ENE 429 Antenna and Transmission lines Theory Lecture 1 Uniform plane waves

Syllabus Dr. Rardchawadee Silapunt, Lecture: 1:30pm-4:20pm Monday, CB41002 Office hours :By appointment Textbook: Applied Electromagnetics by Stuart M. Wentworth (Wiley, 2007)

Homework 20% Midterm exam 40% Final exam 40% Grading Vision: Providing opportunities for intellectual growth in the context of an engineering discipline for the attainment of professional competence, and for the development of a sense of the social context of technology.

Course overview Uniform plane waves Transmission lines Waveguides Antennas

Introduction From Maxwell’s equations, if the electric field is changing with time, then the magnetic field varies spatially in a direction normal to its orientation direction A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation Both fields are of constant magnitude in the transverse plane, such a wave is sometimes called a transverse electromagnetic (TEM) wave.

Maxwell’s equations (1) (2) (3) (4)

Maxwell’s equations in free space  = 0,  r = 1,  r = 1 Ampère’s law Faraday’s law

General wave equations Consider medium free of charge where For linear, isotropic, homogeneous, and time-invariant medium, (1) (2)

General wave equations Take curl of (2), we yield From then For charge free medium

Helmholtz wave equation For electric field For magnetic field

Time-harmonic wave equations Transformation from time to frequency domain Therefore

Time-harmonic wave equations or where This  term is called propagation constant or we can write  =  +j  where  = attenuation constant (Np/m)  = phase constant (rad/m)

Solutions of Helmholtz equations Assuming the electric field is in x-direction and the wave is propagating in z- direction The instantaneous form of the solutions Consider only the forward-propagating wave, we have Use Maxwell’s equation, we get

Solutions of Helmholtz equations in phasor form Showing the forward-propagating fields without time- harmonic terms. Conversion between instantaneous and phasor form Instantaneous field = Re(e j  t  phasor field)

Intrinsic impedance For any medium, For free space

Propagating fields relation where represents a direction of propagation

Propagation in lossless-charge free media Attenuation constant  = 0, conductivity  = 0 Propagation constant Propagation velocity  for free space u p = 3  10 8 m/s (speed of light)  for non-magnetic lossless dielectric (  r = 1),

Propagation in lossless-charge free media intrinsic impedance Wavelength

Ex1 A GHz uniform plane wave is propagating in polyethelene (  r = 2.26). If the amplitude of the electric field intensity is 500 V/m and the material is assumed to be lossless, find a) phase constant b) wavelength in the polyethelene

c) propagation velocity d) Intrinsic impedance e) Amplitude of the magnetic field intensity

Propagation in dielectrics Cause  finite conductivity  polarization loss (  =  ’ -j  ” ) Assume homogeneous and isotropic medium

Propagation in dielectrics Define From and

Propagation in dielectrics We can derive and

Loss tangent A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor

Low loss material or a good dielectric (tan  « 1) If or < 0.1, consider the material ‘low loss’, then and

Low loss material or a good dielectric (tan  « 1) propagation velocity wavelength

High loss material or a good conductor (tan  » 1) In this case or > 10, we can approximate therefore and

High loss material or a good conductor (tan  » 1) depth of penetration or skin depth,  is a distance where the field decreases to e -1 or times of the initial field propagation velocity wavelength

Ex2 Given a nonmagnetic material having  r = 3.2 and  = 1.5  S/m, at f = 3 MHz, find a) loss tangent  b) attenuation constant 

c) phase constant  d) intrinsic impedance

Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity,  = 5.8  10 7 S/m: a) wavelength b) propagation velocity

c) compare these answers with the same wave propagating in a free space