The switched capacitor resistor

Switched-capacitor resistor

The simplest switched capacitor (SC) circuit is the switched capacitor resistor, made of one capacitor C and two switches S₁ and S₂ which connect the capacitor with a given frequency alternately to the input and output of the SC. Each switching cycle transfers a charge q from the input to the output at the switching frequency f. Recall that the charge q on a capacitor C with a voltage V between the plates is given by:

where V is the voltage across the capacitor. Therefore, when S₁ is closed while S₂ is open, the charge transferred from the source to C_S is:

and when S₂ is closed while S₁ is open, the charge transferred from C_S to the load is:

Thus, the charge transferred in each cycle is:

Since a charge q is transferred at a rate f, the rate of transfer of charge per unit time is:

Note that we use I, the symbol for electric current, for this quantity. This is to demonstrate that a continuous transfer of charge from one node to another is equivalent to a current. Substituting for q in the above, we have:

Let us define V, the voltage across the SC from input to output, thus:

We now have a relationship between I and V, which we can rearrange to give an equivalent resistance R:

Thus, the SC behaves like a resistor whose value depends on C_S and f.
The SC resistor is used as a replacement for simple resistors in integrated circuits because it is easier to fabricate reliably with a wide range of values. It also has the benefit that its value can be adjusted by changing the switching frequency. See also: operational amplifier applications.
This same circuit can be used in discrete time systems (such as analog to digital converters) as a track and hold circuit. During the appropriate clock phase, the capacitor samples the analog voltage through switch one and in the second phase presents this held sampled value to an electronic circuit for processing.