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**ENE 428 Microwave Engineering**

Lecture 2 Uniform plane waves RS

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**Propagation in lossless-charge free media**

Attenuation constant = 0, conductivity = 0 Propagation constant Propagation velocity for free space up = 3108 m/s (speed of light) for non-magnetic lossless dielectric (r = 1), RS

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**Propagation in lossless-charge free media**

intrinsic impedance wavelength RS

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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

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**c) propagation velocity**

d) Intrinsic impedance e) Amplitude of the magnetic field intensity v = 2x108 m/s = H = 1.99 A/m RS

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**Propagation in dielectrics**

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

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**Propagation in dielectrics**

From and RS

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**Propagation in dielectrics**

We can derive and RS

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

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**Low loss material or a good dielectric (tan « 1)**

If or < 0.1 , consider the material ‘low loss’ , then and RS

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**Low loss material or a good dielectric (tan « 1)**

propagation velocity wavelength RS

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**High loss material or a good conductor (tan » 1)**

In this case or > 10, we can approximate therefore and RS

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**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

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**Ex2 Given a nonmagnetic material having r = 3. 2 and = 1**

Ex2 Given a nonmagnetic material having r = 3.2 and = 1.510-4 S/m, at f = 30 MHz, find a) loss tangent b) attenuation constant tan = 0.03 = Np/m RS

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**d) intrinsic impedance**

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

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Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity, = 5.8107 S/m: a) wavelength b) propagation velocity = rad/m = 5.36 cm v = 3.22 m/s RS

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

= 1.26x10-6 rad/m = 5000 km v = 3x108 m/s RS

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**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

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**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

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**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 RS

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

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**Currents in conductor From**

Jx or current density decreases as the slab gets thicker RS

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**Currents in conductor For distance L in x-direction**

R is called skin resistance Rskin is called skin-effect resistance For finite thickness, RS

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

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Ex4 A steel pipe is constructed of a material for which r = 180 and = 4106 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 8cost A, where = 1200 rad/s, find: The skin depth The skin resistance = 7.66x10-4 m RS

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c) The dc resistance RS

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**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 RS

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**Example of Poynting theorem in DC case**

Rate of change of energy stored In the fields = 0 RS

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**Example of Poynting theorem in DC case**

From By using Ohm’s law, RS

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**Example of Poynting theorem in DC case**

Verify with From Ampère’s circuital law, RS

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**Example of Poynting theorem in DC case**

Total power W RS

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**Uniform plane wave (UPW) power transmission**

Time-averaged power density W/m2 amount of power for lossless case, W/m2 RS

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**Uniform plane wave (UPW) power transmission**

for lossy medium, we can write intrinsic impedance for lossy medium RS

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