The surge impedance in a transmission line having negligible resistance is:

Explanation:

Surge Impedance in a Transmission Line

Definition: Surge impedance, also known as the characteristic impedance of a transmission line, is a critical parameter in the study of electrical power systems. It is defined as the square root of the ratio of the line's inductance (\(L\)) to its capacitance (\(C\)) per unit length. Mathematically, it is represented as:

\(Z_s = \sqrt{\frac{L}{C}}\)

Here, \(L\) represents the inductance per unit length, and \(C\) represents the capacitance per unit length of the transmission line. Surge impedance is particularly significant in the analysis of transient conditions, such as when a surge or traveling wave propagates along a transmission line.

Formula:

The surge impedance is given by:

\(Z_s = \sqrt{\frac{L}{C}}\)

Where:

• \(L\) = Inductance per unit length (in Henries)
• \(C\) = Capacitance per unit length (in Farads)

Explanation:

When a voltage or current surge propagates along a transmission line, the line's inductance and capacitance determine the surge impedance. The surge impedance is critical in designing transmission lines to minimize reflection and ensure efficient power transfer. Transmission lines are typically designed to have minimal resistance (\(R\)) to enhance efficiency, and in such cases, the surge impedance is primarily dependent on the inductance and capacitance.

Correct Option Analysis:

The correct option is:

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

This option correctly represents the formula for surge impedance in a transmission line with negligible resistance. The surge impedance is determined by the square root of the ratio of inductance (\(L\)) to capacitance (\(C\)). This relationship ensures that the impedance is frequency-independent in the case of lossless transmission lines.

Additional Information

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

Option 1: \(\sqrt{L × C}\)

This option is incorrect. The surge impedance is not the square root of the product of inductance and capacitance. Instead, it is the square root of the ratio of inductance to capacitance.

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

This option is incorrect. The surge impedance is not calculated as the square root of the sum of inductance and capacitance. This formula does not have any physical significance in the context of transmission line surge impedance.

Option 4: \(\sqrt{(L × C) - 1}\)

This option is incorrect. The surge impedance is not calculated as the square root of the product of inductance and capacitance minus one. This formula is not relevant to the calculation of surge impedance and has no basis in electrical engineering principles.

Option 5: (No content provided for this option)

The fifth option was not provided in the statement above and is therefore not relevant for analysis.

Conclusion:

The correct formula for surge impedance in a transmission line with negligible resistance is \(\sqrt{\frac{L}{C}}\), as represented by Option 3. This formula highlights the relationship between the line's inductance and capacitance, which are the key parameters influencing surge impedance. Understanding this concept is crucial for designing efficient transmission lines and managing transient conditions in power systems. Options 1, 2, and 4, on the other hand, misrepresent the relationship between inductance and capacitance and do not correctly describe surge impedance.