What is the surge impedance of a transmission line where L is the inductance and C is the capacitance per unit length of the line? 

Explanation:

Surge Impedance of a Transmission Line

Definition: The surge impedance (also known as the characteristic impedance) of a transmission line is a measure of the inherent impedance that a wave of voltage and current experiences as it travels along the line. It is crucial in the analysis of wave propagation and reflection on transmission lines, especially for high-frequency signals and power transmission.

Formula: The surge impedance \( Z_0 \) of a transmission line where \( L \) is the inductance and \( C \) is the capacitance per unit length of the line is given by:

\[ Z_0 = \sqrt{\frac{L}{C}} \]

This formula derives from the fundamental principles of transmission line theory. It represents the ratio of the inductive reactance to the capacitive reactance per unit length of the line.

Derivation:

To derive the surge impedance, consider the fundamental transmission line equations for a lossless line (ignoring resistance and conductance for simplicity):

\[ \frac{\partial V(x,t)}{\partial x} = -L \frac{\partial I(x,t)}{\partial t} \]

\[ \frac{\partial I(x,t)}{\partial x} = -C \frac{\partial V(x,t)}{\partial t} \]

Here, \( V(x,t) \) is the voltage and \( I(x,t) \) is the current at position \( x \) and time \( t \). The inductance \( L \) and capacitance \( C \) are per unit length.

Taking the partial derivative of the first equation with respect to \( x \) and substituting into the second equation yields a wave equation:

\[ \frac{\partial^2 V(x,t)}{\partial x^2} = LC \frac{\partial^2 V(x,t)}{\partial t^2} \]

A similar equation can be derived for current \( I \). The solution to this wave equation is a traveling wave, and the ratio of voltage to current in such a wave is the surge impedance \( Z_0 \):

\[ Z_0 = \sqrt{\frac{L}{C}} \]

This impedance characterizes the relationship between voltage and current for a traveling wave on the line.

Advantages:

• Surge impedance matching is critical for minimizing reflections in transmission lines, which is essential for efficient power transfer and signal integrity.
• Understanding surge impedance helps in the design of transmission lines and matching networks.

Applications: Surge impedance is widely used in power transmission and distribution systems, telecommunications, and high-frequency signal transmission. It is essential for designing impedance matching networks, analyzing transmission line behavior, and ensuring efficient power transfer.

Correct Option Analysis:

The correct option is:

Option 2: \(\sqrt{\frac{L}{C}}\)

This option correctly represents the formula for the surge impedance of a transmission line. The surge impedance is given by the square root of the ratio of the inductance per unit length \( L \) to the capacitance per unit length \( C \).

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\sqrt{LC}\)

This option is incorrect as it does not represent the surge impedance of a transmission line. The product of \( L \) and \( C \) does not yield the correct impedance value.

Option 3: \(\sqrt{\frac{C}{L}}\)

This option is incorrect. The formula for surge impedance involves the ratio of \( L \) to \( C \), not \( C \) to \( L \). Thus, this option misrepresents the relationship between inductance and capacitance.

Option 4: \(\sqrt{\frac{1}{LC}}\)

This option is also incorrect. The reciprocal of the product of \( L \) and \( C \) does not represent the surge impedance. This expression is related to other parameters in transmission line theory but not to the surge impedance.

Conclusion:

Understanding the surge impedance is crucial for the effective design and analysis of transmission lines. The correct formula for the surge impedance is \(\sqrt{\frac{L}{C}}\), which highlights the importance of the inductance and capacitance per unit length in determining the impedance experienced by a traveling wave on the line. This knowledge is fundamental for minimizing reflections, ensuring efficient power transfer, and maintaining signal integrity in various applications involving transmission lines.