Operational Amplifiers

Integrator

The figure below shows an integrator. The capacitor of this circuit connects the inverting input and the output signal points of the operational amplifier in such a manner that the output potential varies as the time integral of the input potential.

For a capacitor

where i_C is the current through the capacitor, v_C is the voltage across the plates, and C is the capacitance. The inverting input is at a virtual ground (v_n = 0) and currents in the resistor and capacitor are equal (i_in = i_C). Application of Ohm's Law and substitution from the previous equation (i_C = C dv_C/dt) gives

Integrating both sides and rearranging gives

where v₀ is the initial value of output voltage at time t = 0. For the moment assume that v₀ has zero value, when,

where T = RC is the time constant of the integrator. This determines the amplitude scaling which is applied. The relation of the previous equation indicates that a voltage may be integrated with respect to time.