Switched capacitor

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Switched Capacitor (SC) is a circuit made of one capacitor and two switches which connect the capacitor with a given frequency alternately to the input an output of the SC. This simulates the behaviour of a resistor, so SCs are used in integrated circuits instead of resistors. The resistance is set by the frequency.

  S1    S2
    /    /
o--/ ---/ --o
      |
in    |    out
     ===
      | Cs
      |
o-----------o

Often you will find this structure in place of the resistance of an integrator; see Operational amplifier applications. In turn, filters implemented with these integrators are termed Switched capacitor filters.

Let us analyze what happens in this case. Denote by T=1/f 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 Cs has just charged:
Q_s(t) = C_s \cdot V_s(t)\,

2) Because the feedback cap, Cfb, is suddenly charged with that much charge (by the opamp, which seeks a virtual shortcircuit between its inputs):
Q_{fb}(t) = Q_s(t) + Q_{fb}(t-T)\,

Now dividing 2) by Cf:
V_{fb}(t) = \frac {Q_s(t)}{C_{fb}} + V_{fb}(t-T)\,

And inserting 1):
V_{fb}(t) = \frac {C_s}{C_{fb}} \cdot V_s(t) + V_{fb}(t-T)\,

This last equation represents what is going on in Cf -- it increases (or decreases) its voltage each cycle according to the charge that is being "pumped" from Cs (thanks to the op-amp, sure). This is what you must store in your brain.


However, there is a more elegant way to formulate this fact if T is very short. Let us introduce dt\leftarrow T and dV_{fb}\leftarrow V_{fb}(t)-V_{fb}(t-dt) and rewrite the last equation divided by dt:

\frac {dV_{fb}(t)}{dt} = f \frac {C_s}{C_{fb}} \cdot V_s(t)\,

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

V_{OUT}(t) = -V_{fb}(t) = - \frac{1}{\frac{1}{fC_s}C_{fb}} \int V_s(t)dt \,

Note that this is an integrator with an "equivalent resistance" R_{eq} = \frac{1}{fC_s}. 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).

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