# Internship 2019 (Master Level): Mathematical circuit modelling for Antennas

Intership context:
Growing demand in communication bandwidth , triggered by the upcoming 5g standard, call for constant  improvement in the conception of microwave equipments. Among them, as an important building block for any emitting or receiving unit, are the antennas associated with their matching network and channel filters. Usually these elements are conceived separately, on what is called a reference impedance (usually 50 Ohm), in order to be assembled independently during the system’s building phase. In order however to improve the energy efficiency of the whole chain the Factas, Leat  and Xlim teams have been working on the conception of matching filters, realising in one single device the filtering and matching functionality. Matching is a long lasting research problem, with deep connexions to complex analysis, and in particular Nevanlinna-Pick interpolation [1]. To proceed further into the improvement of the energy balance of the whole system a co-conception of filter, matching network, and antenna is currently considered. To do this a proper circuit modelisation of the antenna is needed. Although circuits are usually developed to model the antenna’s input port, circuit approaches describing both, the input as well as the radiating part of the device, have not yet been developed. We however think that based on the expansion of the radiating field into vector spherical harmonics [3, 4] combined with techniques of matrix rational approximation, such models can be systematically derived, as this is for example in the case fore microwave filters [4]. We expect that their computation will give access to a better understanding of the antenna inner functioning as well as pave the way to a complete co-conception synthesis technique

In a nutshell:
The internship will consist in a revue of different techniques used in our labs for the decomposition of radiating fields on the orthogonal basis of spherical harmonics, rational approximation techniques, as well as circuit realisation techniques. Combining these different tools, the derivation of models for antennas of increasing complexity will be considered. The possibility of linking certain property of the derived circuit, with the internal functioning of the antenna will eventually be considered.

Required Skills:
This internship is of prospective nature and might be continued with a PhD, and is therefore meant for students from second of year of masters or third year of engineering school. The Expected skills of the candidate are a strong tastes in mathematical modelling and numerical techniques.

Duration: 4-6 Months,  Location: Sophia-Antipolis

Supervision:
Supervision is ensured by members of the Factas team at Inria and the Leat team at U.C.A.

Contact:
Inria: Fabien Seyfert (fabien.seyfert@inria.fr) and Martine Olivi (martine.olivi@inria.fr)
Leat: Jean-Yves Dauvignac (Jean-Yves.Dauvignac@unice.fr), Fabien Ferrero (fabien.ferrero@unice.fr)

References:
[1] Boundary Nevanlinna–Pick Interpolation with Prescribed Peak Points. Application to Impedance Matching Laurent Baratchart, Martine Olivi, and Fabien Seyfert, SIAM Journal on Mathematical Analysis 2017 49:2, 1131-1165
[2] Modélisation compacte du rayonnement d’antennes ULB en champ proche/champ lointain – mise en application en présence d’interface, These d’Abdellah ROUSSAFI, UCA

[3] Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Jean-Claude Nedelec

[4] Identification of microwave filters by analytic and rational H2 approximation ,M. Olivi, F. Seyfert and J.P. Marmorat Automatica 49(2013) 317-325 , doi: 10.1016/j.automatica.2012.10.005.