ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. E = jωb. H = J + jωd. D = ρ (M3) B = 0 (M4) D = εe
|
|
- Adelia Richards
- 6 years ago
- Views:
Transcription
1 ANTENNAS Vector and Scalar Potentials Maxwell's Equations E = jωb H = J + jωd D = ρ B = (M) (M) (M3) (M4) D = εe B= µh For a linear, homogeneous, isotropic medium µ and ε are contant. Since B =, there exists a vector A such that B = A and ( A) =. A is called the magnetic vector potential. There are infinitely many vectors A that satisfy B = A thus we later need to specify A. (M) can be written as E = -jω A (E + jωa) = () Vector = Vector = (some scalar) () E + jωa = - ϕ ϕ : scalar potential (3) E = - jωa - ϕ (4) (M) can be written as B = µj + jωµεe (5) A = µj + µ ( jωa ϕ) (6) A + ( A) = µj + ω µεa µ ϕ (7)
2 To simplify the equation, choose A as A = - jωµεϕ (8) This is called the Lorentz Condition. This leads to A + ω µεa = - µj D'Alembert's Equation (9) The problem now consists of finding the vector potential A due to a source J. From the knowledge of A, E and H can be determined. Review spherical coordinates, gradient, divergence, curl, and laplacian in spherical coordinates. (Textbook, Appendix A pp ). In spherical coordinates, the solution for the vector potential A(r) is given by jβ r r' J(r')e A(r) =µ dv' () V' r r' In the above equation, V' is the volume defining the source over which the integration is performed. r is the vector from the origin of the coordinate system, O, to the observer and r' is the vector from O to a source point within V'. r' r -r' r observer O V' source Figure From A, use Maxwell's Equations to derive E and H. B = A () E = H ()
3 3 Hertzian Dipole Assume a source point infinitely small in size located at the origin of the coordinate system with elemental length dl, and driven by a current with strength I o in the +z direction. The equation for the current density of such a system is given by Upon substitution in () J(r') = i z I o dlδ(x')δ(y')δ(z') (3) A(r) = i z µi odl exp(-jβr) (4) In spherical coordinates: i z = i r cosθ - i θ sinθ A(r) = (i r cosθ - i θ sinθ) µi odl exp(-jβr) (5) Calculate E and H fields. A r = µ I odl cosθ exp(-jβr) (6) A θ = - µ I odl sinθ exp(-jβr) (7) Using E = H = µ A (8) H r = r sinθ θ (A sin θ) A θ φ = (9) φ H θ = A r r sin θ φ r (ra ) φ = () H φ = r r (ra θ) A r θ () H φ = I odl sinθ exp(-jβr) jβ(+ jβr ) () H, we can derive the components of the electric field.
4 4 = r sinθ = cosθi odl θ (H φ sinθ) jβr e jβr { }jβ + = r r (rh φ) = jβi odl sinθ{ e jβr } jβ + jβr β r (3) (4) (5) (6) In summary, the exact solutions for the fields of an infinitesimal antenna are given by = jβi odl = jβi odl E φ = e jβr cosθ (jβr) jβr + e jβr sin θ + jβr + (jβr) µ ε µ ε (7) (8) H φ = jβi odl e jβr sinθ + (9) jβr H r = H θ = In most practical cases, the observer is located several wavelengths away from the source. This defines a far field region which is the region where the distance from the source to the observer is much larger than the wavelength λ = π β. In this case r >> λ so that βr >>; consequently, the terms varying as /r and /r 3 can be neglected. The far-field solutions for the infinitely small antenna thus become = (3) = µ ε jβi o dl e jβr sinθ (3) H φ = jβi o dl e jβr sinθ (3)
5 5 Note that the ratio /H φ is the characteristic impedance η of the propagation medium. Over a small region, the far field solution is a plane-wave solution since the electric and magnetic fields are in phase, perpendicular to each other and their ratio is the intrinsic impedance η, and are perpendicular to the direction of propagation. However, unlike plane waves, the far field solution is a function of the elevation angle θ, and does not have constant magnitude (/r dependence). Radiation Patterns The graph that describes the far-field strength versus the elevation angle at a fixed distance is called the radiation pattern of the antenna. In general, radiation patterns vary with θ and φ. The distance from the dipole to a point on the radiation pattern is proportional to the field intensity or power density observed in that direction. Figure shows the E-field and power density radiation patterns of a Hertzian dipole. As can be verified these patterns are based on the sinθ and sin θ dependence of the E-field and power density respectively. z θ z θ (a) Figure. (a) Radiation pattern for E field (b) (b) Radiation pattern for power density Time Average Power in Radiation Zone In order to calculate the power radiated in the far field, we need to determine the timeaverage Poynting vector or power density <P>. < P > = Real [ E H* ] = i r Real µ ε H φ (33)
