Deep learning-based wave digital modeling of rate-dependent hysteretic nonlinearities for virtual analog applications

Abstract

Electromagnetic components greatly contribute to the peculiar timbre of analog audio gear. Indeed, distortion effects due to the nonlinear behavior of magnetic materials are known to play an important role in enriching the harmonic content of an audio signal. However, despite the abundant research that has been devoted to the characterization of nonlinearities in the context of virtual analog modeling over the years, the discrete-time simulation of circuits exhibiting rate-dependent hysteretic phenomena remains an open challenge. In this article, we present a novel data-driven approach for the wave digital modeling of rate-dependent hysteresis using recurrent neural networks (RNNs). Thanks to the modularity of wave digital filters, we are able to locally characterize the wave scattering relations of a hysteretic reluctance by encapsulating an RNN-based model into a single one-port wave digital block. Hence, we successfully apply the proposed methodology to the emulation of the output stage of a vacuum-tube guitar amplifier featuring a nonlinear transformer.

1 Introduction

The practice of emulating analog circuits and devices in the context of digital audio effects is known as virtual analog (VA) modeling [1,2,3]. Over the last few years, a lot of research effort has been devoted to deriving efficient and accurate digital implementations of circuit nonlinearities found in analog audio gear, which are well appreciated for their peculiar tonal character by industry professionals. Among such nonlinear phenomena, frequency-dependent saturation effects due to magnetic materials are of particular interest. Indeed, electromagnetic components can be found along the entire analog sound recording chain, which comprises, e.g., guitar pickups, electrodynamic microphones, loudspeaker drivers, electrical transformers, and magnetic tapes. A distinctive characteristic of magnetic materials is rate-dependent hysteresis [4] which affects the magnetic flux density B in response to a variation in the magnetic field H and its gradient [5, 6]. In general, the output of a system exhibiting hysteresis follows different paths with increasing or decreasing inputs, resulting in various loops depending on the past history. For this reason, modeling hysteresis is known to be a challenging task, especially for what concerns discrete-time circuital simulation. Furthermore, modeling the dynamic ferromagnetic effects that modify the hysteresis characteristics depending on the input frequency adds a further aspect of complexity when trying to tackle the problem with conventional VA methods.

In the literature, VA modeling approaches can be divided into two main categories: black-box methods that try to infer the global behavior of a reference circuit from observational input/output data using, e.g., Volterra series [7] or neural networks [8], and white-box methods that emulate the reference circuit by solving the underlying system of ordinary differential equations using, e.g., state-space models [9, 10], port-Hamiltonian methods [11], or wave digital filters (WDFs) [3]. In particular, among white-box approaches, WDFs have recently proved to be an efficient framework for deriving digital representations of electromagnetic circuits [12, 13].

First introduced by A. Fettweis in the 70s to derive digital implementations of passive analog filters [14], WDFs are realized describing a reference analog circuit as an interconnection of wave digital (WD) blocks. This is accomplished by substituting Kirchhoff port variables (port voltages and port currents) with linear combinations of wave variables (incident waves and reflected waves) through the addition of a free parameter at each port called port resistance. Circuit elements and connection networks are dealt with separately as they are described by one-port or multi-port blocks characterized by input/output scattering relations.

Several works in the literature on WDFs are devoted to the modeling of circuit nonlinearities, including diodes [15,16,17,18,19,20,21], transistors [22,23,24,25], and vacuum tubes [26,27,28,29]. Paiva et al. [15] derived an explicit WD description of exponential diodes and diode pairs based on the Lambert function, which was later extended in [30]. A general approach based on the Lambert function to derive closed-form scattering relations for exponential nonlinearities, such as the Shockley diode model or simplifications of the Ebers-Moll model for certain bipolar junction transistor (BJT) amplifier configurations, was later discussed in [17]. D’Angelo et al. [20], in order to reduce computational cost, proposed to reformulate the expressions involving the main branch of the Lambert function in terms of the Wright omega function. A different approach based on one-dimensional Newton-Raphson (NR) solvers was presented in [18] and [19]. Canonical piecewise linear (CPWL) representations of nonlinear functions [31, 32] were also employed to derive explicit WD scalar mappings [12], whereas models of the diode and of triode nonlinearities based on multilayer perceptrons [21, 29] have been recently proposed.

Linear circuits and circuits with up to one nonlinear element, as long as the nonlinearity is characterized by an explicit wave mapping, can be modeled via stable discretization methods without the need of iterative solvers [19, 33]. This is a considerable advantage of WD modeling over VA methods that operate in the Kirchhoff domain, as the latter are typically characterized by systems of implicit equations and entail the use of iterative algorithms [33,34,35]. Moreover, although the use of iterative solvers is still required if multiple nonlinear elements are present in the circuit [36, 37], the modular structure of WDFs and the freedom in selecting the port parameters have proven advantageous in terms of efficiency and robustness compared to iterative methods designed in the Kirchhoff domain [12, 18, 19, 38, 39].

Despite a rich literature on WDFs, only few works to date have focused on the modeling of nonlinearities with memory. For instance, [40] presented an approach based on mutators for the implementation of a class of nonlinear dynamic one-port elements. In [41], instead, the modeling of generic memristors in the WD domain was discussed. Unfortunately, however, none of the existing methods can be readily used for the WD implementation of rate-dependent hysteresis.

Thanks to their excellent nonlinear approximation capabilities [42], deep neural networks have been recently employed for the modeling of hysteretic phenomena in various physical domains [43,44,45,46,47,48,49]. However, whilst neural networks had been previously introduced in the field of WDFs as an alternative methodology to define explicit wave mappings for static nonlinear components, deep learning methods capable of dealing with nonlinearities with memory and input rate dependency have yet to be integrated in a general WDF framework for discrete-time circuit simulation.

In this article, we bridge this gap by studying the modeling and implementation of electromagnetic audio circuits with rate-dependent hysteretic nonlinearities in the WD domain. We locally model a nonlinear reluctance exhibiting hysteresis as a one-port circuit element making use of a recurrent neural network (RNN) [45]. In fact, RNNs are capable of modeling the long-term memory effects that characterize rate-dependent hysteresis. The resulting WD block is trained using wave variables, and it can be readily inserted into multiphysics WD structures [12, 13] in order to implement nonlinear electromagnetic reference circuits. To the best of our knowledge, this constitutes the first example of using RNNs to model the dynamic behavior of circuital blocks in the WD domain. Ultimately, we pursue a hybrid approach, supplementing the white-box WDFs formalism with purely data-driven modules. Notably, the modular structure of WDFs allows us to limit the scope of black-box modeling only to the characterization of specific circuit elements by learning the respective wave scattering relations from measurement data.

