Question 6.21: (a) Find the phase and group velocities for a normal transmi...

(a) From (6.77), we write

\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=\sqrt{\hat{Z} \hat{Y}}=\sqrt{(j \omega \hat{L})(j \omega \hat{C})}=j \omega \sqrt{\hat{L} \hat{C}}

The phase velocity  v_{\phi}   is computed to be

v_{\phi}=\frac{\omega}{\beta}=\frac{1}{\sqrt{\hat{ L } \hat{C}}}

The group velocity   v_{g} is computed to be

v_{g}=\frac{\partial \omega}{\partial \beta}=\frac{1}{\left(\frac{\partial \beta}{\partial \omega}\right)}=\frac{1}{\sqrt{\hat{L} \hat{C}}}

The two velocities are equal in this case and our independent of frequency.

(b) From (6.77), we write

\gamma=\alpha+ j \beta=\sqrt{\hat{Z} \hat{Y}}=\sqrt{\left(\frac{1}{ j \omega \hat{C}}\right)\left(\frac{1}{j \omega \hat{L}}\right)}=- j \frac{1}{\omega \sqrt{\hat{L} \hat{C}}}

The phase velocity  v_{\phi} is computed to be

v _{\phi}=\frac{\omega}{\beta}=-\omega^{2} \sqrt{\hat{ L } \hat{C}}

The group velocity   v_{g} is computed to be

v _{g}=\frac{\partial \omega}{\partial \beta}=\frac{1}{\left(\frac{\partial \beta}{\partial \omega}\right)}=+\omega^{2} \sqrt{\hat{L} \hat{C}}

In this case, the phase and the group velocities are in the opposite direction and both of them depend on frequency.