Scaled Relative Graphs for Nonmonotone Operators with Applications in Circuit Theory

TL;DR

This work introduces scaled relative graphs (SRG) as a graphical framework to analyze incremental gain and phase of nonlinear operators, extending Nyquist-like reasoning to semimonotone and angle-bounded nonmonotone classes. It derives explicit SRG descriptions: $(mu, ho)$-semimonotone operators yield disk-excluded SRGs, while $ heta$-angle-bounded operators produce wedge-shaped SRGs, and establishes geometric containment as a criterion for class membership. The paper then demonstrates that the Ebers–Moll transistor is angle-bounded, enabling semimonotone-based analysis of transistors and tunnel diodes, and shows how Chambolle–Pock iterations can compute the response of nonsmooth, multi-valued common-emitter amplifier circuits with convergence guarantees. These results provide a scalable, graphically verifiable approach to circuit analysis and nonmonotone operator splitting in practical, real-world problems, with future work toward additional nonlinear elements and SRG-guided circuit design.

Abstract

The scaled relative graph (SRG) is a powerful graphical tool for analyzing the properties of operators, by mapping their graph onto the complex plane. In this work, we study the SRG of two classes of nonmonotone operators, namely the general class of semimonotone operators and a class of angle-bounded operators. In particular, we provide an analytical description of the SRG of these classes and show that membership of an operator to these classes can be verified through geometric containment of its SRG. To illustrate the importance of these results, we provide several examples in the context of electrical circuits. Most notably, we show that the Ebers-Moll transistor belongs to the class of angle-bounded operators and use this result to compute the response of a common-emitter amplifier using Chambolle-Pock, despite the underlying nonsmoothness and multi-valuedness, leveraging recent convergence results for this algorithm in the nonmonotone setting.

Figures (8)

• Figure 1: Scaled relative graphs for the classes of monotone ($\mathcal{M}$), hypomonotone ($\mathcal{M}_{-\mu}$), strongly monotone ($\mathcal{M}_{\mu}$), cohypomonotone ($\mathcal{C}_{-\rho}$) and cocoercive operators ($\mathcal{C}_{\rho}$), where $\mu > 0$ and $\rho > 0$. The shaded regions indicate portions of the extended complex plane included in the corresponding SRG.
• Figure 2: Four qualitatively different scaled relative graphs for classes of semimonotone operators, where $\mu > 0$ and $\rho > 0$. The circles in the first and third graphs have radius $\sqrt{1-4\mu\rho}/2|\rho|$, while those in the second and fourth have radius $\sqrt{1+4\mu\rho}/2|\rho|$, as established in \ref{['prop:srgsemi']}.
• Figure 3: The SRG of the class of $\theta$-angle-bounded operators $\mathcal{B}_\theta$ and the construction to relate $\mathcal{B}_\theta + \alpha {\rm id}$ to $\mathcal{S}_{\mu,\rho}$. (a) The SRG of $\mathcal{B}_\theta$. (b) The SRG of $\mathcal{B}_\theta + \alpha {\rm id}$ and the open disk $D(1/2\rho, \sqrt{1-4\mu\rho}/2|\rho|)$ from the proof of \ref{['prop:anglesem']} for some $\rho < 0$ and $\mu > 0$. (c) The SRG of $\mathcal{B}_\theta + \alpha {\rm id}$ and open disks $D(1/2\rho, \sqrt{1-4\mu\rho}/2|\rho|)$ for varying $\rho < 0$, where $\mu$ is defined as in \ref{['SRG:eq:optimal-mu']}.
• Figure 4: NPN transistor. (a) Two-port model. (b) Ebers--Moll model. (c)-(d) Numerical SRG of the NPN transistor $G(T_{\rm NPN})$ for different values of $\alpha_R$ and $\alpha_F$. In both cases, the incremental angle is upper bounded by $135$ degrees.
• Figure 5: Common-emitter amplifier with leakage current.

Theorems & Definitions (22)

• definition 1
• definition 2
• proposition 1
• proposition 2
• definition 3
• proposition 3
• proposition 4
• proposition 5
• proof
• remark 1