Finding attenuation, phase constant, and velocity

  1. 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?
     
  2. jcsd
  3. rude man

    rude man 5,501
    Homework Helper
    Gold Member

    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.
     
  4. 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
     
  5. I used mu=4pix10^-7 and epsilon=8.85e-12
     
  6. rude man

    rude man 5,501
    Homework Helper
    Gold Member

    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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