Ferrite Slabs
`521
`523
`9.4 Ferrite Isolators
`Resonance Isolators
`523
`9.5 Ferrite P hase Shifters
`530
`Nonreciprocal Latching Phase Shifter
`•
`533
`Shifters
`The Gyrator
`9.6 Ferrite Circulators
`535
`Properties of a Mismatched Circulator
`
`511
`512
`
`•
`
`518
`•
`
`•
`
`The Field Displacement Isolator
`
`•
`504
`Point of
`
`•
`
`Propagation
`
`TEmo Modes
`
`527
`
`530
`535
`
`537
`
`•
`
`•
`
`Other Types of Ferrite Phase
`
`Junction Circulator
`
`537
`
`Page 9 of 13
`
`

`

`10
`
`ACTIVE MICROWAVE CIRCUITS
`
`547
`
`Contents
`
`xv
`
`•
`Noise Power and Equi valent
`Measurement of Noise Temperature by the
`Noise Figure
`555
`•
`Noise Figure of a
`
`•
`
`548
`10.1 Noise in Microwave Circuits
`Dynamk Range and Source!'. of Noise
`Noise Tcm1>craturc
`550
`•
`) · -fac tor Method
`553
`Cascaded System
`557
`10.2 Detectors and Mixers
`559
`565 Balanced
`Single-Ended Mixer
`•
`559
`Diode Rccti Hers and Detectors
`57 1
`•
`Intermodulation
`Mi xer
`568
`•
`Other Types of Mi xers
`•
`Products
`574
`Point of Interest: The Spectrum Analyzer
`575
`10.3 PIN Diode Control Circuits
`576
`PIN Diode Phase Shifters
`Single-Pole Switches
`577
`•
`583
`10.4 Microwave Integrated Circuits
`Hybrid Microwave Integrated Circuits
`584
`Integrated Circuits
`584
`10.5 Overview of Microwave Sources
`Solid-State Sources
`589
`•
`
`580
`
`•
`
`Monolithic Microwave
`
`588
`Microwave Tubes
`
`593
`
`11
`
`12
`
`601
`601
`
`• Microwave Bipolar
`
`DESIGN OF MICROWAVE AMPLIFIERS AND
`OSCILLATORS
`600
`11 .1 Characteristics of Microwave Transistors
`Microwave Field Effect Transistors (FETs)
`Transistors
`604
`606
`11.2 Gain and Stability
`Stability
`•
`606
`Two-Port Power Gains
`6 18
`11.3 Single-Stage Transistor Amplifier Design
`Design for Maximum Gain (Conjugate Matching)
`618
`Circles and Design for Specified Gain (Unilateral Device)
`Amplifier Design
`628
`11.4 Broadband Transistor Amplifier Design
`632
`Balanced Amplifiers
`632
`•
`Distributed Amplifiers
`11.5 Oscillator Design
`641
`Transistor
`•
`641
`One-Port Negative Resistance Oscillators
`Oscillators
`644
`•
`Dielectric Resonator Oscillators
`648
`
`612
`
`•
`622
`
`Constant Gain
`•
`Low-Noise
`
`635
`
`INTRODUCTION TO MICROWAVE SYSTEMS
`
`655
`
`I 2. 1 System Aspects of Antennas
`655
`•
`655
`Definite of Important Antenna Parameters
`Antennas
`656
`•
`Antenna Pattern Characteristics
`Efficiency, Gain, and Temperature
`661
`
`Basic Types of
`658
`•
`
`Antenna
`
`Page 10 of 13
`
`

`

`xvi
`
`contents
`
`Pulse Radar
`•
`78
`6
`Cross Section
`
`675
`
`•
`
`Doppler
`
`•
`
`Total Power
`684
`
`12.4
`
`. S steins
`.
`662
`Friis Power Transmission
`C mmumcauon y
`The
`666
`•
`•
`?
`66-
`,., Microwave O
`. Systems
`nd Receivers
`. .
`.
`l 2.-
`C mmun1cauon
`Transnutters a
`Types of o
`•
`Microwave
`. r
`667
`•
`Fom1ula
`.
`663_
`of a Microwave Receive
`Noise Charactenza11on
`670
`. Multiplexed Systems
`frequency-
`672
`3 Radar Systems
`·
`12.
`673
`The Radar Equation
`Radar
`6 77
`•
`Radar
`679
`.
`Radiometry
`679
`r lions of Radiometry
`Theory and App ica
`Th Dicke Radiometer
`Radiometer
`e
`681
`•
`685
`? 5 Microwave Propagation
`L.
`Atmospheric Effects
`685
`•
`Efects
`688
`6E89
`12.6 Other Applications and Topics
`sfer
`T
`ran.
`ncrgy
`Microwave Heating
`689
`•
`Biological Effects and Safety
`Warfare
`691
`
`Ground Effects
`
`687
`
`•
`
`690
`694
`
`•
`
`•
`
`Plasma
`
`Electronic
`
`APPENDICES
`
`697
`
`698
`Prefixes
`A
`698
`Vector Analysis
`B
`700
`Bessel Functions
`C
`Other Mathematical Results
`D
`Physical Constants
`704
`E
`704
`Conductivities for Some Materials
`F
`G Dielectric Constants and Loss Tangents for Some Materials
`H
`Properties of Some Microwave Ferrite Materials
`705
`Standard Rectangular Waveguide Data
`706
`Standard Coaxial Cable Data
`707
`
`703
`
`J
`
`705
`
`INDEX
`
`709
`
`Page 11 of 13
`
`

