# Monodromy Structure and Harmonic Transfer Function in Lossless Transmission Line Circuits

Sébastien Fueyo<sup>a</sup>, Laurent Baratchart<sup>b</sup>, Jean-Baptiste Pomet<sup>b</sup>

<sup>a</sup> CNRS, Gipsa-lab, Grenoble

<sup>b</sup> Inria, Sophia-Antipolis

## Abstract

High-frequency (HF) amplifiers are crucial devices in modern electronics, providing one with a prototypical example of nonstationary, nonlinear systems whose control is crucial to modern telecommunications. In particular, ensuring the local stability of their periodic steady-state responses is critical for reliable performance and safe hardware implementation. While classical methods relying on Floquet theory and harmonic transfer functions (HTFs) are of use to analyze stability in low-frequency settings, these techniques remain underdeveloped for HF amplifiers whose spatial distribution of components and signal propagation delays must be adressed. In this work, we study HF amplifiers with lossless transmission lines modeled as delay systems via telegrapher's equations. Under realistic assumptions (high-frequency dissipation), we show that instabilities can only arise from *eigenvalues* of the monodromy operator (not from more complex spectral elements). Furthermore, the poles that the HTF may have in the right half-plane are the logarithms of the unstable eigenvalues, thereby extending known results from the time-invariant case to the periodic setting; a $H^\infty$ stability criterion is also established. These findings pave the way towards a rigorous foundation for using frequency-domain tools in stability analysis of periodic differential delay systems, and justify the detection of instabilities via HTF pole localization in the right half-plane for HF amplifiers. While the mathematical results were essentially obtained in [3], the present paper intends to explain and advertise them, as well as to numerically illustrate their use on a simple electrical circuit.

*Key words:* High-frequency amplifiers; monodromy operator; harmonic transfer function; periodic solutions; Floquet theory; delay differential equations; transmission lines; infinite-dimensional systems; stability analysis; frequency-domain methods.

## 1 Introduction

An ideal amplifier is a circuit that magnifies the input signal by a gain which is independent of the signal itself. In practice, such behavior is unattainable. The design of amplifiers consists in devising a circuit—built from available elementary components—whose gain remains approximately constant over a prescribed class of input signals.

The available components fall into three categories:

- passive elements, such as resistors, capacitors, and inductors (RLC circuits);

- active nonlinear elements, namely diodes and transistors (which are powered by an external source);

- and transmission lines, which introduce delay that are negligible at low frequencies but not at high frequencies where they result in an infinite-dimensional state space.

Amplifiers are widely used in electronic engineering, be it in mobile phones and multiplexors or relay systems. From a system-theoretic perspective, one expects a periodic forcing of the circuit to yield a periodic steady-state solution, and ensuring local stability of the latter is crucial before undertaking the costly task of physical prototyping. Numerical simulations are therefore used to guarantee that the amplifier is stable enough for the expected periodic response to be observed in practice—and to avoid catastrophic failures, such as circuit burnout.

The standard approach is to linearize the system around the periodic orbit and to analyze the exponential stability at the origin of the linearized system. The latter can be characterized using either the eigenvalues of the monodromy operator (Floquet theory) or, in certain cases, by examining the poles of the so-called harmonic transfer function (HTF) (see e.g. [18]). However, such techniques are underdeveloped in the context of high-frequency (HF) amplifiers.

<footnote>* This paper was not presented at any IFAC meeting. Corresponding author S. Fueyo.</footnote>
In this paper, the term high-frequency (HF) regime refers to the conventional radio-frequency range, with signals between 3 and 30 MHz [9]. In this regime, signal propagation delays along interconnections can no longer be neglected, and the dynamics of HF amplifiers are modeled by telegrapher's equations, a class of hyperbolic PDEs. The resulting system is infinite-dimensional, and its spectral structure and the singularities of the HTF are not fully understood. This paper investigates these properties to provide the foundation for reliable stability analysis and design of HF amplifiers.

More precisely, we address the case of circuits with lossless transmission lines which are known to be equivalent to difference-differential delay systems, see Equation (1). We show—under a dissipativity assumption at high-frequency on the linearized system which is expected to hold for any realistic circuit—that instability can only stem from the existence of eigenvalues off the unit disk for the monodromy operator (see Theorem 6). This result is obtained using compact perturbation theory. We emphasize that the assumed dissipativity is realistic, for at high frequencies capacitors behave like open circuits and inductors like short circuits, hence the only effective devices left in the circuit are positive resistances. Two key challenges arise in this context:

- First, while existing numerical methods are well-developed for simpler classes of delay systems such as retarded ones (see [13]), systems considered in this paper (see (1)) involve more complex delay structures, the numerical analysis of which is significantly more involved, as discussed in [8].

- Second, in practical applications, high-frequency amplifiers often involve a very large number of components making time-domain simulations computationally prohibitive—especially those targeting the spectrum of the monodromy operator. What is typically accessible in practice is the frequency response of certain subsystems and the input-output response of the full amplifier at designated frequencies, obtained by injecting either current or voltage perturbations in the circuit and measuring the corresponding outputs.

Therefore, in situations where only the frequency range of the excitation is known, it is natural to resort to the harmonic transfer function, which can be derived upon combining Fourier series expansion with Laplace transforms, and numerically estimated frequency-wise using so-called harmonic balance techniques [18]. The notion of the HTF was originally introduced heuristically by N. Wereley in the finite-dimensional case [20]. More rigorous mathematical foundations have been developed since then for finite and infinite-dimensional systems, notably through the works of Sandberg et al [16], Möllerstedt et al [14], Vernerey [19] and also in our previous papers [4,3] in the context of hyperbolic PDEs.

As we explain in the present paper, for HF amplifiers

which are dissipative at high-frequency, the HTF admits only poles and no other type of singularity in a right half-plane strictly containing the imaginary axis, and these poles are precisely the logarithms of eigenvalues of the monodromy operator, (Theorem 7). This result extends to the periodic case some time invariant results obtained in [2]. An $H^\infty$ stability criterion is likewise provided, and these theoretical results are validated through a simple illustrative example.

