The settling time Ts of a first-order system is approximately:

• A. A. Ts ≈ τ
• B. B. Ts ≈ 2τ
• C. C. Ts ≈ 3τ
• D. D. Ts ≈ 4τ

Answer: (D)

In the context of a first-order system, the settling time ( T_s ) is typically defined as the time it takes for the system's response to remain within a certain percentage (usually 2% or 5%) of its final value. For first-order systems, the time constant ( \tau ) plays a crucial role in determining the settling time.

The relationship between settling time and the time constant is generally approximated as follows: the settling time ( T_s ) is roughly equal to ( 4τ ). This approximation stems from the exponential nature of the system's response. When analyzing step responses, it takes about four time constants for a first-order system to settle close to its final value, accounting for the exponential decay in the response curve.

Choosing ( 4τ ) as the settling time captures the essence of how the system approaches its steady state, as by this point, the response has effectively reduced to a very small deviation from the final value. Thus, ( T_s ) provides a practical measure for understanding how long a system will take to stabilize after an input change, making it a critical aspect of system analysis and design.