Probability distribution function of crossover frequency of operational amplifiers

For the first time in electronics we represent experimental data for probability distribution function of the crossover frequency of operational amplifiers. We use 400 (200 double) ADA4898-2 low noise amplifiers. We present an innovative method for determination of crossover frequency which requires an USB lock-in with automatic frequency sweep. Our method is based on a differential equation relating the voltage at the output of an operational amplifier $U_0$ and the difference between the input voltages ($U_+$ and $U_-$) which has been derived. The crossover frequency $f_0$ 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 time dependent 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, Proc. of the I.R.E. \textbf{35}(5), 444-452 (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. The exact knowledge of the crossover frequency $f_0$ 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.