For a first-order system, the time constant τ represents:

• A. A. Time to reach 90% of final value
• B. B. Time to reach 63.2% of final value
• C. C. Time to reach peak value
• D. D. Time to settle

Answer: (B)

The time constant, denoted as τ (tau), is a crucial parameter in the analysis of first-order systems, particularly in the context of their transient responses. It is defined as the time required for the system's response to reach approximately 63.2% of its final steady-state value after a step change in input. This characteristic is inherent to the exponential nature of the response in first-order dynamics, where the system gradually approaches its final value rather than reaching it instantaneously.

In a mathematical sense, when a first-order system is subjected to a step input, its output response can be described by an exponential function. At time t = τ, this function indicates that the output reaches about 63.2% of the final value. This threshold is significant because it provides insight into how quickly the system reacts to changes and how it will behave over time, making it a vital concept for engineers and technicians working with dynamic systems.

The other options refer to different aspects of system responses. The time to reach 90% of the final value occurs later in time and is not defined as the time constant. The time to reach peak value applies more to systems that exhibit overshoot, while the time to settle typically encompasses the entire period it takes for the