RS ENE 428 Microwave Engineering Lecture 2 Uniform plane waves.

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RS ENE 428 Microwave Engineering Lecture 2 Uniform plane waves

RS 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),

RS Propagation in lossless-charge free media intrinsic impedance wavelength

RS 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  = 295 rad/m = 2.13 cm

RS c) propagation velocity d) Intrinsic impedance e) Amplitude of the magnetic field intensity v = 2x10 8 m/s  =  H = 1.99 A/m

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

RS Propagation in dielectrics From and

RS Propagation in dielectrics We can derive and

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

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

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

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

RS 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

RS Ex2 Given a nonmagnetic material having  r = 3.2 and  = 1.5  S/m, at f = 30 MHz, find a) loss tangent  b) attenuation constant  tan  = 0.03  = Np/m

RS c) phase constant  d) intrinsic impedance  = 1.12 rad/m  = (1+j0.015) 

RS 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  = rad/m = 5.36 cm v = 3.22 m/s

RS c) compare these answers with the same wave propagating in a free space  = 1.26x10 -6 rad/m = 5000 km v = 3x10 8 m/s

RS 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/  = 

RS 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

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

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

RS Currents in conductor From J x or current density decreases as the slab gets thicker

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

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

RS Ex4 A steel pipe is constructed of a material for which  r = 180 and  = 4  10 6 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  = 7.66x10 -4 m

RS c) The dc resistance

RS The Poynting theorem and power transmission Poynting theorem 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

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

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

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

RS Example of Poynting theorem in DC case Total power W

RS Uniform plane wave (UPW) power transmission Time-averaged power density amount of power for lossless case, W/m 2

RS Uniform plane wave (UPW) power transmission for lossless case, we can write power average in terms of H W/m 2

Uniform plane wave (UPW) power transmission intrinsic impedance for lossy medium for lossy medium, we can write The phasor is Instantaneous magnetic field: which is seen to be out of phase with E(z,t) by the angle  

RS Homework Prob.6.19: In seawater, a propagating electric field is given by V/m Assuming  = 5 S/m,  r ’ = 72,  r ” = 0, find (a)  and β and (b) the instantaneous form of H(z,t) Prob. 6.26: In air,A/m Determine the power passing through a 1.0 m 2 surface that is normal to the direction of propagation.