# Transmission line approimation

1. Oct 15, 2009

### likephysics

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. Oct 16, 2009

### gabbagabbahey

Re: Transmission line approximation

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