General Synthesis Method for Dispersively Coupled Resonator Filters With Cascaded Topologies

This article presents a new synthesis technique for circuits that may include dispersively coupled resonators and admitting an overall cascaded topology. A decomposition technique of the Darlington type is first introduced to split the original response $S$ of the filter, taken as its scattering matrix, into $m$ subresponses $S^{1}, {\dots },S^{m}$ corresponding to each subblock of the cascaded circuital structure. Each individual subresponse $S^{k}$ is then synthesized separately. In this article, the state space equations governing the model of dispersively coupled resonators are detailed. An extension to the case of dispersive coupling of the shortest path rule, which determines the maximum number of finite TZs realizable by a given topology, is then introduced. Congruent transformations that extend the concept of rotations or similarity transformations while preserving the filter response are exploited to reduce the individual synthesis problems to the determination of a basis of vectors verifying certain orthogonality relations. A direct synthesis technique for dispersive building blocks, such as duplets, triplets, and quadruplets, is then given in the form of an orthogonalization procedure used for the computation of the desired basis. This approach is then combined with the aforementioned decomposition technique to produce a versatile algorithm able to synthesize hybrid circuits made of cascaded subblocks of different orders and types that implement each a subset of the overall TZs by means of coupling topologies containing a mixture of dispersive and nondispersive couplings. The first synthesis example is detailed where two dispersive duplets are combined with a classical quadruplet to realize a symmetric 6–4 response. A hardware implementation of the synthesized circuit is presented in combline technology. The second example proposes a slightly more involved coupling topology able to realize 10–8 asymmetric responses by means of four cascaded basic dispersive blocks.