DiPaolo, Franco, Ph.D. Frontmatter Networks and Devices Using Planar Transmission LinesBoca Raton: CRC Press LLC,2000

2000 CRC Press LLC

NetUsing Pl

Franco Di Paolo

works and Devicesanar Transmission Lines

2000 CRC Press LLC

This book contains informatiopermission, and sources are indreliable data and information, bor for the consequences of the

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The consent of CRC Press LLCor for resale. Specic permissi

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Di Paolo, Franco.Networks and dev

p. cm.Includes bibliogrISBN 0-8493-181. Strip transmiss

3. Telecommunicationappliances. I. Title.TK7872.T74 P36 200621.381

32dc21n obtained from authentic and highly regarded sources. Reprinted material is quoted withicated. A wide variety of references are listed. Reasonable efforts have been made to publishut the author and the publisher cannot assume responsibility for the validity of all materialsir use.

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2000 by CRC Press LLC

rary of Congress Cataloging-in-Publication Data

ices using planar transmission lines / Franco di Paolo.

aphical references and index.35-1 (alk. paper)ion lines. 2. Electric linesCarrier transmissionMathematics.Mathematics. 4. Electronic apparatus and

000-008424

CIPNo claim to original U.S. Government worksInternational Standard Book Number 0-8493-1835-1

Library of Congress Card Number 00-008424ted in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

2000 CRC Press LLC

This book has one oprinciples that permit a have been written with chapters.

This book is intendefrequency planar transmiand RF disciplines. Morrecent studies on planaattractive to researchers.

Chapters are dedicati.e., directional couplers,

A special feature is isolators, and circulatorsferrimagnetism. Also protransfer functions of a p

This book is highly rwell as professional desiABSTRACTbjective: to join in one text all the practical information and physicalplanar transmission line device to work properly. The eight appendicesthe aim of helping the reader review the theoretical concepts in the 11

d for microwave engineers studying the design of microwave and radiossion line passive devices in industry, as well as for students in microwavee than 500 up-to-date references make this book a collection of the mostr transmission line devices, a characteristic that also makes this book ed to the analysis of planar transmission lines and their related devices, directional lters, phase shifters, circulators, and isolators. a complete discussion of ferrimagnetic devices, such as phase shifters,, with three appendices completely dedicated to the theoretical aspect ofvided are more than 490 gures to simply and illustrate the inputoutputarticular device, information that is otherwise difcult to nd. ecommended for graduate students in RF and microwave engineering, asgners.

2000 CRC Press LLC

Franco Di Paolo

w

doctorate in Electronic Edi Roma, La Sapienza.

His rst job was wimicrowave circuits for Rresearch engineer at Elechief research engineerDivision, in Rome.

Dr. Di Paolo is authmember. He is an assoSociety, the Ultrasonics, the Circuit and Systems The Author

as born in Rome, Italy, in 1958. He received angineering in 1984 from the Universit degli studi

th Ericsson-Rome, designing wide band RF andX and TX optical networks. He has been a seniorttronica-Rome Microwave Labs. Currently he is at Telit, Microwave Satellite Communication

or of other technical publications and is an IEEEciate of the Microwave Theory and TechniquesFerroelectrics and Frequency Control Society, andSociety.

2000 CRC Press LLC

CHAPTER 1 Fundam

1.1 Generalities1.2 Telegraphist1.3 Solutions of T1.4 Propagation C1.5 Transmission 1.6 Transmission1.7 Consideration1.8 Reection Co1.9 Nonuniform T1.10 Quarter Wave1.11 Coupled Tran1.12 The Smith Ch1.13 Some Exampl1.14 Notes on PlanReferences

CHAPTER 2 Microst

2.1 Geometrical C2.2 Electric and M2.3 Solution Tech2.4 Quasi Static A2.5 Coupled Mod2.6 Full Wave An2.7 Design Equati2.8 Attenuation2.9 Practical ConsReferences

CHAPTER 3 Striplin

3.1 Geometrical C3.2 Electric and M3.3 Solution Tech3.4 Extraction of 3.5 Design Equati3.6 Attenuation3.7 Offset Striplin3.8 Practical ConsReferences

CHAPTER 4 Higher

4.1 Radiation4.2 Surface Wave4.3 Higher Order 4.4 Typical Disco4.5 Bends4.6 Open EndCONTENTSental Theory of Transmission Lines

and Transmission Line Equationsransmission Line Equationsonstant and Characteristic ImpedanceLines with Typical Terminations and Impedance Matrices About Matching Transmission Linesefcients and Standing Wave Ratioransmission Lines Transformerssmission Linesartes Using the Smith Chartar Transmission Line Fabrication

ripsharacteristicsagnetic Field Linesniques for the Electromagnetic Problemnalysis Methodses Analysis Methodalysis Methodons

iderations

esharacteristicsagnetic Field Linesniques for the Electromagnetic ProblemStripline Impedance with a Conformal Transformationons

esiderations

Order Modes and Discontinuities in Strip and Stripline

s

Modesntinuities

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4.7 Gap4.8 Change of Wi4.9 T Junctions4.10 Cross JunctionReferences

CHAPTER 5 Coupled

5.1 Geometrical C5.2 Electric and M5.3 Solution Tech5.4 Quasi Static A5.5 Coupled Mod5.6 Full Wave An5.7 Design Equati5.8 Attenuation5.9 A Particular CReferences

CHAPTER 6 Coupled

6.1 Geometrical C6.2 Electric and M6.3 Solution Tech6.4 Design Equati6.5 Attenuation6.6 A Particular C6.7 Practical ConsReferences

CHAPTER 7 Microst

7.1 Simple Two P7.2 Directional Co7.3 Signal Combi7.4 Directional Fi7.5 Phase Shifters7.6 The Three Po7.7 Ferrimagnetic7.8 Ferrimagnetic7.9 Comparison aReferences

CHAPTER 8 Striplin

8.1 Introduction8.2 Typical Two P8.3 Directional Co8.4 Signal Combi8.5 Directional Fi8.6 Phase Shifters8.7 The Three Po8.8 Ferrimagneticdth

Microstripsharacteristicsagnetic Field Linesniques for the Electromagnetic Problemnalysis Methodses Analysis Methodalysis Methodons

oupled Microstrip Structure: The Meander Line

Striplinesharacteristicsagnetic Field Linesniques for the Electromagnetic Problemons

oupled Stripline Structure: The Meander Lineiderations

rip Devicesort Networksuplersnerslters

rt Circulator Phase Shifters Isolatorsmong Ferrimagnetic Phase Shifters

e Devices

orts Networksuplersnerslters

rt Circulator Phase Shifters

2000 CRC Press LLC

8.9 Ferrimagnetic8.10 Comparison aReferences

CHAPTER 9 Slot Lin

9.1 Geometrical C9.2 Electric and M9.3 Solution Tech9.4 Closed Form 9.5 Connections B9.6 Typical Nonfe9.7 Magnetization9.8 Slot Line Isol9.9 Slot Line Ferr9.10 Coupled Slot References

CHAPTER 10 Coplan

10.1 Geometrical C10.2 Electric and M10.3 Solution Tech10.4 Closed Form 10.5 Closed Form 10.6 Connections B10.7 Typical Nonfe10.8 Magnetization10.9 CPW Isolat10.10 CPW Ferrim10.11 Practical Cons10.12 Coupled CoplReferences

CHAPTER 11 Coplan

11.1 Geometrical C11.2 Electric and M11.3 Solution Tech11.4 Design Equati11.5 Attenuation11.6 Connections B11.7 Use of CPSReferences

APPENDIX 1 Solutio

A1.1 The FundameA1.2 Generalities oA1.3 Finite DiffereA1.4 Image ChargeA1.5 FundamentalsA1.6 Conformal Tr Isolatorsmong Ferrimagnetic Phase Shifters

esharacteristicsagnetic Field Linesniques for the Electromagnetic ProblemEquations for Slot Line Characteristic Impedanceetween Slot Lines and Other Linesrrimagnetic Devices Using Slotlines of Slot Lines on Ferrimagnetic Substratesatorsimagnetic Phase ShiftersLines

ar Waveguidesharacteristicsagnetic Field Linesniques for the Electromagnetic ProblemEquations for CPW Characteristic ImpedanceEquations for CPW Attenuationetween CPW and Other Linesrrimagnetic Devices Using CPW of CPW on Ferrimagnetic Substratesorsagnetic Phase Shiftersiderationsanar Waveguides

ar Stripsharacteristicsagnetic Field Linesniques for the Electromagnetic Problemons

etween CPS and Other Lines

n Methods for Electrostatic Problemsntal Equations of Electrostaticsn Solution Methods for Electrostatic Problemsnce Method Method on Functions with Complex Variablesansformation Method

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A1.7 The Schwarz-References

APPENDIX 2 Wave E

A2.1 IntroductionA2.2 Maxwells EqA2.3 Wave EquatioA2.4 The Propagati

and MagneticA2.5 The Time DepA2.6 Plane Wave DA2.7 Evaluation of A2.8 Waves in Gui

Reference SysA2.9 TE and TMA2.10 TE and TMA2.11 Uniform PlanA2.12 DispersionA2.13 Electrical NetA2.14 Field PenetratReferences

APPENDIX 3 Diffusi

A3.1 Simple AnalyA3.2 Scattering ParA3.3 Conditions onA3.4 Three Port NeA3.5 Four Port NetA3.6 Quality ParamA3.7 Scattering ParReferences

APPENDIX 4 Resona

A4.1 The Intrinsic A4.2 The Quality FA4.3 Elements of FA4.4 Butterworth, CA4.5 Filter GeneratA4.6 Filters with LReferences

APPENDIX 5 Charge

A5.1 IntroductionA5.2 Some ImportaA5.3 Forces WorkinA5.4 Forces WorkinA5.5 Magnetic InduA5.6 Two ImportanA5.7 The FoundatioChristoffel Transformation

quation, Waves, and Dispersion

uations and Boundary Conditionsns in Harmonic Time Dependenceon Vectors and Their Relationships with Electric FieldsendenceenitionsElectromagnetic Energyding Structures with Curvilinear Orthogonal Coordinatestem Modes in Rectangular Waveguide Modes in Circular Waveguide

e Waves and TEM Equations

works Associated with Propagation Modesion Inside Nonideal Conductors

on Parameters and Multiport Devicestical Network Representationsameters and Conversion Formulas Scattering Matrix for Reciprocal and Lossless Networkstworksworkseters for Directional Couplersameters in Unmatched Case

nt Elements, Q, LossesLosses of Real Elementsactor Qilter Theoryhebyshev, and Cauer Low Pass Filtersion from a Normalized Low Passossy Elements

s, Currents, Magnetic Fields, and Forces

nt Relationships of Classic Mechanicsg on Lone Electric Chargesg on Electrical Currentsction Generated by Currentst Relationships of Quantum Mechanicsns of Atom Theory

