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Contents Preface xi Acronyms xiii1 Introduction 1 1.1 Brief History of RF and Microwave Wireless Systems 1 1.2 Frequency Spectrums 3 1.3 Wireless Applications 6 1.4 A Simple System Example 7 1.5 Organization of This Book 82 Review of Waves and Transmission Lines 10 2.1 Introduction 10 2.2 Wave Propagation 12 2.3 Transmission Line Equation 17 2.4 Reﬂection, Transmission, and Impedance for a Terminated Transmission Line 20 2.5 Voltage Standing-Wave Ratio 22 2.6 Decibels, Insertion Loss, and Return Loss 27 2.7 Smith Charts 33 2.8 S-Parameters 39 2.9 Coaxial Lines 41 2.10 Microscript Lines 43 2.11 Waveguides 50 2.12 Lumped Elements 54 2.13 Impedance Matching Networks 55 Problems 63 References 65 vii
viii CONTENTS 3 Antenna Systems 67 3.1 Introduction 67 3.2 Isotropic Radiator and Plane Waves 69 3.3 Far-Field Region 71 3.4 Antenna Analysis 73 3.5 Antenna Characteristics and Parameters 74 3.6 Monopole and Dipole Antennas 80 3.7 Horn Antennas 86 3.8 Parabolic Dish Antennas 88 3.9 Microstrip Patch Antennas 90 3.10 Antenna Arrays and Phased Arrays 98 3.11 Antenna Measurements 104 Problems 104 References 109 4 Various Components and Their System Parameters 111 4.1 Introduction and History 111 4.2 Couplers, Hybrids, and Power Dividers=Combiners 114 4.3 Resonators, Filters, and Multiplexers 118 4.4 Isolators and Circulators 128 4.5 Detectors and Mixers 130 4.6 Switches, Phase Shifters, and Attenuators 134 4.7 Oscillators and Ampliﬁers 139 4.8 Frequency Multipliers and Dividers 143 Problems 145 References 148 5 Receiver System Parameters 149 5.1 Typical Receivers 149 5.2 System Considerations 150 5.3 Natural Sources of Receiver Noise 152 5.4 Receiver Noise Figure and Equivalent Noise Temperature 154 5.5 Compression Points, Minimum Detectable Signal, and Dynamic Range 158 5.6 Third-Order Intercept Point and Intermodulation 161 5.7 Spurious Responses 166 5.8 Spurious-Free Dynamic Range 166 Problems 168 References 171 6 Transmitter and Oscillator Systems 172 6.1 Transmitter Parameters 172 6.2 Transmitter Noise 173
CONTENTS ix 6.3 Frequency Stability and Spurious Signals 176 6.4 Frequency Tuning, Output Power, and Efﬁciency 177 6.5 Intermodulation 180 6.6 Crystal Reference Oscillators 184 6.7 Phase-Locked Oscillators 186 6.8 Frequency Synthesizers 188 Problems 191 References 1947 Radar and Sensor Systems 196 7.1 Introduction and Classiﬁcations 196 7.2 Radar Equation 198 7.3 Radar Equation Including Pulse Integration and System Losses 202 7.4 Radar Cross Section 205 7.5 Pulse Radar 209 7.6 Continuous-Wave or Doppler Radar 212 7.7 Frequency-Modulated Continuous-Wave Radar 216 7.8 Direction Finding and Tracking 222 7.9 Moving-Target Indication and Pulse Doppler Radar 228 7.10 Synthetic Aperture Radar 232 7.11 Practical Radar Examples 233 Problems 236 References 2428 Wireless Communication Systems 243 8.1 Introduction 243 8.2 Friis Transmission Equation 244 8.3 Space Loss 247 8.4 Link Equation and Link Budget 248 8.5 Effective Isotropic Radiated Power and G=T Parameters 252 8.6 Radio=Microwave Links 254 8.7 Satellite Communication Systems 255 8.8 Mobile Communication Systems and Wireless Cellular Phones 258 8.9 Personal Communication Systems and Satellite Personal Communication Systems 262 Problems 270 References 2739 Modulation and Demodulation 274 9.1 Introduction 274 9.2 Amplitude Modulation and Demodulation 275 9.3 Frequency Modulation 279 9.4 Digital Shift-Keying Modulation 280
x CONTENTS 9.5 Bit Error Rate and Bandwidth Efﬁciency 286 9.6 Sampling and Pulse Code Modulation 289 Problems 292 References 29310 Multiple-Access Techniques 294 10.1 Introduction 294 10.2 Frequency Division Multiple Access and Frequency Division Multiplexing 294 10.3 Time Division Multiple Access and Time Division Multiplexing 295 10.4 Spread Spectrum and Code Division Multiple Access 298 References 30311 Other Wireless Systems 304 11.1 Radio Navigation and Global Positioning Systems 304 11.2 Motor Vehicle and Highway Applications 309 11.3 Direct Broadcast Satellite Systems 313 11.4 RF Identiﬁcation Systems 313 11.5 Remote Sensing Systems and Radiometers 317 11.6 Surveillance and Electronic Warfare Systems 320 Problems 328 References 330 Index 333
PrefaceWireless personal mobile and cellular communications are expected to be one of thehottest growth areas of the 2000s and beyond. They have enjoyed the fastest growthrate in the telecommunications industry—adding customers at a rate of 20–30% ayear. Presently, at least six satellite systems are being developed so that wirelesspersonal voice and data communications can be transmitted from any part of theearth to another using a simple, hand-held device. These future systems will providedata and voice communications to anywhere in the world, using a combination ofwireless telephones, wireless modems, terrestrial cellular telephones and satellites.The use of wireless remote sensing, remote identiﬁcation, direct broadcast, globalnavigation, and compact sensors has also gained popularity in the past decade.Wireless communications and sensors have become a part of a consumer’s daily life.All of these wireless systems consist of a radio frequency (RF) or microwave frontend. Although many new wireless courses have been offered at universities and inindustry, there is yet to be a textbook written on RF and microwave wireless systems.The purpose of this book is to introduce students and beginners to the generalhardware components, system parameters, and architectures of RF and microwavewireless systems. Practical examples of components and system conﬁgurations areemphasized. Both communication and radar=sensor systems are covered. Manyother systems, such as, the global positioning system (GPS), RF identiﬁcation(RFID), direct broadcast system (DBS), surveillance, smart highways, and smartautomobiles are introduced. It is hoped that this book will bridge the gap betweenRF=microwave engineers and communication system engineers. The materials covered in this book have been taught successfully at Texas A&MUniversity to a senior class for the past few years. Half of the students are from RFand microwave areas, and half are from communications, signal processing, solid-state, optics, or other areas. The book is intended to be taught for one semester to anundergraduate senior class or ﬁrst-year graduate class with some sections assigned to xi
xii PREFACEstudents for self-study. The end-of-chapter problems will strengthen the reader’sknowledge of the subject. The reference sections list the principal references forfurther reading. Although this book was written as a textbook, it can also be used as a referencebook for practical engineers and technicians. Throughout the book, the emphasis ison the basic operating principles. Many practical examples and design informationhave been included. I would like to thank all of my former students who used my notes in class fortheir useful comments and suggestions. I would also like to thank Mingyi Li, PaolaZepeda, Chris Rodenbeck, Matt Coutant and James McSpadden for critical review ofthe manuscript. Michelle Rubin has done an excellent job in editing and preparingthe manuscript. Taehan Bae has helped to prepared some of the art work. Finally, Iwish to express my deep appreciation to my wife, Suh-jan, and my children, Peterand Nancy, for their patience and support. KAI CHANGFebruary 2000
AcronymsAF array factorAGC automatic gain controlAM amplitude modulatedAMPS advanced mobile phone serviceAPTS advanced public transit systemsASK amplitude shift keyingATIS advanced traveler information systemATMS advanced trafﬁc management systemAUT antenna under testAVCS advanced vehicle control systemAVI automatic vehicle identiﬁcationBER bit error rateBPF bandpass ﬁlterBPSK biphase shift keyingBSF bandstop ﬁlterBW bandwidthCAD computer-aided designCDMA code division multiple accessCEP circular probable errorCMOS complementary MOSCP circularly polarizedCPL cross-polarization levelCRTs cathode ray tubesCT1/2 cordless telephone 1/2CTO cordless telephone OCVO commercial vehicle operationsCVR crystal video receiverCW continuous wave xiii
