DOCSLIB.ORG

Notes on Multi-Layer Impedance

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Multi-Layer Impedance EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN-AB-2003-093 (ABP) The Impedance of Multi-layer Vacuum Chambers Luc Vos Summary Many components of the LHC vacuum chamber have multi-layered walls : the copper coated cold beam screen, the titanium coated ceramic chamber of the dump kickers, the ceramic chamber of the injection kickers coated with copper stripes, only to name a few. Theories and computer programs are available for some time already to evaluate the impedance of these elements. Nevertheless, the algorithm developed in this paper is more convenient in its application and has been used extensively in the design phase of multi-layer LHC vacuum chamber elements. It is based on classical transmission line theory. Closed expressions are derived for simple layer configurations, while beam pipes involving many layers demand a chain calculation. The algorithm has been tested with a number of published examples and was verified with experimental data as well. Geneva, Switzerland October , 2003 1 Introduction For the LHC it has been necessary to compute the impedance of mult-layered vacuum chambers since a major part of the beam pipe belongs to this family : the copper coated cold beam screen, the titanium coated ceramic chamber of the dump kickers, the ceramic chamber of the injection kickers coated with copper stripes, the copper coated µ-metal pipes of the septum magnets, the copper coated ceramic TDI injection collimators, the copper coated cold-warm transition pieces, the copper coated carbon collimators, the NEG coated warm vacuum pipes. The impedance of multi-layer vacuum chamber walls can be found by solving Maxwell’s equations taking into account the proper boundary conditions. The task is tedious and has been done before[1]. A physically transparent alternative exists which is easy to understand and simple to implement. The basic idea is as follows. A particle beam moving with velocity βc, is accompanied by transverse electric and magnetic fields. In free space these fields tend to infinity while they are simply cut by a perfectly conducting vacuum chamber. Both cases are identical from the point of view of the beam. The situation changes drastically when the vacuum chamber is no longer perfectly conducting or has other propagation properties than those of free space. Fields penetrate the inner surface of the vacuum chamber material. The tangential magnetic field is continuous across the inner boundary. The penetrating magnetic field generates a surface current density which requires a longitudinal electric field in the face of a non-zero surface impedance. This field pair (tangential magnetic field and longitudinal electric field) forms a transverse wave in the vacuum chamber material. The problem at hand consists in finding the longitudinal electric field. The transverse wave is launched in a sequence of layers with different but known characteristics. The situation is essentially that of a transmission line comprising different sections, each with a different but known intrinsic impedance and propagation constant. Hence, the input impedance of this structure, which is the longitudinal impedance, can be found in a straightforward manner. The theory of transmission lines implicitly takes into account Maxwell’s equations with the proper boundary conditions. 2 Basic transmission line parameters [2] Each material can be characterised electro-magnetically by three parameters. They are the electric conductivity σm, the permeability µm and the dielectric constant εm. Both µm and εm can be complex. The intrinsic impedance of the material m is : jωµ m Zm = . (1) jωε m +σ m The other parameters that are required are the propagation constants. Zotter specifies without proof the radial propagation constant [3, page 158] ,[6]: 2 2 ω γ Tm = + jωµ m (σ m + jωε m ). (2) βc The derivation of this quantity goes as follows. Consider a beam travelling in vacuum with a speed βc parallel to a material boundary. The intrinsic propagation constants in vacuum and in the material are known quantities : γ0 and γm. In the most general case one can write the transverse propagation constants as : 2 γ 2 = γ 2 − h 2 T 0 s , (3) 2 2 2 γ Tm = γ m −hsm where γT is the transverse propagation constant and hs the longitudinal propagation constant. From the continuity of the magnetic field across the boundary it follows that hs = hsm. Moreover, there is no transverse propagation in the vacuum such that γT = 0 and γ0 = hs = hsm. Hence: 2 j 2 2 2 ω γ Tm = γ m −γ 0 = jωµ m (σ m + jωεm )− . (4) βc 3 Setting-up the transmission line relations Consider a set of parallel layers numbered n = 1, 2 ... from the first vacuum-material boundary onwards. Each set is characterised by its intrinsic impedance Zn, valid for a unit area, and its transverse