The remainder of this manuscript is organized as follows. In Section 2, we provide the theoretical background on WDFs. Section 3 is devoted to the analysis of rate-dependent hysteresis models. In Section 4, we illustrate how to implement reluctances with hysteresis in the WD domain using an RNN-based model. In Section 5, we present the model training procedure. In Section 6, we utilize the proposed WD circuital block for the emulation of the output stage of a vacuum tube guitar amplifier. Conclusions are drawn in Section 7, where future work and applications are also discussed.

2 Background on wave digital filters

The design of WDFs is based on a port-wise description of a reference analog circuit. Circuit elements and topological connection networks are modeled using one- or multi-port WD blocks characterized by scattering relations. This is made possible by substituting each pair of Kirchhoff variables, i.e., port voltage v and port current i, with a pair of wave variables. Although different types of waves exist in the literature on WDFs [14, 35, 50, 51], the most used definition is that of voltage waves [14]

$$\begin{aligned} a = v + Zi\, , \quad b = v - Zi\, , \end{aligned}$$

(1)

where a and b are the incident and reflected waves, respectively, whereas Z is a free parameter called port resistance. The inverse mapping of (1) is given by

$$\begin{aligned} v = \frac{a + b}{2} \, , \quad i = \frac{a - b}{2Z}\, , \end{aligned}$$

(2)

which holds true if and only if $Z \ne 0$. Since it can be arbitrarily selected, Z constitutes a powerful degree of freedom in the port description. Indeed, a proper choice of Z is of fundamental importance for the numerical solution of WD structures [14].

2.1 Linear circuit elements

As a representative example, let us consider a generic linear one-port circuit element as the one in Fig. 1a. A large class of linear one-port circuit elements, including resistors, resistive voltage/current sources, and dynamic elements such as capacitors and inductors discretized using stable methods, can be described by means of the discrete-time Thévenin equivalent model [19]

$$\begin{aligned} v[k] = R_{\textrm{g}}[k]i[k] + V_{\textrm{g}}[k] \, , \end{aligned}$$

(3)

where k is the sampling index, v[k] is the port voltage, i[k] is the port current, $R_{\textrm{g}}[k]$ is a resistive parameter, and $V_{\textrm{g}}[k]$ is a voltage parameter. According to (1), the Thévenin equivalent model can be expressed in the WD domain as follows

$$\begin{aligned} b[k] = \frac{R_{\textrm{g}}[k] - Z[k]}{R_{\textrm{g}}[k] + Z[k]}a[k] + \frac{2Z[k]}{R_{\textrm{g}}[k] + Z[k]}V_{\textrm{g}}[k]\, . \end{aligned}$$

(4)

The instantaneous dependence between b[k] and a[k] can be eliminated by setting $Z[k] = R_{\mathrm{g}}[k]$; in this case, (4) reduces to $b[k] = V_{\mathrm{g}}[k]$, and the linear one port-element is said to be adapted [14] (see Fig. 1b).

Fig. 1

Generic linear one-port element a in the Kirchhoff domain and b in the wave digital domain. The T-shaped stub indicates port adaptation

2.2 Topological connection networks

In the Kirchhoff domain, a N-port topological connection network [51, 52] is characterized by a vector of port voltages $\mathbf{v} = [v_1, \dots , v_N]^{\textrm{T}}$ and a vector of port currents $\mathbf{i} = [i_1, \dots , i_N]^{\textrm{T}}$. Let $\mathbf{v}_{\mathrm{t}}\in \mathbb {R}^{\chi }$ be the vector of independent port voltages and $\mathbf{i}_{\mathrm{l}}\in \mathbb{R}^{\psi }$ be the vector of independent port currents, where $\chi + \psi = N$. Thus, it is possible to write

$$\begin{aligned} \mathbf{v} = \mathbf{Q}^{\textrm{T}}\mathbf{v}_{\textrm{t}} \, , \quad \mathbf{i} = \mathbf{B}^{\textrm{T}}\mathbf{i}_{\textrm{l}}\, , \end{aligned}$$

(5)

where $\mathbf{B}$ is the fundamental loop matrix and $\mathbf{Q}$ is the fundamental cut-set matrix [53]. Given that topological connection networks are lossless and reciprocal, the ortogonality property $\mathbf{Q}\mathbf{B}^{\textrm{T}}=\boldsymbol{0}$ holds true [51, 53]. The matrix $\mathbf{Q}$ of size $\chi \times N$ and the matrix $\mathbf{B}$ of size $\psi \times N$ can be derived performing a tree-cotree decomposition of the reference circuit [54]. In the WD domain, topological connection networks are modeled using N-port junctions characterized by the wave variables

$$\begin{aligned} \mathbf{a} = \mathbf{v} + \mathbf{Z}\mathbf{i} \, , \quad \mathbf{b} = \mathbf{v} - \mathbf{Z}\mathbf{i}\, , \end{aligned}$$

(6)

where $\mathbf{a} = [a_{1}, \dots , a_N]^{\textrm{T}}$ is the vector of waves incident to the junction and $\mathbf{b} = [b_1, \dots , b_N]^{\textrm{T}}$ is the vector of waves reflected by the junction, while $\mathbf{Z} = {\text {diag}}[Z_1, \dots , Z_N]$ is a diagonal matrix having port resistances as diagonal entries. The scattering relation between $\mathbf{a}$ and $\mathbf{b}$ is given by $\mathbf{b} = \mathbf{S}\mathbf{a}$, where $\mathbf{S}\in \mathbb {R}^{N\times N}$ is a scattering matrix that can be computed using either of the two following equivalent equations [51]

$$\begin{aligned} \mathbf{S} = 2\mathbf{Q}^{\textrm{T}}(\mathbf{Q}\mathbf{Z}^{-1}\mathbf{Q}^{\textrm{T}})^{-1}\mathbf{Q}\mathbf{Z}^{-1} - \mathbf{I}\, , \end{aligned}$$

(7)

$$\begin{aligned} \mathbf{S} = \mathbf{I} - 2\mathbf{Z}\mathbf{B}^{\textrm{T}}(\mathbf{B}\mathbf{Z}\mathbf{B}^{\textrm{T}})^{-1}\mathbf{B}\, , \end{aligned}$$

(8)

where $\mathbf{I}$ is the $N\times N$ identity matrix. If $\psi > \chi$, (7) is computationally cheaper than (8). If $\chi > \psi$, the opposite holds true.