xiv ACRONYMSDBS direct broadcast satelliteDC direct currentDECT digital european cordless telephoneDR dynamic rangeDRO dielectric resonator oscillatorDSB double side bandDSSS direct-sequence spread spectrumECCMs electronic courter- countermeasuresECMs electronic countermeasuresEIRP effective isotropic radiated powerEM electromagneticESM electronic support measureEW electronic warfareFCC Federal Communications CommissionFDD frequency division duplexFDM frequency division multiplexingFDMA frequency division multiple accessFETs ﬁeld-effect transistorsFFHSS fast frequency- hopping spread spectrumFHSS frequency-hopping spread spectrumFLAR forward-looking automotive radarFMCW frequency-modulated continous waveFM frequency modulatedFNBW ﬁrst-null beamwidthFSK frequency shift keyingGEO geosynchronous orbitGMSK Gaussian minimum shift keyingGPS global positioning systemGSM global system for mobile communicationG=T receiver antenna gain to system noise temperature ratioHBTs heterojunction bipolar transistorsHEMTs high-electron-mobility transistorsHPA high-power ampliﬁerHPBW half-power beamwidthHPF high-pass ﬁlterIF intermediate frequencyIFM instantaneous frequency measurementIL insertion lossIMD intermodulation distortionIM intermodulationIMPATT impact ionization avalanche transit time deviceIM3 third-order intermodulationIP3 third-order intercept pointI/Q in-phase/quadrature-phaseIVHS intelligent vehicle and highway system
ACRONYMS xvJCT Japanese cordless telephoneJ/S jammer-to-signalLAN local area networkLMDS local multipoint distribution serviceLEO low earth orbitLNA low-noise ampliﬁerLO local oscillatorLOS line-of-sightLPF low-pass ﬁlterLP linearly polarizedMDS minimum detectable signalMEO medium-altitude orbitMESFETs metal–semiconductor ﬁeld-effect transistorsMIC microwave integrated circuitMLS microwave landing systemMMIC monolithic microwave integrated circuitsMSK minimum shift keyingMTI moving target indicatorNMT Nordic mobile telephoneOQPSK offset-keyed quadriphase shift keyingPAMELA pricing and monitoring electronically of automobilesPA power ampliﬁerPAE power added efﬁciencyPCM pulse code modulationPCN personal communication networksPCS personal communication systemsPDC personal digital cellularPHS personal handy phone systemPLL phase-locked loopsPLO phase-locked oscillatorsPM phase modulationPN pseudonoisePRF pulse repetition frequencyPSK phase shift keying8-PSK 8-phase shift keying16-PSK 16-phase shift keyingQAM quadrature amplitude modulationQPSK quadriphase shift keyingRCS radar cross sectionRF radio frequencyRFID radio frequency identiﬁcationRL return lossSAR synthetic aperture radarSAW surface acoustic waveSEP spherical probable error
xvi ACRONYMSSFDR spurious-free dynamic rangeSFHSS slow frequency- hopping spread spectrumSLL sidelobe levelsSMILER short range microwave links for european roadsSNR signal-noise ratioSOJ stand-off jammerSPDT single pole, double throwSPST single pole, single throwSP3T single pole, triple throwSQPSK staggered quadriphase shift keyingSS spread spectrumSS-CDMA spread spectrum code division multiple accessSSJ self-screening jammerSSMI special sensor microwave imagerSTC sensitivity time controlTACS total access communication systemTDM time division multiplexingTDMA time division multiple accessTE transverse electricTEM transverse electromagneticTM transverse magneticTOI third-order intercept pointT/R transmit/receiveTWTs traveling-wave tubesUHF ultrahigh frequenciesVCO voltage-controlled oscillatorVHF very high frequencyVLF very low frequencyVSWR voltage standing-wave ratioWLANs wireless local-area networks
RF and Microwave Wireless Systems. Kai Chang Copyright # 2000 John Wiley & Sons, Inc. ISBNs: 0-471-35199-7 (Hardback); 0-471-22432-4 (Electronic)CHAPTER ONEIntroduction1.1 BRIEF HISTORY OF RF AND MICROWAVE WIRELESS SYSTEMSThe wireless era was started by two European scientists, James Clerk Maxwell andHeinrich Rudolf Hertz. In 1864, Maxwell presented Maxwells equations byunifying the works of Lorentz, Faraday, Ampere, and Gauss. He predicted thepropagation of electromagnetic waves in free space at the speed of light. Hepostulated that light was an electromagnetic phenomenon of a particular wavelengthand predicted that radiation would occur at other wavelengths as well. His theorywas not well accepted until 20 years later, after Hertz validated the electromagneticwave (wireless) propagation. Hertz demonstrated radio frequency (RF) generation,propagation, and reception in the laboratory. His radio system experiment consistedof an end-loaded dipole transmitter and a resonant square-loop antenna receiveroperating at a wavelength of 4 m. For this work, Hertz is known as the father ofradio, and frequency is described in units of hertz (Hz). Hertzs work remained a laboratory curiosity for almost two decades, until ayoung Italian, Guglielmo Marconi, envisioned a method for transmitting andreceiving information. Marconi commercialized the use of electromagnetic wavepropagation for wireless communications and allowed the transfer of informationfrom one continent to another without a physical connection. The telegraph becamethe means of fast communications. Distress signals from the S.S. Titanic made agreat impression on the public regarding the usefulness of wireless communications.Marconis wireless communications using the telegraph meant that a ship was nolonger isolated in the open seas and could have continuous contact to report itspositions. Marconis efforts earned him the Nobel Prize in 1909. In the early 1900s, most wireless transmission occurred at very long wavelengths.Transmitters consisted of Alexanderson alternators, Poulsen arcs, and spark gaps.Receivers used coherers, Fleming valves, and DeForest audions. With the advent ofDeForests triode vacuum tube in 1907, continuous waves (CW) replaced spark gaps, 1
2 INTRODUCTIONand more reliable frequency and power output were obtained for radio broadcastingat frequencies below 1.5 MHz. In the 1920s, the one-way broadcast was made topolice cars in Detroit. Then the use of radio waves for wireless broadcasting,communications between mobile and land stations, public safety systems, maritimemobile services, and land transportation systems was drastically increased. DuringWorld War II, radio communications became indispensable for military use inbattle®elds and troop maneuvering. World War II also created an urgent need for radar (standing for radio detectionand ranging). The acronym radar has since become a common term describing theuse of re¯ections from objects to detect and determine the distance to and relativespeed of a target. A radars resolution (i.e., the minimum object size that can bedetected) is proportional to wavelength. Therefore, shorter wavelengths or higherfrequencies (i.e., microwave frequencies and above) are required to detect smallerobjects such as ®ghter aircraft. Wireless communications using telegraphs, broadcasting, telephones, and point-to-point radio links were available before World War II. The widespread use of thesecommunication methods was accelerated during and after the war. For long-distancewireless communications, relay systems or tropospheric scattering were used. In1959, J. R. Pierce and R. Kompfner envisioned transoceanic communications bysatellites. This opened an era of global communications using satellites. The satelliteuses a broadband high-frequency system that can simultaneously support thousandsof telephone users, tens or hundreds of TV channels, and many data links. Theoperating frequencies are in the gigahertz range. After 1980, cordless phones and FIGURE 1.1 Summary of the history of wireless systems.
