Applications of Linear Graph Theory to Lumped Models of Solid-State Devices

Linear graph theory has received wide application in electrical network theory. This paper extends the well-known methods of network analysis by exploiting the linear relationship between current and carrier density that exists in the lumped (large-signal) model of the bipolar transistor. The lumped model is described briefly and a linear graph is drawn for the lumped model. Relations between the trees and two-trees of this graph and the incrementals α, Ico, and Ieo are then derived. A lumped model for a double emitter transistor is also considered and its graph is drawn. Linear graphs are then used to determine the lumped parameters of a and reverse currents for two conditions: 1) emitter one as the input and emitter two open circuited and 2) emitter one as the input and emitter two shorted. Each condition leads to its own graph which may then be used to find the applicable a and reverse currents. Methods for determining the circuit parameters are fully outlined and some sample calculations are given.