Finding attenuation, phase constant, and velocity

  • #1
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Homework Statement


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


Homework Equations





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?
 

Answers and Replies

  • #2
rude man
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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.
 
  • #3
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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
 
  • #4
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I used mu=4pix10^-7 and epsilon=8.85e-12
 
  • #5
rude man
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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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