`

`CHAPTER 2
`
`ill , I ) -+
`
`••(: , I)
`
`2.1 The Lumped-Element Circuit Model lor a Transmission Line
`
`57
`
`Transmission Line Theory
`
`In 11130v ":i,s rrJn~mi,-.., i,111 hne theory bridges Lhe gap between field analysis and
`bJ.,,, ...rc-u,; th~~. and ,o is of ~ig11itica11t importance i 11 microwave network analysis.
`-1.~ "e " ill ,<'C. the phenomenon ,if wave propagation on transmission lines can be
`appnl1ld1t'd from an cx1cn.,ion of ,ircuit theory or from a specialization of Maxwell's
`equatio,i., : we ,hall prc.-eni ~.>th viewpoints and show how this wave propagation is
`dco-crii:,cd b~ equation, ,·cry ,imilar to those used in Chapter I for plane wave propagation.
`
`2.1
`
`56
`
`~
`
`THE LUMPED-ELEMENT CIRCUIT MODEL FOR A
`TRANSMISSION LINE
`Tu kn difTercocr between circuit theory and transmission line theory is electrical
`s1zt. Circui~ analysis assumes that the physical dimensions of a network are much smaller
`than the electrical wa\'ele.ngth. while transmission lines may be a considerable fraction
`of a wavdengtb. or many wavelengths. in size. Thus a transmission line is a distributed(cid:173)
`parameirr ne1work.. where voltages and currents can vary in magnitude and phase over
`its length.
`N. shown in Figure 2. la. a transmission line is often schematically represented as a
`t~·o-wire Line. i.ince tr.im,mbsion lines (for TEM wave propagation) always have at least
`1~·0 conduetori,. The ,hon piece of line of length Ci.z of Figure 2.1 a can be modeled as
`a lumped-element dn:uit. as ,hown in Figure 2.1 b, where R, L , G, C are per unit length
`quantities ddine-d as follows:
`
`R = , erie, resistance per uni1 length. for both conductors, in 0./m .
`L = series induc:tance per unit length. for both conductors, in Him .
`C = shuni conductance per unit length, in S/m.
`C = shunt capacitanc.e per uni1 length. in F/m.
`and The i.crie, iodu~tance L represe.ms the total self-inductance of the two conductors,
`th
`e_ s_hum caparnance C is due Lo the close proximity of the two conductors. The
`. .
`,encs rcMstaoce /? represent the
`.·.
`d
`..
`. -
`resistance ue to the linue conductivity of the conduc-
`~
`tors, and tile ,hunt conductan . C' · d
`.
`.
`is ue Lo d1elec1m; loss in the material between the
`1.:e
`
`6:
`
`(a)
`
`i (:. +A: . 11
`
`RA:.
`
`Lt,,z
`
`Gllr
`
`Cil.:.
`
`••/, + 6 :. r/
`
`/(:.. I) -+
`
`,,(:.. / )
`
`Az
`
`(b)
`
`FIGURE 2.1 Vo hagc and turrent definitions and equivalenl circuit for an incrcmcn1al length
`of transmission line. (a) Vohage and currem definitions. (bl Lumped-clement
`equivalent circuit
`
`conductors. R and G. therefore. represent loss. A finite length of transmission line can
`be viewed as a cascade of sections of the form of Figure 2. 1 b.
`From the circuit of Figure 2. 1 b, Kirchhoffs vohage law can be applied to give
`
`&i(z, t)
`v(z , t) - R6.zi(z, t) - L6.z-r - -u(z + Ci.z. t) = 0.
`
`~. la
`
`and Kirchhoffs current law leads to
`&v(z + Ci.z, t)
`'(
`) GA
`i z,t - uzv(z+ u z, t) -Cll.z
`&t
`
`A
`
`.
`- ·1(z +Az.t) = 0.
`
`2.lb
`
`Dividing (2. la) and (2. lb) by A z and taking the limit as Ci.z - 0 gives the following
`differential equations:
`
`8v(z,t)
`.
`L &i(z, t)
`~ = -Ri(z, t)- --'lit'
`8i(z, t) = -Gv(z. t) - C&u~, t).
`
`{)z
`
`uo
`
`2.2a
`
`2.2b
`
`These equations are the time-domain form of the transmission line. or telegrapher. equa(cid:173)
`tions.
`For the sinusoidal steady-state condition, with cosine-based phasors. (2.2) simplify to
`
`dV(z) = -(R + jwL)l(z),
`dz
`dl(z) = -(G + jwC)V(z).
`dz
`
`2.3a
`
`2.3b
`
`Page 12 of 13
`
`

