Transmission line secondary coefficients

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Homework Statement

A transmission line has the primary coefficients R= 2 ohm/m, L=8 nH/m, G= 0.5 mS/m and C= 0.23 pF/m. Determine the lines secondary coefficients Zo, α and β at a frequency of 1 GHz.

Homework Equations

ω= 2 π f

See uploaded formulas document

The Attempt at a Solution

I'm not sure if I've approached this right but this is what I've done:

I started by calculating the angular frequency from the given 1GHz frequency

ω= 2 π f = 2π*1exp^9 = ω= 6283185307 rad/s

Using this I went on to calculate Zo

Sqrt ((2+j6283185307*8exp^-9)/(0.5exp^-3+j6283185307*0.23exp^-12))

Zo= 26.4316 + j0.0000327906 or Zo= 26.4316 θ 0.0000711° Ω

then using the above formulas I calculated the following

β=sqrt(1/2*(6283185307^2*8exp^-9*0.23exp^-12-2*0.5exp^-3+sqrt(2^2+6283185307^2*(8exp^-9)^2)*((0.5exp^-3)^2+6283185307^2*(0.23exp^-12)^2))

=1.65952exp^5

β=246.2946 rad m^-1α=sqrt (1/2*(2*0.5exp^-3-6283185307^2*8exp^-9*0.23exp^-12+sqrt(2^2+6283185307^2*(8exp-9)^2)*((0.5exp-3)^2+6283185307^2*(0.23exp^-12)^2))

= 3.24771exp^5

α=482.0029Np m^-1

Am i along the right lines with this? (hope you can understand the above workings...It got complicated with all the brackets etc)

 

[Svein]

Just checking Z₀ using the simplified formula: $Z_{0}\approx \sqrt{\frac{L}{C}}=\sqrt{\frac{8\cdot 10^{-9}}{0.23\cdot 10^{-12}}}=\sqrt{34782.6087 }=186.5$.

 

[rude man]

You need to use symbols instead of numbers until the very end. Keeping track of all those numbers gives me and a lot of other helpers a splitting headache!

I went as far as checking your formula for Z₀ (but not your math) which looks OK.

 

[rude man]

Just checking Z₀ using the simplified formula: $Z_{0}\approx \sqrt{\frac{L}{C}}=\sqrt{\frac{8\cdot 10^{-9}}{0.23\cdot 10^{-12}}}=\sqrt{34782.6087 }=186.5$.

This is not a small-attenuation line. In particular, the requirement G/ωC << 1 is not met. (= 0.35). But I agree, the OP's number looks off.