For an RC differentiating circuit driven by a periodic pulse waveform, which statement correctly captures the two possible regimes relating the pulse width (tw) to the circuit time constant (tau)?

Introduction:
In RC differentiators, performance depends on comparing the input pulse width tw to the circuit time constant tau = R * C. Designers commonly distinguish between "long" pulses (tw much greater than tau) and "short" pulses (tw much less than tau) to anticipate output spike shapes and amplitudes.

Given Data / Assumptions:

• First-order RC differentiator with time constant tau.
• Periodic pulse input with width tw and period Tperiod (not used directly here).
• Rule-of-thumb boundary at ~5 * tau separates long and short regimes.

Concept / Approach:
When tw ≥ 5 * tau, the capacitor has time to charge and discharge substantially within each pulse edge, yielding narrow, well-defined spikes primarily at transitions. When tw < 5 * tau, the circuit operates in a strongly differentiating regime where the output resembles brief spikes with limited decay during the pulse, emphasizing edges over the flat top.

Step-by-Step Reasoning:
1) Define tau = R * C as the characteristic time scale.2) For tw ≥ 5 * tau, treat the pulse as "long"; the capacitor's exponential response largely settles between edges.3) For tw < 5 * tau, treat the pulse as "short"; output is dominated by edge spikes, with minimal settling between edges.4) Hence the two practical conditions are tw ≥ 5 * tau or tw < 5 * tau.

Verification / Alternative check:
Simulate with typical values (e.g., R = 1 kΩ, C = 1 nF → tau = 1 µs). Compare outputs for tw = 10 µs (≥ 5 * tau) versus tw = 0.2 µs (< 5 * tau). The first shows distinct charge/discharge, the second shows sharp, narrow spikes.

Why Other Options Are Wrong:
Options a and b omit the complementary short-pulse regime or duplicate the same inequality; option c covers only short cases; option e is a single value and not a regime.

Common Pitfalls:
Treating 5 * tau as an exact boundary; it is a heuristic. Real component tolerances and loading shift the practical split, but the two regimes remain conceptually valid.

Final Answer:
tw ≥ 5Τ or tw < 5Τ