6 6 < P > = i r η Real βi o dl sin θ (34) The total power radiated P T at a distance r is by definition obtained by integrating the Poynting vector over a sphere of radius r. Total Power = P T = π < P > ds (35) π ds is the elemental surface of radius r and is given by so that ds = i r r sin θ dθ dφ (36) P T = π π η βi o dl r sin 3 θ dθ dφ (37) P T = η βi o dl π π sin 3 θ dθ (38) P T = η 3 βi o dl =π η 3 dl I λ o (39) The directive gain is a figure of merit defined as Directive Gain = Poynting power density AveragePoynting power density over area of sphere of radius r or Directive Gain = < P > P T / (4) For an infinitesimal antenna, we get Directive Gain = η βi o dl η 3 sin θ βi o dl = 3 sin θ (4)
7 7 The directivity is the directive gain in the direction of its maximum value. For an infinitesimal antenna, the direction of maximum value is for θ = π/ and the directivity is.5. Radiation Resistance The radiation resistance of an antenna is defined as the value of the resistor that would dissipate an equal amount of power than the power radiated for the same value of current. Using P T = R rad I o (4) We get For an infinitesimal antenna, we get R rad = P T I o (43) R rad = I o 3 η βi dl o (44) Using β = π λ, and η = π ohms, we obtain R rad = 8π dl λ Ω (45)
ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. D = εe. For a linear, homogeneous, isotropic medium µ and ε are contant.
ANTNNAS Vecto and Scala Potentials Maxwell's quations jωb J + jωd D ρ B (M) (M) (M3) (M4) D ε B Fo a linea, homogeneous, isotopic medium and ε ae contant. Since B, thee exists a vecto A such that B A and
More informationD. S. Weile Radiation
Radiation Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Radiation Outline Outline Maxwell Redux Maxwell s Equation s are: 1 E = jωb = jωµh 2 H = J +
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More information1-2 Vector and scalar potentials
1-2 Vector and scalar potentials z a f a r q a q J y x f Maxwell's equations μ (1) ε (2) ρ (3) (4) - Derivation of E and H field by source current J Approach 1 : take (1) and plug into (2) μ ε μ μ (5)
More informationELE 3310 Tutorial 10. Maxwell s Equations & Plane Waves
ELE 3310 Tutorial 10 Mawell s Equations & Plane Waves Mawell s Equations Differential Form Integral Form Faraday s law Ampere s law Gauss s law No isolated magnetic charge E H D B B D J + ρ 0 C C E r dl
More informationRadiation Integrals and Auxiliary Potential Functions
Radiation Integrals and Auxiliary Potential Functions Ranga Rodrigo June 23, 2010 Lecture notes are fully based on Balanis [?]. Some diagrams and text are directly from the books. Contents 1 The Vector
More informationLinear Wire Antennas. EE-4382/ Antenna Engineering
EE-4382/5306 - Antenna Engineering Outline Introduction Infinitesimal Dipole Small Dipole Finite Length Dipole Half-Wave Dipole Ground Effect Constantine A. Balanis, Antenna Theory: Analysis and Design
More informationECE357H1S ELECTROMAGNETIC FIELDS TERM TEST March 2016, 18:00 19:00. Examiner: Prof. Sean V. Hum
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 2 21 March 2016, 18:00
More informationBasics of Electromagnetics Maxwell s Equations (Part - I)
Basics of Electromagnetics Maxwell s Equations (Part - I) Soln. 1. C A. dl = C. d S [GATE 1994: 1 Mark] A. dl = A. da using Stoke s Theorem = S A. ds 2. The electric field strength at distant point, P,
More informationAntennas and Propagation. Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation : Basic Electromagnetic Analysis Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2 Antenna Theory
