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RS ENE 428 Microwave Engineering Lecture 2 Uniform plane waves

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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),

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RS Propagation in lossless-charge free media intrinsic impedance wavelength

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RS 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 = 295 rad/m = 2.13 cm

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RS c) propagation velocity d) Intrinsic impedance e) Amplitude of the magnetic field intensity v = 2x10 8 m/s = 250.77 H = 1.99 A/m

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

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RS Propagation in dielectrics From and

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RS Propagation in dielectrics We can derive and

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

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

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

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

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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 0.368 times of the initial field propagation velocity wavelength

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RS Ex2 Given a nonmagnetic material having r = 3.2 and = 1.5 10 -4 S/m, at f = 30 MHz, find a) loss tangent b) attenuation constant tan = 0.03 = 0.016 Np/m

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RS c) phase constant d) intrinsic impedance = 1.12 rad/m = 210.74(1+j0.015)

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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 = 117.21 rad/m = 5.36 cm v = 3.22 m/s

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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

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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/ =

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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

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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

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

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

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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

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

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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

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RS c) The dc resistance

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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

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

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

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

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RS Example of Poynting theorem in DC case Total power W

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

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

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