Microwave ElectronicsMeasurement and Materials Characterization L. F. Chen, C. K. Ong and C. P. Neo National University of Singapore V. V. Varadan and V. K. Varadan Pennsylvania State University, USA
Microwave ElectronicsMeasurement and Materials Characterization L. F. Chen, C. K. Ong and C. P. Neo National University of Singapore V. V. Varadan and V. K. Varadan Pennsylvania State University, USA
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ContentsPreface xi1 Electromagnetic Properties of Materials 1 1.1 Materials Research and Engineering at Microwave Frequencies 1 1.2 Physics for Electromagnetic Materials 2 1.2.1 Microscopic scale 2 1.2.2 Macroscopic scale 6 1.3 General Properties of Electromagnetic Materials 11 1.3.1 Dielectric materials 11 1.3.2 Semiconductors 16 1.3.3 Conductors 17 1.3.4 Magnetic materials 19 1.3.5 Metamaterials 24 1.3.6 Other descriptions of electromagnetic materials 28 1.4 Intrinsic Properties and Extrinsic Performances of Materials 32 1.4.1 Intrinsic properties 32 1.4.2 Extrinsic performances 32 References 342 Microwave Theory and Techniques for Materials Characterization 37 2.1 Overview of the Microwave Methods for the Characterization of Electromagnetic Materials 37 2.1.1 Nonresonant methods 38 2.1.2 Resonant methods 40 2.2 Microwave Propagation 42 2.2.1 Transmission-line theory 42 2.2.2 Transmission Smith charts 51 2.2.3 Guided transmission lines 56 2.2.4 Surface-wave transmission lines 73 2.2.5 Free space 83 2.3 Microwave Resonance 87 2.3.1 Introduction 87 2.3.2 Coaxial resonators 93 2.3.3 Planar-circuit resonators 95 2.3.4 Waveguide resonators 97 2.3.5 Dielectric resonators 103 2.3.6 Open resonators 115 2.4 Microwave Network 119 2.4.1 Concept of microwave network 119 2.4.2 Impedance matrix and admittance matrix 119
vi Contents 2.4.3 Scattering parameters 120 2.4.4 Conversions between different network parameters 121 2.4.5 Basics of network analyzer 121 2.4.6 Measurement of reﬂection and transmission properties 126 2.4.7 Measurement of resonant properties 134 References 1393 Reﬂection Methods 142 3.1 Introduction 142 3.1.1 Open-circuited reﬂection 142 3.1.2 Short-circuited reﬂection 143 3.2 Coaxial-line Reﬂection Method 144 3.2.1 Open-ended apertures 145 3.2.2 Coaxial probes terminated into layered materials 151 3.2.3 Coaxial-line-excited monopole probes 154 3.2.4 Coaxial lines open into circular waveguides 157 3.2.5 Shielded coaxial lines 158 3.2.6 Dielectric-ﬁlled cavity adapted to the end of a coaxial line 160 3.3 Free-space Reﬂection Method 161 3.3.1 Requirements for free-space measurements 161 3.3.2 Short-circuited reﬂection method 162 3.3.3 Movable metal-backing method 162 3.3.4 Bistatic reﬂection method 164 3.4 Measurement of Both Permittivity and Permeability Using Reﬂection Methods 164 3.4.1 Two-thickness method 164 3.4.2 Different-position method 165 3.4.3 Combination method 166 3.4.4 Different backing method 167 3.4.5 Frequency-variation method 167 3.4.6 Time-domain method 168 3.5 Surface Impedance Measurement 168 3.6 Near-ﬁeld Scanning Probe 170 References 1724 Transmission/Reﬂection Methods 175 4.1 Theory for Transmission/reﬂection Methods 175 4.1.1 Working principle for transmission/reﬂection methods 175 4.1.2 Nicolson–Ross–Weir (NRW) algorithm 177 4.1.3 Precision model for permittivity determination 178 4.1.4 Effective parameter method 179 4.1.5 Nonlinear least-squares solution 180 4.2 Coaxial Air-line Method 182 4.2.1 Coaxial air lines with different diameters 182 4.2.2 Measurement uncertainties 183 4.2.3 Enlarged coaxial line 185 4.3 Hollow Metallic Waveguide Method 187 4.3.1 Waveguides with different working bands 187 4.3.2 Uncertainty analysis 187 4.3.3 Cylindrical rod in rectangular waveguide 189 4.4 Surface Waveguide Method 190
viii Contents 6.4.1 Surface resistance and surface reactance 268 6.4.2 Measurement of surface resistance 269 6.4.3 Measurement of surface reactance 275 6.5 Near-ﬁeld Microwave Microscope 278 6.5.1 Basic working principle 278 6.5.2 Tip-coaxial resonator 279 6.5.3 Open-ended coaxial resonator 280 6.5.4 Metallic waveguide cavity 284 6.5.5 Dielectric resonator 284 References 2867 Planar-circuit Methods 288 7.1 Introduction 288 7.1.1 Nonresonant methods 288 7.1.2 Resonant methods 290 7.2 Stripline Methods 291 7.2.1 Nonresonant methods 291 7.2.2 Resonant methods 292 7.3 Microstrip Methods 297 7.3.1 Nonresonant methods 298 7.3.2 Resonant methods 300 7.4 Coplanar-line Methods 309 7.4.1 Nonresonant methods 309 7.4.2 Resonant methods 311 7.5 Permeance Meters for Magnetic Thin Films 311 7.5.1 Working principle 312 7.5.2 Two-coil method 312 7.5.3 Single-coil method 314 7.5.4 Electrical impedance method 315 7.6 Planar Near-ﬁeld Microwave Microscopes 317 7.6.1 Working principle 317 7.6.2 Electric and magnetic dipole probes 318 7.6.3 Probes made from different types of planar transmission lines 319 References 3208 Measurement of Permittivity and Permeability Tensors 323 8.1 Introduction 323 8.1.1 Anisotropic dielectric materials 323 8.1.2 Anisotropic magnetic materials 325 8.2 Measurement of Permittivity Tensors 326 8.2.1 Nonresonant methods 327 8.2.2 Resonator methods 333 8.2.3 Resonant-perturbation method 336 8.3 Measurement of Permeability Tensors 340 8.3.1 Nonresonant methods 340 8.3.2 Faraday rotation methods 345 8.3.3 Resonator methods 351 8.3.4 Resonant-perturbation methods 355 8.4 Measurement of Ferromagnetic Resonance 370 8.4.1 Origin of ferromagnetic resonance 370 8.4.2 Measurement principle 371
Contents ix 8.4.3 Cavity methods 373 8.4.4 Waveguide methods 374 8.4.5 Planar-circuit methods 376 References 3799 Measurement of Ferroelectric Materials 382 9.1 Introduction 382 9.1.1 Perovskite structure 383 9.1.2 Hysteresis curve 383 9.1.3 Temperature dependence 383 9.1.4 Electric ﬁeld dependence 385 9.2 Nonresonant Methods 385 9.2.1 Reﬂection methods 385 9.2.2 Transmission/reﬂection method 386 9.3 Resonant Methods 386 9.3.1 Dielectric resonator method 386 9.3.2 Cavity-perturbation method 389 9.3.3 Near-ﬁeld microwave microscope method 390 9.4 Planar-circuit Methods 390 9.4.1 Coplanar waveguide method 390 9.4.2 Coplanar resonator method 394 9.4.3 Capacitor method 394 9.4.4 Inﬂuence of biasing schemes 404 9.5 Responding Time of Ferroelectric Thin Films 405 9.6 Nonlinear Behavior and Power-Handling Capability of Ferroelectric Films 407 9.6.1 Pulsed signal method 407 9.6.2 Intermodulation method 409 References 41210 Microwave Measurement of Chiral Materials 414 10.1 Introduction 414 10.2 Free-space Method 415 10.2.1 Sample preparation 416 10.2.2 Experimental procedure 416 10.2.3 Calibration 417 10.2.4 Time-domain measurement 430 10.2.5 Computation of ε, µ, and β of the chiral composite samples 434 10.2.6 Experimental results for chiral composites 440 10.3 Waveguide Method 452 10.3.1 Sample preparation 452 10.3.2 Experimental procedure 452 10.3.3 Computation of ε, µ, and ξ of the chiral composite samples 453 10.3.4 Experimental results for chiral composites 454 10.4 Concluding Remarks 458 References 45811 Measurement of Microwave Electrical Transport Properties 460 11.1 Hall Effect and Electrical Transport Properties of Materials 460 11.1.1 Direct current Hall effect 461 11.1.2 Alternate current Hall effect 461 11.1.3 Microwave Hall effect 461
x Contents 11.2 Nonresonant Methods for the Measurement of Microwave Hall Effect 464 11.2.1 Faraday rotation 464 11.2.2 Transmission method 465 11.2.3 Reﬂection method 469 11.2.4 Turnstile-junction method 473 11.3 Resonant Methods for the Measurement of the Microwave Hall Effect 475 11.3.1 Coupling between two orthogonal resonant modes 475 11.3.2 Hall effect of materials in MHE cavity 476 11.3.3 Hall effect of endplate of MHE cavity 482 11.3.4 Dielectric MHE resonator 484 11.3.5 Planar MHE resonator 486 11.4 Microwave Electrical Transport Properties of Magnetic Materials 486 11.4.1 Ordinary and extraordinary Hall effect 486 11.4.2 Bimodal cavity method 487 11.4.3 Bimodal dielectric probe method 489 References 48912 Measurement of Dielectric Properties of Materials at High Temperatures 492 12.1 Introduction 492 12.1.1 Dielectric properties of materials at high temperatures 492 12.1.2 Problems in measurements at high temperatures 494 12.1.3 Overviews of the methods for measurements at high temperatures 496 12.2 Coaxial-line Methods 497 12.2.1 Measurement of permittivity using open-ended coaxial probe 498 12.2.2 Problems related to high-temperature measurements 498 12.2.3 Correction of phase shift 500 12.2.4 Spring-loaded coaxial probe 502 12.2.5 Metallized ceramic coaxial probe 502 12.3 Waveguide Methods 503 12.3.1 Open-ended waveguide method 503 12.3.2 Dual-waveguide method 504 12.4 Free-space Methods 506 ∗ 12.4.1 Computation of εr 507 12.5 Cavity-Perturbation Methods 510 12.5.1 Cavity-perturbation methods for high-temperature measurements 510 12.5.2 TE10n mode rectangular cavity 512 12.5.3 TM mode cylindrical cavity 514 12.6 Dielectric-loaded Cavity Method 520 12.6.1 Coaxial reentrant cavity 520 12.6.2 Open-resonator method 523 12.6.3 Oscillation method 524 References 528Index 531
PrefaceMicrowave materials have been widely used in a variety of applications ranging from communicationdevices to military satellite services, and the study of materials properties at microwave frequenciesand the development of functional microwave materials have always been among the most active areasin solid-state physics, materials science, and electrical and electronic engineering. In recent years, theincreasing requirements for the development of high-speed, high-frequency circuits and systems requirecomplete understanding of the properties of materials functioning at microwave frequencies. All theseaspects make the characterization of materials properties an important ﬁeld in microwave electronics. Characterization of materials properties at microwave frequencies has a long history, dating from theearly 1950s. In past decades, dramatic advances have been made in this ﬁeld, and a great deal of newmeasurement methods and techniques have been developed and applied. There is a clear need to have apractical reference text to assist practicing professionals in research and industry. However, we realizethe lack of good reference books dealing with this ﬁeld. Though some chapters, reviews, and bookshave been published in the past, these materials usually deal with only one or several topics in thisﬁeld, and a book containing a comprehensive coverage of up-to-date measurement methodologies is notavailable. Therefore, most of the research and development activities in this ﬁeld are based primarilyon the information scattered throughout numerous reports and journals, and it always takes a great dealof time and effort to collect the information related to on-going projects from the voluminous literature.Furthermore, because of the paucity of comprehensive textbooks, the training in this ﬁeld is usually notsystematic, and this is undesirable for further progress and development in this ﬁeld. This book deals with the microwave methods applied to materials property characterization, and itprovides an in-depth coverage of both established and emerging techniques in materials characterization.It also represents the most comprehensive treatment of microwave methods for materials propertycharacterization that has appeared in book form to date. Although this book is expected to be mostuseful to those engineers actively engaged in designing materials property–characterization methods, itshould also be of considerable value to engineers in other disciplines, such as industrial engineers,bioengineers, and materials scientists, who wish to understand the capabilities and limitations ofmicrowave measurement methods that they use. Meanwhile, this book also satisﬁes the requirement forup-to-date texts at graduate and senior undergraduate levels on the subjects in materials characterization. Among this book’s most outstanding features is its comprehensive coverage. This book discussesalmost all aspects of the microwave theory and techniques for the characterization of the electromagneticproperties of materials at microwave frequencies. In this book, the materials under characterizationmay be dielectrics, semiconductors, conductors, magnetic materials, and artiﬁcial materials; theelectromagnetic properties to be characterized mainly include permittivity, permeability, chirality,mobility, and surface impedance. The two introductory chapters, Chapter 1 and Chapter 2, are intended to acquaint the readers with thebasis for the research and engineering of electromagnetic materials from the materials and microwavefundamentals respectively. As general knowledge of electromagnetic properties of materials is helpfulfor understanding measurement results and correcting possible errors, Chapter 1 introduces the general
xii Prefaceproperties of various electromagnetic materials and their underlying physics. After making a briefreview on the methods for materials properties characterization, Chapter 2 provides a summary ofthe basic microwave theory and techniques, based on which the methods for materials characterizationare developed. This summary is mainly intended for reference rather than for tutorial purposes, althoughsome of the important aspects of microwave theory are treated at a greater length. References are citedto permit readers to further study the topics they are interested in. Chapters 3 to 8 deal with the measurements of the permittivity and permeability of low-conductivitymaterials and the surface impedance of high-conductivity materials. Two types of nonresonant methods,reﬂection method and transmission/reﬂection method, are discussed in Chapters 3 and 4 respectively;two types of resonant methods, resonator method and resonant-perturbation method, are discussed inChapters 5 and 6 respectively. In the methods discussed in Chapters 3 to 6, the transmission lines usedare mainly coaxial-line, waveguide, and free-space, while Chapter 7 is concerned with the measurementmethods developed from planar transmission lines, including stripline, microstrip-, and coplanar line.The methods discussed in Chapters 3 to 7 are suitable for isotropic materials, which have scalar orcomplex permittivity and permeability. The permittivity of anisotropic dielectric materials is a tensorparameter, and magnetic materials usually have tensor permeability under an external dc magnetic ﬁeld.Chapter 8 deals with the measurement of permittivity and permeability tensors. Ferroelectric materials are a special category of dielectric materials often used in microwave electron-ics for developing electrically tunable devices. Chapter 9 discusses the characterization of ferroelectricmaterials, and the topics covered include the techniques for studying the temperature dependence andelectric ﬁeld dependence of dielectric properties. In recent years, the research on artiﬁcial materials has been active. Chapter 10 deals with a specialtype of artiﬁcial materials: chiral materials. After introducing the concept and basic characteristics ofchiral materials, the methods for chirality measurements and the possible applications of chiral materialsare discussed. The electrical transport properties at microwave frequencies are important for the development of high-speed electronic circuits. Chapter 11 discusses the microwave Hall effect techniques for the measurementof the electrical transport properties of low-conductivity, high-conductivity, and magnetic materials. The measurement of materials properties at high temperatures is often required in industry, scientiﬁcresearch, and biological and medical applications. In principle, most of the methods discussed in thisbook can be extended to high-temperature measurements. Chapter 12 concentrates on the measurementof the dielectric properties of materials at high temperatures, and the techniques for solving the problemsin high-temperature measurements can also be applied for the measurement of other materials propertyparameters at high temperatures. In this book, each chapter is written as a self-contained unit, so that readers can quickly getcomprehensive information related to their research interests or on-going projects. To provide a broadtreatment of various topics, we condensed mountains of literature into readable accounts within a text ofreasonable size. Many references have been included for the beneﬁt of the readers who wish to pursuea given topic in greater depth or refer to the original papers. It is clear that the principle of a method for materials characterization is more important thanthe techniques required for implementing this method. If we understand the fundamental principleunderlying a measurement method, we can always ﬁnd a suitable way to realize this method. Althoughthe advances in technology may signiﬁcantly change the techniques for implementing a measurementmethod, they cannot greatly inﬂuence the measurement principle. In writing this book, we tried topresent the fundamental principles behind various designs so that readers can understand the process ofapplying fundamental concepts to arrive at actual designs using different techniques and approaches. Webelieve that an engineer with a sound knowledge of the basic concepts and fundamental principles formaterials property characterization and the ability apply to his knowledge toward design objectives, is
Preface xiiithe engineer who is most likely to make full use of the existing methods, and develop original methodsto fulﬁll ever-rising measurement requirements. We would like to indicate that this text is a compilation of the work of many people. We cannot be heldresponsible for the designs described that are still under patent. It is also difﬁcult to always give propercredits to those who are the originators of new concepts and the inventors of new methods. The names wegive to some measurement methods may not ﬁt the intentions of the inventors or may not accurately reﬂectthe most characteristic features of these methods. We hope that there are not too many such errors and willappreciate it if the readers could bring the errors they discover to our attention. There are many people to whom we owe many thanks for helping us prepare this book. However,space dictates that only a few of them can receive formal acknowledgements. But this should not be takenas a disparagement of those whose contributions remain anonymous. Our foremost appreciation goes toMr. Quek Gim Pew, Deputy Chief Executive (Technology), Singapore Defence Science & TechnologyAgency, Mr. Quek Tong Boon, Chief Executive Ofﬁcer, Singapore DSO National Laboratories, andProfessor Lim Hock, Director, Temasek Laboratories, National University of Singapore, for theirencouragement and support along the way. We are grateful to Pennsylvania State University and HVSTechnologies for giving us permission to include the HVS Free Space Unit and the data in this book.We really appreciate the valuable help and cooperation from Dr. Li Zheng-Wen, Dr. Rao Xuesong, andMr. Tan Chin Yaw. We are very grateful to the staff of John Wiley & Sons for their helpful efforts andcheerful professionalism during this project.L. F. ChenC. K. OngC. P. NeoV. V. VaradanV. K. Varadan
1 Electromagnetic Properties of MaterialsThis chapter starts with the introduction of the the applications of techniques for materials prop-materials research and engineering at microwave erty characterization in various ﬁelds of sciencesfrequencies, with emphasis laid on the signiﬁ- and engineering. The importance of the researchcance and applications of the study of the elec- on the electromagnetic properties of materials attromagnetic properties of materials. The fun- microwave frequencies can be understood in thedamental physics that governs the interactions aspects that follow.between materials and electromagnetic ﬁelds is Firstly, though it is an old ﬁeld in physics,then discussed at both microscopic and macro- the study of electromagnetic properties of mate-scopic scales. Subsequently, we analyze the gen- rials at microwave frequencies is full of academiceral properties of typical electromagnetic materi- importance (Solymar and Walsh 1998; Kittel 1997;als, including dielectric materials, semiconductors, Von Hippel 1995a,b; Jiles 1994; Robert 1988),conductors, magnetic materials, and artiﬁcial mate- especially for magnetic materials (Jiles 1998; Smitrials. Afterward, we discuss the intrinsic proper- 1971) and superconductors (Tinkham 1996) andties and extrinsic performances of electromagnetic ferroelectrics (Lines and Glass 1977). The knowl-materials. edge gained from microwave measurements con- tributes to our information about both the macro-1.1 MATERIALS RESEARCH AND scopic and the microscopic properties of materi-ENGINEERING AT MICROWAVE als, so microwave techniques have been importantFREQUENCIES for materials property research. Though magnetic materials are widely used in various ﬁelds, theWhile technology decides how electromagnetic research of magnetic materials lags far behind theirmaterials can be utilized, science attempts to applications, and this, to some extent, hinders usdecipher why materials behave as they do. The from making full application of magnetic mate-responses of materials to electromagnetic ﬁelds rials. Until now, the electromagnetic propertiesare closely determined by the displacement of of magnetic properties at microwave frequenciestheir free and bounded electrons by electric ﬁelds have not been fully investigated yet, and this isand the orientation of their atomic moments by one of the main obstacles for the development ofmagnetic ﬁelds. The deep understanding and full microwave magnetoelectrics. Besides, one of theutilization of electromagnetic materials have come most promising applications of superconductors isfrom decoding the interactions between materials microwave electronics. A lot of effort has beenand electromagnetic ﬁelds by using both theoretical put in the study of the microwave propertiesand experimental strategies. of superconductors, while many areas are yet to This book mainly deals with the methodology be explored. Meanwhile, as ferroelectric materi-for the characterization of electromagnetic materi- als have great application potential in developingals for microwave electronics, and also discusses smart electromagnetic materials, structures, andMicrowave Electronics: Measurement and Materials Characterization L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan and V. K. Varadan 2004 John Wiley & Sons, Ltd ISBN: 0-470-84492-2
2 Microwave Electronics: Measurement and Materials Characterizationdevices in recent years, microwave ferroelectricity Finally, as the electromagnetic properties ofis under intensive investigation. materials are related to other macroscopic or Secondly, microwave communications are play- microscopic properties of the materials, we caning more and more important roles in military, obtain information about the microscopic orindustrial, and civilian life, and microwave engi- macroscopic properties we are interested in fromneering requires precise knowledge of the elec- the electromagnetic properties of the materials.tromagnetic properties of materials at microwave In materials research and engineering, microwavefrequencies (Ramo et al. 1994). Since World War techniques for the characterization of materialsII, a lot of resources have been put into electromag- properties are widely used in monitoring the fab-netic signature control, and microwave absorbers rication procedure and nondestructive testing ofare widely used in reducing the radar cross sections samples and products (Zoughi 2000; Nyfors and(RCSs) of vehicles. The study of electromagnetic Vainikainen 1989).properties of materials and the ability of tailoring This chapter aims to provide basic knowledgethe electromagnetic properties of composite mate- for understanding the results from microwave mea-rials are very important for the design and devel- surements. We will give a general introductionopment of radar absorbing materials and other on electromagnetic materials at microscopic andfunctional electromagnetic materials and struc- macroscopic scales and will discuss the parameterstures (Knott et al. 1993). describing the electromagnetic properties of mate- Thirdly, as the clock speeds of electronic rials, the classiﬁcation of electromagnetic mate-devices are approaching microwave frequencies, rials, and general properties of typical electro-it becomes indispensable to study the microwave magnetic materials. Further discussions on variouselectronic properties of materials used in elec- topics can be found in later chapters or the refer-tronic components, circuits, and packaging. The ences cited.development of electronic components workingat microwave frequencies needs the electrical 1.2 PHYSICS FOR ELECTROMAGNETICtransport properties at microwave frequencies, MATERIALSsuch as Hall mobility and carrier density; and In physics and materials sciences, electromagneticthe development of electronic circuits work- materials are studied at both the microscopicing at microwave frequencies requires accu- and the macroscopic scale (Von Hippel 1995a,b).rate constitutive properties of materials, such At the microscopic scale, the energy bands foras permittivity and permeability. Meanwhile, the electrons and magnetic moments of the atomselectromagnetic interference (EMI) should be and molecules in materials are investigated, whiletaken into serious consideration in the design of at the macroscopic level, we study the overallcircuit and packaging, and special materials are responses of macroscopic materials to externalneeded to ensure electromagnetic compatibility electromagnetic ﬁelds.(EMC) (Montrose 1999). Fourthly, the study of electromagnetic properties 1.2.1 Microscopic scaleof materials is important for various ﬁelds of sci-ence and technology. The principle of microwave In the microscopic scale, the electrical properties ofremote sensing is based on the reﬂection and a material are mainly determined by the electronscattering of different objects to microwave sig- energy bands of the material. According to thenals, and the reﬂection and scattering proper- energy gap between the valence band and theties of an object are mainly determined by the conduction band, materials can be classiﬁed intoelectromagnetic properties of the object. Besides, insulators, semiconductors, and conductors. Owingthe conclusions of the research of electromag- to its electron spin and electron orbits around thenetic materials are helpful for agriculture, food nucleus, an atom has a magnetic moment. Accordingengineering, medical treatments, and bioengineer- to the responses of magnetic moments to magneticing (Thuery and Grant 1992). ﬁeld, materials can be generally classiﬁed into
Electromagnetic Properties of Materials 3diamagnetic materials, paramagnetic materials, and band. While for some elements, for example carbon,ordered magnetic materials. the merged broadband may further split into separate bands at closer atomic separation. The highest energy band containing occupied126.96.36.199 Electron energy bands energy levels at 0 K in a solid is called the valenceAccording to Bohr’s model, an atom is characterized band. The valence band may be completely ﬁlledby its discrete energy levels. When atoms are or only partially ﬁlled with electrons. The electronsbrought together to constitute a solid, the discrete in the valence band are bonded to their nuclei.levels combine to form energy bands and the The conduction band is the energy band above theoccupancy of electrons in a band is dictated valence energy band, and contains vacant energyby Fermi-dirac statistics. Figure 1.1 shows the levels at 0 K. The electrons in the conduction bandrelationship between energy bands and atomic are called free electrons, which are free to move.separation. When the atoms get closer, the energy Usually, there is a forbidden gap between thebands broaden, and usually the outer band broadens valence band and the conduction band, and themore than the inner one. For some elements, for availability of free electrons in the conduction bandexample lithium, when the atomic separation is mainly depends on the forbidden gap energy. If thereduced, the bands may broaden sufﬁciently for forbidden gap is large, it is possible that no freeneighboring bands to merge, forming a broader electrons are available, and such a material is called an insulator. For a material with a small forbidden energy gap, the availability of free electron in the conduction band permits some electron conduction, and such a material is a semiconductor. In a [Image not available in this electronic edition.] conductor, the conduction and valence bands may overlap, permitting abundant free electrons to be available at any ambient temperature, thus giving high electrical conductivity. The energy bands for insulator, semiconductor, and good conductor are shown schematically in Figure 1.2.Figure 1.1 The relationships between energy bandsand atomic separation. (a) Energy bands of lithium Insulatorsand (b) energy bands of carbon. (Bolton 1992) Source:Bolton, W. (1992), Electrical and Magnetic Properties For most of the insulators, the forbidden gapof Materials, Longman Scientiﬁc & Technical, Harlow between their valence and conduction energy bands [Image not available in this electronic edition.]Figure 1.2 Energy bands for different types of materials. (a) Insulator, (b) semiconductor, and (c) goodconductor. (Bolton 1992). Modiﬁed from Bolton, W. (1992), Electrical and Magnetic Properties of Materials,Longman Scientiﬁc & Technical, Harlow
4 Microwave Electronics: Measurement and Materials Characterizationis larger than 5 eV. Usually, we assume that an conductor, the density of free electrons is on theinsulate is nonmagnetic, and under this assump- order of 1028 m3 . Lithium is a typical example oftion, insulators are called dielectrics. Diamond, a a conductor. It has two electrons in the 1s shellform of carbon, is a typical example of a dielectric. and one in the 2s shell. The energy bands of suchCarbon has two electrons in the 1s shell, two in the elements are of the form shown in Figure 1.1(a).2s shell, and two in the 2p shell. In a diamond, The 2s and 2p bands merge, forming a largethe bonding between carbon atoms is achieved band that is only partially occupied, and under anby covalent bonds with electrons shared between electric ﬁeld, electrons can easily move into vacantneighboring atoms, and each atom has a share in energy levels.eight 2p electrons (Bolton 1992). So all the elec- In the category of conductors, superconduc-trons are tightly held between the atoms by this tors have attracted much research interest. In acovalent bonding. As shown in Figure 1.1(b), the normal conductor, individual electrons are scat-consequence of this bonding is that diamond has tered by impurities and phonons. However, fora full valence band with a substantial forbidden superconductors, the electrons are paired withgap between the valence band and the conduc- those of opposite spins and opposite wave vec-tion band. But it should be noted that, graphite, tors, forming Cooper pairs, which are bondedanother form of carbon, is not a dielectric, but a together by exchanging phonons. In the Bardeen–conductor. This is because all the electrons in the Cooper–Schrieffer (BCS) theory, these Coopergraphite structure are not locked up in covalent pairs are not scattered by the normal mechanisms.bonds and some of them are available for conduc- A superconducting gap is found in superconduc-tion. So the energy bands are related to not only the tors and the size of the gap is in the microwaveatom structures but also the ways in which atoms frequency range, so study of superconductors atare combined. microwave frequencies is important for the under- standing of superconductivity and application of superconductors.SemiconductorsThe energy gap between the valence and conduction 188.8.131.52 Magnetic momentsbands of a semiconductor is about 1 eV. Germaniumand silicon are typical examples of semiconductors. An electron orbiting a nucleus is equivalent to aEach germanium or silicon atom has four valence current in a single-turn coil, so an atom has aelectrons, and the atoms are held together by magnetic dipole moment. Meanwhile, an electroncovalent bonds. Each atom shares electrons with also spins. By considering the electron to be aeach of four neighbors, so all the electrons are small charged sphere, the rotation of the chargelocked up in bonds. So there is a gap between a full on the surface of the sphere is also like a single-valence band and the conduction band. However, turn current loop and also produces a magneticunlike insulators, the gap is relatively small. At moment (Bolton 1992). The magnetic properties ofroom temperature, some of the valence electrons a material are mainly determined by its magneticcan break free from the bonds and have sufﬁcient moments that result from the orbiting and spinningenergy to jump over the forbidden gap, arriving of electrons. According to the responses of theat the conduction band. The density of the free magnetic moments of the atoms in a material to anelectrons for most of the semiconductors is in the external magnetic ﬁeld, materials can be generallyrange of 1016 to 1019 per m3 . classiﬁed into diamagnetic, paramagnetic, and ordered magnetic materials.Conductors Diamagnetic materialsFor a conductor, there is no energy gap between The electrons in a diamagnetic material are all pairedthe valence gap and conduction band. For a good up with spins antiparallel, so there is no net magnetic
Electromagnetic Properties of Materials 5moment on their atoms. When an external magneticﬁeld is applied, the orbits of the electrons change,resulting in a net magnetic moment in the directionopposite to the applied magnetic ﬁeld. It should benoted that all materials have diamagnetism since allmaterials have orbiting electrons. However, for dia-magnetic materials, the spin of the electrons does [Image not available in this electronic edition.]not contribute to the magnetism; while for param-agnetic and ferromagnetic materials, the effects ofthe magnetic dipole moments that result from thespinning of electrons are much greater than the dia-magnetic effect.Paramagnetic materials Figure 1.3 Arrangements of magnetic moments inThe atoms in a paramagnetic material have net various magnetic materials. (a) Paramagnetic, (b) ferro-magnetic moments due to the unpaired electron magnetic, (c) antiferromagnetic, and (d) ferrimagneticspinning in the atoms. When there is no exter- materials. Modiﬁed from Bolton, W. (1992). Electricalnal magnetic ﬁeld, these individual moments are and Magnetic Properties of Materials, Longman Scien-randomly aligned, so the material does not show tiﬁc & Technical, Harlowmacroscopic magnetism. When an external mag-netic ﬁeld is applied, the magnetic moments are ordered arrangement of magnetic dipoles shown inslightly aligned along the direction of the exter- Figure 1.3(b), is quite different from the couplingnal magnetic ﬁeld. If the applied magnetic ﬁeld between atoms of paramagnetic materials, whichis removed, the alignment vanishes immediately. results in the random arrangement of magneticSo a paramagnetic material is weakly magnetic dipoles shown in Figure 1.3(a). Iron, cobalt, andonly in the presence of an external magnetic nickel are typical ferromagnetic materials.ﬁeld. The arrangement of magnetic moments in a As shown in Figure 1.3(c), in an antiferromag-paramagnetic material is shown in Figure 1.3(a). netic material, half of the magnetic dipoles alignAluminum and platinum are typical paramag- themselves in one direction and the other half ofnetic materials. the magnetic moments align themselves in exactly the opposite direction if the dipoles are of theOrdered magnetic materials same size and cancel each other out. Manganese, manganese oxide, and chromium are typical anti-In ordered magnetic materials, the magnetic mo- ferromagnetic materials. However, as shown inments are arranged in certain orders. According to Figure 1.3(d), for a ferrimagnetic material, alsothe ways in which magnetic moments are arranged, called ferrite, the magnetic dipoles have differentordered magnetic materials fall into several subcat- sizes and they do not cancel each other. Magnetiteegories, mainly including ferromagnetic, antiferro- (Fe3 O4 ), nickel ferrite (NiFe2 O4 ), and barium fer-magnetic, and ferrimagnetic (Bolton 1992; Wohl- rite (BaFe12 O19 ) are typical ferrites.farth 1980). Figure 1.3 shows the arrangements Generally speaking, the dipoles in a ferromag-of magnetic moments in paramagnetic, ferromag- netic or ferrimagnetic material may not all benetic, antiferromagnetic, and ferrimagnetic materi- arranged in the same direction. Within a domain, allals, respectively. the dipoles are arranged in its easy-magnetization As shown in Figure 1.3(b), the atoms in a direction, but different domains may have differ-ferromagnetic material are bonded together in such ent directions of arrangement. Owing to the randoma way that the dipoles in neighboring atoms are all orientations of the domains, the material does notin the same direction. The coupling between atoms have macroscopic magnetism without an externalof ferromagnetic materials, which results in the magnetic ﬁeld.
