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Section: New Results

Synthesis and Tuning of broad band microwave filters

Participants : Smain Amari, Magued Bekheit [ RMC, Kingston, Canada ] , Fabien Seyfert.

Some important results have been obtained in order to handle tuning and synthesis of broad band filters. One of the major problems when dealing with wide band filters is the break down of the classical low pass model which relies on a narrow band assumption. We showed however that there exists a unifying “low pass formalism” which is valid in the narrow band as well as in the wide band situations. The latter relies on the following remark. Let S be any inner, real, symmetric (St = S ), rational matrix, which is identity at infinity and has MacMillan degree n . Then the rational matrix Sr defined by:

Im28 ${S_r{(-is^2)}=\mfrac {I\mfenced o=( c=) \mfrac {is+1}{is-1}+S{(s)}}{I+S{(s)}\mfenced o=( c=) \mfrac {is+1}{is-1}}}$(2)

is again an inner, complex, symmetric matrix, which is identity at infinity, and has MacMillan degree n . It can be shown that S is entirely characterised by the knowledge of its reduced “complex” version Sr . Measurements of S on two conjugate frequency bands are mapped to measurements of Sr on a single band, which up to the use of a linear frequency transformation can be cast to the normalized band [-1, 1] . Usual techniques used to recover rational models from low pass responses measured on a single frequency interval can therefore be used to recover high pass responses via the use of the generalized reduced response Sr . Implementation attempts of the latter in the Presto-Hf software were started and encouraging results where obtained for the tuning of an ultra-wide band filter realized with suspended strip lines. Figure 11 shows data and their rational approximation of this 10th order filter (reduced order 5) with a bandwidth ratio of approximately 10% (in collaboration with the university of Ulm, Germany).

Concerning the synthesis of the response of such filters we had already shown that the latter amounts to a Zolotarev problem with a non-polynomial weight (with a square root singularity). For fixed transmission zeros we were however able to derive explicit formulas for the optimal (in the Chebychev sense) filter function Fn :

Im29 ${F_n{(w)}=cosh\mfenced o=( c=) cosh^{-1}\mfenced o=( c=) \mfrac {{\#119983 }^'{(w)}}w+\munderover \# ... ) f_k{(w)},~~f_k{(w)}=\mfrac {\#119983 {(w)}-1/\#119983 {(z_k)}}{1-\#119983 {(w)}/\#119983 {(z_k)}}}$(3)

where Im30 $\#119983 $ is a suitable parabolic frequency transformation and the zk's are prescribed transmission zeros. Recurrence formulas for the practical computation of FN have been derived and implemented as part of the Dedale-HF software package. As for the realization of such responses, first results were obtained with resonant coupling elements 6.8

In collaboration with the RMC and possibly with XLim and ST-Microelectronics (Tours) our goal is now to test the validity of our unified approach on real examples. Data collection campaigns obtained during tuning phases are scheduled. Joint publications about the topic are also in progress.

Figure 11. Bode diagramm of the response of an ultra-wide band filter and its rational approximation of degree 10 - Frequencies are in GHz


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