propagation constant γTn = γn (drop subscript T since only transverse waves are dealt with from this moment onwards) and its length or thickness. The last layer, which can be of any nature (vacuum, conductor, insulator) has infinite length. No reflections occur in this part of the transmission line, hence it loads the previous section with its characteristic impedance. This allows the computation in a piecewise fashion of the input impedance and transfer impedance of each preceding line section. The input impedance of the first section (inverse order, last in computation!) is simply the surface impedance experienced by the beam. The classical formulae for the transfer and input impedances are : Ztn = Zin+1 cosh(γ nsn )+ Zn sinh(γ nsn ) Ztn , (5) Zin = Zn Zn cosh()γ nsn + Zin+1 sinh()γ nsn where Zin, Ztn are the input and transfer impedances of section n and Zin+1, Ztn+1 are the input and transfer impedances of section n+1. The length of section n is sn. The penetration of the electro-magnetic fields through the different layers is defined by the transfer impedances and the input impedances as will be shown below. Consider a flat beam travelling through vacuum between two planar perfect conductors. The electro-magnetic fields travel with a speed βc, no reflections occur. Interrupt the perfect conductors and insert a piece of material with known electro-magnetic and geometric properties and which length is unity. The primary fields that accompany the beam are not altered, namely the tangential magnetic field. Its interaction with the input impedance of the insertion generates a longitudinal electric field. The input impedance is the longitudinal impedance of the vacuum chamber insert. The equivalent transmission line diagram for three consecutive layers is shown in Fig 1. The last layer is always infinitely thick, in this case layer three, and serves as load impedance of the previous section. Since we work with fields, impedances are surface impedances. The magnetic field at the entry of the first transmission line section is given by Ampere’s law : Hi = Hi1 = Jb , (6) 3 where Jb is the planar beam current density . transverse transmission lines Hi1 Z1, γ1, Z2, γ2, Ei1 Z3 Figure 1 : Equivalent transmission line circuit of a vacuum chamber consisting of three layers. The transverse wave will be partially reflected and transmitted at each transmission line junction except at the last one since it is infinitely long. The effect of the partial reflections and transmissions is duly taken into account in the formulae of the input and transfer impedances (Eq.5). The computation proceeds backwards and yields the input impedance Zi1. When needed, the penetration of the magnetic field through the various layers can be found with the transfer impedances Ztn. Indeed, Ei1 Hi1 = , Zi1 Ei1 Zi1 Hi2 = = Hi1 , (7) Zt1 Zt 1 Zi1 Zi 2 Hi3 = Hi1 . Zt1 Zt 2 4 Cylindrical geometry versus planar geometry The input magnetic field in cylindrical geometry is given by : i H = b , (8) i1 2πb where ib is beam intensity and b the radius of the pipe. The beam is located in the center. In free space the magnetic field decreases as r-1, where r is the increasing distance from the beam. Consider two consecutive layers. If the area of the first is unity,then the area of the second is unity increased by ∆1/b, ∆1 being the thickness (length of equivalent transmission line) of the first layer. The increased cross-section of the second layer increases in a similar way the inductance, the capacitance and the conductivity. Hence the characteristic impedance is not changed and the transmission constant increases with ∆1/b. For subsequent layers the transmission constant increases by ∆2/(b+∆1), etc. Obviously, this effect is only 4 significant when ∆1/b is not too small. In practice it turns out that the simple planar geometry is a good approximation for most cylindrical cases. 5 Special cases A few cases with a small number of layers lead to relatively simple closed formulae. 5.1 A single metallic layer with infinite thickness The input impedance is the intrinsic impedance of the metal : jωρ Z1 = Z0 ≈ jωµρ , (9) jωρ + Z 0c where Z0 is the impedance of free space and ρ =1/σ. Eq. 9 is the classical skin depth formula. The approximation is valid for radian frequencies ω c << Z0 ρ . 5.2 A single metallic layer with finite thickness The impedance of the outer layer is Z0. The metal layer has resistivity ρ, thickness s1, and the usual free space value for permeability µ and dielectric constant ε. The intrinsic impedance and the propagation constant are for β = 1: jωρ Z1 = Z0 jωρ + Z 0c . (10) jωµ γ = 1 ρ This yields the following surface impedance : jωρ jωµ 1+ tanh s1 jωρ + Z0c ρ Z = Z . (11) i1 0 jωρ + Z c 0 jωµ 1+ tanh s1 jωρ ρ For ω = 0 the impedance is real and corresponds to a parallel circuit formed by the impedance of vacuum (Z0) and the surface impedance of the metal wall ρ/s1.