6 Microwave Electronics: Measurement and Materials Characterization The crystalline imperfections in a magnetic walls are pinned by crystalline imperfections. Asmaterial have signiﬁcant effects on the magneti- shown in Figure 1.4(b), when an external magneticzation of the material (Robert 1988). For an ideal ﬁeld H is applied, the domains whose orienta-magnetic material, for example monocrystalline tions are near the direction of the external mag-iron without any imperfections, when a magnetic netic ﬁeld grow in size, while the sizes of theﬁeld H is applied, due to the condition of minimum neighboring domains wrongly directed decrease.energy, the sizes of the domains in H direction When the magnetic ﬁeld is very weak, the domainincrease, while the sizes of other domains decrease. walls behave like elastic membranes, and theAlong with the increase of the magnetic ﬁeld, the changes of the domains are reversible. Whenstructures of the domains change successively, and the magnetic ﬁeld increases, the pressure on theﬁnally a single domain in H direction is obtained. domain walls causes the pinning points to giveIn this ideal case, the displacement of domain walls way, and the domain walls move by a series ofis free. When the magnetic ﬁeld H is removed, the jumps. Once a jump of domain wall happens,material returns to its initial state; so the magneti- the magnetization process becomes irreversible. Aszation process is reversible. shown in Figure 1.4(c), when the magnetic ﬁeld H Owing to the inevitable crystalline imperfec- reaches a certain level, all the magnetic momentstions, the magnetization process becomes com- are arranged parallel to the easy magnetizationplicated. Figure 1.4(a) shows the arrangement of direction nearest to the direction of the externaldomains in a ferromagnetic material when no magnetic ﬁeld H. If the external magnetic ﬁeldexternal magnetic ﬁeld is applied. The domain H increases further, the magnetic moments are aligned along H direction, deviating from the easy magnetization direction, as shown in Figure 1.4(d). In this state, the material shows its greatest magne- tization, and the material is magnetically saturated. In a polycrystalline magnetic material, the mag- H netization process in each grain is similar to that in a monocrystalline material as discussed above. However, due to the magnetostatic and magne- tostrictions occurring between neighboring grains, (a) (b) the overall magnetization of the material becomes quite complicated. The grain structures are impor- tant to the overall magnetization of a polycrys- talline magnetic material. The magnetization pro- cess of magnetic materials is further discussed in H H Section 184.108.40.206. It is important to note that for an ordered magnetic material, there is a special temperature called Curie temperature (Tc ). If the temperature is below the Curie temperature, the material is in a magnetically (c) (d) ordered phase. If the temperature is higher thanFigure 1.4 Domains in a ferromagnetic material. the Curie temperature, the material will be in a(a) Arrangement of domains when no external mag- paramagnetic phase. The Curie temperature for ironnetic ﬁeld is applied, (b) arrangement of domains when is 770 ◦ C, for nickel 358 ◦ C, and for cobalt 1115 ◦ C.a weak magnetic ﬁeld is applied, (c) arrangement ofdomains when a medium magnetic ﬁeld is applied,and (d) arrangement of domains when a strong mag- 1.2.2 Macroscopic scalenetic ﬁeld is applied. Modiﬁed from Robert, P. (1988).Electrical and Magnetic Properties of Materials, Artech The interactions between a macroscopic mate-House, Norwood rial and electromagnetic ﬁelds can be generally
Electromagnetic Properties of Materials 7described by Maxwell’s equations: I = jC0wU I ∇·D =ρ (1.1) ∇·B =0 (1.2) C0 U = U0exp(jwt) 90° ∇ × H = ∂D/∂t + J (1.3) ∇ × E = −∂B /∂t (1.4)with the following constitutive relations: U (a) (b) D = εE = (ε − jε )E (1.5) Figure 1.5 The current in a circuit with a capacitor. B = µH = (µ − jµ )H (1.6) (a) Circuit layout and (b) complex plane showing cur- rent and voltage J = σE (1.7)where H is the magnetic ﬁeld strength vector; E, properties of low-conductivity materials. As thethe electric ﬁeld strength vector; B, the magnetic value of conductivity σ is small, we concentrateﬂux density vector; D, the electric displacement on permittivity and permeability. In a general case,vector; J, the current density vector; ρ, the charge both permittivity and permeability are complexdensity; ε = ε − jε , the complex permittivity of numbers, and the imaginary part of permittivitythe material; µ = µ − jµ , the complex perme- is related to the conductivity of the material. Inability of the material; and σ , the conductivity of the following discussion, we analogize microwavethe material. Equations (1.1) to (1.7) indicate that signals to ac signals, and distributed capacitor andthe responses of an electromagnetic material to inductor to lumped capacitor and inductor (Vonelectromagnetic ﬁelds are determined essentially Hippel 1995b).by three constitutive parameters, namely permit- Consider the circuit shown in Figure 1.5(a). Thetivity ε, permeability µ, and conductivity σ . These vacuum capacitor with capacitance C0 is connectedparameters also determine the spatial extent to to an ac voltage source U = U0 exp(jωt). Thewhich the electromagnetic ﬁeld can penetrate into charge storage in the capacitor is Q = C0 U , andthe material at a given frequency. the current I ﬂowing in the circuit is In the following, we discuss the parameters dQ ddescribing two general categories of materials: I= = (C0 U0 ejωt ) = jC0 ωU (1.8)low-conductivity materials and high-conductivity dt dtmaterials. So, in the complex plane shown in Figure 1.5(b), the current I leads the voltage U by a phase angle220.127.116.11 Parameters describing low-conductivity of 90◦ .materials Now, we insert a dielectric material into the capacitor and the equivalent circuit is shown inElectromagnetic waves can propagate in a low- Figure 1.6(a). The total current consists of two parts,conductivity material, so both the surface and inner the charging current (Ic ) and loss current (Il ):parts of the material respond to the electromag-netic wave. There are two types of parameters I = Ic + Il = jCωU + GU = (jCω + G)Udescribing the electromagnetic properties of low- (1.9)conductivity materials: constitutive parameters and where C is the capacitance of the capacitorpropagation parameters. loaded with the dielectric material and G is the conductance of the dielectric material. The lossConstitutive parameters current is in phase with the source voltage U . In the complex plane shown in Figure 1.6(b), theThe constitutive parameters deﬁned in Eqs. (1.5) to charging current Ic leads the loss current Il by a(1.7) are often used to describe the electromagnetic phase angle of 90◦ , and the total current I leads
8 Microwave Electronics: Measurement and Materials Characterization Ic = jwCU I Jc = jwe′E J I Ic Il U C G d d q q Il = GU (a) (b) Jl = we′′EFigure 1.6 The relationships between charging currentand loss current. (a) Equivalent circuit and (b) complex Figure 1.7 Complex plane showing the charging cur-plane showing charging current and loss current rent density and loss current densitythe source voltage U with an angle θ less than 90◦ . and the dielectric power factor is given byThe phase angle between Ic and I is often called cos θe = ε / (ε )2 + (ε )2 (1.14)loss angle δ. We may alternatively use complex permittiv- Equations (1.13) and (1.14) show that for aity ε = ε − jε to describe the effect of dielec- small loss angle δe , cos θ ≈ tan δe .tric material. After a dielectric material is inserted In microwave electronics, we often use relativeinto the capacitor, the capacitance C of the capaci- permittivity, which is a dimensionless quantity,tor becomes deﬁned by εC0 C0 ε ε − jε C= = (ε − jε ) (1.10) εr = = = εr − jεr = εr (1 − j tan δe ) ε0 ε0 ε0 ε0And the charging current is (1.15) where ε is complex permittivity, C0 C0 εr is relative complex permittivity, I = jω(ε − jε ) U = (jωε + ωε ) U ε0 ε0 ε0 = 8.854 × 10−12 F/m is the (1.11) permittivity of free space,Therefore, as shown in Figure 1.6, the current εr is the real part of relative complexdensity J transverse to the capacitor under the permittivity,applied ﬁeld strength E becomes εr is the imaginary part of relative dE complex permittivity, J = (jωε + ωε )E = ε (1.12) dt tan δe is dielectric loss tangent, andThe product of angular frequency and loss factor δe is dielectric loss angle.is equivalent to a dielectric conductivity: σ = ωε . Now, let us consider the magnetic responseThis dielectric conductivity sums over all the dis- of low-conductivity material. According to thesipative effects of the material. It may represent Faraday’s inductance lawan actual conductivity caused by migrating chargecarriers and it may also refer to an energy loss asso- dI U =L , (1.16)ciated with the dispersion of ε , for example, the dtfriction accompanying the orientation of dipoles. we can get the magnetization current Im :The latter part of dielectric conductivity will bediscussed in detail in Section 1.3.1. U Im = −j (1.17) According to Figure 1.7, we deﬁne two parame- ωL0ters describing the energy dissipation of a dielectric where U is the magnetization voltage, L0 ismaterial. The dielectric loss tangent is given by the inductance of an empty inductor, and ω tan δe = ε /ε , (1.13) is the angular frequency. If we introduce an
Electromagnetic Properties of Materials 9 Il given by U q µ µ − jµ µr = = 90° d µ0 µ0 = µr − jµr = µr (1 − j tan δm ) (1.22) Im = −j U where µ is complex complex permeability, w L0 m′r Im I µr is relative complex permeability, (a) (b) µ0 = 4π × 10−7 H/m is the permeabilityFigure 1.8 The magnetization current in a complex of free space,plane. (a) Relationship between magnetization current µr is the real part of relative complexand voltage and (b) relationship between magnetization permeability,current and loss current µr is the imaginary part of the relative complex permeability,ideal, lossless magnetic material with relative tan δm is the magnetic loss tangent, andpermeability µr , the magnetization ﬁeld becomes δm is the magnetic loss angle. U In summary, the macroscopic electric and mag- Im = −j (1.18) netic behavior of a low-conductivity material is ωL0 µr mainly determined by the two complex parame- In the complex plane shown in Figure 1.8(a), ters: permittivity (ε) and permeability (µ). Per-the magnetization current Im lags the voltage mittivity describes the interaction of a materialU by 90◦ for no loss of magnetic materials. with the electric ﬁeld applied on it, while per-As shown in Figure 1.8(b), an actual magnetic meability describes the interaction of a materialmaterial has magnetic loss, and the magnetic loss with magnetic ﬁeld applied on it. Both the elec-current Il caused by energy dissipation during tric and magnetic ﬁelds interact with materials inthe magnetization cycle is in phase with U. By two ways: energy storage and energy dissipation.introducing a complex permeability µ = µ − jµ Energy storage describes the lossless portion ofand a complex relative permeability µr = µr − jµr the exchange of energy between the ﬁeld and thein complete analogy to the dielectric case, we material, and energy dissipation occurs when elec-obtain the total magnetization current tromagnetic energy is absorbed by the material. So both permittivity and permeability are expressed as U jU (µ + jµ ) complex numbers to describe the storage (real part)I = Im + Il = =− jωL0 µr ω(L0 /µ0 )(µ 2 + µ 2 ) and dissipation (imaginary part) effects of each. (1.19) Besides the permittivity and permeability, another parameter, quality factor, is often used to describe Similar to the dielectric case, according to an electromagnetic material:Figure 1.8, we can also deﬁne two parameters εr 1describing magnetic materials: the magnetic loss Qe = = (1.23)tangent given by εr tan δe µr 1 tan δm = µ /µ , (1.20) Qm = = (1.24) µr tan δmand the power factor given by On the basis of the dielectric quality factor Qe and magnetic quality factor Qm , we can get the cos θm = µ / (µ )2 + (µ )2 . (1.21) total quality factor Q of the material: In microwave electronics, relative permeability 1 1 1is often used, which is a dimensionless quantity = + (1.25) Q Qe Qm
10 Microwave Electronics: Measurement and Materials CharacterizationPropagation parameters For a high-conductivity material, we assume σ ωε, which means that the conducting current isThe propagation of electromagnetic waves in a much larger than the displacement current. So,medium is determined by the characteristic wave Eq. (1.29) can be approximated by ignoring theimpedance η of the medium and the wave veloc- displacement current term:ity v in the medium. The characteristic waveimpedance η is also called the intrinsic impedance √ σ ωµσ γ = α + jβ = jω µε = (1 + j)of the medium. When a single wave propagates jωε 2with velocity v in the Z-positive direction, the (1.30)characteristic impedance η is deﬁned as the ratio We deﬁne the skin depth:of total electric ﬁeld to total magnetic ﬁeld at a 1 2Z-plane. The wave impedance and velocity can be δs = = (1.31) α ωµσcalculated from the permittivity and permeabilityof the medium: The physics meaning of skin depth is that, in a µ high-conductivity material, the ﬁelds decay by an η= (1.26) amount e−1 in a distance of a skin depth δs . At ε microwave frequencies, the skin depth δs is a very 1 small distance. For example, the skin depth of a v=√ (1.27) µε metal at microwave frequencies is usually on the order of 10−7 m. From Eqs. (1.26) and (1.27), we can calculate the Because of the skin effect, the utility and behav-wave impedance of free space, η0 = (µ0 /ε0 )1/2 ior of high-conductivity materials at microwave= 376.7 , and the wave velocity in free space, c = frequencies are mainly determined by their surface(µ0 ε0 )−1/2 = 2.998 × 108 m/s. Expressing permit- impedance Zs :tivity and permeability as complex quantities leadsto a complex number for the wave velocity (v), Et µω Zs = Rs + jXs = = (1 + j) (1.32)where the imaginary portion is a mathematical con- Ht 2σvenience for expressing loss. where Ht is the tangential magnetic ﬁeld, Et Sometimes, it is more convenient to use the is the tangential electric ﬁeld, Rs is the surfacecomplex propagation coefﬁcient γ to describe resistance, and Xs is the surface reactance. Forthe propagation of electromagnetic waves in normal conductors, σ is a real number. Accordinga medium: to Eq. (1.32), the surface resistance Rs and the √ ω√ ω surface reactance Xs are equal and they are γ = α + jβ = jω µε = j µ r εr = j n c c proportional to ω1/2 for normal metals: (1.28)where n is the complex index of refraction, where µω Rs = Xs = (1.33)ω is the angular frequency, α is the attenua- 2σtion coefﬁcient, β = 2π/λ is the phase changecoefﬁcient, and λ is the operating wavelength in 18.104.22.168 Classiﬁcation of electromagnetic materialsthe medium. Materials can be classiﬁed according to their macroscopic parameters. According to conductiv-22.214.171.124 Parameters describing high-conductivity ity, materials can be classiﬁed as insulators, semi-materials conductors, and conductors. Meanwhile, materials can also be classiﬁed according to their perme-For a high-conductivity material, for example a ability values. General properties of typical typesmetal, Eq. (1.28) for the complex propagation of materials are discussed in Section 1.3.constant γ should be modiﬁed as When classifying materials according to their √ σ macroscopic parameters, it should be noted that we γ = α + jβ = jω µε 1 − j (1.29) use the terms insulator, semiconductor, conductor, ωε
Electromagnetic Properties of Materials 11and magnetic material to indicate the dominant materials. According to their permeability values,responses of different types of materials. All materials generally fall into three categories: dia-materials have some response to magnetic ﬁelds magnetic (µ < µ0 ), paramagnetic (µ ≥ µ0 ), andbut, except for ferromagnetic and ferrimagnetic highly magnetic materials mainly including ferro-types, their responses are usually very small, magnetic and ferrimagnetic materials. The perme-and their permeability values differ from µ0 by ability values of highly magnetic materials, espe-a negligible fraction. Most of the ferromagnetic cially ferromagnetic materials, are much largermaterials are highly conductive, but we call them than µ0 .magnetic materials, as their magnetic propertiesare the most signiﬁcant in their applications. Forsuperconductors, the Meissner effect shows that 1.3 GENERAL PROPERTIESthey are a kind of very special magnetic materials, OF ELECTROMAGNETIC MATERIALSbut in microwave electronics, people are more Here, we discuss the general properties of typi-interested in their surface impedance. cal electromagnetic materials, including dielectric materials, semiconductors, conductors, magneticInsulators materials, and artiﬁcial materials. The knowledgeInsulators have very low conductivity, usually in the of general properties of electromagnetic materi-range of 10−12 to 10−20 ( m)−1 . Often, we assume als is helpful for understanding the measurementinsulators are nonmagnetic, so they are actually results and correcting the possible errors one maydielectrics. In theoretical analysis of dielectric meet in materials characterization. In the ﬁnal partmaterials, an ideal model, perfect dielectric, is often of this section, we will discuss other descriptionsused, representing a material whose imaginary part of electromagnetic materials, which are importantof permittivity is assumed to be zero: ε = 0. for the design and applications of electromag- netic materials.SemiconductorsThe conductivity of a semiconductor is higher 1.3.1 Dielectric materialsthan that of a dielectric but lower than thatof a conductor. Usually, the conductivities of Figure 1.9 qualitatively shows a typical behaviorsemiconductors at room temperature are in the of permittivity (ε and ε ) as a function of fre-range of 10−7 to 104 ( m)−1 . quency. The permittivity of a material is related to a variety of physical phenomena. Ionic conduction,Conductors dipolar relaxation, atomic polarization, and elec- tronic polarization are the main mechanisms thatConductors have very high conductivity, usually contribute to the permittivity of a dielectric mate-in the range of 104 to 108 ( m)−1 . Metals are rial. In the low frequency range, ε is dominatedtypical conductors. There are two types of special by the inﬂuence of ion conductivity. The variationconductors: perfect conductors and superconduc- of permittivity in the microwave range is mainlytors. A perfect conductor is a theoretical model caused by dipolar relaxation, and the absorptionthat has inﬁnite conductivity at any frequencies. peaks in the infrared region and above is mainlySuperconductors have very special electromagnetic due to atomic and electronic polarizations.properties. For dc electric ﬁelds, their conductivityis virtually inﬁnite; but for high-frequency electro-magnetic ﬁelds, they have complex conductivities. 126.96.36.199 Electronic and atomic polarizations Electronic polarization occurs in neutral atomsMagnetic materials when an electric ﬁeld displaces the nucleus withAll materials respond to external magnetic ﬁelds, respect to the surrounding electrons. Atomic polar-so in a broad sense, all materials are magnetic ization occurs when adjacent positive and negative
12 Microwave Electronics: Measurement and Materials Characterization e′ Dipolar and related relaxation phenomena Atomic Electronic e′′ 0 103 106 109 1012 1015 Microwaves Millimeter Infrared Visible Ultraviolet waves Frequency (Hz)Figure 1.9 Frequency dependence of permittivity for a hypothetical dielectric (Ramo et al. 1994). Source:Ramo, S. Whinnery, J. R and Van Duzer, T. (1994). Fields and Waves in Communication Electronics, 3rd edition,John Wiley & Sons, Inc., New York er B are present, the materials are almost lossless at A+ 2a microwave frequencies. ′ er 2B e In the following discussion, we focus on elec-A+w B 0 a tronic polarization, and the conclusions for elec- A tronic polarization can be extended to atomic polarization. When an external electric ﬁeld is ′′ er applied to neutral atoms, the electron cloud of the atoms will be distorted, resulting in the electronic A−B polarization. In a classical model, it is similar to a 2a w0 − a w0 + a spring-mass resonant system. Owing to the small mass of the electron cloud, the resonant frequency 0 w0 w of electronic polarization is at the infrared regionFigure 1.10 The behavior of permittivity due to elec- or the visible light region. Usually, there are sev-tronic or atomic polarization. Reprinted with permis- eral different resonant frequencies correspondingsion from Industrial Microwave Sensors, by Nyfors, E. to different electron orbits and other quantum-and Vainikainen, P., Artech House Inc., Norwood, MA, mechanical effects. For a material with s differentUSA, www.artechhouse.com oscillators, its permittivity is given by (Nyfors and Vainikainen 1989)ions stretch under an applied electric ﬁeld. Actu-ally, electronic and atomic polarizations are of sim- (ns e2 )/(ε0 ms ) εr = 1 + (1.34)ilar nature. Figure 1.10 shows the behavior of per- s ωs − ω2 + jω2 αs 2mittivity in the vicinity of the resonant frequencyω0 . In the ﬁgure, A is the contribution of higher where ns is the number of electrons per volume withresonance to εr at the present frequency range, and resonant frequency ωs , e is the charge of electron,2B/ω0 is the contribution of the present resonance ms is the mass of electron, ω is the operating angularto lower frequencies. For many dry solids, these frequency, and αs is the damping factor.are the dominant polarization mechanisms deter- As microwave frequencies are far below themining the permittivity at microwave frequencies, lowest resonant frequency of electronic polariza-although the actual resonance occurs at a much tion, the permittivity due to electronic polariza-higher frequency. If only these two polarizations tion is almost independent of the frequency and
Electromagnetic Properties of Materials 13temperature (Nyfors and Vainikainen 1989): tan d ′ er Ns e 2 εr = 1 + 2 (1.35) tan d s ε0 ms ωs er0 Eq. (1.35) indicates that the permittivity εr is a ′ erreal number. However, in actual materials, small er∞and constant losses are often associated with thistype of polarization in the microwave range. wmax w Figure 1.11 The frequency dependence of the com-188.8.131.52 Dipolar polarization plex permittivity according to the Debye relation (Robert 1988). Reprinted with permission from Electrical andIn spite of their different origins, various types Magnetic Properties of Materials by Robert, P., Artechof polarizations at microwave and millimeter-wave House Inc., Norwood, MA, USA, www.artechhouse.comranges can be described in a similar qualitative way.In most cases, the Debye equations can be applied, is inversely proportional to temperature as all thealthough they were ﬁrstly derived for the special movements become faster at higher temperatures.case of dipolar relaxation. According to Debye From Eq. (1.36), we can get the real and imag-theory, the complex permittivity of a dielectric can inary parts of the permittivity and the dielectricbe expressed as (Robert 1988) loss tangent: εr0 − εr∞ εr0 − εr∞ εr = εr∞ + (1.40) εr = εr∞ + (1.36) 1 + β2 1 + jβ εr0 − εr∞ εr = β (1.41)with 1 + β2 εr∞ = limω→∞ εr (1.37) εr0 − εr∞ tan δe = β (1.42) εr0 = limω→0 εr (1.38) εr0 + εr∞ β 2 εr0 + 2 Figure 1.11 shows the variation of complex per- β= ωτ (1.39) mittivity as a function of frequency. At the frequency εr∞ + 2 1 εr0 εr∞ + 2where τ is the relaxation time and ω is the oper- ωmax = · · , (1.43)ating angular frequency. Equation (1.36) indicates τ εr∞ εr0 + 2that the dielectric permittivity due to Debye relax- the dielectric loss tangent reaches its maximumation is mainly determined by three parameters, εr0 , value (Robert 1988)εr∞ , and τ . At sufﬁciently high frequencies, as the 1 εr0 − εr∞period of electric ﬁeld E is much smaller than the tan δmax = · √ (1.44) 2 εr0 εr∞relaxation time of the permanent dipoles, the orien-tations of the dipoles are not inﬂuenced by electric The permittivity as a function of frequency isﬁeld E and remain random, so the permittivity at often presented as a two-dimensional diagram,inﬁnite frequency εr∞ is a real number. As ε∞ is Cole–Cole diagram. We rewrite Eq. (1.36) asmainly due to electronic and atomic polarization, εr0 − εr∞it is independent of the temperature. As at sufﬁ- εr − εr∞ − jεr = (1.45) 1 + jβciently low frequencies there is no phase difference As the moduli of both sides of Eq. (1.45) shouldbetween the polarization P and electric ﬁeld E, εr0 be equal, we haveis a real number. But the static permittivity εr0decreases with increasing temperature because of (εr0 − εr∞ )2 (εr − εr∞ )2 + (εr )2 = (1.46)the increasing disorder, and the relaxation time τ 1 + β2
14 Microwave Electronics: Measurement and Materials Characterization After eliminating the term β 2 using Eq. (1.40), water, the material exhibits a distribution of relax-we get (Robert 1988) ation frequencies. Often an empirical constant, a, is introduced and Eq. (1.36) is modiﬁed into the (εr − εr∞ )2 + (εr )2 = (εr − εr∞ )(εr0 − εr∞ ) following form (Robert 1988): (1.47)Eq. (1.47) represents a circle with its center on εr0 − εr∞ εr = εr∞ + (1.50)the εr axis. Only the points at the top half 1 + (jβa )1−aof this circle have physical meaning as all the where a is related to the distribution of β values,materials have nonnegative value of imaginary part and βa denotes the most possible β value. Theof permittivity. The top half of the circle is called constant a is in the range 0 a < 1. When a = 0,Cole–Cole diagram, as shown in Figure 1.12. Eq. (1.50) becomes Eq. (1.36), and in this case, The relaxation time τ can be determined from there is only single relaxation time. When the valuethe Cole–Cole diagram. According to Eqs. (1.40) of a increases, the relaxation time is distributedand (1.41), we can get over a broader range. εr = β(εr − εr∞ ) (1.48) If we separate the real and imaginary parts of Eq. (1.50) and then eliminate βa , we can ﬁnd that εr = −(1/β)(εr − εr0 ) (1.49) the εr (εr ) curve is also a circle passing through the points εr0 and εr∞ , as shown in Figure 1.13. As shown in Figure 1.12, for a given operating The center of the circle is below the εr axis with afrequency, the β value can be obtained from distance d given bythe slope of a line pass through the pointcorresponding to the operating frequency and the εr0 − εr∞point corresponding to εr0 or εr∞ . After obtaining d= tan θ (1.51) 2the β value, the relaxation time τ can be calculatedfrom β according to Eq. (1.39). where θ is the angle between the εr axis and the In some cases, the relaxation phenomenon line connecting the circle center and the point εr∞ :may be caused by different sources, and the πdielectric material has a relaxation-time spectrum. θ =a (1.52) 2For example, a moist material contains watermolecules bound with different strength. Depend- Similar to Figure 1.12, only the points above the εring on the moisture and the strength of binding axis have physical meaning. Equations (1.51) and (1.52) indicate that the empirical constant a can be calculated from the value of d or θ . ′′er b=1 Slope − 1/b ′′ er Slope b b=∞ er∞ er0 b=0 q ′ er d er∞ er∞ + er0 er0 ′ er 2Figure 1.12 The Cole–Cole presentation for a single Figure 1.13 Cole–Cole diagram for a relaxation-timerelaxation time (Robert 1988). Reprinted with permis- spectrum. Reprinted with permission from Electricalsion from Electrical and Magnetic Properties of Mate- and Magnetic Properties of Materials by Robert, P.,rials by Robert, P., Artech House Inc., Norwood, MA, Artech House Inc., Norwood, MA, USA,USA, www.artechhouse.com www.artechhouse.com
Electromagnetic Properties of Materials 184.108.40.206 Ionic conductivity response of polarization versus electric ﬁeld is non- linear. As shown in Figure 1.14(b), ferroelectricUsually, ionic conductivity only introduces losses materials display a hysteresis effect of polarizationinto a material. As discussed earlier, the dielectric with an applied ﬁeld. The hysteresis loop is causedloss of a material can be expressed as a function by the existence of permanent electric dipoles inof both dielectric loss (εrd ) and conductivity (σ ): the material. When the external electric ﬁeld is ini- σ tially increased from the point 0, the polarization εr = εrd + (1.53) increases as more of the dipoles are lined up. When ωε0 the ﬁeld is strong enough, all dipoles are lined upThe overall conductivity of a material may con- with the ﬁeld, so the material is in a saturationsist of many components due to different conduc- state. If the applied electric ﬁeld decreases from thetion mechanisms, and ionic conductivity is usually saturation point, the polarization also decreases.the most common one in moist materials. At low However, when the external electric ﬁeld reachesfrequencies, εr is dominated by the inﬂuence of zero, the polarization does not reach zero. Theelectrolytic conduction caused by free ions in the polarization at zero ﬁeld is called the remanentpresence of a solvent, for example water. As indi- polarization. When the direction of the electriccated by Eq. (1.53), the effect of ionic conductivity ﬁeld is reversed, the polarization decreases. Whenis inversely proportional to operating frequency. the reversed ﬁeld reaches a certain value, called the coercive ﬁeld, the polarization becomes zero.220.127.116.11 Ferroelectricity By further increasing the ﬁeld in this reverse direc- tion, the reverse saturation can be reached. WhenMost of the dielectric materials are paraelectric. the ﬁeld is decreased from the saturation point, theAs shown in Figure 1.14(a), the polarization of sequence just reverses itself.a paraelectric material is linear. Besides, the For a ferroelectric material, there exists a par-ions in paraelectric materials return to their ticular temperature called the Curie temperature.original positions once the external electric ﬁeld is Ferroelectricity can be maintained only below theremoved; so the ionic displacements in paraelectric Curie temperature. When the temperature is highermaterials are reversible. than the Curie temperature, a ferroelectric material Ferroelectric materials are a subgroup of pyro- is in its paraelectric state.electric materials that are a subgroup of piezo- Ferroelectric materials are very interesting sci-electric materials. For ferroelectric materials, the entiﬁcally. There are rich physics phenomena near [Image not available in this electronic edition.]Figure 1.14 Polarization of dielectric properties. (a) Polarization of linear dielectric and (b) typical hysteresisloop for ferroelectric materials. Modiﬁed from Bolton, W. (1992). Electrical and Magnetic Properties of Materials,Longman Scientiﬁc & Technical, Harlow
16 Microwave Electronics: Measurement and Materials Characterization Silicon and germanium are typical intrinsic semi- Permittivity conductors. An extrinsic semiconductor is obtained by adding a very small amount of impurities to Ferroelectric Paraelectric an intrinsic semiconductor, and this procedure is state state called doping. If the impurities have a higher number of valence electrons than that of the e′′ host, the resulting extrinsic semiconductor is called e′ type n, indicating that the majority of the mobile charges are negative (electrons). Usually the host is Tp Temperature silicon or germanium with four valence electrons, and phosphorus, arsenic, and antimony with ﬁveFigure 1.15 Schematic view of the temperature de- valence electrons are often used as dopants inpendence of a ferroelectric material near its Curie type n semiconductors. Another type of extrinsictemperature semiconductor is obtained by doping an intrinsic semiconductor using impurities with a number ofthe Curie temperature. As shown in Figure 1.15, valence electrons less than that of the host. Boron,the permittivity of a ferroelectric material changes aluminum, gallium, and indium with three valencegreatly with temperature near the Curie tempera- electrons are often used for this purpose. Theture. Dielectric constant increases sharply to a high resulted extrinsic semiconductor is called type p,value just below the Curie point and then steeply indicating that the majority of the charge carriersdrops just above the Curie point. For example, bar- are positive (holes).ium titanate has a relative permittivity on the order Both the free charge carriers and boundedof 2000 at about room temperature, with a sharp electrons in ions in the crystalline lattice haveincrease to about 7000 at the Curie temperature of contributions to the dielectric permittivity ε = ε −120 ◦ C. The dielectric loss decreases quickly when jε (Ramo et al. 1994):the material changes from ferroelectric state to ne e 2paraelectric state. Furthermore, for a ferroelectric ε = ε1 − (1.54) m(v 2 + ω2 )material near its Curie temperature, its dielectricconstant is sensitive to the external electric ﬁeld. ne e 2 v ε = (1.55) Ferroelectric materials have application poten- ωm(v 2 + ω2 )tials in various ﬁelds, including miniature capaci- where ε1 is related to the effects of the boundtors, electrically tunable capacitors and electrically electrons to the positive background, ne is thetunable phase-shifters. Further discussions on fer- density of the charge carriers, v is the collisionroelectric materials can be found in Chapter 9. frequency, ω is the circular frequency, m is the mass of the electron, and (ne e2 /mv) equals the1.3.2 Semiconductors low frequency conductivity σ . At microwave frequency (ω2 v 2 ), for semi-There are two general categories of semiconduc- conductors with low to moderate doping, whosetors: intrinsic and extrinsic semiconductors. An conductivity is usually not higher than 1 S/m, theintrinsic semiconductor is also called a pure semi- second term of Eq. (1.54) is negligible. So the per-conductor or an undoped semiconductor. The band mittivity can be approximated asstructure shown in Figure 1.2(b) is that of an intrin- σsic semiconductor. In an intrinsic semiconductor, ε = ε1 − j (1.56)there are the same numbers of electrons as holes. ωIntrinsic semiconductors usually have high resis- Besides the permittivity discussed above, thetivity, and they are often used as the starting electrical transport properties of semiconductors,materials for fabricating extrinsic semiconductors. including Hall mobility, carrier density, and con-
Electromagnetic Properties of Materials 17ductivity are important parameters in the devel- ﬁeld; while for a perfect conductor, Eq. (1.57) onlyopment of electronic components. Discussions on applies for time-varying magnetic ﬁelds.electrical transport properties can be found in For a superconductor, there exists a critical tem-Chapter 11. perature Tc . When the temperature is lower than Tc , the material is in superconducting state, and at1.3.3 Conductors Tc , the material undergoes a transition from nor- mal state into superconducting state. A materialConductors have high conductivity. If the con- with low Tc is called a low-temperature super-ductivity is not very high, the concept of per- conducting (LTS) material, while a material withmittivity is still applicable, and the value of per- high Tc is called a high-temperature superconduct-mittivity can be approximately calculated from ing (HTS) material. LTS materials are metallicEqs. (1.54) and (1.55). For good conductors with elements, compounds, or alloys, and their criticalvery high conductivity, we usually use penetra- temperatures are usually below about 24 K. HTStion depth and surface resistance to describe the materials are complex oxides and their critical tem-properties of conductors. As the general properties perature may be higher than 100 K. HTS materialsof normal conductors have been discussed earlier, are of immediate interest for microwave applica-here we focus on two special types of conductors: tions because of their very low surface resistance atperfect conductors and superconductors. It should microwave frequency at temperatures that can bebe noted that perfect conductor is only a theoretical readily achieved by immersion in liquid nitrogenmodel, and no perfect conductor physically exists. or with cryocoolers. In contrast to metallic super- A perfect conductor refers to a material within conductors, HTS materials are usually anisotropic,which there is no electric ﬁeld at any frequency. exhibiting strongest superconductive behavior inMaxwell equations ensure that there is also no preferred planes. When these materials are usedtime-varying magnetic ﬁeld in a perfect conduc- in planar microwave structures, for example, thin-tor. However, a strictly static magnetic ﬁeld should ﬁlm transmission lines or resonators, these pre-be unaffected by the conductivity of any value, ferred planes are formed parallel to the surfaceincluding inﬁnite conductivity. Similar to an ideal to facilitate current ﬂow in the required direc-perfect conductor, a superconductor excludes time- tion (Lancaster 1997; Ramo et al. 1994).varying electromagnetic ﬁelds. Furthermore, the The generally accepted mechanism for super-Meissner’s effect shows that constant magnetic conductivity of most LTS materials is phonon-ﬁelds, including strictly static magnetic ﬁelds, are mediated coupling of electrons with opposite spin.also excluded from the interior of a superconduc- The paired electrons, called Cooper pairs, traveltor. From the London theory and the Maxwell’s through the superconductor without being scat-equations, we have tered. The BCS theory describes the electron pair- B = B0 e−z/λL (1.57) ing process, and it explains the general behav- ior of LTS materials very well. However, despitewith the London penetration depth given by the enormous efforts so far, there is no theory 1 that can explain all aspects of high-temperature m 2 λL = (1.58) superconductivity. Fortunately, an understanding µne e2 of the microscopic theory of superconductivity inwhere B is the magnetic ﬁeld in the depth z, B0 HTS materials is not required for the design ofis the magnetic ﬁeld at the surface z = 0, m is the microwave devices (Lancaster 1997; Shen 1994).mass of an electron, µ is permeability, ne is the In the following, we discuss some phenomeno-density of the electron, and e is the electric charge logical theories based on the London equationsof an electron. So an important difference between and the two-ﬂuid model. We will introduce somea superconductor and a perfect conductor is that, commonly accepted theories for explaining thefor a superconductor, Eq. (1.57) applies for both responses of superconductors to electromagnetictime-varying magnetic ﬁeld and static magnetic ﬁelds, and our discussion will be focused on the
18 Microwave Electronics: Measurement and Materials Characterizationpenetration depth, surface impedance, and complex equal, and they are proportional to the square rootconductivity of superconductors. of the operating frequency ω1/2 . If we want to calculate the impedance of a super-18.104.22.168 Penetration depth conductor using Eq. (1.32), the concept of complex conductivity should be introduced. According toThe two-ﬂuid model is often used in analyzing the two-ﬂuid model, there are two types of cur-superconductors, and it is based on the assumption rents: a superconducting current with volume den-that there are two kinds of ﬂuids in a superconduc- sity Js and a normal current with volume densitytor: a superconductive current with a carrier den- Jn . Correspondingly, the conductivity σ also con-sity ns and a normal current with a carrier density sists of two components: superconducting conduc-nn , yielding a total carrier density n = ns + nn . At tivity σs and normal conductivity σn , respectively.temperatures below the transition temperature Tc , The total conductivity of a superconductor is giventhe equilibrium fractions of the normal and the by σ = σs + σn .superconducting electrons vary with the absolute The superconducting conductivity σs is purelytemperature T : imaginary and does not contribute to the loss: 4 nn T 1 = (1.59) σs = (1.63) n Tc jωµλ2 L 4 ns T While the normal conductivity σn contains both =1− (1.60) real and imaginary components and the real part n Tc contributes to the loss: From Eqs. (1.59) and (1.60), we can get the 2relationship between the penetration depth λL and nn qn τ σn = σn1 − jσn2 =temperature T : mn 1 + jωτ 1 4 −2 2 nn qn τ 1 − jωτ T = (1.64) λL (T ) = λL (0) 1 − (1.61) mn 1 + (ωτ )2 Tc where qn is the electrical charge for the normalwith carriers, τ is the relaxation time for electron ms scattering, and mn is the effective mass of the λL (0) = 2 . (1.62) µnqs normal carriers. Therefore, the total conductivitywhere ms and qs are the effective mass and σ of a superconductor is then obtained:electrical charge of the superconductive carriers. 2 nn qn τ 1Eq. (1.62) indicates that the penetration depth has σ = σn + σs = mn 1 + (ωτ )2a minimum value of penetration depth λL (0) at 2T = 0 K. nn qn τ ωτ 1 −j −j (1.65) mn 1 + (ωτ )2 ωµλ2 L22.214.171.124 Surface impedance and complex At microwave frequencies (ωτ 1), Eq. (1.65)conductivity can be simpliﬁed asThe surface impedance is deﬁned as the charac- 2 nn qn τ 1teristic impedance seen by a plane wave incident σ = σ1 − jσ2 = −j (1.66)perpendicularly upon a ﬂat surface of a conductor. mn ωµλ2 LAccording to Eqs. (1.32) and (1.33), the surface where σ1 and σ2 are the real and imaginaryimpedance of normal conductors, such as silver, components of the complex conductivity. Thecopper, or gold, can be calculated from their con- real part of complex conductivity represents theductivity σ . For a normal conductor, the value of loss due to the normal carriers, whereas itsits conductivity σ is a real number, and the sur- imaginary part represents the kinetic energy of theface resistance Rs and the surface reactance Xs are superconductive carriers.
Electromagnetic Properties of Materials 19 From Eqs. (1.32), (1.33) and Eq. (1.66), we can m′, m′′ m′calculate the surface impedance of a superconductor: Zs = Rs + jXs m′′ (d) (e) 1 (a) (b) − jωµ ωµ σ1 2 = =j 1+j (1.67) (c) σ1 − jσ2 σ2 σ2 104 106 108 1010 f (Hz) As usually σ1 σ2 , Eq. (1.67) can be simpli-ﬁed as Figure 1.16 Frequency dependence of permeability Zs = Rs + jXs for a hypothetical ferromagnetic material ωµ σ1 = +j At different frequency ranges, different physics σ2 2σ2 phenomena dominate. In the low frequency range ω2 µ2 λ3 nn qn τ 2 (f < 104 Hz), µ and µ almost do not change = L + jωµλL (1.68) with frequency. In the intermediate frequency 2mn range (104 < f < 106 Hz), µ and µ change 1 nn a little, and for some materials, µ may have Rs = ω2 µ2 λ3 σN L (1.69) 2 n a maximum value. In the high-frequency range Xs = ωµλL (1.70) (106 < f < 108 Hz), µ decreases greatly, while µ increase quickly. In the ultrahigh frequencywhere σN is the conductivity of the superconductor range (108 < f < 1010 Hz), ferromagnetic reso-in its normal state: nance usually occurs. In the extremely high fre- 2 nqn τ quency range (f > 1010 Hz), the magnetic proper- σN = (1.71) ties have not been fully investigated yet. mn 2 4 nn qn τ nn T σn = = σN = σN (1.72) 126.96.36.199 Magnetization and hysteresis loop mn n Tc According to Eqs. (1.67) to (1.72), the two- Figure 1.17 shows the typical relationship betweenﬂuid model leads to the prediction that the the magnetic ﬂux density B in a magnetic materialsurface resistance Rs is proportional to ω2 for and the magnetic ﬁeld strength H . As discussed insuperconductors, which is quite different from theω1/2 frequency dependence for normal conductors. B Saturation1.3.4 Magnetic materials BmAs the penetration depth of metals at microwave Brfrequencies is on the order of a few microns,the interior of a metallic magnetic material doesnot respond to a microwave magnetic ﬁeld.So, metallic magnetic materials are seldom used H Hc 0 Hcas magnetic materials at microwave frequencies.Here, we concentrate on magnetic materials withlow conductivity. Br The frequency dependence of magnetic materi-als is quite complicated (Smit 1971; Fuller 1987), Saturationand some of the underlying mechanisms have notbeen fully understood. Figure 1.16 shows the typ- Figure 1.17 The hysteresis loop for a magneticical magnetic spectrum of a magnetic material. material
20 Microwave Electronics: Measurement and Materials CharacterizationSection 188.8.131.52, at the starting point 0, the domains c-axisare randomly orientated, so the net magnetic ﬂuxdensity is zero. The magnetic ﬂux density Bincreases with the increase of the magnetic ﬁeldstrength H , as the domains close to the directionof the magnetic ﬁeld grow. This continues untilall the domains are in the same direction with themagnetic ﬁeld H and the material is thus saturated. q 60 ° jAt the saturation state, the ﬂux density reachesits maximum value Bm . When the magnetic ﬁeldstrength is reduced to zero, the domains in the Figure 1.19 Preferential directions for a ferroxplanamaterial turn to their easy-magnetization directions material (Smit 1971). Source: Smit, J. (editor), (1971),close to the direction of the magnetic ﬁeld H, and Magnetic Properties of Materials, McGraw-Hill,the material retains a remanence ﬂux density Br . New YorkIf we reverse the direction of the magnetic ﬁeld, be easily achieved, while in the hard-magnetizationthe domains grow in the reverse direction. When direction, high magnetic ﬁeld is required forthe numbers of the domains in the H direction saturation. The magnetic ﬁeld Ha correspondingand opposite the H direction are equal, that is, the to the cross point of the two magnetization curvesﬂux density becomes zero, the value of the applied is called anisotropic ﬁeld.magnetic ﬁeld is called coercive ﬁeld Hc . Further There are two typical types of anisotropiesincrease in the strength of the magnetic ﬁeld in the of magnetic materials: axis anisotropy and planereverse direction results in further growth of the anisotropy for a hexagonal structure. Figure 1.19domains in the reverse direction until saturation in shows the potential directions for a ferrox-the reverse direction is achieved. When this ﬁeld plana material. If the easy-magnetization direc-is reduced to zero, and then reversed back to the tion is along the c-axis, the material has uniaxialinitial direction, we can get a closed hysteresis loop anisotropy, usually described by the anisotropicof the magnetic material. ﬁeld Ha . If the easy-magnetization direction is in In most cases, magnetic materials are anisotropic the c-plane, the material has planar anisotropy.for magnetization. For a hexagonal ferrite, there Planar anisotropy is usually described by theexists an easy-magnetization direction and a hard- anisotropic ﬁelds Hθ and Hϕ , where Hθ is themagnetization direction. As shown in Figure 1.18, magnetic ﬁeld required for turning a domain inin the easy-magnetization direction, saturation can one preferential magnetization direction in the c- plane to another preferential magnetization direc- M tion in the c-plane through the hard-magnetization c-axis, and Hϕ is the magnetic ﬁeld required for Ms turning a domain in one preferential magnetiza- 1 tion direction in the c-plane to another preferential magnetization direction in the c-plane within the 2 easy-magnetization plane. The coercive ﬁeld Hc is an important param- eter in describing the properties of a magnetic material. The value of coercive ﬁeld Hc is mainly governed by two magnetization phenomena: rota- 0 Ha H tion of domain and movement of domain wall. ItFigure 1.18 Magnetization curves for an anisotropic is related to intrinsic magnetic properties, such asmagnetic material. Curve 1 is the magnetization in anisotropic ﬁeld and domain-wall energy, and it isthe easy-magnetization direction and Curve 2 is the also related to the microstructures of the material,magnetization in the hard-magnetization direction such as grain size and domain-wall thickness.
Electromagnetic Properties of Materials 21Besides, the amount and distribution of impurities dB dHin the material also affects the value of the coerciveﬁeld Hc . m0 mrm184.108.40.206 Deﬁnitions of scalar permeability m0 mriAs the relationship between the magnetic ﬂuxdensity B and the magnetic ﬁeld strength H isnonlinear, the permeability is not a constant butvaries with the magnetic ﬁeld strength. Usually, it 0 Hm His not necessary to have a complete knowledge ofthe magnetic ﬁeld dependence of permeability. In Figure 1.21 The dependence of permeability on mag-the mathematical treatment of general applications, netic ﬁeldthe relative permeability is simply a numberdenoted by the symbol µr , but for differentcases, permeability has different physical meaning. theoretical value corresponding to a zero ﬁeld,On the basis of the hysteresis loop shown in and in a strict meaning, it cannot be directlyFigure 1.20, we can distinguish four deﬁnitions measured. Usually, the initial relative permeabilityof scalar permeability often used in materials is determined by extrapolation. In practice, µri isresearch (Robert 1988). often given as the relative permeability measured The initial relative permeability is deﬁned as in a weak ﬁeld lying between 100 and 200 A/m. Figure 1.21 shows the relationship between 1 B (dB/dH ) and H corresponding to the dashed line µri = limH →0 (1.73) µ0 H in Figure 1.20. The (dB/dH ) value point at H = 0It is applicable to a specimen that has never equals the initial permeability discussed above. Atbeen subject to irreversible polarization. It is a the point Hm , which satisﬁes d2 B B = 0, (1.74) dH 2 Slope: m0 mrm the value of (dB/dH ) reaches its maximum value, which is deﬁned as maximum permeability (µ0 µrm ), as shown in Figures 1.20 and 1.21. The value of µrm (Hm, Bm) can be taken as a good approximation of the relative Br Slope: m0 mri permeability for a low-frequency alternating ﬁeld with amplitude Hm . Now, we consider the case when an alternating −Hc Hc H ﬁeld H2 is superimposed on a steady ﬁeld H1 parallel to H2 . If H2 H1 , the hysteresis loop is −Br ∆B simply translated without substantial deformation. If H2 H1 , there will be an eccentric local loop, which is always contained within the main cycle. In the presence of a superimposed steady ﬁeld ∆H H1 , the differential relative permeability ur is deﬁned byFigure 1.20 Deﬁnitions of four scalar permeabili- 1 Bties (Robert 1988). Reprinted with permission from µr = (1.75) µ0 HElectrical and Magnetic Properties of Materials, byRobert, P., Artech House Inc., Norwood, MA, USA, where H is the amplitude of the alternatingwww.artechhouse.com ﬁeld and B is the corresponding variation of the
22 Microwave Electronics: Measurement and Materials Characterizationmagnetic induction. The reversible relative perme- B Bability urr is the value of the differential relativepermeability for an alternating ﬁeld tending to zero 1 B µrr = lim H →0 (1.76) µ0 H H H 01.3.4.3 Soft and hard magnetic materialsAccording to the values of their coercive ﬁelds,magnetic materials can be classiﬁed into soft andhard magnetic materials. Figure 1.22(a) shows a (a) (b)typical hysteresis loop of a soft magnetic material.The term soft is applied to a magnetic material Figure 1.23 Rectangular hysteresis loops. (a) Softthat has a low coercive ﬁeld, so only a small magnetic material and (b) hard magnetic materialmagnetic ﬁeld strength is required to demagnetize hard magnetic materials. Generally speaking, theor reverse the direction of the magnetic ﬂux coercive ﬁeld of a soft magnetic material is less thanin the material. Usually, soft magnetic material ten oersted, while that of a hard magnetic materialhas high permeability. The area enclosed by the is larger than several hundred oersted. It should behysteresis loop is usually small, so little energy is noted that remanence ﬂux density Br is not a criterialost in the magnetization cycle. In a microscopic for the classiﬁcation of soft and hard magneticscale, the domains in a soft magnetic material can materials. A magnetic material with rectangulareasily grow and rotate. Soft magnetic materials hysteresis loop has a relatively high value of Br ,are widely used for electrical applications, such as but high value of Br does not mean high value oftransformer cores. Figure 1.22(b) shows a typical Hc . As shown in Figure 1.23(a) and (b), both softhysteresis curve for a hard magnetic material. A and hard magnetic materials can have rectangularhard magnetic material has a high coercive ﬁeld, so hysteresis loops.it is difﬁcult to demagnetize it. The permeability of For a material with rectangular hysteresis loop,a hard magnetic material is usually small. Besides, when the magnetizing ﬁeld is removed, the ﬂuxa hard magnetic material usually has a large area density almost remains unchanged, so that theenclosed by the hysteresis loop. Hard magnetic remanence ﬂux density is virtually the same as thematerials are often used as permanent magnets. saturation one. This means that, once the material It should be emphasized that the coercive ﬁeld is magnetized, it retains most of the ﬂux densityHc is the criteria for the classiﬁcation of soft and when the magnetizing ﬁeld is switched off. These materials are often used in magnetic recording. 220.127.116.11 Magnetic resonance Magnetic resonance is an important loss mecha- [Image not available in this electronic edition.] nism of magnetic materials, and should be taken into full consideration in the application of mag- netic materials. For most of the magnetic materials, the energy dissipation at microwave frequencies is related to natural resonance and wall resonance.Figure 1.22 Hysteresis loops. (a) Soft magnetic mate- Natural resonancerials and (b) hard magnetic materials. Source: Bolton,W. (1992), Electrical and Magnetic Properties of Mate- As shown in Figure 1.24, under a dc magnetic ﬁeldrials, Longman Scientiﬁc & Technical, Harlow H and ac magnetic ﬁeld h, the magnetic moment M
Electromagnetic Properties of Materials 23 H ferrites and ferromagnetic resonance under the application of external dc magnetic ﬁeld will be discussed in Chapter 8. M The resonance frequency fr of a natural res- onance is mainly determined by the anisotropic ﬁeld of material. For a material with uniaxial anisotropy, the resonance frequency is given by q fr = γ Ha (1.78) h For a material with planar anisotropic anisotropy, the resonance frequency is given byFigure 1.24 Precession of magnetic moment fr = γ (Hθ · Hϕ )1/2 (1.79)makes a precession around the dc magnetic ﬁeld H, There are two typical types of resonances:and the ac magnetic ﬁeld h provides the energy to Lorentzian type and Debye type. It should be indi-compensate the energy dissipation of the precession. cated that, in actual materials, natural resonanceThis is the origin of ferromagnetic resonance, and may be in a type between the Lorentzian one andcan be described by the Gilbert equation: the Debye one. The Lorentzian type occurs when dM λ dM λ is much smaller than one, and it is also called = −γ M × H + M× (1.77) resonant type. From Eq. (1.77), we can get dt M dt χ0where γ = 2.8 MHz/Oe is the gyromagnetic ratio µr = 1 + (1.80)and λ is the damping coefﬁcient. The dc magnetic 1 − (f/fr )2 + j(2λf/f ) rﬁeld H includes external dc magnetic ﬁeld H0 , where χ0 is the static susceptibility of the mate-anisotropic ﬁeld Ha , demagnetization ﬁeld Hd , and rial, fr is the resonance frequency, and f isso on. If H0 = 0, the ferromagnetic resonance is the operation frequency. Figure 1.25(a) shows ausually called natural resonance. In the following typical permeability spectrum of a resonance withtext, we concentrate on natural resonance of Lorentzian type. 12 1.0 10 0.8 8 mr − 1 ′ ′′ mr 6 c′/c0, c′′/c0 c′/c0, c′′/c0 0.6 4 mr − 1 ′ ′′ mr 2 0.4 0 −2 0.2 −4 −6 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.1 1 f /fa f /fa (a) (b)Figure 1.25 Two types of permeability spectrums. (a) Lorentzian type. The results are calculated based onEq. (1.80) with λ = 0.1 and fa = fr . (b) Debye type. The results are calculated based on Eq. (1.81) with fa = fr /λ
24 Microwave Electronics: Measurement and Materials Characterization The Debye type occurs when λ is much largerthan one. The Debye type is also called relaxationtype. From Eq. (1.77), we can get dZ = dt A µr = 1 + (1.81) h 1 + j(λf/fr ) Ms MsFigure 1.25(b) shows a typical permeability spec-trum of Debye type. Z The Snoek limit describes the relationshipbetween the resonant frequency and permeability. Figure 1.26 Mechanism of wall resonanceFor a material with uniaxial anisotropy, we have 2 where δ and D are the thickness and the width of fr · (µr − 1) = γ Ms (1.82) the domain wall respectively. Ms is the magneti- 3 zation within a domain and it equals the saturatedwhere Ms is the saturated magnetization. For magnetization of the material.a material with a given resonance frequency, The movement of domain wall is similar to ahigher saturated magnetization corresponds to forced harmonic movement. So the wall resonancehigher permeability. For a material with planar can be described using spring equation:anisotropy, the Snoek limit is in the form of d2 Z dZ 1/2 mw 2 +β + αZ = 2Ms hejωt (1.85) 1 Hθ dt dt fr · (µr − 1) = γ Ms · (1.83) 2 Hϕ where mw is the effective mass of the domain wall, β is the damping coefﬁcient, α is theEq. (1.83) indicates that planar anisotropy provides elastic coefﬁcient, and h is the amplitude of themore ﬂexibility for the design of materials with microwave magnetic ﬁeld. For a Lorentzian-typeexpected resonant frequency and permeability. resonance, we have A µr = 1 + (1.86)Wall resonance 1 − (f/fβ )2 + j(f/fτ )If a dc magnetic ﬁeld H is applied to a magnetic where the intrinsic vibration frequency fβ ismaterial, the domains in the directions close to given bythe direction of the magnetic ﬁeld grow, while fβ = (α/mw )1/2 (1.87)the domains in the directions close to the opposite and the relaxation frequency fτ is given bydirections of the magnetic ﬁeld shrink. The growthand shrink of domains are actually the movements fτ = α/β (1.88)of the domain wall. If an ac magnetic ﬁeld h For most of the wall resonance, fβ fτ ,is applied, the domain wall will vibrate around Eq. (1.86) becomesits equilibrium position, as shown in Figure 1.26.When the frequency of the ac magnetic ﬁeld A µr = 1 + (1.89)is equal to the frequency of the wall vibration, 1 + j(f/fτ )resonance occurs, and such a resonance is usually Eq. (1.89) represents a Debye-type resonance.called wall resonance. Rado proposed a relationshipbetween the resonance frequency f0 and relative 1.3.5 Metamaterialspermeability µr (Rado 1953): Electromagnetic metamaterials are artiﬁcial struc- 1/2 2δ tures with unique or superior electromagnetic prop- fr · (µr − 1)1/2 = 2γ Ms · (1.84) D erties. The special properties of metamaterials
Electromagnetic Properties of Materials 25come from the inclusion of artiﬁcially fabricated, (LCP) and right- circularly polarized (RCP) waves.extrinsic, low-dimensional inhomogeneities. The LCP and RCP waves propagate with differentdevelopment of metamaterials includes the design velocities and attenuation in a chiral medium.of unit cells that have dimensions commensurate Still, these differential measurements are unable towith small-scale physics and the assembly of the characterize completely the chiral medium. Moreunit cells into bulk materials exhibiting desired recently, (Guire et al. 1990) has studied experi-electromagnetic properties. In recent years, the mentally the normal incidence reﬂection of lin-research on electromagnetic metamaterials is very early polarized waves of metal-backed chiral com-active for their applications in developing func- posite samples at microwave frequencies. Thetional electromagnetic materials. In the following, beginning of a systematic experiment work camewe discuss three examples of metamaterials: chi- from (Umari et al. 1991) when they reported mea-ral materials, left-handed materials, and photonic surements of axial ratio, dichroism, and rotationband-gap materials. of microwaves transmitted through chiral samples. However, in order to characterize completely the chiral composites, the chirality parameter, permit-18.104.22.168 Chiral materials tivity, and permeability have to be determined.Chiral materials have received considerable atten- The chirality parameter, permittivity, and perme-tion during recent years (Jaggard et al. 1979; Mar- ability can be determined from inversion of threeiotte et al. 1995; Theron and Cloete 1996; Hui and measured scattering parameters. The new chiralityEdward 1996) and might have a variety of poten- parameter can be obtained only with the substi-tial applications in the ﬁeld of microwaves, such tution of new sets of constitutive equations (Roas microwave absorbers, microwave antennae, and 1991; Sun et al. 1998),devices (Varadan et al. 1987; Lindell and Sihvola D = εE + βε∇ × E and (1.90)1995). (Lakhtakia et al. 1989) has given a fairlycomplete set of references on the subject. (Bokut B = µH + βµ∇ × H , or (1.91)and Federov 1960; Jaggard et al. 1979; Silver- D = εE + iξ B and (1.92)man 1986; Lakhtakia et al. 1986) have studied thereﬂection and refraction of plane waves at planar H = iξ E + B /µ. (1.93)interfaces involving chiral media. The possibil- Here, ε and µ are the usual permittivity and perme-ity of designing broadband antireﬂection coatings ability respectively, while β and ξ are the chiralitywith chiral materials was addressed by (Varadan parameter that results from the handedness or lacket al. 1987). These researchers have shown that the of inversion symmetry in the microstructure of theintroduction of chirality radically alters in scatter- medium. The values of chirality parameter, per-ing and absorption characteristics. In these papers, mittivity, and permeability vary with frequency,the authors have used assumed values of chirality volume concentration of the inclusions, geometryparameter, permittivity, and permeability in their and size of the inclusion, and the electromagneticnumerical results. properties of the host medium. Further discussion (Winkler 1956; Tinoco and Freeman 1960) have on chiral materials can be found in Chapter 10.studied the rotation and absorption of electro-magnetic waves in dielectric materials contain- 22.214.171.124 Left-handed materialsing a distribution of large helices. Direct andquantitative measurements are made possible with A left-handed material is a material whose perme-the recent advances in microwave components ability and permittivity are simultaneously negative.and measurement techniques. Urry and Krivacic It should be noted that the term “left handed” does(1970) have measured the complex, frequency not refer to either chirality or symmetry breaking.dependent values of (nL − nR ) for suspensions of These other phenomena are often referred to as “leftoptically active molecules, where nL and nR are handed”, but are distinct from the effects that we arethe refractive indices for left- circularly polarized discussing in left-handed materials.
