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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 *v*_{0} is the initial value of output voltage at time
*t* = 0. For the moment assume that *v*_{0} 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.