The 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 from the input to the output at the switching frequency . 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 stored in the capacitor CS is:
When S2 is closed, some of that charge is transferred out of the capacitor, after which the charge that remains in capacitor CS is:
Thus, the charge moved out of the capacitor to the output is:
Because this 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 V be the voltage across the SC from input to output. So:
So the equivalent resistance R (i.e., the voltage–current relationship) is:
Thus, the SC behaves like a lossless resistor whose value depends on capacitance CS and switching frequency 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 (i.e., it is a programmable resistance). 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.
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