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A5.8 The Atom StrA5.9 The PrecessioA5.10 Principles of WReferences

APPENDIX 6 The M

A6.1 IntroductionA6.2 Fundamental A6.3 The DenitionA6.4 Statistics FuncA6.5 Statistic EvaluA6.6 Anisotropy, MA6.7 The Weiss DoA6.8 Application oA6.9 The HeisenbeA6.10 FerromagneticA6.11 AntiferromagnA6.12 FerrimagnetisReferences

APPENDIX 7 The El

A7.1 IntroductionA7.2 The ChemicalA7.3 The Ferrite InA7.4 The PermeabiA7.5 TEM Wave A7.6 Linear Polariz

Magnetized FA7.7 ElectromagneA7.8 ConsiderationA7.9 The BehaviorA7.10 The Quality FA7.11 Losses in FerrA7.12 Isolators, Pha

MagnetizationA7.13 Isolators, Pha

MagnetizationA7.14 Field DisplaceA7.15 The Ferrite inA7.16 Other Uses ofA7.17 Use of FerriteA7.18 Harmonic SigA7.19 Main ResonanReferences

APPENDIX 8 Symbo

A8.1 IntroductionA8.2 Denitions ofA8.3 Operator Deucture in Quantum Mechanicsn Motion of the Atomic Magnetic Momentumave Mechanics.

agnetic Properties of Materials

Relationships for Static Magnetic Fields and Materialss of Materials in Magnetismtions for Particles Distribution in Energy Levelsation of Atomic Magnetic Momentsagnetostriction, Demagnetization in Ferromagnetic Materialsmains in Ferromagnetic Materialsf Weiss Theory to Some Ferromagnetic Phenomenarg Theory for the Molecular Field Materials and Their Applicationsetismm

ectromagnetic Field and the Ferrite

Composition of Ferritesside a Static Magnetic Fieldlity Tensor of FerritesInside an Isodirectional Magnetized Ferriteed, Uniform Plane Wave Inside an Isodirectionalerrite: The Faraday Rotationtic Wave Inside a Transverse Magnetized Ferrites on Demagnetization and Anisotropy of Not Statically Saturated Ferriteactor of Ferrites at Resonanceitesse Shifters, Circulators in Waveguide with Isodirectional

se Shifters, Circulators in Waveguide with Transverse

ment Isolators and Phase Shifters Planar Transmission Lines Ferrite in the Microwave Region Until UHFnal Generation in Ferritece Reduction and Secondary Resonance in Ferrite

ls, Operator Denitions, and Analytical Expressions

Symbols and Abbreviationsnitions and Associated Identities

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A8.4 Delta OperatoA8.5 Delta OperatoA8.6 Delta OperatoA8.7 The DivergenA8.8 Elliptic IntegrReferences

r Functions in a Cartesian Orthogonal Coordinate Systemr Functions in a Cylindrical Coordinate Systemr Functions in a Spherical Coordinate Systemce and Stokes Theorems and Green Identitiesals and Their Approximations

2000 CRC Press LLC

By planar transmissExamples are microstripssome electrical propertydirectional couplers and plines. By network we mperformance beyond inte

While the author hainvolved in this text, a gsuch as mathematical an

Chapter 1 introducesdedicated to microstrip nto the stripline, perhaps tproblems that can be ennuities and higher order the coupled microstrip stripline structure. Chaptdevices, like directional of Chapter 7, and stripltransmission line, i.e., a also studies the most impthe coplanar waveguide, devices employing coplastrips transmission line, wPCB area.

Appendix A1 reviewAppendix A2 introducesto the external propertiemain concepts regardingintroduce only the mainmust not be evaluated asphysical relationships ammagnetic properties of melectromagnetic eld insand some useful relation

To further help the rwhere some particular ispossible we have reporte

Filters, other than plincluded here.

The author hopes thisnetworks and devices anhopes this text will stimu

PREFACEion line we mean a transmission line whose conductors are on planes. and slotlines. By device we mean a component that is capable of having in addition to the obvious RF connecting characteristic. Examples arehase shifters. All the devices we will study are made of planar transmissionean a set of complicated RF transmission lines without any additionalrconnecting capability.s made an effort to explain in a simple way all the theoretical conceptsraduate-level knowledge of electromagnetism and related scientic areas,alysis and physics, is required. all the concepts of the general theory of transmission lines. Chapter 2 isetworks that are widely diffused in planar devices. Chapter 3 is dedicatedhe rst planar transmission line developed. Chapter 4 introduces the maincountered in planar transmission line networks and devices like disconti-modes. Chapter 5 is dedicated to a very important microstrip network, i.e.,structure, while Chapter 6 is the stripline counterpart, i.e., the coupleder 7 is the largest chapter of this text. It introduces the most used microstripcouplers, phase shifters, and more. Chapter 8 is the stripline counterpartine devices are studied. Chapter 9 introduces the slotline, a full planartransmission line with both conductors on the same plane. This chapterortant devices that can be built with slotlines. Chapter 10 is dedicated toanother full planar transmission line. Also in this chapter, the most typicalnar waveguides are studied. Finally, Chapter 11 introduces the coplanarhich is mainly suited for transmitting balanced signals, requiring a small

s the theory of the solution methods for simple electrostatic problems. the most important concepts of wave theory. Appendix A3 is dedicateds of networks, like the [s] parameter matrix. Appendix A4 reviews the resonant circuits. A common note holds for Appendices A5 and A6. These formulas and concepts for a proper understanding of Appendix A7, and an alternative to dedicated texts on physics. Appendix A5 is dedicated toong charges, currents, and magnetic elds. Appendix A6 introduces theaterials. Appendix A7 is dedicated to the most important aspects of theide ferrimagnetic materials. Finally, Appendix A8 reports all the symbolsships used throughout in this text.eader, at the end of each chapter and appendix are additional referencessue is analyzed in more detail. If the reference is difcult to nd, whend alternate texts where the topic under study can be found. anar transmission line devices, are not the goal of this text and are not

text will help the reader understand the world of planar transmission lined will aid in deciding how to choose the proper device. The author alsolate the reader to study and research other new devices.

Franco Di Paolo January 2000

2000 CRC Press LLC

DiPaolo, Franco, Ph.D. Fundamental Theory of Transmission Lines Networks and Devices Using Planar Transmission LinesBoca Raton: CRC Press LLC,2000

2000 CRC Press LLC

Fund

In telecommunicatiosignals can propagate witof the space. The particuthe best compromise existo transmit some frequen

We can divide transm

1. Coupled wires2. Parallel plates3. Coaxial4. Waveguide**

The rst three types while waveguides belongin one of the previous foubelong to type 1 above (

A two conductor tranmission line is characterwhile the potential of thehaving both conductors aof which t.l. to use depeHowever, physical dimeni.e., whether it is best su

Every transmission liand only a fundamental distinguish among lines. dependent phenomena, aagate.*****

* In this text transmission line** Waveguide transmission liAppendix A2.*** Polarization will be studi**** Modes of propagation w***** This multimode propagCHAPTER 1

amental Theory of Transmission Lines

1.1 GENERALITIES

n theory, transmission line means a region of the space where RFh the best compromise between minimum attenuation and available regionlar shape of the transmission line can suggest the frequency range wherets. In fact, depending on the transmission line shape, it will be best suitedcies and not others.ission lines* into four types:

belong to a family usually called two conductor transmission lines (t.l.), to one conductor transmission lines. Other types of t.l. can be includedr types. For example, twisted wires lines, parallel wire lines, and slotlinesslotlines will be studied in Chapter 9).smission line can be balanced or unbalanced. An unbalanced trans-ized as having one conductor xed to a potential, usually the ground one, other conductor moves. A balanced transmission line is characterized ass moving potentials with respect to ground potential. In general, the choicends on the type of the generator or load we have to connect to our line.sions of the t.l. greatly inuence the natural propagation mode of the line,ited for a balanced or unbalanced propagation.ne permits only a fundamental particular polarization*** of the RF eldsmode**** of propagation, and these characteristics can also be used toOf course, polarization and mode of propagation are strongly a frequency-nd at some frequencies other modes than the fundamental one can prop-1

s will be called lines or abbreviated with t.l.nes also are not strongly pertinent to the arguments of this text and will be discussed in

ed in Appendix A2.ill be studied in Appendix A4.ation will be discussed for any transmission line we will study in this text.

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Two sets of equationvoltage v and current

its parallel admittance

transmission line equat

1.2 TELEG

Let us examine Figurof a transmission line. Tcalled hot conductor (reader who is familiar win Figure 1.2.1 with two ccan be dubious about threpresented with an equrepresent any transmissio

Let us dene a positof this coordinate. Let uinductance L for u.l. a

With these assumptioa voltage drop dv give

where the minus sign is means that a positive var

Similarly, we can noproduce a current variati

where the minus sign mwhich is in a direction recognize how v and written more appropriate

These last two equatioand current along a t.l. w

* Microstrip and stripline tran** Generic coupled line theor*** See Appendix A2 for trans exist that can be applied to every transmission line, which relate thei along the t.l. with its series impedance Zs for unit length (u.l.) andYp for u.l. These equations are called telegraphists equations andions and will now be described.

