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ENE 428 Microwave Engineering

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ENE 428Microwave Engineering

Lecture 2 Uniform plane waves

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- 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),

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- intrinsic impedance
- wavelength

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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

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c) propagation velocity

d) Intrinsic impedance

e) Amplitude of the magnetic field intensity

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- Cause
- finite conductivity
- polarization loss ( = ’-j” )

- Assume homogeneous and isotropic medium

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Define

From

and

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We can derive

and

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

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

and

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- propagation velocity
- wavelength

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- In this case or > 10, we can approximate

therefore

and

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- depth of penetration or skin depth, is a distance where the field decreases to e-1or 0.368 times of the initial field
- propagation velocity
- wavelength

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a) loss tangent

b) attenuation constant

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c) phase constant

d)intrinsic impedance

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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

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

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- 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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- 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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- To understand a concept of sheet resistance

from

Rsheet()

sheet resistance

At high frequency, it will be adapted to skin effect resistance

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Therefore the current that flows through the slab at t is

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From

Jxor current density decreases as the slab gets thicker

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For distance L in x-direction

Ris called skin resistance

Rskinis called skin-effect resistance

For finite thickness,

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Current is confined within a skin depth of the coaxial cable

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Ex 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

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c) The dc resistance

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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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Rate of change of energy stored

In the fields = 0

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From

By using Ohm’s law,

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Verify with

From Ampère’s circuital law,

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Total power

W

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- Time-averaged power density

W/m2

amount of power

for lossless case,

W

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for lossy medium, we can write

intrinsic impedance for lossy medium

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