1. Introduction

This work is motivated by the stability analysis of circuits containing active (non-linear) components, such as transistors and diodes, as well as distributed elements like transmission lines. A typical example is that of high-frequency amplifiers. The design of such devices relies today on powerful simulation tools in the frequency domain, like AC or S-parameter simulations [1]. These compute the circuit’s response after it has been linearized around an equilibrium solution obtained via a DC simulation. The obtained equilibrium solution may either be stable, hence physically observable, or unstable, thus physically immaterial. Testing stability is thus mandatory before implementing the circuit. To this effect, a wide variety of methods has been proposed and we refer the reader to [2][3] and their bibliography for a small sample of literature on the subject.

In this paper we focus on closed-loop local stability analysis. This approach, which requires little internal knowledge of the circuit, aroused considerable interest in the microwave community [3][4]. The local stability is studied by computing the response of the circuit to small signal perturbations at its nodes, using AC analysis around a DC solution computed via a DC simulation. For instance, as shown in Figure 1.1, a small sinusoidal current source can input the circuit in parallel at a particular node $k$, and the voltage $V_{k}$ at this node is taken as the output of the system. Then, a sweep of the frequency range is performed and the corresponding partial transfer function is estimated, pointwise in a bandwidth, as being the impedance seen by the current probe at the node under consideration.

Determining the stability of $G\left(s\right)$ then allows to determine whether the equilibrium solution of the circuit is stable.

Figure 1.1 A partial transfer function of circuit $\mathcal{C}$ is determined by connecting a small-signal current source at node $k$ and by determining the circuit’s voltage response to this small-signal current excitation.

A standard definition of stability is: a linear stationary control system is stable when its transfer function belongs to $\mathcal{H}^{\infty}$, the space of bounded analytic functions in the open right half-plane. This is equivalent to require that the system maps inputs of finite energy (i.e. $L^{2}$ signals) to outputs of finite energy [5]. For a rational transfer function to be stable, it is necessary and sufficient that it has no pole in the closed right half-plane including at infinity. Such poles will be called unstable. Because partial frequency responses of circuits containing only lumped elements are rational, they are unstable if and only if they have at least one unstable pole. A common methodology in practice is to approximate the simulated frequency response of the linearized system by a rational function, the poles of which are used to assess the stability of the equilibrium [4]. Namely, the poles of a rational approximant are used to indicate the location of the poles of the true system, and poles lying in the closed right half-plane indicate instability.

It is natural to ask whether similar considerations apply to circuits with distributed elements. This is actually false. It is known there are quotients of quasi-polynomials, which typically represent transfer functions of delay systems, that are unstable though they have no unstable pole (these are neutral systems [6]). In [7] such a circuit was synthesized using resistors, inductors and capacitors, lossless transmission lines and negative resistances. The partial impedance presented by the circuit is

where

It can be shown that $G\left(s\right)$ has no poles in the closed right half-plane nor at infinity (where it has an essential singularity), and still it does not belong to $\mathcal{H}^{\infty}$.

At first, example (1.1) casts doubt on whether assessing the stability of an equilibrium, for active electronic devices, can be achieved upon checking if the linearized circuit has unstable poles. The components used to realize the example are however somewhat unrealistic, requiring lossless components and negative resistances with infinite bandwidth. In practice, no passive component is truly lossless and no active component has gain at all frequencies.

In this paper, we introduce a definition for realistic electronic components (Section 2), which become passive at infinite frequency (as opposed to ideal components). We study the stability of circuits composed of such realistic elements and show that instability without unstable pole can no longer happen: our main result is that a realistic circuit is unstable if and only if it has poles in the closed right half-plane. Moreover, these must be finite in number so that the unstable part of the linearized transfer function of a realistic circuit is demonstrably rational. In retrospect, this justifies to look for unstable poles thereof to check for stability at an operating point. Then, in Section 3, we discuss several component models which are realistic, to show that our definition applies to a broad class of models and that most common element models can easily be made realistic.

Hereafter, given a matrix $M$, we let $M^{t}$ denote its transpose and $M^{*}$ its conjugate transpose. The identity matrix is written $\mathbf{Id}$, irrespective of its size which will be understood from the context. The symbol $\Re$ is used to denote the real part.