Define rise time for a step response and state its typical relationship to the time constant for a first-order system.

Define rise time for a step response and state its typical relationship to the time constant for a first-order system.

• A. Rise time is the time to go from 10% to 90% of the final value; for a first-order system, t_r ≈ 2.2τ.
• B. Rise time is the time to go from 0% to 100% of the final value.
• C. Rise time is not defined for first-order systems.
• D. Rise time is the time to reach 50% of the final value.

Answer: (A)

Rise time measures how quickly a system responds to a step input by looking at the interval it takes for the output to travel from 10% to 90% of its final value. For a first‑order system with step response y(t) = 1 − e^(−t/τ), the final value is 1, so to find the rise time we compute the times when y(t) reaches 0.1 and 0.9. Solving 1 − e^(−t/τ) = 0.1 gives t1 = −τ ln(0.9) ≈ 0.105τ, and solving 1 − e^(−t/τ) = 0.9 gives t2 = −τ ln(0.1) ≈ 2.303τ. The rise time is t2 − t1 ≈ 2.198τ, which is commonly approximated as 2.2τ. This is why the correct choice defines rise time as the 10% to 90% interval and states its typical relationship to the time constant for a first‑order system. Using 0% to 100% isn’t practical for a first‑order response (it would take infinite time to reach the final value exactly), and 50% rise time would not reflect the standard 10–90% definition.

Rise time measures how quickly a system responds to a step input by looking at the interval it takes for the output to travel from 10% to 90% of its final value. For a first‑order system with step response y(t) = 1 − e^(−t/τ), the final value is 1, so to find the rise time we compute the times when y(t) reaches 0.1 and 0.9.

Solving 1 − e^(−t/τ) = 0.1 gives t1 = −τ ln(0.9) ≈ 0.105τ, and solving 1 − e^(−t/τ) = 0.9 gives t2 = −τ ln(0.1) ≈ 2.303τ. The rise time is t2 − t1 ≈ 2.198τ, which is commonly approximated as 2.2τ.

This is why the correct choice defines rise time as the 10% to 90% interval and states its typical relationship to the time constant for a first‑order system. Using 0% to 100% isn’t practical for a first‑order response (it would take infinite time to reach the final value exactly), and 50% rise time would not reflect the standard 10–90% definition.