ENE 325 Electromagnetic Fields and Waves Lecture 11 Uniform Plane Waves 17/01/62

Introduction From Maxwell’s equations, if the electric field http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52 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. 17/01/62

Maxwell’s equations (1) (2) (3) (4) 17/01/62

Maxwell’s equations in free space  = 0, r = 1, r = 1 Ampère’s law Faraday’s law 17/01/62

General wave equations Consider medium free of charge where For linear, isotropic, homogeneous, and time-invariant medium, (1) (2) 17/01/62

General wave equations Take curl of (2), we yield From then For charge free medium 17/01/62

Helmholtz wave equation For electric field For magnetic field 17/01/62

Time-harmonic wave equations Transformation from time to frequency domain Therefore 17/01/62

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) 17/01/62

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 17/01/62

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(ejtphasor field) 17/01/62

Intrinsic impedance For any medium, For free space 17/01/62

Propagating fields relation where represents a direction of propagation 17/01/62

Propagation in lossless-charge free media Attenuation constant  = 0, conductivity  = 0 Propagation constant Propagation velocity for free space up = 3108 m/s (speed of light) for non-magnetic lossless dielectric (r = 1), 17/01/62

Propagation in lossless-charge free media intrinsic impedance Wavelength 17/01/62

Ex1 A 9.375 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 17/01/62

c) propagation velocity d) intrinsic impedance e) amplitude of the magnetic field intensity 17/01/62

Propagation in dielectrics Cause finite conductivity polarization loss ( = ’-j” ) Assume homogeneous and isotropic medium 17/01/62

Propagation in dielectrics Define from and 17/01/62

Propagation in dielectrics We can derive and 17/01/62

Loss tangent A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor 17/01/62

Low loss material or a good dielectric (tan « 1) If , consider the material ‘low loss’ , then and 17/01/62

Low loss material or a good dielectric (tan « 1) propagation velocity wavelength 17/01/62

High loss material or a good conductor (tan » 1) In this case , we can approximate therefore and 17/01/62

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 0.368 times of the initial field propagation velocity wavelength 17/01/62

b) attenuation constant  Ex2 Given a nonmagnetic material having r = 3.2 and  = 1.510-4 S/m, at f = 3 MHz, find a) loss tangent  b) attenuation constant  17/01/62

d) intrinsic impedance c) phase constant  d) intrinsic impedance 17/01/62

Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity,  = 5.8107 S/m: a) wavelength b) propagation velocity 17/01/62

c) compare these answers with the same wave propagating in a free space 17/01/62