Switched capacitor

Switched capacitor is a circuit design technique for discrete time signal processing. It works by moving charges between different capacitors when switches are opened (off) and closed (on). Usually, non-overlapping signals are used to control the switches, so that not all switches are on simultaneously.

Voltage amplification can be achieved by moving a charge from a large capacitor to a small capacitor. Fact|date=May 2008

Voltage amplification can be achieved by repeatedly switching capacitors from a parallel arrangement with regard to the supply to a series arrangement with regards to the load. This arrangement is called a charge pump.

The simplest switched capacitor (SC) circuit is made of one capacitor and two switches which connect the capacitor with a given frequency alternately to the input and 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.

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 $C\_s$ has just charged:

:$Q\_s(t)\; =\; C\_s\; cdot\; V\_s(t),$

2) Because the feedback cap, $C\_\{fb\}$, 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 $C\_f$:

:$V\_\{fb\}(t)\; =\; frac\; \{Q\_s(t)\}\{C\_\{fb\; +\; V\_\{fb\}(t-T),$

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 $C\_f$ -- it increases (or decreases) its voltage each cycle according to the charge that is being "pumped" from $C\_s$ (due to the op-amp).

However, there is a more elegant way to formulate this fact if $T$ is very short. Let us introduce $dtleftarrow\; T$ 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).

See also

* Switched-mode power supply
* Charge pump

References

* Mingliang Liu, "Demystifying Switched-Capacitor Circuits", ISBN 0-7506-7907-7