Generalized Gain and Impedance Expressions for Single-Transistor Amplifiers

TL;DR

This paper derives comprehensive formulas for low-frequency gain and impedance in bipolar and MOS single-transistor amplifiers, accounting for realistic non-idealities and providing a unified, elegant framework for analysis and design.

Contribution

It introduces generalized, mathematically elegant expressions for amplifier parameters that include finite output resistance, current gain, load, and feedback resistors, unifying various cases and revealing deeper circuit relationships.

Findings

Unified expressions for bipolar amplifier gain and impedance.

Revealed reciprocity between base and emitter terminals.

Extended formulas to MOS transistors.

Abstract

This expository manuscript presents generalized expressions for the low-frequency voltage gain and terminal impedances of each of the three fundamental bipolar-amplifier topologies (i.e., common emitter, common base, and common collector). Unlike the formulas that students typically learn and designers typically use, the equations presented in this tutorial assume the most general set of conditions: finite output resistance and base-collector current gain, a load resistor at each non-input terminal of the transistor, and a "feedback" resistor between the base and collector terminals. Although perhaps algebraically complex at first glance, emphasis is placed on mathematical elegance and ease of use -- expressions are formulated in terms of sub-terms that capture important aspects of the circuit's behavior. Similarities in the mathematical structure of the results reveal a deeper conceptual connection between different amplifier topologies and, ultimately, a reciprocity relationship between the base and emitter terminals. Familiar approximate expressions are subsumed as special cases. Tables consolidating the expressions in an organized fashion are provided. Companion results for metal-oxide-semiconductor (MOS) single-transistor amplifiers are also included.