It is worth noting that the infinite-dimensional nature of our model may, in principle, give rise to more general spectral components for the monodromy—such as continuous spectra— and these would give rise to essential singularities of the HTF—which we do not address here. However, Theorems 6 and 7 tell us that such singularities, if any, are stable while Theorem 8 provides us with a rigorous justification for the use of frequency-domain methods to detect instability, specifically by identifying poles of the HTF in the closed right half-plane for HF amplifiers (cf. [7]).

## 2 Governing Equations of Linearized HF amplifier and high Frequency behaviour

In this paper, we consider a high-frequency amplifier composed of resistors, capacitors, inductors, diodes, and transistors, where the transmission delays between components are modeled by lossless transmission lines. When linearized around a periodic trajectory, such HF amplifiers can be written in the form

$$ egin{cases} \frac{dx(t)}{dt} = A_1(t)x(t) + \sum_{i=1}^{N} B_i^1(t)y(t - \tau_i), \\ y(t) = \sum_{i=1}^{N} B_i^2(t)y(t - \tau_i) + A_2(t)x(t), \quad t \ge s, \end{cases} \quad (1) $$

where the delays satisfy $\tau_1 < \dots < \tau_N$ and $s \in \mathbb{R}$ denotes the initial time, while $k, n$ are integers with $x(t) \in \mathbb{R}^k$ and $y(t) \in \mathbb{R}^n$. Without loss of generality, we assume that $T > \tau_N$; otherwise, one may simply consider the smallest integer $p$ such that $pT > \tau_N$ and take $pT$ as the new period. For the class of HF amplifiers under consideration, the governing equations can indeed be reduced to the form (1); a detailed derivation of this fact is provided in the appendix for the convenience of the reader, see Equation (A.6).

System (1) is a periodic linear difference-differential delay system with a real positive period $T$. The system is composed of two states noted $(x(t), y(t))^{\mathrm{t}}$ where the notation designates a column vector with a superscript “t” to mean “transpose”. The component $x(t)$ represents the differentiated state taking values in $\mathbb{R}^k$. The component $y(t)$ is the delayed state assuming values in $\mathbb{R}^n$. The delayed state accounts for the finitely many delays in the system. It is assumed that $A_1(\cdot)$ (resp. $A_2(\cdot)$) is a T-periodic, $C^2$-smooth matrix-valued function of size $k \times k$
(resp. $n \times k$). The family $\{B_i^1(t)\}_{i=0}^N$ (resp. $\{B_i^2(t)\}_{i=1}^N$) consists of $T$-periodic, $C^2$-smooth matrix-valued functions of size $k \times n$ (resp. $n \times n$).

An important feature is the behavior of the linearized system in the high-frequency regime. It is formally obtained by putting $x = 0$, meaning that the system dynamics vanish at very high frequencies. In other words, for our purposes, the high-frequency behavior of the linearized HF amplifier corresponds to the regime where capacitors behave as short circuits and inductors as open circuits, so that only resistive effects remain. The high-frequency system associated with system (1) is thus the following linear $T$-periodic delay-difference system:

$$z(t) = \sum_{i=1}^{N} B_{i}^{2}(t)z(t - \tau_{i}), \quad t \geq s, \qquad (2)$$

where $z(\cdot)$ is defined on $[-\tau_N, \infty]$ with values in $\mathbb{R}^n$.

### 3 Spectrum of Monodromy operators and singularities of the harmonic transfer functions

To analyze the exponential stability of System (1) we consider two key objects, namely its monodromy operator and its harmonic transfer function (HTF). Specifically, we focus on two complementary facets: (i) the correspondence between the spectrum of the monodromy operator and the singularities of the HTF, and (ii) the structure of spectral sets for both objects along with their relation to exponential stability.

#### 3.1 Monodromy operator

Since it involves delayed states $y(t)$, it is simpler to rewrite System (1) in a more compact form as follows. For each $t \ge s$, let us introduce two operators $\mathbf{B}_1(t)$ and $\mathbf{B}_2(t)$, acting on a function $\phi$ defined on $[-\tau_N, 0]$ with values in $\mathbb{R}^n$, by

$$\mathbf{B}_1(t)\phi := \sum_{i=0}^{N} B_i^1(t)\phi(-\tau_i), \qquad (3)$$

$$\mathbf{B}_2(t)\phi := \phi(0) - \sum_{i=1}^{N} B_i^2(t)\phi(-\tau_i). \qquad (4)$$

It may seem pedantic to define $\mathbf{B}_1(t)$ and $\mathbf{B}_2(t)$ on functions since only the values at the $-\tau_i$ and at 0 do matter, but this definition will prove convenient. Putting $y_t(\theta) = y(t + \theta)$ for $\theta \in [-\tau_N, 0]$, System (1) can be expressed in state space form with state $(x(t), y_t)$ as

$$\left\{ \begin{array}{l} \frac{dx(t)}{dt} = A_1(t)x(t) + \mathbf{B}_1(t)y_t \\ \mathbf{B}_2(t)y_t = A_2(t)x(t), \quad t \ge s. \end{array} \right. \qquad (5)$$

