The propagation constant of a transmission line is given as

Question

Examoogle · Accepted Answer

Option A: $j\omega\sqrt{LC}$

Explanation & Concepts:
The propagation constant, denoted by $\gamma$, is a complex quantity that characterizes the change in amplitude and phase of a wave as it travels along a transmission line. It is defined by the fundamental parameters per unit length: resistance $R$, inductance $L$, conductance $G$, and capacitance $C$, given by the expression $\gamma = \sqrt{(R + j\omega L)(G + j\omega C)}$.

Working Principle: For an ideal (lossless) transmission line where resistance $R = 0$ and conductance $G = 0$, the propagation constant simplifies to $\gamma = \sqrt{(j\omega L)(j\omega C)}$. Evaluating this yields $\gamma = \sqrt{j^2\omega^2 LC}$, which reduces to $\gamma = j\omega\sqrt{LC}$. In this state, the real part (attenuation constant $\alpha$) is zero, and the imaginary part (phase constant $\beta$) equals $\omega\sqrt{LC}$.

Key Formulas: $\gamma = \sqrt{(R + j\omega L)(G + j\omega C)}$ — General expression for propagation constant; $\gamma = j\omega\sqrt{LC}$ — Expression for lossless transmission line

Key Points: The propagation constant $\gamma$ is defined as $\gamma = \alpha + j\beta$.; $\alpha$ (Real part) represents attenuation in Nepers per unit length.; $\beta$ (Imaginary part) represents phase shift in radians per unit length.; For an ideal line, $\alpha = 0$, meaning the wave propagates without energy dissipation.

Advantages: Simplifies analysis of signal propagation in high-frequency power or communication systems.; Provides a direct link between physical parameters (L, C) and wave velocity.

Disadvantages: Idealization to a lossless line ignores real-world resistive heating and dielectric losses.; Complexity increases significantly when considering non-uniform lines.

Applications: Power system transient analysis (traveling waves).; Design of telecommunication transmission lines and waveguides.

Additional Info: The velocity of propagation $v$ is given by $v = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}}$.. Option B represents the inverse of the velocity. Options C and D are dimensionally inconsistent for the propagation constant.

Answer: A is correct — For a lossless transmission line, the propagation constant is purely imaginary and is given by $j\omega\sqrt{LC}$.

Exam Tip: Always remember that the propagation constant has units of $m^{-1}$ and deals with the spatial variation of the traveling wave, while impedance deals with the ratio of voltage to current.

Examoogle · Answer

Option B: $j\sqrt{LC}$

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Option C: $j\sqrt{\frac{L}{C}}$

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Option D: $j\omega\sqrt{\frac{C}{L}}$

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