Settling criterion — for a capacitor to “completely” (≈99%) charge during the on-time of a pulse in an RC network, the pulse width should be related to the time constant τ how?

Introduction / Context:In first-order systems, “complete” charging is a practical notion. Engineers often use 5τ as a rule of thumb for ≈99% settling of an exponential response. This item asks how the pulse width (the available charging time) must compare with τ to achieve near-full charging during a pulse interval.

Given Data / Assumptions:

• RC network with time constant τ = R * C.
• Exponential approach to a new level during a pulse.
• “Completely” interpreted as ≈99% (5τ rule).

Concept / Approach:The capacitor voltage follows v_C(t) = V_final + (V_initial − V_final) * exp(−t/τ). After t = 5τ, the residual error is exp(−5) ≈ 0.0067, or about 0.7%. Thus, a pulse that lasts at least 5τ allows the capacitor to get within about 1% of its asymptotic value for that interval. Shorter pulses yield proportionally less settling.

Step-by-Step Solution:

Verification / Alternative check:Simulation or measurement of step response shows 63% at 1τ, 95% at 3τ, ≈99% at 5τ, reinforcing the rule.

Why Other Options Are Wrong:

• “Less than 5τ” or “independent of τ”: would not reach ≈99% reliably.
• “Equal to exactly 5τ”: a useful benchmark, but “greater than or equal” is the safer, more general statement.

Common Pitfalls:Treating “complete” as 100% in finite time; exponentials only approach asymptote, so 5τ is a practical design target.

Final Answer:Greater than or equal to 5 time constants.