2.3 Magnetic/electric junction

As shown in [55], an analogy can be drawn between the magneto-motive force and the electric voltage and between the magnetic flux and the electric current. Consequently, the coupling between magnetic and electric domains can be realized by means of the magnetic/electric (ME) junction [12, 13], shown in Fig. 2a, whose continuous-time constitutive equations are

$$\begin{aligned} \left\{ \begin{array}{l} v(t) = -n_{\textrm{t}}\frac{d\phi }{dt}\\ \mathscr {F}(t) = n_{\textrm{t}}i(t) \end{array}\right. , \end{aligned}$$

(9)

where $\mathscr {F}$ is the magneto-motive force (m.m.f.), $\phi$ is the magnetic flux, and $n_{\textrm{t}}$ is the number of winding turns of an inductive coil. The port facing the electrical subcircuit is called electric port, and the relative signals are marked with the subscript “e,” whereas the port facing the magnetic subsystem is referred to as magnetic port, and the relative signals are marked with the subscript “m.” We set $v_{\textrm{e}} = v$, $v_{\textrm{m}} = \mathscr {F}$, $i_{\textrm{e}} = i$, and $i_{\textrm{m}} = \phi$, and we rewrite (9) to express the electrical variables as functions of the magnetic variables

$$\begin{aligned} \left\{ \begin{array}{l} v_{\textrm{e}}(t) = -n_{\textrm{t}}\frac{di_{\textrm{m}}(t)}{dt} \\ i_{\textrm{e}}(t) = \frac{1}{n_{\textrm{t}}}v_{\textrm{m}}(t) \end{array}\right. . \end{aligned}$$

(10)

In order to implement the ME junction in the discrete-time domain, the time derivative in (10) can be discretized using the Backward Euler method obtaining

$$\begin{aligned} \left\{ \begin{array}{l} v_{\textrm{e}}[k] = -\frac{n_{\textrm{t}}}{T_{\textrm{s}}}(i_{\textrm{m}}[k] - i_{\textrm{m}}[k-1]) \\ i_{\textrm{e}}[k] = \frac{v_{\textrm{m}}[k]}{n_{\textrm{t}}} \end{array}\right. \end{aligned}$$

(11)

where $T_{\textrm{s}}$ is the sampling period, and k is the sampling index. To obtain the WD implementation of the ME junction [12, 13] shown in Fig. 2b, we express the Kirchhoff variables in terms of the wave variables $a_{\textrm{e}}$, $b_{\textrm{e}}$, $a_{\textrm{m}}$, and $b_{\textrm{m}}$ by using (2):

$$\begin{aligned} v_{\textrm{e}}[k]= & {} \frac{a_{\textrm{e}} + b_{\textrm{e}}}{2}, \quad i_{\textrm{e}}[k] = \frac{a_{\textrm{e}} - b_{\textrm{e}}}{2Z_{\textrm{e}}[k]},\end{aligned}$$

(12)

$$\begin{aligned} v_{\textrm{m}}[k]= & {} \frac{a_{\textrm{m}} + b_{\textrm{m}}}{2}, \quad i_{\textrm{m}}[k] = \frac{a_{\textrm{m}} - b_{\textrm{m}}}{2Z_{\textrm{m}}[k]}, \end{aligned}$$

(13)

where $Z_{\textrm{e}}$ and $Z_{\textrm{m}}$ are the free parameters of the electric and the magnetic ports, respectively. By substituting (12) and (13) into (11) and solving for the reflected waves, we obtain the following system of equations

$$\begin{aligned} \left[ \begin{array}{c} b_{\textrm{e}}[k] \\ b_{\textrm{m}}[k] \end{array}\right] = \mathbf{S}_{\text {ME}} \left[ \begin{array}{c} a_{\textrm{e}}[k] \\ a_{\textrm{m}}[k] \end{array}\right] + \mathbf{M}_{\text {ME}} \left[ \begin{array}{c} a_{\textrm{m}}[k-1] \\ b_{\textrm{m}}[k-1] \end{array}\right] , \end{aligned}$$

(14)

where

$$\begin{aligned}&\mathbf{S}_{\text {ME}} = \left[ \begin{array}{cc} -\frac{\frac{2Z_{\textrm{e}}[k]}{n_{\textrm{t}}}-\beta [k]}{\beta [k]} &{} \frac{1}{2}\bigg [\frac{\big (\frac{2Z_{\textrm{e}}[k]}{n_{\textrm{t}}}-\beta [k]\big )^{2}}{\beta [k]}-\beta [k]\bigg ]\\ \frac{2}{\beta [k]} &{} -\frac{\frac{2Z_{\textrm{e}}[k]}{n_{\textrm{t}}}-\beta [k]}{\beta [k]} \end{array}\right] ,\end{aligned}$$

(15)

$$\begin{aligned}&\mathbf{M}_{\text {ME}}= \left[ \begin{array}{cc} \frac{Z_{\textrm{e}}[k]}{T_{\textrm{s}}Z_{\textrm{m}}[k-1]}\frac{1}{\beta [k]} &{} -\frac{Z_{\textrm{e}}[k]}{T_{\textrm{s}}Z_{\textrm{m}}[k-1]}\frac{1}{\beta [k]} \\ -\frac{n_{\textrm{t}}}{T_{\textrm{s}}Z_{\textrm{m}}[k-1]}\frac{1}{\beta [k]} &{} \frac{n_{\textrm{t}}}{T_{\textrm{s}}Z_{\textrm{m}}[k-1]}\frac{1}{\beta [k]} \end{array}\right] , \end{aligned}$$

(16)

and $\beta [k] = \frac{Z_{\textrm{e}}[k]}{n_{\textrm{t}}} + \frac{n_{\textrm{t}}}{T_{\textrm{s}}Z_{\textrm{m}}[k]}$. The two matrices $\mathbf{S}_{\text {ME}}$ and $\mathbf{M}_{\text {ME}}$ need to be recomputed whenever a variation of the two port resistances $Z_{\textrm{e}}$ and $Z_{\textrm{m}}$ occurs.

Fig. 2

Magnetic/electric junction a in the Kirchhoff domain and b in the wave digital domain

In order to efficiently solve WD structures, it is desirable to remove as many implicit relations as possible. As mentioned in Section 2.1, this can be achieved through the adaptation process. Since the diagonal entries of $\mathbf{S}_{\text {ME}}$ are equal, it is possible to make both the electric and the magnetic port reflection-free at the same time by properly setting the free parameters $Z_{\textrm{e}}$ and $Z_{\textrm{m}}$. By imposing the constraint

$$\begin{aligned} \frac{\frac{Z_{\textrm{e}}}{n_{\textrm{t}}} - \frac{n_{\textrm{t}}}{T_{\textrm{s}}Z_{\textrm{m}}[k]}}{\frac{Z_{\textrm{e}}}{n_{\textrm{t}}} + \frac{n_{\textrm{t}}}{T_{\textrm{s}}Z_{\textrm{m}}[k]}} = 0\, , \end{aligned}$$

(17)

we can solve for $Z_{\textrm{e}}$ to obtain the adaptation condition for the electric port

$$\begin{aligned} Z_{\textrm{e}}[k] = \frac{n_{\textrm{t}}^2}{T_{\textrm{s}}Z_{\textrm{m}}[k]}\, , \end{aligned}$$

(18)

or solve for $Z_{\textrm{m}}$ to obtain the adaptation condition for the magnetic port

$$\begin{aligned} Z_{\textrm{m}}[k] = \frac{n_{\textrm{t}}^2}{T_{\textrm{s}}Z_{\textrm{e}}[k]}\, . \end{aligned}$$

(19)

For example, Fig. 3 shows an ME junction adapted using (19). Finally, it is worth pointing out that other discretization methods can be considered for the implementation of the ME junction, as shown in [13].

