In op-amp networks, replacing a resistor with a capacitor changes the operation. What does the placement of that capacitor determine?

Introduction:
Active integrators and differentiators are first-order building blocks for filtering, control, and analog computation. The key idea is that swapping a resistor for a capacitor in the input path or in the feedback path transforms an amplifier into a differentiator or integrator, respectively.

Given Data / Assumptions:

• Ideal op-amp behavior in the frequency range of interest.
• Small-signal linear operation (no slew-rate limiting or saturation).
• Single capacitor substitution relative to the classic inverting topology.

Concept / Approach:
In the inverting configuration, placing the capacitor in the feedback path yields an integrator: Vout is proportional to the time integral of Vin. Placing the capacitor in the input path yields a differentiator: Vout is proportional to the time derivative of Vin. Proper resistor-capacitor networks are added to stabilize real circuits and limit high-frequency noise gain.

Step-by-Step Explanation:
1) For an integrator, use a resistor at the input and a capacitor in feedback.2) The transfer is Vout = −(1 / (R * C)) * ∫ Vin dt, within the op-amp linear region.3) For a differentiator, use a capacitor at the input and a resistor in feedback.4) The transfer is Vout = −R * C * (dVin/dt), within the bandwidth where the ideal model is valid.

Verification / Alternative check:
Examine impedances: For integrator, Zf = 1 / (sC) grows at low frequency, forcing low-frequency gain to be small and high-frequency gain to be larger, consistent with integration. For differentiator, Zin = 1 / (sC) falls with frequency, so high-frequency components dominate, consistent with differentiation.

Why Other Options Are Wrong:
Open- or closed-loop gain: Closed-loop gain is set by network ratios; capacitor placement specifically selects integration vs differentiation behavior.

Saturation or cutoff: Those are operating region terms, not topological functions determined by the capacitor location.

Addition or subtraction: Summing or differencing relates to multi-input resistor networks, not capacitor placement.

Slew rate or bandwidth limit directly: While capacitors affect bandwidth, the principal role in this context is to shape the transfer as an integrator or differentiator.

Common Pitfalls:
Building an ideal differentiator without damping resistors causes excessive high-frequency noise and potential instability. Practical designs include series and parallel resistors to bound gain.

Final Answer:
integration or differentiation