Towards a Simple, and Yet Accurate, Transistor Equivalent Circuit and Its Application to the Analysis and Design of Discrete and Integrated Electronic Circuits

TL;DR

This paper introduces a simple yet accurate transistor equivalent circuit based on the Early effect, enabling improved analysis and design of electronic circuits with better characterization of gain and stability.

Contribution

A novel, simplified transistor equivalent circuit derived from the Early effect, facilitating more accurate circuit analysis and design, including stability and parallel transistor behavior.

Findings

Gain varies almost linearly with base current.

A prototypic Early space summarizes transistor characteristics.

Traditional models may significantly deviate in stability analysis.

Abstract

Transistors are the cornerstone of modern electronics. Yet, their relatively complex characteristics, allied with often observed great parameter variation, remain a challenge for discrete and integrated electronics. Much of transistor research and applications have relied on transistor models, as well as respective equivalent circuits, to be employed for circuit analysis and simulations. Here, a simple and yet accurate transistor equivalent circuit is derived, based on the Early effect, which involves only the voltage $V_a$ and a companion parameter $s$. Equations are obtained for currents and voltages in a common-emitter circuit, allowing the derivation of respective gain functions. These functions are found to exhibit interesting mathematical structure, with gain values varying almost linearly with the base current, allowing the gains to be well characterized in terms of their average and variation values. These results are applied to deriving a prototypic Early space summarizing the characteristics of transistors, enriched with recently experimentally obtained prototypes of NPN and PNP silicon BJTs and alloy germanium transistors. Though a trade-off between gain and linearity is revealed, a band characterized by small values of $V_a$ stands out when aiming at both high gain and low distortion. The Early equivalent model was used also for studying the stability of circuits under voltage supply oscillations, as well as parallel combinations of transistors. In the former case, it was verified that more traditional approaches assuming constant current gain can yield stability factors that deviate substantially from those derived for the more accurate Early approach. The equivalent circuit obtained for parallel combinations of transistors was shown also to closely follow the Early formulation.