The condition that holds good in a distortion-less transmission line is : 

Explanation:

Distortion-less Transmission Line

Definition: A distortion-less transmission line is one where the signal transmitted from one end of the line to the other experiences no distortion. This means that the shape of the signal remains unchanged during transmission. In practical terms, it implies that the transmission line has no frequency-dependent attenuation or phase distortion, ensuring the integrity of the signal over distance.

Condition for Distortion-less Transmission Line: For a transmission line to be distortion-less, it must satisfy a specific condition relating to its primary constants. These primary constants are Resistance (R), Inductance (L), Conductance (G), and Capacitance (C) per unit length of the transmission line. The condition is:

R/L = G/C

This condition ensures that the phase velocity of the signal remains constant across all frequencies, and there is no phase distortion or frequency-dependent attenuation.

Derivation and Explanation:

The primary constants of a transmission line are defined as follows:

• R: Resistance per unit length (ohms per unit length)
• L: Inductance per unit length (henries per unit length)
• G: Conductance per unit length (siemens per unit length)
• C: Capacitance per unit length (farads per unit length)

The propagation constant (γ) of a transmission line is given by:

γ = α + jβ

Where:

• α: Attenuation constant (nepers per unit length)
• β: Phase constant (radians per unit length)

The propagation constant can also be expressed in terms of the primary constants as:

γ = √((R + jωL)(G + jωC))

For a distortion-less transmission line, the attenuation constant (α) must be frequency-independent, and the phase constant (β) must be a linear function of frequency (ω). This implies that the real and imaginary parts of the propagation constant must satisfy specific conditions.

By expanding the expression for the propagation constant, we get:

γ = α + jβ = √((R + jωL)(G + jωC))

Separating the real and imaginary parts, we find that for the line to be distortion-less, the condition R/L = G/C must hold. This ensures that the phase velocity (v_p) is constant and given by:

v_p = 1/√(LC)

and the characteristic impedance (Z_0) is given by:

Z_0 = √(L/C)

Advantages of Distortion-less Transmission Lines:

• Ensures the integrity of transmitted signals, maintaining their original shape.
• Prevents signal degradation over long distances.
• Reduces the need for complex equalization and signal processing at the receiver end.

Applications: Distortion-less transmission lines are crucial in high-speed data communications, such as in telecommunications, networking, and broadcasting. They are also essential in precision measurement systems and other applications where signal integrity is paramount.

Correct Option Analysis:

The correct option is:

Option 1: R/L = G/C

This option correctly represents the condition for a distortion-less transmission line, ensuring no frequency-dependent attenuation or phase distortion