1.2 FREQUENCY SPECTRUMS 3cellular phones became popular and have enjoyed very rapid growth in the past twodecades. Today, personal communication systems (PCSs) operating at higherfrequencies with wider bandwidths are emerging with a combination of variousservices such as voice mail, email, video, messaging, data, and computer on-lineservices. The direct link between satellites and personal communication systems canprovide voice, video, or data communications anywhere in the world, even in themost remote regions of the globe. In addition to communication and radar applications, wireless technologies havemany other applications. In the 1990s, the use of wireless RF and microwavetechnologies for motor vehicle and highway applications has increased, especially inEurope and Japan. The direct broadcast satellite (DBS) systems have offered analternative to cable television, and the end of the Cold War has made many militarytechnologies available to civilian applications. The global positioning systems(GPSs), RF identi®cation (RFID) systems, and remote sensing and surveillancesystems have also found many commercial applications. Figure 1.1 summarizes the history of these wireless systems.1.2 FREQUENCY SPECTRUMSRadio frequencies, microwaves, and millimeter waves occupy the region of theelectromagnetic spectrum below 300 GHz. The microwave frequency spectrum isfrom 300 MHz to 30 GHz with a corresponding wavelength from 100 cm to 1 cm.Below the microwave spectrum is the RF spectrum and above is the millimeter-wavespectrum. Above the millimeter-wave spectrum are submillimeter-wave, infrared,and optical spectrums. Millimeter waves (30±300 GHz), which derive their namefrom the dimensions of the wavelengths (from 10 to 1 mm), can be classi®ed asmicrowaves since millimeter-wave technology is quite similar to that of microwaves.Figure 1.2 shows the electromagnetic spectrum. For convenience, microwave andmillimeter-wave spectrums are further divided into many frequency bands. Figure1.2 shows some microwave bands, and Table 1.1 shows some millimeter-wavebands. The RF spectrum is not well de®ned. One can consider the frequencyspectrum below 300 MHz as the RF spectrum. But frequently, literatures use the RFterm up to 2 GHz or even higher. The Federal Communications Commission (FCC) allocates frequency ranges andspeci®cations for different applications in the United States, including televisions,radios, satellite communications, cellular phones, police radar, burglar alarms, andnavigation beacons. The performance of each application is strongly affected by theatmospheric absorption. The absorption curves are shown in Fig. 1.3. For example, asecure local area network would be ideal at 60 GHz due to the high attenuationcaused by the O2 resonance. As more applications spring up, overcrowding and interference at lowerfrequency bands pushes applications toward higher operating frequencies. Higherfrequency operation has several advantages, including:
1.2 FREQUENCY SPECTRUMS 51. Larger instantaneous bandwidth for greater transfer of information2. Higher resolution for radar, bigger doppler shift for CW radar, and more detailed imaging and sensing3. Reduced dimensions for antennas and other components4. Less interference from nearby applications5. Fast speed for digital system signal processing and data transmission6. Less crowded spectrum7. Dif®culty in jamming (military applications) TABLE 1.1 Millimeter-Wave Band Designation Frequency Range Designation (GHz) Q-band 33±50 U -band 40±60 V -band 50±75 E-band 60±90 W -band 75±110 D-band 110±170 G-band 140±220 Y -band 220±325 FIGURE 1.3 Absorption by the atmosphere in clear weather.
6 INTRODUCTIONThe use of higher frequency also has some disadvantages: 1. More expensive components 2. Higher atmospheric losses 3. Reliance on GaAs instead of Si technology 4. Higher component losses and lower output power from active devices 5. Less accurate design tools and less mature technologies The electron mobility in GaAs is higher than that in silicon. Therefore, GaAsdevices can operate at higher frequencies and speeds. Current silicon-based devicesare commonly used up to 2 GHz. Above 4 GHz, GaAs devices are preferred forbetter performance. However, GaAs processing is more expensive, and the yield islower than that of silicon.1.3 WIRELESS APPLICATIONSTwo of the most historically important RF=microwave applications are communica-tion systems and radar; but there are many others. Currently, the market is driven bythe phenomenal growth of PCSs, although there is also an increased demand forsatellite-based video, telephone, and data communication systems. Radio waves and microwaves play an important role in modern life. Televisionsignals are transmitted around the globe by satellites using microwaves. Airliners areguided from takeoff to landing by microwave radar and navigation systems.Telephone and data signals are transmitted using microwave relays. The militaryuses microwaves for surveillance, navigation, guidance and control, communica-tions, and identi®cation in their tanks, ships, and planes. Cellular telephones areeverywhere. The RF and microwave wireless technologies have many commercial and militaryapplications. The major application areas include communications, radar, navigation,remote sensing, RF identi®cation, broadcasting, automobiles and highways, sensors,surveillance, medical, and astronomy and space exploration. The details of theseapplications are listed below: 1. Wireless Communications. Space, long-distance, cordless phones, cellular telephones, mobile, PCSs, local-area networks (LANs), aircraft, marine, citizens band (CB) radio, vehicle, satellite, global, etc. 2. Radar. Airborne, marine, vehicle, collision avoidance, weather, imaging, air defense, traf®c control, police, intrusion detection, weapon guidance, surveillance, etc. 3. Navigation. Microwave landing system (MLS), GPS, beacon, terrain avoidance, imaging radar, collision avoidance, auto-pilot, aircraft, marine, vehicle, etc.
