Op-amp integrator

An op-amp integrator (Miller integrator) is the inverting configuration with the feedback resistor replaced by a capacitor. Its output is the running time-integral of the input, scaled and inverted:

$v_O(t)=-\frac{1}{RC}\int v_I(t)dt$

Derivation

Keep the input resistor $R_1=R$; replace the feedback element $R_2$ with a capacitor $C$ from the inverting node to the output. Ground the non-inverting input.

Golden rule 2 of the Ideal op-amp model makes the inverting node a virtual ground: $v_{-}=0\text{V}$. The current drawn from the source through $R$ is therefore

$i_R=\frac{v_I-0}{R}=\frac{v_I}{R}$

Golden rule 1 says none of this current enters the op-amp, so all of it flows into the capacitor: $i_C=i_R$. The capacitor’s voltage is the voltage across it; one plate is at the virtual ground ($0\text{V}$) and the other is at $v_O$, so $v_C=0-v_O=-v_O$. The defining law of a capacitor is $i_C=C\frac{d{v}_C}{dt}$, so

$i_C=C\frac{d(-v_O)}{dt}=-C\frac{d{v}_O}{dt}$

Set $i_R=i_C$:

$\frac{v_I}{R}=-C\frac{d{v}_O}{dt}$

Solve for $v_O$ by integrating both sides with respect to time:

$\boxed{v_O(t)=-\frac{1}{RC}\int v_I(t)dt}$

The constant $\tau =RC$ is the integration time constant. A constant input $V_I$ produces a ramp output $v_O=-(V_I/RC)t$ — the textbook signature of integration.

Feedback resistor replaced by $C$ ⇒ gain ∝ 1/frequency (integration).

Output = time integral of input scaled by $-1/RC$.

Frequency-domain view

Replace $C$ by its impedance $Z_C=1/(j\omega C)$ and treat it as the feedback element in the inverting gain formula $-Z_f/Z_{in}$:

$\frac{v_O}{v_I}=-\frac{Z_C}{R}=-\frac{1/(j\omega C)}{R}=-\frac{1}{j\omega RC}$

The magnitude is $1/(\omega RC)$ — it falls with frequency at $-20\text{dB/decade}$ (doubling $\omega$ halves the gain), the Bode plot signature of integration ($1/j\omega$ is the frequency-domain integral operator). Low frequencies are amplified hugely; high frequencies are attenuated.

Magnitude −20 dB/decade; infinite ideal DC gain ⇒ a parallel $R_F$ for DC stabilisation.

The DC-saturation hazard

Set $\omega \to 0$ in $1/(\omega RC)$ and the gain goes to infinity — the ideal integrator has infinite DC gain. In practice that is dangerous: any tiny DC Input offset voltage at the input is integrated forever and ramps the output into the rail (Op-amp output saturation) given enough time. Real integrators always place a large resistor $R_F$ in parallel with $C$. $R_F$ caps the DC gain at the finite (but large) value $-R_F/R$ while leaving the integrating behaviour intact above the corner $1/(R_{F}C)$. The integrator is the workhorse of analog signal processing — PID controllers, active filters, sigma-delta modulators, waveform generators. Swapping which element is the capacitor gives the complementary Op-amp differentiator.