More informationLinear Wire Antennas
Linear Wire Antennas Ranga Rodrigo August 4, 010 Lecture notes are fully based on Balanis?. Some diagrams and text are directly from the books. Contents 1 Infinitesimal Dipole 1 Small Dipole 7 3 Finite-Length
More information3.4-7 First check to see if the loop is indeed electromagnetically small. Ie sinθ ˆφ H* = 2. ˆrr 2 sinθ dθ dφ =
ECE 54/4 Spring 17 Assignment.4-7 First check to see if the loop is indeed electromagnetically small f 1 MHz c 1 8 m/s b.5 m λ = c f m b m Yup. (a) You are welcome to use equation (-5), but I don t like
More informationEECS 117. Lecture 22: Poynting s Theorem and Normal Incidence. Prof. Niknejad. University of California, Berkeley
University of California, Berkeley EECS 117 Lecture 22 p. 1/2 EECS 117 Lecture 22: Poynting s Theorem and Normal Incidence Prof. Niknejad University of California, Berkeley University of California, Berkeley
More informationINSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad Electronics and Communicaton Engineering
INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad - 00 04 Electronics and Communicaton Engineering Question Bank Course Name : Electromagnetic Theory and Transmission Lines (EMTL) Course Code :
More informationElectromagnetic Theorems
Electromagnetic Theorems Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Electromagnetic Theorems Outline Outline Duality The Main Idea Electric Sources
More informationAntenna Theory Exam No. 1 October 9, 2000
ntenna Theory Exa No. 1 October 9, 000 Solve the following 4 probles. Each proble is 0% of the grade. To receive full credit, you ust show all work. If you need to assue anything, state your assuptions
More informationr r 1 r r 1 2 = q 1 p = qd and it points from the negative charge to the positive charge.
MP204, Important Equations page 1 Below is a list of important equations that we meet in our study of Electromagnetism in the MP204 module. For your exam, you are expected to understand all of these, and
More informationElectromagnetic Waves
Electromagnetic Waves Maxwell s equations predict the propagation of electromagnetic energy away from time-varying sources (current and charge) in the form of waves. Consider a linear, homogeneous, isotropic
More informationTECHNO INDIA BATANAGAR
TECHNO INDIA BATANAGAR ( DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING) QUESTION BANK- 2018 1.Vector Calculus Assistant Professor 9432183958.mukherjee@tib.edu.in 1. When the operator operates on
More informationECE 107: Electromagnetism
ECE 107: Electromagnetism Set 7: Dynamic fields Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA 92093 1 Maxwell s equations Maxwell
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 16
ECE 6345 Spring 5 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we calculate the power radiated into space by the circular patch. This will lead to Q sp of the circular patch.
More informationtoroidal iron core compass switch battery secondary coil primary coil
Fundamental Laws of Electrostatics Integral form Differential form d l C S E 0 E 0 D d s V q ev dv D ε E D qev 1 Fundamental Laws of Magnetostatics Integral form Differential form C S dl S J d s B d s
More informationTopic 7. Electric flux Gauss s Law Divergence of E Application of Gauss Law Curl of E
Topic 7 Electric flux Gauss s Law Divergence of E Application of Gauss Law Curl of E urface enclosing an electric dipole. urface enclosing charges 2q and q. Electric flux Flux density : The number of field
More informationUniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation
Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume
More informationMagnetostatics III Magnetic Vector Potential (Griffiths Chapter 5: Section 4)
Dr. Alain Brizard Electromagnetic Theory I PY ) Magnetostatics III Magnetic Vector Potential Griffiths Chapter 5: Section ) Vector Potential The magnetic field B was written previously as Br) = Ar), 1)
More information1 Electromagnetic concepts useful for radar applications
