1. Introduction

Frequency domain simulation methods, like Harmonic Balance (HB) or a DC analysis, impose a structure on the obtained solution of the circuit [28]: The DC analysis only allows for a fixed solution, while HB imposes a frequency grid. Any circuit solution that requires more than the imposed frequencies, e.g. an extra oscillation not on the imposed grid, cannot be represented in the constrained frequency grids of DC and HB. The simulator will still find a valid solution, but the obtained orbit will be locally unstable: it cannot recover from small perturbations and will be physically unobservable in the circuit [29]. It is therefore necessary to perform a local stability analysis on each of the circuit solutions obtained with a DC and HB analysis [28].

Over the years, several methods have been developed to determine the local stability of a circuit solution. Some techniques, like the analysis of the characteristic system [29], require access to the simulator. Open-loop techniques, like the analysis of the normalised determinant [28], require access to the intrinsic device models. These classic techniques are therefore hard to implement in commercial simulators.

Closed-loop stability analysis methods can easily be applied as a post-processing step without any internal knowledge of the circuit and can be used in commercial simulators. This is the reason why they have attracted a large interest lately [28] [13] [7]. A closed-loop local stability analysis performs linearisation of the circuit around the orbit to check the stability thereof: if the linearised circuit has at least one pole in the complex right half-plane, the orbit is unstable. It is moreover assumed that, conversely, the absence of unstable pole implies stability, although no published proof of this fact seems available yet. The question is more subtle than it looks: there exist delay systems which are unstable and still their transfer-function has no unstable pole [24], moreover has an example of an ideal circuit with this property. Nevertheless, it is claimed in [5] that a circuit whose elements are passive at arbitrary high frequencies must indeed have some unstable pole if it is unstable.

The poles of the linearisation around the circuit orbit cannot be obtained directly. Instead a FRF of the linearised circuit is obtained with small-signal simulations on a discrete set of frequencies. The closed-loop stability analysis then aims at determining whether the underlying FRF has a pole in the complex right half-plane. In a pole-zero stability analysis, a rational approximation is fitted on the FRFs. If the rational approximation contains poles in the complex right half-plane, the solution is declared unstable.

Note that the FRF of circuits with distributed elements, like transmission lines, is not rational. Therefore it must be argued that the poles of the computed rational approximant convey information on the poles of the true FRF. This is a delicate issue and a particular instance of a recurring question in approximation theory, namely: what do the singularities of an approximant tell us about the singularities of the approximated function? We observe that no such information can be drawn from the mere quality of approximation in a range of frequencies, since a famous theorem by Runge entails that a continuous function on a segment can be approximated arbitrary well by a proper rational function with prescribed pole location [25]. Thus, for singularity detection, the choice of the approximation algorithm (and not just the fit of the approximant) does matter.

For instance, methods based on linear interpolation, like Padé or multipoint Padé approximation, are famous for generating spurious poles that wander about the domain of analyticity of the approximated function. This phenomenon was intensively studied for meromorphic and branching functions [4] [11] [27], in particular the convergence in capacity of Padé approximants implies that spurious poles have a nearby zero when the order gets large, leading to so-called near pole-zero cancellations (also known as Froissart doublets). Modifications of Padé approximants were proposed to offset this issue [23], but they do not eliminate the problem [6]. Apparently, the theoretically less studied vector fitting method which is a least squares version of linear interpolation, popular today in system analysis, is also prone to producing spurious poles and near cancellations (see [15] for issues on convergence of this method).

In system identification, near cancellations are often ascribed to overmodelling. The terminology suggests an analogy with the stochastic identification paradigm: though measurements may not correspond to a rational transfer function, the basic assumption is that they arise from a well-defined rational system R with added noise. This point of view leads one to postulate the existence of a “correct order” to identify the Frequency Response Function (FRF), i.e. the degree of R, while using a higher degree results in approximating the noise term with inessential, nearly simplifying rational elements. However, if the transfer function is not rational, requirements to keep the degree small conflict with the need to make the approximation error small as well (not to incur undermodelling), thus calling for a compromise akin to the classical trade-off between bias and variance from parametric stochastic identification [16]. To quote [1] ¹: “it is not always trivial to discriminate between overmodelling quasi-cancellations and physical quasi-cancellations that really reflect an unstable behaviour”.

To resolve this issue, the approach proposed in [1] is to cut the frequency band into smaller intervals and use low-order local rational approximations to assess the stability of the FRF on each interval separately. On small enough intervals, rational approximation can be performed accurately in low degree, and if unstable poles occur their physical character is checked by re-modelling the FRF locally around each of them and verifying that the unstable pole remains present in the new model. This procedure is commercially available in the STAN tool [9] [18] and successful applications on several examples are reported in [30] [22] [8] [2].

Still, justifying the above-described technique presently rests on heuristic arguments, and putting it to work is likely to require some know-how since several parameters need to be adjusted adequately. This is why the authors feel that it may be interesting to develop an alternative viewpoint, focusing more on estimating the unstable part of the FRF.

Below, we propose a closed-loop stability analysis method devoid of local models, in which the FRF is projected onto the orthogonal basis of stable and unstable functions. If a significant part of the FRF is projected onto the unstable basis functions, the circuit solution is unstable. Calculating the projection boils down to computing a Fourier transform once the FRF is mapped from the imaginary axis to the unit circle. Using the Fast Fourier Transform (FFT), this can be done fast and in a numerically robust way.

Functional projection onto a stable and unstable basis is a linear operation, simple to implement, and no optimisation step is required. No model-order or maximum approximation error needs to be specified. The parameters in the projection method are the frequency range on which the FRF is determined and the amount of simulation points. When the amount of simulated points is too low, an interpolation error is present in the result of the method. It is shown that the level of this error can easily be estimated and used to correctly choose the amount of needed simulation points.

Once the unstable part of the FRF has been obtained, it is compared to the level of the interpolation error to determine whether the unstable part is significant or not. This final step can be done visually, or a significance threshold can be chosen by the user, both will require some experience with the method.

A final benefit of the projection-based approach is that it may help exploiting the fact that the unstable part is rational in a small-signal stability analysis [5]. The unstable part can therefore be approximated by a rational function without influence of the distributed elements, which are projected onto the stable part of the FRF.

The following of the paper is structured as follows: First, the simulation set-up used to determine the FRF of the linearised circuit is discussed (Section 2). Then, the details of the functional projection are provided (Section 3). In Section 4, the method is applied to four examples: First, an artificial example is considered. Then, the small-signal stability of two amplifiers is investigated and finally, the method is applied to investigate the large-signal stability of a circuit.