Integrating amplifier

An integrating amplifier produces an output proportional to the time integral of its input over an intended operating range. A common circuit uses an operational amplifier, an input resistor $R$ and a feedback capacitor $C$. With the non-inverting input held at zero, ideal negative feedback maintains a virtual ground at the inverting input. Negligible amplifier input current then givesThe initial output records the capacitor's initial charge. Texas Instruments' circuit note derives this inverting relation and explains why real integration is limited by the circuit time constant and amplifier bandwidth.^[ref]

For an original example, choose $R=100\mathrm{k\Omega }$ and $C=10\mathrm{nF}$, so $RC=1\mathrm{ms}$. A constant0.20volt input lasting2milliseconds changes the output by $-0.40\mathrm{V}$. Repeating ten such pulses without intervening input or integrator reset changes it by $-4.0\mathrm{V}$ in the ideal model. The output retains information about accumulated pulse area, rather than only the present input value.

A small offset also accumulates. With the same components, a residual input-equivalent1millivolt gives a slope of $-1\mathrm{V}{\mathrm{s}}^{-1}$. Starting from zero, an idealized negative output limit at5volts would be reached in5seconds. The finite supply range causes amplifier clipping; an ideal integral extending indefinitely is not physically available.

Putting a resistor $R_f$ in parallel with the feedback capacitor supplies a DC path. Under ideal amplifier assumptions its zero-state transfer function isAt frequencies where $|s{R}_{f}C|\gg 1$, this approaches $-1/(sRC)$; at low frequency it instead approaches the finite gain $-R_f/R$. The resistor prevents unrestricted DC accumulation while changing the lowest usable integration frequency.^[ref]

An integrating amplifier can therefore turn a short detector pulse into a measurable accumulated output, but its initial condition, reset, baseline, frequency range and output limits belong to the measurement model. A charge amplifier is a related current-input arrangement in which collected charge is converted through feedback capacitance.