RAPHIST AND TRANSMISSION LINE EQUATIONS

e 1.2.1. In part (a) of this gure we have indicated a general representationhe two long rectangular bars represent two conductors, one of which isor simply hot) and the other cold conductor (or simply cold). Theith microstrip or stripline* circuits should not confuse the representationoupled lines.** Similarly, the reader who knows the waveguide mechanicsis representation, but we know that modes in waveguides can also beivalent transmission line.*** So, Figure 1.2.1 can be used to genericallyn line. ive direction x and take into consideration an innitesimal piece dxs consider the t.l. to be lossless, so that the line will only have a seriesnd a shunt, or parallel, capacitance for u.l.ns, a variation di in the time dt of the series current i will producen by:

(1.2.1)

a consequence of the coordinate system of Figure 1.2.1. This signal alsoiation di of current produces a variation dv that contrasts such di.te that a variation dv in the time dt of the parallel voltage v willon di given by:

(1.2.2)

eans that a positive variation dv of voltage produces a variation di,opposite to the positive one. From the previous two equations we cani can be set as functions of coordinates and time, and so they can bely as:

(1.2.3)

(1.2.4)

ns are called telegraphists equations, and relate time variation of voltageith its physical characteristics as inductance L and capacitance C persmission lines will be studied in Chapter 2 and Chapter 3.y will be studied later in this Chapter.

dv Ldx didt

=

di Cdx dvdt

=

v

xL i

t=

ix

C vt

= smission line equivalents to propagation modes in waveguide.

2000 CRC Press LLC

u.l. From Equations 1.2.current exist. Deriving 1

and inserting 1.2.4 it bec

Similarly, we can ob

3 and 1.2.4 it is possible to obtain two equations where only voltage and.2.3 with respect to coordinate x we have:

(1.2.5)

omes:

(1.2.6)

tain an equation where only current appears:

(1.2.7)

Figure 1.2.1

x

dxI I+dI

I+dII

V V+dV

I

I

V

I+dI

V+dV

I+dI

(Z/2)dx

(Z/2)dx

I

V

I

I+dI

I+dI

Ydx

a)

b)

c)

2

2v

xL i

tix

=

2

2

2

2v

xLC v

t=

2

2

2

2ix

LC it

=

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So, voltage and currewe take into consideratio

it is common practice to

and 1.2.7, for example, b

where v is called proA general solution f

found case by case. The omust satisfy the conditio

A very familiar aspesinusoidal time variation by e

j

, where

isEquation 1.2.6, for exam

which is called the moequation studied in Apobtained substituting vwave equation.

To introduce the trasuppose that the t.l. also

we can write:

Applying the Kirch

* Throughout this text, symboif square brackets are used in e** With the symbol

we w*** Gustav Robert Kirchhoff,nt must satisfy the same equation. Whichever equation, 1.2.6 or 1.2.7, thatn is called a monodimensional generalized wave equation. Since:*

(1.2.8)

set:**

(1.2.9)

ecomes:

(1.2.10)

pagation velocity.or the monodimensional wave equations does not exist, and it must benly general consequence that can occur is that the general solution F(t,x)n:

(1.2.11)

ct assumes the monodimensional generalized wave equation when aexists. In this case the time dependence can be written with a multiplication the angular frequency of voltage or current. With this assumption, theple, becomes:

(1.2.12)

nodimensional wave equation, a particular case of the general wavependix A2. Of course, a similar equation holds for current, and can be with the current i, and in this case it is called the monodimensional

nsmission line equations, let us evaluate part b of Figure 1.2.1. Now, possesses a series resistance R and a parallel conductance Gp so that

(1.2.13)

(1.2.14)

hoff*** voltage loop law at the network in Figure 1.2.1b, we can write:

ls inside square brackets are used to show dimensions. We think that confusion is avoidedquations. Unless otherwise stated, MKSA unit system will be used.

LC m[ ] ( )sec 2

LC v 1 2/

2

2 2

2

21i

x v

it

=

F t x F t x v,( ) ( )

v x v e v ekx kx( ) ( ) ( )= +

Z R j Ls= +

Y G j Cp p= + ill indicate an equality set by denition. German physicist, born in Koenigsberg in 1824 and died in Berlin in 1887.

2000 CRC Press LLC

that is:

Applying the Kirch

that is:

Equations 1.2.16 andtelegraphists equationsand in coupled line casesand currents along the lobtained from the genera

where E is the electric

vector, and n is a versbe of great help for man

1.3 SO

From transmission licurrent are present. Deri

and inserting Equation 1

* See Appendix A2 to see how

v x( (1.2.15)

(1.2.16)

hoff current law at the network in Figure 1.2.1c, we can write:

(1.2.17)

(1.2.18)

1.2.18 are called transmission line equations and, together with the, form a set of equations widely used in all transmission line problems, as will be shown in the next section. Of course, at high frequency, voltagesines are not determined in the same way,* i.e., these quantities are notl relationships:

(1.2.19)

(1.2.20)

eld vector, d is an increment vector, J is the surface current densityor orthogonal to surface S. These equations are very important and willy arguments in this text.

LUTIONS OF TRANSMISSION LINE EQUATIONS

ne equations it is possible to obtain two equations where only voltage andving with respect to x in Equation 1.2.16, we have:

(1.3.1)

.2.18 it becomes:

(1.3.2)

i x Z dx v x dv x i x Z dxs s) = ( ) ( ) + ( ) + ( )[ ] + ( ) ( )2 2

dv xdx

Z i xs

( )= ( )

i x v x Y dx i x di xp( ) = ( ) + ( ) + ( )[ ]

di xdx

Y v xp( )

= ( )

v E da

b

= li J ndS

s

=

d v xdx

Z di xdxs

2

2( )

= ( )

d v Z Y vs p

2

2 = voltages and currents are dened along high frequency transmission lines.

dx

2000 CRC Press LLC

Of course, a similar

The two previous equatiolinear differential equatioequations. The solutionsetting i(x)

ie

kx

. With

and the general solution

Equation 1.3.5 is not are dened as

the solution of 1.3.3 can

All the quantities i

+

The quantity k obtaineare 1/m in MKSA. Nomathematically the same

Of interest is the case

imaginary, as a consequesinus and cosinus, i.e.:

Choosing the best soof the electromagnetic pgoes theoretically to inntransmission lines. The tit decreases in amplitudwhich decreases in ampNote that this procedure

Once the solution ofvariable easily, i.e., currewe can obtain the voltag

* Note that the progressive te

iequation can be obtained for current, i.e.:

(1.3.3)

ns are mathematically equivalent since they are examples of second order ns. In mechanics theory, equations of this type are called harmonic motion of this equation is simple, and with reference to 1.3.3, can be found by this substitution in 1.3.3 we have:

(1.3.4)

is a linear combination of exponentials:

(1.3.5)

the only representation for the solution. Since hyperbolic sinus and cosinus

also be set as a linear combination of hyperbolic sinus and cosinus, i.e.:

(1.3.6)

, i, A, and B are constants, in this case with the unit Ampere.d from Equation 1.3.4 is called the propagation constant, and its unitste that with the insertion of 1.3.4 in 1.3.2 or 1.3.3, these equations are as those in the previous section, i.e., Equation 1.2.12. where the quantity k is imaginary, that is when Zs and Yp are onlynce of 1.3.4. In this case the solution of 1.3.3 is a linear combination of

(1.3.7)

lution between 1.3.5 and 1.3.7 depends on the known boundary conditionsroblem. Exponential solution 1.3.5 is useful when one extreme of the t.l.ity, while hyperbolic solutions are useful when considering limited lengtherm that contains the negative exponential is called progressive,* sincee in the positive direction of x, while the other is called regressive,litude when x decreases in amplitude in the negative direction of x. can also be applied to obtain the solution 1.3.2. 1.3.2 or 1.3.3 is extracted, it is possible to obtain the other electricalnt or voltage. If we employ the exponential solution of 1.3.5 for current,

d idx

Z Y is p

2

2 =

Z Y k k Z Ys p s p= = ( )2 0 5! .

i x i e i ekx kx( ) = ++ ( ) ( )

cosh ( ) ( )x e e senh x e ex x x x

+

2 2

i x( ) = ( ) + ( )Acosh kx Bsenh kx

x A k x B sen k x with k jkj j j( ) = ( ) + ( ) cos .e v, from 1.2.18 which is given by:rm decrease in amplitude when x increases.

2000 CRC Press LLC

If we had used the hy

The quantity:

is called characteristic itransformed in a very we

It is important not toline; in fact, both seriesimpedance. The reciproc

.

Note that with the in

where

It is interesting at thisfor current 1.3.3, we obtaand the exponential exprstudy by resolving the mexponential expression oa minus sign. This sign dhas no inuence in the phfor current or voltage arWhat is always true in thbetween terms, there wisign will depend on the

For the case where lo1.3.11. In fact, for the lo

and using 1.2.9 we have

* Sometimes we will simply (1.3.8)

perbolic solution 1.3.6 for current, from Equation 1.2.18 we would have:

(1.3.9)

(1.3.10)

mpedance of the t.l.* Remembering 1.3.4, the previous equation can bell-known aspect, i.e.:

(1.3.11)

confuse the characteristic impedance with the series impedance of the impedance and shunt admittance of the t.l. compose its characteristical of this quantity is called characteristic admittance and is identied as

troduction of , the Equation 1.3.8 can be written as:

(1.3.12)

(1.3.13)

point to note that from the solution of the monodimensional wave equationined the exponential expression of current with a + sign between termsession for voltage, i.e., 1.3.12, with a minus sign. If we had started ouronodimensional wave equation for voltage, we would have obtained thef voltage with a + sign and the exponential expression for current withiversity for the same equation for current or voltage is only analytical andysical problem. This is because the constants that appear in the expressionse generic, and the sign only depends on the effective physical problem.e general case is that if in one exponential solution there is the sign +ll be the sign in the other exponential solution. In any case, the truecontour conditions of the particular electromagnetic problem.sses can be neglected, useful relationships can be obtained from Equationssless case, 1.3.11 becomes:

(1.3.14)

:

v x i e i e k Ykx kx p( ) = [ ]+ ( ) ( )

v x k Yp( ) = ( ) + ( )[ ] ( )Asenh kx Bcosh kx

k Yp/

( )Z Ys p/ .0 5

v x v e v ekx kx( ) ( ) ( )= +

v i and v i+ +

( )L C/ .0 5named impedance.

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Expressions 1.3.14 aare assumed to be in the

1.4 PROPAGA

The propagation conInserting in that denitio

where k

r

and k

j

are r

L, G

p

, and C are dexact [Neper/m]. The w

Consequently, to extractwords, k is proportionvalue of

in [Neper/m

relationship:

For example, 1 [NepIf now we square Equ

we have:

The ideal lossless lin

In practice, lines are is when the length

o

* Remember that unless other

k RGr p=

k Lj =

2(1.3.15)

nd 1.3.15 are used every time a particular transmission line is studied and most simple case of no losses.

