Differentiator with very long pulses — spike approximation: When the input pulse width is much longer than 5 time constants (T_p ≫ 5τ), can the output of an RC differentiator be considered a pair of narrow spikes at the transitions?

Introduction / Context:
Edge detection is a primary use case for RC differentiators. When a rectangular pulse is very wide compared with the circuit time constant, the differentiator s output between edges is negligible. Recognizing when the spike approximation is valid helps simplify timing analysis in digital interfacing and trigger circuitry.

Given Data / Assumptions:

• RC differentiator: series C, shunt R, output across R.
• Pulse width T_p much larger than 5τ = 5 * R * C.
• Symmetric high and low levels with a return to baseline.

Concept / Approach:
At each transition, dv/dt is large, producing a current impulse through the capacitor and a correspondingly sharp voltage across the resistor. After a few time constants, the capacitor current decays toward zero and the output settles very close to 0 V for the remainder of the pulse plateau. With T_p ≫ 5τ, most of the period is flat at zero output, and the response looks like two spikes per pulse—one positive at the rising edge and one negative at the falling edge.

Step-by-Step Solution:

Verification / Alternative check:
Simulations with τ = 1 ms and pulses of width 50 ms show spikes of a few milliseconds width and nearly zero output elsewhere. Scope captures in labs confirm this classical behavior used for trigger generation.

Why Other Options Are Wrong:

• Incorrect: Contradicts standard differentiator operation for T_p ≫ 5τ.
• True only if the capacitor is leaky / only for certain duty cycles: Leakage and duty cycle adjust amplitudes and baselines but do not negate the spike approximation under the stated condition.

Common Pitfalls:
Using T_p only slightly greater than τ and expecting perfect spikes; forgetting that finite source and load impedances shape spike width and amplitude.

Final Answer:
Correct.