26 Microwave Electronics: Measurement and Materials Characterization All the normal materials are “right handed”,which means that the relationship between the ﬁeldsand the direction of wave vector follows the “right-hand rule”. If the ﬁngers of the right hand representthe electric ﬁeld of the wave, and if the ﬁngers curlaround to the base of the right hand, representingthe magnetic ﬁeld, then the outstretched thumb indi-cates the direction of the ﬂow of the wave energy.However, for a left-handed material, the relationshipbetween the ﬁelds and the direction of wave vector (a) (b)follows the “left-hand rule”. Figure 1.27 Effects of ﬂat plates. (a) Flat plate made Left-handed materials were ﬁrst envisioned in from a normal material and (b) ﬂat plate made from athe 1960s by Russian physicist Victor Veselago of left-handed materialthe Lebedev Physics Institute. He predicted thatwhen light passed through a material with both 2003a, 2003b). Therefore, such a plate can worka negative dielectric permittivity and a negative as a superlens.magnetic permeability, novel optical phenomena Left-handed materials do not exist naturally. Inwould occur, including reversed Cherenkov radi- Veselago’s day, no actual left-handed materialsation, reversed Doppler shift, and reversed Snell were known. In the 1990s, John Pendry of Imperialeffect. Cherenkov radiation is the light emitted College discussed how negative-permittivity mate-when a charged particle passes through a medium, rials could be built from rows of wires (Pendryunder certain conditions. In a normal material, the et al. 1996) and negative-permeability materialsemitted light is in the forward direction, while in a from arrays of tiny resonant rings (Pendry et al.left-handed material, light is emitted in a reversed 1999). In 2000, David Smith and his colleaguesdirection. In a left-handed material, light waves are constructed an actual material with both a neg-expected to exhibit a reversed Doppler effect. The ative permittivity and a negative permeability atlight from a source coming toward you would be microwave frequencies (Smith et al. 2000). Anreddened while the light from a receding source example of a left-handed material is shown inwould be blue shifted. Figure 1.28. The raw materials used, copper wires The Snell effect would also be reversed at the and copper rings, do not have unusual properties ofinterface between a left-handed material and a their own and indeed are nonmagnetic. But whennormal material. For example, light that enters aleft-handed material from a normal material willundergo refraction, but opposite to what is usuallyobserved. The apparent reversal comes aboutbecause a left-handed material has a negative indexof refraction. Using a negative refractive indexin Snell’s law provides the correct description ofrefraction at the interface between left- and right-handed materials. As a further consequence of thenegative index of refraction, lenses made from left-handed materials will produce unusual optics. Asshown in Figure 1.27, a ﬂat plate of left-handedmaterial can focus radiation from a point sourceback to a point. Furthermore, the plate can amplify Figure 1.28 A left-handed material made from wiresthe evanescent waves from the source and thus and rings. This picture is obtained from the homepagethe sub-wavelength details of the source can be for Dr David R. Smith (http://physics.ucsd.edu/∼drs/restored at the image (Pendry 2000; Rao and Ong index.html)
Electromagnetic Properties of Materials 27incoming microwaves fall upon alternating rows of which often refers to the structures exhibiting chi-the rings and wires, a resonant reaction between rality. New descriptive terms have been introducedthe light and the whole of the ring-and-wire array to refer to materials with simultaneously nega-sets up tiny induced currents, making the whole tive permittivity and permeability. “Backward wavestructure “left handed”. The dimensions, geomet- materials” is used to signify the characteristic thatric details, and relative positioning of the wires materials with negative permittivity and permeabil-and the rings strongly inﬂuence the properties of ity reverse the phase and group velocities. “Materi-the left-handed material. als with negative refractive index” emphasizes the However, the surprising optical properties of reversed Snell effect. And “double negative materi-left-handed materials have been thrown into doubt als” is a quick and easy way to indicate that bothby physicists. Some researchers said that the claims the permittivity and permeability of the materialthat left-handed materials could act as perfect are negative.lenses violate the principle of energy conserva-tion (Garcia and Nieto-Vesperinas 2002). Mean- 126.96.36.199 Photonic band-gap materialswhile, some researchers indicated that “negative A photonic band-gap (PBG) material, also calledrefraction” in left-handed materials would breach photonic crystal, is a material structure whosethe fundamental limit of the speed of light (Valanju refraction index varies periodically in space. Theet al. 2002). But other researchers in the ﬁeld periodicity of the refraction index may be in onedefended their claims on left-handed materials. dimension, two dimensions, or three dimensions.The debate should generate some light, and stim- The name is applied since the electromagneticulate better experiments, which would beneﬁt the waves with certain wavelengths cannot propagateunderstanding and utilization of this type of meta- in such a structure. The general properties of amaterials. If the negative refraction and perfect PBG structure are usually described by the rela-lensing of left-handed materials can be proven, tionship between circular frequency and wave vec-left-handed materials could have a wide range of tor, usually called wave dispersion. The wave dis-applications including high-density data storage persion in a PBG structure is analogous to theand high-resolution optical lithography in the semi- band dispersion (electron energy versus wave vec-conductor industry. tor) of electrons in a semiconductor. Figure 1.29(a) Finally, it should be indicated that many research- schematically shows a three-dimensional PBGers in this ﬁeld object to the term “left handed,” structure, which is an array of dielectric spheres 0.7 0.6 0.5 Diamond Frequency 0.4 a 0.3 0.2 0.1 0.0 X U L Γ X W K Wavevector (a) (b)Figure 1.29 A PBG structure formed by dielectric spheres arranged in a diamond lattice. The background is air.The refraction index of the spheres is 3.6 and the ﬁlling ratio of the spheres is 34 %. The frequency is given in unitof c/a, where a is the cubic constant of the diamond lattice and c is the velocity of light in vacuum. (a) Schematicillustration of the three-dimensional structure and (b) wave dispersion of the PBG structure. Theoretical results arefrom (Ho et al. 1990). Source: Ho, K. M., Chan, C. T., and Soukoulis, C. (1990). “Existence of a photonic gap inperiodic dielectric structures”, Phys. Rev. Lett. 65(25), 3152–3155. 2003 The American Physical Society
28 Microwave Electronics: Measurement and Materials Characterizationsurrounded by vacuum. The photonic band of the wings of butterﬂies, and the hairs of a wormlikestructure is shown in Figure 1.29(b). creature called the sea mouse have typical PBG The origin of the band gap stems from the very structures, and their lattice spacing is exactlynature of wave propagation in periodic structures. right to diffract visible light. It should also beWhen a wave propagates in a periodic structure, noted that, although “photonic” refers to light,a series of refraction and reﬂection processes the principle of the band gap applies to all theoccur. The incident wave and the reﬂected wave waves in a similar way, no matter whether they areinterfere and may reinforce or cancel one another electromagnetic or elastic, transverse or longitude,out according to their phase differences. If the vector or scalar (Brillouin 1953).wavelength of the incident wave is of the samescale as the period of the structure, very strong 1.3.6 Other descriptions of electromagneticinterference happens and perfect cancellation may materialsbe achieved. As a result, the wave is attenuated andcannot propagate through the periodic structure. Besides the microscopic and macroscopic param-In a broad sense, the electronic band gaps of eters discussed above, in materials research andsemiconductors, where electron waves propagate engineering, some other macroscopic propertiesin periodic electronic potentials, also fall into are often used to describe materials.this category. Owing to the similarity of PBGsand the electronic band gaps, PBG materials 188.8.131.52 Linear and nonlinear materialsfor electromagnetic waves can be treated assemiconductors for photons. Linear materials respond linearly with externally The ﬁrst PBG phenomenon was observed by applied electric and magnetic ﬁelds. In weakYablonovitch and Gmitter in an artiﬁcial microstruc- ﬁeld ranges, most of the materials show linearture at microwave frequency (Yablonovitch and responses to applied ﬁelds. In the characterizationGmitter 1989). The microstructure was a dielectric of materials’ electromagnetic properties, usuallymaterial with about 8000 spherical air “atoms”. The weak ﬁelds are used, and we assume that theair “atoms” were arranged in a face-centered-cubic materials under study are linear and that the(fcc) lattice. Thereafter, many other structures and applied electric and magnetic ﬁelds do not affectmaterial combinations were designed and fabricated the properties of the materials under test.with superior PBG characteristics and greater man- However, some materials easily show nonlinearufacturability. properties. One typical type of nonlinear material is PBG materials are of great technological and theo- ferrite. As discussed earlier, owing to the nonlinearretical importance because their stop-band and pass- relationship between B and H , if different strengthband frequency characteristics can be used to mold of magnetic ﬁeld H is applied, different valuethe ﬂow of electromagnetic waves (Joannopoulos of permeability can be obtained. High-temperatureet al. 1995). Extensive applications have been superconducting thin ﬁlms also easily show non-achieved using the concept of PBG in various ﬁelds, linear properties. In the characterization of HTSespecially in optoelectronics and optical communi- thin ﬁlms and the development microwave devicescation systems. The PBG is the basis of most appli- using HTS thin ﬁlms, it should be kept in mindcations of PBG materials, and it is characterized by that the surface impedance of HTS thin ﬁlms area strong reﬂection of electromagnetic waves over a dependent on the microwave power.certain frequency range and high transmission out-side this range. The center frequency, depth, and 184.108.40.206 Isotropic and anisotropic materialswidth of the band gap can be tailored by modify-ing the geometry and arrangement of units and the The macroscopic properties of an isotropic mate-intrinsic properties of the constituent materials. rial are the same in all orientations, so they can be It should be noted that PBG structures also exist represented by scalars or complex numbers. How-in the nature. The sparking gem opal, colorful ever, the macroscopic properties of an anisotropic
Electromagnetic Properties of Materials 29material have orientation dependency, and they are sophisticated limits are possible. In an effective-usually represented by tensors or matrixes. Some medium method, we assume the presence of ancrystals are anisotropic because of their crystalline imaginary effective medium, whose properties arestructures. More discussion on anisotropic materi- calculated using general physical principles, such asals can be found in Chapter 8, and further discus- average ﬁelds, potential continuity, average polariz-sion on this topic can be found in (Kong 1990). ability, and so on. Detailed discussion on effective- medium theory can be found in (Choy 1999).220.127.116.11 Monolithic and composite materials To achieve more accurate prediction, numerical methods are often used in predicting the propertiesAccording to the number of constituents, materi- of composite materials. Numerical computation ofals can be classiﬁed into monolithic or composite the effective dielectric constant of discrete randommaterials. A monolithic material has a single con- media is important for practical applications suchstituent. While a composite material has several as geophysical exploration, artiﬁcial dielectrics,constituents, and usually one of the constituents is and so on. In such dielectrics, a propagatingcalled host medium, the others are called inclusions electromagnetic wave undergoes dispersion andor ﬁllers. The properties of a composite material absorption. Some materials are naturally absorptiveare related to the properties and fractions of the owing to viscosity, whereas inhomogeneous mediaconstituents, so the electromagnetic properties of exhibit absorption due to geometric dispersion orcomposites can be tailored by varying the proper- multiple scattering. The scattering characteristicsties and fractions of the constituents. The study of of the individual particles (or the inclusions) inthe electromagnetic properties of composite mate- the composite could be described by a transitionrials has attracted much attention, with the aim of or T-matrix and the frequency-dependent dielectricdeveloping composites with expected electromag- properties of the composite are calculated usingnetic properties. multiple scattering theory and appropriate correla- The prediction of the properties of a composite tion functions between the particles (Varadan andfrom those of the constituents of the composite Varadan 1979; Bringi et al. 1983; Varadan et al.is a long-standing problem for theoretical and 1984; Varadan and Varadan 1985).experimental physics. The mixing laws relating More discussions on monolithic and compos-the macroscopic electromagnetic properties of ite materials can be found in (Sihvola 1999; Nee-composite materials to those of their individual lakanta 1995; Priou 1992; Van Beek 1967). Inconstituents have been a subject of enquiry since the following, we concentrate on the dielec-the end of the nineteenth century. The ability to tric permittivity of composite materials, and wetreat a composite with single effective permittivity mainly discuss dielectric–dielectric composites andand effective permeability is essential to work dielectric-conductor composites.in many ﬁelds, for example, remote sensing,industrial and medical applications of microwaves, Dielectric–dielectric compositesmaterials science, and electrical engineering. The mean-ﬁeld method and effective-medium The host media of composite materials are usuallymethod are two traditional approaches in predict- dielectric materials, and if the inclusions are alsoing the properties of composite materials (Banhegyi dielectric materials, such composites are called1994). In the mean-ﬁeld method, we calculate the dielectric–dielectric composites. The shapes andupper and lower limits of properties represent- structures of the inclusions affect the overalling the parallel and perpendicular arrangements of properties of the composites.the constituents. A practical method is to approxi- A composite with spherical inclusions is the sim-mate the composite structure by elements of ellip- plest and a very important case. Consider a mixturesoidal shape, and various techniques are avail- with a host medium of permittivity ε0 containingable to calculate the composite permittivity. For n inclusions in unit volume, with each of theisotropic composites, closer limits can be calculated inclusions having polarizability α. The permittivityand, depending on morphological knowledge, more ε0 of the host medium can take any value,
30 Microwave Electronics: Measurement and Materials Characterizationincluding complex ones. The effective permittivity So the effective permittivity of this mixture isεeff of a composite is deﬁned as the ratio between εeff − ε0 ε1 − ε0the average electric displacement D and the = f1 (1.98) εeff + 2ε0 ε1 + 2ε0average electric ﬁeld E: D = εeff E . The electricdisplacement D depends on the polarization P in This formula is known as the Rayleigh’s formula.the material, D = ε0 E + P, and the polarization The success of a mixture formula for a compos-can be calculated from the dipole moments p of the ite relies on the accuracy in the modeling of itsn inclusions, P = np. This treatment assumes that real microstructure details. Besides the Rayleigh’sthe dipole moments are the same for all inclusions. formula, several other formulas have been derivedIf the inclusions are of different polarizabilities, the using different approximations of the microstruc-polarization has to be summed by weighting each tural details of the composite. Several other pop-dipole moment with its number density, and the ular formulas for the effective permittivity εeff ofoverall polarization thus consists of a sum or an two-phase nonpolar dielectric mixtures with hostintegral over all the individual inclusions. medium of permittivity ε0 and spherical inclusion The dipole moment p depends on the polariz- of permittivity ε1 with volume fraction f1 are listedability and the exciting ﬁeld Ee : p = αEe . For in the following:spherical inclusions, the exciting ﬁeld Ee is: Ee = Looyenga’s formula:E + P/(3ε0 ). From the above equations, the effec- 1 1 1tive permittivity can be calculated as a function of εeff = f1 ε1 + (1 − f1 )ε03 3 3 (1.99)the dipole moment density nα: Beer’s formula: nα εeff = ε0 + 3ε0 (1.94) 1 1 1 3ε0 − nα εeff = f1 ε1 + (1 − f1 )ε0 2 2 2 (1.100)Equation (1.94) can also be written in the form of Lichtenecher’s formula:the Clausius–Mossotti formulas ln εeff = f1 ln ε1 + (1 − f1 ) ln ε0 (1.101) εeff − ε0 nα = (1.95) εeff + 2ε0 3ε0 In the above formulas (1.98–1.101), the interpar- ticle actions between the inclusions are neglected.If the composite contains inclusions with different The above formulas can be extended to multiphasepolarization, for example N types of spheres composites, but they are not applicable to compos-with different permittivities, Eq. (1.95) should ites with layered inclusions, because they ignorebe modiﬁed into (Sihvola 1989a; Sihvola and the interactions between the different layers in anLindell 1989b): inclusion. The properties of composites with lay- N ered spherical inclusions are discussed in (Sihvola εeff − ε0 ni αi = (1.96) and Lindell 1989b). εeff + 2ε0 i=1 3ε0 Let the permittivity of the background medium Dielectric-conductor compositesbe ε0 , that of the inclusions be ε1 , and the volume In a dielectric-conductor composite, the host me-fraction of the inclusions be f1 . The polarizability dium is a dielectric material, while the inclusions areof this kind of inclusions depends on the ratio conductors. Such composite materials have exten-between the inside and the outside ﬁelds when sive electrical and electromagnetic applications,the inclusions are in a static ﬁeld. According to such as antistatic materials, electromagnetic shields,Sihvola(1989a) and Sihvola and Lindell (1989b), and radar absorbers. Here we do not consider thethe polarizability of a spherical inclusion with frequency dependence of the electromagnetic prop-radius a1 is erties of dielectric-conductor composites, and only ε1 − ε0 consider the static limit (ω → 0). Discussions on α = 4πε0 a1 3 (1.97) ε1 + 2ε0 the frequency dependence of the properties of such
Electromagnetic Properties of Materials 31 e′′ Permittivity e′ Vp Volume concentration (a) (b) (c)Figure 1.30 Percolation in a dielectric-conductor composite. (a) Change of static permittivity near the percolationthreshold, (b) the case when the volume concentration of the ﬁllers is less than the percolation threshold (Vp ), and(c) the case when the volume concentration of the ﬁllers is close to the percolation threshold (Vp ). In (b) and (c),circle denotes inclusions, otherwise the host mediumcomposites can be found in (Potschke et al. 2003) host medium when the volume concentration of theand the references given therein. inclusions is less than and close to the percolation For a dielectric-conductor composite, there threshold respectively.exists a phenomenon called percolation. When the The location of the percolation threshold andvolume concentration of the conductive inclusions the concentration dependence of permittivity andapproaches the percolation threshold, the dielectric conductivity around the threshold depend oncomposite becomes conductive. One can observe the properties of the host medium, conductivea signiﬁcant change in permittivity of the com- inclusions, and the morphology of the composite.posites ﬁlled with conductive inclusions when it Because of the rich physics phenomena nearpercolates. As shown in Figure 1.30(a), near the the percolation threshold, in percolation research,percolation threshold, the real part of permittiv- constructing models of permittivity or conductivityity of the composite increases quickly along with near the percolation threshold is of great theoreticalthe increase of the volume concentration of the importance and application meaning.conductive inclusions and reaches its maximum It should be emphasized that the geometry ofvalue at the percolation threshold; while the imag- the inclusions plays an important role in deter-inary part of permittivity monotonically increases mining the percolation threshold and the electro-with the increase of the volume concentration of magnetic properties of a dielectric-conductor com-the conductive inclusions. The origin of percola- posite. The general geometry of an inclusion istion phenomenon is the connection of the con- elliptic sphere, which, at special conditions, canductive inclusions. Figures 1.30(b) and (c) show be disk, sphere, and needle. In recent years, com-distributions of the conductive inclusions in the posites with ﬁber inclusions have attracted great
32 Microwave Electronics: Measurement and Materials Characterizationattentions (Lagarkov et al. 1998). Fiber can be includes permittivity, permeability, conductivity,taken as a very thin and long needle. The mechan- and chirality. Electromagnetic waves can propa-ical and electrical performance of polymer mate- gate within low-conductivity materials, and therials may be greatly improved by adding carbon propagation parameters for low-conductivity mate-or metal ﬁbers, and the resulted ﬁber-reinforced rials mainly include wave impedance, propaga-composites have a wide range of practical applica- tion constant, and index of refraction. For con-tions due to their unique mechanical, chemical, and ductors and superconductors, the main propagationphysical properties. Fiber-ﬁlled composites present parameters are skin depth and surface impedance.more possibilities of tailoring the dielectric prop- For semiconductors, the intrinsic properties areerties. For example, high values of dielectric con- usually described by their electrical transport prop-stant can be obtained at a low concentration of erties, including Hall mobility, conductivity, andﬁber inclusions, and composites ﬁlled with metal carrier density.ﬁbers possess pronounced microwave dielectricdispersion, which are very important for the devel- 1.4.2 Extrinsic performancesopment microwave absorbing materials. Finally, it should be indicated that percolation As the performances of electromagnetic materi-phenomena also exists in many other systems, als and structures depend on their geometries, thefor example, the superconductivity of metal- performance-related properties are usually extrin-superconductor composites and leakage of ﬂuids sic. The design of functional materials and struc-through porous media. tures is to realize the desired extrinsic perfor- mances based on the intrinsic properties of the raw materials to be used. The performance-related1.4 INTRINSIC PROPERTIES properties could be monitored along the manufac-AND EXTRINSIC PERFORMANCES turing procedures to ensure that the ﬁnal productsOF MATERIALS have the speciﬁed extrinsic performances. There are varieties of extrinsic performances,The properties of materials can be generally clas- and it is difﬁcult to make a systematic classiﬁca-siﬁed into two categories: intrinsic and extrinsic. tion. In the following, we discuss the characteristicIntrinsic properties of a material are independent impedance of a transmission line, the reﬂectivity ofof the size of the material. If the electromagnetic a Dallenbach layer, and the resonance of a dielec-properties of a material are related to the geo- tric resonator.metrical structures and sizes, such properties areextrinsic. 18.104.22.168 Characteristic impedance of a transmission line1.4.1 Intrinsic properties The characteristic impedance of a transmissionMost of the electromagnetic properties discussed line should be taken into serious considerationabove in this chapter are intrinsic, as they are gov- in the design of high-speed circuits. Figure 1.31erned by their respective underlying mechanisms,not by their geometries. This book concentrates on Rsource Z1 Z2 Z3the characterization of intrinsic properties of mate-rials. Here, we make a brief summary of the intrin-sic properties of materials often studied in physics,materials sciences, and microwave electronics. The parameters describing the intrinsic electro-magnetic properties of materials generally include Figure 1.31 Impedance discontinuities in a transmis-constitutive parameters, propagation parameters, sion line. The solid arrows represent the transmissionand electrical transport properties. The constitutive signals and the dashed arrows represent the reﬂec-parameter for low-conductivity materials mainly tion signals
Electromagnetic Properties of Materials 33shows a transmission line connected to a source, where the effective dielectric constant εe isand how the voltage and current signals from given bythe source interact with the transmission line as εr + 1 εr − 1 1they propagate along the transmission line. If the εe = + √ (1.104)characteristic impedance of the transmission line 2 2 1 + 12d/Wchanges, either by a geometry change or a material If we want to fabricate a microstrip transmis-change, some of the signals will be reﬂected. At the sion line of 50 from a substrate with dielec-interface between two segments of transmission tric thickness of 5 mils and dielectric constantlines with characteristic impedances Z1 and Z2 of 5, the line width should be around 9 mils. Ifrespectively, the reﬂectivity is given by the line width varies by ±1 mil, the characteris- Vreﬂected Z2 − Z1 tic impedance would vary by about ±10 %. If the = = . (1.102) dielectric constant varies by ±10 %, the character- Vincident Z2 + Z1 istic impedance would vary by about ±5 %. There- At each interface, there will be a series of fore, the line width is more sensitive to the char-reﬂections. Therefore, the impedance discontinu- acteristic impedance than the dielectric constant ofities distort signal, decrease the signal integrity, the substrate. More discussions on microstrip lineincrease standing waves, increase rise-time degra- and other types of transmission lines will be madedation, and require longer setting time. The way in Chapter 2.to solving these problems is to use the same char-acteristic impedance throughout the transmission 22.214.171.124 Reﬂectivity of a Dallenbach layerline, including traces and connectors. By consider-ing manufacturability, cost, noise sensitivity, and As shown in Figure 1.33, a Dallenbach layer ispower dissipation, the optimum Z0 value for most a homogeneous layer backed by a metal platesystems is in the range of 50 to 80 . and the dissipation of microwave energy is made Microstrip is the most widely used transmis- throughout the layer (Knott et al. 1993). Thesion line in microwave electronics. As shown reﬂectivity of a Dallenbach layer is dependent onin Figure 1.32, the structural parameters of a its thickness, and it is a typical example of extrinsicmicrostrip line are the width W of the microstrip performance. In the design of a Dallenbach layer,and the thickness d of the dielectric substrate. The we should consider the interferences betweencharacteristic impedance of a microstrip depends the reﬂections at different interfaces. When aon the dielectric constant of the substrate and microwave signal is incident to the layer, partthe structural parameters. For a thin substrate of the signal is reﬂected at the interface betweencase (d/W < 1), the characteristic impedance of the free space and the material layer, and thisa microstrip line is given by (Pozar 1998): part of signal is called the ﬁrst reﬂection. Part 120π Z0 = √ (1.103) εe [W/d + 1.393 + 0.667 Incident wave ln(W/d + 1.444)] Metal plate First reflection Second reflection W t d er d Figure 1.33 The structure and working principle of aFigure 1.32 Geometry of a microstrip line Dallenbach layer
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36 Microwave Electronics: Measurement and Materials CharacterizationValanju, P. M. Walser, R. M. and Valanju, A. P. (2002). using chiral composites,” Journal of Wave-Material “Wave refraction in negative-index media: Always Interaction, 2, 71–81. positive and very inhomogeneous”, Physical Review Von Hippel, A. R. (1995a). Dielectrics and Waves, Letters, 88 (18), art. no. 187401, 1–4. Artech House, Boston.Varadan, V. K. and Varadan, V. V. (1979). Acoustic, Von Hippel, A. R. Ed. (1995b). Dielectric Materials and Electromagnetic and Elastic Wave Scattering – Focus Applications, Artech House, Boston. on the T-matrix Approach, Pergamon Press, New Winkler, M. H. (1956). “An experimental investigation York. of some models for optical activity,” Journal ofVaradan, V. V. and Varadan, V. K (1985). Multiple Physical Chemistry, 60, 1656–1659. Scattering of Waves in Random Media and Random Wohlfarth, E. P. (1980). Ferromagnetic Materials: a Rough Surfaces, Pennsylvania State University Press, Handbook on the Properties of Magnetically Ordered Pennsylvania. Substances, Vol. 2, North Holland, Amsterdam.Varadan, V. V. Ma, Y and Varadan, V. K (1984). “Aniso- Yablonovitch, E. and Gmitter, T. J. (1989). “Photonic tropic dielectric properties of media containing non- band structure: the face-centered-cubic case”, Phys- spherical scatterers”, IEEE Transactions on Antennas ical Review Letters, 63 (18), 1950–1953. and Propagation, AP-33, 886–890. Zoughi, R. (2000). Microwave Non-Destructive TestingVaradan, V. K. Varadan, V. V. and Lakhtakia, A. (1987). and Evaluation, Kluwer Academic Publishers, Dor- “On the possibility of designing anti-reﬂection coatings drecht.