Fig. 3

ME junction showing adapted ports symbolically represented by a T-shaped stub. The magnetic port resistance is set according to the adaptation condition given in (19)

2.4 WD structures

WDFs can be organized into tree structures called connection trees. Three types of constitutive blocks can be identified in a WD connection tree: the root, which has no upward-facing ports and can have one or more downward-facing ports; nodes (typically multi-port topological or ME junctions), which have one upward-facing port and one or more downward-facing ports; leaves (typically circuit elements), which have upward-facing ports and no downward-facing ports [12].

A WD structure can be solved without employing iterative solvers only if there are no delay-free loops [14]. Delay-free loops are formed every time instantaneous implicit relations exist among wave variables. Breaking delay-free loops at each upward-facing port makes the structure realizable and computable in the WD domain. This is achieved through adaptation of all the elements and upward-facing junctions except for the root, which has no upward-facing ports.

2.5 Solving WD structures with at most one nonlinear element

Contrary to linear elements, nonlinear elements cannot be adapted as described in Section 2.1 [12, 18, 19, 38, 39]. However, in the WD domain, it is possible to implement circuits with up to one nonlinear element without resorting to iterative solvers as long as the nonlinear element is characterized by an explicit WD mapping. This is accomplished by choosing the nonlinear element as the root of the connection tree and adapting all nodes and leaves. As done in [12] for solving electromagnetic circuits characterized by an arbitrary number of WD topological junctions, we divide the WD structure in levels, where level $\ell =1$ contains only the root and level $\ell =L$ contains only leaves, and we index each node/leaf on the $\ell$-th level with the subscript u. At each sampling step k, the computational flow comprises four stages, depicted in Fig. 4:

• 1 Leaves scattering stage: the waves reflected by the leaves are computed using their scattering relations (4).

• 2 Forward scattering stage: the wave reflected by each node of level $\ell$ is computed and propagated towards level $\ell -1$, until the root is reached. The generic reflected wave is computed as

  $$\begin{aligned} b_{\ell , u, n}[k] = \mathbf{s}_{\ell , u, n}[k]\mathbf{a}_{\ell , u}[k]\, , \end{aligned}$$

  where $\mathbf{s}_{\ell ,u,n}$ is the nth row of the scattering matrix $\mathbf{S}_{\ell ,u}[k]$ corresponding to the port with index n facing level $\ell -1$. In case the considered node is a ME junction, the reflected wave is computed using one of the scalar scattering relations in (14), depending on which port (electric or magnetic) faces level $\ell -1$.

• 3 Root scattering stage: the wave reflected by the root is computed according to the WD scattering relation f characterizing the nonlinear element

  $$\begin{aligned} b_{1, 1}[k] = f_{1, 1}(a_{1, 1}[k]). \end{aligned}$$

• 4 Backward scattering stage: the waves scattered by the nodes in the $\ell$th level are computed and propagated towards level $\ell +1$. The computation starts at level $\ell =2$ and ends at $\ell =L-1$. The vector of reflected waves is evaluated as

  $$\begin{aligned} \mathbf{b}_{\ell , u}[k] = \mathbf{S}_{\ell , u}[k]\mathbf{a}_{\ell , u}[k]. \end{aligned}$$

  In case the considered node is a ME junction, the reflected wave is computed using one of the scalar scattering relations in (14), depending on which port (electric or magnetic) faces level $\ell +1$.

Fig. 4

WD connection tree computational flow. Black elements are nodes, white elements are the leaves. The root is the red element

3 Background on hysteresis modeling

In the literature, two main categories of hysteresis models can be found: physical and phenomenological. Physical models are based on the physics laws governing the target system [56, 57]. Typically, such models are mathematically complex and require detailed knowledge of the physical properties of the materials. Phenomenological models, instead, make use of conventional system identification techniques and often lack a direct physical interpretation [58, 59]. Most phenomenological models approximate the whole hysteretic nonlinearity by weighting elementary hysteretic operators, known as hysterons, characterized by a simple mathematical description [60,61,62]. Such models are mostly empirical and rely on acquired experimental data [60].

The most popular operator-based phenomenological model is the Preisach model [63], which is used in a wide range of applications. The Preisach model is a rate-independent model, which means that the variation rate of the input signal within a given range does not affect the shape of the hysteresis loop. If instead the output hysteresis depends both on the value of the input and on the speed at which it changes, we call it rate-dependent hysteresis or dynamic hysteresis. Although rate-dependent extensions of the Preisach model exist [64, 65], they entail solving computationally-intensive parameter identification problems, which make their use in practical applications very challenging.

More recently, starting from the definition of the Preisach model, new phenomenological methods based on neural networks have been introduced in the literature. Such methods model rate-dependent hysteretic nonlinearities in a data-driven fashion, relying on physical measurements of ferromagnetic or ferroelectric materials [44,45,46, 49]. Specifically, Farrokh et al. [44] proposed a multi-layer feedforward neural network called extended Preisach neural network (XPNN), based on newly defined hysteron-like neurons, which proved to be capable of simulating both rate-independent and rate-dependent hysteresis loops. Chen et al. [46] proved that diagonal recurrent neural networks (dRNNs) are able to realize the superposition of a number of rate-dependent hysterons. Moreover, Amodeo et al. [49] employed multilayer nonlinear autoregressive exogenous neural networks (NARX) for both quasi-static and dynamic hysteretic modeling of iron-dominated magnets. In the following, we focus on the recently proposed Preisach-RNN [45] which showed promising results for predicting the behavior of dynamic hysteresis in ARMCO pure iron.

3.1 Preisach-RNN

The Preisach-RNN [45] is a rate-dependent implementation of the Preisach model based on a single-layer RNN [66, 67].

