# Master equation for operational amplifiers: stability of negative differential converters, crossover frequency and pass-bandwidth

## TL;DR

This paper revisits the master equation for operational amplifiers from 1947, demonstrating its relevance for analyzing stability, crossover frequency, and pass-bandwidth in modern circuits, with practical examples and a simple measurement method.

## Contribution

It applies the classical master equation to contemporary operational amplifier analysis, including stability criteria and frequency response, providing new insights and practical tools.

## Key findings

- Stability criterion for negative impedance converters derived.
- Method for determining crossover frequency demonstrated.
- Master equation shown to be useful in modern software simulations.

## Abstract

The time dependent master equation from the seminal article by Ragazzini, Randall and Russell [J. R. Ragazzini, R. H. Randall and F. A. Russell, "Analysis of Problems in Dynamics by Electronic Circuits", Proc. of the I.R.E., Vol. 35, pp. 444--452, (1947)] is recovered as necessary tool for the analysis of contemporary circuits with operational amplifiers. This equation gives the relation between time dependent the output voltage $U_0(t)$ and the difference between the input voltages ($U_{+}(t)$ and $U_{-}(t)$). The crossover frequency $f_0$ is represented the time constant $\tau_{_0}$ of this equation. The work of the master equation is illustrated by two typical examples: a) the stability criterion of the devices with negative impedance converters, which we consider as a new result b) the frequency dependence of the amplifiers with operational amplifiers given in the technical specifications without citations of time dependent equation. A simple circuit for determination of $f_0$ is suggested and the method is illustrated by determination of crossover frequency for the low-noise and high speed ADA4898 operational amplifier. It is concluded that for an exact calculation of the pass bandwidth of amplifiers with active filters the 70 years old master equation is a useful technique implicitly included in the contemporary software. The frequency dependent formulae for the amplification coefficient of inverting and non-inverting amplifiers are given for the case of non-zero conductivity between the inputs of the operational amplifiers.