2 Microwave Theory and Techniques for Materials CharacterizationThis chapter discusses the basic microwave theory surface impedance of high-conductivity materials.and techniques for the characterization of elec- We do not discuss the detailed structures of ﬁx-tromagnetic materials. The methods for materials tures and detailed algorithms for the calculationproperties characterization generally fall into non- of materials properties, which will be discussedresonant methods and resonant methods; and, cor- in Chapters 3 to 8. The characterization of chi-respondingly, we mainly discuss two microwave ral materials will be discussed in Chapter 10 andphenomena: microwave propagation based on the measurement of microwave electrical transportwhich the nonresonant methods are developed and properties will be discussed in Chapter 11.microwave resonance based on which the resonant The microwave methods for materials charac-methods are developed. In our discussion, both the terization generally fall into nonresonant methodsﬁeld approach and the line approach are used in and resonant methods. Nonresonant methods areanalyzing electromagnetic structures. In the ﬁnal often used to get a general knowledge of electro-part of this chapter, we introduce the concept of magnetic properties over a frequency range, whilemicrowave network and discuss the experimental resonant methods are used to get accurate knowl-techniques for characterizing propagation and res- edge of dielectric properties at single frequency oronance networks. several discrete frequencies. Nonresonant methods and resonant methods are often used in combi- nation. By modifying the general knowledge of2.1 OVERVIEW OF THE MICROWAVE materials properties over a certain frequency rangeMETHODS FOR THE CHARACTERIZATION obtained from nonresonant methods with the accu-OF ELECTROMAGNETIC MATERIALS rate knowledge of materials properties at several discrete frequencies obtained from resonant meth-There have been many extensive review papers ods, accurate knowledge of materials properties(Afsar et al. 1986; Baker–Jarvis et al. 1993; Guil- over a frequency range can be obtained.lon 1995; Krupka and Weil 1998; Weil 1995; In the following, we discuss the working princi-Zaki and Wang 1995) on the microwave methods ples of various measurement methods. It should befor materials property characterization, and there noted that the examples given below are used onlyare also several monographs on special measure- to illustrate the basic conﬁgurations of the mea-ment methods and for special purposes (Musil and surement methods, and they are not necessarily theZacek 1986; Nyfors and Vainikainen 1989; Zoughi conﬁgurations with highest accuracy and sensitivi-2000). In this section, we focus on the basic prin- ties. As will be discussed in later chapters, there areciples for the measurement of the permittivity and many practical considerations in the developmentpermeability of low conductivity materials and the of actual measurement methods.Microwave Electronics: Measurement and Materials Characterization L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan and V. K. Varadan 2004 John Wiley & Sons, Ltd ISBN: 0-470-84492-2
38 Microwave Electronics: Measurement and Materials Characterization Incident wave 188.8.131.52 Reﬂection methods Sample In reﬂection methods, electromagnetic waves are directed to a sample under study, and the properties of the material sample are deduced from the reﬂection coefﬁcient at a deﬁned reference plane. Usually, a reﬂection method can only measure one Transmitted wave parameter, either permittivity or permeability. Two types of reﬂections are often used in Reflected wave materials property characterization: open-circuit reﬂection and short-circuit reﬂection, and the corresponding methods are called open-reﬂectionFigure 2.1 Boundary condition for material character-ization using a nonresonant method method and shorted reﬂection method. As coax- ial lines can cover broad frequency bands, coax- ial lines are often used in developing measure-2.1.1 Nonresonant methods ment ﬁxtures for reﬂection methods. Detailed dis- cussions on reﬂection methods can be found inIn nonresonant methods, the properties of materials Chapter 3.are fundamentally deduced from their impedanceand the wave velocities in the materials. As shownin Figure 2.1, when an electromagnetic wave prop- Open-reﬂection methodagates from one material to another (from freespace to sample), both the characteristic wave Figure 2.2 shows the basic measurement conﬁg-impedance and the wave velocity change, result- uration of an open – reﬂection method. In actualing in a partial reﬂection of the electromagnetic applications, the outer conductor at the open endwave from the interface between the two materials. is usually fabricated into a ﬂange to provide suit-Measurements of the reﬂection from such an inter- able capacitance and ensure the repeatability offace and the transmission through the interface can sample loading (Li and Chen 1995; Stuchly andprovide information for the deduction of permittiv- Stuchy 1980), and the measurement ﬁxture isity and permeability relationships between the two usually called the coaxial dielectric probe. Thismaterials. method assumes that materials under measurement Nonresonant methods mainly include reﬂection are nonmagnetic, and that interactions of the elec-methods and transmission/reﬂection methods. In tromagnetic ﬁeld with the noncontacting bound-a reﬂection method, the materials properties are aries of the sample are not sensed by the probe.calculated on the basis of the reﬂection from the To satisfy the second assumption, the thickness ofsample, and in a transmission/reﬂection method,the material properties are calculated on thebasis of the reﬂection from the sample and the Sampletransmission through the sample. Nonresonant methods require a means of direct-ing the electromagnetic energy toward a material,and then collecting what is reﬂected from the mate-rial, and/or what is transmitted through the mate-rial. In principle, all types of transmission lines canbe used to carry the wave for nonresonant methods,such as coaxial line, hollow metallic waveguide,dielectric waveguide, planar transmission line, andfree space. In this section, we use hollow metallicwaveguide or coaxial line as examples. Figure 2.2 Coaxial open-circuit reﬂection
Microwave Theory and Techniques for Materials Characterization 39 Sample High- conductivity materialFigure 2.3 Coaxial short-circuit reﬂectionthe sample should be much larger than the diame- Figure 2.4 Reﬂection method for the measurement ofter of the aperture of the open-ended coaxial line, surface impedance of high-conductivity materialsand, meanwhile, the material should have enoughloss. that both the inner and the outer conductors of the open end of the coaxial line have goodShorted reﬂection method electrical contact with the sample. This methodFigure 2.3 shows a coaxial short-circuit reﬂection. has been used for the measurement of the surfaceIn this method, the sample under study is usually resistance of high-temperature superconductingelectrically short, and this method is often used thin ﬁlms (Booth et al. 1994).to measure magnetic permeability (Guillon 1995;Fannis et al. 1995). In this method, the permittivityof the sample is not sensitive to the measurement 184.108.40.206 Transmission/reﬂection methodsresults, and in the calculation of permeability, the In a transmission/reﬂection method, the materialpermittivity is often assumed to be ε0 . under test is inserted in a piece of transmission line, and the properties of the material are deduced onReﬂection method for surface impedance the basis of the reﬂection from the material and themeasurement transmission through the material. This method is widely used in the measurement of the permittiv-Besides their applications in the measurement of ity and permeability of low conductivity materials,permittivity and permeability of low-conductivity and it can also be used in the measurement ofmaterials, reﬂection methods are also used for the the surface impedance of high-conductivity materi-measurement of the surface impedance of high- als. Detailed discussions on transmission/reﬂectionconductivity materials. As shown in Figure 2.4, methods can be found in Chapter 4.the high-conductivity material under study contactsthe open end of a coaxial line. Microwaveradiation can propagate into some extent of the Permittivity and permeability measurementhigh-conductivity material. The complex surfaceimpedance of the sample can be extracted from the Transmission/reﬂection method can measure bothcomplex reﬂection coefﬁcient. As the penetration permittivity and permeability of low conductivitydepth of high-conductivity materials is small, materials. Figure 2.5 shows the conﬁguration ofmicrowave cannot go deep into the sample. This coaxial transmission/reﬂection method. The char-method does not require a very thick sample, but acteristic impedance of the piece of transmissionthe thickness of the sample should be several line loaded with the sample is different fromtimes larger than the penetration depth. Because that of the transmission line without the sample,there are electrical currents ﬂowing between the and such difference results in special transmis-inner and the outer conductors of the coaxial sion and reﬂection properties at the interfaces.line through the sample, this method requires The permittivity and permeability of the sample
40 Microwave Electronics: Measurement and Materials Characterization 2.1.2 Resonant methods Z0 Z Z0 Resonant methods usually have higher accuracies and sensitivities than nonresonant methods, and they are most suitable for low-loss samples. e0, m0 e, m e0, m0 Resonant methods generally include the resonator method and the resonant-perturbation method. The resonator method is based on the fact thatFigure 2.5 Coaxial transmission/reﬂection method the resonant frequency and quality factor of a dielectric resonator with given dimensions are determined by its permittivity and permeability.are derived from the reﬂection and transmission This method is usually used to measure low-coefﬁcients of the sample-loaded cell (Weir 1974; loss dielectrics whose permeability is µ0 . TheNicolson and Rose 1970). resonant-perturbation method is based on resonant- perturbation theory. For a resonator with given electromagnetic boundaries, when part of the electromagnetic boundary condition is changed bySurface impedance measurement introducing a sample, its resonant frequency and quality factor will also be changed. From theReﬂection/transmission method can also be used changes of the resonant frequency and qualityfor the measurement of surface impedance of high- factor, the properties of the sample can be derived.conductivity thin ﬁlms. As shown in Figure 2.6,the thin ﬁlm under study forms a quasi-short circuitin a waveguide transmission structure. From the 220.127.116.11 Resonator methodratio of the transmitted power to the incident This method is often called dielectric resonatorpower and the phase shift across the thin ﬁlm, method. It can be used to measure the permittivitythe surface impedance of the thin ﬁlm can be of dielectric materials and the surface resistance ofdeduced. However, this method is only suitable conducting materials. More detailed discussions onfor thin ﬁlms whose thickness is less than the the resonator method can be found in Chapter 5.penetration depth of the sample, and requiresa measurement system with very high dynamicrange. This method has been used to study the Measurement of dielectric permittivitymicrowave surface impedance of superconducting In a resonator method for dielectric property mea-extrathin ﬁlms (Dew–Hughes 1997; Wu and Qian surement, the dielectric sample under measurement1997). serves as a resonator in the measurement circuit, and dielectric constant and loss tangent of the sam- ple are determined from its resonant frequency and quality factor (Kobayashi and Tanaka 1980). Sample Figure 2.7 shows the conﬁguration often used in the dielectric resonator method. In this conﬁg- uration, the sample is sandwiched between two conducting plates, and the resonant properties of this conﬁguration are mainly determined by the properties of the dielectric cylinder and the two pieces of the conducting plates. In the measure- Waveguide ment of the dielectric properties of the dielectric cylinder, we assume the properties of the conduct-Figure 2.6 Transmission/reﬂection method for surface ing plates are known. The TE011 mode is oftenimpedance measurements selected as the working mode, as this mode does
Microwave Theory and Techniques for Materials Characterization 41 D factor of the cavity are changed subsequently. The wall-loss perturbation method is usually used to Z measure the surface resistance of conductors. In the material perturbation method, the introduction of the material into a cavity causes changes in the Sample resonant frequency and quality factor of the cavity. L The material perturbation method is also called Conducting the cavity-perturbation method, and is suitable plates for measuring low-loss materials. More detailed Y discussions on resonant-perturbation methods can X be found in Chapter 6.Figure 2.7 A dielectric cylinder sandwiched betweentwo conducting plates Permittivity and permeability measurement In the cavity perturbation method, the samplenot have a transverse electric ﬁeld between the under study is introduced into an antinode ofsample and the conducting plates. Therefore, a the electric ﬁeld or magnetic ﬁeld, dependingsmall gap between the sample and the plates does on whether permittivity or permeability is beingnot greatly affect the measurement results. This measured. As shown in Figure 2.8, if the samplemethod can be used to measure high dielectric under study is introduced into place A withconstant (Cohn and Kelly 1966), low loss (Krupka maximum dielectric ﬁeld and minimum magneticet al. 1994), and anisotropic materials (Geyer and ﬁeld, the dielectric properties of the sample canKrupka 1995). be characterized; if the sample is inserted into place B with maximum electric ﬁeld and minimum magnetic ﬁeld, the magnetic properties of theSurface resistance measurement sample can be characterized.The conﬁguration shown in Figure 2.7 can also beused for the measurement of the surface resistance Surface resistance measurementof conductors. If the dielectric properties of thedielectric cylinder are known, from the quality fac- As shown in Figure 2.9, in this method, the endtor of the whole resonant structure, we can calcu- wall of a hollow metallic cavity is replaced by thelate the surface resistance of the conducting plates.18.104.22.168 Resonant-perturbation methodWhen a sample is introduced into a resonator, Athe resonant frequency and quality factor of theresonator will be changed, and the electromagnetic Bproperties of the sample can be derived from thechanges of the resonant frequency and qualityfactor of the resonator. Generally speaking, thereare three types of resonant perturbations: cavityshape perturbation, wall-loss perturbation, andmaterial perturbation. Cavity shape perturbation is Figure 2.8 Cylindrical cavity (TM010 mode) foroften used to adjust the resonant frequency of a measurement of materials properties using reso-cavity. In the wall-loss perturbation method, part nant-perturbation method. Position A is for permittiv-of the cavity wall is replaced by the sample under ity measurement and position B is for permeabilitystudy, and the resonant frequency and quality measurement
42 Microwave Electronics: Measurement and Materials Characterization types of methods are microwave propagation and microwave resonance. In this section, we discuss microwave propagation, and microwave resonance will be discussed in Section 2.3. Cavity In this section, we start with transmission-line theory, and then we introduce transmission Smith Charts, which are powerful tools in microwave the- ory and engineering. Afterward, we discuss three Conducting categories of transmission lines widely used in sample materials property characterization: guided trans- mission lines, surface-wave transmission lines, and free space.Figure 2.9 Cavity perturbation method for the mea- 2.2.1 Transmission-line theorysurement of surface resistance of conductors In this part, line approach will be used to analyze transmission structures. We use equivalent circuitsconducting sample under test. With the knowledge to represent transmission structures, and discussof the geometry and the resonant mode of the the propagation of equivalent voltage and currentcavity, the surface resistance of the conducting along transmission structures.sample can be derived from the change of thequality factor due to the replacement of the endwall. 22.214.171.124 General properties of transmission For better understanding and application of var- structuresious methods for materials property characteriza-tion, in the following two sections, we discuss We consider a general cylindrical metallic trans-two important microwave phenomena: microwave mission structure whose cross sections do notpropagation, which is the basis for nonresonant change in z-direction. As shown in Figure 2.10,methods, and microwave resonance, which is the there are two types of metallic transmission lines,basis for resonant methods. In the last section of one is a single hollow metallic tube and the otherthis chapter, we discuss microwave network, which consists of two or more conductors.is often used in microwave theoretical analysis and The propagation of an electromagnetic waveexperimental measurement. along a transmission structure can be analyzed Field approach and line approach are often using Maxwell’s equations. From Eqs. (1.1)–(1.4),used in microwave theory and engineering. In the we can get the wave equations for electric ﬁeld Eﬁeld approach, we analyze the distributions of theelectric and magnetic ﬁelds. In the line approach,we use equivalent circuits to represent microwave Z Zstructures. In the following discussions, both theﬁeld and the line approaches are used.2.2 MICROWAVE PROPAGATIONAs discussed above, there are two general types (a) (b)of methods for materials property characterization:nonresonant methods and resonant methods, and Figure 2.10 Two types of cylindrical metallic trans-the microwave phenomena related to these two mission lines. (a) Hollow tube and (b) two metal bars
Microwave Theory and Techniques for Materials Characterization 43and magnetic ﬁeld H : and such a wave is also called H wave. If Hz = 0, the electromagnetic wave is called transverse mag- ∇ 2E + k2E = 0 (2.1) netic (TM) wave, and such a wave is also called E ∇ H + k H = 0, 2 2 (2.2) wave. If Ez = 0 and Hz = 0, the electromagnetic wave is called transverse electromagnetic (TEM)where k = 2π/λ is the wave number and λ is the wave. In the following, we discuss the generalwavelength. properties of TE, TM, and TEM waves. In the transmission structure, the electromag-netic ﬁelds can be decomposed into the transverse TE wavecomponents (ET and HT ) and the axial components(Ez and Hz ): For TE wave (Ez = 0), from Eqs. (2.9) and (2.10), we have E = ET + Ez (2.3) jkz HT = − 2 ∇ H 2 T z (2.11) H = HT + Hz (2.4) k − kz jωµ ωµWe can also revolve the ∇ operator into the ET = z × ∇T Hz = − ˆ z × HT (2.12) ˆtransverse part ∇T and axial part ∇Z : k2 − kz 2 kz ∇ = ∇T + ∇Z (2.5) ˆ Equation (2.12) indicates that ET , HT , and z are perpendicular to each other. If we deﬁne the wavewith impedance of TE wave as ∂ ωµ∇Z = z ˆ (2.6) ηTE = , (2.13) ∂z kz ∂ ∂ Eq. (2.12) can be rewritten as∇T = x ˆ +y ˆ (rectangular coordinates) ∂x ∂y ET = −ηTE z × HT ˆ (2.14) (2.7) ∂ ˆ1 ∂ The wave impedance for TE wave can be∇T = r ˆ +φ (cylindrical coordinates) rewritten as ∂r r ∂φ (2.8) √ ω εµ µ k ˆ ηTE = = η (2.15)where z , x , y , r , and φ are the unit vectors along ˆ ˆ ˆ ˆ kz ε kztheir corresponding axes. with the wave impedance of plane wave From Eqs. (2.1)–(2.8), we can get the propaga-tion equations for metallic waveguides µ η= (2.16) ε (k 2 − kz )HT = −jωεz × ∇T Ez − jkz ∇T Hz 2 ˆ (2.9) and the wave number √ (k 2 − kz )ET = −jωµˆ × ∇T Hz − jkz ∇T Ez , 2 z k = ω εµ (2.17) (2.10) TM wavewhere kz is the wave number in Z-direction.Equations (2.9) and (2.10) show the relationships For TM wave (Hz = 0), from Eqs. (2.9) andbetween the transverse and axial ﬁelds. If Ez and (2.10), we haveHz are known, other components of electromag- jkznetic waves can be calculated based on Ez and Hz . ET = − ∇ E 2 T z (2.18) k2 − kz There are three types of electromagnetic waveswith special Ez and Hz . If Ez = 0, the electromag- jωµ ωε HT = − z × ∇T Ez = ˆ z × ET ˆ (2.19)netic wave is called transverse electric (TE) wave, k2− kz 2 kz
44 Microwave Electronics: Measurement and Materials Characterization ˆEquation (2.19) also indicates that ET , HT , and zare perpendicular to each other. Also, we deﬁnethe wave impedance of TM wave: kz ηTM = (2.20) (a) (b) (c) ωεSimilarly, the wave impedance for TM wave can Figure 2.11 Transmission structures for TE and TMbe rewritten as waves. (a) Rectangular waveguide, (b) circular waveg- uide, and (c) ridged waveguide kz ηTM = η (2.21) kTEM waveFor TEM wave (Ez = Hz = 0), from Eqs. (2.9)and (2.10), we have (a) (b) (c) (k − 2 2 kz )HT =0 (2.22) Figure 2.12 Transmission structures for TEM waves. (k 2 − kz )ET = 0 2 (2.23) (a) Two parallel lines, (b) coaxial line, and (c) striplineAs ET = 0 and HT = 0, from Eqs. (2.22) and Static electric ﬁeld cannot be built within a hollow(2.23), we have tube, and TEM wave also cannot propagate in a k = kz (2.24) hollow tube. But TE wave and TM wave couldEquation (2.24) indicates that the wave number of propagate in hollow tubes, and some hollow tubeTEM wave propagating in the z-direction is equal examples are shown in Figure 2.11. TEM wavesto the wave number of that of a plane wave. Also, can only propagate in a transmission structure withthe wave impedance for TEM wave is equal to that at least two conductors, and some examples offor plane wave: TEM transmission lines are shown in Figure 2.12. ηTEM = η. (2.25) 126.96.36.199 Propagation equationsAlso we have The above discussion indicates that, in the cross ET = −ηTEM z × HT ˆ (2.26) section of a TEM transmission line, the distribution of the electromagnetic ﬁeld is the same as ˆEquation (2.26) shows that ET , HT , and z are that of the static ﬁeld, so we can introduceperpendicular to each other. the concepts of equivalent voltage, equivalent Meanwhile, it can be proven that (Pozar 1998), current, and equivalent impedance. Therefore, wefor TEM wave, its ﬁeld ET at a cross section can can use the “line” method to analyze the TEMbe expressed by a scalar potential : transmission line. However, to ensure that this method is applicable, the geometrical length l ET = −∇T (2.27) of the transmission line should be comparableWe can further get that for a source-free transmis- to the wavelength λ, which means l/λ ≥ 1, andsion line the current and voltage distribution along the ∇T = 0 2 (2.28) transmission line are not uniform. In engineering, such a transmission line is called long line.Equation (2.28) indicates that if a TEM wave It should be indicated that in the following dis-can propagate in a transmission line, the static cussion, we use TEM transmission line. Actually,ﬁeld can also be established there and vice versa. TE transmission line and TM transmission line can
Microwave Theory and Techniques for Materials Characterization 45 Zg Vg ϳ ZL ∆z R1∆z L1∆z G1∆x C1∆z (a) (b)Figure 2.13 Equivalent of a transmission line. (a) Equivalent circuit for a short line element and (b) equivalentcircuit for a long line. In the ﬁgure, Zg and Vg are the impedance and driving voltage of the generator, and ZL isthe impedance of the loadalso be analyzed in this way, provided suitable i (z, t) i (z + ∆z, t)equivalent current and equivalent voltage are intro- R1∆z L1∆zduced.Equivalent circuit of transmission line G1∆z C1∆zOwing to the distribution properties of a transmis-sion line, the voltage and current on a long line are ∆z zfunctions of both time and positions. The distribu-tions of the voltage and current are mainly deter- Figure 2.14 A short transmission line with length zmined by the shape, dimension, and the properties at position zof the conductors and dielectrics. In following dis-cussion, we assume that the cross section of the voltage and current at the two ends of the elementtransmission structure does not change along with be: v(z, t), i(z, t) and v(z + z, t), i(z + z, t).its axis. According to the Kirchhoff’s law, we have We divide a long line into many short lineelements with length z ( z λ), and represent − v(z, t) = v(z + z, t) − v(z, t)the short line elements with effective parameters: ∂i(z, t) = R1 z · i(z, t) + L1 zR1 z, L1 z, G1 z, and C1 z, with R1 , L1 , G1 ∂tand C1 representing the resistance, inductance, (2.29)conductance and capacitance of the line element − i(z, t) = i(z + z, t) − i(z, t)respectively. The transmission line can therefore ∂v(z, t)be represented by an equivalent circuit, as shown = G1 z · v(z, t) + C1 z ∂tin Figure 2.13. (2.30) Letting z → 0, we can get the famous Telegra-Telegrapher equations pher equationsConsider a short line element with length z and ∂v(z, t) ∂i(z, t) − = R1 i(z, t) + L1 (2.31)starting point at z, as shown in Figure 2.14. Let the ∂z ∂t
46 Microwave Electronics: Measurement and Materials Characterization ∂i(z, t) ∂v(z, t) From Eqs. (2.33) and (2.37), we can get − = G1 v(z, t) + C1 (2.32) ∂z ∂t 1 1 The above Telegrapher equations can be rewrit- I (z) = V0+ e−γ z + − V0− eγ z , (2.45) Zc Zcten as dV (z) where Zc is the characteristic impedance of the − = Z1 I (z) (2.33) transmission line: dz dI (z) − = Y1 V (z) (2.34) Z1 Z1 1 dz Zc = = = , (2.46) γ Y1 Ycwith v(z, t) = V (z)ejωt (2.35) where Yc is the characteristic admittance of the i(z, t) = I (z)e jωt (2.36) transmission line. By rewriting Eq. (2.45) as Z1 = R1 + jωL1 (2.37) V0+ −γ z V0− γ z Y1 = G1 + jωC1 . (2.38) I (z) = I0+ e−γ z + I0− eγ z = e + e Zc −ZcEquations (2.33) and (2.34) indicate that, in the (2.47)transmission line, the change of voltage is due to we can get V0+the series impedance Z1 and the change of current = Zc (2.48)is due to the parallel admittance Y1 . I0+ V0− = −Zc (2.49)Characteristic parameters of a transmission line I0−Following propagation equations can be obtained The negative sign in Eq. (2.49) is totally due to thefrom Eqs. (2.33) and (2.34): deﬁnitions of voltage, current, and the direction of z, and it does not indicate negative impedance. d2 V In summary, Eqs. (2.42) and (2.45) are the = γ 2V (2.39) dz2 solutions of the Telegrapher Equations, and they d2 I indicate that there are voltage waves and current = γ 2I (2.40) waves propagating in +Z direction and −Z dz2 direction. As the propagation constant γ is awith the transmission constant given by complex number, these waves attenuate along the transmission line. From Eq. (2.46), we have γ = Y1 Z1 = α + jβ (2.41) γ = α + jβ = Y1 Z1The general solutions for Eqs. (2.39) and (2.40)are = (R1 + jωL1 )(G1 + jωC1 ) (2.50) −γ z V = V0+ e + V0− e γz (2.42) with −γ z I = I0+ e + I0− e , γz (2.43) 1where V0± and I0± are constants that should be (R1 + ω2 L2 )(G2 + ω2 C1 ) 2 1 1 2 α= 2 (2.51)deﬁned by the boundary conditions. Equations(2.42) and (2.43) indicate that there may exist − (ω2 L1 C1 − R1 G1 )waves propagating along the +z direction and the−z direction, respectively. The phase velocity is 1 (R1 + ω2 L2 )(G2 + ω2 C1 ) 2 2 1 1given by β= 2 (2.52) ω vp = (2.44) + (ω2 L1 C1 − R1 G1 ) β
Microwave Theory and Techniques for Materials Characterization 47According to Eq. (2.46), the characteristic imped- Zc becomesance of the transmission line is given by L1 j R1 G1 L1 Zc = 1− − ≈ R1 + jωL1 L11 − j(R1 /ωL1 ) C1 2 ωL1 ωC1 C1 Zc = = G1 + jωC1 1 − j(G1 /ωC1 ) C1 (2.63) (2.53)In the following, we discuss three typical con-ditions: high loss, low loss, and no loss. In No-loss transmission linemicrowave electronics, most of the actual condi- This is an ideal case. At microwave frequencies,tions can be approximated as these conditions. no actual transmission lines can be strictly no loss. However, if the transmission line is made of good conductors and low-loss dielectric, and theHigh-loss transmission line transmission line is not very long, we can neglectFor a transmission line with high loss, we assume its loss. For a no-loss transmission line, ωL1 R1 (2.54) R1 = 0, (2.64) ωC1 G1 (2.55) G1 = 0. (2.65)From Eqs. (2.50)–(2.53), we can get From Eqs. (2.50)–(2.53), we can get α≈ R1 G1 (2.56) α=0 (2.66) β≈0 (2.57) β = ω L1 C1 (2.67) R1 L1 Zc ≈ (2.58) Zc = (2.68) G1 C1So the electromagnetic waves attenuate quickly, In this condition, Eqs. (2.42) and (2.43) becomeand cannot propagate in such a transmission line.Matched load can be taken as such a kind of V (z) = V0+ e−jβz + V0− ejβz (2.69) −βztransmission line. I (z) = I0+ e + I0− e jβz V0+ −jβz V0− jβz = e + e (2.70)Low-loss transmission line Zc −ZcFor a low-loss transmission line, we assume Equations (2.69) and (2.70) indicate that both the voltage and current waves can propagate along +Z ωL1 R1 (2.59) direction and −Z direction. As the characteristic impedance is a real number, the current and voltage ωC1 G1 (2.60) are in phase. This is the most common situationFrom Eqs. (2.50)–(2.53), we can get in microwave engineering, and is usually called lossless transmission line or ideal transmission line. 1 C1 L1 α≈ R1 + G1 (2.61) 2 L1 C1 188.8.131.52 Reﬂection and impedance β ≈ ω L1 C1 (2.62) As discussed above, in a uniform transmissionEquation (2.62) indicates that β is almost indepen- line, both the voltage wave and current wave havedent of R1 and G1 . The characteristic impedance two components propagating along +z direction
48 Microwave Electronics: Measurement and Materials Characterization Input impedance Zc ZL On the basis of the deﬁnition of reﬂection coefﬁ- Z cient ( ), the total voltage V (z) and total current I (z) along a transmission line can be expressed asFigure 2.15 A transmission line connected to a loadwith impedance ZL V (z) = V+ (1 + ) (2.74) I (z) = I+ (1 − ) (2.75)and −z direction, and these two componentscan be called incident wave and reﬂection wave. The relationships between the total voltage V (z)In nonresonant methods for materials property and total current I (z) are described by the inputcharacterization, the sample under study is loaded impedance Zi and the input admittance Yi :to a transmission line. We will discuss how theload to a transmission line affects the relationship 1 V V+ (1 + )between the incident wave and reﬂected wave. Zi = = = (2.76) Yi I I+ (1 − ) Sometimes, we use normalized impedance:Voltage reﬂection coefﬁcient ZiAs shown in Figure 2.15, a load with impedance z= , (2.77)ZL is connected to a piece of transmission line with Zclength l. In analyzing the reﬂection properties, the where Zc is the characteristic impedance of theorigin of the axis is chosen at the place of load, and transmission line, and normalized admittance:the positive direction of the axis is from the loadto the generator, while the positive direction of Yicurrent is still from the generator to the load. The y= , (2.78) Ycrelationship between the voltage and the current isdetermined by the loading impedance: where Yc is the characteristic admittance of the VL transmission line. The relationships between the ZL = (2.71) reﬂection, impedance, and admittance are listed in IL Table 2.1. The voltage reﬂection coefﬁcient represents the The reﬂection coefﬁcient is periodical withvoltage ratio between the reﬂected voltage V− and period λg /2, and so are the impedance andincident voltage V+ : V− V0− e−jβz V0− −j2βz Table 2.1 Relationships between reﬂection coefﬁcient = = = e (2.72) and impedance and admittance V+ V0+ e+jβz V0+ Unnormalized NormalizedEquation (2.72) indicates that the reﬂection coefﬁ-cient is related to the position along the z-axis. As 1+ 1+ Relationship between Zi = Zc z=the origin of the axis is chosen at the position of the impedance and 1− 1−load, the reﬂection at the load is: L = V0− /V0+ . reﬂection Zi − Zc z−1 = =The reﬂection coefﬁcient at a position (z) is Zi + Zc z+1 −j2kZ z = Le (2.73) 1− 1− Relationship between Yi = Yc y= admittance and 1+ 1+It is clear that the amplitude of the reﬂection Yc − Yi 1−y reﬂection = =coefﬁcient does not change along a uniform Yc + Yi 1+ytransmission line.
Microwave Theory and Techniques for Materials Characterization 49 lg lg 4 2 Short Short Open ShortFigure 2.16 Impedance inversion along a transmission line. λg is the wavelength along the transmission lineadmittance. We can ﬁnd the inversion property of From Eqs. (2.82) and (2.83), we can getthe impedance: ZL + jZc tan kz z Zi = Zc (2.84) 1 Zc + jZL tan kz z z(z) = , (2.79) z(z + λg /4) zL + j tan kz z zi = (2.85) 1 + jzL tan kz zwhere z(z) is the normalized impedance at positionz, and z(z + λg /4) is the normalized impedance at YL + jYc tan kz z Yi = Yc (2.86)position (z + λg /4). As shown in Figure 2.16, if a Yc + jYL tan kz ztransmission line is loaded with a short, it is open, y L + j tan kz zseen at the position λg /4 away from the short, and yi = (2.87) 1 + jy L tan kz zis short again seen at the position λg /2 away fromthe short. The above equations indicate that if the load is purely reactive, the input impedance at any position is also purely reactive. We consider twoOther expressions for input impedance special cases. For a short loading (zL = 0), we haveIf we write the total voltage and total current in zi = j tan kz z (2.88)following forms For an open loading (zL = j∞), we haveV (z) = (V0+ + V0− ) cos kz z + j(V0+ − V0− ) sin kz z, 1 (2.80) zi = −j (2.89) tan kz z V0+ − V0− V0+ + V0−I (z) = cos kz z + j sin kz z, Zc Zc 184.108.40.206 Typical working states of transmission (2.81) linesthe input impedance is given by The distributions of current and voltage are (V0+ + V0− ) cos kz z determined by the properties of the loading. + j(V0+ − V0− ) sin kz z Zi (z) = (2.82) Here, we discuss three typical working states [(V0+ − V0− )/Zc ] cos kz z of transmission lines: pure travelling wave, pure + j[(V0+ + V0− )/Zc ] sin kz z standing wave, and mixed wave.As the input impedance at z = 0 is the impedanceof the load ZL , we have Pure travelling wave V0+ + V0− In this state, there is no reﬂection wave: = 0, ZL = Zc (2.83) V0+ − V0− V (z) = V+ , Zi = Zc , and zi = 1. The total voltage
50 Microwave Electronics: Measurement and Materials Characterizationis the incident voltage, and the input impedance load is short (ZL = 0), open (ZL = ∞), or purelyat any position in the transmission line equals reactive (ZL = jX), the transmission line is in purethe characteristic impedance of the transmission standing-wave state.line. The ratio between the voltage and current Figure 2.17 shows the standing wave in a trans-equals to Zc , and the current and voltage are in mission line terminated by a short. In this state,phase. ZL = 0 and L = −1. According to Eqs. (2.80) As the input impedance at any cross section and (2.81), we haveat the transmission line equals to Zc , the loadimpedance is also Zc , and such a load is called 2πa matching load. In microwave electronics, the V (z) = j2V0+ sin z (2.90) λmatching loads for coaxial line and waveguide are 2V0+ 2πnot an actual resistance, but a kind of material or I (z) = cos z (2.91)structure which can absorb all the energy from the Zc λgenerator. As shown in Figure 2.17(b), the phase differ- ence between the voltage and current is 90◦ .Pure standing wave Figure 2.17(c) shows the distributions of theIn this state, the load does not absorb any energy, amplitudes of the voltage and current along theall the energy is reﬂected: | | = 1. When the transmission line. l 7l 3l 5l l 3l l l 0 8 4 8 2 8 4 8 (a) t3 v, i t5 t2, t4 t4 t1, t5 t3 z 0 t2 t1 (b) V V , I I z (c) Z z l 3l l l 0 4 2 4 (d) (e)Figure 2.17 Standing wave of a shorted transmission line. (a) Short terminated transmission line, (b) instantdistributions of voltage (full line) and current (dashed line), (c) amplitudes of voltage and current, (d) the impedancedistribution along the transmission line, and (e) lumped equivalent circuits at typical positions
Microwave Theory and Techniques for Materials Characterization 51 Equation (2.88) indicates that the impedance ofthe transmission line shorted at the end is sure reac- ZLtive. As shown in Figure 2.17(d), the impedanceperiodically appears as inductive reactance and V ∼ (1 + Γ)capacitive reactance along z-direction with periodλ/2. Figure 2.17(e) shows the equivalent lumped-element circuits at several typical positions. Apiece of shorted transmission line with length lessthan λ/4 is equivalent to a lumped inductance, anda piece of shorted transmission line with length I ∼ (1 − Γ)equal to λ/4 is equivalent to a lumped parallel LCresonant circuit. A piece of shorted transmissionline with length more than λ/4 but less than λ/2is equivalent to a lumped capacitor, and a pieceof shorted transmission line with length equal toλ/2 is equivalent to a lumped series LC reso- Figure 2.19 The voltage and current distributionsnant circuit. along a loaded transmission line at the mixed state Figure 2.19 indicates that the distributions ofMixed wave voltage and current along a transmission line inIn most of the transmission lines, some of the mixed state are periodical with period λ/2. Theenergy is absorbed by the load and some of the voltage V is in the range of V+ (1 + | |) to V+ (1 −energy is reﬂected. This state is a mixture of | |), while current is in the range of I+ (1 + | |)pure travelling wave and pure standing wave. The to I+ (1 − | |), with I+ = V+ /Zc .current and voltage along the line are −j2kz z V = V+ (1 + Le ) (2.92) 2.2.2 Transmission Smith charts V+ −j2kz z In microwave engineering, it is often required to I= (1 − Le ), (2.93) Zc transform the impedance to different positions at the transmission line. Such transformation is usu-where L is the reﬂection coefﬁcient at the ally complicated (Eqs. (2.83)–(2.87)), while theloading, and = L exp(−j2kz z) is the reﬂection calculation of reﬂection coefﬁcient is relativelycoefﬁcient at position z. As L is a constant, the simple (Eqs. (2.72) and (2.73)). As there are corre-voltage and current can be drawn using vector sponding relationships between the impedance androtation method, as shown in Figure 2.18. reﬂection coefﬁcient, it is helpful to draw charts to represent the corresponding relationships, so that the transformation between impedance and reﬂec- tion can be easily conducted. Γ Actually, there are two types of charts, square chart and circle chart, for this purpose. In a square −Γ chart, the magnitude and phase of the reﬂection V ∼ (1 + Γ) coefﬁcient are plotted in rectangular normalized impedance coordinates, while in a circle chart, the real part and imaginary part of the normalized impedance are plotted on polar coordinates of I ∼ (1 − Γ) the reﬂection coefﬁcient. The equal radius curve represents the magnitude of reﬂection, while equalFigure 2.18 Vector forms of the voltage and current angle line represents the phase of reﬂection. As
52 Microwave Electronics: Measurement and Materials Characterizationthe reﬂection coefﬁcient cannot be larger than As z = r + jx and = u + jv, where r = R/Zcunity, the reﬂection coefﬁcient is always within the is the normalized resistance and x = X/Zc is theunit circle of the polar coordinate. In fact, these normalized reactance, Eq. (2.94) can be rewrittentwo kinds of charts indicate the transformation asrelationships between the impedance complexplane and reﬂection complex plane. Usually, the 1 + (u + jv) 1 − u2 − v 2 + j2v r + jx = =second chart is more often used and is often called 1 − (u + jv) (1 − u)2 + v 2Smith chart. (2.95) Smith chart provides great convenience for As the real parts and the imaginary parts at the twotransmission-line calculation. In the following sides of Eq. (2.95) should be equal, respectively,discussion, we concentrate on Smith chart. There we haveare two kinds of Smith charts: impedance Smith 1 − (u2 − v 2 )chart and admittance Smith chart. We will discuss r= (2.96) (1 − u)2 + v 2the impedance Smith chart ﬁrst, and the admittanceSmith chart can then be transformed from the 2v x= (2.97)impedance Smith chart. (1 − u)2 + v 2 The impedance Smith chart is based on220.127.116.11 Impedance Smith chart Eqs. (2.96) and (2.97). In the following, we discuss several sets of special lines in the impedance SmithThe impedance Smith chart is formed by plotting chart, including r circles, x circles, ρ circles, andthe impedance on the reﬂection polar plane. θ lines.Figure 2.20 shows the Smith chart and square chartin the impedance plane. As r ≥ 0 and ≤ 1, thetransformation between the square chart and Smith r circleschart is a transformation between the right halfplane in the impedance plane and the unit circle We rewrite Eq. (2.96) into the following form:area in the reﬂection plane. 2 2 The relationship between the reﬂection coefﬁ- r 1 u− + v2 = (2.98)cient and the normalized impedance is: 1+r 1+r 1+ Equation (2.98) represents a circle in the reﬂec- z= (2.94) tion plane with center (r/(r + 1), 0) and radius 1− jx jv 2 x = 0.5 x=1 x=2 1 r r = 0.5 r=1 r=2 u 0 1 2 3 −1 −2 x = −0.5 x = −2 x = −1 (a) (b)Figure 2.20 Normalized impedance charts. (a) Chart on impedance plane and (b) Smith chart (chart on reﬂectionplane)
Microwave Theory and Techniques for Materials Characterization 531/(r + 1). Different r values correspond to differ- 1+| | ρ= (2.100)ent circles which form a series of tangent circles 1−| |with the tangent point (1, 0). When r = 0, the cir-cle center is at the original point and the radius is 1, The points with the same standing-wave coefﬁ-representing a pure reactive state. When r → ∞, cient ρ form a circle whose center is the origin ofthe center is at (1, 0) and the radius becomes zero, the reﬂection plane. Different ρ values representso the circle becomes a point (1, 0). Figure 2.21(a) a series of circles having the same center. Whenshows a series of r circles. r > 1, ρ = r and when r < 1, ρ = (1/r). A series of ρ circles are shown in Figure 2.22(c).x circlesEquation (2.97) can be modiﬁed into the following θ linesform: According to the expression: = +j = 2 2 1 1 | |ejθ , it is clear that θ = tan−1 ( / ). As shown (u − 1) + v − 2 = (2.99) x x in Figure 2.22(c), the θ lines are a series of straight lines passing through the center. Usually,Equation (2.99) represents a circle in the reﬂection θ lines are not shown in Smith chart, while theplane with center (1, 1/x) and radius 1/x. Differ- corresponding θ values are labeled at the outermostent x values correspond to different circles which circle. Sometimes the electric length is labeledform a series of tangent circles with the tangent instead of θ values.point (1, 0). The center for the circle correspond- By combing the circles and lines in Figure 2.22,ing to x = 0 is at (1, ∞), and its radius is ∞. The we can get a full Smith chart. As shown inreal axis of the impedance plane represents a pureresistive state (x = 0). A circle with x > 0 is on Figure 2.22, to make the chart clear and simple,the upper half, representing an inductive reactance ρ circles and θ lines are usually not shown.state. A circle with x < 0 is at the lower half, rep- Smith chart is often used to ﬁnd the impedanceresenting a capacitive reactance state. For the cases of a point at a transmission from the impedancex = ±∞, the center of the circle is at (1, 0) and of another point at the transmission line. Forthe radius is 0, so the circles become a point (1, 0). a lossless transmission line, the module of theFigure 2.21(b) shows a series of x circles. reﬂection coefﬁcient does not change, and only the phase angle changes. So the normalized impedance points at different positions on the transmissionρ circles line are on a ρ circle. When we know theThe standing-wave coefﬁcient ρ can be calculated impedance at one point, the impedance at anotherfrom the reﬂection coefﬁcient : point can be obtained by rotating the point along jΓ′′ jΓ′′ jΓ′′ +1 90° +0.5 +2 135° 45° r =0 0.5 1 2 ∞ x=0 ∞ 180° q = 0° q B A Γ′ Γ′ 1 Γ′ −0.5 −2 −1 (a) (b) (c)Figure 2.21 Typical circles and lines at the plane. (a) r circles, (b) x circles, (c) ρ circles and θ lines
54 Microwave Electronics: Measurement and Materials Characterization 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.10 0.39 90 0.36 100 80 0.35 9 0.40 0.1 0.0 6 1.0 0.9 70 0.8 110 1.2 1 0.3 0.4 4 0.1 8 1.4 0.0 0.7 7 2 60 0.3 0.4 120 3 1.6 0.1 0.6 7 0.0 8 1.8 3 0.2 0.3 0.4 0 50 2 13 0.5 2.0 06 0. 19 0. 44 0. 31 0. e tiv 0.4 0.4 tec ro 40 0 5 14 0.2 pr 0.0 0 5 0.3 x 0.4 xx 0 xx 0 3. 0.6 xx xx x 4 0.2 nt 0.3 0.0 30 6 0.2 150 ne 1 0.4 po 0.8 9 com 4.0 ty 1.0 yvi 0.22 0.03 160 .47 0.28 acit 5.0 0 0.2 cap 1.0 20 a rd G 0.8 0.02 0.23 0.48 0.27 ths tow 0.6 90Waveleng 1.0 10 170 0.1 0.4 0.240.01 0.49 0.26 0.2 20 50 0.5 2.0 3.0 4.0 0.6 0.7 0.8 1.0 1.2 1.8 0.3 1.6 1.4 0.4 5.0 0.1 0.9 0.2 20 10 0.25 0.25 0 0 0 0 0 Intective reactions (xxxx)conductans component 50 0.2 20 0.24 0.01 0.49 0.26 -10 0.4 -170 0.1 10 0.6 0.23 0.02 0.48 0.27 8 0. -20 0.03 60 -1 0.2 0.22 1.0 5.0 0.47 0.28 0.1 4.0 20 4 0.2 -30 0 0.0 -15 6 0.3 0.2 1 0.4 9 xx 30 x 3.0 xx 5 0.2 xx 0.0 0 5 xx 0.3 0.4 -4 40 x 0 0.4 xx 0 -1 xx xx xx 40 06 0. xx 19 0. xx 44 0. xx 31 0. xx -5 xx 0.5 30 2.0 xx 0 xxx -1 07 0.1 xx 0. 8 xxx 50 0.3 1.8 xxx 3 0.4 2 0.6 0.1 1.6 8 -60 -120 0.0 7 0.7 2 0.3 1.4 0.4 0.1 3 9 0.8 -110 0.0 -70 6 1.2 0.9 1.0 1 0.3 0 .10 0.4 -80 0.15 4 -100 -90 0.11 0.14 0.35 0.40 0.12 0.13 0.39 0.36 0.38 0.37Figure 2.22 A Smith chart. Source: Pozar, D. M. (1998). Microwave Engineering, 2nd ed., John Wiley & Sons,Inc., New Yorkthe ρ circle. In the rotation, it should be noted that direction of the z-axis is from the load to the signalthe θ value is calculated starting from the positive generator. axis. The θ value increases along the counterclockwise direction, while it decreases along the 18.104.22.168 Admittance Smith chartclockwise direction. As shown in Figure 2.15, ina transmission line, the zero position is chosen at In some cases, it is more convenient to usethe end of the transmission line, and the positive admittance. For a complex admittance y = g + jb,
Microwave Theory and Techniques for Materials Characterization 55its normalized conductance is g = GZc , and its complex plane, an admittance Smith chart can benormalized susceptance is b = BZc . According to obtained by rotating the whole impedance SmithTable 2.1, the relationship between admittance and chart by 180◦ , as shown in Figure 2.23. So a Smithreﬂection coefﬁcient is chart can be used as an impedance Smith chart or 1− 1 + (− ) an admittance Smith chart. y= = (2.101) However, it is necessary to give special attention 1+ 1 − (− ) to some special points and lines on the two Smith Equations (2.94) and (2.101) indicate that the charts. The main differences of the two Smithrelationship between y and (− ) are the same as charts are listed in Table 2.2.those between Z and . So the g lines and b lines Smith charts are powerful tools widely used inare also two groups of orthogonal tangent circles, microwave engineering. With known impedance orand they have the same shapes as r lines and x admittance, we can calculate the reﬂection coefﬁ-lines. As every point on the impedance Smith chart cient and standing-wave coefﬁcient. We can alsocan be converted into its admittance counterpart by calculate the impedance or admittance from thetaking a 180◦ rotation around the origin of the standing-wave coefﬁcient and position of voltage Toward load Toward load x=1 b = −1 r=1 g=1 −1 1 −1 0 0 1 x = −1 b=1 Toward generator Toward generator (a) (b)Figure 2.23 Two types of Smith charts. (a) Impedance Smith chart and (b) admittance Smith chart Table 2.2 Comparison between the impedance Smith chart shown in Figure 2.23(a) and the admittance Smith chart shown in Figure 2.23(b) Impedance Smith chart Admittance Smith chart Point “1” Open point, Open point, = 1, r = ∞, x = ∞ = 1, g = 0, b = 0 Point “−1” Short point Short point = −1, r = 0, x = 0 = −1, g = ∞, b = ∞ Point “0” Matching point Matching point = 0, r = 1, x = 0 = 0, g = 1, b = 0 Line “−1” to “0” Voltage nodes Voltage nodes r < 1, x = 0 g > 1, b = 0 Line “0” to “1” Voltage antinodes Voltage antinodes r > 1, x = 0 g < 1, x = 0 Upper half circle Inductive impedance Inductive admittance x>0 b<0 Lower half circle Capacitive impedance Capacitive admittance X<0 b>0