Load more

Recommended publications
  • Non-Ionizing Radiation, Part 2: Radiofrequency Electromagnetic Fields Volume 102
    NON-IONIZING RADIATION, PART 2: RADIOFREQUENCY ELECTROMAGNETIC FIELDS VOLUME 102 This publication represents the views and expert opinions of an IARC Working Group on the Evaluation of Carcinogenic Risks to Humans, which met in Lyon, 24-31 May 2011 LYON, FRANCE - 2013 IARC MONOGRAPHS ON THE EVALUATION OF CARCINOGENIC RISKS TO HUMANS 1. EXPOSURE DATA 1.1 Introduction a wave (Fig. 1.1) and is related to the frequency according to λ = c/f (ICNIRP, 2009a). This chapter explains the physical principles The fundamental equations of electromag- and terminology relating to sources, exposures netism, Maxwell’s equations, imply that a time- and dosimetry for human exposures to radiofre- varying electric field generates a time-varying quency electromagnetic fields (RF-EMF). It also magnetic field and vice versa. These varying identifies critical aspects for consideration in the fields are thus described as “interdependent” and interpretation of biological and epidemiological together they form a propagating electromagnetic studies. wave. The ratio of the strength of the electric-field component to that of the magnetic-field compo- 1.1.1 Electromagnetic radiation nent is constant in an electromagnetic wave and is known as the characteristic impedance of the Radiation is the process through which medium (η) through which the wave propagates. energy travels (or “propagates”) in the form of The characteristic impedance of free space and waves or particles through space or some other air is equal to 377 ohm (ICNIRP, 2009a). medium. The term “electromagnetic radiation” It should be noted that the perfect sinusoidal specifically refers to the wave-like mode of case shown in Fig.
    [Show full text]
  • PHYS 352 Electromagnetic Waves
    Part 1: Fundamentals These are notes for the first part of PHYS 352 Electromagnetic Waves. This course follows on from PHYS 350. At the end of that course, you will have seen the full set of Maxwell's equations, which in vacuum are ρ @B~ r~ · E~ = r~ × E~ = − 0 @t @E~ r~ · B~ = 0 r~ × B~ = µ J~ + µ (1.1) 0 0 0 @t with @ρ r~ · J~ = − : (1.2) @t In this course, we will investigate the implications and applications of these results. We will cover • electromagnetic waves • energy and momentum of electromagnetic fields • electromagnetism and relativity • electromagnetic waves in materials and plasmas • waveguides and transmission lines • electromagnetic radiation from accelerated charges • numerical methods for solving problems in electromagnetism By the end of the course, you will be able to calculate the properties of electromagnetic waves in a range of materials, calculate the radiation from arrangements of accelerating charges, and have a greater appreciation of the theory of electromagnetism and its relation to special relativity. The spirit of the course is well-summed up by the \intermission" in Griffith’s book. After working from statics to dynamics in the first seven chapters of the book, developing the full set of Maxwell's equations, Griffiths comments (I paraphrase) that the full power of electromagnetism now lies at your fingertips, and the fun is only just beginning. It is a disappointing ending to PHYS 350, but an exciting place to start PHYS 352! { 2 { Why study electromagnetism? One reason is that it is a fundamental part of physics (one of the four forces), but it is also ubiquitous in everyday life, technology, and in natural phenomena in geophysics, astrophysics or biophysics.
