In a first-order system's step response, what fraction of the final change is reached after one time constant?

In a first-order system's step response, what fraction of the final change is reached after one time constant?

• A. 50%
• B. 75%
• C. 63%
• D. 100%

Answer: (C)

A first-order system responds to a step input with a curve described by y(t) = y_final (1 − e^(−t/τ)). Here, τ is the time constant that sets the rate of approach to the final value. After one time constant, t = τ, giving y(τ) = y_final (1 − e^(−1)) ≈ y_final × 0.632. Since e^(−1) ≈ 0.368, about 63% of the final change has occurred. This is the characteristic fraction for a single time constant.

A first-order system responds to a step input with a curve described by y(t) = y_final (1 − e^(−t/τ)). Here, τ is the time constant that sets the rate of approach to the final value. After one time constant, t = τ, giving y(τ) = y_final (1 − e^(−1)) ≈ y_final × 0.632. Since e^(−1) ≈ 0.368, about 63% of the final change has occurred. This is the characteristic fraction for a single time constant.