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Finding attenuation, phase constant, and velocity

by DODGEVIPER13
Tags: attenuation, constant, phase, velocity
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DODGEVIPER13
#1
Oct25-13, 07:48 PM
P: 663
1. The problem statement, all variables and given/known data
Find the attenuation constant alpha, phase constant β, and phase velocity v if the conductivity of the material is σ=ωε the material parameters are μr=1, εr =2.5, and the wavelength in free space is λ naught = 30cm


2. Relevant equations



3. The attempt at a solution
So using some big hairy equations I found alpha to be equal to 9.53 nepers/m. The part I am having trouble with is the beta part? I start out using this equation β=sqrt((με/2)(1+sqrt(1+(σ/ωε)^2))). I have tried a ton of different ways to arrive at this equation β=(ω/c)(sqrt(εr))(sqrt((1+sqrt(2))/(2))) what do I do?
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rude man
#2
Oct26-13, 02:46 AM
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I can't reproduce your equations, sorry.

If we assume an E wave polarized in the y direction (propagation along x direction), the y component of E is

Ey = Emexp(jωt +/-Γx)
where Em = constant and
Γ is a (very!) complex number including ω, μ, ε, and σ. I leave it to you to obtain or derive this relationship. It will be in your textbook somewhere I'm sure.

Then, Γ = α + jβ so the answer to your problem is the imaginary part of Γ.
DODGEVIPER13
#3
Oct26-13, 10:46 AM
P: 663
I went through it using the gamma equation I found in my book gamma= alpha+jbeta and then jomega(sqrt(mu(epsilon)))(1-j(sigma/((omega)(epsilon)))) once I had gamma I uses gamm=alpha+jbeta and found for beta using alpha I got 22.976

DODGEVIPER13
#4
Oct26-13, 10:46 AM
P: 663
Finding attenuation, phase constant, and velocity

I used mu=4pix10^-7 and epsilon=8.85e-12
rude man
#5
Oct26-13, 02:30 PM
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Quote Quote by DODGEVIPER13 View Post
I went through it using the gamma equation I found in my book gamma= alpha+jbeta and then jomega(sqrt(mu(epsilon)))(1-j(sigma/((omega)(epsilon)))) once I had gamma I uses gamm=alpha+jbeta and found for beta using alpha I got 22.976
You have the right equation for Gamma.

You don't need alpha to get beta. Alpha is the real part of Gamma and beta is the imaginary part.

I did not check your numbers. What did you wind up with for alpha and beta in terms of omega, epsion, mu, sigma?


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