    [Show full text]
  • (Rees Chapters 2 and 3) Review of E and B the Electric Field E Is a Vector
    ELECTROMAGNETISM SUMMARY (Rees Chapters 2 and 3) Review of E and B The electric field E is a vector function. E q q o If we place a second test charged qo in the electric field of the charge q, the two spheres will repel each other. The strength of the radial force Fr is proportional to the product of the charges and inversely proportional to the distance between them squared. 1 qoq Fr = Coulomb's Law 4πεo r2 -12 where εo is the electric permittivity (8.85 x 10 F/m) . The radial component of the electric field is the force divided by the strength of the test charge qo. 1 q force ML Er = - 4πεo r2 charge T2Q The magnetic field B is also a vector function. Magnetic fields are generated by charges in motion. For example, a current traveling through a wire will produce a magnetic field. A charge moving through a uniform magnetic field will experience a force that is proportional to the component of velocity perpendicular to the magnetic field. The direction of the force is perpendicular to both v and B. X X X X X X X X XB X X X X X X X X X qo X X X vX X X X X X X X X X X F X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X F = qov ⊗ B The particle will move along a curved path that balances the inward-directed force F⊥ and the € outward-directed centrifugal force which, of course, depends on the mass of the particle.
    [Show full text]
  • Module 2: Fundamental Behavior of Electrical Systems 2.0 Introduction
    Module 2: Fundamental Behavior of Electrical Systems 2.0 Introduction All electrical systems, at the most fundamental level, obey Maxwell's equations and the postulates of electromagnetics. Under certain circumstances, approximations can be made that allow simpler methods of analysis, such as circuit theory, to be employed. However, the problems associated with EMC usually involve departures from these approximations. Therefore, a review of fundamental concepts is the logical starting point for a proper understanding of electromagnetic compatibility. 2.1 Fundamental quantities and electrical dimensions speed of light, permittivity and permeability in free space The speed of light in free space has been measured through very precise experiments, and extremely accurate values are known (299,792,458 m/s). For most purposes, however, the approximation c 3x108 m/s is sufficiently accurate. In the International System of units (SI units), the constant known as the permeability of free space is defined to be 7 0 4 ×10 H/m. The constant permittivity of free space is then derived through the relationship 1 c o µo and is usually expressed in SI units as 1 1 8.854×10 12 ×10 9 F/m. 0 2 36 c µo Both the permittivity and permeability of free space have been repeatedly verified through experiment. wavelength in lossless media Wavelength is defined as the distance between adjacent equiphase points on a wave. For an electromagnetic wave propagating in a lossless medium, this is given by 2-2 v f where f is frequency. In free space v=c, and in a simple lossless medium other than free space, the velocity of propagation is v 1 µ , where r 0 and µ µr µ0 .
    [Show full text]
  • 6.013 Electromagnetics and Applications, Chapter 2
    Chapter 2: Introduction to Electrodynamics 2.1 Maxwell’s differential equations in the time domain Whereas the Lorentz force law characterizes the observable effects of electric and magnetic fields on charges, Maxwell’s equations characterize the origins of those fields and their relationships to each other. The simplest representation of Maxwell’s equations is in differential form, which leads directly to waves; the alternate integral form is presented in Section 2.4.3. The differential form uses the vector del operator ∇: ∂ ∂ ∂ ∇≡ xˆ + yˆ + zˆ (2.1.1) ∂x ∂y ∂z where xˆ , yˆ , and zˆ are defined as unit vectors in cartesian coordinates. Relations involving ∇ are summarized in Appendix D. Here we use the conventional vector dot product1 and cross product2 of ∇ with the electric and magnetic field vectors where, for example: E = xˆEx + yˆEy + zˆEz (2.1.2) ∂E ∂Ey ∂E ∇•E ≡ x + + z (2.1.3) ∂x ∂y ∂z We call ∇•E the divergence of E because it is a measure of the degree to which the vector field⎯E diverges or flows outward from any position. The cross product is defined as: ⎛ ∂Ez ∂Ey ⎞ ⎛ ∂Ex ∂Ez ⎞ ⎛ ∂Ey ∂Ex ⎞ ∇×E ≡ xˆ ⎜ − ⎟ + yˆ ⎜ − ⎟ + zˆ ⎜− ⎟ ⎝ ∂y ∂z ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂x ∂y ⎠ xˆ yˆ zˆ (2.1.4) =det ∂ ∂x ∂ ∂y ∂ ∂z E x E y E z which is often called the curl of E . Figure 2.1.1 illustrates when the divergence and curl are zero or non-zero for five representative field distributions. 1 The dot product of A and B can be defined as A • B = AxxB+ AyBy+ AzBz= AB cos θ , where θ is the angle between the two vectors.