56 Microwave Electronics: Measurement and Materials Characterizationnode. Using Smith chart, we can realize impedancetransformation at two points on a transmission zline, and we can also design impedance match-ing. Besides its applications in transmission lines, bas will be discussed in Section 2.4.7, Smith charts a xcan also be used in analyzing resonant structures. 0 ϕ r2.2.3 Guided transmission lines Figure 2.24 The structure of coaxial lineHere, we discuss several typical kinds of transmis-sion lines often used in materials property char- central conductor and outer conductor is air, theacterization, including coaxial lines, planar trans- coaxial line is usually called coaxial air line.mission lines, and hollow metallic waveguides. In materials property characterization, coaxial airCoaxial lines and planar transmission lines can lines are often used, and the toroidal samples undersupport TE mode, TM mode, and TEM mode or test are inserted in the space between the centralquasi-TEM mode, while hollow metallic waveg- conductor and outer conductor.uides cannot support TEM mode, but can support Coaxial lines can support TEM, TE, and TMTE or TM modes. modes, and TEM mode is its fundamental mode. In materials property characterization, both the As in most of the microwave applications, singleequivalent lumped parameters and the ﬁeld distri- mode is required, most of the coaxial lines workbutions of transmission lines are important. In the at the TEM mode. In the following discussion, wefollowing discussions, line approach and the ﬁeld focus on TEM mode.approach are used in combination. According to Eq. (2.28), for a coaxial line shown in Figure 2.25, its potential function22.214.171.124 Coaxial line satisﬁes the two-dimensional Laplace’s equation:As shown in Figure 2.24, a coaxial line mainly 1 ∂ ∂ 1 ∂2 ∇T 2 = r + =0 (2.102)consists of a central conductor with diameter a r ∂r ∂r r 2 ∂ϕ 2and an outer conductor with inner diameter b. As the potential function does not change withFor coaxial cables used in microwave circuits, ϕ(∂ /∂φ = 0), Eq. (2.102) becomesthe space between the central conductor and outerconductor is ﬁlled with a dielectric material, such 1 d d r =0 (2.103)as Teﬂon. If the dielectric material between the r dr dr l 2 Signal Φ = V0 Power Signal Power Φ=0 (a) (b)Figure 2.25 Field distributions of TEM mode in a coaxial line. (a) Field distribution at a transverse crosssection and (b) the ﬁeld distribution along the z-axis. Modiﬁed from Ishii, T. K. (1995). Handbook of MicrowaveTechnology, vol 1, Academic Press, San Diago, CA, 1995
Microwave Theory and Techniques for Materials Characterization 57The general solutions for Eq. (2.103) is where Rs is the surface resistance of the conduc- tor and λ0 is the free-space wavelength. If the (r) = C1 ln r + C2 (2.104) conductor used is copper at 20 ◦ C, the conductor attenuation αc can be calculated using the follow-According to the boundary conditions: (a) = V0 ing equation (Chang 1989):and (b) = 0, we can get √ √ V0 9.5 × 10−5 f (a + b) εr C1 = (2.105) αc = ln(a/b) ab ln(b/a) C2 = −C1 ln b (2.106) (dB/unit length), (2.115)So the electric and magnetic ﬁelds of a TEM wave where f is the operating frequency.propagating in +z direction are ∂ −jkz 1 V0 126.96.36.199 Planar transmission line ET = −ˆ r e =r ˆ e−jkz ∂r r ln(b/a) As the characteristics of a planar transmission line (2.107) can be controlled by the dimensions in a single 1 1 ε V0 HT = z × Er e−jkz = ϕ ˆ ˆ e−jkz plane, the circuit fabrication can be conveniently η r µ ln(b/a) carried out by photolithography and photoetching (2.108) techniques. The application of these techniques atThe ﬁeld distributions are shown in Figure 2.25. microwave frequencies has led to the development The characteristic impedance Zc is deﬁned by of microwave integrated circuits. As will be V02 V0 2P discussed in Chapter 7, planar transmission lines Zc = = = 2, (2.109) are also used in materials property characterization. 2P I I As shown in Figure 2.26, three types of pla-where I is the current ﬂowing in the coaxial line nar transmission lines are often used in microwaveand P is the power transmitted by the line. For a electronics and materials characterization: stripline,coaxial line, its characteristic impedance is microstrip, and coplanar waveguide. The stripline shown in Figure 2.26(a) has an advantage that the η b 1 µ b Zc = ln = ln (2.110) radiation losses are negligible. The propagation in 2π a 2π ε a a stripline is in pure TEM mode, and striplineAs the ﬁlling medium in a coaxial line is usually circuits are usually quite compact. The problemdielectric and the wave impedance of free space is with stripline is the difﬁculty of construction.377 , we have Usually, two substrates are required to be sand- wiched together, and the air gaps between the sub- 60 b strates may cause perturbation to the impedance. Zc = √ ln ( ) (2.111) εr a Microstrip line, shown in Figure 2.26(b), is the most widely used planar transmission structures. The attenuation of a coaxial line consists of Usually the propagation mode on a microstripconductor attenuation αc and dielectric attenuation circuit is quasi-TEM. For the development high-αd : density microstrip circuits, thin substrates are often α = αc + αd (2.112)with 4.34Rs 1 + (b/a) αc = (dB/unit length) 2bη ln(b/a) (a) (b) (c) (2.113) √ tan δ Figure 2.26 Cross-sectional views of three types of αd = 27.3 εr (dB/unit length), λ0 transmission planar lines. (a) Stripline, (b) microstrip, (2.114) and (c) coplanar waveguide
58 Microwave Electronics: Measurement and Materials Characterization of Zc . In the following, we discuss two special cases. Grounds Thin central conductor (t/b 1) For a stripline with a thin central conductor, if b t Central w the width of the control conductor w is much conductor larger than the distance between the ground plate and the central conductor (b/2), the ﬁeld betweenFigure 2.27 Structure of a stripline the central conductor and the ground plate is uniform except the ﬁelds at the edges. Usingused to maintain reasonable impedance and to conformal mapping techniques, we can get anreduce the coupling between different parts of the appropriate equation for the stripline with zero-circuit. As shown in Figure 2.26(c), the circuit line thickness central conductor (t = 0):and the grounding of a coplanar waveguide are onthe same plane, and the wave propagation mode is 120π 2 Zc = √ ( ) (2.118)also quasi-TEM. 8 εr cosh−1 e(πw)/(2b) As the central conductor of an actual stripline hasStripline certain thickness, Eq. (2.118) can be modiﬁed intoAs shown in Figure 2.27, a stripline consists of 120π 2 (1 − t/b)upper and down grounding plates, and the central Zc = √ (2.119) 8 εr cosh−1 e(πw)/(2b)conductor. Between the grounding plates and thecentral conductor is air or dielectric materials.This structure can be taken as a derivation from Thick central conductora coaxial line, by cutting the outer conductor intotwo pieces and ﬂattening them. Usually, the ﬁlling If the condition (t/b 1) cannot be satisﬁed, themedium is dielectric (µr = 1), and the dimensions calculation of the distributed capacitance becomesof the transverse cross section (b and w deﬁned in complicated. We consider two conditions: wideFigure 2.27) of a stripline are much less than the central conductor and narrow central conductor.wavelength. If the width of the central conductor satisﬁes The fundamental propagation mode for a the condition w/(b − t) ≥ 0.35, we can assumestripline is TEM. For the TEM wave propagating that the ﬁeld at right side and left side do notin a stripline, the phase velocity is interfere. As shown in Figure 2.28, the distributed capacitance C1 mainly consists of two parallel- 1 c plate capacitors and four edge capacitors: vp = √ =√ , (2.116) L1 C1 εr C1 = 2Cp + 4Cf (2.120)where εr is the dielectric constant of the ﬁllingmedium, C1 and L1 are distributed capacitance and withinductance, respectively, and c is the speed of light. 0.0885εr w The characteristic impedance is given by Cp = (pF/cm) (2.121) (b − t)/2 L1 1 0.0885εr 2 1 Zc = = (2.117) Cf = ln +1 C1 vp C1 π 1 − t/b 1 − t/b 1 1As the calculations of phase velocity and dis- − − 1 ln −1tributed capacitance are quite complicated, it is dif- 1 − t/b (1 − t/b)2ﬁcult to give a general equation for the calculation (pF/cm). (2.122)
Microwave Theory and Techniques for Materials Characterization 59 w Here we make some explanations on two ratio units: “dB” and “neper”. The deﬁnitions for “dB” C′ f Cp C′ and “neper” are f t b dB = 10 · log10 (power ratio) C′ f Cp C′ f = 20 · log10 (voltage ratio) (2.128)Figure 2.28 The distribution capacitance of stripline neper = ln (voltage ratio). (2.129)with thick central conductor The conversion relations between dB and neper areTherefore the characteristic impedance can becalculated using neper = dB × 0.115129255, (2.130) 94.15 dB = neper × 8.685889638. (2.131) Zc = ( ) √ w/b Cf εr + 1 − t/b 0.0885εr Microstrip (2.123)If the central conductor is narrow, the interference As shown in Figure 2.29, a microstrip line consistsbetween the ﬁelds at the two edges cannot be of a strip conductor and a ground plane separatedneglected. We may take the central conductor as by a dielectric substrate. It can be taken asa cylinder by introducing equivalent diameter a transformation of coaxial line by cutting the outer conductor and ﬂattening it. As the dielectric w t 4πw t 2d= 1+ 1 + ln + 0.51π constant of the substrate is usually high, the ﬁeld 2 w t w is concentrated near the substrate. (2.124) In a strict meaning, the wave propagating on aand the characteristic impedance can be calculated microstrip line is not a pure TEM wave, nor a sim-using the following equation: ple TE wave or TM wave. The wave propagating 60 4b on a microstrip is in a quasi-TEM mode. Accu- Zc = √ ln ( ) (2.125) rate determination of the wave propagation on a εr πd microstrip line requires intense numerical simula- Similar to coaxial line, the attenuation also con- tions. But in engineering design, we may take thesists of conductor attenuation and dielectric attenu- wave in a microstrip line as TEM wave, and useation: α = αc + αd . An approximate expression for the quasi-static method to calculate the distributedattenuation resulting from conductor surface resis- capacitance, and then calculate its propagation con-tance is (Ramo et al. 1994) stant, wavelength, and characteristic impedance. Rs πw/b + ln (4b/πt) αc = ηb ln 2 + πw/2b (nepers/unit length) (2.126)Equation (2.126) is valid for w > 2b and t <b/10. Approximations for other dimensions can be erfound in (Hoffmann 1987). The attenuation caused xby dielectric loss is (Collin 1991) W t πεr π εr tan δαd = = (nepers/unit length), er h y λ0 εr λ0 (2.127)where λ0 is the free-space wavelength. Figure 2.29 Geometry of a microstrip
60 Microwave Electronics: Measurement and Materials Characterization er = 1 c er vp = √ (2.134) εeff Z0 1 Zc = √ c = , (2.135) εeff vp C1 where c is the speed of light, C1 is the distributed 0 capacitance of the microstrip, and Zc is the (a) (b) characteristic impedance of the microstrip when the ﬁlling medium is air. er = 1 eeff In most cases, the thickness of the strip is negli- gible (t/ h ≤ 0.005). The characteristic impedance and effective permittivity can be calculated using appropriate equations. If we deﬁne the relative strip er ′ width w u= , (2.136) h (c) (d) the effective dielectric constant and characteristicFigure 2.30 The concept of effective dielectric con- impedance are given by (Ishii 1995)stant. (a) Microstrip fully ﬁlled with air, (b) microstrip εr + 1 εr − 1 1fully ﬁlled with dielectric with permittivity εr , (c) εeff = + √microstrip partially ﬁlled with dielectric with permittiv- 2 2 1 + 12/uity εr , and (d) microstrip fully ﬁlled with dielectric withpermittivity εeff + 0.041(1 − u)2 (for u ≤ 1) (2.137) εr + 1 εr − 1 1 εeff = + √ In the analysis of microstrip lines using quasi- 2 2 1 + 12/ustatic method, we introduce the concept of (for u > 1) (2.138)the effective dielectric constant, as shown in 60 8Figure 2.30. If the ﬁlling medium is air (εr = 1), Z0 = √ ln + 0.25u ( )as shown in Figure 2.30(a), the microstrip line εeff ucan support the TEM wave, and its phase veloc- (for u ≤ 1) (2.139)ity equals the speed of light c. If the transmission 120π 1system is fully ﬁlled with a dielectric material with Zc = √ ( ) εeff 1.393 + u + ln(u + 1.4444)εr > 1, as shown in Figure 2.30(b), the microstrip (for u > 1) (2.140)can support TEM wave, and its phase velocity: √ Actually, the thickness of strip conductor affects vp = c/ εr . (2.132) the transmission properties of the microstrip line. We assume t < h and t < w/2. If the thickness ofIf a microstrip line is partially ﬁlled with a the strip t is not negligible, the effective dielectricdielectric material with dielectric constant εr , constant should be modiﬁed (Ishii 1995):as shown in Figure 2.30(c), we introduce theconcept of effective dielectric permittivity εeff εeff (t) = εeff − δεeff (2.141)to calculate the transmission parameters of thetransmission line: wavelength λg , phase velocity with tvp , and characteristic impedance Zc : δεeff = (εr − 1) √ (2.142) 4.6h u λ0 We should also introduce a concept of effective λg = √ (2.133) εeff relative strip width ueff :
Microwave Theory and Techniques for Materials Characterization 61 1.25t 4πw Coplanar waveguide ueff = u + 1 + ln πh t As shown in Figure 2.31, in a coplanar waveguide, (for u ≤ 1/(2π)) (2.143) all the conductors are on the top surface of a 1.25t 2h dielectric substrate. Similar to the microstrip, the ueff = u + 1 + ln πh t fundamental mode of propagation in the coplanar (for u > 1/(2π)) (2.144) waveguide is a quasi-TEM mode. As shown in Figure 2.31(b), the pattern of the electric ﬁeld in The attenuation factor of microstrip consists of the space above the substrate is the same as in thedielectric loss factor and conductor loss factor: substrate if the thickness of the strip is negligibleα = αc + αd (Ishii 1995). The dielectric loss factor and the substrate is thick enough. Therefore, weis given by can get the effective dielectric constant: εr εeff − 1 tan δ εr + 1αd = 27.3 √ , (dB/unit length) εeff = , (2.150) εr − 1 εeff λ0 2 (2.145)and the conductor loss factor can be calculated by and the phase velocity is given by Rs 32 − u2 1αc = 1.38A eff vp = √ (2.151) hZc 32 + u2 µε0 εeff eff (dB/unit length) (for u ≤ 1) (2.146) By assuming that the thickness of the conductors Rs Zc εeff 0.667ueff is zero, ground conductors are inﬁnitely wide,αc = 6.1 × 10−5 A ueff + and the substrate has inﬁnite thickness, we have h 1.444 + ueff the following approximate formulas (Ramo et al. (dB/unit length) (for u > 1) (2.147) 1994):with η0 a Zc = √ ln 2 ( ) 1 1 2B π εeff w A=1+ 1 + ln , (2.148) ueff π t for 0 < w/a < 0.173 (2.152) h for u ≥ (1/2π) √ −1 B= , (2.149) πη0 1 + w/a 2πw for u ≤ (1/2π) Zc = √ ln 2 √ ( ) 4 εeff 1 − w/awhere Rs is the surface resistance of the conductor. for 0.173 < w/a < 1 (2.153) a w s sd er Dielectric (a) (b)Figure 2.31 Coplanar line. (a) Structural dimensions and (b) ﬁeld distributions. The solid lines represent electricﬁeld and the dashed lines represent magnetic ﬁeld
62 Microwave Electronics: Measurement and Materials Characterization 0.1 A (nepers/ohm) s = 0.02 d 0.01 0.1 0.5 2 0.001 10 0.01 0.1 1.0 10 w/dFigure 2.32 Factor for calculation of conductor loss in coplanar waveguide (Hoffmann 1987). Source: M¨ ller, uE.: Wellenwiderstand und Mittlere Dielektrizit¨ tskonstante von koplanaren Zwei-und Dreidrahtleitungen auf einem adielektrischen Tr¨ ger und deren Beeinﬂussung durch Metallw¨ nde. Dissertation, Techn. Universit¨ t Stuttgart (1977) a a aThe relations for coplanar waveguides with thick dielectric constant. More discussions on this topicconductor and thin substrate are very compli- can be found in (Ramo et al. 1994).cated (Hoffmann 1987; Gupta et al. 1979). The attenuation factor of a coplanar waveguide 188.8.131.52 Hollow metallic waveguidesconsists of dielectric loss factor and conductor lossfactor: α = αc + αd . The attenuation from con- Hollow metallic waveguides are widely used inductor loss factor in the coplanar waveguide can microwave engineering and materials propertybe calculated from the following equation (Ramo characterization. We ﬁrst introduce the parameterset al. 1994): describing the propagation properties of hollow metallic waveguides, and then discuss two types Rs √ αc = A εeff (nippers/unit length) (2.154) of hollow metallic waveguides: rectangular waveg- d uide and circular waveguide. Finally, we introduceThe value of A can be found from Figure 2.32 on a transition between a circular waveguide and athe condition that all the conductors have the same rectangular waveguide.Rs and have thickness satisfying t > 3δ, whereδ is the penetration depth of the conductor. If Propagation parametersd > a, the dielectric loss factor can be calculatedusing (Hoffmann 1987) As shown in Figure 2.33, a hollow metallic √ waveguide refers to a straight metal tube which has πf εeff 1 − 1/εeff αd = tan δ (nepers/m) inﬁnite length and whose cross section does change c 1 − 1/εr along the z-axis. The wave propagation in a general (2.155) hollow metallic waveguide can be described by Finally, it should be noted that in the above dis-cussions on microstrip and coplanar waveguide, E (u1 , u2 , z, t) = C1 E (u1 , u2 )ejωt−γ z (2.156)we do not consider the dispersion of the effective H (u1 , u2 , z, t) = C2 H (u1 , u2 )ejωt−γ z (2.157)dielectric constant. In a strict meaning, the oper-ating frequency also affects the value of effective γ = −(k − 2 2 2 kc ), (2.158)
Microwave Theory and Techniques for Materials Characterization 63 called the cutoff frequency fc , and its correspond- y ing wavelength is called the cutoff wavelength λc . u2 = constant z The relationship between kc and λc is u1 = constant 2π kc = (2.166) x λc e, m Both kc and λc are related to the transverse ﬁeld distribution in the waveguide. The transmissionFigure 2.33 A general hollow metallic waveguide requirement can be described as k > kc , λ < λc , and Eq. (2.160) can be rewritten aswhere E and H are the electric and magnetic ﬁelds 2propagating in the waveguide along the Z-axis, and 2π λu1 and u2 are two orthogonal coordinates perpen- β= 1− (2.167) λ λcdicular to Z-axis. The propagation parameter γ isrelated to wave frequency, medium properties, and For a TEM wave, as β = k, the cutoff wavelengthﬁeld distributions. is inﬁnity, so TEM waves with any frequency If the frequency is high enough (k > kc ), then γ satisfy the propagation requirement.is an imaginary number: In the following, we discuss parameters often used in describing the propagation properties γ = jβ (2.159) of hollow metallic waveguides, including phase velocity, group velocity, and the wave impedanceswith for TE and TM waves. β = k 1 − (kc /k)2 (2.160) According to the deﬁnition, phase velocity is theSo Eqs. (2.156) and (2.157) can be rewritten as velocity of the movement of phase planes. From Eqs. (2.161) and (2.162), for a certain phase plane E (u1 , u2 , z, t) = C1 E (u1 , u2 )ej(ωt−βz) (2.161) moving along the z-axis, following requirement is satisﬁed: H (u1 , u2 , z, t) = C2 H (u1 , u2 )e j(ωt−βz) (2.162) ωt − βz = constant (2.168)Equations (2.161) and (2.162) show that β rep- From Eq. (2.168), we haveresents the phase change of a unit length alongz-axis, and is usually called the phase constant. d dz (ωt − βz) = ω − β =0 (2.169) If the frequency is low (k < kc ), γ becomes a dt dtreal number: So the phase velocity is γ = α = kc 1 − (k/kc )2 (2.163) dz ω vp = = (2.170)So Eqs. (2.156) and (2.157) become dt β E (u1 , u2 , z, t) = C1 E (u1 , u2 )e−αz ejωt (2.164) From (2.167), we have ω v v H (u1 , u2 , z, t) = C2 H (u1 , u2 )e−αz ejωt (2.165) vp = = = , β 1 − (λ/λc )2 1 − (λ/λc )2Equations (2.164) and (2.165) indicate that the (2.171)phase of E and H does not change with z-axis, where v is the velocity of electromagnetic wave:and the ﬁelds decrease along the z-axis. So the ωwave is in a cutoff state. v= λ = fλ (2.172) There is a critical state between the transmis- 2πsion state and cutoff state: k = kc and so γ = 0. Equation (2.171) indicates that the phase velocityThe frequency corresponding to the critical state is of TE or TM wave is frequency-dependent, while
64 Microwave Electronics: Measurement and Materials Characterizationfor TEM wave, the phase velocity does not change Equation (2.178) can also be written aswith frequency: E = 2E0 cos( ω · t − βz) cos(ω0 t − β0 z) √ vp = ω/k = 1/ εµ. (2.173) (2.179) It is an amplitude-modulated wave with two cosine The waveguide wavelength refers to the wave- factors. The factor cos(ω0 t − β0 z) corresponds tolength of a wave propagating along a waveguide. the transmission of the wave group along z-axis,For TEM wave, the waveguide wavelength and the factor cos( ωt − βz) corresponds to the change of magnitude along the z-axis. The λg = v/f (2.174) information transmitted is the change of magnitude (envelope) along the z-axis. So the velocity ofSo it is the same as the wavelength when the TEM information transmission is the velocity of thewave propagates in a free space ﬁlled with the envelope transmission. For a plane in the envelope,same medium. However, for TE and TM wave,the waveguide wavelength λg is given by ω · t − βz = constant (2.180) vp λ Differentiating Eq. (2.180) with t gives λg = = , (2.175) f 1 − (λ/λc )2 dz ω = (2.181)where λ is the wavelength of wave propagation dt βin free space ﬁlled with the same medium. By taking ω → 0, we can get the group velocityEquation (2.175) indicates that λg is related tothe shape and size of the waveguide, and the dωpropagation mode in the waveguide. vg = (2.182) dβ Equation (2.171) indicates that the phase veloc-ity may be greater than the speed of light. Actually, For TEM mode, as β = k = ω(εµ)1/2 , we havephase velocity is deﬁned to a single frequency andendless signal (−∞ < t < +∞), and such a signal dω 1 vg = =√ = vp (2.183)does not transmit any information. Information is dβ εµtransmitted through modulation, and the speed ofthe transmitting information is the speed of trans- However, for TE and TM modes, as β 2 = −γ 2 =mitting the information component in a modulated k 2 − kc = ω2 εµ − kc , we have 2 2wave. A modulated wave is not a single frequencywave but a group of waves with different frequen- dω β λ 2cies, so its transmission velocity is called the group vg = = √ =v 1− (2.184) dβ k εµ λcvelocity. Here we discuss a simple example. We assume It is clear that the group velocity is less than thethat the wave group consists of two signals with speed of light. From Eqs. (2.171) and (2.184), wethe same magnitude and very close frequencies and ﬁnd thatphase constants: vp · v g = v 2 (2.185) E1 = E0 ej((ω0 + ω)t−(β0 + β)z) (2.176) Figure 2.34 shows the relationship between group j((ω0 − ω)t−(β0 − β)z) E2 = E0 e (2.177) velocity, phase velocity, and frequency of electro- magnetic waves propagating along a hollow metal-So the modulated wave is lic waveguide. Now, we discuss the wave impedances of TEE = E1 + E2 = 2E0 cos( ω · t − βz)ej(ω0 t−β0 z) and TM waves. As discussed earlier, the wave (2.178) impedance is deﬁned as the ratio between the
Microwave Theory and Techniques for Materials Characterization 65 v/c Rectangular waveguide vp /c Figure 2.35 shows a rectangular waveguide with width a and height b. Rectangular waveguides can transmit TE and TM modes. Usually, two Cut-off subscripts m and n are used to specify TE or TM range modes, so the propagation mode is often denoted v=c 1 as TEmn or TMmn . The subscript “m” indicates the number of changing cycles along the width vg /c a, while the subscript “n” indicates the number of 0 1 changing cycles along the height b. f / fc The ﬁeld components of a TEmn wave areFigure 2.34 Relationship between phase velocity, γmn mπ mπ nπ Hx = A 2 sin x cos (2.190)group velocity, and frequency kc a a b γmn nπ mπ nπ Hy = A 2 cos x sin (2.191)transverse electric ﬁeld and magnetic ﬁeld. For TE kc b a bmode at the transmission state, we have mπ nπ √ Hz = A cos x cos (2.192) jωµ ωµ µ/ε a b ZTE = = = Ex = ZTE Hy (2.193) γ β 1 − (λ/λc )2 ZTEM Ey = −ZTE Hx (2.194) = (2.186) 1 − (λ/λc )2 Ez = 0 (2.195)So, the impedance is pure resistance at transmis- The constant A is related to the power of the wave.sion state. But in the cutoff state, γ = α, the wave The parameters γmn , kc , and other parameters areimpedance is an inductive reactance: listed in Table 2.3. The ﬁeld components of a TMmn wave are: jωµ ZTE = (2.187) γmn mπ mπ nπ α Ex = B 2 cos x sin (2.196) kc a a b Similarly, the wave impedance for TM modes atthe transmission state is 2 γ µ λ ZTM = = 1− jωε ε λc 2 λ = ZTEM 1 − (2.188) λcIt is pure resistance. But for the cutoff state, the bwave impedance is a capacitive reactance: y z x α ZTM = (2.189) jωε a In the following, we discuss two typical types Figure 2.35 Rectangular waveguide. Source: Ramo,of waveguides: rectangular waveguide and circular S. Whinnery, J. R. and Van Duzer, T. (1994). Fieldswaveguide, which are widely used in materials and Waves in Communication Electronics, 3rd ed., Johnproperty characterization. Wiley & Sons, Inc., New York
66 Microwave Electronics: Measurement and Materials Characterization Table 2.3 Properties of empty rectangular waveguide TEmn mode TMmn mode mπ 2 nπ 2 mπ 2 nπ 2 Cutoff wave number, kc + + a b a b Propagation constant, γmn kc − k0 2 2 kc − k0 2 2 λ0 λ0 Guided wavelength, λg 1 − (kc /k0 )2 1 − (kc /k0 )2 λ0 λ0 Group velocity, vg c c λg λg λg λg Phase velocity, vp c c λ0 λ0 jk0 η0 jγmn η0 Wave impedance, Z − γmn k0 2 2Rs b fc Attenuation for TEmn modes 1+ (Ramo et al. 1994) bη 1 − (fc /f )2 a f 2 fc (b/a)((b/a)m2 + n2 ) + 1− f (b2 m2 /a 2 + n2 ) (nepers/unit length) (n = 0) 2 Rs 2b fc 1+ bη 1 − (fc /f )2 a f (nepers/unit length) (n = 0) 2Rs m2 (b/a)3 + n2 Attenuation for TMmn modes (nepers/unit length) (Ramo et al. 1994) bη 1 − (fc /f )2 m2 (b/a)2 + n2 γmn nπ mπ nπ The constants A and B in Eqs. (2.190)–(2.201) Ey = B 2 sin x cos (2.197) kc b a b affect the amplitude of the ﬁelds, but do not affect mπ nπ the ﬁeld distribution. The ﬁeld distributions of Ez = B sin x sin (2.198) several typical TE and TM modes are shown in a b −1 Figure 2.36. Hx = Hy (2.199) Figure 2.37 shows the sequence in which var- ZTM ious modes come into existence as the operation 1 frequency increases for aspect ratio b/a equals 1 Hy = Hx (2.200) ZTM and 0.5. In the cutoff range, no mode can propa- Hz =0 (2.201) gate in the waveguide. TE10 is the most often used mode, and in most cases we should ensure wave-The constant B is related to the power of the wave. guides work in single mode state. In microwaveThe parameters γmn , kc , and other parameters are engineering, waveguides with b/a = 0.5 is morelisted in Table 2.3. widely used.
TE10 TE11 TE21 3 3 3 2 1 1 2 1 y y 2 y z z z x 1 x 1 1 3 2 3 2 3 2 TE20 TM11 TM21 3 2 3 3 2 2 1 1 1 y z y z 1 x 1 x 1 3 2 3 2 3 2Figure 2.36 Summary of wave types of rectangular waveguides. Electric ﬁeld lines are shown solid and magnetic ﬁeld lines are dashed (Ramo et al.1994). Source: Ramo, S. Whinnery, J. R. and Van Duzer, T. (1994). Fields and Waves in Communication Electronics, 3rd ed., John Wiley & Sons, Inc.,New York Microwave Theory and Techniques for Materials Characterization 67
68 Microwave Electronics: Measurement and Materials Characterization TE12 TE10 TE 20 TE 21 TE 01 TM11 TE 02 TM12 Cutoff TE11 TM 21 b/a = 1 range fc /(fc)TE10 0 1 2 3 (a) TE 01 TE10 TE 20 TE11 Cutoff TM11 b/a = 1/2 range fc /(fc)TE10 0 1 2 3 (b)Figure 2.37 Relative cutoff frequencies of rectangular guides (Ramo et al. 1994). Modiﬁed from Ishii, T. K.(1995). Handbook of Microwave Technology, vol 1, Academic Press, San Diago, CA, 1995; Ramo, S. Whinnery,J. R. and Van Duzer, T. (1965). Fields and Waves in Communication Electronics, John Wiley & Sons, Inc., NewYorkCircular waveguide a 2 µni Hϕ = −Anγni Jn r sin nϕ (2.203)As shown in Figure 2.38, in the analysis of a circu- µni alar waveguide, it is more convenient to use cylin- µni Hz = −AJn r cos nϕ (2.204)drical coordinate (r, ϕ, z). The dimension of a cir- acular waveguide is its radius a. Circular waveguide Er = ZTE Hϕ (2.205)can transmit TE and TM modes. Usually, the prop-agation mode in a circular waveguide is denoted as Eϕ = −ZTE Hr (2.206)TEni or TMni . The subscript “n” indicates the num- Ez = 0 (2.207)ber of changing periods in ϕ-direction, while thesubscript “i” indicates the number of the changing The constant A is related to the microwave powerperiods in r-direction. transmitted in the waveguide, Jn is nth-order The ﬁeld components of a TEni wave are Bessel function, and µni is the ith root of Jn . Some a µni characteristic parameters are listed in Table 2.4.Hr = −Aγni Jn r cos nϕ (2.202) The ﬁeld components of a TMni wave are µni a a vni Er = −Bγni Jn r cos nϕ (2.208) vni a Z n a 2 vni Eϕ = −B γni Jn r sin nϕ (2.209) r vni a vni Ez = BJn r cos nϕ (2.210) a 1 ϕ Hr = − Eϕ (2.211) X ZTM r 1 Hϕ = Er (2.212) ZTMFigure 2.38 Circular waveguide Hz = 0, (2.213)
Microwave Theory and Techniques for Materials Characterization 69 Table 2.4 Properties of empty rectangular waveguide TEni mode TMni mode µni vni Cutoff wave number, kc a a (i = 0, 1, 2, 3, . . .) (i = 1, 2, 3, . . .) Propagation constant, γni kc − k0 2 2 kc − k0 2 2 λ0 λ0 Guided wavelength, λg 1 − (kc /k0 )2 1 − (kc /k0 )2 λ0 λ0 Group velocity, vg c c λg λg λg λg Phase velocity, vp c c λ0 λ0 jk0 η0 jγni η0 Wave impedance, Z − γni k0 2 Rs 1 kc n2 Attenuation for TEni modes + (Ishii 1995) aη0 1 − (kc /k0 )2 k0 (kc a)2 − i 2 (nepers/unit length) 2 Rs k0 Attenuation for TMni modes (nepers/unit length) (Ishii 1995) aη0 k0 − kc 2 2where vni is the is the ith root of Jn , and B the roots for J0 (x) and J1 (x) are the same: µ0i =is related to the microwave power transmitted in v1i . Therefore, TE0i and TM1i have the samethe waveguide. Some characteristic parameters are wavelength, and this is called E-H degeneration.listed in Table 2.4. Figure 2.40 shows the cutoff frequencies for Figure 2.39 shows the ﬁeld distributions of different modes of circular waveguides. Similarseveral typical circular waveguide modes. We can to rectangular waveguides, there is a cutoff rangesee that along the ϕ direction, the ﬁeld changes where no mode can propagate, and the fundamentalin a sinuous way, and the number n indicates the mode is TE11 mode. In the design and selection ofperiod number in the range of 0 to 2π. In the circular waveguides, we should ensure single moderadius direction, the ﬁeld changes according to requirement.Bessel function or differentiated Bessel functions,and i indicates the number of zeros along the Transition from rectangular waveguide to circularradius (0 < r < a). waveguide In circular waveguides, there exist degenerationphenomena. There are two kinds of degenerations: In microwave engineering, we often use the primepolar degeneration and E-H degeneration. For a modes of rectangular and circular waveguides. TheTEni or a TMni (n = 0), mode there are two kinds prime mode of rectangular waveguide is TE10 ,of ﬁeld distributions that have the same shape, and that of circular waveguide is TE11 . Owingbut their polarization planes are perpendicular to to the E-H degeneration of circular waveguides,each other. Such degeneration is called polar rectangular waveguides are more widely useddegeneration. Meanwhile, because while circular waveguides are often used in antennas, polarization attenuators, ferrite isolators, J0 (x) = −J 1 (x), (2.214) and circulators.
70 Wave type TM01 TM02 TM11 TE01 TE12 Field distributions in cross-sectional plane, at plane of maximum Distributions Distributions transverse fields below along below along this plane this plane Field distributions along guide Field components Es1Er1Hf Es1Er1Hf Es1Er1Ef1Hr1Hf Hs1Hr1Ef Hs1Hr1Hf1Er1Ef present pal or p′al 2.405 5.52 3.83 3.83 1.84 (kc)al 2.405 5.52 3.83 3.83 1.84 a a a a a (lc)al 2.61a 1.14a 1.04a 1.64a 3.41a ( fc)al 0.383 0.877 0.609 0.609 0.293 a me a me a me a me a me Attenuation due to Rs 1 Rs 1 Rs 1 Rs ( fc/f )2 Rs 1 Microwave Electronics: Measurement and Materials Characterization fc 2+ 0.420 imperfect-conductors ah 1 − ( f /f )2 ah 1 − ( f /f )2 ah 1 − ( f /f )2 f c c c ah 1 − ( f /f )2 c ah 1 − ( f /f )2 cFigure 2.39 Summary of wave types of circular waveguides. Electric ﬁeld lines are shown solid and magnetic ﬁeld lines are dashed (Ramo et al. 1994).Source: Ramo, S. Whinnery, J. R. and Van Duzer, T. (1994). Fields and Waves in Communication Electronics, 3rd ed., John Wiley & Sons, Inc., NewYork
Microwave Theory and Techniques for Materials Characterization 71 Cutoff TE11 TE 21 TE 01 TE 31 TE 41 TE12 range fc /( fc)TE11 0 1 2 3 TM 01 TM11 TM 21 TM 02Figure 2.40 Relative cutoff frequencies of waves in a circular guide (Ramo et al. 1994). Source: Ramo, S.Whinnery, J. R. and Van Duzer, T. (1994). Fields and Waves in Communication Electronics, 3rd ed., John Wiley& Sons, Inc., New York Field matching and impedance matching The objective of transition design is to make the transformation between different types of transmission structures as efﬁcient as possible. TE11 To obtain good transformation, two requirements TE10 should be satisﬁed: impedance matching and ﬁeld matching. A practical approach to realize the efﬁcientFigure 2.41 Transition between a rectangular waveg- ﬁeld transition from one transmission structureuide and a circular waveguide to another is to smoothly and gradually change the physical boundary conditions. A transition example is the transition between rectangular and As the ﬁeld distributions of rectangular TE10 circular waveguides, as shown in Figure 2.41.and circular TE11 are similar, it is easy to realize Here, we discuss the transitions between differentthe transition between them. Figure 2.41 shows an types of transmission structures.example for the transition between a rectangular Waveguide, coaxial line, and microstrip are threewaveguide and a circular waveguide. types of transmission structures often used in microwave engineering and materials characteri- zation, and their electric ﬁeld and magnetic ﬁeld184.108.40.206 Transitions between different types distributions are shown in Figure 2.42. To ensureof transmission lines ﬁeld matching, it is necessary to transform theIn building microwave measurement circuits, it ﬁeld geometry by reshaping the transmission struc-is necessary to make transitions between differ- tures. Figure 2.43 illustrates the evolution pro-ent types of transmission structures, for example, cedure of modifying a coaxial transmission linetransitions between waveguide and coaxial line, into a microstrip line by cutting the coaxial lineand transitions between waveguide and microstrip along the longitudinal direction and unfolding it.line. The function of a microwave transition is to Meanwhile, the transition between two transmis-couple the electromagnetic wave in one type of sion structures must provide impedance matchingtransmission structure into another. Meanwhile, atransition between two different transmission struc-tures transforms the electromagnetic ﬁeld distribu-tions in one transmission structure to conform tothe boundary conditions of another transmissionstructure. Here, we discuss the basic requirements (a) (b) (c)for the design of transitions and then give severaltransition examples. Detailed analysis and more Figure 2.42 Field distributions of three types of trans-examples can be found in (Izadian and Izadian mission structures. (a) Rectangular waveguide, (b)1988). coaxial line, and (c) microstrip
72 Microwave Electronics: Measurement and Materials Characterization A Cut here A′ (a) (b) Figure 2.44 Magnetic-dipole approach for the tran- sition between a rectangular waveguide and coaxialFigure 2.43 The evolution from a coaxial line to a line (Izadian and Izadian 1988). Reprinted with permis-microstrip line. (a) Coaxial line and (b) microstrip sion from Microwave Transition Design, by Izadian, J. S.line. Reprinted with permission from Microwave Tran- and Izadian, S. M. (1988). Artech House Inc., Norwood,sition Design, by Izadian, J. S. and Izadian, S. M. MA, USA. www.artechhouse.com(1988). Artech House Inc., Norwood, MA, USA.www.artechhouse.com and the shield of the coaxial line is connected tobetween the two transmission structures to reduce the bottom side of the waveguide wall. At thethe reﬂection at the transition and improve the tran- connection region, the height of the waveguide issition efﬁciency. To achieve both ﬁeld matching decreased to achieve an impedance value close toand impedance matching, most of the transitions that of the coaxial line (50 ), and so the ﬁrstuse step transition or continuous taper transition critical translation is made between coaxial lineapproaches. and waveguide. Quarter-wavelength waveguide In the following, we discuss two typical tran- sections between the normal waveguide and thesitions: transition between rectangular waveguide connection region are often used to make aand coaxial line, and transition between coaxial transition between the normal waveguide and 50line and microstrip line. waveguide. Figure 2.45 shows an electric probe approach for making a transition between a rectangularTransition between rectangular waveguide waveguide and a coaxial line. In this approach,and coaxial line the central conductor of the coaxial line extends into the waveguide, but is not connected to theRectangular waveguide is a non-TEM wave trans- opposite waveguide wall. The central conductormission structure, which only supports TE or TM acts as a small monopole antenna exciting themodes, so a rectangular waveguide may be taken as propagation mode in the rectangular waveguide.a high-pass or band-pass structure, while a coaxialline supports a TEM mode and can be used fromdc. The upper frequency limit of coaxial lines isthe increase of loss of the transmission line andthe higher order modes. C Two classic approaches have been used in the Bdesign of transition between a waveguide and acoaxial line: the electric probe and the magnetic Aprobe. The coaxial line is usually 50 Ohms, butthe impedance of the TE10 mode in a rectangular (a) (b)waveguide is usually several hundred Ohms. So, the Figure 2.45 Electric-probe approach for a transitiondesign procedure is to get an optimum impedance between a rectangular waveguide and a coaxial line.matching by changing the location, height, and (a) Side view and (b) front view (Izadian and Izadiandiameter of the electric or magnetic probe. 1988). Reprinted with permission from Microwave Figure 2.44 shows a transition in magnetic probe Transition Design, by Izadian, J. S. and Izadian, S.approach. The inner conductor of the coaxial line M. (1988). Artech House Inc., Norwood, MA, USA.is connected to the top side of the waveguide wall, www.artechhouse.com
Microwave Theory and Techniques for Materials Characterization 73The impedance matching can be achieved by Meanwhile, to obtain better transition efﬁciency,varying the dimensional parameters, including the width of the microstrip must be close to theoff-center position of the probe A, probe length diameter of the coaxial central pin.B, and probe position in the waveguide C. Figure 2.46(b) shows a right-angle transition between a coaxial line and a microstrip line. Such a transition can be fabricated by drilling a hole inTransition between coaxial line the microstrip substrate. The pin of the connectorand microstrip line is inserted through the hole and connected to the microstrip circuit and the shield of the connectorThe transitions between coaxial and microstrip is connected to the ground of the microstrip line.line are the most frequently used transitions in This kind of transition is often used in antennas,microwave electronics. In materials property char- but it requires high fabrication techniques.acterization, these types of transitions are often Figure 2.47 shows another example of transitionrequired in developing planar circuit method, between a coaxial line and a microstrip line.which will be discussed in Chapter 7. The transi- In such a transition, the central conductor oftions between coaxial line and microstrip lines are the coaxial line goes gradually from the axisoften required to have broad working frequency position to the strip of the microstrip, and therange, high return loss, and low insertion loss. grounding plate of the microstrip is connected The basic principle for a transition between to the outer conductor of the coaxial line. Thecoaxial line and microstrip line has been shown ﬁeld distributions at different cross sections of thein Figure 2.43. As microstrip circuits are usu- transition structure indicate that the ﬁelds changeally hosted in a casing, the transition between gradually, and so high transition efﬁciency can bemicrostrip line and coaxial line is often designed achieved using this transition structure.in the package casing. In the development of suchtransitions, commercially available standard con- 2.2.4 Surface-wave transmission linesnectors, including the central pins and shields, areoften used. In a transition structure, the central pin Besides the guided transmission lines discussed inis connected to the microstrip circuit, and the shield Section 2.2.3, there exists a class of open-boundaryis connected to the wall of the casing, as shown structures which can also be used in guidingin Figure 2.46(a). It is preferable to choose a con- electromagnetic waves. Such structures are capablenector with a dielectric insulator whose thickness of supporting a mode that is closely bound to theis close to the thickness of the microstrip substrate, surfaces of the structures. The ﬁeld distributionsand the dielectric insulator of the connector and the of the electromagnetic waves on such structuresmicrostrip substrate have close dielectric constant. are characterized by an exponential decay away Coaxial connector with flange Microstrip Microstrip Substrate Grounding Substrate Grounding Housing base plate Coaxial connector with flange (a) (b)Figure 2.46 Transitions from coaxial line to microstrip. (a) Straight transition and (b) right angle transition
74 Microwave Electronics: Measurement and Materials Characterization Dielectric Microstrip line Coaxial line Housing base plate Field distributions at different cross sectionsFigure 2.47 A transition between a coaxial line and a microstrip line (Modiﬁed from Hoffmann, R. K. (1987).Handbook of Microwave Integrated Circuits, Artech House, Norwood, MA, 1987. 2003 IEEEfrom the surface and having the usual propagation high permittivity and extremely low loss dielec-function exp(±jβz) along the axis of the structure. tric materials. But the use of high permittivitySuch an electromagnetic wave is called a surface materials may result in very small size of the sur-wave, and the structure that guides this wave is face waveguide and severe fabrication toleranceoften called a surface waveguide. One of the most requirements.characteristic properties of a surface wave is that In the following, we mainly discuss the sur-it does not have low-frequency limit. face waves at dielectric interfaces, dielectric Sometimes, surface waveguides are also called slabs, rectangular dielectric waveguides, cylindri-dielectric waveguides, as in most cases, the key cal dielectric waveguides, and coaxial surface-components consisting a surface waveguide are wave transmission structures.dielectrics. In a surface waveguide, the wave trav-els because of the total internal reﬂections at theboundary between two different dielectric materi- 220.127.116.11 Dielectric interfaceals. Figure 2.48 shows a cross section of a gener- The simplest surface waveguide structure is aalized surface waveguide. The conductor loss in a dielectric interface between two dielectric materi-surface waveguide is usually very low, while the als with different dielectric permittivities as shownloss due to the curvature, junction, and disconti- in Figure 2.49. For an electromagnetic wave inci-nuities, and so on, may be quite large. The loss dent on the interface, we have Snell’s laws ofof a dielectric waveguide can be decreased using reﬂection and refraction: Dielectric θi = θr (2.215) er = e2 k1 sin θi = k2 sin θt , (2.216) er = e1 where k1 and k2 are the wave-numbers in the two Conductor dielectric media, given by boundary √ ki = ω µ0 εi (i = 1, 2), (2.217) where ω is the operating frequency. Other param-Figure 2.48 Cross section of a generalized dielectric eters are deﬁned in Figure 2.49. In the followingwaveguide discussion, we assume ε1 > ε2 .
Microwave Theory and Techniques for Materials Characterization 75 x where T is the transmission coefﬁcient, β is the propagation constant, and ηi is the wave impedance given by (Er, Hr) (Et, Ht) ηi = µ0 /εi (i = 1, 2) (2.224) From the boundary condition, we can get qr qt qi z β = k1 sin θi = k1 sin θr (2.225) α= β 2 − k2 = 2 k1 sin2 θi − k2 . 2 2 (2.226) (Ei, Hi) The reﬂection and transmission coefﬁcients can Region 1 Region 2 then be obtained (Pozar 1998): (e1, m0) (e2, m0) (jα/k2 )η2 − η1 cos θi = (2.227) (jα/k2 )η2 + η1 cos θiFigure 2.49 Geometry for a plane wave obliquely 2η2 cos θiincident at the interface between two dielectric regions T = (2.228) (jα/k2 )η2 + η1 cos θi From Eq. (2.216), we have The magnitude of is unity as it is of the form (a − jb)/(a + jb), so all the incident power is sin θt = ε1 /ε2 sin θi (2.218) reﬂected.Equation (2.218) indicates that when the incident Equations (2.222) and (2.223) indicate that theangle θi increases from 0◦ to 90◦ , the refraction transmitted wave propagates in the x-directionangle θt will increase, in a faster rate, from 0◦ to along the interface, while it decays in the z-90◦ . At a critical incident angle θc deﬁned by direction. As the ﬁeld is tightly bound to the inter- face, the transmitted wave is called surface wave. sin θc = ε2 /ε1 , (2.219) From Eqs. (2.222) and (2.223), we can calculate the complex Poynting vector (Pozar 1998):θt = 90◦ . When the incident angle is equal to orlarger than the critical angle, the transmitted wave |E0 |2 |T |2 jα βdoes not propagate into region 2. St = Et × Ht∗ = ˆ z +x ˆ η2 k2 k2 When θi > θc , the angle θt loses its physical exp(−2αz) (2.229)meaning deﬁned in Figure 2.49. We write theincident ﬁelds as Equation (2.229) indicates that no real power ﬂow Ei = E0 (x cos θi − z sin θi ) ˆ ˆ occurs in the z-direction. The real power ﬂow in the x-direction is that of the surface wave exp[−jk1 (x sin θi + z cos θi )] (2.220) ﬁeld, which decays exponentially with distance E0 into region 2. So, even though no real power is Hi = y exp[−jk1 (x sin θi + z cos θi )] ˆ (2.221) transmitted into region 2, a nonzero ﬁeld does exist η1 there in order to satisfy the boundary conditions atWhen θi > θc , the transmitted ﬁelds are usually the interface.expressed as jα β 18.104.22.168 Dielectric slab Et = E0 T x − z exp(−jβx) exp(−αz) ˆ ˆ k2 k2 (2.222) Surface waves can propagate on dielectric slabs, E0 T including ungrounded and grounded dielectric Ht = ˆ y exp(−jβx) exp(−αz), (2.223) η2 slabs.
