Integrator settling — “5τ regardless of pulses” claim: “The steady-state condition of an RC integrator is reached after 5 time constants regardless of how many input pulses occur in that interval.” Evaluate this statement.

Introduction / Context:
The “5τ rule” is a convenient guideline for first-order responses to a step input: after about five time constants, a system is essentially settled near its final value. However, applying this rule blindly to RC integrators driven by pulse trains can be misleading. The steady-state level depends on waveform shape, duty cycle, and repetition rate, not just elapsed time.

Given Data / Assumptions:

• RC integrator with output across the capacitor.
• Input: not a single step but pulses that may repeat within the 5τ window.
• No saturation; linear behavior assumed.

Concept / Approach:
For a single step, the capacitor voltage approaches its final value as v_C(t) = V_final − (V_final − V_initial) * exp(−t/τ). After ~5τ, the error is under 1%. For a pulse train, the output is the convolution of the input with the RC exponential; the final steady-state depends on the periodic input’s average and duty cycle. Multiple pulses within 5τ continually perturb the capacitor, preventing it from reaching the same level as a single uninterrupted step would. Therefore, “steady-state after 5τ regardless of pulses” is incorrect.

Step-by-Step Solution:

Verification / Alternative check:
Simulate with τ = 10 ms and a 2 ms wide pulse repeating every 4 ms. Even after many multiples of 5τ, the waveform remains a periodic ripple around a duty-cycle-dependent mean, not a step-like flat level. This contradicts the claim.

Why Other Options Are Wrong:

• Correct: Misapplies the 5τ rule beyond its domain.
• True only for single-step inputs / Undetermined without duty cycle: These clarifications highlight the real dependency; the original statement’s universality is incorrect.

Common Pitfalls:
Using “5τ” as a magic number for every situation; forgetting that different inputs produce different steady-state behaviors even in the same RC network.

Final Answer:
Incorrect.