The traditional mathematical formulation of the Preisach model [63, 68] can be written as

$$\begin{aligned} \hat{y}(t) = \iint _{\alpha \ge \beta }\mu (\alpha , \beta )\gamma _{\alpha \beta }(u(t))d\alpha d\beta \, , \end{aligned}$$

(20)

where $\hat{y}(t)$ is the model output at time t, u(t) is the model input at time t, whereas $\mu (\alpha ,\beta )$ is the density function that weights the elementary rectangular hysteresis operators $\gamma _{\alpha \beta }$, also called Preisach hysterons [46, 47], with $\alpha$ and $\beta$ being the ascending and descending switching thresholds, respectively. Applying the following change in coordinates [69]

$$\begin{aligned} r=\frac{\alpha -\beta }{2},\quad \nu =\frac{\alpha +\beta }{2},\quad \hat{\mu } = \mu (\nu +r, \nu -r), \end{aligned}$$

(21)

the Preisach half-plane $\{(\alpha , \beta )\,|\,\alpha \ge \beta \}$ is mapped onto the half-plane $\{(r, \nu )\,|\,r>0,\, \nu \in \mathbb {R}\}$, where the boundary between the $+1$ and $-1$ regions is described by the curve $\nu = P(u(t))$, known as Play Operator. This allows us to rearrange (20) as

$$\begin{aligned} \hat{y}(t) = \int _{0}^{\infty } \rho (r, P(u(t)))dr\, , \end{aligned}$$

(22)

where the function $\rho$ is defined as

$$\begin{aligned} \rho (r, P(u(t))) = \int _{-\infty }^{P(u(t))}\hat{\mu }(r, \nu )d\nu \,\ - \int _{P(u(t))}^{\infty }\hat{\mu }(r, \nu )d\nu \, . \end{aligned}$$

(23)

Equation (22) can be approximated by considering only M operators, yielding

$$\begin{aligned} \hat{y}(t) = \sum _{j=1}^{M} \varphi _{j}P_{j}(u(t))\, , \end{aligned}$$

(24)

where $P_{j}(u(t))$ is the jth Play Operator, and $\varphi _{j}$ represents its density function. In the literature, (24) is commonly referred to as Prandtl-Ishlinskii model [44, 70]. Figure 5 shows a Play Operator, which is defined as

$$\begin{aligned} P_{j}(u(t))= & {} \max (u(t) - r_{j},\nonumber \\{} & {} \qquad \min (u(t) + r_{j}, P_{j}(u(t-1)))), \end{aligned}$$

(25)

$$\begin{aligned} P_{j}(u(0))= & {} \max (u(0) - r_{j},\nonumber \\{} & {} \qquad \min (u(0) + r_{j}, \kappa _{0})), \end{aligned}$$

(26)

where $\kappa _0$ is the initial condition for the operator and $r_j$, which represents the discrete counterpart of r, is defined as

$$\begin{aligned} r_{j} = \frac{j-1}{M}\left( \max (u(t)) - \min (u(t))\right) \, , \end{aligned}$$

(27)

for $j = 1, 2, \dots , M$.

Fig. 5

Play Operator $P_j(u)$

The idea behind the Preisach-RNN, shown in Fig. 6, is to model the density function $\varphi _j$ in (24) in the discrete-time domain using an RNN. Furthermore, RNNs allow us to model rate-dependent hysteresis, extending the traditional Preisach model to this case. The hidden state of a U-node RNN at sample k is computed as

$$\begin{aligned} \mathbf{h}[k] = f_{\textrm{h}}\left( \mathbf{W}_{\textrm{xh}}\mathbf{x}[k] + \mathbf{W}_{\textrm{hh}}\mathbf{h}[k-1] + \mathbf{b}_{\textrm{h}}\right) \, , \end{aligned}$$

(28)

where $\mathbf{h}[k]$, $\mathbf{h}[k-1] \in \mathbb {R}^{U}$ are the current and previous hidden states. The scalar output at sample k is computed as

$$\begin{aligned} \hat{y}[k] = f_{\textrm{o}}\left( \mathbf{w}^{\textrm{T}}_{\textrm{hy}}\mathbf{h}[k] + b_{\textrm{y}}\right) . \end{aligned}$$

(29)

The input vector $\mathbf{x}[k] \in \mathbb {R}^{M+2}$ is built concatenating the input signal u[k], the input derivative $\dot{u}[k]$, and M Play Operators $P_{j}(u[k])$ with $j=1,...,M$. $\mathbf{W}_{\textrm{xh}} \in \mathbb {R}^{U \times (M+2)},\,\mathbf{W}_{\textrm{hh}} \in \mathbb {R}^{U\times U},\,\mathbf{w}_{\textrm{hy}}\in \mathbb {R}^{U}$, $\mathbf{b}_{\textrm{h}}\in \mathbb {R}^{U}$, and $\ b_{\textrm{y}}\in \mathbb {R}$ are the network weights and biases. Finally, the hidden state activation function $f_{\textrm{h}}$, which is applied element-wise to the output of each hidden neuron, is the hyperbolic tangent, and the output activation function $f_{\textrm{o}}$ is a linear activation, i.e., $f_{\textrm{o}}(x) = x$.

Fig. 6

Diagram of a Preisach-RNN architecture

4 Wave digital hysteretic nonlinearities

The conventional approach to develop a magnetic equivalent circuit, employed, for example, in [71], is based on the analogy between magneto-motive force $\mathscr {F}$ and electric voltage v and between the magnetic flux $\phi$ and electric current i [55]. In the rest of this manuscript, with an abuse of nomenclature, we will generally refer to both pairs of v-i variables and $\phi$-$\mathscr {F}$ variables at the circuit ports as variables in the Kirchhoff domain, in order to distinguish them from the corresponding variables in the WD domain. The electrical variables v and i in linear resistors are related by the Ohm’s law $v=Ri$, whereas the two magnetic variables are related by an equivalent linear and instantaneous Ohm-like law

$$\begin{aligned} \mathscr {F} = \mathscr {R}\phi , \end{aligned}$$

(30)

known as Hopkinson’s law, which relates $\mathscr {F}$ and $\phi$ through the reluctance parameter $\mathscr {R}$, which is analogous to the electrical resistance. Since, as a first approximation, higher-order magnetic effects can be considered negligible in the audio frequency band, we can assume $\phi$ to be uniform along cross sections, and thus that the topology of the magnetic equivalent circuit can be directly derived from the magnetic structure: each winding of $n_{\textrm{t}}$ turns is represented by an ideal magnetic voltage generator $\mathscr {F} = n_{\textrm{t}}i$, whose polarity is given by the sign convention depicted in Fig. 7, where i is the winding current. The magnetic path is represented by an equivalent reluctance, possibly nonlinear, whose value depends on the geometry and physical properties of the magnetic material. Although (30) refers to the linear case, in the following we will consider a nonlinear mapping between $\phi$ and $\mathscr {F}$ to address rate-dependent hysteresis.

