Continuously equivalent networks and their application

The design of filters and amplifiers with minimum element distribution, minimum number of elements, minimum sensitivity to parameter change, etc., is important today. Since the synthesis problem usually has an infinite number of solutions, it is desirable to be able to transform a given design into a network with the same transfer function but with fewer elements or better element distribution, etc. The techniques of equivalent network theory, which apply a congruence transformation to a network matrix in order to generate equivalent networks, are used to do this. An extension of the Cauer formulation is discussed which, by applying the transformation directly to the element parameter matrices, places the elements of the transformed network directly in evidence. The transformation is then used to transform to an equivalent network whose elements differ from those in the original network by only an incremental amount. In the limiting case, a set of differential equations for the elements of the equivalent network results, in arbitrary inputs. The method of steepest descent is applied to choose these inputs in order to force the network to converge on the optimal design.