Switched capacitor


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


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