1.4 A SIMPLE SYSTEM EXAMPLE 7 4. Remote Sensing. Earth monitoring, meteorology, pollution monitoring, forest, soil moisture, vegetation, agriculture, ®sheries, mining, water, desert, ocean, land surface, clouds, precipitation, wind, ¯ood, snow, iceberg, urban growth, aviation and marine traf®c, surveillance, etc. 5. RF Identi®cation. Security, antitheft, access control, product tracking, inventory control, keyless entry, animal tracking, toll collection, automatic checkout, asset management, etc. 6. Broadcasting. Amplitude- and frequency-modulated (AM, FM) radio, TV, DBS, universal radio system, etc. 7. Automobiles and Highways. Collision warning and avoidance, GPS, blind- spot radar, adaptive cruise control, autonavigation, road-to-vehicle commu- nications, automobile communications, near-obstacle detection, radar speed sensors, vehicle RF identi®cation, intelligent vehicle and highway system (IVHS), automated highway, automatic toll collection, traf®c control, ground penetration radar, structure inspection, road guidance, range and speed detection, vehicle detection, etc. 8. Sensors. Moisture sensors, temperature sensors, robotics, buried-object detection, traf®c monitoring, antitheft, intruder detection, industrial sensors, etc. 9. Surveillance and Electronic Warfare. Spy satellites, signal or radiation monitoring, troop movement, jamming, antijamming, police radar detectors, intruder detection, etc. 10. Medical. Magnetic resonance imaging, microwave imaging, patient monitor- ing, etc. 11. Radio Astronomy and Space Exploration. Radio telescopes, deep-space probes, space monitoring, etc. 12. Wireless Power Transmission. Space to space, space to ground, ground to space, ground to ground power transmission.1.4 A SIMPLE SYSTEM EXAMPLEA wireless system is composed of active and passive devices interconnected toperform a useful function. A simple example of a wireless radio system is shown inFig. 1.4. The transmitter operates as follows. The input baseband signal, which could bevoice, video, or data, is assumed to be bandlimited to a frequency fm. This signal is®ltered to remove any components that may be beyond the channels passband. Themessage signal is then mixed with a local oscillator (LO) signal to produce amodulated carrier in a process called up-conversion since it produces signals atfrequencies fLO fm or fLO À fm which are normally much higher than fm . Themodulated carrier can then be ampli®ed and transmitted by the antenna. When the signal arrives at the receiver, it is normally ampli®ed by a low-noiseampli®er (LNA). The LNA may be omitted from some systems when the receivedsignal has enough power to be mixed directly, as may occur in short-distance
8 INTRODUCTION FIGURE 1.4 Block diagram of a simpli®ed wireless radio system.communication links. The mixer then produces a signal at a frequency fIF fm orfIF À fm in a process called down-conversion since fIF is chosen to be much lowerthan fLO . The signal is ®ltered to remove any undesired harmonic and spuriousproducts resulting from the mixing process and is ampli®ed by an intermediate-frequency (IF) ampli®er. The output of the ampli®er goes to a detector stage wherethe baseband signal fm , which contains the original message, is recovered. To perform all of these functions, the microwave system relies on separatecomponents that contribute speci®c functions to the overall system performance.Broadly speaking, microwave components can be classi®ed as transmission lines,couplers, ®lters, resonators, signal control components, ampli®ers, oscillators,mixers, detectors, and antennas.1.5 ORGANIZATION OF THIS BOOKThis book is organized into 11 chapters. Chapter 2 reviews some fundamentalprinciples of transmission lines and electromagnetic waves. Chapter 3 gives a briefoverview of how antennas and antenna arrays work. Chapter 4 provides a discussion
1.5 ORGANIZATION OF THIS BOOK 9of system parameters for various components that form building blocks for a system.Chapters 5 and 6 are devoted to receiver and transmitter systems. Chapter 7introduces the radar equation and the basic principles for pulse and CW radarsystems. Chapter 8 describes various wireless communication systems, includingradios, microwave links, satellite communications, and cellular phones. It alsodiscusses the Friis transmission equation, space loss, and link budget calculation.A brief introduction to modulation techniques and multiple-access methods is givenin Chapters 9 and 10. Finally, Chapter 11 describes other wireless applications,including navigation systems, automobile and highway applications, direct broadcastsystems, remote sensing systems, RF identi®cation systems, and surveillancesystems.
RF and Microwave Wireless Systems. Kai Chang Copyright # 2000 John Wiley & Sons, Inc. ISBNs: 0-471-35199-7 (Hardback); 0-471-22432-4 (Electronic)CHAPTER TWOReview of Waves andTransmission Lines2.1 INTRODUCTIONAt low RF, a wire or a line on a printed circuit board can be used to connect twoelectronic components. At higher frequencies, the current tends to concentrate on thesurface of the wire due to the skin effect. The skin depth is a function of frequencyand conductivity given by 1=2 2 ds 2:1 omswhere o 2pf is the angular frequency, f is the frequency, m is the permeability,and s is the conductivity. For copper at a frequency of 10 GHz, s 5:8 Â 107 S=mand ds 6:6 Â 10À5 cm, which is a very small distance. The ®eld amplitude decaysexponentially from its surface value according to eÀz=ds , as shown in Fig. 2.1. The®eld decays by an amount of eÀ1 in a distance of skin depth ds . When a wire isoperating at low RF, the current is distributed uniformly inside the wire, as shown inFig. 2.2. As the frequency is increased, the current will move to the surface of thewire. This will cause higher conductor losses and ®eld radiation. To overcome thisproblem, shielded wires or ®eld-con®ned lines are used at higher frequencies. Many transmission lines and waveguides have been proposed and used in RF andmicrowave frequencies. Figure 2.3 shows the cross-sectional views of some of thesestructures. They can be classi®ed into two categories: conventional and integratedcircuits. A qualitative comparison of some of these structures is given in Table 2.1.Transmission lines and=or waveguides are extensively used in any system. They areused for interconnecting different components. They form the building blocks of10
2.1 INTRODUCTION 11 FIGURE 2.1 Fields inside the conductor. FIGURE 2.2 The currrent distribution within a wire operating at different frequencies.many components and circuits. Examples are the matching networks for an ampli®erand sections for a ®lter. They can be used for wired communications to connect atransmitter to a receiver (Cable TV is an example). The choice of a suitable transmission medium for constructing microwavecircuits, components, and subsystems is dictated by electrical and mechanicaltrade-offs. Electrical trade-offs involve such parameters as transmission line loss,dispersion, higher order modes, range of impedance levels, bandwidth, maximumoperating frequency, and suitability for component and device implementation.Mechanical trade-offs include ease of fabrication, tolerance, reliability, ¯exibility,weight, and size. In many applications, cost is an important consideration. This chapter will discuss the transmission line theory, re¯ection and transmission,S-parameters, and impedance matching techniques. The most commonly usedtransmission lines and waveguides such as coaxial cables, microstrip lines, andrectangular waveguides will be described.
12 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.3 Transmission line and waveguide structures.2.2 WAVE PROPAGATIONWaves can propagate in free space or in a transmission line or waveguide. Wavepropagation in free space forms the basis for wireless applications. Maxwellpredicted wave propagation in 1864 by the derivation of the wave equations.Hertz validated Maxwells theory and demonstrated radio wave propagation in the
