Question

For a lossless line, the surge is purely resistive and the propagation constant is pure imaginary.

(a) True

(b) False

Short answer

Therefore, the correct option is (a): True.

Step-by-step solution

Determine the expression circuit constant for the transmission line,

The impedance of the transmission line given by,

$\mathbf{z}\mathbf{=}\mathbf{R}\mathbf{+}\mathbf{j\omega L}\frac{\mathbf{\Omega }}{\mathbf{m}}$ …… (1)

Here, Ris the resistance, L is the inductance, $\mathit{\omega }$ is the angular frequency.

The admittance of the transmission line is given by.

$\mathbf{y}\mathbf{=}\mathbf{G}\mathbf{+}\mathbf{j\omega C}\frac{\mathbf{S}}{\mathbf{m}}$ …… (2)

Here, G is the conductance andC is the capacitance.

The expression for the surge impedance is given by,

${\mathbf{Z}}_{\mathbf{c}}\mathbf{=}\sqrt{\frac{\mathbf{z}}{\mathbf{y}}}$ …… (3)

The expression for the propagation constant is given by,

$\mathbf{\gamma }\mathbf{=}\sqrt{\mathbf{ZY}}$ …… (4)

Determine the surge and propagation constant.

For the lossless line, the resistance of the transmission line is zero.

Substitute 0 for R into equation (1).

$\begin{array}{l}z=0+j\omega L\\ z=j\omega L\end{array}$

For the lossless line, the conductance of the transmission line is zero.

Substitute 0 for G into equation (1).

$\begin{array}{l}y=0+j\omega C\\ y=j\omega C\end{array}$

Calculate the surge impedance.

Substitute$j\omega L$ for z and$j\omega C$ for y into equation (3).

$\begin{array}{l}{Z}_c=\sqrt{\frac{j\omega L}{j\omega C}}\\ Z_c=\sqrt{\frac{L}{C}}\end{array}$

Hence, the surge for the lossless transmission line is$\sqrt{\frac{L}{C}}$ which is purely resistive.

Calculate the propagation constant.

Substitute$j\omega L$ for z and$j\omega C$ for y into equation (4).

$\begin{array}{l}\gamma =\sqrt{\left(j\omega L\right)\left(j\omega C\right)}\\ \gamma =\sqrt{j^2{\omega }^{2}LC}\\ \gamma =j\omega \sqrt{LC}\end{array}$

Hence the propagation constant is$j\omega \sqrt{LC}$ which is purely imaginary.

Therefore, the given statement is true.