Fig. 7

Sign convention of magnetic voltage generators. a Coil wound counter-clockwise. b Coil wound clockwise. The figure is taken from [12]

While the reluctance defines the constitutive relation between the magnetic variables $\phi$-$\mathscr {F}$, the pair B-H, where B is the flux density and H is the magnetic field, is related by the permeability. In fact, as a first approximation, the relation between B and H can be defined by the formula $B = \mu _0\mu _r H$, where $\mu _0$ is the vacuum permeability and $\mu _r$ is the relative permeability. Moreover, we can convert the B-H curve of the material into its $\phi$-$\mathscr {F}$ representation as follows

$$\begin{aligned} \phi = B\Lambda , \quad \mathscr {F}=H\Gamma \, , \end{aligned}$$

(31)

where $\Lambda$ and $\Gamma$ are the cross-section and the length of the magnetic path expressed in meters, respectively [12, 13, 55]. Therefore, if we consider the magnetic structure to be made of a magnetic material with a B-H characteristic exhibiting rate-dependent hysteresis, the equivalent nonlinear reluctances that model the different magnetic paths across the magnetic material are going to exhibit hysteresis. The dynamic nonlinearity is thus confined into the constitutive equation of the reluctance $\mathscr {R}$, i.e., the relation between $\phi$ and $\mathscr {F}$.

In the Kirchhoff discrete-time domain, we can recast the problem of modeling the constitutive equation of a reluctance with rate-dependent hysteresis into a nonlinear regression problem, i.e.,

$$\begin{aligned} \hat{\phi }[k] = g(\mathscr {F}[k], \mathscr {F}[k-1], \dots , \mathscr {F}[0]\,;\,\theta _{\textrm{KD}})\, , \end{aligned}$$

(32)

where the mapping g is modeled by a suitable RNN. The set of parameters $\theta _{\textrm{KD}}$ is obtained by training the network in the Kirchhoff domain to predict the current value of the magnetic flux $\phi [k]$ given the input time-series $\mathscr {F}[k], \mathscr {F}[k-1], \dots , \mathscr {F}[0]$. RNNs are particularly suitable for modeling non-instantaneous nonlinearities, because they are fed with input time-series data and use recurrent connections to implement an infinite dynamic response.

The strategy employed to avoid the shortcomings of learning long-term dependencies from excessively long time-series [72] is to limit the length of the input sequence. Each time-series is thus split into sequences of length K, which are then sequentially fed to the RNN. To ensure long-term memory, however, we propagate the hidden states of the recurrent layers between consecutive sequences, according to a cross-batch statefulness paradigm. Hence, the rate-dependent hysteresis nonlinear regression problem in (32) can be rewritten as

$$\begin{aligned} \hat{\phi }[k] = g(\mathscr {F}[k], \dots , \mathscr {F}[k-K+1]\,;\,\theta '_{\textrm{KD}})\, , \end{aligned}$$

(33)

where $\theta '_{\textrm{KD}}$ are obtained by training a stateful RNN.

A similar approach can be adopted in the WD domain by properly converting Kirchhoff variables into wave variables according to (1). This yields

$$\begin{aligned} \hat{b}[k] = g(a[k], \dots , a[k-K+1]\,;\,\theta '_{\textrm{WD}})\, , \end{aligned}$$

(34)

where $\theta '_{\textrm{WD}}$ is the set of parameters obtained by training the neural network to predict the reflected wave b[k] given an input sequence composed of the incident wave a[k] and its $K-1$ previous values, thus obtaining an explicit scattering relation. This relation can be then used to implement the one-port WD realization of a nonlinear reluctance with rate-dependent hysteresis.

It is interesting to note that the newly defined WD block shares some common characteristics with WD models of linear dynamic elements such as capacitors and inductors [19], as their behavior depends on (buffers of) past samples of wave variables.

In the following, we implement the rate-dependent hysteretic mapping g using a Preisach-RNN architecture (see Section 3.1) with $U=32$ hidden units and $M=8$ Play Operators.

5 Model training and evaluation

We hereby present the training procedure of the WD hysteretic block for the specific application scenario described in Section 6. In particular, we will assume that a single nonlinear reluctance component is sufficient to model the characteristics of the constitutive magnetic material. Let us consider the magnetic components to be made of Pearlitic Steel R260, whose measurements are used to determine the B-H hysteretic behavior under consideration.

Presented in [73], the available dataset [74] consists of magnetic measurements obtained driving the system with a triangular input current sampled at 100 kHz having 11 different input frequencies, i.e., 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000 Hz. For each frequency, the curves include measurements of two periods of the triangular input signal H and the corresponding flux density B: the first period contains the first magnetization curve, whereas the second period fully describes the main hysteresis loop. The first magnetization curve is defined as the B-H curve branch that describes the material magnetization process starting from a state of no magnetization ($B=0$). A suitable measurement procedure would involve a complete de-magnetization of the magnetic material between two consecutive acquisitions, leading to the same initial condition $H[0]=B[0]=0$, regardless of the input frequency. However, each hysteresis loop in the available dataset is characterized by a different initial magnetization value B[0], probably due to incomplete de-magnetization. In fact, in order to obtain a coherent dataset, it would be desirable to have each hysteresis curve characterized by the same B[0]. With the aim of reducing possible inconsistencies during training, we preprocess the dataset by excluding the first period of each curve and circularly shifting the result by a quarter of a period. As shown in Fig. 8, this ensures that all training examples start from the same rate-independent value, i.e., the magnetic saturation value. Furthermore, we discard the 0.5 Hz measurements due to discontinuities in the resulting B-H curve. We repeat periods to let all remaining curves match the duration of two periods of the lowest frequency curve (1 Hz); this corresponds to fixing the duration of the measurement session to two seconds for all input rates and allows us to balance the training set across frequencies. To transform the B-H pair of variables into WD variables a-b, it is first necessary to convert the Pearlitic Steel R260 curves into the corresponding $\phi$-$\mathscr {F}$ curves, by using the equalities in (31). Then, since $\mathscr {F}$ is the magnetic equivalent of voltage and $\phi$ is the equivalent of current, the dataset can be expressed in the WD domain via a transformation similar to (1), i.e., [12]

$$\begin{aligned} a = \mathscr {F} + Z\phi , \quad b = \mathscr {F} - Z\phi \, , \end{aligned}$$

(35)

where the free parameter Z is fixed and can be set according to the adaptation condition of the junction port to which the reluctance is connected. The inverse transformation of (35) is given by

$$\begin{aligned} \mathscr {F} = \frac{a + b}{2} \, , \quad \phi = \frac{a - b}{2Z}\, . \end{aligned}$$

(36)

