Time Response of Reactive Circuits Questions

• 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.
• Where to measure — RC differentiator output node: In a standard passive RC differentiator, is the output correctly taken across the capacitor, or across the resistor?
• RC integrator fault diagnosis — zero output reading: “If the output of an RC integrator is zero volts, the capacitor might be open.” Decide whether this is a reliable diagnostic conclusion for the standard topology (output across the capacitor).
• RC pulse response with duty cycle — a repetitive rectangular pulse (50% duty) is applied to the input of an RC “integrator.” If one time constant τ is less than one-fifth of the pulse width (τ < PW/5), will the capacitor be able to essentially charge and discharge to its final values each half-cycle (i.e., reach ≈99% within 5τ)?
• RL pulse shaping — in a basic RL differentiator (step/pulse shaping network), is the output conventionally taken across the inductor to emphasize rapid changes (spikes at edges)?
• RC integrator under pulses — when a repetitive-pulse waveform drives an RC integrator, does the output waveshape depend on the relationship between the time constant τ and the pulse duty cycle/width?
• 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?
• Repair for missing schematic — “If the pulse width were cut in half in the given RC pulse circuit, the voltage across the resistor at the end of the pulse would be ______.” Without R, C, τ, and the original pulse width, can a unique numeric value be selected?
• Repair for missing values — “If the pulse source has an internal resistance of 80 Ω in the given circuit, it will take ______ for the output voltage to decrease to zero.” Can a time-to-zero be specified for a first-order RC without knowing C (and topology)?
• Repair for missing figure — “The capacitor voltage at the beginning of the second pulse is ______. Assume the capacitor is initially uncharged.” Without the component values, period, and τ relative to the pulse timing, can we determine the exact voltage?
• Exponential decay reality check — after the rising edge in a first-order RC pulse network, how long does it take for the voltage across the resistor to “decrease to zero”? (Interpret “zero” as a practical threshold; without τ the value is not unique.)
• Fault effect on time constant — in an RC integrator or differentiator, which component fault most directly decreases the effective time constant τ = R * C (thus collapsing the time-based behavior)?
• Topology swap thought experiment — starting with a classic RC integrator (series resistor feeding a capacitor to ground, output across the capacitor), if the positions of the components are swapped (series capacitor feeding a resistor to ground, output across the resistor), what does the new circuit implement?
• RC charging (repaired for solvability): An RC low-pass (first-order) is driven by two identical rectangular pulses from 0 V to 12 V, applied back-to-back with no gap. Each pulse width equals 5 time constants (5τ), so the node sees a total “high” duration of 10τ before returning low. What is the output voltage across the capacitor at the end of the second pulse (assume it started at 0 V and the op-amp/buffer, if present, is ideal)?
• Foundations — what do we call the characteristic time measure for first-order charging/discharging (e.g., in RC circuits) that sets the exponential response speed?
• RC settling time (repaired for solvability): A first-order RC output is considered “at maximum value” for practical purposes when it has reached about 99.3% of its final level (i.e., after 5 time constants). If the circuit’s time constant is 243.9 µs, how long does it take to reach this practical maximum?