Electromagnetic concepts useful for radar applications The scattering of electromagnetic waves by precipitation particles and their propagation through precipitation media are of fundamental importance
More information( z) ( ) ( )( ) ω ω. Wave equation. Transmission line formulas. = v. Helmholtz equation. Exponential Equation. Trig Formulas = Γ. cos sin 1 1+Γ = VSWR
Wave equation 1 u tu v u(, t f ( vt + g( + vt Helmholt equation U + ku jk U Ae + Be + jk Eponential Equation γ e + e + γ + γ Trig Formulas sin( + y sin cos y+ sin y cos cos( + y cos cos y sin sin y + cos
More informationElectromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used
Electromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used B( t) E = dt D t H = J+ t D =ρ B = 0 D=εE B=µ H () F
More informationTheory of Electromagnetic Fields
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK Abstract We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to
More informationSmall Antennas and Some Antenna Parameters
Unit 2 Small Antennas and Some Antenna Parameters 2.1 The Fundamental Source of Radiated Energy We have seen from Maxwell s equations that a time-varying B field (or H field) implies a time-varying E field
More informationA Dash of Maxwell s. In Part 4, we derived our third form of Maxwell s. A Maxwell s Equations Primer. Part 5 Radiation from a Small Wire Element
A Dash of Maxwell s by Glen Dash Ampyx LLC A Maxwell s Equations Primer Part 5 Radiation from a Small Wire Element In Part 4, we derived our third form of Maxwell s Equations, which we called the computational
More informationLinear Wire Antennas Dipoles and Monopoles
inear Wire Antennas Dipoles and Monopoles The dipole and the monopole are arguably the two most widely used antennas across the UHF, VHF and lower-microwave bands. Arrays of dipoles are commonly used as
More information1 Chapter 8 Maxwell s Equations
Electromagnetic Waves ECEN 3410 Prof. Wagner Final Review Questions 1 Chapter 8 Maxwell s Equations 1. Describe the integral form of charge conservation within a volume V through a surface S, and give
More informationEELE 3331 Electromagnetic I Chapter 3. Vector Calculus. Islamic University of Gaza Electrical Engineering Department Dr.
EELE 3331 Electromagnetic I Chapter 3 Vector Calculus Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 2012 1 Differential Length, Area, and Volume This chapter deals with integration
More informationCoordinates 2D and 3D Gauss & Stokes Theorems
Coordinates 2 and 3 Gauss & Stokes Theorems Yi-Zen Chu 1 2 imensions In 2 dimensions, we may use Cartesian coordinates r = (x, y) and the associated infinitesimal area We may also employ polar coordinates
More informationECE 546 Lecture 02 Review of Electromagnetics
C 546 Lecture 0 Review f lectrmagnetics Spring 018 Jse. Schutt-Aine lectrical & Cmputer ngineering University f Illinis jesa@illinis.edu C 546 Jse Schutt Aine 1 Printed Circuit Bard C 546 Jse Schutt Aine
More informationSolutions: Homework 5
Ex. 5.1: Capacitor Solutions: Homework 5 (a) Consider a parallel plate capacitor with large circular plates, radius a, a distance d apart, with a d. Choose cylindrical coordinates (r,φ,z) and let the z
More informationIII. Spherical Waves and Radiation
III. Spherical Waves and Radiation Antennas radiate spherical waves into free space Receiving antennas, reciprocity, path gain and path loss Noise as a limit to reception Ray model for antennas above a
More informationELE3310: Basic ElectroMagnetic Theory
A summary for the final examination EE Department The Chinese University of Hong Kong November 2008 Outline Mathematics 1 Mathematics Vectors and products Differential operators Integrals 2 Integral expressions
More informationxy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.
Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density
More informationCBE 6333, R. Levicky 1. Orthogonal Curvilinear Coordinates
CBE 6333, R. Levicky 1 Orthogonal Curvilinear Coordinates Introduction. Rectangular Cartesian coordinates are convenient when solving problems in which the geometry of a problem is well described by the
More informationLecture notes for ELECTRODYNAMICS.
