Estimating unstable poles in simulations of microwave circuits

International audienceThe impedance of a microwave circuit has an infinite number of poles due to the distributed elements. This complicates locating those poles with a rational approximation. In this paper, we propose an algorithm to locate the unstable poles of a circuit with distributed elements. The proposed method exploits the fact that a realistic circuit can only have a finite number of unstable poles. We first determine the unstable part whose poles coincide with the unstable poles of the circuit. A rational approximation of the unstable part is used to estimate the unstable poles. Having the ability to trace a circuit's poles as a function of the circuit parameters is a useful design tool. Pole tracking techniques have been used for the design of oscillators [1], [2], to stabilise power amplifiers [3]-[5] and during the design of frequency dividers [1], [6]. The core of a pole tracking tool is a robust automatic algorithm to estimate the poles of a circuit. To determine the poles of a circuit, a two-step procedure is followed. First, an impedance Z(jω) of the circuit is determined at discrete frequency points between 0 and f max with an AC simulation (Fig. 1). Then, the poles of Z(jω) are determined in a post-processing step. In lumped circuits, Z(jω) is a rational function, so a rational approximation can be used to determine the circuit poles. The impedance presented by a circuit with distributed elements, like transmission lines, is non-rational but still meromorphic 1 [7]. This makes estimating the poles of a microwave circuit more difficult. A good fit of Z(jω) can be obtained with a high-order rational function, but the rational approximation will contain spurious poles that do not correspond to poles of the underlying function [8]. To circumvent this problem, another approach proposed in [9], is to compute low-order rational approximants of the circuit's response restricted to small frequency intervals. This local version of the rational approximation scheme, yields precise estimates of poles when these are close enough to the imaginary axis. In this paper, we propose an algorithm that can estimate the unstable poles of a circuit without performing a rational approximation of a non-rational function. The proposed technique exploits the fact that the equilibrium solution of a realistic circuit 2 can only have a finite amount of poles in the right half-plane [10]

Extraction Techniques for Generic Cascaded Synthesis

International audienceThe aim of this paper is to demonstrate that extraction techniques can be efficiently combined with coupling matrix synthesis techniques. An extension of a recently presented method based on a functional decomposition of the scattering matrix is introduced in order to add dangling resonators based on non resonating nodes among possible buildings blocks. This leads to a complete analytical procedure for hybrid realisations including classical N-tuplets, frequency depending couplings, constant phase shifts and NRNs. After detailing some elementary concepts of functional extraction techniques for NRN's, a synthesis and design example is presented to illustrate the approach

A Systematic Algorithm for the Design of Hybrid Waveguide-Microstrip Transversal Microwave Filters

A novel algorithm to design hybrid waveguide-microstrip transversal microwave filters is proposed in this study. The idea consists in isolating each resonator involved in the filter structure, to synthesise individually its frequency response, both in resonant frequency and bandwidth. These separated optimisations lead to the dimensions of the filtering structure to meet the requirements of a certain, specified transfer function. The novel algorithm introduces an iterative procedure in order to synthesise the direct coupling term between source and load, which is always present in the hybrid filter structure. The whole algorithm is illustrated by means of a design example, where a very good agreement between the ideal specified transfer function and the final frequency response of the designed hybrid filter is obtained. A second different design example, which has been manufactured and tested, is also included for validation.Ministerio de Educación y Ciencia, Ref. TEC2010-21520-C04-0

General Synthesis Method for Dispersively Coupled Resonator Filters With Cascaded Topologies

International audienceThis 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 S¹,...,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

Analysis and design of cross-coupled, folded e-plane filters with asymmetric responses

Cross-coupled folded metal insert filters with asymmetric responses are introduced. The selectivity of these filters is further improved by exploiting the source-load coupling to generate up to n finite transmission zeros for a filter of order n. Two 4 th order filters with 2 and 5 finite transmission zeros, respectively are presented to demonstrate the significantly improved response of these filters over their conventional counter part

A compact and broadband 90-degree waveguide twist transformer for integrated waveguide subsystems

A novel 90-degree waveguide twist is presented. The design features compactness by utilizing a central triple-mode quarter-wave transformer region between input and output waveguides of different cross sections. Return-loss values in excess of 30 dB are obtained over an operating bandwidth close to 20 percent. Larger bandwidths are obtained by using additional quarter-wave steps preceding and/or succeeding the central transformer region. Several different design examples at 4, 6, 19 and 26 GHz are presented for this unique 90- degree twist transformer approach. Experimental verification is provided in the 6 and 19 GHz range

Design of dual-mode ridge cavity filters

International audienceA variety of dual-mode ridge cavity filters are presented. The resonances that exist in the dual-mode cavities with all the perturbations present are used to represent and design the filter. Different arrangements which produce similar field distributions are used. Since the degenerate modes of a lossless dual-mode cavity can not be coupled but only rotated, or have their field distributions changed in a complex way, different mechanisms are used to affect such rotations. The first consists in adding standard perturbations or ldquocouplingrdquo elements. The second version consists in physically rotating an almost square ridge. Good results are obtained

A New Approach to Canonical Dual-Mode Cavity Filter Design

International audienceThis paper presents a new approach to the design of canonical dual-mode cavity filters within representation theory. Instead of the folded canonical coupling matrix, which results from using the resonances of the empty cavities as a basis, a representation based on the resonances of the cavities with all perturbations present is used. The resulting coupling matrix takes the form of a cul-de-sac configuration. For canonical dual-mode filters with a symmetric response, it allows direct design of the entire structure, except of the input cavity that requires optimization. A sixth-order filter with two symmetrically located transmission zeros is presented to document the validity of the approach. A sixth-order filter with two transmission zeros above the passband is also presented

Theory of Coupled Resonator Microwave Bandpass Filters of Arbitrary Bandwidth

International audienceThis paper presents a new approach to the design of canonical dual-mode cavity filters within representation theory. Instead of the folded canonical coupling matrix, which results from using the resonances of the empty cavities as a basis, a representation based on the resonances of the cavities with all perturbations present is used. The resulting coupling matrix takes the form of a cul-de-sac configuration. For canonical dual-mode filters with a symmetric response, it allows direct design of the entire structure, except of the input cavity that requires optimization. A sixth-order filter with two symmetrically located transmission zeros is presented to document the validity of the approach. A sixth-order filter with two transmission zeros above the passband is also presented