TION CONSTANT AND CHARACTERISTIC IMPEDANCE

stant k dened by Equation 1.3.4 is in general a complex number.n the general expression 1.2.13 for Zs 1.2.14 and Yp we have:

(1.4.1)

eal numbers. Remember that in the previous equation the quantities R,ened for t.l. so that the dimension of k is [1/m]* more theoreticallyord Neper reminds us that k appears in an exponential form ekx. k we have to perform an operation of natural logarithm ln. In otheral to the natural logarithm of the signal amplitude along the t.l. From a], it is simple to calculate the value of dB in [dB/m] using the obvious

er/m] = 8.686 [dB/m].ation 1.4.1 and equate real with real and imaginary with imaginary terms,

(1.4.2)

(1.4.3)

es are those where R = 0 = Gp, and in this case from 1.4.2 and 1.4.3:

(1.4.4)

never without losses. So, the practical approximation to the lossless casef the t.l. is so that:

(1.4.5)

= Lv Cv1

k R j L G j C k jkp r j= +( ) +( )[ ] + 0 5.

dB e= ( )20 * log

LC RG LC RC LGp p + ( ) + +( ) 2 22 2 2

0 5 0 5

2.

.

C RG RG LC RC LGp p p + ( ) + +( ) 22 2 2

0 5 0 5

2.

.

k j LCr j= = ( )0 0 5and k .

l l / /R and Gpwise stated we will use the MKSA system unit.

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When losses cannot b

purpose, let us evaluate current and voltage and

The mean power W

and 1.4.6 becomes:

or, using 1.3.13:

The mean power W

and, remembering Equat

The decrease along

which with 1.4.8 gives:

or, using 1.4.8 valuated

In the most general

conductor loss and oneconductor that contains th

i (e neglected it is possible to simply obtain an expression for kr. For thisthe case of a very long t.l., so that we can only use progressive terms forwrite:

(1.4.6)

t transmitted along the line will be:

(1.4.7)

(1.4.8)

r dissipated in R and Wg dissipated in Gp are given by:

(1.4.9)

ion 1.3.13, the total mean power Wdt dissipated will be:

(1.4.10)

x of Wt will be equal to Wdt, so we can write:

simply for x = 0 and 1.4.10:

(1.4.11)

case, kr is given by the sum of two quantities, one dependent on the dependent on the dielectric loss, i.e., the medium that surrounds thee e.m. eld. These two quantities are indicated with c and d, and so:

(1.4.12)

x i e e v x v e er

k xjjk x

r

k xjjk x) = ( ) =+ + ( ) ( ) ( ) ( )and

W v x i xt ( ) ( )[ ]Re * 2

W v i et rk x

=+ + ( )2 2

W v et rk x

=+ 2 2 2( )

W R i and W G vr g p= ( ) = ( )+ +2 22 2

W G R vdt p= +( ) +2 2 2

=dW dx Wt dt

k W Wr dt t= 2

k G Rr p= +( ) 2

kr c d= +

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where:

where W

c

is the mean

in the dielectric. Appendsame expression, while c

A more general dengates inside a medium w

1.

r

= relative perme2.

r

= relative permi3. g = conductivity

In this case, the prop

where:

0

r

Note that from the tw

and the second denition

For some simple tranL

s

, and capacitance C

relationships between genFrom 1.4.15 and 1.4.14

then k is purely imagtransmission lines can beit is shown that for any mline. Not considering gyimaginary until

c

is a

quantity, independent of

phenomena associated w

is often characterized by

* See Appendix A2 for other ** The subscript a recalls th*** The relative relationships

**** Gyromagnetic materialsstudied in the following chapte***** Dielectric polarizabilitythe references at the end of thi(1.4.13)

power dissipated in the conductors and Wd is the mean power dissipatedix A2 shows that for any TEM, t.l. dielectric losses are governed by theonductor losses are in general different.ition of the propagation constant can be obtained when the signal propa-ith the following characteristics:

abilityttivity

agation constant is given by:*

(1.4.14)

c jg/ 0 r ** ar j a j r rr j r j (1.4.15)

o previous denitions it follows:

(1.4.16)

of 1.4.15 becomes:

(1.4.17)

smission lines, for example, the coaxial cable, the equivalent inductance can be simply related to and .*** The reader interested in theeral transmission line theory and wave propagation can read Appendix A2.it is simple to recognize that if the medium is lossless, i.e., rj = g = 0,inary, as in the case of 1.4.4. Other coincidences between waves and obtained remembering the wave theory, as given in Appendix A2, whereode of propagation it is possible to associate an equivalent transmissionromagnetic dielectrics,**** from Equation 1.4.14 we note that k isreal quantity. Note that the dielectric constant r is in general a complexthe presence of a dielectric conductivity g, since rj is due to a dampingith the dielectric polarizability.1,2,3***** Using this concept, a dielectric a tangent delta tan, (also called a loss tangent) dened as:

expressions of propagation constant.e signicance absolute. among Ls, C, , and for coaxial cable are given in Appendix A2. will be studied in Appendix A7, while devices working with gyromagnetic materials arers.

c c t d d tW W and W W= =2 2

k jc

= ( ) 0 5.

ar rr aj rjand 0 0

c ar ajj g +( ) is assumed to be known to the reader. Fundamentals about this argument can be found ins chapter.

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At

wave frequencie

Sometimes the so-ca

We want to conclude

and imaginary parts. Thi

while for a lossless trans

While all used transmfollowing chapters we wand the propagation cann

1.5 TRAN

Quite often t.l. are teto another line, then the scases of simple terminatiespecially for tuning pur

We will study cases are at the end. Since we aform for current and vol

a. Terminations aOur environment is a

x = 0 at the beginning ofa1. OPEN circuit a

The current at x

The value of Bcondition. If wevaluated for x

Inserting 1.5.1 athe t.l.:

* See Appendix A7 and, amo(1.4.18)

s, usually aj g and tan assumes the well-known expression:

(1.4.19)

lled power factor is used, indicated with sen . this section noting that impedance can also be decomposed into reals means that inserting 1.2.13 and 1.2.14 into 1.3.11, in general we have:

(1.4.20)

mission line from 1.3.14 we have (L/C)0.5, i.e., it is a real quantity.ission lines can be practically considered to have real impedances, in theill study other transmission* lines where the impedance can be imaginaryot take place.

SMISSION LINES WITH TYPICAL TERMINATIONS

rminated with short or open circuits. In both cases if this line is in shunthort or open terminated t.l. is called a stub. It is important to study suchons since stubs are frequently employed in planar transmission line devices,poses.where these terminations are at the beginning of the t.l., and when theyre evaluating limited length transmission lines, we will use the hyperbolictage.

t the INPUT of the Line transmission line of length with a longitudinal axis x with origin the line.t the INPUT = 0 will be zero, while the voltage is known. From 1.3.6 we have:

(1.5.1)

cannot be dened with only the condition i(0) = 0. We need to have a furthere introduce the condition A = !0 in the hyperbolic voltage expression 1.3.9= 0 we have:

(1.5.2)

nd 1.5.2 in 1.3.6 and 1.3.9 we have the expression of voltage and current along

tan Im Re ( ) ( ) +( )c c aj arg

tan rj rr

= +r jj

i A and0 0 0( ) = ! B = any finite value

B v= ( )! 0 ng others, chapters 7 and 8 where ferrimagnetic devices are studied.

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a2. SHORT circuit The voltage at

To dene the va1.3.6 evaluated

Inserting 1.5.5 athe t.l.:

a3. GENERAL termIn this case, a voof voltage and can use the supa2. So, for this

b. Terminations aTo have a simple ex

variable:

to the hyperbolic expres

This transformation positive direction of x1.5.12, i.e., applying the(1.5.3)

(1.5.4)

at the INPUTx = 0 will be zero, while the current is known. From 1.3.9 we have:

(1.5.5)

lue of A we introduce the condition B = !0 in the hyperbolic current expressionfor x = 0 we have:

(1.5.6)

nd 1.5.6 in 1.3.6 and 1.3.9 we have the expression of voltage and current along

(1.5.7)

(1.5.8)

ination at the INPUTltage v(0) and a current i(0) are present at the input. To have the expressioncurrent along the t.l., as a function of the general termination at the input, weerposition effect principle and apply the solutions of the previous points a1 andcase we have:

(1.5.9)

(1.5.10)

t the OUTPUT of the Linepression for the constant A and B we will apply the transformation

(1.5.11)

sion of current, and write:

(1.5.12)

corresponds to having the new axis origin at the end of the t.l. and the in the opposite direction with respect to the previous case a. Deriving 1.2.18, we have:

(1.5.13)

i x v senh kx( ) = ( ) ( )0 v x v osh kx( ) = ( ) ( )0 c

v B and A0 0 0( ) = =! any finite value

A i= ( )! 0

i x i kx( ) = ( ) ( )0 coshv x i senh kx( ) = ( ) ( ) 0

i x i kx v senh kx( ) = ( ) ( ) ( ) ( )0 0cosh v x i senh kx v kx( ) = ( ) ( ) + ( ) ( ) 0 0 cosh

x xl

i x i x( ) = ( ) + ( ) ( )Acosh kx Bsenh kx l

v x v x = + Asenh kx Bcosh kx l( ) ( ) ( )[ ] ( )

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With 1.5.12 and 1.5.

b1. OPEN circuit aThe current at x

The value of Bcondition. If wevaluated for x

Inserting 1.5.14along the t.l.:

b2. SHORT circuit The voltage at

To dene the va1.5.12 evaluated

Inserting 1.5.18along the t.l.:

b3. GENERAL termIn this case, a expression of vinput, we can upoints b1 and b

i x(

v x =(

i

v13 we can repeat the previous points a1 through a3.

t the OUTPUT = 0 will be zero, while the voltage is known. From 1.5.12 we have:

(1.5.14)

cannot be dened with only the condition i(0) = 0. We need to have a furthere introduce the condition A = !0 in the hyperbolic voltage expression 1.5.13 = 0 we have:

(1.5.15)

and 1.5.15 in 1.5.12 and 1.5.13 we have the expression of voltage and current

(1.5.16)

(1.5.17)

at the OUTPUTx = 0 will be zero while the current is known. From 1.5.13 we have:

(1.5.18)

lue of A we introduce the condition B=!0 in the hyperbolic current expression for x = 0 we have:

(1.5.19)

and 1.5.19 in 1.5.12 and 1.5.13 we have the expression of voltage and current

(1.5.20)

(1.5.21)

ination at the OUTPUTvoltage v(0) and a current i(0) are present at the output. To have theoltage and current along the t.l., as a function of the general termination at these the superposition effect principle and apply the solutions of the previous2. So, for this case we have:

(1.5.22)

(1.5.23)

i x A) ( ) != = = = =0 0 0l and B any finite value

B v x v= =( ) ( )! 0 l

i x x v( ) = ( )[ ] ( )senh k l l v x v( ) = ( ) ( )[ ]l lcosh k x

v x B and A) = =( ) = =0 0 0l ! any finite value

A i x i x= =( ) =( )! 0 l

i x i k x( ) = ( ) ( )[ ]l lcoshv x i senh k x( ) = ( ) ( )[ ] l l

x i k x senh k x v( ) = ( ) ( )[ ] + ( )[ ] ( )l l l lcosh x i senh k x v k x( ) = ( ) ( ) + ( ) ( ) l l l lcosh[ ] [ ]

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b4. Input impedancIt is useful to hand of nite va

Calculating the

Of course, if wt.l. Y(0) give

Particular casesb4a. Input imp

In this cas

b4b. Input impIn this cas

b4c. Input impA t.l. is sa

Then, in t

When the1.4.4 that

In this casby:

* A completely matched line he with known termination at the OUTPUTave the input impedance Z(0) when the load impedance Z( ) is knownlue, i.e., when:

(1.5.24)

ratio of 1.5.23 with 1.5.22, both evaluated for x = 0, and using 1.5.24 we have:

(1.5.25)

e do the reciprocal of 1.5.24, we can also calculate the input admittance of then by:

(1.5.26)

of 1.5.25, or 1.5.26, are when:edance with open circuited linee Z( ) = and from 1.5.25:

(1.5.27)

edance with short circuited linee Z( ) = 0 and from 1.5.25:

(1.5.28)

edance with matched terminated lineid to be matched if the load impedance* Z is so that:

(1.5.29)

his case Z( ) = and from 1.5.25:

(1.5.30)

lines can be approximated with the ideal case of zero losses we know fromk jk j, and:

(1.5.31)

(1.5.32)

e the expressions 1.5.25, 1.5.27, and 1.5.28 assume a very simple aspect given

v i Zl l l( ) ( ) ( )

Z Z osos Z sen

02

( ) = ( ) + ( ) ( )( ) + ( ) ( ) senh k c h kc h k h k

l l l

l l l

Y Y osos Y sen

02

( ) = ( ) + ( ) ( )( ) + ( ) ( )

senh k c h kc h k h k

l l l

l l l

Zoc

0( ) ( ) cotgh kl

Zsc

0( ) ( ) tgh kl

Zl

Z 0( )

senh jk x jsen k xj j( ) ( )cosh cosjk x k xj j( ) ( )as both source Zg and load Z impedance equal to .

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The last three expresand matching networks.

1.6 T

With transmission mrelates input and output

Let us examine Figuexcitated at one extremeorigin of the x axis codistance x from the or

from which (using 1.3.1

Vi

Vi(x)(1.5.33)

(1.5.34)

(1.5.35)

sions are widely used in many transmission line networks such as lters

RANSMISSION AND IMPEDANCE MATRICES

atrices we have a representation of the transmission line that simplyline excitations.re 1.6.1 a, where a t.l. of length and characteristic impedance is with voltage vi and current i i. The excitation extreme is set as theordinate. We want to evaluate the voltage v(x) and current i(x) at aigin. We can write:

(1.6.1)

(1.6.2)

3) we have:

(1.6.3)

Zj g Z

jZ tg02

( ) = ( ) + ( )+ ( ) ( )

t k

kj

j

l l

l l

Z j co goc

0( ) ( ) t k jlZ j gsc

0( ) ( ) t k jl

v v vi +

i i ii + +

vi v

vi vi i i i

=

=

++

2 2

x

Ii Iu(x)

Vu(x)

a)

b)

0

0

Iu

Vu

x

Ii(x)Figure 1.6.1

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From 1.3.5 and 1.3.1

Collecting together t

Eq. 1.6.4 becomes:

Inserting 1.6.3 into 11.6.6 we have:

Equations 1.6.7 and

where the square matrix

Let us consider part current i i(x), with vuWe have:

So, inverting* the Eq

where the square matrix

* Matrix inversion operation c3 and using the previous relationships we have:

(1.6.4)

erms with i i and vi and remembering that:

(1.6.5)

(1.6.6)

(1.6.7)

.3.12, collecting together terms with ii and vi, and applying 1.6.5 and

(1.6.8)

1.6.8 can be written as:

(1.6.9)

is indicated with Tf and called the forward transmission matrix, i.e.:

(1.6.10)

b of Figure 1.6.1 and attempt to evaluate the input voltage vi(x) and and iu known. This can be simply done by looking at Equation 1.6.9.

(1.6.11)

uation 1.6.10 we have:

(1.6.12)

is indicated with Tr and called the reverse transmission matrix, i.e.:

i x i v e i v i eu

i i kx i i kx( ) = + +

2 2( ) ( )

e e kxkx kx( ) ( ) cosh+ = ( ) 2e e senh kxkx kx( ) ( ) = ( ) 2

i x i kx v senh kxu i i( ) = ( ) ( ) ( )cosh

v x v kx i senh kxu i i( ) = ( ) ( )cosh

v x

i xkx senh kx

senh kx kxv

iu

u

i

i

( )( )

=

( ) ( ) ( ) ( )

coshcosh

Tkx senh kx

senh kx kxf=

( ) ( ) ( ) ( )

coshcosh

v x

i xT

v

i Tv x

i xv

iu

u

fi

if

u

u

i

i

( )( )

=

( )( )

=

1

v x

i xkx senh kx

senh kx kxv

ii

i

u

u

( )( )

=

( ) ( )( ) ( )

coshcosh

an be found in many mathematical books.

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It is very important orientation along the t.l. have not changed the orieis no input or output of we do not dene a posiforward or reverse transmboth give the voltage anis no excitation, but TfTr is relative to the opused to obtain Tf, i.eoperation.

The reader who knowcan recognize how Tr iused to dene the chain

An interesting appliclines,** is the case of a i i . The input voltage

and from the forward tra

Another matrix that [Z] dened as:

where:

* See Appendix A3 for ACBD** This example will be used

v ju= (1.6.13)

to say that transmission matrices are always relative to a well-denedand a well-dened orientation of currents. Note that to dene the Tr, wentation in Figure 1.6.1b with respect to that in Figure 1.6.1a. In fact, therea line until we do not dene a positive direction for it. In other words, iftive direction for a transmission line, there is no reason to speak aboutission matrix. With these concepts clear, we can say that Tf and Tr

d current at the opposite extreme of the t.l., i.e., the extreme where there is relative to the same direction dened as positive along the t.l. whileposite direction. Note that Tr can be obtained with the same procedure., using transmission line theory, instead of using the inversion matrix

s the chain or ABCD matrix representation of a two-port network*s the ABCD matrix of our transmission line. Voltage and current directionmatrix are the same as those we used to dene Tr.ation of the transmission matrix, which is useful when analyzing coupledlossless t.l. open terminated and excited by a current generator of valuevi will be:

(1.6.14)

nsmission matrix the output voltage vu will be:

(1.6.15)

is sometimes used in transmission line problems is the impedance matrix

(1.6.16)

Tkx senh kx

senh kx kxr=

( ) ( )( ) ( )

coshcosh

v j ii i= ( ) cotg k jl

i j i en j i eni i i( ) ( ) ( ) ( ) cotg k k s k s kj j j jl l l lcos

ZZ ZZ Z( ) =

11 12

21 22

Z V I for I

Z V I for I

Z V I for I

Z V I for I

11 1 1 2

22 2 2 1

12 1 2 1

21 2 1 2

0

0

0

0

=

=

=

= matrix denition. in Chapter 6

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Subscripts 1 and 2device, it is only necessaZ12 and Z21.

Since to evaluate Zsimply be obtained from

To evaluate the other

To obtain V2 we u

which, with insertion of

Performing the ratio

and all the matrix [Z]

1.7 CONSIDE

As we said before, trawill discuss operating bdened. The operating bfrequency characteristicsoperating bandwidth of ais included between the athis value. Bandwidth is

a. Ratio of Bandwidth and minimum frequency

where with the symbol indicate ports. Note that since the transmission line is a linear reciprocalry to evaluate one term between Z11 and Z22 and one term between

11 or Z22 we have an open circuit at one extreme, these values can 1.5.27 as follows:

(1.6.17)

parameters, we extract I1 from the previous equation:

(1.6.18)

se Tf and write:

(1.6.19)

1.6.18 becomes:

(1.6.20)

of 1.6.20 with 1.6.18 we have:

(1.6.21)

parameters are now dened.

RATIONS ABOUT MATCHING TRANSMISSION LINES

nsmission lines are quite often used for impedance matching. This sectionandwidth of such matching. First of all, the term bandwidth will beandwidth of a device is the frequency interval where some, or all of its are evaluated as acceptable for the device purpose. For instance, the band pass lter is the frequency interval where the value of its attenuationttenuation at center frequency and a number of dB, typically 1 or 3, below usually indicated in three manners:

Limits If we indicate with fh and f l, respectively, the maximum of the operating bandwidth, then:

we indicate an equality by denition.

Z V I gh k Z V I11 1 1 22 2 2

( ) = cot l

I V tgh ki1 ( )l

V V k I senh k2 1 1= ( ) ( )cosh l l

V V k senh k tgh k2 1= ( ) ( ) ( )[ ]cosh l l l

Z k Z21 12= ( ) =cosech l

n f fh

1

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b. Fractional It is de

or:

if in percent. The relat

or:

if in percent.

c. Octave Each octaof 6 GHz the operating the bandwidth is 12 GHupper frequency of the bo of octave, the resultinrelationship:

For instance, the mum3h = 12 and so on.

