# Transmission Line Coefficients & Definitions

## Homework Statement

(a) State what is meant by a ‘distortionless’ and a ‘lossless’ transmission line.

(b) A transmission line has the primary coefficients as given below. Determine the line’s secondary coefficients Zo, α and β at a frequency of 1 GHz.
R = 2 Ω/m
L = 8 nH/m
G=0.5 mS/m
C=0.23 pF/m

## The Attempt at a Solution

I've seen some old threads with this question, but it's more the workings, methodology and definitions - if someone could be so kind to have a look over.
a)
Distortionless:
The transmission line in order to be 'distortionless' must both attenuate all signal frequencies in the same proportion and shift them in time by the same amount.
Loseless:
A transmission line is known as loseless when there are no energy loses along the line due to coeffecients R & G. This occurs when both of the aforementioned coefficients equal 0.

b)
So:
R &= 2 Ω/m
L &= 8 nH/m = 8x10^{-9} H/m
G=0.5 mS/m = 0.0005 S/m
C &=0.23 pF/m = 0.23*10^{-12} F/m

ω=2πf = 1x10^{9} *2π = 6.283*10^{9)

Due to high frequency the formulas:
β = ω√(LC)
= 6.283*10^{9} * √ ( 8*10^{-9) * 0.23*10^{-12}

α = R/2 * √(C/L)+G/*√(L/C)
= 2/2 * √(0.23*10^{-12}/8*10^{-9})+0.0005/2*√(8*10^{-9}/0.23*10^{-12})
= 1 * 0.00536+0.00025*186.501
= 0.05198714308
= 51.987 mNepers m-1

Zo = √((R+jωL)/ (G+jωC))
= √((2+j6.283*10^{9}*8*10^{-9}) / (0.0005+j6.283*10^{9}*0.23*10^{-12}))
= √ ( (2+j16π) / (0.0005+j(0.00046π) )
= √ (31491.630+j9511.802)
= 179.427+j26.506Ω

I appreciate any and all help. If anyone else is doing this question and wants advice I can try to help also.

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rude man
Homework Helper
Gold Member
This all looks good.

An exact value of α = 0.5(R/R0 + G/G0)
where R0 = Re{Z0} and G0 = Re{Y0} and Y0 = 1/Z0
which checks with your computation (within a few %; I approximated Z0 = R0 and G0 = 1/R0).

Jason-Li
This all looks good.

An exact value of α = 0.5(R/R0 + G/G0)
where R0 = Re{Z0} and G0 = Re{Y0} and Y0 = 1/Z0
which checks with your computation (within a few %; I approximated Z0 = R0 and G0 = 1/R0).
Thank you very much for the feedback.

berkeman
This all looks good.

An exact value of α = 0.5(R/R0 + G/G0)
where R0 = Re{Z0} and G0 = Re{Y0} and Y0 = 1/Z0
which checks with your computation (within a few %; I approximated Z0 = R0 and G0 = 1/R0).
Hi again Rudeman,

Was looking over my old working and found that in the final line of working is incorrect, unless I'm just being an idiot!

= √ (31491.630+j9511.802)
= 179.427+j26.506Ω

Just to update this post.

rude man
Homework Helper
Gold Member
Hi again Rudeman,

Was looking over my old working and found that in the final line of working is incorrect, unless I'm just being an idiot!

= √ (31491.630+j9511.802)
= 179.427+j26.506Ω

Just to update this post.
Well, I checked that last computation and found it correct. So you don't meet the qualifications for idiot. Sorry!

As I said, I had previously checked your answer against my approximate one and found it correct also.
So I'm not sure what problems are still extant?