Lecture notes for 640-343 ELECTRODYNAMICS. 1 Summary of Electrostatics 1.1 Coulomb s Law Force between two point charges F 12 = 1 4πɛ 0 Q 1 Q 2ˆr 12 r 1 r 2 2 (1.1.1) 1.2 Electric Field For a charge distribution:
More informationFundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis
Fundamentals of pplied Electromagnetics Chapter - Vector nalsis Chapter Objectives Operations of vector algebra Dot product of two vectors Differential functions in vector calculus Divergence of a vector
More informationNotes 3 Review of Vector Calculus
ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present
More informationThe most fundamental antenna is the incremental dipole as pictured in Figure 1. a Z. I o δh. a X. Figure 1. Incremental dipole
. Chapter 13 Antennas Features Used crossp( ), dotp( ), real( ), conj( ), Í, NewProb,, Polar graphs Setup 1 NewFold ant setmode("complex Format", "Polar") This chapter describes how to perform basic antenna
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 17
ECE 634 Intermediate EM Waves Fall 16 Prof. David R. Jacson Dept. of ECE Notes 17 1 General Plane Waves General form of plane wave: E( xz,, ) = Eψ ( xz,, ) where ψ ( xz,, ) = e j( xx+ + zz) The wavenumber
More informationModern Physics. Unit 6: Hydrogen Atom - Radiation Lecture 6.3: Vector Model of Angular Momentum
Modern Physics Unit 6: Hydrogen Atom - Radiation ecture 6.3: Vector Model of Angular Momentum Ron Reifenberger Professor of Physics Purdue University 1 Summary of Important Points from ast ecture The magnitude
More informationYell if you have any questions
Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 efore Starting All of your grades should now be posted
More informationElectromagnetic Waves Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space
Electromagnetic Waves 1 1. Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space 1 Retarded Potentials For volume charge & current = 1 4πε
More informationMIDSUMMER EXAMINATIONS 2001
No. of Pages: 7 No. of Questions: 10 MIDSUMMER EXAMINATIONS 2001 Subject PHYSICS, PHYSICS WITH ASTROPHYSICS, PHYSICS WITH SPACE SCIENCE & TECHNOLOGY, PHYSICS WITH MEDICAL PHYSICS Title of Paper MODULE
More informationElectromagnetism. Christopher R Prior. ASTeC Intense Beams Group Rutherford Appleton Laboratory
lectromagnetism Christopher R Prior Fellow and Tutor in Mathematics Trinity College, Oxford ASTeC Intense Beams Group Rutherford Appleton Laboratory Contents Review of Maxwell s equations and Lorentz Force
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 1
EE 6340 Intermediate EM Waves Fall 2016 Prof. David R. Jackson Dept. of EE Notes 1 1 Maxwell s Equations E D rt 2, V/m, rt, Wb/m T ( ) [ ] ( ) ( ) 2 rt, /m, H ( rt, ) [ A/m] B E = t (Faraday's Law) D H
More informationYell if you have any questions
Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 Before Starting All of your grades should now be posted
More informationELECTROMAGNETIC WAVE PROPAGATION EC 442. Prof. Darwish Abdel Aziz
ELECTROMAGNETIC WAVE PROPAGATION EC 442 Prof. Darwish Abdel Aziz CHAPTER 6 LINEAR WIRE ANTENNAS INFINITESIMAL DIPOLE INTRODUCTION Wire antennas, linear or curved, are some of the oldest, simplest, cheapest,
More informationEECS 117 Lecture 19: Faraday s Law and Maxwell s Eq.
University of California, Berkeley EECS 117 Lecture 19 p. 1/2 EECS 117 Lecture 19: Faraday s Law and Maxwell s Eq. Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS
More informationShort Wire Antennas: A Simplified Approach Part I: Scaling Arguments. Dan Dobkin version 1.0 July 8, 2005
Short Wire Antennas: A Simplified Approach Part I: Scaling Arguments Dan Dobkin version 1.0 July 8, 2005 0. Introduction: How does a wire dipole antenna work? How do we find the resistance and the reactance?
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationIntroduction to Electromagnetic Theory
Introduction to Electromagnetic Theory Lecture topics Laws of magnetism and electricity Meaning of Maxwell s equations Solution of Maxwell s equations Electromagnetic radiation: wave model James Clerk
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More informationAperture Antennas 1 Introduction
1 Introduction Very often, we have antennas in aperture forms, for example, the antennas shown below: Pyramidal horn antenna Conical horn antenna 1 Paraboloidal antenna Slot antenna Analysis Method for.1
More informationPhys 122 Lecture 3 G. Rybka
Phys 122 Lecture 3 G. Rybka A few more Demos Electric Field Lines Example Calculations: Discrete: Electric Dipole Overview Continuous: Infinite Line of Charge Next week Labs and Tutorials begin Electric
More informationBasics of Wave Propagation
Basics of Wave Propagation S. R. Zinka zinka@hyderabad.bits-pilani.ac.in Department of Electrical & Electronics Engineering BITS Pilani, Hyderbad Campus May 7, 2015 Outline 1 Time Harmonic Fields 2 Helmholtz
More informationPart IB Electromagnetism
Part IB Electromagnetism Theorems Based on lectures by D. Tong Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationMath 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
More informationElectromagnetic optics!