Of course, if B is

After these importanhave:ned as:

ionship between the denition in point a and b is:

ve is a multiplication by 2. For instance, if from a minimum frequencybandwidth extends for one octave, it means that the upper frequency ofz; if the operating bandwidth extends for two octaves, it means that theandwidth is 24 GHz, and so on. If one half an octave is added to a numberg multiplication number moh depends on o according to the following

ltiplication factor m1h for one octave and a half is 3 while m2h = 6,

known, the n may be obtained very simply by:

t denitions, let us rewrite Equation 1.4.12 evaluated for zero losses. We

(1.7.1)

B f ff fh

h

+2 1

1

B f ff fh

h%

+2 1001

1

B nn

+2 1

1

B nn

% +

2 11100

m for ooh

o o o= + [ ] =+2 2 2 2 1 2 31( ) , , , ...

nBB

=+

22

k k j jj = ( ) = 0 5 2.

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where is the signal w

In Figure 1.7.1 the shWe see how any value ofFrom this point of viewsignal is xed, this gureof the stubs when varyinon the exact required vavs. . This means thatstubs, where n is an inthe stub is xed while transformation ratio thatvariations in frequency cmatching.

Other important matformula, i.e., Equation 1

* Unless otherwise stated, in inside a transmission line.avelength along the line.* With 1.7.1, Equations 1.5.34 and 1.5.35 become:

(1.7.2)

(1.7.3)

ape of | Zoc(0)/ | and | Zsc(0)/ | is represented as a function of /. input impedance can be obtained, i.e., positive, negative, zero, or innite., stubs work like transformers. If we assume that the wavelength of the represents the possible impedance values that we can report at the inputg the t.l. length . If we want to operate in a region where toleranceslue of are permitted, we have to work in a region with a small slope (2n+1)/4 for open circuited stubs and n/2 for short circuitedteger number. The same conclusions hold if we assume that the length ofthe wavelength of the signal is varied. In other words, the higher the is needed, the lower the resulting operating bandwidth is, since smallause a large change in the reported impedance, which results in a mis-

ching characteristics can be obtained from the general input impedance.5.25, which, with 1.7.1 becomes:

(1.7.4)

Figure 1.7.1

Z joc

0 2( ) ( ) cotg lZ jsc

0 2( ) ( ) tg l

Zj sen Z

jZ sen02 22 2

2

( ) = ( ) + ( ) ( )( ) + ( ) ( )

l l l

l l l

cos

costhis text signal wavelength will always be relative to a guided case, i.e., the signal travels

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Let us evaluate the f

a. Line Length Equal t

which, inserted in 1.7.4

This means that whathe termination impedan

b. Line Length Equal t

which, inserted in 1.7.4

This relationship rep1.7.8. This characteristicwavelength from a s= , then from the prevshorted to ground. Of couthis type of ltering is naalthough in the next secare near to ltering prop

c. Line Terminated Wit

which, inserted in 1.7.4

This equation meansthe characteristic impedaimpedance. Note that sibroadest possible bandw

1.8 REFLEC

In the previous paraoperating bandwidth. In ollowing cases:

o Integer Number of Half Wavelength In this case we have:

(1.7.5)

gives:

(1.7.6)

tever the characteristic impedance of a t.l. is, if Equation 1.7.5 holds,ce Z() is reported at the input of the transmission line.

o an Odd Number of Quarter Wavelength In this case we have:

(1.7.7)

gives:

(1.7.8)

resents the most useful effect of a stub that can be realized according to is also used to do simple lters. Suppose we need to remove a tone ofignal passing in a t.l. If we insert a stub open terminated, i.e., with Z()ious equation we have Z(0) = 0, which means that the desired signal isrse, problems arise if the signal to be shorted possesses a bandwidth sincerrowband, as we said previously. Filter theory is not the topic of this text,tions we will quite often study networks, which have characteristics thaterties.

h Matched Load In this case we have:

(1.7.9)

gives:

(1.7.10)

that whichever is the length of the t.l. when the termination is equal tonce of the line, the input impedance is always equal to this characteristicnce 1.7.10 is independent of frequency, the matching condition is theidth relationship for a transmission line.

TION COEFFICIENTS AND STANDING WAVE RATIO

l = =n with n 2 1 2 3, , ...

Z Z0( ) ( )l

l = +( ) =2 1 4 1 2 3n with n , , ...

Z Z0 2( ) ( ) l

Z l( )

Z 0( ) graphs we have shown how a matched transmission line has the widestaddition, matching condition is also helpful, which will now be discussed.

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Suppose that a transmissiof impedance Rg = andcase the t.l. is completelZ(x) function of x g

Taking into account and assuming that only p

while if only regressive

So, Equation 1.8.1 giwave exists. The negaticonventional sign for v+in 1.8.2 and 1.8.3 as shox, while current is a loif this explanation workparallel and longitudiconvenient to refer to ththe termination point, wvoltage v(x) must be adoesnt happen, it meansif the t.l. is matched, noa monodirectional wavematched load in any secwe insert a matched loatransmission line, we havis comparable to a transZ = /2, which is not thtransmission line is said

After such introductiload Z. We can denterminals and the follow

vo

volta

cu

curre

All these parameterson line of characteristic impedance is fed at one extreme by a generator at the other extreme is terminated by a load of impedance Z = . In thisy matched, i.e., matched at both ends. It is useful to dene an impedanceiven by:

(1.8.1)

expressions 1.3.5 for i(x) and 1.3.12 for v(x), remembering 1.3.13,rogressive terms exist, then:

(1.8.2)

terms exist, then:

(1.8.3)

ves the characteristic impedance as a result, only if a monodirectionalve sign in 1.8.3 has no physical effect, since it comes out from the, v, i+, and i. It is common practice to explain the different signswing that voltage is a parallel quantity and doesnt change sign withngitudinal quantity and does change sign with direction of x. However,s to explain the signs in 1.8.2 and 1.8.3, sometimes the assumptions onnal quantities can lead to error. To avoid such a possibility, it is alwayse general expressions of v(x) and i(x). Equation 1.8.1 is also true athere the load is connected. At this coordinate, 1.8.1 means that all thet the load terminals, and all the current i(x) must pass inside it. If this that at the termination there is not the proper impedance, i.e., Z . So, reection exists, and consequently, in a matched transmission line, only exists. These results could lead one to think it is possible to connect ation of the line without affecting the matching. This is not true. In fact, ifd Z = , for instance, at the middle coordinate x = xh of a matchede the half line on the right report in parallel to Z = . This situationmission line with impedance and length xh terminated with a loade matching condition for the t.l. The case when reections exist inside ato be a standing wave phenomena.on, let us suppose that the line of length x is terminated by a generice a current it passing inside the load and a voltage vt between itsing parameters:

ltage reection coefcient: (1.8.4)

ge transmission coefcient: (1.8.5)

rrent reection coefcient: (1.8.6)

nt transmission coefcient: (1.8.7)

Z x v x i x( ) ( ) ( )

Z x( )

Z x( ) .

v

v vr p

T v vt pv

i +i i

T i iti + are, in general, complex quantities.

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At the termination cterminals must satisfy th

where:

The quantities vp aor reected waves, i.e., Equation 1.8.8 with 1.8.

Indicating with Zn

Summing and subtra

Using 1.8.6, the ratio

Dividing 1.8.11 by

To extract v and

or

Summing and subtrato that used for 1.8.14 anoordinate, current i t passing across the load and voltage vt at itse following relationships:

(1.8.8)

(1.8.9)

(1.8.10)

nd vr are respectively the amplitudes of the progressive and regressive,traveling in the positive and negative direction of x. Since Z vt /i t ,10 becomes:

(1.8.11)

the value of Z normalized to , the previous equation becomes:

(1.8.12)

cting the previous equation to 1.8.9 we have, respectively:

(1.8.13)

of the two equations in 1.8.12 becomes:

(1.8.14)

i+ and using 1.8.7 and 1.8.14, we have:

(1.8.15)

Tv we can begin to rewrite 1.8.9 as:

(1.8.16)

(1.8.17)

cting the previous equation to 1.8.8, we can proceed in a manner similar

v v vt p r= +

i i it = ++

v i and v ip r +

Z i i itl = +

Z i i in

tl =

+

Z i i and Z i in t n tl l+( ) = ( ) = + 1 2 1 2

i nn

ZZ

=

+

11

l

l

TZ Zi

i

n n

=

=

+

1 21

l l

v Z v vt p rl =

v Z v vtn

p rl = d 1.8.15, obtaining:

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It is simple to evaluadened. Indicating with avalue and considering po

The situation is differesistive parts. Passive ccan possess negative rescase, let us suppose thatof negative value, i.e., Zpositive, i.e., Zg Rg. N

from which we see that

then |v | > 1. TheoreticaThe use of negative r

oscillator circuits. Oscilltopic is not treated in thisin references. 4,5,6,7,8,9

* We will soon return to this (1.8.18)

(1.8.19)

te the limits value for all transmission and reection coefcients we have subscript M or m, respectively, the maximum and minimum parametersitive values for the normalized impedance,* we have:

rent when the terminating impedances assume negative values for theiromponents always have positive values of resistance, but active devicesistances under particular conditions of bias and loading network. In this an active device possesses an input impedance Zi purely resistive andi Ri, and that the source impedance Zg is also purely resistive andormalizing these impedances to Rg, from 1.8.18 we have:

if:

lly, if Zin = 1, then |v | = .esistance presented by an active device is one of the main foundations ofators are one of the most attractive devices of all electronic circuits. This text, but the interested reader can refer to the articles and books indicated

v

n

n

iZ

Z=

+ l

l

11

TZZ

Z Tv v

n

n

n i= + = +=1

21

ll

l

v M n n

v m n

v M n

v m n

i M n n

i m n

i M n

v m n

for Z or Z

for Z

T for Z

T for Z

for Z or Z

for Z

T for Z

T for Z

= = =

= =

= =

= =

= = =

= =

= =

= =

1 0

0 1

2

0 0

1 0

0 1

2 0

0

l l

l

l

l

l l

l

l

l

v

in

inin i g

ZZ

with Z R R= +

11

1 1 < +Z Zin inassumption regarding positive values for terminating impedance.

