# Switched capacitor

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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:

:$Q_s\left(t\right) = C_s cdot V_s\left(t\right),$

2) Because the feedback cap, $C_\left\{fb\right\}$, is suddenly charged with that much charge (by the opamp, which seeks a virtual shortcircuit between its inputs):

:$Q_\left\{fb\right\}\left(t\right) = Q_s\left(t\right) + Q_\left\{fb\right\}\left(t-T\right),$

Now dividing 2) by $C_f$:

:$V_\left\{fb\right\}\left(t\right) = frac \left\{Q_s\left(t\right)\right\}\left\{C_\left\{fb + V_\left\{fb\right\}\left(t-T\right),$

And inserting 1):

:$V_\left\{fb\right\}\left(t\right) = frac \left\{C_s\right\}\left\{C_\left\{fb cdot V_s\left(t\right) + V_\left\{fb\right\}\left(t-T\right),$

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 $dV_\left\{fb\right\}leftarrow V_\left\{fb\right\}\left(t\right)-V_\left\{fb\right\}\left(t-dt\right)$ and rewrite the last equation divided by dt:

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

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

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

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

* Switched-mode power supply
* Charge pump

References

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

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