Time constant in first-order dynamics: by definition, the time constant is the time required for the measured (output) variable to reach what fraction of its ultimate change after a step input?

Introduction / Context:
First-order systems (e.g., thermowells, level tanks, RC circuits) are characterized by a single time constant that determines how quickly the output approaches a new steady state after a step input. Clear definitions avoid confusion when comparing dynamics or tuning controllers.

Given Data / Assumptions:

• System: first-order, linear, time-invariant.
• Input: step change; Output: measured variable y(t).
• Ultimate change: the final steady-state difference y(∞) − y(0).

Concept / Approach:
The step response of a first-order element is y(t) = y(∞) − [y(∞) − y(0)] * exp(−t/τ). At t = τ, exp(−1) ≈ 0.368; therefore y(τ) − y(0) = 0.632 * [y(∞) − y(0)]. Hence the time constant τ is the time for the output to achieve 63.2% of its total change. Transportation lag and dead time are separate concepts representing pure delay without dynamic smoothing and should not be conflated with τ.

Step-by-Step Solution:
Write the standard first-order step response equation.Evaluate at t = τ to find the fraction of ultimate change (0.632).Identify the definition in the provided options.

Verification / Alternative check:
Plotting the response shows a prominent 63.2% point at one time constant; at two time constants the response reaches 86.5%, and at four it is 98.2%, which aligns with rule-of-thumb settling estimates.

Why Other Options Are Wrong:
Transportation lag/dead time: Pure delays; not equal to τ in general.Controlled variable phrasing can be ambiguous; the precise definition uses the measured output.50% error reduction corresponds to ln(2) * τ, not τ itself.

Common Pitfalls:
Confusing lag with delay; using 50% instead of 63.2%; assuming the definition depends on controller action rather than plant dynamics.

Final Answer:
The measured variable reaches 63.2% of its ultimate change