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It is possible to dedene:

p

pow

where transmitted wt,

Note that inserting th

These coefcients ca

where Ti* is the comp

Inserting this last equ

and from 1.8.23:

Another parameter ofAw dened by:

Reections and transus rewrite Equation 1.8.8

which, with the denitio

Tw= (

4Rene transmission and reection coefcients using power. In this case we

ower reection coefcient: (1.8.20)

er transmission coefcient: (1.8.21)

progressive wp, and reected wr powers must satisfy:

(1.8.22)

e previous equation in 1.8.20 we have:

(1.8.23)

n easily be obtained from the previous ones. In fact, for Tw we have:

(1.8.24)

lex conjugate of Ti. From 1.8.15 and 1.8.19 we have:

(1.8.25)

ality in Equation 1.8.23 we have:

(1.8.26)

(1.8.27)

ten used, especially in lter network theory, is the power attenuation factor

(1.8.28)

mission coefcients can also be obtained using admittances. To do that let as:

w

r pw w

T w ww

t p

w w wp t r +

w

r p p t pw

w w w w w T ( ) = 1

T T Tw v i ( )Re

ZZ Z

ZZ Z

Z ZZ Z

n

n n

n

n n

n n

n n+ ) +( )

=

( )+( ) +( ) =

+( )+( ) +( )

1 14

1 12

1 1Re

*

l

l l

l

l l

l l

l l

w i v

2 2

w v= 1 2

w

p tw v

w w T = ( )1 1 1 2

i Y i it l = + n of normalized load admittance Y n Y / it becomes:

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Summing and subtra

Calculating the ratio

and directly from 1.8.30

Rewriting 1.8.9 as:

and summing and subtra

It is interesting to ocomplex numbers if the icoefcients are always reas indicated by 1.8.26 an

After dening these v (x) along x has atake as the origin of axeswe choose the left side.

If v v (0) is the r

The positive sign of of vp (x) vpekx has to bthe denition in 1.8.4 of(1.8.29)

cting this expression to 1.8.9 we have:

(1.8.30)

(1.8.31)

between 1.8.31 with 1.8.30 we have:

(1.8.32)

:

(1.8.33)

cting this expression to 1.8.8 we can proceed as we did before, obtaining:

(1.8.34)

(1.8.35)

bserve that voltage, current reection, and transmission coefcients arempedances or admittances are complex. Power reection and transmissional numbers, since they are related to the modulus of reection coefcient,d 1.8.27.parameters, it is very interesting to show that the reection coefcient simpler expression than Z(x) as we have seen in Section 1.7. Let us x, the point where the load Z is connected, and as negative directionThis situation is indicated in Figure 1.8.1. eection coefcient of the load, we can write:

(1.8.36)

the exponential is due to the fact that the negative sign in the exponentiale changed by the negative direction of propagation along x. Generalizing

i Y i itnl =

+

i Y itn

1 1 2+( ) = +li Y it

n1 1 2l ( ) =

i

n

n

ii

+ =

+l

l

11

i

tn

n

ii

+ = +

21

l

l

v v vtn

p rl =

v

r

pn

n

v

v

=

+

11

l

l

T vvv

t

pn

=

+

21 l

v x v x v erv

pv

p kx( ) ( ) ( ) ( ) = 0 0 v and inserting in it the dependence with x, we have:

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which is clearly a simpmoves along x with twwe use the Smith* chart

The value of 1.8.37 aimpedance n(x) preseIn fact, from the express

The impedance and vcourse, similar to (x)

Another important palso indicated briey wi

where | v |M and | v |mFrom 1.8.39 we can recoobtained from v , andusing 1.8.10, i.e.:

which, with the denitio

Since v is a comp

* Denition and use of the Sm

(x)

-x(1.8.37)

ler expression than (x). From 1.8.37 we can recognize how v(x)o times the dependence of v(x). This result will be very useful when to study matching problems.lso lies in the fact that from v (x), it is possible to have the normalizednted by the t.l. at the coordinate x, still obtaining a simpler expression.ion 1.8.18 of v we have:

(1.8.38)

oltage reection coefcient along x are strictly related from 1.8.18. Of and v , v (x) is, in general, a complex number.arameter is the voltage standing wave ratio, abbreviated with VSWR,th SWR. This is dened by:

(1.8.39)

indicate respectively the maximum and minimum of voltage modulus.gnize that the VSWR is always a real number. Also this parameter can be can be set as function of coordinate x. Let us start to rewrite 1.3.12

(1.8.40)

n of v becomes:

(1.8.41)

lex number, we can write:

(1.8.42)

Figure 1.8.1

0

(0)

ZNegative axis

v

r

pv

p kx

p kx vkxx

v x

v x

v e

v ee( ) ( )( )

( ) ( )

=0 0 2

n

v

v

xx

x( ) ( )( )=

+

11

VSWR v vM m

v x v e v ep kx r kx( ) ( ) ( )= +

v x v e v ep kxv

p kx( ) ( ) ( )= +

v v

jve

ith chart will be studied in Section 1.12.

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Assuming a losslessimaginary parts, 1.8.41 b

whose modulus is:

Using expression 1.7

where it is simple to recIn other words, every intminimum of 1.8.45 corre

So, from the denitio

Maximum VSWRM aand |v |m have been giv

The measure of VSWis known if its value is hvalue of the load impedaambiguous, due to presesimple to show that thedenormalizing expressioif Z > , then:

v x vp( ) = ([cos

v x( t.l., (i.e., k jkj), inserting 1.8.42 in 1.8.41 and separating real andecomes:

(1.8.43)

(1.8.44)

.1 for kj, Equation 1.8.44 can be written as:

(1.8.45)

ognize how the modulus of v(x) moves along x with a period /2.eger of half wavelength | v(x) | assumes the same value. Maximum andspond to the value of 1 or 1 for the cosinus. Consequently, we have:

(1.8.46)

(1.8.47)

n of VSWR we have:

(1.8.48)

nd minimum VSWRm value of VSWR can simply be evaluated since |v |Men before. Therefore, we have:

(1.8.49)

(1.8.50)

R is sometimes used to determine the value of a load impedance when itigher or lower than the reference impedance. If it is not known that thence is higher or lower than the reference impedance, then the measure isnce in 1.8.48 of the modulus of v . Given a normalized load Zn it is associated |v | is also obtained for another load Z n = 1/Z n . So,n 1.8.18 for v and inserting it into 1.8.48, we have that if v > 0, i.e.,

(1.8.51)

k x k x jv sen k x sen k xj v j v p v j v j) + +( )] + +( ) ( )[ ]cos

v x v k xpv v j v( ) = + + +( )[ ]1 2 22 0 5 cos .

v xpv v v) = + + ( ) +[ ]{ }1 2 2 22 0 5 cos .

v x vM

pv( ) = +( )1

v x vm

pv( ) = ( )1

VSWR vv

+

11

VSWR whenm v v m= =1 0

VSWR whenM v v M= = 1

Z VSWRl =

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while if v < 0, i.e., if Z

We can conclude thissince impedances and Vcoefcient has an impor

1

A nonuniform transmlongitudinal coordinate. parallel admittance Yp ,of coordinates. Nonunifoprole, as indicated in Fof x. The reader who islines did not make the estriplines. The two conhot one, then the other is line. In the most generatransmission line theory ance must be constant alour case. In this text, Mthem in Appendix A2. Ctexts on electromagnetismMaxwells equations, the

If we impose the restis only a small percent dthe study of nonuniform

* James Clark Maxwell, Engl < , then:

(1.8.52)

section noting that the measure of the reection coefcient is very useful,SWR are related to this parameter. We will show later how the reectiontant role in the Smith chart.

.9 NONUNIFORM TRANSMISSION LINES

ission line is a line where characteristic impedance is a function of itsIf we want to use the concepts of line series impedance Zs and line as we did in Section 1.2, in this case these quantities are also a functionrm transmission lines are usually generically represented with a taperedigure 1.9.1, just to remind one that characteristic impedance is a function familiar with the representation with the technology of planar transmissionrror of thinking of Figure 1.9.1 as the case of two coupled microstrip orductors indicated in Figure 1.9.1 just means that if one conductor is thethe cold one as we used in Section 1.2 to describe the uniform transmissionl case, i.e., without restrictions about the shape of the t.l., the generalcannot be used in this case, since for this theory the characteristic imped-ong the line. Only the applications of Maxwells* equations are correct inaxwells equations are assumed to be known, but we have summarizedomplete explanations of these fundamental equations are found in a lot of.10,11,12 To explain how these equations are important, we say that without whole of electromagnetism would still be an obscure physics argument.

riction that for a coordinate increment dx, the new impedance Z(x+dx)ifferent than Z(x), then we can apply the theory used in Section 1.2 to transmission lines. So, in our case we can write:

(1.9.1)

ish physicist, born in Edinburgh in 1831, died in Cambridge in 1879.

Figure 1.9.1

Z VSWRl =

x

Generic non uniform line.

dv xdx

Z x i xs

( ) ( ) ( )=

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Indicating the derivat

Inserting in this equafrom 1.9.2, we have:

Of course, a similar

In contrast with the simple solution v(x) orcontrary to the simple explines. Simple solutions cexample is the exponeadmittance can be writte

where the constant s iand 1.9.5 become:

which are simple second

respectively in 1.9.7 and(1.9.2)

ive operation with a prime sign, and again deriving 1.9.1 it becomes:

(1.9.3)

tion the value of i(x) dervied from 1.9.1 and the value of i(x) derived

(1.9.4)

equation can be obtained for current, resulting in:

(1.9.5)

case of uniform transmission lines, now there does not exist a general i(x) for the second order nonlinear differential Equations 1.9.4 or 1.9.5,onential or hyperbolic solutions 1.3.5 or 1.3.6 for the uniform transmissionan only be found for particular expressions for Zs(x) and Yp(x). Anntial lossless t.l., i.e., a line where its series impedance and paralleln as:

(1.9.6)

s related to the geometrical shape of the t.l. With 1.9.6, Equations 1.9.4

(1.9.7)

(1.9.8)

order linear differential equations. Setting

(1.9.9)

1.9.8 we have:

(1.9.10)

di xdx

Y x v xp( ) ( ) ( )=

= + [ ]v x Z x i x Z x i xs s( ) ( ) ( ) ( ) ( )

=v xZ xZ x

v x Z x Y x v xss

s p( )( )( ) ( ) ( ) ( ) ( ) 0

=i xY xY x

i x Z x Y x i xpp

s p( )( )( ) ( ) ( ) ( ) ( ) 0

Z x j Le and Y x j Ces

sx

psx( ) ( )

+ =v x sv x LCv x( ) ( ) ( )2 0

+ + =i x si x LCi x( ) ( ) ( )2 0

v x ve i x iek kvx ix( ) ( )

ks s LC

v=

( )2 2 0 542

.

s s LC ( )2 2 0 54 . (1.9.11)ki = 2

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Note that kv and affected by losses. In pa

these quantities are purelThe frequency:

is called the cutoff freqhave propagation, also if

It is possible to have the variation of (x). with the variation of itsimpedance of the t.l. at thThe impedance Zi + dZinput impedance Zi wFrom Equation 1.5.33 w

with m the mean valuthat dx is so small that So, the previous equatio

Now, expand the lastvalue of dx. We have:

which, inserted in 1.9.14

From 1.8.38, denorm

which, inserted in 1.9.15

Zk i are never completely imaginary, which means that propagation isrticular, for:

y real, which means that there will be a complete attenuated propagation.