TABLE 2.1 Transmission Line and Waveguide Comparisons Useful Frequency Potential for Range Impedance Cross-Sectional Power Active Device Low-Cost Transmission Line (GHz) Range (O) Dimensions Q-Factor Rating Mounting Production Rectangular waveguide 300 100±500 Moderate to large High High Easy Poor Coaxial line 50 10±100 Moderate Moderate Moderate Fair Poor Stripline 10 10±100 Moderate Low Low Fair Good Microstrip line 100 10±100 Small Low Low Easy Good Suspended stripline 150 20±150 Small Moderate Low Easy Fair Finline 150 20±400 Moderate Moderate Low Easy Fair Slotline 60 60±200 Small Low Low Fair Good Coplanar waveguide 60 40±150 Small Low Low Fair Good Image guide 300 30±30 Moderate High Low Poor Good Dielectric line 300 20±50 Moderate High Low Poor Fair13
14 REVIEW OF WAVES AND TRANSMISSION LINESlaboratory in 1886. This opened up an era of radio wave applications. For his work,Hertz is known as the father of radio, and his name is used as the frequency unit. Let us consider the following four Maxwell equations: r HÁE ~ Gauss law 2:2a e ~ @B HÂE À ~ Faradays law 2:2b @t ~ @D ~ HÂH ~ J Amperes law 2:2c @t H Á B 0 flux law ~ 2:2d ~ ~ ~where E and B are electric and magnetic ®elds, D is the electric displacement, H is ~ ~ is the conduction current density, e is the permittivity, and rthe magnetic intensity, J ~is the charge density. The term @D=@t is displacement current density, which was ®rstadded by Maxwell. This term is important in leading to the possibility of wavepropagation. The last equation is for the continuity of ¯ux. We also have two constitutive relations: D e0 E P eE ~ ~ ~ ~ 2:3a B m0 H M m H ~ ~ ~ ~ 2:3bwhere P and M are the electric and magnetic dipole moments, respectively, m is the ~ ~permeability, and e is the permittivity. The relative dielectric constant of the mediumand the relative permeability are given by e er 2:4a e0 m mr 2:4b m0where m0 4p Â 10À7 H=m is the permeability of vacuum and e0 8:85 Â10À12 F=m is the permittivity of vacuum. With Eqs. (2.2) and (2.3), the wave equation can be derived for a source-freetransmission line (or waveguide) or free space. For a source-free case, we haveJ r 0, and Eq. (2.2) can be rewritten as~ HÁE 0 ~ 2:5a H Â E ÀjomH ~ ~ 2:5b H Â H joeE ~ ~ 2:5c HÁH 0 ~ 2:5d
2.2 WAVE PROPAGATION 15Here we assume that all ®elds vary as e jot and @=@t is replaced by jo. The curl of Eq. (2.5b) gives H Â H Â E ÀjomH Â H ~ ~ 2:6Using the vector identity H Â H Â E H H Á E À H2 E and substituting (2.5c) into ~ ~ ~Eq. (2.6), we have H H Á E À H2 E Àjom joeE o2 meE ~ ~ ~ ~ 2:7Substituting Eq. (2.5a) into the above equation leads to H2 E o2 meE 0 ~ ~ 2:8aor H2 E k 2 E 0 ~ ~ 2:8b pwhere k o me propagation constant. Similarly, one can derive H2 H o2 meH 0 ~ ~ 2:9Equations (2.8) and (2.9) are referred to as the Helmholtz equations or waveequations. The constant k (or b) is called the wave number or propagation constant,which may be expressed as p 2p f o k o me 2p 2:10 l v vwhere l is the wavelength and v is the wave velocity. In free space or air-®lled transmission lines, m m0 and e e0 , we have p pk k0 o m0 e0 , and v c 1= m0 e0 speed of light. Equations (2.8) and(2.9) can be solved in rectangular, cylindrical, or spherical coordinates. Antennaradiation in free space is an example of spherical coordinates. The solution in a wavepropagating in the ~ direction: r ~r E r; y; f E y; feÀjkÁ~ ~ ~ 2:11
16 REVIEW OF WAVES AND TRANSMISSION LINESThe propagation in a rectangular waveguide is an example of rectangular coordinateswith a wave propagating in the z direction: E x; y; z E x; yeÀjkz ~ ~ 2:12Wave propagation in cylindrical coordinates can be found in the solution for acoaxial line or a circular waveguide with the ®eld given by E r; f; z E r; feÀjkz ~ ~ 2:13 From the above discussion, we can conclude that electromagnetic waves canpropagate in free space or in a transmission line. The wave amplitude varies withtime as a function of e jot . It also varies in the direction of propagation and in thetransverse direction. The periodic variation in time as shown in Fig. 2.4 gives the FIGURE 2.4 Wave variation in time and space domains.
2.3 TRANSMISSION LINE EQUATION 17frequency f , which is equal to 1=T , where T is the period. The period length in thepropagation direction gives the wavelength. The wave propagates at a speed as v fl 2:14Here, v equals the speed of light c if the propagation is in free space: v c f l0 2:15l0 being the free-space wavelength.2.3 TRANSMISSION LINE EQUATIONThe transmission line equation can be derived from circuit theory. Suppose atransmission line is used to connect a source to a load, as shown in Fig. 2.5. Atposition x along the line, there exists a time-varying voltage v x; t and current i x; t.For a small section between x and x Dx, the equivalent circuit of this section Dxcan be represented by the distributed elements of L, R, C, and G, which are theinductance, resistance, capacitance, and conductance per unit length. For a losslessline, R G 0. In most cases, R and G are small. This equivalent circuit can beeasily understood by considering a coaxial line in Fig. 2.6. The parameters L and Rare due to the length and conductor losses of the outer and inner conductors, whereas i(x, t) Source v(x, t) Load ∆x x x + ∆x x L∆ x R∆ x i(x, t) i(x + ∆ x, t) = v(x, t) C∆ x G∆ x v(x + ∆ x, t) ∆x x x + ∆x x x + ∆x FIGURE 2.5 Transmission line equivalent circuit.
18 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.6 L, R, C for a coaxial line.C and G are attributed to the separation and dielectric losses between the outer andinner conductors. Applying Kirchhoffs current and voltage laws to the equivalent circuit shown inFig. 2.5, we have v x Dx; t À v x; t Dv x; t À R Dxi x; t @i x; t À L Dx 2:16 @t i x Dx; t À i x; t Di x; t À G Dxv x Dx; t @v x Dx; t À C Dx 2:17 @tDividing the above two equations by Dx and taking the limit as Dx approaches 0, wehave the following equations: @v x; t @i x; t ÀRi x; t À L 2:18 @x @t @i x; t @v x; t ÀGv x; t À C 2:19 @x @tDifferentiating Eq. (2.18) with respect to x and Eq. (2.19) with respect to t gives @2 v x; t @i x; t @2 i x; t ÀR ÀL 2:20 @x2 @x @x @t @2 i x; t @v x; t @2 v x; t ÀG ÀC 2:21 @t @x @t @t 2
2.3 TRANSMISSION LINE EQUATION 19By substituting (2.19) and (2.21) into (2.20), one can eliminate @i=@x and @2 i= @x @t.If only the steady-state sinusoidally time-varying solution is desired, phasor notationcan be used to simplify these equations [1, 2]. Here, v and i can be expressed as v x; t ReV xe jot 2:22 i x; t ReI xe jot 2:23where Re is the real part and o is the angular frequency equal to 2pf . A ®nalequation can be written as d 2 V x À g2 V x 0 2:24 dx2Note that Eq. (2.24) is a wave equation, and g is the wave propagation constant givenby g R joL G joC1=2 a jb 2:25where a attenuation constant in nepers per unit length b phase constant in radians per unit length:The general solution to Eq. (2.24) is V x V eÀgx VÀ egx 2:26Equation (2.26) gives the solution for voltage along the transmission line. Thevoltage is the summation of a forward wave (V eÀgx ) and a re¯ected wave (VÀ egx )propagating in the x and Àx directions, respectively. The current I x can be found from Eq. (2.18) in the frequency domain: I x I eÀgx À IÀ egx 2:27where g g I V ; IÀ V R joL R joL ÀThe characteristic impedance of the line is de®ned by 1=2 V VÀ R joL R joL Z0 2:28 I IÀ g G joC
20 REVIEW OF WAVES AND TRANSMISSION LINESFor a lossless line, R G 0, we have p g jb jo LC 2:29a r L Z0 2:29b C o 1 Phase velocity vp f lg p 2:29c b LCwhere lg is the guided wavelength and b is the propagation constant.2.4 REFLECTION, TRANSMISSION, AND IMPEDANCE FOR ATERMINATED TRANSMISSION LINEThe transmission line is used to connect two components. Figure 2.7 shows atransmission line with a length l and a characteristic impedance Z0 . If the line islossless and terminated by a load Z0, there is no re¯ection and the input impedanceZin is always equal to Z0 regardless of the length of the transmission line. If a load ZLFIGURE 2.7 Terminated transmission line: (a) a load Z0 is connected to a transmission linewith a characteristic impedance Z0 ; (b) a load ZL is connected to a transmission line with acharacteristic impedance Z0 .