1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals
More informationUnit-1 Electrostatics-1
1. Describe about Co-ordinate Systems. Co-ordinate Systems Unit-1 Electrostatics-1 In order to describe the spatial variations of the quantities, we require using appropriate coordinate system. A point
More informationAppendix: Orthogonal Curvilinear Coordinates. We define the infinitesimal spatial displacement vector dx in a given orthogonal coordinate system with
Appendix: Orthogonal Curvilinear Coordinates Notes: Most of the material presented in this chapter is taken from Anupam G (Classical Electromagnetism in a Nutshell 2012 (Princeton: New Jersey)) Chap 2
More informationPhysics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves
Physics 331 Lecture 9 February 13, 019 LAST TIME: Finished mirrors and aberrations, more on plane waves Recall, Represents a plane wave having a propagation vector k that propagates in any direction with
More information2nd Year Electromagnetism 2012:.Exam Practice
2nd Year Electromagnetism 2012:.Exam Practice These are sample questions of the type of question that will be set in the exam. They haven t been checked the way exam questions are checked so there may
More informationPlane Wave: Introduction
Plane Wave: Introduction According to Mawell s equations a timevarying electric field produces a time-varying magnetic field and conversely a time-varying magnetic field produces an electric field ( i.e.
More informationTime-harmonic form Phasor form. =0 (1.11d)
Chapter 2 Wave in an Unbounded Medium Maxwell s Equations Time-harmonic form Phasor form (Real quantity) (complex quantity) B E = Eˆ = jωbˆ (1.11 a) t D= ρ Dˆ = ρ (1.11 b) D H = J + Hˆ = Jˆ+ jωdˆ ( 1.11
More informationChapter 2 Basics of Electricity and Magnetism
Chapter 2 Basics of Electricity and Magnetism My direct path to the special theory of relativity was mainly determined by the conviction that the electromotive force induced in a conductor moving in a
More informationUniform Plane Waves. Ranga Rodrigo. University of Moratuwa. November 7, 2008
Uniform Plane Waves Ranga Rodrigo University of Moratuwa November 7, 2008 Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, 2008 1 / 51 Summary of Last Week s Lecture Basic Relations
More informationTime-Varying Systems; Maxwell s Equations
Time-Varying Systems; Maxwell s Equations 1. Faraday s law in differential form 2. Scalar and vector potentials; the Lorenz condition 3. Ampere s law with displacement current 4. Maxwell s equations 5.
More informationr,t r R Z j ³ 0 1 4π² 0 r,t) = 4π
5.4 Lienard-Wiechert Potential and Consequent Fields 5.4.1 Potential and Fields (chapter 10) Lienard-Wiechert potential In the previous section, we studied the radiation from an electric dipole, a λ/2
More informationMP204 Electricity and Magnetism
MATHEMATICAL PHYSICS SEMESTER 2, REPEAT 2016 2017 MP204 Electricity and Magnetism Prof. S. J. Hands, Dr. M. Haque and Dr. J.-I. Skullerud Time allowed: 1 1 2 hours Answer ALL questions MP204, 2016 2017,
More informationPhysics Lecture 07
Physics 2113 Jonathan Dowling Physics 2113 Lecture 07 Electric Fields III Charles-Augustin de Coulomb (1736-1806) Electric Charges and Fields First: Given Electric Charges, We Calculate the Electric Field
More informationPhysics Lecture 40: FRI3 DEC
Physics 3 Physics 3 Lecture 4: FRI3 DEC Review of concepts for the final exam Electric Fields Electric field E at some point in space is defined as the force divided by the electric charge. Force on charge
More information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics
More informationLecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics Nick Fang
Outline: A. Electromagnetism B. Frequency Domain (Fourier transform) C. EM waves in Cartesian coordinates D. Energy Flow and Poynting Vector E. Connection to geometrical optics F. Eikonal Equations: Path
More informationChap. 1 Fundamental Concepts
NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays
More information5 Electromagnetic Waves
