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

The well forgotten 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., 35, pp. 11–19, (1947)] is recovered as indispensable tool for the analysis of contemporary circuits with operational amplifiers. This equation gives the relation between time dependent the output voltage U0(t) and the difference between the input voltages (U+(t) and U−(t)). The crossover frequency f0 parameterizes the time constant τ0 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. A simple circuit for determination of f0 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 indispensable.