2.4 REFLECTION, TRANSMISSION, AND IMPEDANCE 21(ZL could be real or complex) is connected to the line as shown in Fig. 2.7b andZL T Z0 , there exists a re¯ected wave and the input impedance is no longer equal toZ0 . Instead, Zin is a function of frequency ( f ), l, ZL , and Z0 . Note that at lowfrequencies, Zin % ZL regardless of l. In the last section, the voltage along the line was given by V x V eÀgx VÀ egx 2:30A re¯ection coef®cient along the line is de®ned as G x: reflected V x V egx V G x À Àgx À e2gx 2:31a incident V x V e Vwhere VÀ GL G 0 V reflection coefficient at load 2:31bSubstituting GL into Eqs. (2.26) and (2.27), the impedance along the line is given by V x eÀgx GL egx Z x Z0 Àgx 2:32 I x e À GL egxAt x 0, Z x ZL . Therefore, 1 GL ZL Z 0 2:33 1 À GL ZL À Z0 GL jGL jejf 2:34 ZL Z0To ®nd the input impedance, we set x Àl in Z x. This gives egl GL eÀgl Zin Z Àl Z0 2:35 egl À GL eÀglSubstituting (2.34) into the above equation, we have ZL Z0 tanh gl Zin Z0 2:36 Z0 ZL tanh gl
22 REVIEW OF WAVES AND TRANSMISSION LINESFor the lossless case, g jb, (2.36) becomes ZL jZ0 tan bl Zin Z0 Z0 jZL tan bl Zin l; f ; ZL ; Z0 2:37Equation (2.37) is used to calculate the input impedance for a terminated losslesstransmission line. It is interesting to note that, for low frequencies, bl % 0 andZin % ZL . The power transmitted and re¯ected can be calculated by the following: jV j2 Incident power Pin 2:38 Z0 jVÀ j2 Reflected power Pr Z0 jV j2 jGL j2 jGL j2 Pin 2:39 Z0 Transmitted power Pt Pin À Pr 1 À jGL j2 Pin 2:402.5 VOLTAGE STANDING-WAVE RATIOFor a transmission line with a matched load, there is no re¯ection, and the magnitudeof the voltage along the line is equal to jV j. For a transmission line terminated witha load ZL, a re¯ected wave exists, and the incident and re¯ected waves interfere toproduce a standing-wave pattern along the line. The voltage at point x along thelossless line is given by V x V eÀjbx VÀ ejbx V eÀjbx 1 GL e2jbx 2:41Substituting GL jGL jejf into the above equation gives the magnitude of V x as Â À ÁÃ1=2 jV xj jV j 1 jGL j2 À 4jGL j sin2 bx 1 f 2 2:42Equation (2.42) shows that jV xj oscillates between a maximum value ofjV j 1 jGL j when sin bx 1 f 0 (or bx 1 f np) and a minimum value 2 2of jV j 1 À jGL j when sin bx 1 f 1 (or bx 1 f mp À 1 p). Figure 2.8 2 2 2shows the pattern that repeats itself every 1 lg . 2
2.5 VOLTAGE STANDING-WAVE RATIO 23 FIGURE 2.8 Pattern of voltage magnitude along line. The ®rst maximum voltage can be found by setting x Àdmax and n 0. Wehave 2bdmax f 2:43The ®rst minimum voltage, found by setting x Àdmin and m 0, is given by 2bdmin f p 2:44The voltage standing-wave ratio (VSWR) is de®ned as the ratio of the maximumvoltage to the minimum voltage. From (2.42), jVmax j 1 jGL j VSWR 2:45 jVmin j 1 À jGL jIf the VSWR is known, jGL j can be found by VSWR À 1 jGL j jG xj 2:46 VSWR 1The VSWR is an important speci®cation for all microwave components. For goodmatching, a low VSWR close to 1 is generally required over the operating frequencybandwidth. The VSWR can be measured by a VSWR meter together with a slotted
24 REVIEW OF WAVES AND TRANSMISSION LINESline, a re¯ectometer, or a network analyzer. Figure 2.9 shows a nomograph of theVSWR. The return loss and power transmission are de®ned in the next section. Table2.2 summarizes the formulas derived in the previous sections.Example 2.1 Calculate the VSWR and input impedance for a transmission lineconnected to (a) a short and (b) an open load. Plot Zin as a function of bl. FIGURE 2.9 VSWR nomograph.
2.5 VOLTAGE STANDING-WAVE RATIO 25TABLE 2.2 Formulas for Transmission LinesQuantity General Line Lossless Line p pPropagation constant, g a jb R joL G joC jo LC p w 2pPhase constant, b Im g o LC v lAttenuation constant, a Re g 0 s r R joL LCharacteristic impedance, Z0 G joC C ZL cosh gl Z0 sinh gl ZL cos bl jZ0 sin blInput impedance, Zin Z0 Z0 Z0 cosh gl ZL sinh gl Z0 cos bl jZL sin blImpedance of shorted line Z0 tanh gl jZ0 tan blImpedance of open line Z0 coth gl ÀjZ0 cot bl ZL sinh al Z0 cosh al 2 Z0Impedance of quarter-wave line Z0 Z0 sinh al ZL cosh al ZL ZL cosh al Z0 sinh alImpedance of half-wave line Z0 ZL Z0 cosh al ZL sinh al ZL À Z0 ZL À Z0Re¯ection coef®cient, GL ZL Z0 ZL Z0 1 jGL j 1 jGL jVoltage standing-wave ratio 1 À jGL j 1 À jGL j (VSWR)Solution (a) A transmission line with a characteristic impedance Z0 is connected toa short load ZL as shown in Fig. 2.10. Here, ZL 0, and GL is given by ZL À Z0 GL À1 1ej180 jGL jejf ZL Z0Therefore, jGL j 1 and f 180 . From Eq. (2.45), 1 jGL j VSWR I 1 À jGL j Reflected power jGL j2 Pin Pin Transmitted power 1 À jGL j2 Pin 0
26 REVIEW OF WAVES AND TRANSMISSION LINESThe input impedance is calculated by Eq. (2.37), ZL jZ0 tan bl Zin Z0 Z0 jZL tan bl jZ0 tan bl jXinThe impedance Zin is plotted as a function of bl or l, as shown in Fig. 2.10. It isinteresting to note that any value of reactances can be obtained by varying l. For this FIGURE 2.10 Transmission line connected to a shorted load.
2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSS 27reason, a short stub is useful for impedance tuning and impedance matchingnetworks. (b) For an open load, ZL I, and ZL À Z0 GL 1 1ej0 jGL je jf ZL Z0Therefore, jGL j 1 and f 0 . Again, 1 jGL j VSWR I and Pr Pin 1 À jGL jThe input impedance is given by ZL jZ0 tan bl Zin Z0 ÀjZ0 cot bl jXin Z0 jZL tan bland is plotted as a function of bl or l as shown in Fig. 2.11. Again, any value ofreactance can be obtained and the open stub can also be used as an impedance tuner. j2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSSThe decibel (dB) is a dimensionless number that expresses the ratio of two powerlevels. Speci®cally, P2 Power ratio in dB 10 log10 2:47 P1where P1 and P2 are the two power levels being compared. If power level P2 ishigher than P1 , the decibel is positive and vice versa. Since P V 2 =R, the voltagede®nition of the decibel is given by V2 Voltage ratio in dB 20 log10 2:48 V1The decibel was originally named for Alexander Graham Bell. The unit was used asa measure of attenuation in telephone cable, that is, the ratio of the power of thesignal emerging from one end of a cable to the power of the signal fed in at the otherend. It so happened that one decibel almost equaled the attenuation of one mile oftelephone cable.
28 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.11 Transmission line connected to an open load.2.6.1 Conversion from Power Ratios to Decibels and Vice VersaOne can convert any power ratio (P2 =P1 ) to decibels, with any desired degree ofaccuracy, by dividing P2 by P1, ®nding the logarithm of the result, and multiplying itby 10.
