The Synthesis of Minimum Sensitivity Networks

Abstract

The synthesis of networks with minimum sensitivity to element tolerances is studied from a computer viewpoint. The theory of equivalent networks is used to generate a sequence of networks whose transfer functions are identical to that of a given network but whose elements differ from one network to the next by an incremental amount.In the limit, differential equations result whose solution at any value of the independent variable give the elements of an equivalent network. Similarly, differential equations for the sensitivity of the transfer func- 1.Introduction:The advent of thin film and integrated circuit techniques has changed some of the criteria by which networks are evaluated. In particular, the new techniques place less emphasis on the number of elements in a network but require designs which are fairly insensitive to changes in the element values, due to the difficulty of maintaining tolerances at this stage in the development of the thin film techniques. Several authors have recently considered the design of insensitive networks [1][2][3][4] from various viewpoints. In this paper, it is shown that the theory of equivalent networks is very useful in the synthesis of networks insensitive to large element tolerances.Cauer in 1929 showed that by means of a congruence transformation, one physically realizable network could be generated from another in such a way that specified driving point and/or transfer functions were held invariant. 5 ' 6 This approach to network synthesis has been discussed by many authors and with the exception of the minimum-inductance transformation in filter theory has not realized its apparent potentialities.Useful results from this theory can be derived by using the concept of continuously equivalent network theory together witha digital computer for implementation. In this paper, this approach is used.Given a network with the desired transfer function (synthesized by any of the known schemes), the theory of equivalent networks is used to generate another network with the same transfer function but with elements differing from those of the original network by an incremental amount. In the limit, differential equations result whose solution at any value of the independent variable give the elements of an equivalent network. In this wayp it is possible to generate many networks equivalent to a given network but having widely differing element values while insuring at all times that no elements become negative. In addition, differential equations for the sensitivities of the transfer function to changes in element values are derived and an efficient computational algorithm j derived which allows rapid realization of minimal sensitive networks on a computer. In an example 30 element network, computing times of several minutes were found. Equivalent Network Theory:Consider a network with n independent node-pairs. Such a network is described by a set of n equations of the form The theory becomes more useful if we imagine a transformation from a given network to one whose eleme...

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