Capacitor behavior with long pulses: For a periodic pulse train where pulse width and the time between pulses are each ≥ 5 time constants, how does the capacitor charge and discharge each period?

Introduction / Context:Pulse-response in RC networks depends on the relation between time constants and pulse timing. If both the on-time and off-time are sufficiently long, the capacitor reaches its steady limits in each half of the cycle.

Given Data / Assumptions:

• Pulse width ≥ 5τ and spacing ≥ 5τ (τ = RC).
• Standard first-order RC behavior.
• Ideal rectangular pulses.

Concept / Approach:A first-order RC charges toward the applied level with the law v(t) = Vfinal * (1 - e^{-t/RC}). By t = 5τ, the exponential term e^{-5} ≈ 0.0067, meaning the capacitor is within 0.67% of its final value—effectively fully charged or discharged for practical purposes.

Step-by-Step Reasoning:During pulse (≥ 5τ): v_C ≈ V_high (fully charged)Between pulses (≥ 5τ): v_C ≈ 0 or the low reference (fully discharged)Therefore, each period contains both full charge and full discharge events

Verification / Alternative check:Using 1 - e^{-5} ≈ 0.9933 indicates a charge to 99.33% of the final value. Similarly, discharge leaves only 0.67%, which is practically zero.

Why Other Options Are Wrong:

• Partial charge/discharge options contradict the ≥ 5τ condition.

Common Pitfalls:Misinterpreting “fully” as exactly 100%. In engineering practice, ≥ 5τ is considered effectively full for time-domain design and timing calculations.

Final Answer:fully charge and fully discharge during each period of the input waveform