ECE 546 Lecture 02 Review of Electromagnetics Spring

ECE 546 Lecture 02 Review of Electromagnetics Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jschutt@emlab. uiuc. edu ECE 546 – Jose Schutt-Aine 1

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

Maxwell’s Equations Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field ECE 546 – Jose Schutt-Aine 3

Constitutive Relations Permittivity e: Farads/m Permeability m: Henries/m Free Space ECE 546 – Jose Schutt-Aine 4

Continuity Equation ECE 546 – Jose Schutt-Aine 5

Electrostatics Assume no time dependence Poisson’s Equation if no charge is present Laplace’s Equation ECE 546 – Jose Schutt-Aine 6

Integral Form of ME ECE 546 – Jose Schutt-Aine 7

Boundary Conditions ECE 546 – Jose Schutt-Aine 8

Free Space Solution Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field ECE 546 – Jose Schutt-Aine 9

Wave Equation can show that ECE 546 – Jose Schutt-Aine 10

Wave Equation separating the components ECE 546 – Jose Schutt-Aine 11

Wave Equation Plane Wave (a) Assume that only Ex exists Ey=Ez=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-Aine 12

Plane Wave Solution solution where propagation constant In the time domain solution ECE 546 – Jose Schutt-Aine 13

Plane Wave Characteristics where propagation constant In free space ECE 546 – Jose Schutt-Aine 14

Solution for Magnetic Field If we assume that then intrinsic impedance of medium ECE 546 – Jose Schutt-Aine 15

Time-Average Poynting Vector Poynting vector W/m 2 time-average Poynting vector W/m 2 We can show that ECE 546 – Jose Schutt-Aine 16

Material Medium or s: conductivity of material medium (W-1 m-1) since then ECE 546 – Jose Schutt-Aine 17

Wave in Material Medium g is complex propagation constant a: associated with attenuation of wave b: associated with propagation of wave ECE 546 – Jose Schutt-Aine 18

Wave in Material Medium Solution: decaying exponential Complex intrinsic impedance Magnetic field ECE 546 – Jose Schutt-Aine 19

Wave in Material Medium Phase Velocity: Wavelength: Special Cases 1. Perfect dielectric air, free space and ECE 546 – Jose Schutt-Aine 20

Wave in Material Medium 2. Lossy dielectric Loss tangent: ECE 546 – Jose Schutt-Aine 21

Wave in Material Medium 3. Good conductors Loss tangent: ECE 546 – Jose Schutt-Aine 22

Material Medium a attenuation PEC b propagation - h dp H, E Examples 0 0 0 supercond copper Good conductor finite Poor conductor Perfect dielectric Ice finite air 0 finite ECE 546 – Jose Schutt-Aine 23

Radiation - Vector Potential Assume time harmonicity ~ (1) (2) (3) (4) ECE 546 – Jose Schutt-Aine 24

Radiation - Vector Potential Using the property: : vector potential ECE 546 – Jose Schutt-Aine 25

Vector Potential Since a vector is uniquely defined by its curl and its divergence, we can choose the divergence of A Lorentz condition 26 ECE 546 – Jose Schutt-Aine 26

Vector Potential D’Alembert’s equation 27 ECE 546 – Jose Schutt-Aine 27

Vector Potential Three-dimensional free-space Green’s function Vector potential From A, get E and H using Maxwell’s equations 28 ECE 546 – Jose Schutt-Aine 28

Vector Potential For infinitesimal antenna, the current density is: Calculating the vector potential, In spherical coordinates, 29 ECE 546 – Jose Schutt-Aine 29

Vector Potential Resolving into components, 30 ECE 546 – Jose Schutt-Aine 30

E and H Fields Calculate E and H fields 31 ECE 546 – Jose Schutt-Aine 31

E and H Fields 32 ECE 546 – Jose Schutt-Aine 32

E and H Fields 33 ECE 546 – Jose Schutt-Aine 33

E and H Fields 34 ECE 546 – Jose Schutt-Aine 34

Far Field Approximation Note that: ECE 546 – Jose Schutt-Aine 35

Far Field Approximation Characteristics of plane waves • • Uniform constant phase locus is a plane Constant magnitude Independent of q Does not decay Similarities between infinitesimal antenna far field radiated and plane wave (a) E and H are perpendicular (b) E and H are related by h (c) E is perpendicular to H ECE 546 – Jose Schutt-Aine 36

Poynting Vector Time-average Poynting vector or TA power density E and H here are PHASORS ECE 546 – Jose Schutt-Aine 37

Time-Average Power Total power radiated (time-average) ECE 546 – Jose Schutt-Aine 38

Time-Average Power ECE 546 – Jose Schutt-Aine 39

Directivity For infinitesimal antenna, ECE 546 – Jose Schutt-Aine 40

Directivity: gain in direction of maximum value Radiation resistance: From we have: For infinitesimal antenna: ECE 546 – Jose Schutt-Aine 41

Radiation Resistance For free space, (for Hertzian dipole) 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 Pr when the current in the resistance is equal to the maximum current along the antenna A high radiation resistance is a desirable property for an antenna ECE 546 – Jose Schutt-Aine 42