First-order transient rule of thumb — after exactly one time constant (τ) in a first-order RL or RC response, by what percentage has the transient changed, and what percentage remains?

Introduction / Context:
First-order circuits (RL and RC) exhibit exponential responses to steps. Engineers routinely estimate settling using the time constant τ. A classic benchmark: after one τ, the exponential has moved 63.2% of the way from its initial value toward its final value, leaving 36.8% of the excursion to go. This aids quick design checks without a calculator.

Given Data / Assumptions:

• First-order system with step input; no oscillation.
• Time constant τ = L/R for RL or τ = R*C for RC.
• We are asking for normalized percentages independent of the specific values.

Concept / Approach:
Standard exponential form: y(t) = y_final + (y_initial − y_final) * exp(−t/τ). At t = τ, exp(−1) ≈ 0.3679. Therefore, the remaining difference from final is 36.79% (≈ 36.8%), and the amount of change completed is 1 − 0.3679 ≈ 0.6321 or 63.21% (≈ 63.2%). These percentages hold for rising or falling responses (signs invert but magnitudes match).

Step-by-Step Solution:

Verification / Alternative check:
At t = 2τ, remaining ≈ 13.5%; at t = 3τ, ≈ 5.0%; at t = 5τ, ≈ 0.67%. These checkpoints form the common “1τ = 63.2%” mnemonic and the “5τ ≈ settled” rule.

Why Other Options Are Wrong:

• 36.8% changed / 63.2% remaining: Reversed.
• 13.5% or 86.5% pairs: Those correspond to t = 2τ (13.5% remaining) and its complement, not 1τ.

Common Pitfalls:
Confusing “remaining” with “changed,” or mixing the percentages for 1τ and 2τ. Always derive from exp(−t/τ) to avoid memory slips.

Final Answer:
63.2% changed, 36.8% remaining after one time constant.