Modeling nonlinear wave digital elements using the Lambert function

A large class of transcendental equations involving exponentials can be made explicit using the Lambert W function. In the last fifteen years, this powerful mathematical tool has been extensively used to find closed-form expressions for currents or voltages in circuits containing diodes. Until now almost all the studies about the W function in circuit analysis concern the Kirchhoff (K) domain, while only few works in the literature describe explicit models for diode circuits in the Wave Digital (WD) domain. However explicit models of NonLinear Elements (NLEs) in the WD domain are particularly desirable, especially in order to avoid the use of iterative algorithms. This paper explores the range of action of the W function in the WD domain; it describes a procedure to search for explicit wave mappings, for both one-port and multi-port NLEs containing diodes. WD models, describing an arbitrary number of different parallel and anti-parallel diodes, a transformerless ring modulator and some BJT amplifier configurations, are derived. In particular, an extended version of the BJT Ebers-Moll model, suitable for implementing feedback between terminals, is introduced.