(1.9.12)

uency, and represents the minimum frequency that must be overcome to attenuated.an equation for a nonuniform t.l. that doesnt require any limitation aboutThis equation relates the variation of reection coefcient along the line characteristic impedance. To do that, let us dene with Zi the inpute coordinate x, and Zi + dZi the impedance at the coordinate x + dx.i can be regarded as the load impedance for the line length dx, whosee want to evaluate. For simplicity we will suppose the t.l. to be lossless.e have:

(1.9.13)

e of the characteristic impedance in the element dx. We now supposetg(kjdx) k jdx and that the product of innitesimal terms can be neglected.n becomes:

(1.9.14)

term in the McLaurin series, stopping at the second term due to the small

results in:

(1.9.15)

alized and applied to our case, we have:

(1.9.16)

< ( )s LC2 0 5.

f s LCc= ( )4 0 5 .

Z xj tg k dx Z dZ

j Z dZ tg k dxim j m i i

m i i j( ) = ( ) + +( )

+ +( ) ( )

2

x j k dx Z dZ jZ k dxi m j m i i m i j( ) = + +( )[ ] +[ ] 2 1

m i j m i j mjZ x k dx jZ x k dx+ ( )[ ] ( )[ ] 1 1 1

dZ xdx

jk Z xi j i m m( )

= ( ) [ ]2

Z x xxi

v

v

m( ) = + ( ) ( )

11

results in:

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Deriving the Equatio

Inserting 1.9.17 into

Similar to 1.9.4 or 1.9is also said to be a Ricca

If we still apply the simply evaluate the voltaLet us examine Figure 1a possible approximationvariation d moving f

* I.F. Riccati, Italian mathema(1.9.17)

n 1.9.16 with respect to the coordinate x we have:

(1.9.18)

1.9.18 we have:

(1.9.19)

.5, a general simple solution of 1.9.19 doesnt exist. The previous equationti* equation.same restriction used above for variation of with coordinate, we cange reection coefcient at the input of the nonuniform transmission line..9.2. Part a represents the original nonuniform t.l., while part b indicates, made with steps of uniform lines. We can observe how an impedancerom x at x + dx will generate a variation dv given by:

(1.9.20)

Figure 1.9.2

dZ xdx

jk xx

ij

v

v

m

( )=

+

11

( )( )

dZ xdx

x

x

ddx x

d xdx

i v

v

m m

v

v( )=

+

+[ ]

11

21 2

( )( ) ( )

( )

d xdx

jk x x ddx

vj v

v m ( ) ( ) ( ) ln= ( )[ ]2 1

2

2

d ddv

+( ) +( ) +

dx

x

x

a)

b)

Z Ys p YpZs +dZ

+dYs

ptician, born in Venezia in 1676 and died in Treviso in 1754.

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Assuming d is ne

Now, suppose the t.reection coefcient d

With respect to Equadistance x is considereas shown in Figure 1.8.1

If we assume the muslow variation of withe continuous sum of th

where is the length will use the previous for

The argument of theall RF and microwave detheory of this topic. We since the specialization planar transmission line

As we said in Sectiotransmission line is ideaelectrical length of the lsituation is indicated in Fand an electrical length

* Directional coupler will be gligible with respect to 2 , the previous equation becomes:

(1.9.21)

l. is lossless. According to 1.8.37, at the input of the section dx, thevi, will be:

(1.9.22)

tion 1.8.37, now the exponential changes sign, because in this theory thed an absolute sign while for 1.8.37 the distance x has a negative sign,.ltiple reections along the t.l. are negligible, which still means there is ath the coordinate, we can evaluate the input reection coefcient v ase previous terms, i.e.:

(1.9.23)

of the nonuniform t.l. When we study tapered directional couplers* wemula to dene the coupling for such important devices.

1.10 QUARTER WAVE TRANSFORMERS

quarter wave transmission line transformers is of great importance invices. For this reason, we think it is necessary to go more deeply into thewill only consider the ideal case of TEM, lossless transmission lines,of the general theory is carried out in the following chapters for everyused as a transformer.n 1.7, the matching between a load and a source using a quarter wavelly perfect only at the design frequency, i.e., at that frequency where theine is a quarter wavelength of the signal guided by the t.l. The operatingigure 1.10.1. The line is characterized by a characteristic impedance , and will be evaluated as lossless.

d dd

v

( )[ ]

2 2

ln

d x d evi v

jk xj ( ) = 2

v

j xvi

j xe d x ed

dxdx= ( ) ( )[ ] 2

0

2

0

12

l l ln

Z

Zg ,studied in Chapter 7 and Chapter 8.

Figure 1.10.1

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In most practical caVSWR = 1, but usually iof this less stringent reqinstead we have an operdetermine the frequencyformed very easily, remetransmission line of imp

and the voltage reectionare:

where Zg is the impedhave represented the twoR, where R Z /Zg. frequency where the tranwhere the transformer isratio R. For frequencieratio. This is simple to uone from 1.7.8, it followvalue, and no mismatchi

At this point, remembthat no theoretical reasonsection can transform thby the other quarter wavnumber of s sections ocase, the only requiremfollowing relationships:

i.e., an odd number s

i.e., an even number s their values are normalizses it is not required that the matching be exactly perfect, i.e., with at is required that the VSWR be lower than a xed value. The consequenceuirement is that we no longer have a sole frequency for matching, butating bandwidth where the matching can be accepted. So, it is useful to characteristics of the single quarter wave transformer. This can be per-mbering expression 1.5.33, which gave us the input impedance Zi of aedance and electrical length terminated in a load Z. We have:

(1.10.1)

coefcient and VSWR at its input, for the situation in Figure 1.10.1

(1.10.2)

(1.10.3)

ance of the generator which feeds the transformer. In Figure 1.10.2 we previous equations vs. the normalized frequency fn for two values offn is given by the ratio of the general variable frequency f and thatsmission line is /4 long. We see how the matching is exact for frequencies an odd multiple of a quarter wavelength, regardless of the transformers outside the designed one, the VSWR value depends on the transformernderstand. Remember that if the load impedance is equal to the sources that the impedance of the transformer transmission line is equal to thisng occurs.ering what we have said in the previous sections, the reader should realize exists not to realize the matching using more than one section since eache impedance in a value that can still be transformed to the desired valueelength section. This situation is represented in Figure 1.10.3, where af the quarter wave transformer are connected in series. In the most generalents between the impedances of the transmission line are given by the

(1.10.4)

(1.10.5)

of sections, or

(1.10.6)

Z Z jjZi =+ ( )+ ( )

l

l

2 tantan

=

+

Z ZZ Z

i g

i g

VSWR = +

11

1 1= =R for s

, , ..

1 3 5

2 4 6

2

1

3 5 7L ss

sR for s

= =

, , ..

1 3 5

2 4 6

1 1 2 4 6L ss

Rfor s = =of sections. The symbol over the name of the impedances means thated to the system measurement impedance, usually 50 Ohm. It is clear that

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in the case of a multiple since it is theoretically ponly requirements stated

The rst study on thR.E. Collin.13,14,15,16,17 In hin the matching bandwidof Chebyshev polynomipolynomials together wi

gZquarter wavelength transmission line transformer, the design is not uniqueossible to choose the impedances in an innite number of ways, with the by Equations 1.10.5 and 1.10.6.e problem to design the network indicated in Figure 1.10.3 was made byis work, Collin studied the case of synthesizing the network with a VSWRth of Chebyshev shape. In Appendix A4 we have reported the expressionsals, together with their shape. Here we report the rst three Chebyshevth the recursion formula, as follows:

(1.10.7)

Figure 1.10.2

T x x

T x x

T x x x

T x xT x T xn n

1

22

33

4 1 2

2 1

4 3

2

( ) =( ) = ( ) = ( ) = ( ) ( )

s

s 20 1

321

Z

,,,,

s-1Figure 1.10.3

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It is known that Chebsuch as lters, directionagreatest possible useful Conversely, after xing in a network with the lohigher value of ripple, thwhen using a transformewider useful bandwidth of VSWR has been denthree sections of quarter this gure it is evident thof sections is increased.

To see how the synthpolynomials, we will folthat the reection coefcto different transitions mof the network in Figure

Then, it is assumed tof the transformer, so th

* See Appendix A4 for Cheby

vswr is3(

vswr is

2

1.6

1.4

1.2

1

1.8

0.3 0.4 0.5

= 0yshev polynomials* can be used in the design of signal handling networksl couplers, transmission line transformers. The resulting network has thebandwidth, after xing an acceptable ripple in the desired bandwidth.an operating bandwidth, the design with Chebyshev polynomials resultswest possible ripple in the bandwidth. It is also known that accepting ae bandwidth increases. Applying these concepts to our case, we nd thatr as indicated in Figure 1.10.3, with s 2, it is possible to have a muchwith respect to the single quarter wave transformer, when a desired valueed. In Figure 1.10.4 we have indicated the input VSWR of one, two, andwave transmission line transformer, with an impedance ratio R = 2. Fromat we have a large increase in the operating bandwidth when the number

esis of the network in Figure 1.10.3 can be performed using Chebyshevlow the original simplied theory of Collin. In this theory, it is assumedient value between each transition is so small that multiple reections dueay be neglected. In other words, the reection coefcient at the input 1.10.3 can be written as:

(1.10.8)

hat the reection coefcients are symmetrical with respect to the middleat we may write:

(1.10.9)

Figure 1.10.4

vswr is1( )

)

2( )

1.266

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

+ + + + + +

12

24

36

12 1 2e e e e ej j j

s

j ss

j s