Optimum Design of Single-Stage Gyrator-RC Filters with Prescribed Sensitivity

Abstract

The design of low-frequency filters by means of RC feedback around active elements has long been known. The advent of the transistor has renewed interest in this field and has necessitated a more exact synthesis procedure. This is because in the classical design the active element is assumed to be an ideal voltage amplifier with infinite input impedance, zero output impedance and zero reverse transmission. The transistor satisfies the dual ideal assumptions to a considerably lesser degree than the vacuum tube. The sensitivity of the transistor parameters to temperature also requires more attention to be paid to the effect of parameter variations on the filter response. The paper extends the classical theory in several directions: 1) A synthesis procedure is developed in which the finite input and output impedances and reverse transmission of the active element are taken into account in an exact manner. 2) The freedom that exists is used to optimize the synthesis so as to permit design with a "least active" element. 3) Expressions are developed for the sensitivity of the filter to the four low-frequency active parameters and to the passive parameters. It is presumed that the design specifications include a statement of the extent of active and passive parameter variation and the tolerances on the filter response. The procedure for satisfying such specifications is an integral part of the design procedure. 4) Under the constraint of fixed source and load impedances and desired parameter insensitivity, the design achieves maximum gain. An example which includes all the above features is worked out in detail. The principal limitations of the design procedure are: 1) It is restricted to a transfer function of second degree so that if sharper cut-offs are needed, several basic sections must be used, separated by isolating stages. 2) It is best suited to achieve low-pass or moderate band-pass characteristics.