Likewise, setting $z_t(\theta) = z(t + \theta)$ for $\theta \in [-\tau_N, 0]$, System (2) can be written in the following functional form:

$$\mathbf{B}_2(t)z_t = 0, \quad t \ge s. \qquad (6)$$

Systems (5) and (6) require initial data defined on an interval of length $\tau_N$ in order to define a solution for $t \ge s$. In this work, we restrict attention to continuous solutions. Consequently, the initial data must be continuous and satisfy an appropriate boundary condition to guarantee existence and uniqueness of solutions on each compact time interval. Formally, for $s \in \mathbb{R}$ we introduce the following spaces:

$$\bullet C_s := \{(x, \phi) \in \mathbb{R}^k \times C^0([-\tau_N, 0], \mathbb{R}^n) | \mathbf{B}_2(s)\phi = A_2(s)x\},$$

$$\bullet \tilde{C}_s := \{(0, \phi) \in \{0_k\} \times C^0([-\tau_N, 0], \mathbb{R}^n) | \mathbf{B}_2(s)\phi = 0\},$$

where $C^0([-\tau_N, 0], \mathbb{R}^n)$ denotes the space of continuous functions defined on the interval $[-\tau_N, 0]$ with values in $\mathbb{R}^n$. We define the following norms:

$$\bullet \text{ the Euclidean norm } \| \cdot \| \text{ on } \mathbb{R}^k \text{ or } \mathbb{R}^n;$$

$$\bullet \text{ for } (x, \phi) \in C_s \text{ or } \tilde{C}_s, \\ \qquad \| (x, \phi) \|_{C^0} = \|x\| + \sup_{\theta \in [-\tau_N, 0]} \|\phi(\theta)\|.$$

For $i$ and $j$ integers, we put $0_{i,j}$ (resp. $I_{i,j}$) for the zero (resp. identity) matrix of size $i \times j$. By abuse of notation, we shall write $0_i$ or $0$ (resp. $I_i$ or $I$) for $0_{i,1}$ (resp. $I_{i,i}$). It follows from the periodicity of the systems under consideration that $C_{s+T} = C_s$ and $\tilde{C}_{s+T} = \tilde{C}_s$. We define the solution operator $U(t, s) : C_s \to C_t$ by $U(t, s)(x_0, \phi) = (x(t), y_t)$, where $(x(t), y_t)$ is the unique solution of system (5) equal to $(x_0, \phi) \in C_s$ at time $s$. Periodicity implies that $U(t + T, s + T) = U(t, s)$. Similarly, the operator $T_D(t, s) : \tilde{C}_s \to \tilde{C}_t$ denotes the solution operator of System (6) with initial condition $\phi \in \tilde{C}_s$ at time $s$. A simple, recursive argument shows that both $U(t, s)$ and $T_D(t, s)$ are well-defined.

**Definition 1** For $s \in \mathbb{R}$, the monodromy operators are given by

$$V(s) := U(s + T, s), \qquad \tilde{V}(s) := T_D(s + T, s).$$

We also introduce the notion of $C^0$-exponential stability for each of the systems previously defined:

**Definition 2** Systems (5) and (6) are said to be $C^0$-exponentially stable if there exist constants $\gamma, \gamma_D > 0$ and
$K, K_D > 0$ such that, for all $t \ge 0$,

$$ \|U(t, s)(x_0, \phi)\|_{C^0} \le K e^{-\gamma(t-s)} \|(x_0, \phi)\|_{C^0}, \\ (x_0, \phi) \in C_s, \qquad (7) $$

$$ \|T_D(t, s)(x_0, \phi)\|_{C^0} \le K_D e^{-\gamma_D(t-s)} \|(x_0, \phi)\|_{C^0}, \\ (x_0, \phi) \in \tilde{C}_s. \qquad (8) $$

General properties of periodic dynamical systems tell us that the exponential stability of Systems (5) and (6) is completely determined by the spectral radius of their respective monodromy operators. Specifically, the next lemma follows at once from [6, Lemma 4.2].

**Proposition 3** (1) System (6) is $C^0$-exponentially stable if and only if the spectral radius of the monodromy operator $\tilde{V}(s)$ is strictly less than one.

(2) System (5) is $C^0$-exponentially stable if and only if the spectral radius of the monodromy operator $V(s)$ is strictly less than one.

Now, classical techniques from neutral differential systems, e.g. [11,10], can be adapted here to show that the monodromy operator of System (5) is a compact perturbation of the one of System (6). This property is a cornerstone of the present approach and we prove it below.

**Lemma 4** Let $\tilde{\mathbf{B}}_2(s) : (x_1^t, \phi^t)^t \mapsto (x_1^t, (\mathbf{B}_2(s)\phi)^t)^t$. There exists (infinitely many choices of) $n+k$ independent $C^1$-smooth functions defined on $[-\tau_N, 0]$, valued in $\mathbb{R}^{n+k}$, such that when arranged into a $(n+k) \times (n+k)$ matrix $\tilde{\Phi}$, they satisfy the relation $\tilde{\mathbf{B}}_2(s)\tilde{\Phi} = I_{k+n}$. The operator $T_D(t, s)(I_{k+n} - \tilde{\Phi}\tilde{\mathbf{B}}_2(s))$ has the same spectrum as $T_D(t, s)$ except possibly for the extra element $\{0\}$. Moreover, for all real $t, s$ such that $t \ge s$, it holds that

$$ U(t, s) = T_D(t, s)(I_{k+n} - \tilde{\Phi}\tilde{\mathbf{B}}_2(s)) + K(t, s), \qquad (9) $$

where $K(t, s)$ is a compact operator from $C_s$ to $C_t$. In particular, $U(t, s)$ is a compact perturbation of $T_D(t, s)$.