    [Show full text]
  • Light As Electromagnetic Waves
    EE485 Introduction to Photonics Introduction Nature of Light “They could but make the best of it and went around with woebegone faces, sadly complaining that on Mondays, Wednesdays, and Fridays, they must look on light as a wave; on Tuesdays, Thursdays, and Saturdays, as a particle. On Sundays they simply prayed.” The Strange Story of the Quantum Banesh Hoffmann, 1947 Geometrical (ray) optics Lih Y. Lin 2 History of Optics Quantum optics • Geometrical optics (Ray optics) E-M wave optics Ray optics – Enunciated by Euclid in Catoptrics, 300 B.C. • Early 1600: First telescope by Galileo Galilei • End of the 17th century: Light as wave to explain reflection and refraction, by Christian Huygens • 1704: Corpuscular nature of light (light as moving particles) to explain refraction, dispersion, diffraction, and polarization, by Issac Newton • Early 1800: Interference experiment by Thomas Young – light is wave • Maxwell equation (1864) ― Light as electromagnetic waves, by James Clerk Maxwell • How about emission and absorption? • Quantum theory ― Light as photons – 1900: Max Plank – quantum theory of light – 1905: Albert Einstein – photoelectric effect experiment, light behaves as particles with energies E = h – 1925-1935: de Broglie –quantum mechanics explaining the wave-particle duality of light • 1950s: Communication and information theory • 1960: First laser Lih Y. Lin 3 Topics we plan to cover • Light as electromagnetic waves • Polarized light • Superposition of waves and interference • Diffraction • Photon and laser basics • Laser operation • Nonlinear optics and light modulation Lih Y. Lin 4 Electromagnetic Spectrum Optical frequencies Lih Y. Lin 5 EE485 Introduction to Photonics Light as Electromagnetic Waves 1. Wave equations 2.
    [Show full text]
  • Electromagnetic Fields and Waves
    Lecture 2 Review 2 More on Maxwell Wave Equations Boundary Conditions Poynting Vector Transmission Line A. Nassiri - ANL Maxwell’s equations in differential form ∇.D = ρ Gauss' law for electrostatics ∇.B = 0 Gauss' law for magnetostatics ∂D ∇ × H = J + Ampere' s law dt ∂B ∇ × E = − Faraday' s law dt ∂ρ ∇.J = − Equation of continuity ∂t D = εE Varying E and H fields are coupled B = µH Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 2 Electromagnetic waves in lossless media - Maxwell’s equations Constitutive relations Maxwell ∂D D = εE = ε ε E ∇× H = J + r o dt B = µH = µ µ H ∂B r o ∇×E = − dt J = σE SI Units ∇ = ρ • J Amp/ metre2 .D • D Coulomb/metre2 • H Amps/metre • B Tesla ∇.B = 0 Weber/metre2 Volt-Second/metre2 Equation of continuity • E Volt/metre • ε Farad/metre • µ Henry/metre ∂ρ • σ Siemen/metre ∇.J = − ∂t Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 3 Wave equations in free space • In free space – σ=0 ⇒J=0 – Hence: ∂D ∂D ∇× H = J + = dt dt ∂B ∇×E = − dt – Taking curl of both sides of latter equation: ∂B ∂ ∂ ∇×∇× E = −∇× = − ∇× B = −µ ∇× H ∂t ∂t o ∂t ∂ ∂D = −µo ∂t ∂t ∂ 2E ∇×∇× E = −µoε ∂t 2 Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 4 Wave equations in free space cont. ∂ 2E ∇×∇×E = −µoε ∂t 2 • It has been shown (last week) that for any vector A where is the Laplacian operator 2 Thus: ∇×∇× A = ∇∇.A − ∇ A ∂ 2 ∂ 2 ∂ 2 ∇2 = + + ∂x2 ∂y 2 ∂z 2 2 2 ∂ E ∇∇.E − ∇ E = −µoε ∂t 2 There are no free charges in free space so ∇.E=ρ=0 and we get 2 2 ∂ E ∇ E = µoε ∂t 2 A three dimensional wave equation Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 5 Wave equations in free space cont.