76 Microwave Electronics: Measurement and Materials Characterization X For an antisymmetrical TM mode, at x = 0 plane, x=d e0, m0 we have Hy = 0, (2.232) Y Z er e0, m0 so we can put a magnetic wall at the x = 0 plane. x = −d For TE modes, we have opposite conclusions. For a symmetrical TE mode, we can put a magnetic wall at the x = 0 plane; and for an antisymmetricalFigure 2.50 Cross section of an ungrounded dielectric TE mode, we can put an electrical wall at the x = 0slab plane. The cutoff wavelength for both TMn and TEnUngrounded dielectric slab modes are given byAn ungrounded dielectric slab is also called sym- 2d nmetrical dielectric slab due to its structural sym- = (n = 0, 1, 2, 3, . . .) (2.233)metry. Figure 2.50 shows an ungrounded dielectric λc 2(εr − 1)1/2slab with a thickness 2d, and at the regions x > d Even values of n(0, 2, 4, . . .) correspond toand x < −d, the medium is air. We assume that even TM or TE modes, while odd values ofthe dielectric loss of the slab is negligible and the n(1, 3, 5, . . .) correspond to odd TM or TE modes.dielectric constant of the slab is εr . For a plane Equation (2.233) indicates that for an ungroundedwave propagating from the slab to the interface dielectric slab, the ﬁrst even mode (n = 0) has nobetween the dielectric and air, if the incident angle low-frequency cutoff.satisﬁes √ θi > sin−1 (1/ εr ), (2.230) Grounded dielectric slabsthe wave energy will be totally reﬂected, resultingin surface wave propagation. Figure 2.51 shows a dielectric slab grounded by We assume that the dielectric slab is inﬁnitely a metal plate. A grounded dielectric slab withwide, the electromagnetic ﬁeld does not change thickness d can be taken as a special casealong the y-direction, and the propagation factor of ungrounded dielectric slab with thickness 2dalong the z-direction is exp(−jβz). According to as shown in Figure 2.50, with an electric wallMaxwell’s equations and the boundary conditions, placed at the plane x = 0. Detailed discussionit can be veriﬁed that there are two types of surface on grounded dielectric slab can be found inwaves: TM modes with components Hy , Ex , and (Pozar 1998).Ez , and TE modes with components Ey , Hx , and The surface waves propagating on a groundedHz . Detailed discussions on TM and TE modes can dielectric slab can also classiﬁed into TM and TEbe found in (Collin 1991). modes. The cutoff wavelength for TMn mode is Owing to the symmetrical structure of thedielectric slab, the surface waves also fall intosymmetrical modes and antisymmetrical modes. xFor a symmetrical TM mode, as the distribution ofHy along x-direction is symmetrical for the plane Dielectric e0x = 0, we have ∂Hy d e0 er = 0. (2.231) z ∂x x =0Equation (2.231) indicates that the tangent electric Ground planeﬁeld component along the x = 0 plane equals zero,so we can put an electric wall at the x = 0 plane. Figure 2.51 Geometry of a grounded dielectric slab
Microwave Theory and Techniques for Materials Characterization 77given by surface waves on dielectric waveguides usually requires numerical techniques, among which the 2d n = (n = 0, 1, 2, . . .), (2.234) mode-matching method is often used. Detailed λc (εr − 1)1/2 discussion on rectangular dielectric waveguidewhile the cutoff wavelength for TEn mode is given can be found in (Ishii 1995; Goal 1969). In theby: following, we discuss the propagation constants of isolated rectangular dielectric waveguides and 2d 2n − 1 image guides. = (n = 1, 2, 3, . . .) (2.235) λc 2(εr − 1)1/2Equations (2.234) and (2.235) indicate that the Isolated rectangular waveguideorder of propagation for the TMn and TEn modesis TM0 , TE1 , TM1 , TE2 , TM2 , . . .. Figure 2.52 shows the geometrical structure of an isolated rectangular waveguide and its ﬁeld distributions along the x-direction and y-direction.22.214.171.124 Rectangular dielectric waveguide The axis of the dielectric waveguide is along the z-direction, and dimensions along the x-directionA rectangular dielectric waveguide can be taken and y-direction are 2a and 2b, respectively.as a modiﬁcation from a dielectric slab, by The propagation constant for the surface wavelimiting the width of the slab. Corresponding to along the rectangular waveguide is given by (Ishiiungrounded and grounded dielectric slabs, we have 1995)isolated dielectric waveguides and image guides.The determination of propagation properties of kz = (εr k0 − kx − ky )1/2 2 2 2 (2.236) eo y z 2b x er 2a (a) Ey , Hx Ey , Hx cos (kx x) cos (ky y) cos (kx a) e−kxax cos (ky b) e−kyay 2a x 2b y (b) (c)Figure 2.52 Rectangular dielectric waveguide and its ﬁeld distributions (Ishii 1995). (a) Geometrical structure,(b) ﬁeld distribution along x-direction, and (c) ﬁeld distribution along y-direction. Source: Ishii, T. K. (1995).Handbook of Microwave Technology, Vol 1, Academic Press, San Diago, CA, 1995
78 Microwave Electronics: Measurement and Materials Characterization y b er eo x 2a Ground planeFigure 2.53 Conﬁguration of an image guide (Ishii 1995). Source: Ishii, T. K. (1995). Handbook of MicrowaveTechnology, Vol 1, Academic Press, San Diago, CA, 1995with εre (y) = εr − (ky /k0 )2 (2.244) −1 mπ 1 and the value of ky is the solution of the following kx = 1+ (2.237) 2a a[(εr − 1)k0 − ky ] 2 2 set of equations (Ishii 1995): −1 nπ 1 tan(ky b) = εre (x)ky0 /ky (2.245) ky = 1+ (2.238) 2b εr b[(εr − 1)k0 ]1/2 ky = εre (x)k0 − kz 2 2 2 (2.246)kx0 = (εr − 2 1)k0 − 2 kx − 2 ky (2.239) ky0 = [εre (x) − 1]k0 − ky 2 2 2 (2.247)ky0 = (εr − 2 1)k0 − 2 ky , (2.240) εre (x) = εr − (kx /k0 )2 (2.248)where kx and ky , kx0 and ky0 are the trans-verse propagation constants inside and outside the Dielectric microstripdielectric waveguide respectively, and k0 is thefree-space propagation constant. Figure 2.54 shows the geometry of a dielectric microstrip, which is a modiﬁed image guide with a dielectric slab interposed between a dielectricRectangular image guide ridge and the grounding plane. In this structure,A rectangular image guide can be taken as a the dielectric constant of the ridge (εr2 ) is usuallymodiﬁcation from the grounded dielectric slab by greater than that of the substrate (εr1 ). The ﬁeldslimiting the width of the slab. Figure 2.53 shows are thus mostly conﬁned to the area around thethe conﬁguration of an image guide whose axis is dielectric ridge, resulting in low attenuation. Onalong the z-direction. The width of the dielectric the basis of this basic geometry, many variationsis 2a while the height of the dielectric is b. can be made for different purposes. The propagation constant of a surface wave onan image guide is also given by Eq. (2.236), where Dielectric 2the value of kx is the solution of the following setof equations (Ishii 1995): Conductor er2 Dielectric 1 tan(kx a) = kx0 /kx (2.241) er1 2 kx = 2 εre (y)k0 − 2 kz (2.242) kx0 = kz − k0 2 2 2 = [εre (y) − 1]k0 − kx 2 2 (2.243) Figure 2.54 Geometry of dielectric microstrip
Microwave Theory and Techniques for Materials Characterization 7126.96.36.199 Cylindrical dielectric waveguide withFigure 2.55(a) shows a cylindrical dielectric wave- kc = n 2 k0 − β 2 = h 2 2 2 (r < a) (2.250) 1guide whose cross section is a circle with radiusa. The dielectric constant of the cylinder is εr1 , kc = n2 k0 − β 2 = −p 2 (r > a) 2 2 2 (2.251)and that of the environment is εr2 . In some cases, √ ni = εri (i = 1, 2) (2.252)the dielectric cylinder is covered with a layerof another dielectric material, and such a struc- By assumingture is often used in optical communications,and is usually called optical cable, as shown Ez Hz = AB R(r) (ϕ), (2.253)in Figure 2.55(b). For optical cables, usually therefraction index n = (εr )1/2 is used. Usually, the from Eq. (2.249), we can getrefraction index of the core n1 is larger thanthat of the cover n2 . Both dielectric cylinders d2 + n2 =0 (2.254)and optical cables can support surface waves. As dϕ 2electromagnetic ﬁelds decay quickly in the cover d2 R dRlayer along the r-direction, if the cover layer is r2 2 +r + (h2 r 2 − n2 )R = 0 (r < a) dr drthick enough, the ﬁelds outside the cover can be (2.255) 2neglected and we can assume that the cover layer 2d R dR r +r − (p r + n )R = 0 (r > a) 2 2 2has inﬁnite thickness. Therefore, for the propa- dr 2 drgation of surface waves, optical cables shown in (2.256)Figure 2.55(b) can be taken as a dielectric cylinder Equations (2.254)–(2.256) indicate that the lon-shown in Figure 2.55(a). In the following discus- gitudinal ﬁeld components are in the followingsion, we concentrate on the surface waves propa- forms:gating along a dielectric cylinder. Ez = An Jn (hr) exp(jnϕ) exp(−jβz) (r < a) As shown in Figure 2.55(a), we assume that the (2.257)axis of the dielectric cylinder is along the z-axis,and the propagation factor of electromagnetic wave Hz = Bn Jn (hr) exp(jnϕ) exp(−jβz) (r < a)along the z-direction is exp(−jβz). The longitudi- (2.258)nal ﬁeld components Ez (r, ϕ) and Hz (r, ϕ) satisfy Ez = Cn Kn (pr) exp(jnϕ) exp(−jβz) (r > a)the following equation: (2.259) ∂2 1 ∂ 1 ∂2 Hz = Dn Kn (pr) exp(jnϕ) exp(−jβz) (r > a), Ez Hz + Ez Hz + 2 2 Ez Hz (2.260) ∂r 2 r ∂r r ∂ϕ where An , Bn , Cn , and Dn are amplitude constants, + kc Ez Hz = 0 2 (2.249) Jn (hr) is the ﬁrst type Bessel function, and Kn (pr) is the second type modiﬁed Bessel function. According to wave propagation equations, we can get the transverse ﬁeld components (Er , Eϕ , Hr , and Hϕ ) from the longitudinal ﬁeld components z (Ez and Hz ). According to the boundary conditions at r = a, we can determine the relative amplitudes of the er2 er1 0 x n2 ﬁeld components and get the eigenvalue equation: j r 2 k1 Jn (u1 ) k 2 K (u2 ) Jn (u1 ) Kn (u2 ) n1 + 2 n + (a) (b) u1 Jn (u1 ) u2 Kn (u2 ) u1 Jn (u1 ) u2 Kn (u2 ) 2Figure 2.55 Cylindrical surface waveguides. (a) Di- 1 1 = n2 β 2 2 + 2 , (2.261)electric cylinder and (b) optical cable u1 u2
80 Microwave Electronics: Measurement and Materials Characterizationwhere 2πa n2 − n2 ki2 = ωεri ε0 µ0 (i = 1, 2) (2.262) λc,0i = 1 2 , (2.266) v0iand the two parameters (u1 = ha and u2 = pa)satisfy the following equation: where v0i (i = 1, 2, 3, . . .) is the root of zero order Bessel function. TM01 and TE01 modes have the u2 + µ2 = (n2 − n2 )(k0 a)2 1 2 1 2 (2.263) longest cutoff wavelength: From Eqs. (2.261) and (2.263), we can calculatethe values of u1 and u2 from which we can further 2πa n2 − n2 1 2get the values of h, p, and β. The results obtained λc,01 = (2.267) 2.405are related to the value of n. For Eq. (2.261), when n = 0, the right-hand side It should be indicated that pure TM or TE modesvanishes, and one of the two factors should be are possible only if the ﬁeld is independent of theequal to zero. Actually, the two factors are the angular coordinate (n = 0). As the radius of the rodeigenvalue equations for the axially symmetric TM increases, the number of TM and TE modes alsoand TE modes, respectively: increases. When the ﬁeld depends on the angular 2 k1 Jn (u1 ) k 2 K (u2 ) coordinate (n = 0), pure TM or TE modes no + 2 n =0 (TM modes) longer exist. All modes with angular dependence u1 Jn (u1 ) u2 Kn (u2 ) (2.264) are a combination of a TM and a TE mode, and are Jn (u1 ) Kn (u2 ) classiﬁed as hybrid EH or HE modes, depending + =0 (TE modes) on whether the TM or TE mode predominates, u1 Jn (u1 ) u2 Kn (u2 ) (2.265) respectively. For hybrid EHni and HEni modes,TM0i and TE0i modes are degenerate, and their the solutions for Eq. (2.261) are quite complicated,cutoff wavelength is given by and usually numerical methods are needed. 1.5 1.4 1.3 b/k0 HE11 TE 01 TM 01 1.2 1.1 0.613 1.0 0.2 0.4 0.6 0.8 1.0 1.2 2a/ l0Figure 2.56 Ratio of β to k0 for the ﬁrst three surface-wave modes on a polystyrene rod with εr = 2.56 (Collin1991, p722). Source: Collin, R. E. (1991). Field Theory of Guided Waves, 2nd ed., IEEE Press, Piscataway, NJ,1991. 2003 IEEE
Microwave Theory and Techniques for Materials Characterization 81 All the hybrid modes (n = 0), with the exception move away the half below the conducting plane,of the HE11 mode, exhibit cutoff phenomena and such a structure is usually called cylindricalsimilar to those of the axially symmetric modes. image guide. As the electromagnetic energy isFor n = 1, the cutoff condition for HE1i mode concentrated on the space close to the dielectricis J1 (u1 ) = 0, and the cutoff wavelength for material, such structure does not require very wideHE11 is inﬁnity. Since the HE11 mode has no conducting plane. In practical applications, thelow-frequency cutoff, it is the dominant mode. conducting plane is also used as a support to theFigure 2.56 shows the relationship between β and image guide.λ0 of the three lowest surface wave modes (HE11 ,TM01 , TE01 ) of a polystyrene rod in air. It is clearthat if the diameter of the rod is less than 0.613λ0 , 188.8.131.52 Coaxial surface-wave transmissiononly the HE11 mode can propagate. structure Figure 2.57 shows the ﬁeld distribution of HE11 As shown in Figure 2.60, a conducting cylindermode. The ﬁeld distribution of HE11 mode is covered with a dielectric layer can also supportquite similar to that of TE11 mode in a circular surface waves. Among the possible propagationwaveguide. So the HE11 mode of a dielectric rod modes, the TM01 one has no low-frequency cutoff.can be excited using a circular waveguide, as Coaxial surface-wave structures have the advan-shown in Figure 2.58. tage that Maxwell’s equations for the structures As shown in Figure 2.59, if we place an can be solved rigorously. Usually, a perfectly con-inﬁnitely large ideal conducting plane at the ducting cylinder and a lossless dielectric coatingcenter of the dielectric cylinder, as the electric are assumed.ﬁeld is perpendicular to the conducting plane, In a cylindrical coordinate system, if wethe ﬁeld distribution is not affected. So we can assume longitudinal ﬁeld components in the form G(r, ϕ) = R(r) exp(jvϕ), we have the following differential equation (Marincic et al. 1986): d2 R 1 dR v2 + · + ε r k0 − β 2 − 2 2 R=0 dr 2 r dr r (2.268) Equation (2.268) has to be solved in two regions: the dielectric region and the outer space. In the dielectric region, εr is the relative permittivity ofFigure 2.57 Field distribution of HE11 mode the dielectric, while in the outer region εr = 1. Circular waveguide Dielectric cylinder E K TE11 HE11 H Conical impedance matchFigure 2.58 Excitation of HE11 mode surface wave on a dielectric cylinder by a circular waveguide in TE11mode (Musil and Zacek 1986). Reprinted from Musil, J. and Zacek, F. (1986). Microwave Measurements of ComplexPermittivity by Free Space Methods and Their Applications with permission from Elsevier, Amsterdam
82 Microwave Electronics: Measurement and Materials Characterization and the characteristic equation for TE modes is: J1 (ua)Y0 (ub) − Y1 (ua)J0 (ub) wK0 (wb) −u = J1 (ua)Y1 (ub) − Y1 (ua)J1 (ub) K1 (wb) (2.273) Equation (2.272) gives a solution for the wave that has no low-frequency cutoff, and in fact itFigure 2.59 Cylindrical image guide represents the eigenvalue equation for the TM0m modes. The lowest-order mode is TM01 , which closely resembles coaxial line TEM mode in the y r Dielectric Metal dielectric region. This type of wave is known as the Sommerfeld–Goubau wave (Goubau 1950). b Equation (2.273) is the eigenvalue equation for the j TE0m modes. If v = 0, the boundary conditions for x z TE or TM modes cannot be satisﬁed, while those a er for hybrid HE and EH modes can be satisﬁed. The TE0m and all hybrid modes have low-frequency cutoff. Usually, TM01 mode is the prime mode. TheFigure 2.60 Cross section of a coaxial surface-wave ﬁeld components of TM01 mode in the dielectricguide and free-space region are (Marincic et al. 1986) Eϕ (r) = 0, Hr (r) = 0 (2.274)The solutions for Eq. (2.268) are either Besselfunctions of the ﬁrst and second kind, Jv (x) and In the dielectric region (a ≤ r ≤ b)Yv (x), or modiﬁed Bessel functions of the ﬁrst andsecond kind, Iv (x) and Kv (x). The selection of Ez (r) = AJ0 (ur) + BY0 (ur) (2.275)solutions depends on the sign of (k0 εr − β 2 ). If this 2 βterm is positive, the solutions are Bessel functions Er (r) = j [AJ1 (ur) + BY1 (ur)] (2.276) uof the ﬁrst and second kind. In the opposite case, ωεr ε0the solutions are the modiﬁed Bessel functions. Hϕ (r) = j [AJ1 (ur) + BY1 (ur)] (2.277) It can be shown that the phase coefﬁcient β must ulie between the limits (Marincic et al. 1986): In the free space region (r ≥ b) √ 1 ≤ β/k0 ≤ εr (2.269) Ez (r) = CK 0 (wr) (2.278) βHere, we introduce following two parameters u Er (r) = −j CK 1 (wr) (2.279)and w: w ωε0 Hϕ (r) = −j CK 1 (wr) (2.280) u2 = k0 εr − β 2 2 (2.270) w w 2 = β 2 − k0 2 (2.271) The constants A, B, and C satisfy the following relations:If β satisﬁes Eq. (2.269), the parameters u and ware real. B = −A[J0 (ua)/Y0 (ua)] (2.281) It can be shown that (Collin 1991) the charac- J0 (ub)Y0 (ua) − J0 (ua)Y0 (ub) C=A (2.282)teristic equation for TM modes is K0 (wb)Y0 (ua) u J0 (ua)Y0 (ub) − Y0 (ua)J0 (ub) wK0 (wb) The surface waves on a coaxial surface waveguide = , are usually launched and received using horns. εr Y0 (ua)J1 (ub) − J0 (ua)Y1 (ub) K1 (wb) (2.272) Figure 2.61 shows an example for launching
Microwave Theory and Techniques for Materials Characterization 83 Magnetic field Electric field Coaxial line Launching horn Coaxial surface waveguideFigure 2.61 Launching of surface waves on a coaxial surface waveguide. Modiﬁed from Friedman, M. andFernsler, R. F. (2001). “Low-loss RF transport over long distance”, IEEE Transactions on Microwave Theory andTechniques, 49 (2), 341–348. 2003 IEEEsurface waves on a coaxial surface waveguide energy from one point to another without signiﬁ-(Friedman and Fernsler 2001). cant loss or dispersion. As shown in Figure 2.62, In the design of a coaxial surface waveguide, it antennas can be taken as a transition from a trans-is important to know the radius that determines the mission line to free space. In actual applications, acontour through which a certain speciﬁed amount piece of transmission line is used to transport elec-of power is transmitted. Another important factor is tromagnetic energy from the signal source to thethe cross section through which a certain speciﬁed antenna, or from the antenna to the receiver. Soamount of power is transmitted. Discussions on antennas can be classiﬁed into transmitting anten-these two issues can be found in (Marincic et al. nas and receiving antennas. Sometimes, an antenna1986). is used as both transmitting and receiving antenna.2.2.5 Free spaceFree space is an important wave transmissionscheme in communication and materials research.In radar and satellite communication, electromag-netic waves propagate through free-space. In mate-rials property characterization, free-space providesmuch ﬂexibility in studying electromagnetic mate-rials under different conditions. In this section, weﬁrst introduce antennas as transitions from guidedlines to free-space, and then we discuss two typesof electromagnetic waves in free-space: parallelelectromagnetic beams and focused electromag-netic beams. a b c d184.108.40.206 Antenna as transition Figure 2.62 Antenna as a transition device. (a) Source, (b) transmission line, (c) antenna, and (d) radiatedAntennas are designed to efﬁciently radiate elec- free-space wave (Balanis 1997). Source: Balanis, C.tromagnetic energy into free-space, while transmis- A. (1997), Antenna Theory: Analysis and Design, Johnsion lines are designed to efﬁciently transport the Wiley, New York
84 Microwave Electronics: Measurement and Materials Characterization (a) (b) (c)Figure 2.63 Conﬁgurations of typical aperture antennas. (a) Rectangular waveguide, (b) pyramidal horn, and (c)conical horn (Balanis 1997). Source: Balanis, C. A. (1997), Antenna Theory: Analysis and Design, John Wiley,New York Figure 2.63 shows three types of aperture anten- in Chapter 10, circularly polarized electromagneticnas often used in microwave electronics and the waves are often used.characterization of electromagnetic materials. An Radiation pattern is one of the most impor-open-end hollow pipe, such as rectangular waveg- tant properties of an antenna. For a transmittinguide, can be taken as an antenna, and such antennas antenna, the pattern is a graphical plot of the powercan be used in materials property measurement or ﬁeld strength radiated by the antenna in differ-using reﬂection method. In materials characteriza- ent angular directions. Based on the principle oftion, pyramidal horn and conical horn are often reciprocity, the transmitting and receiving radiationused in reﬂection and transmission measurements patterns of a given antenna are the same. Omnidi-in free-space. Besides, antennas sometimes are rectional and pencil-beam patterns are two typi- cal polar patterns. In an omnidirectional pattern,covered with a dielectric material to protect them the antenna radiates or receives energy equally infrom hazardous conditions of the environment. In all directions, while in a pencil-beam pattern, theaddition, as will be discussed later, sometimes antenna radiates or receives energy mainly in onedielectric lens is used to control the beam shape. direction. A typical pencil-beam pattern consists of For the characterization of electromagnetic a main lobe and a number of side lobes. The levelproperties of materials, important electromagnetic of the side lobes in a pencil-beam pattern must beparameters describing antennas mainly include kept to a minimum as the energy in the side lobes ispolarization, radiation pattern, power gain, band- wasted when the antenna is transmitting radiationswidth, and reciprocity. In the following, we make and the side lobes can pick up noise and interfer-a brief discussion on these parameters, and detailed ence when the antenna is receiving radiations.discussions on them can be found in (Balanis Two gain parameters are related to the radiation1997; Connor 1989; Wait 1986). pattern: power gain and directive gain. Power gain Generally speaking, the electromagnetic wave G is normally deﬁned in the direction of maximumradiated from an antenna is in elliptical polar- radiation per unit area asization. Linear and circular polarizations are two Pspecial cases of the general form of elliptical polar- G= , (2.283)ization. In experiments, linear polarizations are P0further classiﬁed into vertical or horizontal polar- where P is the power radiated by the antenna underizations. Circular polarization can be taken as a study, while P0 is the power radiated by a referencecombination of vertical and horizontal polariza- antenna. A reference antenna usually is assumedtions. In materials property characterization, elec- to be lossless and radiates equally in all directions.tromagnetic waves are usually linearly polarized The power gain G is expressed as a pure numberthough other types of polarization may be used for or in dB. Equation (2.283) assumes that the inputspeciﬁc purposes on occasions. For example, in the power to the given antenna is the same as the inputmeasurement of chirality, which will be discussed power to the reference antenna.
Microwave Theory and Techniques for Materials Characterization 85 If the antenna is lossless, its directive gain D is by the edges of the sample are negligible. How-deﬁned by ever, in actual experiments, the dimensions of sam- Pmax ples are limited. In some cases, we need to mea- D= , (2.284) Pave sure samples with small transverse dimensions, for example, in the order of a few centimeters. There-where Pmax is the maximum power radiated per fore, to achieve accurate measurement results, weunit solid angle, while Pave is the average power should control the transverse cross section of theradiated per unit solid angle. From their deﬁnitions, beam. To ensure that the algorithm is applica-G is slightly less than D. ble for practical measurements, following require- Working bandwidth is also an important param- ments should be satisﬁed:eter describing an antenna. Three expressions forthe working frequency band are often used. For D >> h >> λ0 , (2.287)an antenna with the lowest working frequency f1and the highest working frequency f2 , the absolute where D stands for the transverse dimensions ofbandwidth of the antenna is the plate or specimen, h denotes the transverse dimensions of the probing beam, λ0 is the wave- length of the electromagnetic wave. f = f2 − f1 (2.285) To satisfy the requirements of Eq. (2.287), two kinds of microwave beams are often used: parallelThe relative bandwidth is microwave beams and focused microwave beams. As shown in Figure 2.64, a parallel microwave f f2 − f1 =2 (2.286) beam can be achieved by using specially designed f f2 + f1 lenses or reﬂectors. The function of the lens and the reﬂector is to transform the spheri-The bandwidth ratio is expressed as: f2 : f1 . cal wave from a point-source feed to a plane Reciprocity is another important consideration in wave.selecting antennas for communication and materi- Though a lens and a reﬂector may have theals characterization. The properties of a transmit- same functions in producing parallel beams, theting antenna are very similar to those of a receiving positions of the point-source feeds for a lens and aantenna because of the theorem of reciprocity. It reﬂector are different. As shown in Figure 2.64,is often assumed that the antenna parameters, such the point-source feed for a lens is located onas polar power gain and bandwidth, are the same one side of the lens and the generated planewhenever the antenna is used for transmitting or wave emerges from the other side of the lens,receiving electromagnetic waves. while for a reﬂector system, the point-source feed and the generated plane wave lie on the same side of a reﬂector. As the point-source feed is220.127.116.11 Parallel microwave beam placed at the region for the outgoing wave, the output parallel beam is disturbed by the sourceThe control of the beam shape is important for feed. So in materials property characterization,microwave communication and materials property we often use plane waves generated by dielectriccharacterization (Musil and Zacek 1986). In free- lenses. However, in some special cases such asspace measurements, the sample under test is often the measurement of radar cross section (RCS), asput between the transmitting and receiving anten- the object to be measured is large, the requirednas. In the development of the algorithms for parallel beam should also be large. As it ismaterials property characterization, we assume that impractical to build a very large lens, usuallyall the energy of the plane electromagnetic wave reﬂectors are used to produce parallel beams,interacts with the sample, and the sample is inﬁ- and the perturbation of the source feed can benite in the direction perpendicular to the direc- minimized by proper arrangement of the sourcetion of wave propagation so that the diffractions feed and the reﬂector.
86 Microwave Electronics: Measurement and Materials Characterization Lens Plane wave Reflector Plane wave Point-source feed F F Point-source feed f f (a) (b)Figure 2.64 The generation of a parallel microwave beam by means of (a) a lens and (b) a reﬂector. Source:Musil, J. and Zacek, F. (1986). Microwave Measurements of Complex Permittivity by Free Space Methods and TheirApplications, Elsevier, Amsterdam Aperture of focussing 1.5h0 1.5h0 antenna h0 Focus Z1 0 Z2 Z (|Z1| + |Z2|) (E/E0) = Constant fFigure 2.65 Parameters describing a focused microwave beam. Source: Musil, J. and Zacek, F. (1986). MicrowaveMeasurements of Complex Permittivity by Free Space Methods and Their Applications, Elsevier, Amsterdam It should be noted that the focus length f of a into a focus point, and its geometry is determinedlens or a reﬂector may be different for different by the spatial distribution of the electric ﬁeldfrequencies. If we want to generate parallel beams strength in the beam (Musil and Zacek 1986).using the conﬁgurations shown in Figure 2.64 at As shown in Figure 2.65, a focused microwavefrequencies far separated, it may be necessary to beam is usually described by three parameters:adjust the position of the source feed for different focus distance f that determines the position offrequencies. the focal plane (z = 0), beam width h0 at the focal plane, and beam depth that is the distance between the two planes (z = z1 ) and (z = z2 ),18.104.22.168 Focused microwave beam where the beam width increases by 50 % comparedFor the characterization of small samples, focused to the beam width at the focal plane (z = 0).microwave beams are often required. A focused It should be noted that the magnitudes used inmicrowave beam can be produced by focusing deﬁning these parameters are determined at amicrowave energy transmitted from an antenna certain level of ﬁeld intensity with respect to the
Microwave Theory and Techniques for Materials Characterization 87ﬁeld intensity on the axis of the focused beam. and line approach. After introducing the basicThe beam width increases with the decrease of the parameters describing a resonator, we use equiva-ﬁeld strength level at which we determine the beam lent circuits to analyze the general properties ofwidth. microwave resonance, then we discuss the ﬁeld Beam width determines minimum transverse distributions of several types of resonators oftendimensions a specimen should have. If the sam- used in materials characterization, including coax-ples have large transverse dimensions, the beam ial resonators, planar-circuit resonators, waveguidewidth determines the spatial resolution for the resonators, dielectric resonators, and open res-measurement of local inhomogeneities. Generally onators.speaking, a system with smaller wavelength λ0 ,smaller focus distance l, and larger radius of thelens aperture has smaller beam width. The small- 22.214.171.124 Resonant frequency and quality factorest beam width that can be obtained in practice ata level of −10 dB is approximately 1.5λ0 (Muzil Resonance is related to energy exchange, and1986). electromagnetic resonance can be taken as a The phase distribution in a focused microwave phenomenon when electric energy and magneticbeam is also important for materials property energy can periodically change totally from onecharacterization using focused microwave beams. to the other. If the resonance is lossless, the sumIn the place near the focal plane z = 0, the of electric energy and magnetic energy does notelectromagnetic ﬁeld can be taken as a plane change with time:wave. So we use focused microwave beams innonresonant methods; we should ensure that the We (t) + Wm (t) = W0 (2.288)sample under test is placed close to the focal plane(z = 0). Therefore beams with larger depths are Resonant frequency is the frequency when thefavorable. electric energy can be totally changed to magnetic In summary, for materials characterization, we frequency, and vice versa. Resonant frequency f0hope to have minimum beam-width and meanwhile is the most important parameter for a resonator.a maximum beam depth. Practical conﬁgurations The electric energy and magnetic energy canusing focused microwave beams in materials be calculated from the ﬁeld distributions in theproperty characterization will be given in later resonator. The electromagnetic ﬁeld distributionchapters. and the resonant frequency can be found by solving the wave functions with certain boundary conditions:2.3 MICROWAVE RESONANCE ∇ 2E + k2E = 0 (2.289)2.3.1 Introduction ∇ H + k H = 0, 2 2 (2.290)The resonant methods for materials property char-acterization are based on microwave resonance. where k 2 = ω2 εµ, and ε and µ are the permittivityGenerally speaking, there are two kinds of reso- and permeability of the medium in the resonator.nant structures: transmission type, which is made For an ideal lossless resonator, k is a series offrom transmission structures, and non-transmission discrete real numbers k1 , k2 , . . ., and they aretype, such as ring resonators and sphere resonators. called the eigenvalues of Eqs. (2.289) and (2.290).In the following discussions, we focus on trans- The resonant frequencies can be calculated frommission type resonators, such as rectangular res- the eigenvalues:onator, cylindrical resonator, coaxial resonator, and cmicrostrip resonator. fi = ki (i = 1, 2, . . .), (2.291) 2π Similar to microwave transmission, microwaveresonance can be studied in both ﬁeld approach where c is the speed of light in free space.
88 Microwave Electronics: Measurement and Materials Characterization Quality factor is deﬁned as where Pc is the energy dissipation due to conductor loss, Pd is the energy dissipation due to the W W Q0 = 2π = ω0 , (2.292) dielectric loss, Q0c and Q0d are the quality PL T0 PL factors of the resonator if we only consider thewhere W is the total energy storage in the cavity, conductor loss and dielectric loss, respectively. IfPL is the average energy dissipation within the the conductivity of the dielectric medium withincavity, and T0 is the resonant period. At resonance, the cavity is σ , we havethe total energy storage equals the maximum 1electric energy or maximum magnetic energy: Pd = σ |E|2 dV (2.298) 2 V ε So the value of Q0d can be calculated from W = We,max = |E|2 dV 2 V 1 ω0 · εr |E|2 dV µ ω0 W 2 = |H |2 dV = Wm,max (2.293) Q0d = = V 2 Pd 1 V σ |E|2 dVThe above integration is made over the whole res- 2 Vonator. ω0 εr 1 = = , (2.299) Here we consider a hollow metallic cavity ﬁlled σ tan δwith a dielectric medium. If the medium within the where εr is the dielectric constant of the mediumcavity is lossless the energy dissipation power is in the cavity, σ = ω0 εr , and tan δ = εr /εr .caused by the cavity wall: The relationship between the energy storage and PL = Rs 2 x Ž |H | dS S t 2 (2.294) energy dissipation can also be described by the attenuation parameter α, which is related to the attenuation rate of a resonator after the source isThe above integration is made over the whole removed. By deﬁning E = E0 e−αt , we havecavity wall. The surface resistance of the cavitywall Rs in Eq. (2.294) is given by: W = W0 e−2αt , (2.300) ωµ1 δ where W0 is the energy storage when t = 0. So the Rs = , (2.295) energy dissipation is given by 2where µ1 is the permeability of the conductor, δ is dW PL = − = 2αW (2.301)the penetration depth of the conductor. According dtto Eqs. (2.292)–(2.295), we have From Eqs. (2.292) and (2.301), we can get PL ω0 |H |2 dV α= = (2.302) 2 2W 2Q0 Q0 = · V (2.296) δ |Ht |2 dS From Eqs. (2.300) and (2.302), we have S ω0 − t If we also consider the dielectric loss of the W = W0 e Q0 (2.303)medium in the cavity, the value of Q0 will be Equation (2.303) indicates that the higher the qual-decreased: ity factor, the slower the resonance attenuation. ω0 W ω0 W Q0 = = PL Pc + Pd 126.96.36.199 Equivalent circuits of resonant structure 1 As shown in Figure 2.66, depending on the = [Pc /(ω0 W ) + Pd /(ω0 W )] selection of reference plane, a resonator can be 1 represented by a series equivalent circuit or a = , (2.297) parallel equivalent circuit. If the reference plane 1/Q0c + 1/Q0d
Microwave Theory and Techniques for Materials Characterization 89 R L I + + ϳ V I C ϳ V C L R − − (a) (b)Figure 2.66 Equivalent RLC circuits for a resonator. (a) Series circuit and (b) parallel circuit |U(w)| |I | G 1 |I| Current source √2 G I ϳ C G L V U w w1 w0 w2 ∆w (a) (b)Figure 2.67 Frequency response of a resonator. (a) Parallel equivalent circuit of a resonator. The relationshipbetween conductance G and resistance R is: G = 1/R. (b) Characteristic frequencies on a resonant curveis chosen at a place where the electric ﬁeld So we can get the resonant frequency:integration is zero (correspondingly the magneticﬁeld integration there is the maximum), the 1 ω0 = √ (2.306)resonator can be represented by a series RLC LCcircuit, and if the reference plane is chosenat a plane where the electric ﬁeld integration From Eqs. (2.304)–(2.306), we can getis the maximum (correspondingly the magneticﬁeld integration there is zero), the resonator can 1 1 U= =be represented by a parallel RLC equivalent G + jC[ω − 1/(ωLC)] G + jC ωcircuit. The following discussions focus on parallel (2.307)equivalent circuits. with As shown in Figure 2.67, when a current source ω = 2(ω − ω0 ) (2.308)is connected to the resonator, a voltage U is builtacross the resonator. The voltage U will change If ω = ±G/C, the voltage amplitude decreases √with the change of frequency: to 1/ 2 of the maximum value: I |I | |U (ω0 )| U= (2.304) |U (ω)| = √ = √ (2.309) G + jωC + 1/(jωL) 2G 2For a parallel circuit, the voltage reaches its highestvalue at resonance. From Eq. (2.304), we have From Eq. (2.309), we can get 1 1 jω0 C + =0 (2.305) U (ω)U ∗ (ω) = U (ω0 )U ∗ (ω0 ) (2.310) jω0 L 2
90 Microwave Electronics: Measurement and Materials CharacterizationAs the energy dissipation at G is given by resonator equals the magnetic ﬁeld energy in the resonator. 1 Pd = U U ∗ G, (2.311) 1 2 We = Wm = CU U ∗ (2.317) 4it also decreases to half of the maximum value. Actually, Eq. (2.317) can be used as a criteria toConsider the average energy storage at the capac- determine the resonant frequency of a resonator.itor and inductor: For a parallel equivalent circuit, if the working 1 frequency is higher than the resonant frequency We = U U ∗C (2.312) (ω > ω0 ), the electric ﬁeld energy is larger than 4 ∗ the magnetic ﬁeld energy; if the working frequency 1 ∗ 1 U U is lower than the resonant frequency (ω < ω0 ), Wm = IL IL L = L 4 4 jωL jωL the magnetic ﬁeld energy is higher than the 1 1 1 ω2 electric ﬁeld energy. For a series equivalent circuit, = U U ∗ = CU U ∗ 0 (2.313) contrary conclusions can be obtained. 4 ω2 L 4 ω2Equation (2.312) indicates that the energy storageat the capacitor decreases to half of its maximum 188.8.131.52 Coupling to external circuitvalue. As the frequency (ω) is close to theresonant frequency (ω0 ), Eq. (2.313) indicates that An actual resonator is always coupled to externalthe energy storage at the inductor also decreases circuits. Through coupling, the source providesto half the value of its maximum value. energy to the resonator, and such a procedure The difference between two half-power frequen- is usually called excitation. A resonator cancies (ω1 = ω0 − G/(2C) and ω1 = ω0 + G/(2C)) also provide energy to an external load throughis called the half-power bandwidth: coupling, and such a procedure is usually called loading. G The coupling mechanisms often used generally ω = ω2 − ω1 = (2.314) fall into three categories: electrical coupling, mag- C netic coupling, and mixed coupling. In electricalSometimes, the half power bandwidth is described coupling, an electrical dipole, usually a coaxialas needle made from the central conductor of a coax- ω 1 G ial line, is inserted into the place with maximum f = = (2.315) electric ﬁeld. In magnetic coupling, a magnetic 2π 2π C dipole, usually a coaxial loop made by connect-It is clear that the narrower the half-power ing the central conductor to the outer conductorbandwidth, the better the frequency selectivity of is often placed at the place with maximum mag-the resonator. netic ﬁeld. A typical example of mixed coupling The quality of the resonator can also be deﬁned is the coupling iris between a hollow cavity andas a metallic waveguide. In the equivalent circuits, a coupling structure is often represented by a trans- ω0 C (1/2)U U ∗ C W former. Q= = ω0 = ω0 ∗G = ω0 ω G (1/2)U U Pd (2.316)Comparing Eqs. (2.292) and (2.316) indicates Cavity coupled to one transmission linethat the two deﬁnitions are consistent, and inexperiments, Eq. (2.316) is more often used. Figure 2.68 shows the equivalent circuits of a Equations (2.312) and (2.313) indicate that at resonator coupled to a transmission line. Asresonant frequency, the electric ﬁeld energy in the shown in Figure 2.68(b), if the position of the
Microwave Theory and Techniques for Materials Characterization 91reference plane Ts is suitably chosen, the res- The coupling coefﬁcient β describes the relation-onator is represented by a parallel RLC cir- ship between the energy dissipation in the cavitycuit, and the coupler is represented by an ideal Pd and the energy dissipation of the external circuittransformer with transforming ratio (1 : n). As Pe , and indicates the extent to which the externalshown in Figure 2.68(c), the parallel RLC cir- circuit is coupled to the resonator:cuit can be transformed to Ts plane through thetransformer: C = n2 C, RP = RP /n2 , and L = 1 U2L/n2 . Pe 2 Zc RP R After the transformation, the resonant frequency β= = = 2 = P = rP , Pd 1 U2 n Zc Zcand quality factor of the equivalent circuit do not 2 2 RP /nchange: (2.320) 1 1 1 where U is voltage applied to the resonator.ω0 = √ = =√ = ω0 Equation (2.320) indicates that the coupling coef- LC (L/n2 )(n2 C) LC ﬁcient equals the normalized resonance resistance (2.318) of the cavity transformed to the plane Ts . At a crit- RP ical coupling state (β = 1), the energy dissipationQ0 = ω0 C RP = ω0 (n2 C) = ω0 CRP = Q0 n2 of the external circuit equals the energy dissipa- (2.319) tion in the cavity (Pe = Pd ). At an over-coupling state (β > 1), the energy dissipation of the exter- A nal load is larger than the energy dissipation in the cavity (Pe > Pd ). At an under-coupling state Transmission line (β < 1), the energy dissipation of the external load Resonant Coupler cavity is less than the energy dissipation in the cavity (Pe < Pd ). The external quality factor of a resonator, (a) coupled to an external circuit, describes the energy A dissipation in the external circuit: Ts A W Qe = ω0 , (2.321) Zc L RP C Pe (b) Ts 1:n A where W is the energy storage in the res- onator and Pe is the energy dissipation in the Ts external circuit coupled to the resonator. From Eqs. (2.297), (2.320), and (2.321), we can get Zc L′ R′ P C′ (c) Q0 β= (2.322) Qe Ts So the coupling coefﬁcient equals the ratio between 1 l′ r′P c′ (d) the intrinsic quality factor and external quality factor of the resonator. In experiments, loaded quality factor QL is oftenFigure 2.68 A resonant cavity coupled to a transmis- used:sion line and its equivalent circuits. (a) A cavity coupled W QL = ω0 (2.323)to a transmission line, (b) an equivalent circuit of the P0 + Pecavity with coupling structure and transmission line, (c)the equivalent circuit transformed to the Ts plane, (d) The relationships between the loaded quality factornormalized equivalent circuit QL , intrinsic quality factor Q0 , external quality
92 Microwave Electronics: Measurement and Materials Characterization A Matching Matching source load Zc1 Coupler 1 Coupler 2 Zc2 (a) A Ts1 A Ts2 im Zc1 L C Zc2 (b) RP 1 : n1 A n2 :1Figure 2.69 (a) A resonator coupled to two transmission lines and (b) its equivalent circuitfactor Qe , and coupling coefﬁcient β are The coupling coefﬁcients of the two couplers are 1 β Q0 RP QL = Q0 = Qe (2.324) β1 = = 2 (2.329) 1+β 1+β Qe1 n1 Zc1 1 1 1 Q0 RP = + (2.325) β2 = = 2 (2.330) QL Q0 Qe Qe2 n2 Zc2 The loaded quality factorCavity coupled to two transmission lines 1Figure 2.69(a) shows a resonator coupled to two QL = ω0 C 1/RP + 1/(n2 Zc1 ) 1 + 1/(n2 Zc2 ) 2transmission lines. One transmission line is con-nected to a matching source and the other trans- 1 = Q0 (2.331)mission line is connected to a matching load. The 1 + β1 + β2equivalent circuit of the resonator is shown in 1 1 1 1Figure 2.69(b), and the two transformers represent = + + (2.332) QL Q0 Qe1 Qe2the two couplers. Using a similar method used in analyzing the Equations (2.331) and (2.332) are the extensionscavity coupled to one transmission line, we can of Eqs. (2.324) and (2.325), respectively. We canget the parameters describing the resonant and analyze a resonator coupled to more transmissioncoupling properties of the resonator. The intrinsic lines in a similar way, and similar conclusions canquality factor Q0 of the resonator is given by be drawn. In the following, we discuss several typical Q0 = ω0 CR P (2.326) types of resonators, including coaxial resonators,The external quality factors for the two transmis- planar circuit resonators, waveguide resonators,sion lines are dielectric resonators, surface wave resonators, Qe1 = ω0 Cn2 Zc1 (2.327) and open resonators. As will be discussed in 1 Chapters 5 and 6, these resonators are often used Qe2 = ω0 Cn2 Zc2 2 (2.328) in the characterization of materials properties
Microwave Theory and Techniques for Materials Characterization 93 l = (2n − 1) l l = (2n − 1) l l=nl 4 4 2 (a) (b) (c)Figure 2.70 Three typical types of coaxial resonators. (a) Half-wavelength resonator, (b) quarter wavelengthresonator, and (c) capacitor-loaded resonatorusing resonator methods and resonant-perturbationmethods. 2a 2b2.3.2 Coaxial resonatorsCoaxial resonators are made from coaxial trans- lmission lines. As shown in Figure 2.70, thereare three typical types of coaxial resonators: Figure 2.71 Field distribution of a half-wavelengthhalf-wavelength resonator, quarter wavelength res- resonator (n = 1). The solid lines stand for electriconator, and capacitor-loaded resonator. ﬁelds. The dots stand for the magnetic ﬁelds coming out from the paper, while the crosses stand for magnetic ﬁeld going into the paper184.108.40.206 Half-wavelength resonatorAs shown in Figure 2.70(a), a half-wavelength the relationship between the resonant wavelengthcoaxial resonator is a segment of coaxial line λ0 and the length of the resonator l:with length l terminated at two ends. Usually, acoaxial resonator works at TEM mode. To avoid λ0 l=n (2.336)the possible resonance along ϕ direction, following 2requirements should be satisﬁed: Equation (2.336) indicates that the length of the π(a + b) < λmin , (2.333) resonator is integral times of the half wavelength, so such a resonator is called half-wavelengthwhere a and b are the radius of the inner and outer resonator, and its ﬁeld distribution is shown inconductors respectively, and λmin is the shortest Figure 2.71.wavelength corresponding to the highest working In microwave engineering, the waveform factorfrequency. is often used, which is deﬁned as: Q0 δ/λ0 , where The ﬁeld distributions can be calculated accord- δ is the penetration depth of the conductor.ing to the boundary conditions: Waveform factor is only related to the geometry, size, and working mode of the resonator. The 2E0 Er = −j sin βz (2.334) waveform factor of a half-wavelength resonator r (n = 1) is given by 2E0 Hϕ = Y0 cos βz, (2.335) δ 1 r Q0 = (2.337) λ0 l 1 + (b/a)where E0 = V0 / ln(b/a), Y0 = (ε/µ) , and β = 1/2 4+ · b ln(b/a)nπ/ l (n = 1, 2, 3, . . .). As β = 2π/λ0 , we can get
94 Microwave Electronics: Measurement and Materials CharacterizationIt can be proven that when (b/a) = 3.6, the The quality factor of a quarter-wavelengthwaveform factor has the highest value. The quality resonator can be calculated from the waveformfactor can be easily calculated from the waveform factor, which is given byfactor. δ 1 1 Q0 = · (2.339) λ0 4 l 1 + (b/a) 4+220.127.116.11 Quarter-wavelength resonator 2b ln(b/a)As shown in Figure 2.70(b), in a quarter-wave- Similar to half-wavelength resonator, the wave-length resonator, one end is shorted, and the form factor has the highest value at (b/a) = 3.6.other end is open. As an open load also causestotal reﬂection, pure standing wave is also built.According to the boundary conditions, we have: 18.104.22.168 Capacitor-loaded resonatorβl = (2n+1)π/2 (n = 0, 1, 2, . . .), so the relation-ship between the length of the resonator l and In a capacitor-loaded resonator, shown in Figureresonant wavelength λ0 is given by 2.70(c), the two ends of the coaxial line are shorted, but at one end of the resonator, there λ0 is a small gap between the inner conductor and l = (2n + 1) (n = 0, 1, 2, . . .) (2.338) 4 the short plate. As shown in Figure 2.73(a), the gap between the inner conductor and short plateEquation (2.338) indicates that the length of the is equivalent to a lumped capacitor, so such aresonator is odd number times of the quarter resonator can be regarded as a hybrid resonatorwavelength. The ﬁeld distribution of a quarter- with both lumped and distributed elements. Thewavelength resonator is shown in Figure 2.72. ﬁeld distribution of the basic mode is shown in In an actual structure, an open end has some Figure 2.73(b). At the gap between the inner con-radiation loss. To minimize the radiation loss, ductor and the short plate, electric ﬁeld dominates,usually the outer conductor is extended, forming and the magnetic ﬁeld is very weak.a segment of cutting-off TM01 circular waveguide. The susceptance B1 at plane AA looking into theHowever, owing to the capacitance introduced by left is B1 = − cot(βl)/Zc , with β = ω/c, where Zcextending the outer conductor, the inner conductor is the characteristic impedance of the coaxial lineis usually made a bit shorter than the quarter and c is the speed of light. The susceptance B2 ofwavelength. the lumped capacitor C0 is B2 = ωC0 . At resonance, B1 + B2 = 0: therefore we have Zc ωC0 = cot(βl) (2.340) A A C0 l A′ l l dFigure 2.72 The ﬁeld distribution of quarter wave- A′length resonator (n = 0). The solid lines stand for elec- (a) (b)tric ﬁelds. The dotes stand for the magnetic ﬁelds com-ing out from the paper, while the crosses stand for Figure 2.73 Capacitor-loaded resonator. (a) Equi-magnetic ﬁeld going into the paper valent circuit and (b) ﬁeld distribution
Microwave Theory and Techniques for Materials Characterization 95 BZc cotbl ZcwC0 (a) (b) (c) O w01 w02 w03 w Figure 2.76 Electric couplings. (a) Excitation of quar- ter-wavelength resonator using a needle dipole, (b) excitation of half-wavelength resonator, and (c) exci- tation of capacitor-loaded resonator. The small metal plates on the tips of the needles in (b) and (c) are usedFigure 2.74 Graphical method for solving Eq. (2.340) to increase the couplingsEquation (2.340) can be solved by numerical is perpendicular to the magnetic ﬁeld distributionmethod and graphical method. The graphical of the mode. The loop is equivalent to a magneticmethod is shown in Figure 2.74. We draw the dipole, and the coupling can be adjusted by adjust-graph BZ c = cot(βl) and BZ c = Zc ωC0 . Their ing the orientation and position of the loop.crossing points are the solutions of Eq. (2.340),and each point corresponds to a resonant mode. Electric coupling As shown in Figure 2.76, electric coupling is22.214.171.124 Coupling to external circuit often achieved by a needle made from the innerCoupling method is an important consideration conductor of a coaxial line. The needle should bein the design of resonators. Here we introduce put at a place where electric ﬁeld dominates, andmagnetic coupling and electric coupling often used the orientation of the needle should be parallel tofor coaxial resonators. the electric ﬁeld. As shown in Figures 2.76(b) and (c), adding a small metal plate on the tip of a coupling needle may increase the coupling.Magnetic couplingFigure 2.75 shows a magnetic coupling by a co- 2.3.3 Planar-circuit resonatorsaxial loop often used in coaxial resonators. Theloop is placed at the place where the magnetic The planar circuits used in microwave electronicsﬁeld dominates, and usually the plane of the loop mainly include stripline, microstrip, and coplanar. As microstrip is most widely used in microwave integrated circuits, we concentrate our discussion on microstrip circuits, and similar conclusions can be obtained for other types of planar circuits. The application of planar resonators in materials characterization will be discussed in Chapter 7. In the following, we will make a brief discussion on the three types of microstrip resonators shown in Figure 2.77: straight ribbon resonator, ring resonator, and circular resonator. However, it should be noted that the equations given below are approximate and can only be used for estimation.Figure 2.75 Excitation of a half-wavelength resonator To make accurate analysis, numerical methods areusing a magnetic coupling often required.