**PROOF.** The existence of $\Phi \in (C^1([-\tau_N, 0], \mathbb{R}^n))^n$ satisfying $\mathbf{B}_2(s)\Phi = I_n$ is straightforward. We can thus define the matrix $\tilde{\Phi}$ as:

$$ \tilde{\Phi} := \begin{bmatrix} I_k & 0_{k,n} \\ 0_{n,k} & \Phi \end{bmatrix} \qquad (10) $$

and it follows that $\tilde{\mathbf{B}}_2(s)\tilde{\Phi} = I_{k+n}$. To see that the spectrum of $T_D(t, s)(I_{k+n} - \tilde{\Phi}\tilde{\mathbf{B}}_2(s))$ consists of the spectrum of $T_D(t, s)$ possibly up to the adjunction of $\{0\}$, we observe that the following decomposition holds:

$$ C_s = \tilde{C}_s \oplus W(s) \qquad (11) $$

with $W(s) := \{(x, \Phi A_2(s)x) | x \in \mathbb{R}^k\}$, because $\tilde{C}_s$ is the kernel of $\tilde{\mathbf{B}}_2(s)$. As $T_D(t, s)(I_{k+n} - \tilde{\Phi}\tilde{\mathbf{B}}_2(s))$ is zero on $W(s)$ and is equal to $T_D(t, s)$ on $\tilde{C}_s$, we get the result. Let now $(x_0, \phi_0) \in C_s$, so that we may write: $K(t, s)(x_0, \phi_0) = (x(t), w_t)$, where $w$ satisfies the equation:

$$ \left\{ \begin{array}{l} w_s = \Phi A_2(s)x_0 \\ w(t) = \sum_{i=1}^{N} B_i^2(t)w(t - \tau_i) + A_2(t)x(t), \text{ for } t > s, \end{array} \right. \qquad (12) $$

and $x(t)$ is the solution of the equation (5) with initial data $(x_0, \phi_0)$. Hence, $x(\cdot)$ is continuously differentiable. Since we assumed that $A_2(t)$, $B_i^2(t)$ for all $i = 1, \dots, N$ and $\Phi$ are $C^1$-smooth, a fortiori $w(\cdot)$ is Lipschitz continuous in each compact interval. Thus by Ascoli's theorem, the operator $K(t, s)$ is compact from $C_s$ to $C_t$. Moreover, since operators of finite rank are compact so is $T_D(t, s)\tilde{\Phi}\tilde{\mathbf{B}}_2(s)$, therefore $U(t, s)$ is a compact perturbation of $T_D(t, s)$, as wanted.

Lemma 4 yields spectral information on $V(s)$ knowing some on $\tilde{V}(s)$. Indeed, operators that differ by a compact one have the same essential spectrum [12, Ch. V, Thm. 5.35]. As all realistic circuits are dissipative at sufficiently high frequency and therefore get associated to an exponentially stable high-frequency system (see Appendix A), we make the following assumption on $\tilde{V}(s)$:

**Assumption 5** System (6) is $C^0$ exponentially stable. In particular the spectral radius of the monodromy operators $\tilde{V}(s)$ is strictly less than one, by Proposition 3.

We can now state a theorem that sheds light on the structure of the monodromy for systems of the form (1) when they dissipate energy at high frequencies; to the best of our knowledge it is new for such delay systems, and it applies in particular to HF circuits like amplifiers.

**Theorem 6** Let $\rho(\tilde{V}(s))$ be the spectral radius of $\tilde{V}(s)$. Under Assumption 5 holds $\rho(\tilde{V}(s)) < 1$, moreover the monodromy operator $V(s)$ has finitely many eigenvalues outside each disk $D(0, \rho)$ of radius $\rho > \rho(\tilde{V}(s))$ centered at 0, and no other spectral element there.

**PROOF.** By Lemma 4 we have $V(s) = \tilde{V}(s) + A$ with $A$ a compact operator, and since a compact perturbation may only introduce additional points that cannot accumulate except on the essential spectrum, $V(s)$ has finitely many spectral elements off $D(0, \rho)$, see [12, Ch. IV, Thm. 5.35 & Pb. 5.37]. In another connection, if $\zeta I - \tilde{V}(s)$ is invertible then $\zeta I - V(s) = \zeta I - \tilde{V}(s) - A$ is Fredholm of index 0 [12, Thm. 5.26], hence $\zeta$ must be an eigenvalue if a spectral element of $V(s)$ at all.
Theorem 6 reduces the local stability check of periodic solutions in HF amplifiers (in the the $C^0$-norm) to checking for unstable eigenvalues of the monodromy. As mentioned already, the dissipativity Assumption 5 is not restrictive for it merely reflects the physical intuition that circuits dissipate energy at high frequencies. From an electronic engineering standpoint, estimating the spectrum of the monodromy operator through time-domain simulations is generally impractical, and the only available information comes from numerical estimates of the system's response to pure frequencies, using harmonic balance techniques [18]. This warrants the frequency domain approach taken in subsequent sections.

## 3.2 Harmonic transfer function

The notion of Harmonic Transfer Function (HTF) pertains to control of linear periodic systems. It arises in the context of circuits via the response of the linearized system along a periodic trajectory (System (5)) to current or voltage perturbations. Typically, one injects small current perturbations arranged in a vector $i$ at some set of nodes, and one observes the voltage perturbations $v$ thus created between certain ports; see Figure 1.

<figure><img src="image_1.png" /><figcaption>Fig. 1. Current perturbation with voltage response.</figcaption></figure>

The familiar transfer function for stationary systems gets generalized in the periodic case into the harmonic transfer function. This is a function $\mathbf{H}(p)$ of one complex variable $p$, valued in the space $\mathcal{L}(\ell^2(\mathbb{Z}))$ of bounded operators $\ell^2(\mathbb{Z}, \mathbb{C}) \to \ell^2(\mathbb{Z}, \mathbb{C})$ rather than $\mathbb{C}$; here

$$ \ell^2(\mathbb{Z}, \mathbb{C}) := \left\{ z = (z_j)_{j \in \mathbb{Z}} : z_j \in \mathbb{C}^d, \sum_{j=-\infty}^{+\infty} |z_j|^2 < \infty \right\}. $$

