Wave Digital Modeling of Nonlinear 3-terminal Devices for Virtual Analog Applications

Abstract

We propose a novel modeling method for circuits containing arbitrary nonlinear 3-terminal devices, which operates in the wave digital (WD) domain. This approach leads to the definition of a general and flexible WD model for 3-terminal devices, whose number of ports varies from 1 to 6. The generality of the method is confirmed by the fact that the WD models of 3-terminal devices already discussed in the literature can be seen as particular cases of the model that we present here. As examples of applications of our method, we develop WD models of the three most widespread types of transistors in audio circuitry, i.e., the MOSFET, the JFET and the BJT. These models are here designed to be used in Virtual Analog audio applications; therefore, their derivation is aimed at minimizing computational complexity while avoiding implicit relations between port variables, as far as possible. Proposed MOSFET and JFET models are characterized by third-order polynomial equations; hence, explicit closed-form wave scattering relations are obtained. On the other hand, the Ebers–Moll model describing the BJT results in transcendental equations in the WD domain that cannot be solved analytically. In order to cope with this problem, we propose a modified Newton–Raphson (NR) method for solving the implicit Ebers–Moll equations in the WD domain. Such iterative method exhibits a significantly higher robustness and convergence rate with respect to the traditional NR method, without compromising its efficiency. Finally, WD implementations of some audio circuits containing transistors are discussed.

Appendix A

Let us consider a $3^{rd}$ grade equation in the canonical form

$$\begin{aligned} \xi x^3 + \delta x^2 + \nu x + \rho =0 \end{aligned}$$

where $\xi \ne 0$ and x is the unknown variable. The 3 solutions in closed form are

$$\begin{aligned}&x_1= s + t - \frac{\delta }{3\xi } \\&x_2=-\frac{1}{2} (s+t) -\frac{\delta }{3\xi } +\frac{ \root \of {3}}{2} (s-t) j \\&x_3=-\frac{1}{2} (s+t) -\frac{\delta }{3\xi } -\frac{ \root \of {3}}{2} (s-t) j \end{aligned}$$

where j is the imaginary unit and

$$\begin{aligned}&s= \root 3 \of {\frac{r}{2}+\sqrt{\frac{q^3}{27} +\frac{r^2}{4}}},\,\,\,\,\,\,\,\,\, t= \root 3 \of {\frac{r}{2}-\sqrt{\frac{q^3}{27} +\frac{r^2}{4}}},\\&q=\frac{3\xi \nu -\delta ^2}{3\xi ^2},\,\,\,\,\,\,\,\,\, r=\frac{9\xi \delta \nu - 27\xi ^2 \rho -2\delta ^3}{27\xi ^3}. \end{aligned}$$