96 Microwave Electronics: Measurement and Materials Characterization method has higher accuracy and sensitivity than l r1 R the straight ribbon resonator method. r2 Most of the ring resonators work at TMmn0 , and the prime mode is TM110 . The resonant condition for TMm10 is (a) (b) (c) π(r1 + r2 ) = mλg , (2.343)Figure 2.77 Three types of microstrip resonators: where r1 and r2 are the inner and outer radius of the(a) straight ribbon resonator, (b) ring resonator, and(c) circular resonator ring respectively. As λg = λ0 /(εeff )1/2 , where εeff is the effective dielectric constant of a microstrip line, we have126.96.36.199 Straight ribbon resonator π(r1 + r2 ) √A straight ribbon resonator shown in Figure 2.77(a) λ0 = εeff (2.344) mis a segment of microstrip line with two openends, and the relationship between the length of To avoid higher order modes, usually the dimen-the microstrip line l and the resonant wavelength sions of the ring resonator should satisfy the fol-λg is given by lowing requirements: r2 − r1 λg < 0.05 (2.345) l=n (n = 1, 2, 3, . . .) (2.341) r2 + r1 2 In actual structures, the ﬁelds extend slightly 188.8.131.52 Circular resonatorbeyond the ends of the microstrip line on bothends. The ﬁeld effects can be represented by a As shown in Figure 2.77(c), a circular resonatorgrounding capacitor Cend at each end, and Cend can can be taken as a special case of a ring resonatorbe transformed to a length of transmission line l. when r2 = 0. Its resonant modes are also TMmno ,So the resonant condition becomes and the prime mode is TM110 mode. λg As the structure of the circular resonator is quite l+2 l =n (n = 1, 2, 3, . . .). (2.342) simple, and has high quality factor, it is widely 2 used as resonant element in microwave electronicThe values of Cend and l can be calculated circuits, and is also often used in materials propertyusing some empirical equations, and can also be characterization.measured experimentally. Straight ribbon resonators are a type of transmis-sion type resonators made from their correspond- 184.108.40.206 Coupling to external circuiting transmission line, microstrip. However, owing Here, we discuss the coupling methods forto the radiations at the two open ends, the quality microstrip half-wavelength resonator, while thefactors of these types of resonators are usually not methods can be extended for other kinds of pla-very high. nar resonators. More discussions on the coupling of planar circuits can be found in (Chang 1989).220.127.116.11 Ring resonator Figure 2.78 schematically shows three types of coupling methods for the half wavelength res-A ring resonator shown in Figure 2.77(b) does not onators: capacitive coupling, parallel-line cou-have open ends, so the radiation loss is greatly pling, and tap coupling. In the following, we focusdecreased. Usually, a ring resonator has higher on the coupling mechanisms of these methods.quality factor than a straight ribbon resonator. However, it should be indicated that to get accurateMainly due to its high quality factor, in mate- knowledge about the coupling properties, numeri-rials property characterization, the ring resonator cal simulations are often needed.
Microwave Theory and Techniques for Materials Characterization 97 q l0 l0 2 2 1 4 q S S 2 3 l0 2 (a) (b) (c)Figure 2.78 Three types of coupling methods. (a) Capacitive coupling, (b) parallel-line coupling, and (c) tapcoupling (Chang 1989). Source: Chang, K. (1989). Handbook of Microwave and Optical Components, Vol. 1, JohnWiley, New York As shown in Figure 2.78(a), capacitive coupling coefﬁcient is proportional to cos2 θ . In practicalis achieved by the capacitance resulted from the structures, to minimize the effects of the T-junctionfringing ﬁelds at the open ends of the resonator at the taping point to the resonant properties ofand the coupling port. The impedance of the cou- the resonator, the input line is often tapered, orpling ports is usually chosen as the characteristic a quarter-wavelength transformer is used (Changimpedance of the external transmission line. The 1989).coupling coefﬁcient is closely related to the widthof the gap (S) between the open ends of the res-onator and the coupling port. Generally speaking, 2.3.4 Waveguide resonatorsthe coupling coefﬁcient increases with the decreaseof the gap. However, when the gap is very small, Waveguide resonators are widely used in materialsa small variation in the gap width may result in property characterization, especially in cavity per-a large change in the coupling coefﬁcient, so this turbation methods. A waveguide resonator is usu-method requires high fabrication accuracy. ally made by shorting the two ends of a segment of As shown in Figure 2.78(b), a parallel-line waveguide. According to the types of waveguidescoupler consists of a coupled line of an electrical from which the resonators are made, two kindslength θ with two open ports (2 and 4). The port of waveguide resonators are often used: rectangu-3 is connected to an open-circuited line of an lar cavity resonators and circular cavity resonators.electrical length (π – θ ), forming the resonator, Corresponding to the wave propagation modes inand the input port (port 1) is connected to a waveguides, resonant modes include TE and TMtransmission line with characteristic impedance Zc . modes.The coupling coefﬁcient is mainly determined bythe electrical length θ and the spacing between theparallel lines S. Compared to capacitive coupling, 18.104.22.168 Rectangular cavity resonatorcoupled line coupling has more ﬂexibility inadjusting the coupling coefﬁcient. Figure 2.79 shows a rectangular cavity resonator As shown in Figure 2.78(c), a tap coupling is made from a rectangular waveguide by shorting theachieved by directly connecting the coupling port two ends. Its structural dimensions include widthto the resonator at an appropriate position. The a, height b, and length c. For a rectangular cavitycoupling coefﬁcient is mainly related to the electri- resonator, there are two groups of resonant modescal length θ between the open end of the resonator (TEmnp and TMmnp ), which correspond to the TEmnand taping point. Theoretically, if the taping point and TMmn propagation modes in a rectangularis at the center of the resonator, the coefﬁcient is waveguide respectively. The ﬁrst two subscriptszero because the center point is at the electric ﬁeld m and n come from the wave propagation modesnode. The smaller the θ value, the stronger the cou- TEmn and TMmn , and they represent the changingpling. Theoretical analysis shows that the coupling cycles along the x and y directions. The last
98 Microwave Electronics: Measurement and Materials Characterization a H y b E x z c y x zFigure 2.79 Structure of a rectangular cavity resonator Figure 2.80 Field distribution of TE101 rectangular cavitysubscript p represents the changing cycles alongthe z direction. A special resonant mode, TE101 mode is widely used in the characterization of electromagnetic materials. From Eqs. (2.346)–(2.352), we can getTE resonant modes the ﬁeld components of TE101 mode:On the basis of our discussions on TE propagation Aωµa π π Ey = −2 sin x sin z (2.353)modes of rectangular waveguides and the boundary π a cconditions, we can get the ﬁeld distributions of the Aa π πTEmnp mode: Hx = j sin x cos z (2.354) c a c 2A pπ mπ mπ π π Hx = j · · sin x Hz = −j2A cos x sin z (2.355) 2 kc c a a a c Ex = Ez = Hy = 0 (2.356) nπ pπ × cos y cos z (2.346) b c The ﬁeld distribution is shown in Figure 2.80. 2A pπ mπ mπ The resonant wavelength for TEmnp mode res- Hy =j 2 · · cos x onator can be calculated: kc c b a nπ pπ 2 × sin y cos z (2.347) λ0 = (2.357) b c m 2 n 2 p 2 mπ + + Hz = −j2A cos x a b c a So the resonant wavelength for TE101 mode is nπ pπ × cos y sin z (2.348) given by b c 2ac ωµ nπ mπ λ0 = √ (2.358) Ex = 2A 2 · cos x a 2 + c2 kc b a Equation (2.358) indicates that, if b is the shortest nπ pπ × sin y sin z (2.349) among the three sides (a, b, and c), TE101 mode b c has the largest resonant wavelength, so it is the ωµ mπ mπ lowest mode. Ey = −2A 2 · sin x kc a a It should be indicated that, for a rectangular nπ pπ cavity resonator with a = b, TE101 and TE011 are × cos y sin z (2.350) degenerate modes. They have the same resonant b c Ez =0 (2.351) frequency and the ﬁeld distribution pattern, but their ﬁelds are perpendicular to each other. Aswith will be discussed in Chapter 11, these two degen- mπ 2 nπ 2 erate modes can be used in the measurement of kc = 2 + (2.352) microwave Hall effects. a b
Microwave Theory and Techniques for Materials Characterization 99 The quality factor of the resonance can beestimated by (Ishii 1995) H E δ abcQ = λ0 4 3 m 2 n 2 m 2 n 2 p 2 2 + + + a b a b c y × m 2 p 2 m 2 n 2 2 ac + + x a c a b z n p 22 m 2 n 2 2 + bc + + Figure 2.81 The ﬁeld distribution of TM111 mode b c a b p 2 m 2 n 2 ωε mπ mπ + ab + Hy = −j2B · cos x c a b 2 kc a a (2.359) nπ pπSo the waveform factor for TE101 mode is given by × sin y cos z (2.365) b c 3 Hz = 0, (2.366) δ b (a 2 + c2 ) 2 Q0 = · (2.360) where kc can be calculated from Eq. (2.352). As an λ0 2 2b(a 3 + c3 ) + ac(a 2 + c2 ) example, the ﬁeld distribution of TM111 is shown Figure 2.81. For TMmnp mode, the resonant wavelength canTM resonant modes be calculated from Eq. (2.357), and the waveform factor can be calculated according to followingAccording to the properties of TM wave propaga- equation (Ishii 1995):tion modes and the boundary conditions, we can 3get the ﬁeld components of the TMmnp mode: m 2 n 2 2 + δ abc a b −2B pπ mπ mπ Q0 = · Ex = · · cos x λ0 4 m 2 n 2 2 kc c a a b(a + c) + a(b + 2c) a b nπ pπ (2.367) × sin sin (2.361) b c −2B pπ nπ mπ TE101 /TM110 mode Ey = 2 · · sin x kc c b a nπ pπ As discussed earlier, a rectangular cavity working × cos y sin z (2.362) at TE101 mode can be made by shorting both ends b c of a segment of rectangular waveguide working mπ Ez = 2B sin x at the TE10 mode with length λg /2. As shown in a Figure 2.82, if we change the coordinate system nπ pπ × sin y cos z (2.363) from xyz to XYZ, the TE101 mode in the xyz b c coordinate system becomes TM110 mode in the ωε nπ mπ XYZ coordinate system. For a rectangular cavity Hx = j2B 2 · sin x kc b a resonator, its TE or TM modes with zero subscripts nπ pπ can always be taken as TM or TE modes, × cos y cos z (2.364) respectively, in another coordinate system. b c
100 Microwave Electronics: Measurement and Materials Characterization TE resonant modes y(Z) On the basis the boundary conditions, we can get the ﬁeld components of the TEnip resonant modes in a cylindrical cavity with radius a and length l: 2A pπ pπ Hr = −j · J (kc r) cos nϕ cos z b(C) kc l n l (2.368) z(X) 2An pπ pπ c(A) x(Y) Hϕ = j 2 · Jn (kc r) sin nϕ cos z a(B) kc r l l (2.369)Figure 2.82 A TE101 mode rectangular cavity in the pπ Hz = −j2AJn (kc r) cos nϕ sin z (2.370)xyz coordinate system can be taken as a TM110 mode lrectangular cavity in the XYZ coordinate system 2Aωµn pπ Er = 2r Jn (kc r) sin nϕ sin z (2.371) kc l22.214.171.124 Cylindrical cavity resonator 2Aωµn pπ Eϕ = 2 Jn (kc r) cos nϕ sin z (2.372)Figure 2.83 shows a cylindrical cavity resonator kc r lmade from a segment of circular waveguide by Ez = 0 (2.373)shorting the two ends. Cylindrical cavity res-onators usually have higher quality factors than withcorresponding rectangular resonators. The struc-tural parameters of a cylindrical cavity resonator µni kc = , (n = 0, 1, 2, . . . ; i = 1, 2, 3, . . .)include the radius a and the length l. The resonant a (2.374)modes of a cylindrical cavity resonator include It should be noted that p should be a positiveTEnip and TMnip , corresponding to TEni and TMni integer. If p equals zero, all the ﬁeld componentspropagation modes in a cylindrical waveguide. The in Eqs. (2.368)–(2.373) become zero.ﬁrst two subscripts n and i come from the wave The resonant wavelength for a TEnip mode ispropagation modes TEni and TMni , representing given bythe changing cycles in ϕ-direction and r-directionrespectively, and the subscript p represents the 1changing cycles in z-direction. λ0 = (2.375) µni 2 p 2 + 2πa 2l z The waveform factor for a TEnip mode resonator can be calculated (Ishii 1995): a δ Q0 λ0 l 2 2 3/2 l pπa 1− µni + µni l f = 3 2 pπ 2 2a pπa x 2π µ2 + ni + (1 − r) 2 l µniFigure 2.83 Structure of a cylindrical cavity resonator (2.376)
Microwave Theory and Techniques for Materials Characterization 101 Two TE modes are widely used in microwaveengineering and materials property characteriza-tion: TE011 and TE111 . If l > 2.1a, TE111 mode isthe basic mode. From Eqs. (2.368)– (2.373), we Hcan get the ﬁeld components for TE111 mode byletting n = 1, j = 1, p = 1, and µ11 = 1.841. Theﬁeld distribution of TE111 mode is schematically Eshown in Figure 2.84. As TE111 mode has degener-ate modes with electric ﬁelds perpendicular to eachother, this mode is often used in the measurementof microwave Hall effect. Detailed discussions onthe measurement of microwave Hall effect can befound in Chapter 11. Figure 2.85 Field distribution of TE011 mode From Eqs. (2.368)–(2.373), we can get the ﬁeldcomponents of TE011 mode: high quality factor, and is often used in frequency ωµa 3.832 π meters. Eϕ = −2A J1 r sin z (2.377) 3.832 a l a π 3.832 π TM resonant modes Hr = j2A · J1 r cos z 3.832 l a l According to the boundary conditions, we can get (2.378) the ﬁeld components of a TMnip mode: 3.832 π Hz = −j2AJ0 r sin z (2.379) 2B pπ pπ a l Er = − · J (kc r) cos nϕ sin z kc l n l Hϕ = Er = Ez = 0 (2.380) (2.381) 2Bn pπ pπ As shown in Figure 2.85, in a TE011 mode Eϕ = 2 · Jn (kc r) sin nϕ sin zcavity resonator, all the electric ﬁelds are in ϕ- kc r l ldirection. Near the sidewall, magnetic ﬁeld is in (2.382)z-direction, and near the end wall, magnetic ﬁeld is pπin r-direction. So all the electric currents are in ϕ- Ez = 2BJn (kc r) cos nϕ cos z (2.383) ldirection, and there is no current ﬂowing between 2Bωεn pπthe sidewall and end wall. Therefore this mode Hr = −j 2 Jn (kc r) sin nϕ cos z kc r ldoes not require good electric contact between thesidewall and end wall. This mode usually has very (2.384) 2Bωµ pπ Hϕ = −j 2 Jn (kc r) cos nϕ cos z kc l (2.385) with H vni kc = (n = 0, 1, 2, 3, . . . ; i = 1, 2, 3, . . .) E a (2.386) The resonant wavelength of a cylindrical cavity in TMnip mode is given by 1 λ0 = (2.387) vni 2 p 2Figure 2.84 Field distribution of TE111 mode + 2πa 2l
102 Microwave Electronics: Measurement and Materials CharacterizationThe waveform factor a TMnip mode can be Eq. (2.375) and (2.387) can be rewritten ascalculated from (Ishii 1995) 1 λ0 = (2.392) pπa 2 uni 2 p 2 + 2 vni + δ l 2πa 2l Q0 = (2.388) λ0 2a 2π 1 + Equation (2.392) can be further modiﬁed as l 2 The TM010 mode is often used in materials cuni 2 cp 2 2a (2af0 )2 = + , (2.393)property characterization. By letting n = 0, I = 1, π 2 land p = 0, we can get the ﬁeld distributions ofTM010 mode from Eqs. (2.381)–(2.385): where f0 is the resonant frequency, c is the speed of light. So there are linear relationships v01 between (2af0 )2 and (2a/l)2 with the tangent Ez = 2BJ0 r (2.389) a (cp/2)2 and the cross value (cuni /π)2 . These lines 2Bωεa v01 2Bωεa v01 form the mode chart, as shown in Figure 2.87.Hϕ = −j J0 r =j J1 r v01 a v01 a The mode chart can be used in the following (2.390) three ways. First, if we know the diameter and resonant mode of the cavity, we can get the Er = Eϕ = Hr = Hz = 0 (2.391) relationship between the resonant frequency and The ﬁeld distribution of TM010 mode is shown the length of the cavity. Second, if we knowin Figure 2.86. As the electric ﬁeld at the central the diameter and the length of the cavity, wepart of the cavity is quite uniform, in the resonant- can obtain the resonant frequencies for differentperturbation method for permittivity measurement, working modes. Third, we can build a workingthe dielectric sample under test is often placed at rectangular from which we can check whether thethe central part of the cavity. 20 × 108 (2af )2, (MHz − cm)2 TE TE 012 112 TM212 112Mode chart 1 11 TM TE M 1T 01 1 21In the design of cavity resonators, usually we 1 TE 1 11 01 TM TMshould ensure cavities working in a single mode 15 × 108state. The mode chart is often used to check how TM110many modes may exist in a cylindrical cavity. (2afmax)2 Mode chart is drawn on the basis of Eqs. (2.375) (2afmax)2and (2.387). If we use uni to represent µni and vni , 10 × 108 TM010 5 × 108 H E 0 2 4 6 (2a/l)2 (2a/l)2 (2a/l)2 fmax fmax Figure 2.87 Resonant mode chart for a cylindrical cavity. Modiﬁed from Pozar, D. M. (1998). Microwave Engineering, 2nd ed., John Wiley & Sons, Inc., NewFigure 2.86 Field distribution of TM010 mode York
Microwave Theory and Techniques for Materials Characterization 103 Table 2.5 Changes in resonant frequency due to materials perturbation and cavity shape perturbation. In the table, ω0 is the resonant frequency before perturbation and ω is the resonant frequency after the perturbation Materials perturbation Cavity shape perturbation Dielectric or Metal magnetic materials Inward Outward Electric ﬁeld ω < ω0 ω < ω0 ω < ω0 ω > ω0 dominates (εr > 1) Magnetic ﬁeld ω > ω0 ω < ω0 ω > ω0 ω < ω0 dominates (µr > 1)cavity is in single-mode state. For a resonator with 126.96.36.199 Coupling to external circuita given working mode, diameter (2a), and workingfrequency range (fmin and fmax ), there are two Aperture coupling is often used for couplingcross points between the resonant line and two between a waveguide and a cavity. Usually, thehorizontal lines (2afmin )2 and (2afmax )2 : (2a/ l)2 waveguide works at TE10 mode, whose magnetic fminand (2a/ l)2 . The two horizontal lines (2afmin )2 ﬁeld and electric current distribution is shown in fmaxand (2afmax )2 and two vertical lines (2a/ l)2 Figure 2.88. The coupling is mainly based on the fminand (2a/ l)2 form a working rectangular. If there continuity of the magnetic ﬁeld or electric current. fmaxis no other resonant line in the working rectangular, Figure 2.89 shows two examples of the excita-the cavity works at a single mode state. tion of resonant cavities using a rectangular waveg- If different resonant modes have the same uide. In Figure 2.89(a), TM010 resonator is excitedresonant line, for example TE011 and TM111 , these by the magnetic ﬁeld, while in Figure 2.89(b),modes are degenerate. In the design of resonators, TE011 resonator is excited by the current along thespecial methods are needed to eliminate degenerate inside wall of the waveguide.modes, which will be discussed in later chapters. 2.3.5 Dielectric resonators188.8.131.52 Resonant perturbation Figure 2.90 shows several typical dielectric res-In some cases, the boundary conditions of a onators, including spherical dielectric resonator,cavity may change slightly, or the properties of cylindrical dielectric resonator, ring dielectricthe medium in the cavity may change slightly. resonator, and rectangular dielectric resonator.In materials property characterization, we mayintroduce a small sample into a resonant cavity. jSuch changes may result in small changes in the Hresonant frequency and quality factor of the cavity. The changes in resonant frequency and qualityfactor can be approximately obtained on the basisof the resonant-perturbation theory. Resonant-perturbation theory focuses on the change ofthe energy due to the perturbation, not the bchange of the electromagnetic ﬁeld distribution.Some useful conclusions about the change of athe resonant frequency due to different typesof resonant perturbations are listed in Table 2.5.Further discussions on resonant perturbation can Figure 2.88 Distributions of magnetic ﬁeld and elec-be found in Chapter 6. tric current of a rectangular waveguide at TE10 mode
104 Microwave Electronics: Measurement and Materials Characterization TE 011 circular TM 010 circular H cavity cavity H EH a E j E TE10 rectangular waveguide TE10 rectangular waveguide (a) (b)Figure 2.89 Aperture coupling using rectangular waveguide. (a) Excitation of TM010 mode and (b) excitation ofTE011 mode In a closed metallic cavity, all the electro- magnetic ﬁelds are conﬁned within the space enclosed by the metallic cavity; however, there is ﬁeld fringing or leakage from the boundaries of a dielectric resonator. The resonators shown in Figure 2.90 are isolated dielectric resonators without any support. Isolated dielectric resonators are convenient for theoretical analysis. However, (a) (b) in actual cases, dielectric resonators should be supported or shielded. Owing to ﬁeld fringing and leakage, the support or shield affects the resonant properties of the dielectric resonator. Usually, a dielectric resonator with support or shield is called a shielded dielectric resonator. The dielectric permittivity εr of the dielectric material is an important parameter in determining the resonant properties of a dielectric resonator. For a given resonant frequency, the higher the dielectric (c) (d) constant, the smaller the dielectric resonator, and the microwave energy is more concentrated withinFigure 2.90 Typical conﬁgurations of dielectric res-onators. (a) Sphere, (b) cylinder, (c) ring, and (d) rec- the dielectric resonator and the effects of the exter-tangular nal circuit to the dielectric resonator become less. For microwave dielectric resonators, the dielectricDielectric resonators also fall into transmission constant is usually in the range of 10–100. If thetype, such as cylinder resonator and rectangu- dielectric constant is very large, the dielectric res-lar resonator, and non-transmission type, such as onator becomes very small, and the fabrication ofsphere resonator. In the following discussion, we dielectric resonators becomes very difﬁcult. Thereconcentrate on transmission-type dielectric res- are a type of resonators working at whispering-onators, especially the cylindrical dielectric res- gallery modes, which have relatively large sizes.onators. A transmission-type dielectric resonator is Such resonators are often used in the characteriza-made from its corresponding dielectric waveguide, tion of high-dielectric-constant materials.and a cylindrical dielectric resonator is actually a In the following, we start with the resonantsegment of cylindrical dielectric waveguide with properties of isolated dielectric resonators, fol-two open ends. lowed by the resonant properties of shielded
Microwave Theory and Techniques for Materials Characterization 105 7000 M1, f1 6000 L M2, f2 f (MHz) f3 ( Next higher observed resonance) D= 5000 0.325″ f2 er = 95 f1 4000 3000 0.100 0.200 0.300 0.400 0.500 D = L = 0.325 L (in.)Figure 2.91 First three resonant frequencies versus lengths of a dielectric cylinder of constant diameter (Cohn1968). Source: Cohn, S. B. (1968). “Microwave bandpass ﬁlters containing high-Q dielectric resonators”, IEEETransactions on Microwave Theory and Techniques, 16 (4), 218–227. 2003 IEEEdielectric resonators. We then introduce the tech- the waveguide mode and δ is the non-integer rationiques for realizing the couplings between dielec- 2L/λg < 1. As shown in Figure 2.92, for a distanttric resonators and external circuits, and ﬁnally the observer, this mode appears as a magnetic dipole,resonant properties of whispering-gallery dielectric so this mode is also called magnetic dipole mode.resonators will be discussed. For a given dielectric resonator, its accurate resonant frequency of TE01δ mode can be calculated only by complicated numerical procedures. For an184.108.40.206 Isolated dielectric resonators appropriate estimation of the resonant frequencyCylindrical dielectric resonators are widely used in of the isolated dielectric resonator, the followingmicrowave electronics and materials property char- simple formula can be used (Kajfez and Guillonacterizations. Exact analyses of cylindrical dielec- 1986):tric resonators are quite complicated, and approx-imation techniques and experimental methods are 34 aoften used. Figure 2.91 shows some experimental f = + 3.45 (GHz), (2.394) a εr Lresults for the resonant frequencies of cylindricaldielectric resonators with same diameter but withdifferent lengths. The curve labelled f1 represents where a and L are the radius and length of thethe TE mode with zero axial electric ﬁeld, and the dielectric resonator, εr is the dielectric constant ofcurve labelled f2 represents the TM mode with zero the material. In Eq. (2.394), the unit for frequencyaxial magnetic ﬁeld. is GHz, and the unit for a and L is mm. The In most cases, the length L of the dielectric accuracy for Eq. (2.394) is about 2 % in the rangeresonator is chosen to be less than the diameter of 0.5 < (a/L) < 2 and 30 < εr < 50.(D = 2a) of the dielectric resonator. In microwave For the case with L greater than D, theelectronics and materials characterization, the prime mode has an equivalent magnetic dipoleprime resonance of interest is usually the lowest moment transverse to the axis, as shown inorder mode. From Figure 2.91, the lowest mode for Figure 2.93 (Cohn 1968). Resonant properties forcylinder dielectric resonator with L < D is TE01δ cylindrical resonators with L = D can be foundmode, where the ﬁrst two subscript integers denote in (Tsuji et al. 1984).
106 Microwave Electronics: Measurement and Materials Characterization z E L H D (a) (b) (c)Figure 2.92 Prime mode (TE01δ ) of a cylindrical dielectric resonator with L < D. (a) Dimensions of the resonator,(b) top view of the ﬁeld distribution, and (c) three-dimensional view of magnetic ﬁeld H H Parallel-plate dielectric resonator D As shown in Figure 2.94, a parallel-plate dielec- E E tric resonator consists of an isotropic cylindrical dielectric between two parallel conducting plates L perpendicular to the axis of the dielectric cylinder. (a) (b) It is a symmetrical structure, and in theoretical analysis of its TE01δ mode, the following assump- A–A′ A tions are made. Firstly, the dielectric cylinder H H can be treated as a lossless cylindrical dielectric E waveguide with length L excited in TE01 mode. H Secondly, the z-dependence of the z-component A′ E magnetic ﬁeld (Hz ) for |z| ≥ L/2 can be described by sinh α(L1 + (L/2) ± z). Thirdly, the cross- sectional ﬁeld distribution of Hz for |z| ≥ (L/2) is (c) (d) the same as for the TE01 mode in cylindrical dielec- tric waveguide at the “cutoff” frequency. We fol-Figure 2.93 Prime mode of a cylindrical dielectric res- low the method proposed by Pospieszalski (1979).onator with L > D. (a) Dimensions of the resonator, We deﬁne the normalized resonant frequency F0 :(b) side view of the ﬁeld distributions, (c) top viewof the ﬁeld distributions, and (d) A–A cross view πD √of the ﬁeld distributions. Source: Cohn, S. B. (1968). F0 = εr , (2.395)“Microwave bandpass ﬁlters containing high-Q dielec- λ0tric resonators”, IEEE Transactions on Microwave The- where λ0 is the free-space wavelength correspond-ory and Techniques, 16 (4), 218–227 2003 IEEE ing to the resonant frequency f0 , D is the diame- ter, and εr is the relative dielectric constant. From220.127.116.11 Shielded dielectric resonators the ﬁrst assumption, F0 satisﬁes the following equations:In actual applications, dielectric resonators are εr − 1 F0 = (u2 + w2 ) 2 (2.396)often supported or enclosed by metal shields, and εrdifferent ways of shielding may result in different J1 (u) K1 (w)resonant properties. In this part, we discuss sev- =− , (2.397) uJ0 (u) wK0 (w)eral typical shielding methods, including parallelplates, asymmetrical parallel plates, cut-off waveg- where Jn is the Bessel function of the ﬁrst kinduide, and closed shields. of nth order and Kn is the modiﬁed Hankel
Microwave Theory and Techniques for Materials Characterization 107function of nth order. The parameters w and u e1 Dielectric rare deﬁned by e0 = er resonator e0 2 z w 2 F0 1 (a) = β2 − (2.398) D e1 a a a εr 2 L1 L L1 Conducting u 2 F0 Conducting plate = −β , 2 (2.399) plate a a Hzwhere β is the propagation constant of the mode incylindrical dielectric waveguide and a is the radiusof dielectric cylinder. The second assumption indicates that α and βsatisfy the following transcendental equation: (b) z L β L cot β = tanh αL1 . (2.400) 2 2 αAccording to the third assumption, α is given by Figure 2.94 Parallel-plate dielectric resonator at TE01δ mode. (a) Conﬁguration of the resonator and (b) distri- 2 ρ01 2 1 F0 bution of magnetic ﬁeld Hz along z-axis (Pospieszalski α2 = − (2.401) 1979). Source: Pospieszalski, M. W. (1979). “Cylin- a εr a drical dielectric resonators and their applications inwhere ρ01 is the ﬁrst root of J0 . From Eqs. (2.398)– TEM line microwave circuits”, IEEE Transactions on(2.401), we can get Microwave Theory and Techniques, 27 (3), 233–238 2003 IEEE L tan F0 − u2 2 D The present structure can be regarded as modiﬁed 1 2 from the one shown in Figure 2.94 by inserting a ρ01 − 2 F dielectric layer in one side of the cylindrical dielec- εr 0 = tric resonator, and the thickness values at the two L1 1 2 F0 − u2 tanh 2 2 · F ρ01 − 2 sides are not equal. This structure is often used in D εr 0 microwave integrated circuits, and is also called (2.402) microstrip dielectric resonator. The normalized resonant frequency F0 can be On the basis of similar assumptions and with thecomputed from the set of Eqs. (2.396), (2.397) and similar analysis method, we can get (Pospieszalski(2.402) as a function of (D/L), (L1 /L), and εr . 1979)Figure 2.95 shows the TE01δ mode chart, which isoften used in the design of cylindrical dielectric L 2 F0 − u2 2resonators. D 1 2 As will be discussed later, in materials property ρ01 − 2 F εr 0characterizations, usually the conducting plates = tan−1directly shorts the dielectric resonator (L1 = 0). So F0 − u2 tanh 2 2 L1 · ρ01 − 2 1 2 Fthe resonator is in TE011 mode, and it corresponds D εr 0to the line (S1 = 0) in Figure 2.95. 1 2 ρ01 − 2 F εr 0Asymmetrical parallel-plate dielectric resonator + , L2 εp 2 2 F0 − u2 tanh 2 · 2 ρ01 − F0Figure 2.96 shows an asymmetrical dielectric res- D εronator structure with parallel conducting plates. (2.403)
108 Microwave Electronics: Measurement and Materials Characterization (f0·D)2.er e1 [GHz cm]2 e0 = er L1 L L1 S1 = 0.0 4·103 L εr > 500 e0 e1 S1 = 1 L εr = 10 S1 = 0.05 D Conducting S1 = 0.1 3·103 plates Exact S1 = 0.2 Solution TE011 mode S1 = 0.4 2·103 S1 = 0.7 S1 8 1·103 D 2 L 0.5 1.0 2 3 4 5 6 7 8 9 10 11 12 13 14 15Figure 2.95 The TE01δ mode chart for cylindrical dielectric resonator between parallel plates (Pospieszalski1979). Source: Pospieszalski, M. W. (1979). “Cylindrical dielectric resonators and their applications in TEM linemicrowave circuits”, IEEE Transactions on Microwave Theory and Techniques, 27 (3), 233–238. 2003 IEEE where εp is the relative dielectric constant of Substrate e1 the dielectric layer (substrate). From the set of e0 e0 = er Eqs. (2.396), (2.397), and (2.403), F0 can be found e2 as a function of (D/L), (L1 /L), (L2 /L), εr , and εp . e0 = ep There are approximate and straightforward a (a) z equations for the design of this kind of dielec- D e1 tric resonators (Ishii 1995). To design a cylindrical Metallic plate Metallic plate e2 dielectric resonator with resonant frequency f0 , we L Dielectric should determine its diameter D and length L. The 2 resonator ﬁrst step is to select D satisfying L2 L L1 5.4 5.4 Hz √ ≤D≤ √ , (2.404) k0 ε r k0 ε p (b) where k0 = (ε0 µ0 )1/2 is the free-space propagation constant. The second step is to determine the length L according to Eq. (2.405): x ∆L 1 α1 L= tan−1 coth α1 L1 β βFigure 2.96 An asymmetrical parallel-plate dielectricresonator. (a) Conﬁguration of the dielectric resonator α2 + tan−1 coth α2 L2 (2.405)and (b) distribution of magnetic ﬁeld Hz along z-axis βfor the TE01δ mode (Pospieszalski 1979). Source:Pospieszalski, M. W. (1979). “Cylindrical dielectric withresonators and their applications in TEM line microwavecircuits”, IEEE Transactions on Microwave Theory andTechniques, 27 (3), 233–238. 2003 IEEE α1 = h 2 − k0 2 (2.406)
Microwave Theory and Techniques for Materials Characterization 109 1.0 0.9 d = 0.25 R 0.8 h = 0.1 h =∞ R d =0 0.7 R d =∞ R 0 .6 h =0 R R 2pR l0 h = 0.2 d = 0.50 0.5 R R er = 37.0 er = 37.0 eg = 2.1 0.4 eg = 2.1 h = 0.2 d=∞ R R 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.70.8 1.0 t 2RFigure 2.97 Generalized normalized design curves for microstrip dielectric resonators (Chang 1989). Source:Chang, K. (1989). Handbook of Microwave and Optical Components, Vol. 1, John Wiley, New Yorkα2 = h 2 − k0 ε p 2 (2.407) Magnetic zβ= k0 ε r − h 2 2 (2.408) wall er = 1 2 y0h= · 2.405 + D 2.405(1 + 2.43/y0 + 0.291y0 ) (2.409) r O q L 2 D E0y0 = k0 (εr − 1) − 2.4052 (2.410) 2 er > 1 Figure 2.97 shows a generalized normalized D = 2a0design curve for asymmetrical parallel-plate dielec-tric resonators. In the ﬁgure, λ0 is the wavelength er = 1in free space, t is the height of the dielectric res-onator, R is the radius of the dielectric resonator,d is the distance of air gap between the dielec- Figure 2.98 Dielectric cylinder in magnetic-wall waveguide boundary. Source: Cohn, S. B. (1968).tric resonator and metal plate, h is the thickness “Microwave bandpass ﬁlters containing high-Q dielec-of the substrate, εr is the dielectric constant of tric resonators”, IEEE Transactions on Microwave The-the resonator, and εg is the dielectric constant of ory and Techniques, 16 (4), 218–227. 2003 IEEEthe substrate. Figure 2.97 shows the case whenεr = 37.0 and εg = 2.1. There are two sets of Dielectric resonator in cutoff magnetic-wallcurves. One set of curves are for the condition of waveguide(h/R = 0.2), and they are indicated by d/R = 0,0.25, 0.50, and ∞ respectively. Another set of Figure 2.98 shows a dielectric cylinder containedcurves are for the condition of (d/R = ∞), and in a contiguous magnetic-wall waveguide. In thethey are indicated by h/R = 0, 0.1, 0.2, and ∞, dielectric region, the waveguide is above its cutoffrespectively. frequency, while in the air regions, the waveguide
110 Microwave Electronics: Measurement and Materials Characterizationis below the cut-off frequency. At resonance,a standing wave exists in the dielectric region, qwhile exponentially attenuating waves exist in Conductive rair regions. The equivalent circuit is a piece of walltransmission line with length L, terminated at bothends by reactance equal to the pure imaginarycharacteristic impedance of the cut-off air-ﬁlled Dielectricwaveguide. According to Cohn’s mode (Cohn 1968), we canget the ﬁeld distributions for TE01δ mode: J1 (kc r) L R1 R2Eφ = 2E0 cos βd z |z| ≤ (2.411) Ground J1 (p01 ) 2 conductor J1 (kc r) βd L fEφ = 2E0 cos J1 (p01 ) 2 Figure 2.99 Semispherical dielectric resonator in a −αa (|z|−L/2) L semispherical metal cavity ×e |z| ≥ (2.412) 2 βd J1 (kc r) L p01 cHr = j 2E0 sin βd z |z| ≤ (2.413) f2 = · (2.420) ωµ J1 (p01 ) 2 a 2π αa J1 (kc r) Because of the magnetic-wall waveguide, theHr = ±j 2E0 cos βd z resonant frequency of this conﬁguration is about ωµ J1 (p01 ) 10 % lower than those of an isolated dielectric L resonator and a parallel-plate dielectric resonator × e−αa (|z|−L/2) |z| ≥ (2.414) 2 with same dimensions and dielectric constant.where p01 2.405 Dielectric resonator in closed metal shieldskc = = (2.415) a a Dielectric resonators can be enclosed in closed metal shields, and the study of the resonant (2πf )2 εr p01 2 properties of such resonant structures is importantβd = ε r k 0 − kc = 2 2 − (2.416) c2 a for the design of dielectric resonators and materials 2 property characterization. p01 2 2πf A spherical dielectric resonator enclosed in aαa = k c − k0 = 2 2 − (2.417) a c metal sphere is often used in studying the effects of a metal shield to the resonant properties of aThe upper sign in Eq. (2.414) applies at z = L/2 resonator, because the space between the dielectricand the lower at z = −L/2. The matching tangent resonator and the shield can be described by oneﬁelds at z = ±L/2 lead to parameter: radius. Imai and Yamamoto studied βd L αa the resonant properties a semi-spherical dielectric tan = (2.418) resonator in a semi-spherical conductive shield, as 2 βd shown in Figure 2.99 (Imai and Yamamoto 1984).This equation can be solved numerically for the The properties of a semi-spherical resonator atresonant frequency. As αa and βd should be real TE011 mode are similar with those for TE021numbers, the possible frequency range is from f1 spherical resonator. One advantage of using ato f2 , where semi-spherical dielectric resonator is that the p01 c ground conductor is also a support to the dielectric f1 = · √ (2.419) a 2π εr resonator, and no other support is needed.
