Chapter 11, Problem 1RQ

To determine

Choose the correct option from the given choices to show that the given statements are not true of the line parameters R, L, G, and C.

Answer to Problem 1RQ

The correct option from the given choices is Options (c), (d), and (e).

Explanation of Solution

Calculation:

A transmission line is described in terms of its line parameters, which are its resistance per unit length R, conductance per unit length due to the dielectric medium separating the conductors G, inductance per unit length L, and capacitance per unit length C.

• The line parameters R, L, G, and C are not lumped or discrete. Rather, the parameters are distributed as shown in figure (Distributed parameters of a two-conductor transmission line) in the textbook. That is, the parameters are uniformly distributed along the entire length of the line in which R and L are series elements and G and C are shunt elements.
• $G\ne \frac{1}{R}$, where R is the ac resistance per unit length of the conductors comprising the line and G is the conductance per unit length due to the dielectric medium separating the conductors.
• For each line, $LC=\mu \epsilon$ and $\frac{G}{C}=\frac{\sigma }{\epsilon }$.

The characteristic impedance $Z_{\text{o}}$ of the line is the ratio of the positively traveling voltage wave to the current wave at any point on the line.

Consider the general expression for characteristic impedance.

$Z_{\text{o}}=\frac{\gamma }{G+j\omega C}$

$Z_{\text{o}}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}=R_{\text{o}}+j{X}_{\text{o}}$.

Here,

$\gamma$ is the propagation constant,

$R_{\text{o}}$ is the real part of $Z_{\text{o}}$, and

$X_{\text{o}}$ is the imaginary part of $Z_{\text{o}}$.

The propagation constant $\gamma$ and characteristic impedance $Z_{\text{o}}$ are the important properties of the line because they both depend on the line parameters R, L, G, and C and the frequency of operation.

Therefore, from the above explanation, the given statements $G=\frac{1}{R}$, $LC=\mu \epsilon$ and $RG=\sigma \epsilon$ and both R and G depend on the conductivity of the conductors forming the line are not true of the line parameters R, L, G, and C. Hence, the correct options are Options (c), (d), and (e).

Conclusion:

Thus, the correct option from the given choices is Options (c), (d), and (e).