The HTF relates the frequencies of the input to the corresponding frequencies of the output. Unlike the classical transfer function which describes the input-output relation at a single complex frequency $p$, the HTF takes into account not only the frequency $p$ of the input but also its harmonics; i.e., frequencies of the form $p + ik\omega$ where $k \in \mathbb{Z}$ and $\omega := 2\pi/T$. Formally, we can write

$$ V(p) = \mathbf{H}(p) \mathbb{I}(p), \quad (13) $$

with

$$ V(p) = (V(p - ik\omega))_{k \in \mathbb{Z}}^{\mathbf{t}} \quad \text{and} \quad \mathbb{I}(p) = (\mathcal{I}(p - ik\omega))_{k \in \mathbb{Z}}^{\mathbf{t}}, $$

where $\mathcal{I}(p)$ and $V(p)$ are the Laplace transforms:

$$ V(p) := \int_{s}^{+\infty} e^{-pt} v(t) dt, \quad \mathcal{I}(p) := \int_{s}^{+\infty} e^{-pt} i(t) dt. $$

Establishing existence of the HTF and its connection to the monodromy operator is beyond the scope of this paper: we refer the reader to [3]. Still, two points are worth emphasizing. First, the HTF is a holomorphic operator-valued function which is uniformly bounded on the half-plane $\{p \in \mathbb{C} | \Re(p) > \gamma\}$ for $\gamma$ large enough. Second, its singularities are intimately connected to the spectrum of the monodromy operator, thereby justifying why exponential stability of the linearized system around a periodic trajectory–therefore also local stability of the original, nonlinear circuit–can be assessed by checking for unstable poles of the harmonic transfer function. Hereafter, we formulate more precise statements thereof.

In view of Theorem 6, let $(\zeta_j)_{j \in \mathbb{N}}$ enumerate the eigenvalues of $V(s)$ outside $D(0, \rho(\tilde{V}(s)))$. Put $z_{j,k} = \frac{\ln(\zeta_j) + 2ik\pi}{T}$, $j \in \mathbb{N}$, $k \in \mathbb{Z}$, for their complex logarithms.

**Theorem 7 (Structure of the HTF)** Let $\rho(\tilde{V}(s))$ denote the spectral radius of the monodromy operator $\tilde{V}(s)$. For $\gamma > \ln(\rho(\tilde{V}(s))/T)$, under Assumption 5, the harmonic transfer function $\mathbf{H}$ extends to a meromorphic operator-valued function

$$ \mathbf{H} : \{p \in \mathbb{C} | \Re(p) > \gamma\} \to \mathcal{L}(\ell^2(\mathbb{Z})), $$

whose poles all lie in the discrete set $\{z_{j,k}, j \in \mathbb{N}, k \in \mathbb{Z}\}$.

To base a stability criterion for System (5) on the singularities of the HTF via Theorem 7, we need to know that, for each $j$, there is at least one $k$ such that $z_{j,k}$ is indeed a pole of $\mathbf{H}$. This is certainly the case if current perturbations and voltage measurements (cf. Figure 1) are performed at every node of the circuit. However, this is computationally unfeasible; instead, one relies on a limited number of input-output experiments at well-chosen nodes, hoping they would detect internal instability in the system. The underlying properties of the system under such input-output experiments are detectability and stabilizability. One can surmise these generically hold for some appropriate notion of a generic circuit, but discussing the matter would take us too far afield and we shall merely assume that the input-output measurements do keep track of internal instability in the system, so that the poles of $\mathbf{H}$ faithfully reflect the unstable spectrum of the monodromy $V$ if any.

**Theorem 8 (Stability criterion)** If, for each $j$, there is at least one $k$ such that $z_{j,k}$ is indeed a pole of $\mathbf{H}$, then the absence of poles of $\mathbf{H}$ in the right half-plane

$$ \{p \in \mathbb{C} | \Re(p) \ge 0\} \quad (14) $$
is equivalent to the exponential stability of System (5) and the HTF is uniformly bounded there in $\mathcal{L}(\ell^2(\mathbb{Z}))$-norm.

Theorem 8 yields a $H^\infty$ stability criterion:

**Corollary 9 (Stability $H^\infty$) Assumptions as in Theorem 8, System (5) is exponentially stable if and only if**

$$ \sup_{\Re(p)>0} \|\mathbf{H}(p)\|_{\mathcal{L}(\ell^2(\mathbb{Z}))} < \infty, \quad (15) $$

in which case

$$ \sup_{\omega \in \mathbb{R}} \|\mathbf{H}(i\omega)\|_{\mathcal{L}(\ell^2(\mathbb{Z}))} = \sup_{\Re(p)>0} \|\mathbf{H}(p)\|_{\mathcal{L}(\ell^2(\mathbb{Z}))}. \quad (16) $$

**PROOF.** Clearly, (15) implies that the HTF has no poles in $\{p \in \mathbb{C} | \Re(p) \ge 0\}$, so exponential stability is ensured by Theorem 8. Conversely, if System (5) is exponentially stable then $\mathbf{H}$ is uniformly bounded in $\{p \in \mathbb{C} | \Re(p) \ge 0\}$, by Theorem 8. Moreover, by the operator-valued Fatou theorem (see, e.g., [15, Theorem C, Section 4.8]), the supremum of $\|\mathbf{H}(\cdot)\|_{\mathcal{L}(\ell^2(\mathbb{Z}))}$ over the imaginary axis coincides with the supremum in the right half-plane, so that (16) holds.

To recap, checking the local exponential stability of a linear periodic system like (5), whose high-frequency approximation (6) is itself exponentially stable, reduces to ensuring that $\mathbf{H}$ has no poles in the closed right half-plane, and such poles may only align on finitely many vertical lines, even though the system in question is infinite-dimensional. This is still challenging of course, as the convergence of poles in rational or meromorphic approximation is rather subtle and interpolation-based algorithms generate spurious poles. We do not touch upon such issues here, but let us simply mention that pole-detection heuristics for circuit stability check have been developed already [17]. What precedes warrants them.

# 4 A simple example

In this section, we illustrate the previous theoretical results on a classical nonlinear circuit containing a lossless transmission line. We derive the constitutive relations, linearize around a periodic solution, and apply the just developed framework to analyze both the monodromy operator and the associated harmonic transfer function. Then, stability properties and the structure of the harmonic transfer function unfold, in light of numerical simulations, for specific choices of the circuit parameters.

