## Section: New Results

### Detection of the instability of amplifiers

Participants : Laurent Baratchart, Sylvain Chevillard, Martine Olivi, Fabien Seyfert.

This work is conducted in collaboration with Jean-Baptiste Pomet from the McTao team. It is a continuation of a collaboration with CNES and the University of Bilbao.The goal is to help developing amplifiers, in particular to detect instability at an early stage of the design.

Currently, electrical engineers from the University of Bilbao, under contract with CNES (the French Space Agency), use heuristics to diagnose instability before the circuit is physically implemented. We intend to set up a rigorously founded algorithm, based on properties of transfer functions of such amplifiers which belong to particular classes of analytic functions.

In non-degenerate cases, non-linear electrical components can be replaced by their first order approximation when studying stability to small perturbations. Using this approximation, diodes appear as perfect negative resistors and transistors as perfect current sources controlled by the voltages at certain points of the circuit.

In previous years, we had proved that the class of transfer functions which can be realized with such ideal components and standard passive components (resistors, selfs, capacitors and transmission lines) is rather large since it contains all rational functions in the variable and in the exponentials thereof. This makes possible to design circuits that are unstable, although they have no pole in the right half-plane. This remains true even if a high resistor is put in parallel of the circuit, which is rather unusual. These pathological examples are unrealistic, though, because they assume that non-linear elements continue to provide gain even at very high frequencies. In practice, small capacitive and inductive effects (negligible at moderate frequencies) make these components passive for very high frequencies.

In 2013, we showed that under this simple assumption that there are small inductive and capacitive effects in active components, the class of transfer functions of realistic circuits is much smaller than in previous situation. Our main result is that a realistic circuit is unstable if and only if it has poles in the right half-plane. Moreover, there can only be finitely many of them. Besides this result, we also generalized our description of the class of transfer functions achievable with ideal components, to include the case of transmission lines with loss. An article is currently being written on this subject.