    [Show full text]
  • Lecture 23 – Electromagnetism Spring 2013 Semester Matthew Jones Midterm Exam
    Physics 42200 Waves & Oscillations Lecture 23 – Electromagnetism Spring 2013 Semester Matthew Jones Midterm Exam: Date: Wednesday, March 6 th Time: 8:00 – 10:00 pm Room: PHYS 203 Material: French, chapters 1-8 Electromagnetism • Previous examples of wave propagation: – String with tension and mass per unit length – Elastic rod with Young’s modulus and density – Sound propagating in air… – Waves in water… – Etc… • The rest of the course is devoted to the study of one very important example of wave phenomena: Forces on Charges • Coulomb’s law of electrostatic force: Charles-Augustin de Coulomb (1736 - 1806) #" # • The magnitude of the attractive/repulsive force is = where 1 = = 8.99 10 4 and therefore " " = 8.85 10 (This constant is called the “permittivity of free space”) Electric Field • An electric charge changes the properties of the space around it. – It is the source of an “electric field”. – It could be defined as the “force per unit charge”: " $ = %&'( = 4$ – Quantum field theory provides a deeper description… • Gauss’s Law: # 1 ) *+ $ ,- = /01/23 . Electric “flux” through surface S Magnetic Field • Moving charges (ie, electric current) produce a magnetic field: 6 ,7 5 = ) 4 A moving charge in a magnetic field experiences a force: Lorentz force law:= 4 8 5 = 4$ 9 48 5 Gauss’s Law for Magnetism • Electric charges produce electric fields: # ) *+ $ ,- = /01/23 . • But there are no “magnetic charges”: 2 ) *+ 5 ,- = 0 . Magnetic “flux” through surface S Ampere’s Law • An electric current produces a magnetic field that curls around it: : : (Close, but not quite the whole story) Faraday’s Law of Magnetic Induction • Magnetic flux: ;< = =.
    [Show full text]
  • Analysis of Frequency Selective Surfaces with Ferrite Substrates
    University of Central Florida STARS Retrospective Theses and Dissertations 1996 Analysis of frequency selective surfaces with ferrite substrates Youcheng Liu University of Central Florida Part of the Electrical and Computer Engineering Commons Find similar works at: https://stars.library.ucf.edu/rtd University of Central Florida Libraries http://library.ucf.edu This Doctoral Dissertation (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Liu, Youcheng, "Analysis of frequency selective surfaces with ferrite substrates" (1996). Retrospective Theses and Dissertations. 2996. https://stars.library.ucf.edu/rtd/2996 ANALYSIS OF FREQUENCY SELECTIVE SURFACES WITH FERRITE SUBSTRATES YOUCHENG LIU B.S.E.E.,Shanghai Jiao-Tong University, 1982 M.S.E.E.,Shanghai Jiao-Tong University, 1990 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering College of Engineering University of Central Florida Orlando, Florida Spring Term 1996 ABSTRACT Frequency selective surfaces (FSS) have numerous applications in several microwave and optical systems. Most frequency selective surface structures have one or more dielectric substrates. In this work, instead of traditional dielectric substrates, ferrite substrates are used under different dc bias conditions. By using ferrite materials, one can change the spectral properties of these structures without physically altering them. An applied magnetic field (dc bias) on the ferrite substrate changes its properties and hence the electrical dimensions of the elements comprising the periodic structure.
