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likephysics
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Transmission line approximation
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
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]
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.
Homework Statement
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
Homework 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]
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.
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