The wave variables are then rescaled in $[-1, 1]$. In order to accomplish this, the values $a_{\text {min}}$, $a_{\text {max}}$, $b_{\text {min}}$, $b_{\text {max}}$ are estimated from the WD data, and used to scale the wave variables according to

$$\begin{aligned} \tilde{a}[k]= & {} 2\cdot \frac{a[k] - a_{\text {min}}}{a_{\text {max}} - a_{\text {min}}} - 1,\nonumber \\ \tilde{b}[k]= & {} 2\cdot \frac{b[k] - b_{\text {min}}}{b_{\text {max}} - b_{\text {min}}} - 1, \end{aligned}$$

(37)

where $\tilde{a}[k]$ is the scaled incident wave, and $\tilde{b}[k]$ is the scaled reflected wave. It is then possible to scale back $\tilde{a}[k]$ and $\tilde{b}[k]$ into their original range using the following equations

$$\begin{aligned} a[k]= & {} a_{\text {min}} + \frac{\tilde{a}[k] +1}{2}\cdot (a_{\text {max}} - a_{\text {min}}),\nonumber \\ b[k]= & {} b_{\text {min}} + \frac{\tilde{b}[k] +1}{2}\cdot (b_{\text {max}} - b_{\text {min}}). \end{aligned}$$

(38)

Fig. 8

a Exemplificative H and B curves contained in the measurement dataset before time shift. b The corresponding H and B curves after circular time shift. The curves are starting from positive saturation values

Given the audio application scenario, data are resampled at a common audio sampling frequency, i.e., $f_{\textrm{s}}=48$ kHz. Input sequences $\tilde{\varvec{a}}[k] = \tilde{a}[k], \dots , \tilde{a}[k-K+1]$ are obtained by shifting a rectangular window of length $K=20$ over the input waves with unitary hop-size and assigning the corresponding ground truth value $\tilde{b}[k]$ to each of them. The pairs $(\tilde{\varvec{a}}[k], \tilde{b}[k])$ are assembled in batches containing one sequence for each input rate included in the training set; this is done to let all training frequencies contribute to each optimization step. The Preisach-RNN described in Section 4 is implemented in Python using Pytorch [75] and comprises 1441 trainable parameters. It is trained to minimize the following loss function defined in the Kirchhoff domain:

$$\begin{aligned} \mathcal {L}= & {} \mathcal {E}(\mathscr {F}, \hat{\mathscr {F}}) + \mathcal {E}(\phi , \hat{\phi }) \nonumber \\= & {} \mathcal {E}\!\left( \frac{a+b}{2}, \frac{a+\hat{b}}{2}\right) + \mathcal {E}\!\left( \frac{a-b}{2Z}, \frac{a-\hat{b}}{2Z}\right) , \end{aligned}$$

(39)

where

$$\begin{aligned} \mathcal {E}\left( y, \hat{y}\right) = \frac{\sum _k{(y[k] - \hat{y}[k])^ 2}}{\sum _k{y^2[k]}} \end{aligned}$$

(40)

is the normalized mean squared error (NMSE), whereas a[k] and b[k] are obtained from the scaled network inputs $\tilde{a}[k]$ and outputs $\tilde{b}[k]$ through (38). Notably, the loss function in (39) comprises two NMSE terms, one for $\hat{\mathscr {F}}$ and one for $\hat{\phi }$, as both depend on the predicted wave $\hat{b}$.

To evaluate the model, we perform leave-one-out cross-validation (LOOCV). Namely, we train ten different Preisach-RNNs, each time selecting nine of the ten frequencies as training set and using the remaining one for evaluation. Each training consists of ten epochs using Adam [76] and a learning rate of $10^{-4}$, and it is run on a single NVIDIA TITAN V with 12 GB of RAM. The results are reported in Table 1. Figure 9a shows the model predictions for the test curve at 20 Hz in the WD domain, whereas Fig. 9b shows the corresponding Kirchhoff variables obtained applying the inverse wave transformation in (36). Despite the limited amount of available data, LOOCV shows an average NMSE in the order of magnitude of $10^{-4}$ for both the WD variables and the Kirchhoff domain variables. These results suggest that the network may exhibit good generalization properties when used to predict waves with an input rate that was not included in the training set. In turn, this gives us confidence that the proposed rate-dependent hysteresis model, trained on the entire measurement dataset, could be successfully applied in a discrete-time circuital simulation scenario.

Table 1 Leave-one-out cross-validation results

In the next section, we will describe the use of the proposed WD hysteretic block for the emulation of the output stage of a vacuum tube guitar amplifier.

Fig. 9

a Predictions of a Preisach-RNN in the WD domain with $U = 32$ hidden units and $M=8$ Play Operators (blue) vs. the ground truth (orange). Input rate: $f=20 \textrm{Hz}$. b WD predictions transformed into Kirchhoff domain variables by means of (36)

6 Example of application

As a reference circuit, let us consider the push-pull output stage of a vacuum tube guitar amplifier shown in Fig. 10. Let us assume that the output stage consists only of the nonlinear three-winding audio transformer directly driving the loudspeaker.¹ The secondary side of the transformer is connected to the speaker, modeled by means of the series between the resistor $R_{\textrm{L}}$ and the inductor $L_{\textrm{L}}$. Without loss of generality, we consider two identical input signals $V_{\textrm{in}1}$ and $V_{\textrm{in}2}$ and, thus, two primary windings with $n_{\textrm{t}1}$ and $n_{\textrm{t}2}$ turns, respectively. The secondary side of the transformer presents a single winding with $n_{\textrm{t}3}$ turns. $R_{\textrm{p}1}$ and $R_{\textrm{p}2}$ are the primary coil resistances, whereas $R_{\textrm{s}}$ corresponds to the secondary coil. Let us assume that the transformer in Fig. 10 has a UI geometry. We also assume that the three windings are positioned on the core structure, as depicted in Fig. 11.

Fig. 10

A possible output stage of vacuum tube guitar amplifier

Fig. 11

Core magnetic structure under consideration

The core geometry is taken from one of GRAU GmbH datasheets [77], and all of its dimensions are reported in Fig. 12.

Fig. 12

Core geometry of the output transformer. All dimensions are in millimeters ($10^{-3}\ \textrm{m}$). a Front view. b Side view

The magnetic equivalent circuit of the UI core structure shown in Fig. 11 is derived according to Section 4. The result is shown in Fig. 13, where we notice the m.m.f. sources modeling the three windings with $n_{\textrm{t}1}$, $n_{\textrm{t}2}$, and $n_{\textrm{t}3}$ turns, respectively, as well as the nonlinear reluctance $\mathscr {R}$ modeling the magnetic material. Given its specific topology and disregarding the effect of eddy currents or other higher-order effects, a single nonlinear reluctance is, in fact, enough to model the whole magnetic core [13, 71]. The geometric parameters of the magnetic path of the considered UI core are $\Gamma = 240\,\text {mm}$ and $\Lambda = 400\,\text {mm}^{2}$. The circuit parameters are summarized in Table 2.