5 Electromagnetic Waves 5.1 General Form for Electromagnetic Waves. In free space, Maxwell s equations are: E ρ ɛ 0 (5.1.1) E + B 0 (5.1.) B 0 (5.1.3) B µ 0 ɛ 0 E µ 0 J (5.1.4) In section 4.3 we derived
More informationremain essentially unchanged for the case of time-varying fields, the remaining two
Unit 2 Maxwell s Equations Time-Varying Form While the Gauss law forms for the static electric and steady magnetic field equations remain essentially unchanged for the case of time-varying fields, the
More informationReflection/Refraction
Reflection/Refraction Page Reflection/Refraction Boundary Conditions Interfaces between different media imposed special boundary conditions on Maxwell s equations. It is important to understand what restrictions
More informationA Review of Radiation and Optics
A Review of Radiation and Optics Abraham Asfaw 12 aasfaw.student@manhattan.edu May 20, 2011 Abstract This paper attempts to summarize selected topics in Radiation and Optics. It is, by no means, a complete
More informationAntennas and Propagation
Antennas and Propagation Ranga Rodrigo University of Moratuwa October 20, 2008 Compiled based on Lectures of Prof. (Mrs.) Indra Dayawansa. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation
More informationE&M. 1 Capacitors. January 2009
E&M January 2009 1 Capacitors Consider a spherical capacitor which has the space between its plates filled with a dielectric of permittivity ɛ. The inner sphere has radius r 1 and the outer sphere has
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationAntenna Theory (Engineering 9816) Course Notes. Winter 2016
Antenna Theory (Engineering 9816) Course Notes Winter 2016 by E.W. Gill, Ph.D., P.Eng. Unit 1 Electromagnetics Review (Mostly) 1.1 Introduction Antennas act as transducers associated with the region of
More informationWorked Examples Set 2
Worked Examples Set 2 Q.1. Application of Maxwell s eqns. [Griffiths Problem 7.42] In a perfect conductor the conductivity σ is infinite, so from Ohm s law J = σe, E = 0. Any net charge must be on the
More informationLECTURE 18: Horn Antennas (Rectangular horn antennas. Circular apertures.)
LCTUR 18: Horn Antennas (Rectangular horn antennas. Circular apertures.) 1 Rectangular Horn Antennas Horn antennas are popular in the microwave bands (above 1 GHz). Horns provide high gain, low VSWR (with
More informationPoynting Vector and Energy Flow W14D1
Poynting Vector and Energy Flow W14D1 1 Announcements Week 14 Prepset due online Friday 8:30 am PS 11 due Week 14 Friday at 9 pm in boxes outside 26-152 Sunday Tutoring 1-5 pm in 26-152 2 Outline Poynting
More informationSummary: Curvilinear Coordinates
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10 1 Summary: Curvilinear Coordinates 1. Summary of Integral Theorems 2. Generalized Coordinates 3. Cartesian Coordinates: Surfaces of Constant
More informationELECTRICITY AND MAGNETISM
THIRD EDITION ELECTRICITY AND MAGNETISM EDWARD M. PURCELL DAVID J. MORIN Harvard University, Massachusetts Щ CAMBRIDGE Ell UNIVERSITY PRESS Preface to the third edition of Volume 2 XIII CONTENTS Preface
More informationElectromagnetic Field Theory 1 (fundamental relations and definitions)
(fundamental relations and definitions) Lukas Jelinek lukas.jelinek@fel.cvut.cz Department of Electromagnetic Field Czech Technical University in Prague Czech Republic Ver. 216/12/14 Fundamental Question
More information3.2 Numerical Methods for Antenna Analysis
ECEn 665: Antennas and Propagation for Wireless Communications 43 3.2 Numerical Methods for Antenna Analysis The sinusoidal current model for a dipole antenna is convenient because antenna parameters can
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : ELECTROMAGNETIC FIELDS SUBJECT CODE : EC 2253 YEAR / SEMESTER : II / IV UNIT- I - STATIC ELECTRIC
More informationExam 4 Solutions. a. 1,2,and 3 b. 1 and 2, not 3 c. 1 and 3, not 2 d. 2 and 3, not 1 e. only 2
Prof. Darin Acosta Prof. Greg Stewart April 8, 007 1. Which of the following statements is true? 1. In equilibrium all of any excess charge stored on a conductor is on the outer surface.. In equilibrium
More information