For a passive RC integrator circuit, the input is connected to a resistance while the output voltage is taken from across a capacitor being the exact opposite to the RC Differentiator Circuit. The capacitor charges up when the input is high and discharges when the input is low.
In Electronics, the basic series connected resistor-capacitor (RC) circuit has many uses and applications from basic charging/discharging circuits to high-order filter circuits. This two component passive RC circuit may look simple enough, but depending on the type and frequency of the applied input signal, the behaviour and response of this basic RC circuit can be very different.
Thus the rate of charging or discharging depends on the RC time constant, τ = RC. Consider the circuit below.
RC Integrator
For an RC integrator circuit, the input signal is applied to the resistance with the output taken across the capacitor, then VOUT equals VC. As the capacitor is a frequency dependant element, the amount of charge that is established across the plates is equal to the time domain integral of the current. That is it takes a certain amount of time for the capacitor to fully charge as the capacitor can not charge instantaneously only charge exponentially.
Therefore the capacitor current can be written as:
This basic equation above of iC = C(dVc/dt) can also be expressed as the instantaneous rate of change of charge, Q with respect to time giving us the following standard equation of: iC = dQ/dt where the charge Q = C x Vc, that is capacitance times voltage.
The rate at which the capacitor charges (or discharges) is directly proportional to the amount of the resistance and capacitance giving the time constant of the circuit. Thus the time constant of a RC integrator circuit is the time interval that equals the product of R and C.
Since capacitance is equal to Q/Vc where electrical charge, Q is the flow of a current (i) over time (t), that is the product of i x t in coulombs, and from Ohms law we know that voltage (V) is equal to i x R, substituting these into the equation for the RC time constant gives:
RC Time Constant
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