ENE 429 Antenna and Transmission lines Theory Lecture 2 Uniform plane waves

Review Wave equations Time-Harmonics equations where

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)

Transverse ElectroMagnetic wave (TEM) http://www.edumedia.fr/a185_l2-transverse-electromagnetic-wave.html

Solutions of Helmholtz equations The instantaneous forms of the solutions The phasor forms of the solutions incident wave reflected wave

Attenuation constant  Attenuation constant determines the penetration of the wave into a medium Attenuation constant are different for different applications The penetration depth or skin depth,  is the distance z that causes to reduce to z = 1  z = 1/  = 

Good conductor At high operation frequency, skin depth decreases A magnetic material is not suitable for signal carrier A high conductivity material has low skin depth

Currents in conductor To understand a concept of sheet resistance from Rsheet () sheet resistance At high frequency, it will be adapted to skin effect resistance

Currents in conductor Therefore the current that flows through the slab at t   is

Currents in conductor From Jx or current density decreases as the slab gets thicker

Currents in conductor For distance L in x-direction R is called skin resistance Rskin is called skin-effect resistance For finite thickness,

Currents in conductor Current is confined within a skin depth of the coaxial cable

Ex1 A steel pipe is constructed of a material for which r = 180 and  = 4106 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total current I(t) carried by the pipe is 8cost A, where  = 1200 rad/s, find: The skin depth The skin resistance

c) The dc resistance

The Poynting theorem and power transmission Total power leaving the surface Joule’s law for instantaneous power dissipated per volume (dissi- pated by heat) Rate of change of energy stored In the fields Instantaneous poynting vector

Example of Poynting theorem in DC case Rate of change of energy stored In the fields = 0

Example of Poynting theorem in DC case From By using Ohm’s law,

Example of Poynting theorem in DC case Verify with From Ampère’s circuital law,

Example of Poynting theorem in DC case Total power W

Uniform plane wave (UPW) power transmission Time-averaged power density W/m2 amount of power W for lossless case, W/m2

Uniform plane wave (UPW) power transmission for lossy medium, we can write intrinsic impedance for lossy medium

Uniform plane wave (UPW) power transmission from W/m2 Question: Have you ever wondered why aluminum foil is not allowed in the microwave oven?

Polarization UPW is characterized by its propagation direction and frequency. Its attenuation and phase are determined by medium’s parameters. Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.

Linear polarization Consider in free space, At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave”

Linear polarization A pair of linearly polarized wave also produces linear polarization At z = 0 plane At t = 0, both linearly polarized waves Have their maximum values

More generalized linear polarization More generalized of two linearly poloraized waves, Linear polarization occurs when two linearly polarized waves are in phase out of phase

Elliptically polarized wave Super position of two linearly polarized waves that If x = 0 and y = 45, we have

Circularly polarized wave occurs when Exo and Eyo are equal and Right hand circularly polarized (RHCP) wave Left hand circularly polarized (LHCP) wave

Circularly polarized wave Phasor forms: for RHCP, for LHCP, from Note: There are also RHEP and LHEP

Ex2 Given ,determine the polarization of this wave

Ex3 The electric field of a uniform plane wave in free space is given by , determine The magnetic field intensity

c) d) Describe the polarization of the wave