Bipolar transistor circuit analysis using the Lambert W-function

The generalized diode equation describes conduction in a diode with series resistance. An analytical solution for the generalized diode equation has been elusive; however, one was found based on the transcendental equation w=ln(x/w). The solution of this equation; w=W(x), is traditionally referred to as the Lambert W-function. This function provides a long sought after natural continuity between exponential diode and linear resistor behavior. The W-function also describes more general circuits consisting of a diode or bipolar transistor with local linear negative or positive feedback. The properties of W(x) are reviewed and several iterative methods for its calculation are compared. Three approximations for the W function are derived which can simplify bipolar circuit analysis and design. The practical utility of the proposed solutions are demonstrated in four circuits along with experimental confirmation: a common emitter amplifier with an emitter or collector feedback resistor, Schmitt trigger threshold temperature compensation, bandgap stabilized current source, and a novel current-efficient laser driver.