`

`58
`
`Chapter 2: Transmission Line Theory
`
`Note the similarity in 1l1e fonn of (2.3) and Maxwell's curl equations of ( 1.41a) and
`
`(1.41b).
`
`Wave Propagation on a Transmission Line
`
`The two equations of (2.3) can be solved s imultaneously to give wave equations for
`I '(z) and / (:):
`
`dlV:z) - ,l\f(z) - 0,
`d::-
`
`21()- 0
`d1l(:::)
`~ - - y z - ,
`
`where
`
`) =a+ j 6 = /(R+ jwL)(G + jwC)
`
`2.4a
`
`2.4b
`
`2.5
`
`is the complex propagation constant, which is a function of frequency. Traveling wave
`solutions to (2.4) can be found as
`V(z) = V/e-'F + vo- e'l'',
`I(z) = r: e_,,, + I;; e ,,,,
`where the e-1 = term l\epresents wave propagation in the +z direction, and the e'l'' tenn
`represents wave propagation in the -z direction. Applying (2.3a) to the voltage of (2.6a)
`gives the current on the line:
`J(z) =
`
`""f
`R + jwL
`
`O
`
`2.6a
`
`2.6b
`
`•
`
`[v+ e- ')'Z - v-e-.•]
`
`O
`
`Comparison with (2.6b) shows that a characteristic impedance, Zo, can be defined as
`
`R+ ·
`Zo = _ _ JwL
`"I
`
`R+jwL
`G+jwC'
`
`2.7
`
`to relate the voltage and current on the line as
`v,,+ - Z, - - vo(cid:173)
`lt -
`lo .
`o -
`
`Then (2.6b) can be rewritten in the following form:
`,
`v +
`v-
`·1(z) = -2-e-'l'' -
`- 0- e-r•.
`Zo
`Zo
`Converting back to the time domain, the voltage waveform can be expressed as
`v(z, t) = IV/I cos(wt - {3z + tp+)e-"'
`+ iv.-1 cos(wt + /3z + r)e'",
`
`2.8
`
`2.9
`
`~ -
`
`... ± is the phase angle of the complex voltage V/. Using arguments similar to
`where 'I'
`•
`.
`.
`·n Section 1.4, we find that the wavelength on the lme 1s
`those 1
`
`2.2 Field Analysis of Transmission Lines
`
`59
`
`and the phase velocity is
`
`The Losslesa Line
`
`A=h
`(J ,
`
`w
`Vp: f3 : J..f.
`
`2.10
`
`2. 11
`
`T he above solution was for a general transmission line. including loss effects. and it
`was seen that the propagation constant and characteristic impedance were complex. In
`many practical cases, however, the loss of the line is very small and so can be neglected.
`resulting in a simplification of the above results. Setting R = G == 0 in (2.5) gives the
`propagation constant as
`
`or
`
`'Y =o + j(J =jwFG,
`(J = w../Lc,
`o =O.
`
`2.12a
`
`2.12b
`
`As expected for the lossless case, the attenuation constant a is zero. The characteristic
`impedance of (2. 7) reduces to
`
`Zo=~,
`
`2.13
`
`which is now a real number. The general solutions for voltage and current on a lossless
`transmission line can then be written as
`V ( z) = V.,+ e- i/l• + v0-ei11•,
`v+
`.
`v-
`.
`J(z) = -Le- 1/J• - _2...e3/J•
`Zo
`Zo
`.
`
`2.14b
`
`2. 14a
`
`The wavelength is
`
`and the phase velocity is
`
`271"
`2ir
`>-.=e=w/LC'
`
`w
`I
`Vp = f3 = .;re·
`
`2.15
`
`2. 16
`
`2.2
`
`FIELD ANALYSIS OF TRANSMISSION LINES
`In this section we will rederi,•e the time-harmonic fonn of the telegrapher's equations.
`staning with Maxwell's equations. We will begio by deriviog the uansmission line
`
`Page 13 of 13
`
`