Design of single-voltage amplifier active filters for minimum open-loop gain sensitivity

Abstract

The pole sensitivity with reference to the open-loop gainbandwidth product <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|\omega_{a}|</tex> of the amplifier for a large class of single-voltage amplifier <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> networks is examined. Both finite gain and operational amplifier networks are considered under the assumption that the amplifier openloop gain effectively has a one-pole rolloff frequency characteristic. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p=\omega_{0}(-1/2Q+j\sqrt{1-1/4Q^{2}})</tex> a nominal pole of the transfer function, general expressions for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dp/d(1/ \omega_{a}), dQ/d(1/ \omega_{a})</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d \omega_{0}/d(1 / \omega_{a})</tex> are presented in terms of certain parameters of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> network. It is shown that these parameters may generally be found by inspection from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> openand short-circuit time constants and the high-frequency current gain of the passive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> network. The second-order transfer function case is considered in detail and the optimum choices of the pertinent passive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> network parameters for minimum sensitivity magnitude under various constraints are presented. A simple expression for the minimum possible magnitude of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dp / d(1 / \omega_{a})</tex> for a given second-order transfer function is derived. Finally, the application of these criteria to existing synthesis schemes in order to minimize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dp/d(1/ \omega_{a})</tex> or achieve other sensitivity properties, such as zero <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> sensitivity, is discussed.