Chapter 2, Problem 13P

In addition to not dissipating power, a lossless line has two important features: (1) it is dispersionless (u_p is independent of frequency); and (2) its characteristic impedance Z₀ is purely real. Sometimes, it is not possible to design a transmission line such that R′ ≪ ωL′ and G′ ≪ ωC′ but it is possible to choose the dimensions of the line and its material properties so as to satisfy the condition

R′C′ = L′G′ (distortionless line)

Such a line is called a distortionless line, because despite the fact that it is not lossless, it nonetheless possesses the previously mentioned features of the lossless line. Show that for a distortionless line.

$\alpha =R^{\prime }\sqrt{\frac{C^{\prime }}{L^{\prime }}}=\sqrt{R^{\prime }{G}^{\prime }}$,

$\beta =\omega \sqrt{L^{\prime }{C}^{\prime }}$,

$Z_0=\sqrt{\frac{L^{\prime }}{C^{\prime }}}$.