Answer 3
Yes, they are exactly equivalent, assuming the switching frequency (f) is much higher than the frequency components of V1 and V2. This equivalence is the basis for the technology known as "switched-capacitor filtering."

The current through the resistor on the left (a) is given by Ohm's Law:

[1]

The current through the circuit on the right (b) occurs in discrete packets. Each time the switch cycles from left to right, the voltage on the capacitor changes from V1 to V2. The amount of charge transferred is proportional to the voltage difference and the size of the capacitor:

[2]

When the switch cycles back to the left, the same amount of charge is transferred back into the capacitor. If the switching happens often enough, you can assume the current flow is continuous:

[3]

Combining this equation with equation 1 shows that:

[4]

A switched-capacitor filter replaces each resistor in a conventional analog filter network with a capacitor and a SPDT switch, as shown above. Because the capacitor transfers a fixed amount of charge between two circuit nodes on each cycle of the clock, it functions as a precise amount of conductance that's directly proprotional to the clock frequency. (Or, think of it as a resistance that's inversely proportional to the clock frequency.) Multiple such elements within a network track closely, making it possible to build high-order filters that maintain their performance over a wide range of frequencies.

The big advantage is that these are easy to build on an IC. It's difficult to make precise large-value resistors in silicon, but easy to make precise (and well-matched) small-value capacitors. For example, if you need a resistance of 100 kilohms, and your switching frequency is 1 MHz, the capacitor required is just 10 pF.

The worst drawback to these circuits is that they are discrete-time, so you do need to think about sampling issues such as aliasing. However, these issues tend to be minimized by the fact that the clock frequency is usually several hundred times the signal frequencies of interest.

Contributor: Dave Tweed

Published May 2003