Transmission Line Parameters Explained

This question asks us to identify the correct statements about the parameters of transmission lines, specifically characteristic impedance ($Z_o$), phase constant ($\beta$), and phase velocity ($V_p$). Let's analyze the fundamental formulas and then evaluate each option.

Key Transmission Line Formulas

Transmission lines are characterized by four primary parameters per unit length: resistance ($R$), inductance ($L$), conductance ($G$), and capacitance ($C$). These parameters determine the line's behavior.

Characteristic Impedance ($Z_o$)

The characteristic impedance ($Z_o$) represents the ratio of the voltage to the current for a traveling wave on an infinitely long, lossless transmission line. The general formula, derived from the transmission line equations, is:

$ Z_o = \sqrt{\frac{R+j\omega L}{G+j\omega C}} $

Where:

• $R$ is the resistance per unit length
• $L$ is the inductance per unit length
• $G$ is the conductance per unit length
• $C$ is the capacitance per unit length
• $\omega$ is the angular frequency ($\omega = 2\pi f$)
• $j$ is the imaginary unit

For a lossless line ($R=0$ and $G=0$), the formula simplifies significantly to:

$ Z_o = \sqrt{\frac{j\omega L}{j\omega C}} = \sqrt{\frac{L}{C}} $

Phase Constant ($\beta$)

The phase constant ($\beta$) describes how the phase of the signal changes with distance along the line. It is the imaginary part of the propagation constant ($\gamma$). The general formula for the propagation constant is:

$ \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} = \alpha + j\beta $

Where $\alpha$ is the attenuation constant.

For a lossless line ($R=0$ and $G=0$), the propagation constant becomes:

$ \gamma = \sqrt{(j\omega L)(j\omega C)} = \sqrt{-\omega^2 LC} = j\omega\sqrt{LC} $

Comparing this to $\gamma = \alpha + j\beta$, we find that for a lossless line, $\alpha = 0$ and:

$ \beta = \omega\sqrt{LC} $

Phase Velocity ($V_p$)

The phase velocity ($V_p$) is the speed at which a point of constant phase (a wave crest, for example) travels along the line. It is related to the angular frequency ($\omega$) and the phase constant ($\beta$) by:

$ V_p = \frac{\omega}{\beta} $

Using the phase constant formula for a lossless line ($\beta = \omega\sqrt{LC}$), we get:

$ V_p = \frac{\omega}{\omega\sqrt{LC}} = \frac{1}{\sqrt{LC}} $

Analysis of Options

Analysis of Option A

Statement: characteristic impedance $Z_o = \sqrt{\frac{R+j\omega L}{G+j\omega C}}$

This statement correctly represents the general formula for the characteristic impedance of a transmission line, considering all four primary parameters ($R, L, G, C$). Therefore, statement A is correct.

Analysis of Option B

Statement: phase constant $\beta = \omega\sqrt{LC}$

This statement provides the formula for the phase constant in a lossless transmission line ($R=0, G=0$). This is a commonly used and important formula in transmission line theory. Therefore, statement B is considered correct in the typical context.

Analysis of Option C

Statement: characteristic impedance $Z_o = \sqrt{\frac{R+j\omega C}{G+j\omega L}}$

This formula incorrectly swaps the roles of $L$ and $C$ and mixes them with $R$ and $G$ in the numerator and denominator. The correct formula involves $L$ in the numerator and $C$ in the denominator. Therefore, statement C is incorrect.

Analysis of Option D

Statement: phase constant $\beta = L\sqrt{\omega C}$

This formula is dimensionally incorrect. The phase constant $\beta$ has units of radians per meter. The formula $\beta = \omega\sqrt{LC}$ is dimensionally correct ( $\omega$ is rad/s, $L$ is H, $C$ is F; $\sqrt{H \cdot F}$ has units of s/m, so $\omega\sqrt{LC}$ is (rad/s)*(s/m) = rad/m). The formula $L\sqrt{\omega C}$ does not yield the correct units or the correct relationship. Therefore, statement D is incorrect.

Analysis of Option E

Statement: phase velocity $V_p = \frac{1}{\sqrt{LC}}$

This statement gives the phase velocity for a lossless transmission line. As derived above, $V_p = \frac{\omega}{\beta}$ and for a lossless line $\beta = \omega\sqrt{LC}$, which leads to $V_p = \frac{1}{\sqrt{LC}}$. Therefore, statement E is correct.

Conclusion

Based on the analysis, the correct statements are A, B, and E. Statement A is the general formula for characteristic impedance. Statements B and E provide the phase constant and phase velocity, respectively, for the important case of a lossless transmission line.