Predicting the characteristic impedance of a TEM guide
One important parameter of a TEM waveguide is its characteristic impedance $Z_{c}=\left(L^{\prime} / C^{\prime}\right)^{1 / 2}$, where $L^{\prime}$ and $C^{\prime}$ are, respectively, the inductance and the capacitance per meter of guide (Prob. 33-3).
In designing such lines it is important to predict the values of $L^{\prime}$ and of $C^{\prime} .$ If the geometry is such that these quantities are difficult to calculate, as in Fig. 33-6(a), one can perform the following measurements on a resistance-sheet analog.
Show that, if the material has a conductivity $\sigma$ and a thickness $s$, and if the permittivity of the dielectric is $\epsilon$, then $R_{1} C^{\prime}=\epsilon / s \sigma$.
(b) One can measure $L^{\prime}$ as in Fig. 33-6(c) by measuring the resistance $R_{2}$ between electrodes $C$ and $D$.
Show that $R_{2} L^{\prime}=\mu_{0} / s \sigma$. Thus $Z_{c}=\left(\mu_{0} / \epsilon\right)^{1 / 2}\left(R_{1} / R_{2}\right)^{1 / 2}$.
(a) One can find the value of $C^{\prime}$ by cutting out a sheet of resistive material in the shape of the cross section of the dielectric as in Fig. 33-6(b) and by measuring the resistance $R_{1}$ between electrodes $A$ and $B$. See Prob. 9-10.