Lossy media	Feb. 19, 2007


+ Wave equation for lossy media in phasor form (pp. 287-295): [equation] where [equation] and [equation] is the complex permittivity with the imaginary part involving conductivity [equation] that denotes Ohmic loss. [equation] [equation] ,
i.e. attenuation coefficient [equation] ([equation] )
and phase factor [equation] (rad/m).
Note -- [equation] is equal to 8.69 dB/m (p.289 footnote).

+ Plane wave in lossy media: e.g.
[equation] ([equation] , [equation] ) [equation]
time varying form [equation]
Note -- [equation] equiv. to [equation] in lossless medium [equation] .
- In terms of phasors, it is best to use exponential form. Consistently, convert all imaginary terms inside the argument of the exponent to radian (avoid degrees), e.g. [equation] .
- For conversion into time varying form, group all the phases (exponent with j) into one group and group all the real amplitudes into another group.
- To identify a plane wave, group all (real and imaginary) exponents with spatial dependency into one group as propagation factor (e.g. [equation] ) and remaining terms make up the complex amplitude.

+ TEM wave in lossy media (Sect. 7.3):
[equation] & [equation] are valid but wave impedance [equation] is complex.
Example 2.4: Plane wave in lossy medium
Example 2.5: Maxwell's eqn method applied to a standing wave
Example 2.6: Plane wave method applied to a standing wave

+ Determine of material properties:
[equation] [equation] insulator or good dielectric
[equation] [equation] good conductor
Otherwise, semi-conductor or lossy dielectric.
Flow chart in determining wave parameters
Recall [equation] . Generally, [equation]
where [equation] accounts for conductor and dielectric losses.
Loss tangent [equation] .
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HW #9	Due 3/1/07


1. Given [equation] for a plane wave. Find frequency in Hz, [equation] , polarization direction, propagation direction, phase velocity, wavelength inside the medium and [equation] if [equation] . Also find phasor [equation] in exponential form.
2. Solve the problem 2. in HW 8 with plane wave method.
3. Problem 6.18 (page 271). Notice that method should be used to find [equation] (plane wave or Maxwell's eqn) and what coordinates is used in this problem (r in the text book is my [equation] ). Use the formula for k to compute k.
Extra-Credit
Plot the solutions for the following wave equations using ``PDEplot'' of Maple: a) [equation] , b) [equation] , c) 2 [equation] for initial input [equation] . Comment on the plots (see http://scylla.ceas.uwm.edu/361/maple.html)


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Last Modified: February 17, 2007
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