Switched-Capacitor Circuits [Education

Abstract

The first publication of a switched-capacitor resistor occurred nearly 150 years ago in James Clark Maxwell’s pioneering book A Treatise on Electricity and Magnetism [1] . He pointed out that the average current through a periodically inverted capacitor C remains the same when it is replaced by a resistor of value ${R} = {T}/{2}{C}$ , where ${T}$ is the period of inversions. He used this equivalence to give a method for measuring capacitance using a circuit employing a battery and galvanometer. The next publication came nearly a century later, when D. L. Fried proposed the idea of sampled-data analog filters [2] , containing only switches, capacitors and (if necessary) amplifiers. His paper showed how to realize the equivalent of a resistor using two switches and a capacitor. A motivation for using such circuits may be found from the history of analog filters. These were developed for telephony, and used initially resistors, capacitors and inductors. Inductors were bulky and lossy, and were replaced at the earliest opportunity by alternative circuits using amplifiers. The resulting active-RC filters were realized by discrete elements: capacitors, resistors and integrated-circuit amplifiers. With the development of integrated-circuit technology, there was strong motivation to put these filters on a single-substrate IC. However, the absolute values of resistors and capacitors could only poorly be controlled by the fabrication process: errors of 20-30% were common. Since the errors of resistors and capacitors were not tracking each other, the errors of time constants given by RC products were unacceptably high. This made their frequency responses unpredictable. Trimming could be used to tune such filters, but this was expensive. When the resistors were replaced by their switched-capacitor (SC) equivalents, the RC time constants were replaced by time constants of the form ${TC}_{1}/{C}_{2}$ . Since the switching period ${T}$ can be accurately controlled, as can the ratio of on-chip capacitors, the SC filters (SCFs) could be implemented with high accuracy. The design of such filters initially was based on those of active-RC ones, but it was soon recognized that they can be more effectively designed directly in the sampled-data domain, in terms of the ${z}$ variable, similarly to digital filters. Although other design techniques exist, the most popular one constructs higher-order filters as a cascade of lower-order sections, such as biquadratic filter or “biquad” shown in Figure 1 .