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.

中文翻译：

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用于虚拟模拟应用的非线性 3 端器件的波形数字建模

我们为包含任意非线性 3 端器件的电路提出了一种新的建模方法，该方法在波形数字 (WD) 域中运行。这种方法导致为 3 端设备定义了一个通用且灵活的 WD 模型，其端口数从 1 到 6 不等。 已经讨论过的 3 端设备的 WD 模型证实了该方法的通用性在文献中可以看作是我们在此介绍的模型的特殊情况。作为我们方法的应用示例，我们开发了音频电路中三种最广泛使用的晶体管类型的 WD 模型，即 MOSFET、JFET 和 BJT。这些模型设计用于虚拟模拟音频应用；所以，它们的推导旨在最小化计算复杂度，同时尽可能避免端口变量之间的隐式关系。建议的 MOSFET 和 JFET 模型以三阶多项式方程为特征；因此，获得了明确的封闭形式的波散射关系。另一方面，描述 BJT 的 Ebers-Moll 模型导致 WD 域中的超越方程无法解析求解。为了解决这个问题，我们提出了一种改进的 Newton-Raphson (NR) 方法来求解 WD 域中的隐式 Ebers-Moll 方程。这种迭代方法相对于传统的 NR 方法表现出明显更高的鲁棒性和收敛速度，而不会影响其效率。最后，讨论了一些包含晶体管的音频电路的 WD 实现。