Integrators

The Ideal Integrator

The ideal integrator, shown in Fig. 1., makes use of an ideal operational amplifier with $A_{v}\rightarrow \infty$, $R_{i}\rightarrow \infty$, and $R_{o}=0$. The current through the resistor, $i_{R}$, can be expressed as:

$i_{R}={\frac {v_{i}}{R}}=i_{C}=-C\cdot {\frac {\partial v_{o}}{\partial t}}$

(1)

Thus, we can write the integrator output voltage, $v_{o}$, as:

$v_{o}=-{\frac {1}{RC}}\int v_{i}\cdot dt$

(2)

In the Laplace domain:

${\frac {v_{i}\left(s\right)}{R}}=-sC\cdot v_{o}\left(s\right)$

(3)

Or equivalently:

${\frac {v_{o}\left(s\right)}{v_{i}\left(s\right)}}=-{\frac {1}{s\cdot RC}}=-{\frac {1}{s\cdot \tau }}=-{\frac {\omega _{0}}{s}}$

(4)