ECE 546 – Jose Schutt-Aine1 ECE 546 Lecture 02 Review of Electromagnetics Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois

ECE 546 – Jose Schutt-Aine2 Electromagnetic Quantities Electric field (Volts/m) Magnetic field (Amperes/m) Electric flux density (Coulombs/m 2 ) Magnetic flux density (Webers/m 2 ) Current density (Amperes/m 2 ) Charge density (Coulombs/m 2 )

ECE 546 – Jose Schutt-Aine3 Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field Maxwell’s Equations

ECE 546 – Jose Schutt-Aine4 Constitutive Relations Permittivity  : Farads/m Permeability  : Henries/m Free Space

ECE 546 – Jose Schutt-Aine5 Continuity Equation

ECE 546 – Jose Schutt-Aine6 Electrostatics Assume no time dependence  Poisson’s Equation  Laplace’s Equationif no charge is present

ECE 546 – Jose Schutt-Aine7 Integral Form of ME

ECE 546 – Jose Schutt-Aine8 Boundary Conditions

ECE 546 – Jose Schutt-Aine9 Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field Free Space Solution

ECE 546 – Jose Schutt-Aine10 Wave Equation  can show that

ECE 546 – Jose Schutt-Aine11 Wave Equation separating the components

ECE 546 – Jose Schutt-Aine12 Wave Equation  Plane Wave (a) Assume that only E x exists  E y =E z =0 (b) Only z spatial dependence  This situation leads to the plane wave solution In addition, assume a time-harmonic dependence then

ECE 546 – Jose Schutt-Aine13 Plane Wave Solution solution where In the time domain propagation constant solution

ECE 546 – Jose Schutt-Aine14 Plane Wave Characteristics where In free space propagation constant

ECE 546 – Jose Schutt-Aine15 Solution for Magnetic Field If we assume that then If we assume that then intrinsic impedance of medium

ECE 546 – Jose Schutt-Aine16 Time-Average Poynting Vector We can show that Poynting vector W/m 2 time-average Poynting vector W/m 2

ECE 546 – Jose Schutt-Aine17  : conductivity of material medium (  -1 m -1 ) Material Medium or sincethen

ECE 546 – Jose Schutt-Aine18 Wave in Material Medium  is complex propagation constant  associated with attenuation of wave  associated with propagation of wave

ECE 546 – Jose Schutt-Aine19 Wave in Material Medium decaying exponential Solution: Magnetic field Complex intrinsic impedance

ECE 546 – Jose Schutt-Aine20 Wave in Material Medium Phase Velocity: 1. Perfect dielectric Special Cases Wavelength: air, free space and

ECE 546 – Jose Schutt-Aine21 Wave in Material Medium 2. Lossy dielectric Loss tangent:

ECE 546 – Jose Schutt-Aine22 Wave in Material Medium 3. Good conductors Loss tangent:

ECE 546 – Jose Schutt-Aine23  attenuation  propagation  dpdp H, EExamples PEC-000supercond Good conductor finite copper Poor conductorfinite Ice Perfect dielectric 0 finite air Material Medium

ECE 546 – Jose Schutt-Aine24 Radiation - Vector Potential (1) (2) (3) (4) Assume time harmonicity ~

ECE 546 – Jose Schutt-Aine25 Radiation - Vector Potential : vector potential Using the property:

ECE 546 – Jose Schutt-Aine26 Since a vector is uniquely defined by its curl and its divergence, we can choose the divergence of A Lorentz condition Vector Potential

ECE 546 – Jose Schutt-Aine27 D’Alembert’s equation Vector Potential

ECE 546 – Jose Schutt-Aine28 From A, get E and H using Maxwell’s equations Three-dimensional free-space Green’s function Vector potential Vector Potential

ECE 546 – Jose Schutt-Aine29 For infinitesimal antenna, the current density is: Calculating the vector potential, In spherical coordinates, Vector Potential

ECE 546 – Jose Schutt-Aine30 Resolving into components, Vector Potential

ECE 546 – Jose Schutt-Aine31 Calculate E and H fields E and H Fields

ECE 546 – Jose Schutt-Aine32 E and H Fields

ECE 546 – Jose Schutt-Aine33 E and H Fields

ECE 546 – Jose Schutt-Aine34 E and H Fields

ECE 546 – Jose Schutt-Aine35 Note that: Far Field Approximation

ECE 546 – Jose Schutt-Aine36 Uniform constant phase locus is a plane Constant magnitude Independent of  Does not decay Characteristics of plane waves Similarities between infinitesimal antenna far field radiated and plane wave (a)E and H are perpendicular (b)E and H are related by  (c)E is perpendicular to H Far Field Approximation

ECE 546 – Jose Schutt-Aine37 Time-average Poynting vector or TA power density E and H here are PHASORS Poynting Vector

ECE 546 – Jose Schutt-Aine38 Total power radiated (time-average) Time-Average Power

ECE 546 – Jose Schutt-Aine39 Time-Average Power

ECE 546 – Jose Schutt-Aine40 For infinitesimal antenna, Directivity

ECE 546 – Jose Schutt-Aine41 Directivity: gain in direction of maximum value Radiation resistance: From we have: For infinitesimal antenna: Directivity

ECE 546 – Jose Schutt-Aine42 For free space, The radiation resistance of an antenna is the value of a fictitious resistance that would dissipate an amount of power equal to the radiated power P r when the current in the resistance is equal to the maximum current along the antenna (for Hertzian dipole) A high radiation resistance is a desirable property for an antenna Radiation Resistance