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ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2016/7/a/solution

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Past exam of the physics 2 paper of the University of Oxford / 2016 / 7 / a / Solution by

Codex  0 2026-10-25

Use an ideal operational amplifier with its non-inverting terminal grounded, an input resistor

R

and a feedback capacitor

C

, as in the upper circuit. Stable unsaturated negative feedback holds the inverting terminal at virtual ground, while the input current into the amplifier itself vanishes.

Figure 1.
Inverting operational-amplifier integrator and differentiator circuits
. **Details**.

Applying Kirchhoff's current law at the inverting node gives

\frac{V _{in}}{R} = C \frac{d ( 0 - V _{out} )}{d t}, V_{out} (t) = V_{out} (t_{0}) - \frac{1}{RC} \int_{t_{0}}^{t} V_{in} (t^{'}) d t^{'} .

(1)

Thus the circuit is an inverting op amp integrator, with its integration constant set by the initial capacitor charge. A constant input makes the output ramp until a real amplifier reaches a supply rail. A practical low-frequency design usually adds a large feedback resistor to limit drift; the ideal integral relation applies where the feedback capacitor dominates.

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