## 4.1 Derivation of the equations and theoretical results

Let us consider the classical Brayton circuit [5] described in Fig. 2, consisting of a T-periodic voltage source $p(t)$,

<figure><img src="image_2.png" /><figcaption>Fig. 2. Nonlinear circuit containing a lossless transmission line</figcaption></figure>

a nonlinear diode $i = g(v)$, a resistor $R$, a capacitor $C_1$, and a lossless transmission line with characteristic parameters $\mathcal{L}$ and $\mathcal{C}$. Though perhaps the simplest possible circuit with lossless transmission lines, it is highly nontrivial already. Using the method of characteristics to eliminate the line variables and applying Kirchhoff's law, the dynamics reduces to a neutral-type delay differential equation with state $x_1(t)$, $t \ge 0$:

$$ \frac{d}{dt}(x_1(t) - ax_1(t-r)) = f(t, x_1(t), x_1(t-r)), \quad (17) $$

with $r = 2\sqrt{\mathcal{L}\mathcal{C}}$,

$$ f(t, x, y) = \frac{\mathcal{R}}{C_1} [-ay + Z_0p(t - r/2) - x] - Z_0p'(t - r/2) \\ - \frac{1}{C_1} g(-ay + Z_0p(t - r/2) + x), \quad (18) $$

and constants $a, Z_0, \mathcal{R}$ defined as in

$$ \mathcal{R} = \sqrt{\mathcal{C}/\mathcal{L}} \quad Z_0 = \frac{1}{1 + R\mathcal{R}}, \quad a = \frac{1 - R\mathcal{R}}{1 + R\mathcal{R}} = 2Z_0 - 1. \quad (19) $$

This formulation is standard, see for instance [5] or [10, Introduction Chapter].

We may suppose that $T > r$ (otherwise we would have just to consider the smallest integer $n$ such that $nT > r$ and take as a period of the system $nT$). We also assume there is a T-periodic solution $\bar{x}_1(t)$. Existence and uniqueness of periodic solutions under periodic forcing for systems like (17) are discussed in [21]. Linearizing around the periodic trajectory, we obtain:

$$ \frac{d}{dt}(\tilde{x}_1(t) - a\tilde{x}_1(t-r)) = B(t)\tilde{x}_1(t) + C(t)\tilde{x}_1(t-r), \quad (20) $$

with

$$ B(t) = -\frac{1}{C_1} [\mathcal{R} + G(t)], \quad C(t) = \frac{a}{C_1} [-\mathcal{R} + G(t)], $$

where

$$ G(t) = g'(-a \bar{x}_1(t - r) + Z_0 p(t - r/2) + \bar{x}_1(t)). $$
Defining $w(t) = \tilde{x}_1(t) - a\tilde{x}_1(t - r)$, System (20) can be rewritten as a difference-delay differential system:

$$ \left\{ \begin{array}{l} \frac{d}{dt} w(t) = B(t)w(t) + (aB(t) + C(t))\tilde{x}_1(t-r), \\ \tilde{x}_1(t) = w(t) + a\tilde{x}_1(t-r), \end{array} \right. \qquad (21) $$

which is amenable to the approach developed in this paper. Indeed, the high frequency behavior of the linearized circuit is given by the simple difference delay system $z(t) = az(t-r)$ and, by a classical result [10, Theorem 3.5, chapter 9], the spectral radius of the monodromy operator is $|a|^{T/r}$. Hence, Assumption 5 is satisfied since $|a| < 1$, so the monodromy operator of System (20) has finitely many eigenvalues outside any disk of radius larger than $|a|^{T/r}$, which determines its exponential stability by Proposition 3 and Theorem 6.

As for the HTF, it is obtained by perturbing the linearized circuit with a current $i$ at one end of the line (see Fig. 3). We let $i \in L^2([0, +\infty), \mathbb{R})$, the space of square-integrable functions on $[0, +\infty)$ with values in $\mathbb{R}$.

<figure><img src="image_3.png" /><figcaption>Fig. 3. Linearised circuit around the periodic solution disturbed by a current $i$.</figcaption></figure>

The state response to the current perturbation $i$ is the voltage $v := \tilde{x}_1$ at the same point, so Kirshoff's law and the method of characteristics together yield the following input-output system:

$$ \left\{ \begin{array}{l} \frac{d}{dt} (\tilde{x}_1(t) - a \tilde{x}_1(t-r)) = B(t) \tilde{x}_1(t) \\ + C(t) \tilde{x}_1(t-r) - \frac{i(t)}{C_1}, \end{array} \right. \qquad (22) \\ v(t) = \tilde{x}_1(t), $$

with $\tilde{x}_1(\cdot)$ a real-valued function of a single variable.

We now provide an explicit representation of the HTF, which maps the input $i$ to $v$ via (22). It relies on the Fourier expansions of the $T$-periodic coefficients:

$$ B(t) = \sum_{k \in \mathbb{Z}} B_k e^{ik\omega_0 t}, \quad C(t) = \sum_{k \in \mathbb{Z}} C_k e^{ik\omega_0 t}, \quad \omega_0 = \frac{2\pi}{T}. $$

Substituting the latter into (22) and taking Laplace transforms, we can integrate termwise and identify with

Equation (13) to obtain the desired representation in the form of a doubly infinite matrix:

$$ \mathbf{H}(p) = \frac{1}{C_1} M^{-1}(p), \quad p \in \mathbb{C}, $$

where the entries of $M$, indexed by $m, n \in \mathbb{Z}$, are

$$ M_{m,n}(p) = (p + in\omega_0)(1 - a e^{-(p+in\omega_0)r}) \delta_{m,n} - B_{n-m} - C_{n-m} e^{-(p+in\omega_0)r}, $$

and $\delta_{m,n}$ denotes the Kronecker delta.

