## Section: New Results

### Circuit realisations of filter responses: determination of canonical forms and exhaustive computations of constrained realisations

Participants : Smain Amari, Jean Charles Faugère [ EPI SALSA, INRIA Rocquencourt ] , Stéphane Bila [ XLim, Limoges ] , Fabien Seyfert.

Groomed by industrial users like Thales Alenia-Space, we made some progress in the analysis
of the realizations of 2x2 lossless scattering systems whose scattering response
(S_{i, j}) satisfies the so-called *auto-reciprocal* condition
S_{1, 1} = S_{2, 2} . It was shown that auto-reciprocal inner responses admit a canonical circuit
realisation of the form of Fig. 7 . The length difference (m-l ) of the two antennas of Fig. 7 is equal to the Cauchy index on the imaginary axes of the filter function to
be realised. Surprisingly enough this form appears to be central in the new modal framework
S.Amari is currently developing on dual mode filters ( [34] ). It was shown that the classical folded
form can be advantageously replaced by the latter yielding a design procedure with nearly no tuning required (all
the physical dimensions of the filter can be computed exactly from the circuit parameters): a paper has been published
on this topic [19] . In future work, we will focus on the practical implementation of
this analysis within the software Dedale-HF
5.6 .

We also made some progress on the problem of circuit realisations with mixed type (inductive or capacitive) coupling elements. An algebraic formulation of the synthesis problem of circuits with mixed type elements has been obtained which relies on a set of two matricial equations. As opposed to the classical low pass case with frequency independent couplings the unknown is no longer a similarity transform but a general non-singular matrix acting on two coupling matrices: the capacitive and the inductive one. First results were obtained in this field allowing the exact synthesis of filters with resonating coupling elements, see 8 . Applications of this technique to synthesise extremely compact filters, with sharp responses, is being studied in collaboration with the Royal Military College (Canada). Note however, that the filter orders for which this synthesis is computationally tractable, for the moment, is modest (no more than 5 or 6). Further developments, focusing in particular on an efficient algebraic formulation of the problem, are needed in order to convince engineers of its relevancy when compared to generic local optimisation techniques. The state of our work was presented at Rome (European Microwave Conference) and Toulouse (CNES-ESA filter workshop) while a publication is currently being reviewed.