In a first-order dynamic system, the time constant is defined as the time required to reach what percent of the final output?

In a first-order dynamic system, the time constant is defined as the time required to reach what percent of the final output?

• A. 63% of the final output
• B. 100% of the final output
• C. 50% of the final output
• D. 10% of the final output

Answer: (A)

For a first-order system responding to a step input, the output follows y(t) = y∞(1 − e^(−t/τ)). This means the time constant τ is the time it takes for the output to reach about 63.2% of its final steady-state value. Specifically, at t = τ, y(τ) = y∞(1 − e^(−1)) ≈ 0.632 y∞, so the response has reached roughly 63% of the final value. It isn’t 100% because the exponential approach never finishes in finite time; it’s not 50% or 10% because those percentages occur at shorter times (50% at t = τ ln 2 ≈ 0.693τ, 10% at t ≈ 0.105τ). In practice, several τs are used to get very close to the final value.

For a first-order system responding to a step input, the output follows y(t) = y∞(1 − e^(−t/τ)). This means the time constant τ is the time it takes for the output to reach about 63.2% of its final steady-state value. Specifically, at t = τ, y(τ) = y∞(1 − e^(−1)) ≈ 0.632 y∞, so the response has reached roughly 63% of the final value. It isn’t 100% because the exponential approach never finishes in finite time; it’s not 50% or 10% because those percentages occur at shorter times (50% at t = τ ln 2 ≈ 0.693τ, 10% at t ≈ 0.105τ). In practice, several τs are used to get very close to the final value.