By Theorem 7, for $\gamma > \ln(a)/r$, the HTF $\mathbf{H}$ associated with the input $i$ and the output $v$ extends to a meromorphic operator-valued function on

$$ \{p \in \mathbb{C} \mid \Re(p) > \gamma\}, $$

whose poles all belong to the discrete set of eigenvalues of the monodromy operator of the linearized system (20). Subsequently, under detectability and stabilizability assumptions, Theorem 8 ensures the exponential stability of the linearized circuit provided that the harmonic transfer function has no poles in the closed right half-plane $\{p \in \mathbb{C} \mid \Re(p) \ge 0\}$.

We illustrate below the stability of the Brayton circuit in Figure 3 by numerically spotting both the spectra of the monodromy operator and the poles of the HTF, for a particular periodic solution.

## 4.2 Numerical validation

We choose the circuit parameters as $R = 0.5$, $C_1 = 1.0$, $\mathcal{L} = \mathcal{C} = 0.1$, which yields $a \approx 0.333$, $Z_0 \approx 0.667$, and $r = 0.2$. The input voltage is choosen to be a high-frequency sinusoid $p(t) = 0.01 \sin(20t)$, and the diode is modeled by the smooth cubic nonlinearity $g(v) = v + 0.1v^3$. Using a standard numerical solver for neutral delay differential equations, the evolution of the system is simulated starting from zero initial conditions. After a transient, the solution converges to a periodic regime corresponding to $\bar{x}_1(t)$, which is shown over several periods of the input excitation $p(t)$, see Fig. 4. Next, we consider $\bar{x}_1(t)$ as the periodic solution (after the transient) and investigate its local stability through the monodromy operator and the HTF.

### 4.2.1 Stability through the monodromy operator

Illustrating Theorem 6, Fig. 5 indicates that the monodromy operator of the circuit has a single eigenvalue $\zeta_1$ outside the disk of radius $a^{T/r}$. This eigenvalue is estimated to be real and approximately equal to 0.3783, which implies that System (20) is exponentially stable.
<figure><img src="image_4.png" /><figcaption>Fig. 4. Numerical simulation of the circuit showing the evolution of $\tilde{x}_1(t)$ with respect to the input $p(t)$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 5. Eigenvalues of the monodromy operator for the linearized neutral delay system around the numerically computed T-periodic trajectory. The delay interval $[0, r]$ was discretized into $M = 1300$ subdivisions and the system was integrated over one period using the 'RK45' ODE solver.</figcaption></figure>

### 4.2.2 Stability through the harmonic transfer function

From the structure of the monodromy operator, the expected poles of the HTF in $\{p \in \mathbb{C} | \Re(p) > \ln(a)/r\}$ are the complex logarithms of the eigenvalue $\zeta_1 \approx 0.3783$, divided by the period of the system. As we already mentioned, estimating these poles from pointwise measurements of the HTF is an active research topic facing the recurring challenge of singularity detection from rational (ot meromorphic) approximation. Exact or least squares interpolation methods are attractive since they proceed linearly from pointwise data, but they are hampered by the occurrence of spurious poles. Below, we take advantage of the simplicity of the circuit to follow a direct approach: we compute the zeros of the inverse of the HTF determinant and compare them with the poles inferred from the eigenvalues of the monodromy operator, as illustrated in Fig. 6. According to Theorem 7, these poles should be close to the complex logarithm of the monodromy eigenvalues divided by $T$. Our numerical results confirm this behavior, thereby providing a practical validation of Theorem 7. Minor discrepancies between the HTF poles and the logarithms of the monodromy eigenvalues are attributed to numerical approximations.

Since the harmonic transfer function is infinite-dimensional, all computations have been performed on a truncated version thereof. We define the truncated

<figure><img src="image_6.png" /><figcaption>Fig. 6. Poles from HTF approximation (blue, $N = 3$, found via root-finding of $\det M(p)$) and monodromy (red, monodromy discretized with $M = 1300$, repeated via complex logarithm).</figcaption></figure>

harmonic transfer function, $\mathbf{H}_N(p)$, of order $N \ge 0$ as

$$ (\mathbf{H}_N(p))_{m,n} = \begin{cases} \mathbf{H}(p)_{m,n}, & |m|, |n| \le N, \\ 0, & \text{otherwise,} \end{cases} \quad (23) $$

which provides a finite-dimensional approximation suitable for numerical computations.

Moreover, since the linearized circuit is stable, we can estimate the supremum of the HTF in the closed right half-plane $\{p \in \mathbb{C} | \Re(p) \ge 0\}$ based on Corollary 9. In practice, this supremum is approximated numerically along the imaginary axis by

$$ \sup_{\Re(p)>0} \|\mathbf{H}(p)\|_{L(\ell^2(\mathbb{Z}))} \approx \max_j \in \omega_j \|\mathbf{H}_N(i\omega_j)\|_{L(\ell^2(\mathbb{Z}))}, $$

where $\omega_j$ denotes a sufficiently fine frequency grid, whose resolution increases with the truncation order $N$. As shown in Fig. 7, the supremum of the truncated harmonic transfer function $\mathbf{H}_N$ stabilizes as $N$ increases.

<figure><img src="image_7.png" /><figcaption>Fig. 7. Supremum of $\mathbf{H}_N$ for truncation orders $N = 1, 2, 5, 10, 20$, with operator norm computed on a 25N-point grid over $\omega_j \in [-2\pi N/T, 2\pi N/T]$.</figcaption></figure>

## 5 Conclusion

In this work, we studied the local exponential stability of periodic steady-state solutions in high-frequency amplifiers with lossless transmission lines. The stability of
the T-periodic orbit was characterized through the spectrum of the monodromy operator. Under realistic dissipation assumptions, we proved (Theorem 6) that possible instabilities arise only from a finite set of its eigenvalues, which correspond exactly to the poles of the harmonic transfer function in the right half-plane (Theorem 7). Hence, the absence of HTF poles for $\mathbb{R}(p) \ge 0$ ensures local exponential stability (Theorem 8), leading to an $H^\infty$ stability criterion (Corollary 9). The results were illustrated numerically on a simple circuit.

