Current-voltage relations and equivalent circuits of transistors at high injection levels

Abstract

A theory is given of the modifications of the large signal current-voltage relations as well as of the small signal parameters under high injection conditions in the forward active mode. Though a brief discussion is presented of a variety of effects taking place at high current levels, we restrict ourselves mainly to the consequences of the electric field and the ohmic drops in the active base region. The differences and inconsistencies between the J n = 0 or Webster approach, the ambipolar or Ebner-Gray approach, and the div E = 0 or Spenke approach, are pointed out, as well as the connection with other theories such as the charge-control model of Gummel and Poon. Using the basic ambipolar flow equations, the Ebers-Moll equations are generalized and by a variational procedure the low frequency parameters of the common base T-circuit and the common emitter pi-circuit are found. For d.c. the limiting high injection characteristics go as exp( qV EB /2 kT); the mobility dependent asymmetry for PNP and NPN transistors, characterized by the ratio m = ( μ n − μ p )/( μ n + μ p ), as found by Ebner and Gray for transistors with zero base current, is shown not to apply to regular transistors with finite I B . However, the Ebner-Gray quantity m remains in the a.c. parameters. An extensive analytical high frequency analysis is made, using the Green's function method. For a special model simple formulas are ultimately found for the junction input impedance (equation (4.44)) and for the conductivity-modulation bulk impedance (equation (4.52)). The latter confirms completely the Kohn-Nonnenmacher model (Fig. 5) and all elements are calculated. On the collector side the Early effects are incorporated in the modified Ebers-Moll model.