RC differentiator classification: An RC differentiator behaves as which type of linear filter with respect to frequency response?

Introduction / Context:
Understanding the frequency-domain identity of time-domain circuits helps bridge intuition between transient and steady-state behavior. An RC differentiator emphasizes rapid changes (high-frequency components) and attenuates slow variations (low-frequency components).

Given Data / Assumptions:

• Passive RC differentiator: capacitor in series with input, resistor to ground, output across the resistor.
• Small-signal linear analysis.
• No active gain elements.

Concept / Approach:
The transfer function magnitude increases with frequency over the useful operating region. At low frequency, the capacitor's reactance is high, blocking current and yielding near-zero output—characteristic of high-pass filters. At high frequency, the capacitor reactance drops, passing more signal to the output.

Step-by-Step Reasoning:
Impedances: Zc = 1/(j * 2 * pi * f * C), Zr = R.Voltage divider: Vout/Vin = Zr / (Zr + Zc) = R / (R + 1/(j * 2 * pi * f * C)).As f → 0, |Zc| → ∞ ⇒ Vout ≈ 0 (rejects low frequency).As f increases, |Zc| decreases ⇒ Vout increases (passes high frequency).

Verification / Alternative check:
Time-domain view: A differentiator produces spikes at edges (high dV/dt), which are inherently high-frequency components; thus, it acts as a high-pass element for signals.

Why Other Options Are Wrong:

• Low-pass: Corresponds to the RC integrator output across C.
• Band-pass / band-stop: Require at least second-order (or more complex) behavior; a single RC differentiator is first-order high-pass.

Common Pitfalls:
Confusing integrator and differentiator placements (output across C vs R). Remember: differentiator output is across R and behaves as a high-pass filter.

Final Answer:
high-pass filter