Manhattan equation for the operational amplifier

Abstract

A differential equation relating the voltage at the output of an operational amplifier U0 and the difference between the input voltages (U+ and U−) has been derived. The crossover frequency f0 is a parameter in this operational amplifier master equation. The formulae derived as a consequence of this equation find applications in thousands of specifications for electronic devices but as far as we know, the equation has never been published. Actually, the master equation of operational amplifiers can be found in 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, no. 5, pp. 11–19, (1947); Eq. (6), Eq. (7), Eq. (32)], but for more than 70 years it was not analyzed and cited in journals and specifications of operational amplifiers. During World War II, John Ragazzini was involved in the Manhattan Project [“John Ragazzini, 76, Educator and Engineer”,The New York Times, November 24, 1988] working on significant projects in the field of electronics and therefore it would be deservedly to say that the master equation we propose is “Manhattan equation” for operational amplifiers. As it is well forgotten, this equation can be considered as new. As an illustrative example of the Manhattan equation, the crossover frequency of a dozen ADA4898-2 operational amplifiers has been measured. The exact knowledge of the crossover frequency f0 is necessary when we need to precisely determine the non-ideal effects of operational amplifiers. For instance, in cases when there is a need of an exact calculation of the pass bandwidth of amplifiers with active filters, the Manhattan equation is indispensable.