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A new family of wave-digital vacuum tube triode models is presented. These models are inspired by the triode model by Cardarilli , which provides realistic simulation of the triode's transconductance behavior, and hence high accuracy in saturation conditions. The triode is modeled as a single memoryless nonlinear three-port wave digital filter element in which the outgoing wave variables are computed by locally applying the monodimensional secant method to one or two port voltages, depending on whether the grid current effect is taken into account. The proposed algorithms were found to produce a richer static harmonic response, introducing comparable or less aliasing and requiring approximately 50% less CPU time than previous models. The proposed models are suitable for real-time virtual analog circuit simulation.