Section: New Results

Stability of amplifiers

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

This work is performed under contract with CNES-Toulouse and the University of Bilbao. The goal is to help designing 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 detect instability before an amplifying circuit is physically built. Our goal is 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 in the small signal regime. Using this approximation, diodes appear as negative resistors and transistors as current sources controlled by the voltage at certain nodes of the circuit.

Over the last three years, we studied several features of transfer functions of amplifying electronic circuits:

• We characterized the class of transfer functions which can be realized with ideal components linearized active components, together with standard passive components (resistors, inductors, capacitors and transmission lines). It is exactly the field of rational functions in the complex variable and in the hyperbolic cosines and identity-times-hyperbolic sines of polynomials of degree 2 with real negative roots.

• We introduced a realistic notion of stability, by terming stable a circuit whose transfer function belongs to $H^{\infty }$, as long a sufficiently high resistor is added in parallel to that circuit.

• We constructed unstable circuits having no pole in the right half-plane, which came as a surprise to our partners.

• In order to circumvent these pathological examples, we introduced a realistic hypothesis that there are small inductive and capacitive effects to active components. Our main result is that a realistic circuit without poles on the imaginary axis is unstable if and only if it has poles in the right half-plane. Moreover, there can only be finitely many of them.

This year, we were led to modify our definition of stability, taking a hint from scattering theory. We say that a transfer function $Z$ is stable whenever $(R-Z)/(R+Z)$ belongs to $H^{\infty }$ with uniformly bounded $H^{\infty }$-norm for all $R$ large enough. Equivalently, this means that the circuit can amplify signals but not require an unbounded amount of energy from the primary power circuit. This new definition is really about energy, hence is more natural. Also, it allows us a unified characterization in the corner case where instabilities are located on the imaginary axis. We obtained this way a nice characterization: $Z$ is stable if and only if it has no pole in the open right half plane, while each pole it may have on the imaginary axis is simple and has a residue with strictly positive real part. We published a research report [23] and an article is being written to report on our results.