Switched Capacitor - The Parasitic Sensitive Integrator

The Parasitic Sensitive Integrator

Often switched capacitor circuits are used to provide accurate voltage gain and integration by switching a sampled capacitor onto an op-amp with a capacitor in feedback. One of the earliest of these circuits is the parasitic-Sensitive integrator developed by the Czech engineer Bedrich Hosticka. Let us analyze what happens in this case. Denote by the switching period. Recall that in capacitors charge = capacitance x voltage. Then, at the instant when S1 opens and S2 closes, we have the following:

1) Because has just charged:

2) Because the feedback cap, is suddenly charged with that much charge (by the opamp, which seeks a virtual short circuit between its inputs):

Now dividing 2) by :

And inserting 1):

This last equation represents what is going on in - it increases (or decreases) its voltage each cycle according to the charge that is being "pumped" from (due to the op-amp).

However, there is a more elegant way to formulate this fact if is very short. Let us introduce and and rewrite the last equation divided by dt:

Therefore, the op-amp output voltage takes the form:

Note that this is an integrator with an "equivalent resistance" . This allows its on-line or runtime adjustment (if we manage to make the switches oscillate according to some signal given by e.g. a microcontroller).