    [Show full text]
  • Electromagnetics Volume 2
    ELECTROMAGNETICS STEVEN W. ELLINGSON VOLUME 2 ELECTROMAGNETICS VOLUME 2 Publication of this book was made possible in part by the Virginia Tech University Libraries’ Open Education Initiative Faculty Grant program: http://guides.lib.vt.edu/oer/grants e Open Electromagnetics Project, https://www.faculty.ece.vt.edu/swe/oem Books in this series Electromagnetics, Volume 1, https://doi.org/10.21061/electromagnetics-vol-1 Electromagnetics, Volume 2, https://doi.org/10.21061/electromagnetics-vol-2 ELECTROMAGNETICS STEVEN W. ELLINGSON VOLUME 2 Copyright © 2020 Steven W. Ellingson This textbook is licensed with a Creative Commons Attribution Share-Alike 4.0 license: https://creativecommons.org/licenses/by-sa/4.0. You are free to copy, share, adapt, remix, transform, and build upon the material for any purpose, even commercially, as long as you follow the terms of the license: https://creativecommons.org/licenses/by-sa/4.0/legalcode. This work is published by Virginia Tech Publishing, a division of the University Libraries at Virginia Tech, 560 Drillfield Drive, Blacksburg, VA 24061, USA ([email protected]). Suggested citation: Ellingson, Steven W. (2020) Electromagnetics, Vol. 2. Blacksburg, VA: Virginia Tech Publishing. https://doi.org/10.21061/electromagnetics-vol-2. Licensed with CC BY-SA 4.0. https://creativecommons.org/licenses/by- sa/4.0. Peer Review: This book has undergone single-blind peer review by a minimum of three external subject matter experts. Accessibility Statement: Virginia Tech Publishing is committed to making its publications accessible in accordance with the Americans with Disabilities Act of 1990. The screen reader–friendly PDF version of this book is tagged structurally and includes alternative text which allows for machine-readability.
    [Show full text]
  • The Development and Improvement of Instructions
    ANALYSIS, DESIGN, AND OPERATION OF A SPHERICAL INVERTED-F ANTENNA A Thesis by JACOB JEREMIAH MCDONALD Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2009 Major Subject: Electrical Engineering ANALYSIS, DESIGN, AND OPERATION OF A SPHERICAL INVERTED-F ANTENNA A Thesis by JACOB JEREMIAH MCDONALD Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Approved by: Chair of Committee, Gregory Huff Committee Members, Ray James Robert Nevels Steve Wright Head of Department, Costas Georghiades May 2009 Major Subject: Electrical Engineering iii ABSTRACT Analysis, Design, and Operation of a Spherical Inverted-F Antenna. (May 2009) Jacob Jeremiah McDonald, B.S., Texas A&M University Chair of Advisory Committee: Dr. Gregory Huff This thesis presents the analysis, design, and fabrication of a spherical inverted-F antenna (SIFA). The SIFA consists of a spherically conformal rectangular patch antenna recessed into a quarter section of a metallic sphere. The sphere acts as a ground plane, and a metal strip shorts the patch to the metallic sphere. The SIFA incorporates planar microstrip design into a conformal spherical geometry to better meet the design constraints for integrated wireless sensors. The SIFA extends a well-established technology into a new application space, including microsatellites, mobile sensor networks, and wireless biomedical implants. The complete SIFA design depends on several parameters, several of which parallel planar design variables. A modified transmission line model determines the antenna input impedance based on the sphere’s inner and outer radii, the patch length and width, short length and width, and feed position.
    [Show full text]
  • On the Superposition and Elastic Recoil of Electromagnetic Waves
    Forum for Electromagnetic Research Methods and Application Technologies (FERMAT) On the Superposition and Elastic Recoil of Electromagnetic Waves Hans G. Schantz Q-Track Corporation (Email: [email protected]) Abstract—Superposition demands that a linear combination Alternate interpretations are possible, but appear less likely. of solutions to an electromagnetic problem also be a solution. Although laid out in the context of transmission lines, one This paper analyzes some very simple problems – the may readily extend the conclusions of this paper to constructive and destructive interferences of short impulse electromagnetic waves of arbitrary shape and those in free voltage and current waves along an ideal free-space space. We may also interpret electromagnetic elastic recoil transmission line. When voltage waves constructively interfere, as each wave reflecting from the impedance discontinuity the superposition has twice the electrical energy of the that the superposition imparts on free space. This paper individual waveforms because current goes to zero, converting demonstrates that different mathematical models are magnetic to electrical energy. When voltage waves destructively available to describe electromagnetic behavior. We may interfere, the superposition has no electrical energy because it transforms to magnetic energy. Although the impedance of the choose among models by considering their physical individual waves is that of free space, a superposition of waves implications. This paper further notes applications of may exhibit arbitrary impedance. Further, interferences of electromagnetic recoil phenomena for quantum mechanics, identical waveforms allow no energy transfer between opposite near-field electromagnetic ranging, and for field-diversity ends of a transmission line. The waves appear to recoil antenna systems.
    [Show full text]