2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSS 29 From Eq. (2.47), we can ®nd the power ratio in decibels given below: Power Ratio dB Power Ratio dB 1 0 1 0 1.26 1 0.794 À1 1.58 2 0.631 À2 2 3 0.501 À3 2.51 4 0.398 À4 3.16 5 0.316 À5 3.98 6 0.251 À6 5.01 7 0.2 À7 6.31 8 0.158 À8 7.94 9 0.126 À9 10 10 0.1 À 10 100 20 0.01 À 20 1000 30 0.001 À 30 107 70 10À7 À 70As one can see from these results, the use of decibels is very convenient to representa very large or very small number. To convert from decibels to power ratios, thefollowing equation can be used: Power ratio 10dB=10 2:492.6.2 Gain or Loss RepresentationsA common use of decibels is in expressing power gains and power losses in thecircuits. Gain is the term for an increase in power level. As shown in Fig. 2.12, anampli®er is used to amplify an input signal with Pin 1 mW. The output signal is200 mW. The ampli®er has a gain given by output power Gain in ratio 200 2:50a input power output power Gain in db 10 log10 23 dB 2:50b input powerNow consider an attenuator as shown in Fig. 2.13. The loss is the term of a decreasein power. The attenuator has a loss given by input power Loss in ratio 2 2:51a output power input power Loss in db 10 log10 3 dB 2:51b output power
30 REVIEW OF WAVES AND TRANSMISSION LINESThe above loss is called insertion loss. The insertion loss occurs in most circuitcomponents, waveguides, and transmission lines. One can consider a 3-dB loss as aÀ 3-dB gain. For a cascaded circuit, one can add all gains (in decibels) together and subtract thelosses (in decibels). Figure 2.14 shows an example. The total gain (or loss) is then Total gain 23 23 23 À 3 66 dBIf one uses ratios, the total gain in ratio is Total gain 200 Â 200 Â 200 Â 1 4,000,000 2In general, for any number of gains (in decibels) and losses (in decibels) in acascaded circuit, the total gain or loss can be found by GT G1 G2 G3 Á Á Á À L1 L2 L3 Á Á Á 2:52 FIGURE 2.12 Ampli®er circuit. FIGURE 2.13 Attenuator circuit. FIGURE 2.14 Cascaded circuit.
2.6 DECIBELS, INSERTION LOSS, AND RETURN LOSS 312.6.3 Decibels as Absolute UnitsDecibels can be used to express values of power. All that is necessary is to establishsome absolute unit of power as a reference. By relating a given value of power to thisunit, the power can be expressed with decibels. The often-used reference units are 1 mW and 1 W. If 1 milliwatt is used as areference, dBm is expressed as decibels relative to 1 mW: P in dBm 10 log P in mW 2:53Therefore, the following results can be written: 1 mW 0 dBm 10 mW 10 dBm 1 W 30 dBm 0:1 mW À10 dBm 1 Â 10À7 mW À70 dBmIf 1 W is used as a reference, dBW is expressed as decibels relative to 1 W. Theconversion equation is given by P in dBW 10 log P in W 2:54From the above equation, we have 1 W 0 dBW 10 W 10 dBW 0:1 W À10 dBWNow for the system shown in Fig. 2.14, if the input power Pin 1 mW 0 dBm,the output power will be Pout Pin G1 G2 G3 À L 0 dBm 23 dB 23 dB 23 dB À 3 dB 66 dBm; or 3981 WThe above calculation is equivalent to the following equation using ratios: Pin G1 G2 G3 Pout L 1 mW Â 200 Â 200 Â 200 4000 W 2
32 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.15 Two-port component.2.6.4 Insertion Loss and Return LossInsertion loss, return loss, and VSWR are commonly used for component speci®ca-tion. As shown in Fig. 2.15, the insertion loss and return loss are de®ned as Pin Insertion loss IL 10 log 2:55a Pt Pin Return loss RL 10 log 2:55b PrSince Pr jGL j2 Pin , Eq. (2.55b) becomes RL À20 log jGL j 2:56The return loss indicates an input mismatch loss of a component. The insertion lossincludes the input and output mismatch losses and other circuit losses (conductorloss, dielectric loss, and radiation loss).Example 2.2 A coaxial three-way power divider (Fig. 2.16) has an input VSWR of1.5 over a frequency range of 2.5±5.5 GHz. The insertion loss is 0.5 dB. What are thepercentages of power re¯ection and transmission? What is the return loss indecibels?Solution Since VSWR 1:5, from Eq. (2.46) VSWR À 1 0:5 jGL j 0:2 Pr jGL j2 Pin 0:04Pin VSWR 1 2:5The return loss is calculated by Eq. (2.55b) or (2.56): Pin RL 10 log À20 log jGL j 13:98 dB Pr
2.7 SMITH CHARTS 33 FIGURE 2.16 Three-way power divider.The transmitted power is calculated from Eq. (2.55a): Pin IL 10 log 0:5 dB Pt 0:89Pin PtAssuming the input power is split into three output ports equally, each output portwill transmit 29.7% of the input power. The input mismatch loss is 4% of the inputpower. Another 7% of power is lost due to the output mismatch and circuit losses. j2.7 SMITH CHARTSThe Smith chart was invented by P. H. Smith of Bell Laboratories in 1939. It is agraphical representation of the impedance transformation property of a length oftransmission line. Although the impedance and re¯ection information can beobtained from equations in the previous sections, the calculations normally involvecomplex numbers that can be complicated and time consuming. The use of theSmith chart avoids the tedious computation. It also provides a graphical representa-tion on the impedance locus as a function of frequency. De®ne a normalized impedance Z x as Z x Z x R x jX x 2:57 Z0
34 REVIEW OF WAVES AND TRANSMISSION LINESThe re¯ection coef®cient G x is given by G x Gr x jGi x 2:58From Eqs. (2.31a) and (2.32), we have Z x 1 G x Z x 2:59 Z0 1 À G xTherefore 1 Gr jGi R x jX x 2:60 1 À Gr À jGiBy multiplying both numerator and denominator by 1 À Gr jGi , two equations aregenerated: 2 2 R 1 Gr À G2 2:61a 1R i 1R 2 2 1 1 Gr À 1 Gi À 2 2:61b X XIn the Gr À Gi coordinate system, Eq. (2.61a) represents circles centered at (R= 1 R; 0) with a radii of 1= 1 R. These are called constant R circles.Equation (2.61b) represents circles centered at (1; 1=X ) with radii of 1=X . They are called constant X circles. Figure 2.17 shows these circles in the Gr À Gi plane.The plot of these circles is called the Smith chart. On the Smith chart, a constant jGjis a circle centered at (0, 0) with a radius of jGj. Hence, motion along a losslesstransmission line gives a circular path on the Smith chart. From Eq. (2.31), we knowfor a lossless line that G x GL e2gx GL ej2bx 2:62 Hence, given ZL , we can ®nd Zin at a distance Àl from the load by proceeding at an angle 2bl in a clockwise direction. Thus, Zin can be found graphically. The Smith chart has the following features: 1. Impedance or admittance values read from the chart are normalized values. 2. Moving away from the load (i.e., toward the generator) corresponds to moving in a clockwise direction. 3. A complete revolution around the chart is made by moving a distance l 1 lg along the transmission line. 2 4. The same chart can be used for reading admittance.
2.7 SMITH CHARTS 35 FIGURE 2.17 Constant R and X circles in the re¯ection-coef®cient plane. 5. The center of the chart corresponds to the impedance-matched condition since G x 0. 6. A circle centered at the origin is a constant jG xj circle. 7. Moving along the lossless transmission line is equivalent to moving along the constant jG xj circle. 8. For impedance reading, the point (Gr 1; Gi 0) corresponds to an open circuit. For admittance reading, the same point corresponds to a short circuit. 9. The impedance at a distance 1 lg from Z L is equal to Y L . 4 10. The VSWR can be found by reading R at the intersection of the constant jGj circle with the real axis. The Smith chart can be used to ®nd (1) GL from ZL and vice-versa; (2) Zin from ZL and vice-versa; (3) Z from Y and vice-versa; (4) the VSWR; and (5) dmin anddmax . The Smith chart is also useful for impedance matching, ampli®er design,oscillator design, and passive component design. The Smith charts shown in Fig. 2.17 are called Z-charts or Y -charts. One can readthe normalized impedance or admittance directly from these charts. If we rotate theY -chart by 180 , we have a rotated Y -chart, as shown in Fig. 2.18b. The combinationof the Z-chart and the rotated Y -chart is called a Z±Y chart, as shown in Fig. 2.18c. On the Z±Y chart, for any point A, one can read ZA from the Z-chart and YA from therotated Y -chart.
