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Operational Amplifiers


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.

Integrator circuit
Integrator circuit.

For a capacitor

Capacitor current

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

Integrator voltage

Integrating both sides and rearranging gives

Integrator output

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

Integrator output

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.

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