RC-Amplifier Resonators for Active Filters

Abstract

Active filters may consist of a cascade of isolated resonators, with each resonator supplying a single pair of complexconjugate poles. This paper supplies a detailed analysis of some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> -amplifier resonators whose characteristics are well suited for practical filters. The calculations are facilitated by a set of sensitivity identities. One resonator, taken from the Sallen and Key catalog, is shown to have extremely low passive sensitivities: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S_{Z}^{Q} = \pm \frac{1}{6}</tex> for all passive elements. Active <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> multiplication is used to extend the usefulness of the circuit. Analysis of the dualintegrator feedback resonator shows that it is notably insensitive to amplifier parasitics. Finally, the dual-integrator concept is realized in a resonator that uses only two amplifiers. The several resonators supply a range of useful pole- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> 's from low values to several hundred.