Microwave Theory and Techniques for Materials Characterization 111 In Figure 2.99, the radius of the dielectric res- According to the incremental frequency rule, theonator is R1 , and that of the metal sphere is R2 . quality factor Qc related to conductor losses of theWhen R2 gradually increases while R1 keeps con- shield is large when the slope df/dR2 (frequencystant, the change of the resonant frequency of the vs. the size of the cavity) is small (Kajfez andresonant structure is shown in Figure 2.100. In Guillon 1986). Therefore, in the region b, Qc isregion a, R2 is slightly larger than R1 , and the large and so the effect of the conductor loss to theresonant frequency changes rapidly as a function overall quality factor Q of the resonant structureof R2 . When R2 is approaching R1 , the resonant can be neglected. Therefore in this region Q isfrequency approaches that of a cavity fully ﬁlled mainly determined by the losses in the dielectric.with a dielectric material. In region b, the resonant In materials property characterization, usuallyfrequency is almost independent of R2 . In region cylindrical dielectric resonators are used, andc, the resonant frequency again changes as a func- the cylindrical dielectric resonators are oftention of R2 . When R2 is much larger than R1 , the enclosed in metallic cylinders. The conclusionsresonant structure can be taken as a semi-spherical for spherical resonators discussed above candielectric resonator mounted on a ground plane, be extended to cylindrical dielectric resonators.which is discussed in (Collin 1992). Figure 2.101 shows two typical conﬁgurations. In Resonant frequency a b c R1 Radius of the metal shieldFigure 2.100 Resonant frequency of the semispherical dielectric resonator. (a) Transient mode, (b) dielectricresonator mode, and (c) hollow cavity mode b b a L h a h L (a) (b)Figure 2.101 Cylindrical resonators in cylindrical cavities. (a) The dielectric resonator is at the center of thecavity. (b) The dielectric resonator contacts the end wall of the cavity
112 Microwave Electronics: Measurement and Materials CharacterizationFigure 2.101(a), the dielectric resonator is at thecenter of the cavity, and the contribution of theconductor losses of the metal shield to the overallquality factor of the resonant structure is small. Electric dipole Magnetic dipoleThis conﬁguration is often used in characteriz- for TM modes for TE modesing the dielectric permittivity of dielectric cylin-ders. In Figure 2.101(b), one end of the dielec-tric cylinder directly contacts the end-wall of theresonant cavity. In this conﬁguration, the surface Electric dipole Magnetic dipoleresistance of the end-wall contacting the dielec- for TM modes for TE modestric cylinder affects the overall quality factor ofthe resonant structure. This conﬁguration is often Figure 2.102 Electric and magnetic dipole probes forused in characterizing the surface resistance of couplings of dielectric TE and TM resonant modesconductors. Kobayashi et al. made discussions on the qual-ity factors of the circularly symmetric TE0 modesfor dielectric cylinders placed between two parallelconductor plates and in a closed conductorshield (Kobayashi et al. 1985). The techniquesallowing separate estimation of the quality factorsdue to radiation, conductor, and dielectric lossesare proposed. The conclusions obtained are help-ful for realizing high-quality dielectric resonatorsand for the measurement of dielectric properties of Figure 2.103 Typical WGM resonance in a dielectriclow-loss materials. resonator. The dark regions indicate areas of high energy density18.104.22.168 Coupling to external circuit resonant structures, typically characterized by highGenerally speaking, all kinds of coupling methods values of the azimuthal index n. The name iscan be used for dielectric resonators. The coupling applied due to the similarity with Lord Rayleigh’sprobes often used for dielectric resonators include observations of whispers that would travel aroundelectric dipole, magnetic dipole, microstrip line, the inside wall of St. Catherine’s cathedral inand waveguide. It should be noted that, for a England in early 20th century. As shown incoupling probe, different coupling positions and Figure 2.103, WGM resonators exhibit a multiplic-different orientations result in different resonant ity of sharp resonances and they are natural candi-modes. As shown in Figure 2.102, in materials dates for the realization of an ultra wideband res-property characterization, electric dipole probe and onant structure. Since WGM dielectric resonatorsmagnetic dipole probe are often used in couplings are running essentially in the azimuthal direction,to dielectric resonators. Using these kinds of they offer the possibility of new microwave devices,probes, the coupling coefﬁcients can be easily such as directional ﬁlters and power combiners withadjusted. Detailed discussions on the coupling very low reﬂection coefﬁcient. In materials prop-methods for dielectric resonators can be found erty characterization, WGM dielectric resonatorsin (Kajfez and Guillon 1986). are often used in characterizing extremely low- loss dielectrics, high-dielectric-constant materials,22.214.171.124 Whispering-gallery dielectric resonator anisotropic dielectrics, and ferrites. In a WGM dielectric resonator, due to theWhispering-gallery mode (WGM) dielectric res- total internal reﬂections, the electromagnetic waveonators form a class of rotationally invariant bounces around inside the dielectric/air interface.
Microwave Theory and Techniques for Materials Characterization 113For given boundary conditions, resonance occurs side of the cylindrical boundary of the rod (Crosonly at certain wavelengths (λ) given by and Guillon 1990). The waves move essentially √ in the plane of the circular cross section. As nλ = πd εr µr , (2.421) shown in Figure 2.104(a), most of the energy iswhere the integer n is the azimuthal index, d conﬁned between the cylindrical boundary (r = a)is the diameter of the crystal, εr is the relative and an inner modal caustic (r = ac ). Near thispermittivity, and µr is the relative permeability. region, the electromagnetic ﬁelds are evanescent.However, as will be indicated later, this relation The modal energy conﬁnement can be explainedis only true at high frequency where n is large. from a ray optics point of view. As shown inAt lower frequencies, the electrical path length is Figure 2.104(b), a ray is totally reﬂected at theextended by reﬂections from an internal caustic dielectric–air interface; it is then tangent to ansurface (Cros and Guillon 1990). inner circle called the caustic. Thus the ray moves WGM in a dielectric rod can be described as merely in a small region within the rod near thecomprising waves running against the concave rod boundaries. Dielectric rod Caustic Evanescent field (a) (b)Figure 2.104 WGM in a dielectric rod. (a) Electric ﬁeld variation as a function of the radius, (b) representationby ray optic techniques of WGM propagation in cylindrical rod (Cros and Guillon 1990). Source: Cros, D. andGuillon, P. (1990). “Whispering gallery dielectric resonator modes for W-band devices”, IEEE Transactions onMicrowave Theory and Techniques, 38 (11), 1667–1674. 2003 IEEE 2a 2a >> h 2a h h 2a1 (a) (b)Figure 2.105 Two types of WGM dielectric resonators. (a) Cylindrical dielectric resonator and (b) planardielectric resonator. Source: Cros, D. and Guillon, P. (1990). “Whispering gallery dielectric resonator modes forW-band devices”, IEEE Transactions on Microwave Theory and Techniques, 38 (11), 1667–1674. 2003 IEEE
114 Microwave Electronics: Measurement and Materials Characterization As shown in Figure 2.105, WGM can be the propagation constant along the Z-axis is veryexcited in cylindrical dielectric resonators and small and the unwanted modes leak out axiallyplanar dielectric resonators. In the cylindrical and can be absorbed without perturbation. Last,dielectric resonator shown in Figure 2.105(a), they offer a high level of integration. They can beits WGM is characterized by a second energy easily integrated into planar circuits.conﬁnement (axial) ensured by the enlargement of As shown in Figure 2.106, the WGM of dielec-the resonator radius in the central zone. In this tric resonators can be classiﬁed as WGEnml andarea, the modes propagate axially with a small WGHnml . In a WGEnml mode, the electric ﬁeld ispropagation constant and they decay exponentially essentially transversal, while in a WGHnml mode,away from the discontinuity of the resonator the electric ﬁeld is essentially axial. The integer nradius. In Figure 2.105(b), WGM is excited in a denotes the azimuthal variations, m the radial vari-thin dielectric disk with the diameter 2a much ations, and l the axial ones. As an example, thelarger than its thickness h. Such resonators have ﬁeld distributions of WGE800 and WGE810 modesbeen extensively investigated because they are are shown in Figure 2.107.compatible with millimeter-wavelength integratedcircuits. WGM dielectric resonators are very interesting Hz Hθ Ezfor a number of reasons (Cros and Guillon 1990). EθFirst, their dimensions are relatively large, even Er Hrin the millimeter-wavelength band. Second, thequality factors are very high: the unloaded qualityfactor of a WGM dielectric resonator is limitedonly by the value of the loss tangent of thematerial used to realize the dielectric resonator, (a) (b)and the radiation losses are negligible. This is Figure 2.106 Electromagnetic ﬁelds of WGE andan important feature of WGM compared with the WGH modes in planar dielectric resonator. (a) WGEconventional TE or TM modes, whose unloaded mode and (b) WGH mode. Source: Cros, D. andquality factor depends not only on the material Guillon, P. (1990). “Whispering gallery dielectric res-loss tangent but also on the metallic shields in onator modes for W-band devices” IEEE Transac-which they are enclosed. Third, WGM resonators tions on Microwave Theory and Techniques, 38 (11),have good suppression of spurious modes because 1667–1674. 2003 IEEE (a) (b)Figure 2.107 Field distributions of WGM. (a) Electric ﬁeld amplitude for WGE800 mode and (b) electric ﬁelddistribution of WGE810 mode (Niman 1992). Source: Niman, M. J. (1992). “Comments on ‘Whispering gallerydielectric resonator modes for W-band devices’ ”, IEEE Transactions on Microwave Theory and Techniques, 40 (5),1035–1036. 2003 IEEE
Microwave Theory and Techniques for Materials Characterization 115 Dielectric 2.3.6 Open resonators resonator 126.96.36.199 Concept of open resonator H E As discussed earlier, free space can also be taken as a type of transmission line. We can also make res- Magnetic Electric dipole dipole onators from free-space transmission line by termi- nating it with two parallel metal plates, forming aFigure 2.108 Whispering-gallery modes excited by Fabry–Perot resonator, as shown in Figure 2.111.electric and magnetic dipole. Source: Cros, D. and Guil- In millimeter-wave and sub millimeter-wave fre-lon, P. (1990). “Whispering gallery dielectric resonator quency range, the conductor loss is a main partmodes for W-band devices”, IEEE Transactions on of energy dissipation of a resonant cavity. Com-Microwave Theory and Techniques, 38 (11), 1667–1674. pared to closed resonators, open resonators may 2003 IEEE have higher quality factors because the conductor loss is decreased as some of the conductor walls WGM dielectric resonators can be excited in are moved away.various ways. As shown in Figure 2.108, in the We assume the parallel plates in Figure 2.111 tolow microwave frequency range we use an electric be inﬁnite in extent, and a TEM mode standingdipole or a magnetic dipole. Using this type wave is built between the plates. From theof excitation we can obtain stationary WGM. boundary conditions, we can get the resonantIn the high microwave frequency range we can frequency f0 : cnuse dielectric image waveguides or microstrip f0 = , (2.422)transmission lines as shown in Figure 2.109. Using 2dthis method, we can build travelling WGM. where d is the distance between the two parallel Finally, it should be indicated that WGM plates, n is the mode number (n = 1, 2, 3, . . .), ccould also be built in other types of reso- is the speed of light. Equation (2.422) indicatesnant structures. Figure 2.110 shows a travelling that the resonant frequency can be adjusted bymicrostrip resonator (Harvey 1963). If the length changing the distance between the two plates. Inof the microstrip ring is a multiple of the wave- an actual Fabry–Perot resonator, usually one of thelength, absorption can be observed from the reﬂectors is movable so that the resonant frequencytransmission line, indicating the occurrence of a can be changed continuously. This is an attractiveresonance. Correspondingly, we may also take the advantage for materials property research.resonance in a microstrip ring resonator discussed The quality factor Q0 of the resonator can bein Section 188.8.131.52 as a stationary microstrip WGM. estimated by H field Microstrip E field line w RD Dielectric εx line RD hε εx εx εx Ground plane (a) (b)Figure 2.109 Whispering gallery modes excited by (a) dielectric image waveguide and (b) microstrip line. Source:Cros, D. and Guillon, P. (1990). “Whispering gallery dielectric resonator modes for W-band devices”, IEEETransactions on Microwave Theory and Techniques, 38 (11), 1667–1674. 2003 IEEE
116 Microwave Electronics: Measurement and Materials Characterization Microstrip loop Figure 2.112 Leakage of Fabry–Perot resonator due Microstrip to limited size of the plates. Reprinted from Musil, transmission line J. and Zacek, F. (1986). Microwave Measurements ofFigure 2.110 A traveling-wave microstrip resonator Complex Permittivity by Free Space Methods and Their Applications, Permission from Elsevier, Amsterdam r1 r1 z r2 d dFigure 2.111 An ideal Fabry–Perot resonator (a) (b) Figure 2.113 Resonator geometries with planar and πnη0 spherical reﬂectors (Musil and Zacek 1986). Reprinted Q0 = , (2.423) 4Rs from Musil, J. and Zacek, F. (1986). Microwave Measurements of Complex Permittivity by Free Spacewhere n is the mode number, η0 is the intrinsic Methods and Their Applications, Permission fromimpedance of the free space, and Rs is the Elsevier, Amsterdamsurface resistance of the reﬂector. Equation (2.423)indicates that the quality factor increases with the inside the resonator has a well-known Gaussianincrease of the mode number. As n is often several distribution. This fact can be used to simulate free-thousand or more, very high quality factors could space ﬁeld conditions, quite similar with the casebe achieved. of focused beams described in Section 2.2.5. An open resonator consisting of two spherical184.108.40.206 Stability requirements reﬂectors with radii of r1 and r2 can support a stable mode if the following condition isEquation (2.423) gives the quality factor of an met (Pozar 1998):ideal case: the two plates are inﬁnite and they d dare strictly parallel. However, in an actual case 0≤ 1− 1− ≤1 (2.424)as shown in Figure 2.112, the sizes of the plates r1 r2are limited, and the two plates may be not On the basis of Eq. (2.424), a stability diagramstrictly parallel, microwave energy will be radiated. can be drawn, as shown in Figure 2.114. In theTherefore, the actual quality factor is less than stability diagram, the stable regions are shadowed.what Eq. (2.423) gives. There are three special conditions, representing As shown in Figure 2.113, signiﬁcant improve- parallel-plane resonator, concentric resonator, andments can be made if one or both of the reﬂect- symmetrical confocal resonator respectively.ing plane plates is replaced by a concave reﬂec- Parallel plate can be taken as a spherical reﬂectortor (Musil and Zacek 1986). In these cases, the with inﬁnite radius. The point corresponding to aﬁeld is focused in a smaller volume and the parallel plate resonator (d/r1 = d/r2 = 0) is at therequirements for arranging the two reﬂectors are boundary between the stable region and unstablenot very high. Furthermore, if one or both reﬂec- region. If there are any irregularities, the systemtors have a concave spherical surface, the ﬁeld becomes unstable.
Microwave Theory and Techniques for Materials Characterization 117 d/r1 ﬁeld intensity with respect to the ﬁeld intensity on the axis of the resonator. 220.127.116.11 Coupling to external circuit Concentric In a microwave Fabry–Perot resonator, coupling may be achieved by quasi-optical methods or typi- 1 cal microwave methods. Figure 2.116 diagrammat- Symmetrical confocal d/r2 ically shows two common geometries for coupling 1 in a quasi-optical manner: through one or both mir- Parallel rors or normal to the resonator axis (Clarke and plane Rosenberg 1982). In the through-mirror method, mirrors of the open resonator are perforated metal plates, the RF signal is injected by means of a colli- mating lens fed by a small horn placed at its focus,Figure 2.114 Stability diagram for open resonators and a similar lens-and-horn arrangement is used to collect the small signals passing through the sys- tem. The method shown in Figure 2.116(b) utilizes For a confocal resonator (r1 = r2 = d), the cor- a dielectric beam splitter set at 45◦ to the res-responding point is at the boundary between the onator axis, and usually the beam splitter is madestable region and unstable region. For a concen- of polyethylene. In this system, the quality-factortric resonator (r1 = r2 = (d/2)), the correspond- of the resonator can be continuously adjusted bying point is at the stability boundary. So these rotating the incident polarization. The quality fac-two kinds of resonators are also not very sta- tor reaches its maximum value when the electricble. To increase the stability, confocal and con- vector lies in the plane of incidence and reachescentric resonators are often modiﬁed into near- its minimum value when the electric vector is per-confocal and near-concentric geometries, as shown pendicular to the plane of incidence. Meanwhile,in Figure 2.115. the quality factor of the resonator is also affected When open resonators are used in materi- by the thickness of the beam splitter.als property characterization, we should con- Fabry–Perot resonators can also be coupledsider the beam-width in the resonator, as the using typical microwave methods, such as the cou-beam-width determines the minimum dimensions pling apertures in the mirror reﬂectors. Figure 2.117of samples that can be measured. Similar to shows an open resonator coupled to waveguidesfocused microwave beam, in the determination of through apertures. The details of the coupling aper-beam width, we should deﬁne a certain level of ture are shown in Figure 2.118. It comprises a h(z) 0 h(z) 0 z z r0 r0 (a) (b)Figure 2.115 Modiﬁcation of confocal and concentric resonators. (a) Near-confocal resonator and (b) near-con-centric resonator (Musil and Zacek 1986). Reprinted from Musil, J. and Zacek, F. (1986). Microwave Measurementsof Complex Permittivity by Free Space Methods and Their Applications, Permission from Elsevier, Amsterdam
118 Microwave Electronics: Measurement and Materials Characterization (a) (b)Figure 2.116 Two types of quasi-optical coupling methods. (a) Through-mirror method and (b) normal-to-axismethod. Modiﬁed from Clarke, R. N. and Rosenberg, C. B. (1982). “Fabry–Perot and open resonators at microwaveand millimeter wave frequencies, 2–300 GHz”, Journal of Physics E: Scientiﬁc Instruments, 15 (1), 9–24 Mirror Mirror Output Input Concave ConcaveFigure 2.117 An open resonator coupled to waveguides through apertures standard TE10 mode feed-waveguide butted into a rectangular recess in the back surface of one reﬂec- tor. Loops of magnetic ﬁeld emanate from the aper- ture which thus acts as a magnetic dipole radiating into a half-space (Clarke and Rosenberg 1982). The coupling coefﬁcient is mainly determined by the diameter of the coupling hole and the thickness of the wall. Measurement of dielectric properties is a nat- ural application for open resonators since easily measured parameters of an open resonator, includ- ing the resonant frequency f , quality factor Q,Figure 2.118 Structure of a coupling aperture (Clarke and resonator length d, are simply related to theand Rosenberg 1982). Source: Clarke, R. N. andRosenberg, C. B. (1982). “Fabry–Perot and open res- dielectric constant and loss tangent of the mate-onators at microwave and millimeter wave frequencies, rial included in the resonator. Actually, materials2–300 GHz”, Journal of Physics E: Scientiﬁc Instru- property characterization was one of the ﬁrst appli-ments, 15 (1), 9–24, by permission of The Institute of cations for microwave open resonators, and open-Physics resonator methods are presently among the most
Microwave Theory and Techniques for Materials Characterization 119sensitive for low-loss dielectric measurements at i2 a2 b2millimeter-wave frequencies (Cullen 1983). More T2discussions on the application of open resonators i1 T1 v2in materials property characterization can be found a1in Chapter 5. b1 v1 Tn in vn an2.4 MICROWAVE NETWORK bnAs mentioned earlier, ﬁeld method and line method Figure 2.119 Two deﬁnitions of a networkare two important methods in microwave the-ory and engineering. In the ﬁeld method, thedistributions of electric ﬁeld and magnetic ﬁeld used for different sets of physical parameters. Forare analyzed. In the line method, the microwave example, impedance and admittance matrixes areproperties of the transmission lines or reso- used to describe the relationship between voltagenant structures are represented by their equiv- and current, while scattering parameters are used toalent lumped elements. Network approach is describe the relationships between the input wavesdeveloped from the line method. In the net- and output waves.work approach, we do not care the distribu- In the following discussion, we focus on two-tions of electromagnetic ﬁelds within a microwave port networks, and the conclusions obtained forstructure, and we are only interested in how two-port networks can be extended to multiportthe microwave structure responds to external networks. We will discuss impedance matrix,microwave signals. admittance matrix, and scattering matrix, and we In this section, we ﬁrst introduce the concept of will also discuss the conversions between thesemicrowave network and the parameters describing parameters.microwave networks. We then introduce networkanalyzer, followed by a discussion on the methodsfor the measurements of reﬂection, transmission, 2.4.2 Impedance matrix and admittanceand resonant properties of microwave networks. matrix 18.104.22.168 Unnormalized impedance and admittance matrixes2.4.1 Concept of microwave networkThe concept of microwave network is developed There are two types of impedance matrixesfrom the transmission line theory, and is a powerful and admittance matrixes: unnormalized ones andtool in microwave engineering. Microwave net- normalized ones. Unnormalized impedance andwork method studies the responses of a microwave admittance describe the relationships betweenstructure to external signals, and it is a com- unnormalized voltage and unnormalized current.plement to the microwave ﬁeld theory that ana- For a two-port network as shown in Figure 2.120,lyzes the ﬁeld distribution inside the microwave we havestructure. [V ] = [Z][I ] (2.425) Two sets of physical parameters are often used in [I ] = [Y ][V ], (2.426)network analysis. As shown in Figure 2.119, oneset of parameters are voltage V (or normalized where [V ] is the unnormalized voltage: [V ] =voltage v) and current I (or normalized current [V1 , V2 ]T , [I ] is the unnormalized current: [I ] =i). The other set of parameters are the input [I1 , I2 ]T , and the impedance matrix iswave a (the wave going into the network) andthe output wave b (the wave coming out of Z11 Z12 [Z] = (2.427)the network). Different network parameters are Z21 Z22
120 Microwave Electronics: Measurement and Materials Characterization I1 I2 with: √ + + Zc1 0 Port Time-port Port [ Zc ] = √ (2.435) V1 network V2 0 Zc2 1 2 − − √ Yc1 0 [ Yc ] = √ , (2.436) 0 Yc2Figure 2.120 A two-port network with voltages and where Zc1 and Zc2 are the characteristic impedancescurrents deﬁned of the transmission lines connected to port 1 and port 2 respectively; Yc1 and Yc2 are the characteristicFrom Eq. (2.425), we know that if Ii = 0 (i = j ) admittances of the transmission lines connected to Vj Vi port 1 and port 2 respectively. It is clear that Zjj = , Zij = (i = 1, 2; j = 1, 2). Ij Ij Zc Yc =  (2.437) (2.428)Equation (2.428) indicates that Zjj is the input For single components in Eqs. (2.433) andimpedance at port j when the other port is open, (2.434), we haveand Zij (i = j ) is the transition impedance from jport to i port when port i is open. Zij The admittance matrix [Y ] in Eq. (2.426) is in zij = Zij Yci Ycj = (2.438) Zci Zcjthe form of Yij Y11 Y12 yij = Yij Zci Zcj = (2.439) [Y ] = (2.429) Yci Ycj Y21 Y22 When i = j ,Similarly, Yjj is the input admittance at port j Ziiwhen the other port is shorted, and Yij (i = j ) is zii = (2.440) Zcithe transition admittance from j port to i port whenport i shorted. From Eqs. (2.425) and (2.426), we Yii yii = (2.441)can get the relationship between [Y ] and [Z] Yci [Z][Y ] =  (2.430) Equations (2.440) and (2.441) agree with the deﬁnitions in transmission line theory.22.214.171.124 Normalized impedance and admittancematrixes 2.4.3 Scattering parametersNormalized impedance matrix [z] and normalized As shown in Figure 2.121, the responses of aadmittance matrix [y] deﬁnes the relationships network to external circuits can also be describedbetween the normalized currents and normalized by the input and output microwave waves. Thevoltages: input waves at port 1 and port 2 are denoted as a1 and a2 respectively, and the output waves [v] = [z][i] (2.431) from port 1 and port 2 are denoted as b1 and b2 respectively. These parameters (a1 , a2 , b1 , and b2 ) [i] = [y][v] (2.432) may be voltage or current, and in most cases, we doFrom Eqs. (2.425), (2.426), (2.431), and (2.432), not distinguish whether they are voltage or current.we can get The relationships between the input wave [a] and output wave [b] are often described by [z] = [ Yc ][Z][ Yc ] (2.433) scattering parameters [S]: [y] = [ Zc ][Y ][ Zc ] (2.434) [b] = [S][a], (2.442)
Microwave Theory and Techniques for Materials Characterization 121 anda1 a2 a = 1 (v + i) 2 (2.450) Port Two-port Port 1 network 2 b = 1 (v − i). 2 (2.451)b1 b2 From Eqs. (2.448)–(2.451), we can get the rela- tionships between [S], [z], and [y] as listed inFigure 2.121 A two-port network with “a”s and “b”s Table 2.6. If we know any one of [S], [z], anddeﬁned [y], from Table 2.6, we can ﬁnd the other two. As will be discussed later, in experiments, usu-where [a] = [a1 , a2 ]T , [b] = [b1 , b2 ]T , and the ally the scattering parameters are measured, andscattering matrix [S] is in the form of other parameters including impedance parameters and admittance parameters can be calculated from S11 S12 [S] = (2.443) scattering parameters. S21 S22For a scattering parameter Sij , if ai = 0 (i = j ),from Eq. (2.442), we have 2.4.5 Basics of network analyzer bj Network analyzer is one of the most important Sjj = (j = 1, 2) (2.444) tools for analyzing analog circuits. By measur- aj ing the amplitudes and phases of transmission and bi Sij = (i = j ; i = 1, 2; j = 1, 2) (2.445) reﬂection coefﬁcients of an analog circuit, a net- aj work analyzer reveals all the network characteris-Equation (2.444) shows that when port j is tics of the circuit. In microwave engineering, net-connected to a source and the other port is work analyzers are used to analyze a wide varietyconnected to a matching load, the reﬂection of materials, components, circuits, and systems.coefﬁcient at port j is equal to Sjj : In most of the methods, which will be discussed in later chapters, for materials property character- bj ization, measurements are conducted by network j = Sjj = (2.446) aj analyzers. In this part, we discuss the basic princi- ple of network analyzers and the error correctionEquation (2.445) shows that when port j is techniques. More detailed information about net-connected to a source, and port i is connected to work analyzers can be found in (Ballo 1998).a matching load, the transmission coefﬁcient fromport j to port i is equal to Sij : bi 126.96.36.199 Principle of network analyzers Tj →i = Sij = (2.447) aj Network analyzers are widely used to measure the four elements in a scattering matrix: S11 ,2.4.4 Conversions between different network S12 , S21 , and S22 . As shown in Figure 2.122, aparameters network analyzer mainly consists of a source, signal separation devices, and detectors. Basically,From the relationships between different physical a network analyzer can measure the four wavesparameters, we can get conversions between differ- independently: two forward traveling waves a1 andent network parameters (Frickey 1994). From the a2 , and two reverse traveling waves b1 and b2 .deﬁnitions of normalized voltage and current, input The scattering parameters can then be obtained byand output waves, we have the combinations of these four waves according to Eqs. (2.444) and (2.445). The four detectors, v =a+b (2.448) labeled by a1 , a2 , b1 , and b2 are used to measure i =a−b (2.449) the four corresponding waves respectively, and the
Microwave Theory and Techniques for Materials Characterization 123reﬂections of the device under test (DUT) that are measurements are made, the effects of the system-not sensed at the incident wave detector. Direc- atic errors are mathematically removed from thetivity errors are due to leakage signals that are measurement results. The two main types of errorsensed at the reﬂected wave detector but are not corrections that can be done are response correc-reﬂected from the DUT. Cross-talk errors are due tions and vector corrections. Response calibrationto leakage signals that are sensed at the transmitted is simple to perform, but only corrects for a few ofwave detector without passing through the DUT. the possible systematic error terms. Response cal-The frequency response errors arise from path loss, ibration is essentially a normalized measurementphase delay, and detector response. These errors where a reference trace is stored in memory, andare caused by imperfections in the measurement subsequent measurement data is divided by thissystems. Most of these errors do not vary with memory trace. A more advanced form of responsetime, so they can be characterized through cal- calibration is open/short averaging for reﬂectionibration and mathematically removed during the measurements using broadband diode detectors.measurement process. In this case, two traces are averaged together to The dynamic ranges of a measurement system derive the reference trace. Vector-error correctionare limited by systematic errors. The dynamic requires the network analyzer can measure bothrange for reﬂection measurements is mainly lim- magnitude and phase data. Vector-error correctionited by directivity, while the dynamic range for can account for all major sources of systematictransmission measurement is mainly limited by error, and requires more calibration standards. Itnoise ﬂoor or cross talk. should be noted that response calibration can be performed on a vector network analyzer, in which case we store a vector reference trace in memory,Random errors so that we can display normalized magnitude orRandom errors are usually unpredictable and phase data. This is not the same as vector-errorcannot be removed by calibration. The possible correction, because we cannot remove the indi-sources of random errors include instrument noise, vidual systematic errors, all of which are vectorswitch repeatability, and connector repeatability. quantities.Random errors can be minimized by making Vector-error correction is a process of character-measurements several times and taking the average izing systematic vector error-terms by measuringvalues. known calibration standards, and then removing the effects of these errors from subsequent mea- surements. In microwave electronics and materialsDrift errors property characterization, one-port and two-portDrift errors are mainly caused by the change of measurements are often used. In the following,working conditions of the measurement system we discuss one-port and two port calibrations.after a calibration has been done. Temperature More discussions on vector-error corrections canvariation is one of the main sources for drift be found in (Ballo 1998).errors. In experiments, we should try to keepthe working conditions as close to the calibration One-port calibrationconditions as possible. To remove drift errors,further calibrations are needed. In reﬂection methods for materials property char- acterization, one-port reﬂection measurements are188.8.131.52 Corrections of systematic errors required. One-port calibration can measure and remove three systematic error terms in one-The systematic errors of a network analyzer can port measurements: directivity, source match, andbe corrected by calibration. This process com- reﬂection tracking.putes the systematic errors from measurements Figure 2.123 shows the equivalent circuits of anon known reference standards. When subsequent ideal case and an error adapter. The relationship
124 Microwave Electronics: Measurement and Materials Characterization Error adapter Ideal 1 RF in RF in S11a S11m S11a S11m ED ES ERTFigure 2.123 Model for one-port calibration. In the ﬁgure, S11m is the measured S11 value, S11a the actual S11value, ED the directivity, ERT the reﬂection tracking, and ES the source match. Source: Ballo, D. (1998). NetworkAnalyzer Basics, Hewlett-Packard Company. Santa Rosa, CAbetween the actual S-parameter S11a and the errors relating to signal leakage, source, and load;measurement result S11m is given by impedance mismatches relating to reﬂections; and frequency response errors caused by reﬂection and ERT S11a S11m = ED + (2.452) transmission tracking within the test receivers. The 1 − ES S11a full two-port error model includes all six of theseIn order to get the three systematic error terms terms for the forward direction and the same sixso that the actual reﬂection S-parameters can be terms in the reverse direction, for a total of 12 errorderived from our measurements, it is necessary terms. So full two-port calibration is often referredto create three equations with three unknowns to as 12-term error correction.and solve them simultaneously. So three known The relationships between the actual device S-standards, for example, a short, an open, and a Z0 parameters and the measured S-parameters areload should be measured. Solving these equations given by (Ballo 1998)will yield the systematic error terms. S11m − ED S22m − ED 1+ ES ERT ERTTwo-port calibration S21m − EX S12m − EX − ELTwo-port error correction accounts for all the ETT ETT S =major sources of systematic errors. There are two 11a S11m − ED S22m − EDtypes of two-port calibrations often used: full 1+ ES 1+ ES ERT ERTtwo-port calibration and through-reﬂect-line (TRL) S21m − EX S12m − EXcalibration. These two calibration methods are − EL EL ETT ETTbased on similar models, and we focus on full two-port calibration. (2.453) S21m − EXFull two-port calibration ETT S22m − ED × 1+ (ES − EL )All the systematic error terms are measured and ERTremoved by full two-port calibration. After full S21a = S11m − ED S22m − EDtwo-port calibration, the network analyzer can be 1+ ES 1+ ES ERT ERTused for both reﬂection and transmission mea- S21m − EX S12m − EXsurements. The error model for a two-port device −EL ELis shown in Figure 2.124. There are six types ETT ETTof systematic errors: directivity and cross-talk (2.454)
Microwave Theory and Techniques for Materials Characterization 125 Port 1 EX Port 2 S21a ETT b2 a1 ES ED S11a S22a a2 b1 EL ERT S12a (a) Port1 Port2 ERT S21a a1 b2 EL′ ED′ S11a S22a ES′ b1 a2 ETT′ S12a EX′ (b)Figure 2.124 Systematic errors of network analyzer. (a) Forward model. In the ﬁgure, ED is the forwarddirectivity; ES , forward source match; ERT , forward reﬂection tracking; EL , forward load match; ETT , forwardtransmission tracking; and EX , forward isolation and (b) reverse model. In the ﬁgure, ED is the reverse directivity;ES , reverse source match; ERT , reverse reﬂection tracking; EL , reverse load match; ETT , reverse transmissiontracking; and EX , reverse isolation. Source: Ballo, D. (1998). Network Analyzer Basics, Hewlett-Packard Company.Santa Rosa, CA S12m − EX To obtain the 12 terms of systematic error terms, ETT 12 independent measurements on known standards S22m − ED are required. Usually, four known standards, × 1+ (ES − EL ) short-open-load-thru (SOLT) are used, and thus ERTS12a = this calibration is also called SOLT calibration. S11m − ED S22m − ED 1+ ES 1+ ES Some standards are measured multiple times, for ERT ERT example, the thru standard is usually measured S21m − EX S12m − EX four times. Once the systematic error terms have −EL EL ETT ETT been characterized, the actual device S-parameters (2.455) can be derived from the measured S-parameters S22m − ED S11m − ED using Eqs. (2.453)– (2.456). It should be noted 1+ ES that each actual S-parameter is a function of ERT ERT S21m − EX S12m − EX all four measured S-parameters, so the network −EL analyzer must make a forward and reverse sweep ETT ETTS22a = to update any one of the four S-parameters. S11m − ED S22m − ED 1+ ES 1+ ES ERT ERT S21m − EX S12m − EX TRL calibration −EL EL ETT ETT To perform a two-port calibration, there are several (2.456) choices based on the type of calibration standards
126 Microwave Electronics: Measurement and Materials Characterizationto be used. Besides the full two-port calibration from the sample and/or the transmission throughdiscussed above, TRL calibration is also widely the sample are measured for the calculation ofused. The name comes from the three standards materials properties. In the following, we dis-used: thru-reﬂect-line (TRL). TRL calibration is cuss one-port method and two-port method fora two-port calibration technique primarily used in reﬂection and transmission measurements. In one-noncoaxial systems, such as waveguide, ﬁxtures, port and two-port measurements, the measurementand wafer probing. It solves for the same 12 results may be affected by some unwanted signals,error terms as the full two-port calibration, using such as the reﬂections from the mismatch betweena slightly different error model. One advantage of the connectors in coaxial-line measurement andTRL calibration is that the required standards can the reﬂections from ground in free-space measure-be easily designed, fabricated, and characterized. ment. So we will also discuss the time-domain For a four-receiver network analyzer as shown techniques, which are often used in reﬂection andin Figure 2.122, true TRL calibration can be transmission measurement to eliminate unwantedperformed, while for a three-receiver analyzer, the signals.version of TRL calibration is usually called TRL*.There are several variations of TRL calibration, 184.108.40.206 One-port methodsuch as Line-Reﬂect-Line (LRL), Line-Reﬂect-Match (LRM), Thru-Reﬂect-Match (TRM), and One-port method is mainly for the reﬂectionall these calibration methods share error models measurement. In this method, the single portsimilar to the ones for TRL calibration. More transmits signals, and meanwhile receives thediscussions on TRL calibration can be found signals reﬂected from the sample under study. Thein (Focus Microwaves Inc. 1994) and (Metzger S-parameter measured is S11 or S22 .1995). The dynamic range of reﬂection measurements is mainly limited by the directivity of the measure-220.127.116.11 Improvement of the accuracy for materials ment port. To improve the measurement accuracyproperty characterization and sensitivity, it is usually required to conduct one-port calibration, which requires three knownNetwork analyzers are widely used in materi- standards, as discussed in Section 18.104.22.168. How-als property characterization. As discussed earlier, ever, in some cases, when we do not require highthere are two types of materials property char- accuracy and sensitivity, one known standard, suchacterization methods: resonant methods and non- as “short” or “air”, may be enough to conduct aresonant methods. Resonant methods usually have simple calibration.higher accuracy and sensitivity, but require higherfrequency stability. To improve the accuracy and 22.214.171.124 Two-port methodsensitivity of resonant methods, the primary choiceis using synthesized source, and the secondary is Two-port method is suitable for the transmis-error correction. Nonresonant methods can char- sion/reﬂection method for materials property char-acterize materials over a frequency range. For acterization. In a two-port method, one port trans-nonresonant methods, the primary choice is error mits signal and the other port receives the sig-correction, and the secondary is using synthesized nal from the sample under study. As shown insource. Figure 2.125, two-port method can be used for transmission/reﬂection measurement and bi-static2.4.6 Measurement of reﬂection and reﬂection or scattering measurement.transmission properties The dynamic range of a two-port method is mainly limited by the noise ﬂoor of theIn nonresonant methods for materials property signal and the cross talk between the twocharacterization, the sample under test is inserted ports. Transmission measurement usually requiresin a segment of transmission line, and the reﬂection two-port calibration discussed in Section 126.96.36.199.
Microwave Theory and Techniques for Materials Characterization 127 Sample Sample Port 1 Port 2 Port 1 Port 2 (a) (b)Figure 2.125 Two-port measurements. (a) Transmission/reﬂection measurement and (b) bistatic reﬂection orscattering measurementIf we do not need accurate measurement, in Fundamentalssome cases a “through” calibration may beenough. For bistatic reﬂection or scattering mea- The responses of a linear and invariant networksurement, special calibrations and measurement to electromagnetic waves can be represented intechniques are needed (Knott 1993; Knott et al. the time domain by its impulse response h(t) or1993). in the frequency domain by its transfer function H (f ). Fundamentally speaking, h(t) and H (f ) give the same information, and they can be188.8.131.52 Time-domain techniques transformed from one to the other through theThe Fourier transform techniques provide a method Fourier transform:for transforming data between frequency domain ∞and time domain. In network analyzers, usually a H (f ) = h(t) exp(−j2πf t) dt (2.457) −∞type of fast Fourier transform (FFT) known as theChirp Z Transform is used, which permits users to It is clear that the data measured in the frequency“zoom in” on a speciﬁc time (distance) range of domain by a network analyzer can be transformedinterest. Here, we discuss the main properties of to the time domain using inverse Fourier trans-the transform process and how the various pro- form:cessing options can be used to obtain optimum ∞results. More discussions on the applications of h(t) = H (f ) exp(−j2πf t) df (2.458)time-domain techniques in network analyzers can −∞be found in (Anritsu Company 1998; Rohde &Schwarz 1998). Impulse response and step response By means of the inverse Fourier transform, themeasurement results in the frequency domain can In time domain, besides the impulse response,be transformed to the time domain. The time- signals can also be represented by the stepdomain results give clear representations of the response, which can be obtained by integration ofcharacteristics of the device under test (DUT). the impulse response h(t). Figure 2.126 shows aFor instance, the faults in cables can be directly stepped coaxial line and its time-domain responseslocalized. Furthermore, special time domain ﬁlters, in impulse and step forms. In general, the relation-called gates, can be used to suppress unwanted ship between the measured reﬂection coefﬁcientsignal components such as multireﬂections. The S11 and the impedance Z is given bymeasured data “gated” in the time domain can be Z − Z0transformed back to the frequency domain and an S11 = , (2.459) Z + Z0S-parameter representation without the unwantedsignal components can be obtained as a function where Z0 is the characteristic impedance. Inof frequency. Figure 2.126, the characteristic impedance is 50 .
128 Microwave Electronics: Measurement and Materials Characterization I II III IV (a) Impulse 50 Ω 25 Ω 75 Ω 50 Ω Step (b) (a) (b) Figure 2.127 Ideal low-pass responses associated with (c) (a) shunt capacitance and (b) series inductance Figure 2.127 shows the low-pass responses associated with a series inductance and a shuntFigure 2.126 Time-domain responses (S11 in low-pass capacitance. The meaning of “low-pass” willmode) of a stepped coaxial line. (a) Impedance of astepped coaxial line, (b) impulse response, and (c) step be discussed later. Figure 2.127(a) shows theresponse response of a shunt capacitance. Before the capac- itance, the impedance of the line is equal to the reference impedance (Z0 ). The capacitor ﬁrst actsFrom time-domain responses, we can locate the as a short and is responsible for the negative edgeimpedance discontinuities along a transmission of the step response. The capacitor charges upline. Furthermore, the step representation clearly gradually. When the capacitor is fully charged, itshows the variation of the impedance along the acts as an open. The parallel connection consist-coaxial line. There is no reﬂection in the regions ing of capacitor and resistor is then equivalent to(I) and (IV), so the zero value in Figure 2.126(c) a single resistor. Since the value of the resistor inrepresents the reference impedance Z0 . Positive the example is equal to the reference impedancevalues stand for higher impedances than the (Z0 ), no reﬂection will occur due to the match-reference impedance (Z > Z0 ) and negative values ing. Thus the step response again attains zero. Thefor lower impedances (Z < Z0 ). characteristic of the impulse response can be imag- Mathematically, these two forms of representa- ined by a differentiation of the step response. In ations are equivalent. They can be converted into similar way, we can understand the response of aneach other by differentiation or integration. It inductance in series, as shown in Figure 2.127(b).is recommended to use the step response if theimpedance characteristics of the DUT are of inter- Finite pulse widthest, while, in most of the other cases, the impulseresponse should be used, especially for the deter- Mathematically, in time domain, an inﬁnitely widemination of discontinuities. frequency range is assumed. By Fourier transform, Now, we consider the time domain responses of inﬁnitely narrow Dirac pulses can be obtained.reactive DUTs. As the signal components of the However, in actual Fourier transform, as thestep response occurring later in time correspond to frequency span is limited, the frequency domainlower frequency components down to DC, only data are multiplied by a rectangular weightingthe low-frequency behavior of the DUT has an function that takes the value 1 for the actualeffect on the later part of the step response. A frequency range of the network analyzer andcapacitor now reacts like an interruption whereas which is otherwise zero. This multiplication in thean inductor for low frequencies is similar to a frequency domain corresponds to a convolution ofthrough connection. ideal Dirac pulses with a si function in the time
Microwave Theory and Techniques for Materials Characterization 129domain: amplitudes to the left and right of the main pulse sin x are given by si(x) = , (2.460) x sin(3π/2) = −0.212 (2.462)and the width T of the si impulses is inversely 3π/2proportional to the frequency span F of the This corresponds to a side-lobe suppression offrequency range: −13.46 dB. Similar to the width of the impulse, the 2 side lobes can also be reduced by suitable windows T = (2.461) in the frequency domain, as will be discussed later. F The principal property of time domain tech- Aliasniques for most microwave applications is reso-lution, which reﬂects the ability to locate a spe- The actual measurement results in the frequency domain are not obtained continuously with fre-ciﬁc signal in the presence of other signals. The quency but only at a ﬁnite number of discreteabove discussion indicates that the basic limi- frequency points. The frequency discrete measure-tation of resolution is inversely related to data ments can be regarded as a modiﬁed continuouscollection bandwidth in the frequency domain. spectrum by multiplying the continuous spectrumA rule of thumb: resolution is on the order of with a comb function in the frequency domain.150 mm/[frequency span (GHz)]. For example, a In the time domain, this corresponds to a con-10-GHz frequency span will provide resolution of volution of the time response with a periodicabout 15 mm. Besides, it should be noted that reso- Dirac impulse sequence. This results in the aliaslution is also inﬂuenced by the processing method effect of a frequent repetition of the original timeand window selection which will be discussed response, as shown in Figure 2.129. The time inter-later. val t between the repetitions in the time domain Figure 2.128(a) shows another characteristic of response is called the alias-free range, which issi pulses: the occurrence of side lobes to the related to the frequency step width f in the fre-left and right of the main pulse. In experiments, quency domain:side lobes are perceived as interference. Accordingto the si function, the highest (negative) side t = 1/ f. (2.463) CH1 S11 LIN Re Ref 0 U CH1 S11 LIN Re Ref 0 U 1U 1U 200.0 mU/ 200.0 mU/ −1 U −1 U START 0 s 200 ps/ STOP 2 ns START 0 s 200 ps/ STOP 2 ns (a) (b)Figure 2.128 The time-domain responses of a shorted line. (a) Widened si impulse due to ﬁnite span( F = 4 GHz). The pulse width t is about 500 ps and (b) the width of si pulse is halved in low-pass mode.The pulse width t is about 250 ps. Source: Rohde & Schwarz (1998). Time domain measurements using vectornetwork analyzer ZVR, Application Note 1EZ44 0E, Olaf Ostwald, Munich