Question 6.18: Find the complex propagation constant if the circuit element...
Find the complex propagation constant if the circuit elements satisfy the ratio \frac{\hat{R}}{\hat{L}}=\frac{\hat{G}}{\hat{C}}. Interpret the propagation of such a signal that propagates on this line.
The complex propagation constant (6.77) can be written as
\gamma=\alpha+ j \beta=\pm \sqrt{\hat{Z} \hat{Y}}=\pm \sqrt{(\hat{R}+j \omega \hat{L})(\hat{G}+j \omega \hat{C})} (6.77)
\gamma=\alpha+j \beta=\pm \sqrt{(\hat{R}+j \omega \hat{L})(\hat{G}+j \omega \hat{C})}=\pm \sqrt{(\hat{R}+j \omega \hat{L})\left(\frac{\hat{R} \hat{C}}{\hat{L}}+j \omega \hat{C}\right)}=\pm \sqrt{\frac{\hat{C}}{\hat{L}}}(\hat{R}+j \omega \hat{L})
In this case, the attenuation constant \alpha and the phase velocity \frac{\omega}{\beta} are independent of frequency. This implies that there will be no distortion of a signal as it propagates on this transmission line. There will only be a constant attenuation of the signal. The characteristic impedance of this transmission line
Z_{C}=\sqrt{\frac{\hat{Z}}{\hat{Y}}}=\sqrt{\frac{\hat{R}+j \omega \hat{L}}{\hat{G}+j \omega \hat{C}}}=\sqrt{\frac{\hat{R}+j \omega \hat{L}}{\frac{\hat{R} \hat{C}}{\hat{L}}+j \omega \hat{C}}}=\sqrt{\frac{\hat{L}}{\hat{C}}}
is also independent of frequency. This transmission line is called a “distortionless line.”