Manhattan equation for the operational amplifier
Preprint
English
OPEN
Mishonov, Todor M.
;
Danchev, Victor I.
;
Petkov, Emil G.
;
Gourev, Vassil N.
;
Dimitrova, Iglika M.
;
Varonov, Albert M.
(2018)

Subject:
Electrical Engineering and Systems Science  Signal Processing
A differential equation relating the voltage at the output of an operational amplifier $U_0$ and the difference between the input voltages ($U_{+}$ and $U_{}$) has been derived. The crossover frequency $f_0$ is a parameter in this operational amplifier master equation. The formulas 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. Vol. 35, no. 5, pp. 1119, (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 ADA48982 operational amplifiers has been measured. The exact knowledge of the crossover frequency $f_0$ is necessary when we need to precisely determine the nonideal 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.

References
(4)
[1] Technical Specification ADA4817: Low Noise, 1 GHz FastFET Op Amps, Rev. D, Figs. 53, 54, Eqs. (4) http://www.analog.com/static/importedfiles/data sheets/ADA48171 48172.pdf.
[2] K. G. McConnell and W. F. Riley, “Tools of tensometry” in A. S. Kobayashi, Handbook on experimental mechanics, (PrenticeHall Inc., Seattle, Washington, 1987), chap. 3, Fig. 3.6b, Eq. (3.12).
[3] H. W. Bode,“Relations between attenuation and phase in feedback amplifier design,” Bell System Technical Journal, 19, 1940, pp. 421 454, http://cdn14.21dianyuan.com/download.php?id=73990.
[4] W. Jung, “Op Amp History”, http://www.analog.com/media/en/ trainingseminars/designhandbooks/OpAmpApplications/SectionH. pdf; page H.16.

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