# Linearized active circuits:

transfer functions and stability

Laurent Baratchart, Sylvain Chevillard, Adam Cooman,

Martine Olivi, and Fabien Seyfert

*We study the properties of electronic circuits after linearization around a fixed operating point in the context of closed-loop stability analysis. When distributed elements, like transmission lines, are present in the circuit it is known that unstable circuits can be created without poles in the complex right half-plane. This undermines existing closed-loop stability analysis techniques that determine stability by looking for right half-plane poles.*

*We observed that the problematic circuits rely on unrealistic elements with an infinite bandwidth. In this paper, we therefore define a class of realistic linearized components and show that a circuit composed of realistic elements is only unstable with poles in the complex right half-plane. Furthermore, we show that the amount of right half-plane poles in a realistic circuit is finite, even when distributed elements are present.*

*In the second part of the paper, we provide examples of component models that are realistic and show that the class includes many existing models, including ones for passive devices, active devices and transmission lines.*

- 1. Introduction
- 2. Stability of linearized circuits under realistic assumption
- Proofs
- Discussion
- 3. Realistic lumped passives
- 4. Realistic transmission lines
- 5. Realistic active devices
- 6. Example:
- 7. Generalization to lossy transmission lines with frequency-dependent parameters
- Conclusion
- Appendices
- Bibliography

This research was funded in part by the CNES through grant R&T RS 10/TG1-019. We also wish to thank Juan-Mari Collantes, from Universidad del Pais Vasco Euskal Herriko Unibertsitatea, for introducing the authors to electronic circuits and providing them with patient advice during long discussions on the subject.