We thus establish a unified framework that combines Floquet theory with frequency-domain methods in this infinite-dimensional setting arising from lossless transmission lines, and provide rigorous justification for pole-search algorithms from input-output frequency data or computation of a $H^\infty$ criterion. The method actually applies to any linearized delay-differential system along a periodic trajectory, provided that its differentiated state is finite-dimensional, its delays finitely many, and its high frequency behaviour exponentially stable. Future work should explore several avenues: extending the analysis to circuits with weakly non-dissipative active elements or essential spectral components; developing efficient numerical schemes to locate HTF poles and compute $H^\infty$ norms; and generalizing our results to lossy transmission lines.

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## A Realistic Linearized HF Amplifiers

In this appendix, we consider realistic high-frequency amplifier circuits modeled as networks of *multiport subsystems* interconnected by lossless transmission lines. Each multiport represents a portion of the circuit with a finite number of terminals and may include nonlinearities or frequency-dependent effects from both passive and active elements. We focus on the linearized behavior around a given periodic trajectory, yielding a network of linear time-periodic systems describing the multiports and their interconnections.

**Multiport.** A multiport is formally represented as a graph whose vertices are junction nodes and edges are
branches connecting two terminals. Let $I_p$ and $V_p$ denote the vectors of currents and voltages at the external ports of multiport $p$, and $z_p$ the vector of internal variables. Under standard modeling assumptions, its linearized behavior is described by

$$ \left\{ \begin{array}{l} V_p(t) = A_p(t)I_p(t) + B_p(t)z_p(t), \\ \frac{dz_p}{dt} = C_p(t)I_p(t) + D_p(t)z_p(t), \end{array} \right. \qquad (A.1) $$

with T-periodic, twice continuously differentiable matrices $A_p, B_p, C_p, D_p$.

**High-frequency dissipativity.** All real components dissipate energy at high frequency. A multiport is high-frequency dissipative if

$$ A_p(t) + A_p(t)^* \geq \alpha I, \qquad (A.2) $$

for some $\alpha > 0$, independent of $t$. This ensures $A_p$ is invertible and $A_p^{-1}$ is bounded. Moreover, a network composed of high-frequency dissipative multiports is itself dissipative [2].

**Transmission Lines.** Edges of the network are lossless transmission lines with capacitance $C_k > 0$ and impedance $L_k > 0$, satisfying the Telegrapher's equations

$$ C_k \frac{\partial v_k}{\partial t} = - \frac{\partial i_k}{\partial x}, \quad L_k \frac{\partial i_k}{\partial t} = - \frac{\partial v_k}{\partial x}, \qquad (A.3) $$

with characteristic delay $\tau_k = \sqrt{L_k C_k}$ and impedance $K_k = \sqrt{C_k / L_k}$. The resolved boundary form is

$$ \left\{ \begin{array}{l} v_k(t, 0) = x_k(t) + y_k(t - \tau_k), \\ i_k(t, 0) = K_k[x_k(t) - y_k(t - \tau_k)], \\ v_k(t, 1) = x_k(t - \tau_k) + y_k(t), \\ i_k(t, 1) = K_k[x_k(t - \tau_k) - y_k(t)]. \end{array} \right. \qquad (A.4) $$

**Coupling at Multiport Nodes.** We now consider the complete network obtained by interconnecting all multiports through the transmission lines. Each node imposes algebraic and differential constraints coupling the line boundary signals to the internal multiport variables. Let $\hat{V}_p$ and $\hat{I}_p$ denote the voltages and currents at the line boundaries, and $\hat{z}_p(t)$ the stacked vector of internal variables. Then, combining (A.1) with these constraints, the global network dynamics can be written as

$$ \left\{ \begin{array}{l} \hat{V}_p(t) = \hat{A}_p(t) \hat{I}_p(t) + \hat{A}_x(t) \hat{z}_p(t), \\ \frac{d\hat{z}_p(t)}{dt} = \hat{B}_p(t) \hat{I}_p(t) + \hat{B}_x(t) \hat{z}_p(t). \end{array} \right. \qquad (A.5) $$

Substituting the resolved line expressions (A.4) for $\hat{V}_p(t)$ and $\hat{I}_p(t)$, we obtain a difference-differential system with

individual delays for each line:

$$ \left\{ \begin{array}{l} \mathbf{x}(t) = \mathfrak{A}(t) \mathbf{x}_{\tau}(t) + \mathbf{A}_{\mathbf{z}}(t) z(t), \\ \dot{\mathbf{z}}(t) = \mathbf{A}_{\mathbf{z}}(t) \mathbf{x}_{\tau}(t) + \mathbf{B}_{\mathbf{z}}(t) z(t), \end{array} \right. \qquad (A.6) $$

where $\mathbf{x}(t) = (x_1(t), \dots, x_N(t), y_1(t), \dots, y_N(t))^t$ and $\mathbf{x}_{\tau}(t) = (x_1(t-\tau_1), \dots, x_N(t-\tau_N), y_1(t-\tau_1), \dots, y_N(t-\tau_N))^t$. The matrices $\mathfrak{A}, \mathbf{A}_{\mathbf{z}}, \mathbf{A}_{\mathbf{z}},$ and $\mathbf{B}_{\mathbf{z}}$ are T-periodic and twice continuously differentiable. Because all multiports are high-frequency dissipative, it follows from Theorem 3.7 in [1] that the high-frequency behavior of the linearized HF amplifier is exponentially stable. More precisely, the system governing the delayed port signals

$$ \mathbf{x}(t) = \mathfrak{A}(t) \mathbf{x}_{\tau}(t) \qquad (A.7) $$

is exponentially stable. Hence, such high-frequency amplifiers fall within the theoretical framework developed in the main body of the paper.