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Transmission line approimation

  1. Oct 15, 2009 #1
    Transmission line approximation

    1. The problem statement, all variables and given/known data
    In the derivation of the approximate formulas of \gamma and Z0 for low loss lines, all terms containing the second and higher order powers of R/wL and G/wC were neglected in comparison with unity. (R/wL<<1 and G/wC<<1)
    gamma=jw*sqrt(LC)*sqrt(1+R/jwL)*sqrt (1+G/jwC)
    approximated to
    gamma = jw*sqrt(LC)*(1+R/2jwL)*sqrt(1+G/2jwC)

    gamma is the propagation constant which is equal to alpha+j beta

    At lower frequencies, better approximation may be required. find new formulas for \gamma and Z0 for low loss lines that retain terms containing (R/wL)^2 and (G/wL)^2


    2. Relevant equations
    Required result is
    alpha = sqrt(LC/2)*(R/L+G/C)*[1-(1/8w^2)*(R/L-G/C)^2]
    beta = w*sqrt (LC)*[1+(1/8w^2)*(R/L-G/C)^2]


    3. The attempt at a solution
    I tried expanding the term
    sqrt(1+R/jwL) using square root expansion :
    1+(1/2)*R/jwL-(1/8)*(R/jwL)^2
    did the same for sqrt (1+G/jwC)
    I am unable to get the desired result. Any help.
    FYI, this is prob 9.7 in cheng.
     
    Last edited: Oct 15, 2009
  2. jcsd
  3. Oct 16, 2009 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Re: Transmission line approximation

    That looks fine to me....what do you get when you multiply everything out?
     
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