A new statistical multiparameter measure of sensitivity is presented. The new measure realistically accounts for component tolerances and component tolerance correlations and is shown to be particularly useful in the design of integrated circuits. For illustration the measure is used to analyze several high- Q low-sensitivity networks. It is shown that for high- Q networks the new measure is readily interpreted in terms of the Q and center frequency sensitivities of the circuit elements. The measure is also generalized for high- Q analysis to account for the error in realization due to the ampliler pole as well as the element variations. This enables an evaluation of the circuit response versus center frequency to be performed.

http://ieeexplore.ieee.org/iel5/8147/23414/01083355.pdf

Multiparameter sensitivity in active RC networks

A comparative study of single-amplifier filters, fifteen different two-amplifier circuits, and the state-variable three-amplifier configuration is given. The figures-of-merit used are the passive sensitivities, the \omega_0 gain-sensitivity products, the Q gain-sensitivity products, and the \omega_0 and Q deviations due to the amplifiers' limited bandwidth. In addition, practical matters such as input impedance and component spread are considered. Among the circuits considered a particular twoamplifier resonator emerges as the "best" biquadratic section. Some of the interesting results include a unified expression, applicable to a large class of low-sensitivity circuits, for the \omega_0 deviation due to the amplifier's limited frequency response. In this expression, which applies for all active filters which have low \omega_0 active sensitivities, the only configurationdependent quantities are the Q gain-sensitivity products.

Sensitivity and frequency limitations of biquadratic active filters

A sensitivity analysis for a general second-order multiamplifier RC-active network is described. The conditions for minimized sensitivity to amplifier gain-bandwidth product (GBWP) are established after first formulating the transfer function gain dependence under the assumption of a single-pole representation of the complex amplifier gain. On the basis of a perturbation technique, approximate expressions are presented for the evaluation of the selectivity and frequency sensitivities. By distinguishing a general form of denominator decomposition, the methods available for realization of high Q's with improved sensitivity are explored. Three special cases of the decomposition are identified; two of these, in common with the general case, suggest the use of twin-T null networks with their attendant high passive sensitivities, but the third gives rise to an additional group of circuits 'realizable as double first-order all-pass sections. By comparison with some previous circuits which require matched amplifiers to achieve a low sensitivity, the proposed configurations are shown to be unconditionally insensitive to variations of the GBWP of the amplifiers.

RC-active networks with reduced sensitivity to amplifier gain bandwidth product

A unified approach is presented for generating configurations for circuits containing one operational amplifier together with RC elements and capable of realizing a general second-order voltage transfer function. Systematic application of the generation procedure leads to various single amplifier biquads with the known circuits, as well as some new ones, obtained as special cases. The insight gained from the generalized approach as well as the results of a general sensitivity analysis (including the effects of the amplifier limited bandwidth) offer a method for categorizing and classifying single amplifier filters. The different filter classes are illustrated and compared with the aid of extensive design tables. General equations for the effect of the amplifier limited bandwidth are also given.

Generation and classification of single amplifier filters

Abstracr-A class of positive-feedback active RC filters which realize high stable Q's is presented. This class generalizes the earlier circuits of Tarmy and Ghausi and of Moschytz. The performance of the circuits presented is analyzed using a one-pole rolloff model for the operational amplifier. The Q and wO variations due to one-pole rolloff are tabulated for different filters. It is shown that the positive-feedback filters have lower Q sensitivity to the gain-bandwidth product than the state-variable filter. The theoretical behavior of the circuit Q is experimentally verified.

http://ieeexplore.ieee.org/iel5/31/23433/01083859.pdf

Comparison of a class of active filters with respect to the operational-amplifier gain-bandwidth product

Expressions relating Q and pole-sensitivities of a. third-order filter to the readily calculable denominator polynomial coefficient sensitivities are derived. Use of these expressions for the sensitivity analysis of two new third-order active RC filters are illustrated.

Sensitivity analysis of third-order filters†

A new active RC bandpass filter containing two resistors, two capacitors, and two voltage amplifiers is presented. The Q sensitivities of the filter with respect to the passive elements can be made zero, and the Q sensitivities with respect to the gains of the amplifiersare less than one half.

http://ieeexplore.ieee.org/iel5/5/31127/01449453.pdf

Extremely low sensitivity active RC filter

Pole-polynomial decomposition forms the basis for classifying active RC filters into ten groups. This classification serves two purposes. First, the gain and sensitivity properties are directly related to the type of decomposition. Thus the optimum gain and sensitivity values for any filter structure can be easily determined by simply finding the classification to which the filter belongs. Second, active filters have been developed for two new types of polynomial decomposition for which no active filters previously existed. These new circuits have very good gain-sensitivity properties.

Gain and sensitivity limitations of active RC filters

A root locus with respect to the active parameter for any second-order active RC filter is circular with center on the real axis. It is shown that for any pole pair Q and pole frequency \omega_0 , the circular root locus centered at -2Q \omega_0 on the real axis and of radius \omega_{0}\sqrt{4Q^{2}-1} exhibits zero Q -sensitivity and zero gain-sensitivity product (GSP) with respect to the active parameter. Furthermore, realization techniques to achieve zero Q -sensitivity and zero GSP for prescribed Q and \omega_0 are outlined. Experimental results agree closely with the theoretical expectations.

M. A. Soderstrand

Papers

Sensitivity analysis of third-order filters†

Extremely low sensitivity active RC filter

Gain and sensitivity limitations of active RC filters

Design of active RC filters with zero gain-sensitivity product