36 REVIEW OF WAVES AND TRANSMISSION LINES Therefore, this chart avoids the necessity of moving Z by 1 lg (i.e., 180 ) to ®nd 4 . The Z±Y chart is useful for impedance matching using lumped elements.YExample 2.3 A load of 100 j50 O is connected to a 50-O transmission line. Useboth equations and the Smith chart to ®nd the following: (a) GL ; (b) Zin at 0:2lgaway from the load; (c) YL ; (d) the VSWR; and (e) dmax and dmin .Solution (a) ZL 100 j50 Z ZL L 2 j Z0 ZL À Z0 ZL À 1 1 j GL 0:4 0:2j ZL Z0 ZL 1 3 j 0:44727 From the Smith chart shown in Fig. 2.19, ZL is located by ®nding the intersection of two circles with R 2 and X 1. The distance from the center to ZL is 0.44, andthe angle can be read directly from the scale as 27 . (b) ZL jZ0 tan bl Zin Z0 Z0 jZL tan bl 2p l 0:2lg ; bl Â 0:2lg 0:4p lg tan bl 3:08 2 j j3:08 Zin 0:496 À j0:492 1 j 2 j Â 3:08 From the Smith chart, we move ZL on a constant jGj circle toward a generator by adistance 0:2lg to a point Z in . The value of Zin can be read from the Smith chart as in is located at the intersection of two circles with R 0:5 and0:5 À j0:5, since Z X À0:5. The distance of movement (0:2lg ) can be read from the outermost scalein the chart. (c) ZL 2 j 1 1 YL 0:4 À j0:2 ZL 2 j Y 0:4 À j0:2 YL YL Y0 L 0:008 À j0:004 Z0 50
2.7 SMITH CHARTS 37FIGURE 2.18 Different Smith charts: (a) Z- or Y -chart; (b) rotated Y -chart; and (c) Z±Ychart.
38 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.19 Smith chart for Example 2.3. From the Smith chart, YL is found by moving ZL a distance 0:25lg toward the L is located opposite ZL through the center. This can begenerator. Therefore, Y proved by the following: l in l g ZL jZ0 tan bl Z 4 Z0 jZL tan bl Z 1 0 YL ZL ZL (d) GL 0:44727 jGL j 0:447 1 jGL j 1 0:447 VSWR 2:62 1 À jGL j 1 À 0:447
2.8 S-PARAMETERS 39 The VSWR can be found from the reading of R at the constant jGj circle and the realaxis. (e) From Eq. (2.43), we have f f dmax l 2b 4p gNow f 27 dmax 0:0375lg From the Smith chart, dmax is the distance measured by moving from ZL through aconstant jGj circle to the real axis. This can be proved by using Eq. (2.42). The ®rstdmax occurs when bx 1 f 0 or 2bx f 0. Since G x GL e2jbx 2jGL je j 2bxf , the ®rst dmax occurs when G x has zero phase (i.e., on the realaxis). The parameter dmin is just 1 lg away from dmax . 4 j2.8 S-PARAMETERSScattering parameters are widely used in RF and microwave frequencies forcomponent modeling, component speci®cations, and circuit design. S-parameterscan be measured by network analyzers and can be directly related to ABCD, Z-, andY -parameters used for circuit analysis [1, 2]. For a general N -port network as shownin Fig. 2.20, the S-matrix is given in the following equations: P Q P Q b1 P Q a T b2 U S11 S12 S13 F F F S1N T 1 U T U T S21 S22 S23 F F F S2N UT a2 U T b3 U T UT U T UT F F F F U T a3 U 2:63 T F U R F F F F F F F ST F U F R F S F R F S F SN 1 SN 2 SN 3 F F F SNN bN aNor b Sa 2:64where a1 ; a2 ; F F F ; aN are the incident-wave voltages at ports 1; 2; F F F ; N , respec-tively and b1 ; b2 ; F F F ; bN are the re¯ected-wave voltages at these ports. S-parametersare complex variables relating the re¯ected wave to the incident wave. The S-parameters have some interesting properties [1, 2]: 1. For any matched port i, Sii 0. 2. For a reciprocal network, Snm Smn . 3. For a passive circuit, jSmn j 1.
40 REVIEW OF WAVES AND TRANSMISSION LINES FIGURE 2.20 An N -port network. 4. For a lossless and reciprocal network, one has, for the ith port, N N Ã jSni j2 Sni Sni 1 2:65 n1 n1or jS1i j2 jS2i j2 jS3i j2 Á Á Á jSii j2 Á Á Á jSNi j2 1 2:66Equation (2.65) states that the product of any column of the S-matrix with theconjugate of this column is equal to 1. Equation (2.65) is due to the powerconservation of a lossless network. In Eq. (2.65), the total power incident at theith port is normalized and becomes 1, which is equal to the power re¯ected at the ithport plus the power transmitted into all other ports. A two-port network is the most common circuit. For a two-port network, asshown in Fig. 2.21, the S-parameters are given by b1 S11 a1 S12 a2 2:67a b2 S21 a1 S22 a2 2:67b
2.9 COAXIAL LINES 41FIGURE 2.21 Two-port network and its S-parameters, which can be measured by using anetwork analyzer.The S-parameters can be de®ned in the following: b1 S11 G1 reflection coefficient at port 1 with a2 0 a1 a2 0 b S21 2 T21 transmission coefficient from port 1 to 2 with a2 0 a1 a2 0 b2 S22 G2 reflection coefficient at port 2 with a1 0 a2 a1 0 b S12 1 T12 transmission coefficient from port 2 to port 1 with a1 0 a 2 a1 0The return loss can be found by a 1 RL 20 log 1 20 log in dB b S 2:68 1 11The attenuation or insertion loss is given by a 1 IL a 20 log 1 20 log b S in dB 2:69 2 21The phase shift of the network is f phase of S21 .2.9 COAXIAL LINESOpen-wire lines have high radiation losses and therefore are not used for high-frequency signal transmission. Shielded structures such as coaxial lines are
42 REVIEW OF WAVES AND TRANSMISSION LINEScommonly used. Cable TV and video transmission cables are coaxial lines. Coaxiallines are frequently used for interconnection, signal transmission, and measurements. A coaxial line normally operates in the transverse electromagnetic (TEM) mode(i.e., no axial electric and magnetic ®eld components). It has no cutoff frequency andcan be used from direct current (DC) to microwave or millimeter-wave frequencies.It can be made ¯exible. Presently, coaxial lines and connectors can be operated up to50 GHz. The upper frequency limit is set by the excessive losses and the excitationof circular waveguide transverse electric (TE) and transverse magnetic (TM) modes.However, improved manufacturing of small coaxial lines will push the operatingfrequency even higher. Since the coaxial line is operated in the TEM mode, the electric and magnetic®elds can be found in the static case. Figure 2.22 shows a coaxial transmission line.The electric potential of the coaxial line can be found by solving the Laplaceequation in the cylindrical coordinates. The Laplace equation for electric potential is H2 F 0 2:70Solving this equation by meeting the boundary conditions at the inner and outerconductors, we have [1, 2] ln r=ro F r V0 2:71 ln ri =ro FIGURE 2.22 Coaxial line with inner and outer conductor radii ri and ro .