Table 2 Values of the parameters of the circuit in Fig. 10

Fig. 13

Equivalent circuit model of the magnetic structure shown in Fig. 11

Once we derived the magnetic subcircuit, we connect it to the electrical subcircuits by means of the ME junctions (introduced in Section 2.3), thus obtaining a modular multiphysics model [12, 13]. The multiphysics model of the reference circuit in Fig. 10 is shown in Fig. 14. The magnetic domain is represented by the central subcircuit, and it is coupled to the electrical subcircuits by means of three ME junctions.

Fig. 14

Output stage of a vacuum tube guitar amplifier including a multiphysics transformer model

The WD realization of the circuital model in Fig. 14 is shown in Fig. 15. The scattering matrix $\mathbf{S}_{\textrm{s}1}$ of the WD 4-port junction that embeds the topological information related to the circuit at the secondary side of the transformer can be computed substituting the fundamental loop matrix $\mathbf{B}_{\textrm{s}1} = \begin{bmatrix}-1&1&1&1 \end{bmatrix}$ into (8). Port 1 is connected to the electric port of junction $M_1/E_3$, resistor $R_{\textrm{s}}$ is connected to port 2, while resistor $R_{\textrm{L}}$ and inductor $L_{\textrm{L}}$ are connected to port 3 and 4, respectively. All these circuit elements are linear and can be thus adapted as described in Section 2.1. The scattering matrix $\mathbf{S}_{\textrm{s}2}$ of the WD 4-port junction related to the magnetic subcircuit is again obtained from (8) but considering $\mathbf{B}_{\textrm{s}2} = \begin{bmatrix}1&-1&-1&1\end{bmatrix}$. Ports 2, 3, and 4 are connected to the magnetic port of junction $M_1/E_1$, $M_1/E_2$, and $M_1/E_3$, respectively. The resistive voltage sources $V_{\textrm{in}1}$ and $V_{\textrm{in}2}$ at the primary side of the transformer are also linear, and are connected to the electric ports of junction $M_1/E_1$ and $M_1/E_2$, respectively. Finally, the WD one-port block related to the nonlinear reluctance $\mathscr {R}$ is connected to port 1 of the magnetic topological junction as shown in Fig. 15.

Fig. 15

WD structure implementing the output stage of a vacuum tube guitar amplifier in a multiphysics fashion. The T-shaped stubs indicate port adaptation

The reference circuit contains a single nonlinear one-port element, which means that it is possible to solve the WD structure by employing the algorithm described in Section 2.5 and illustrated in Fig. 4. Since, as a first approximation, the two generators are assumed to be identical, we set $V_{\textrm{in}1} = V_{\textrm{in}2} = A\cos (2\pi k f_0 / f_{\textrm{s}})$, where k is the sampling index and $f_{\textrm{s}}$ is the sampling frequency. We choose the frequency $f_0 = 50$ Hz. We set $A=250$ V, a value high enough to force the transformer to reach core saturation, in accordance to the available measurement dataset [74].

Before the main WD simulation loop, we run an initialization phase to define the initial hidden state of the WD Preisach-RNN. Namely, we set the signals $V_{\textrm{in}1} = V_{\textrm{in}2} = A$, i.e., the first value of the sinusoidal input signal, and we run the discrete-time simulation for two seconds. Both the input buffer and the hidden states of the WD Preisach-RNN are populated using the values obtained at the end of the initialization loop. The initialization of incident and reflected wave variables at the ports of ME junctions is also performed in the same way.

6.1 Results

In this subsection, we discuss the numerical results obtained from the simulation of the WD structure shown in Fig. 15. The simulation of a single input period takes on average 784 ms on a laptop-mounted Intel Core i5-1240P 1.70 GHz CPU. Figure 16 shows the voltage $v_{Z_{\textrm{L}}}$ across the series between resistor $R_{\textrm{L}}$ and inductor $L_{\textrm{L}}$, which models the loudspeaker connected to the secondary side of the transformer, whereas Fig. 17 shows the operation points on the nonlinear reluctance curve visited during the simulation. Being proportional to the derivative of the magnetic flux, the voltage $v_{Z_{\textrm{L}}}$ exhibits sharp peaks associated to magnetic core transitions from a positive magnetic flux saturation region to a negative saturation region, and vice versa. It is difficult to quantify the accuracy of the simulation results shown in Fig. 16, due to the fact that there is no easy way to simulate such a rate-dependent nonlinearity within existing circuit simulation software such as LTspice or Mathworks Simscape. However, referring to Fig. 17, we may state that the hysteretic curve is correctly visited throughout the discrete-time simulation, raising our confidence as far as the accuracy of the proposed method is concerned. The predicted curve reacts to different input frequencies $f_0$ with an hysteresis that is in fact comparable to the physical measurements contained in the dataset presented in Section 5.

Fig. 16

Voltage across the series of resistor $R_{\textrm{L}}$ and inductor $L_{\textrm{L}}$

Fig. 17

Hysteresis characteristics of reluctance $\mathscr {R}$ visited during the WD simulation

7 Conclusions

The modeling and discrete-time circuit simulation of magnetic hysteresis is a notoriously challenging task, especially due to its rate-dependent nature. For this reason, despite the pervasive presence of magnetic components in analog audio gear, circuits with hysteretic elements are usually not tabled for virtual analog applications. In this manuscript, we explored, for the first time, the possibility of using an RNN-based architecture to model hysteretic nonlinear elements in the WD domain. By properly converting the training data expressed as Kirchhoff variables into wave variables, we defined a data-driven WD circuital block that encapsulates a neural network capable of modeling reluctances with rate-dependent hysteresis. Thus, we successfully employed the proposed WD block for the emulation of the output stage of a vacuum tube guitar amplifier, where the nonlinear transformer is modeled in a multiphysics fashion. More in general, this work does not only constitute the first example of using RNNs to model rate-dependent hysteresis behaviors in the WD domain but also a first step into the exploration of deep learning-based solutions for the WD modeling of nonlinearities with memory for virtual analog applications.

Future work may concern refining the proposed model by considering a dataset of magnetic measurements at input rates spanning the entire audio bandwidth that includes minor hysteresis loops. A noteworthy extension would also be integrating the WD hysteresis block into audio circuits with multiple nonlinearities, which can be then efficiently emulated in the WD domain by exploiting iterative techniques, such as the hierarchical scattering iterative method introduced in [12, 13]. We envision scenarios where data-driven methods could be further developed to supplement the WDF framework in the characterization of circuit nonlinearities directly from experimental measurements. This is done with the full awareness that, in the future, the availability of more efficient simulation algorithms and of more computational power will lead the way towards the real-time implementation of increasingly complex audio circuits.