Revisiting and Generalizing Barkhausen’s Equality

Abstract

A study of the conditions for sustaining signals in a loop shows that loop equations are essentially fixed-point equations over a space of functions, with the loop performing a mapping on that space of functions. When the space of functions is specified, one can derive particular conditions for the loop has a solution. Barkhausen conditions fall in this category. Loops composed of two subsystems are in the first place analyzed. The purpose of the chapter is to put into a general perspective the problems of loops, showings the general conditions that must be satisfied. The analysis aims to clarify several perspectives on and the framework of loops operation.

Appendix

Historical note. Barkhausen has proposed what is better named but less known as Barkhausen equation, written by him as SDRi = 1, which translates to $G\beta = 1$ or $HG = 1$ in the notations used in this chapter. Barkhausen was also interested in the starting of oscillations (we name the topic today “stability of oscillations”), as proved by the manuscript pictured in [Eugen-Georg Woschni, The Life-Work of Heinrich Barkhausen http://www2.mst.ei.tum.de/ahmt/publ/symp/2004/2004_075.pdf]. The oscillation problem has been the subject of Brakhausen’s doctoral thesis, Das Problem Der Schwingungserzeugung: MitBesondererBerucksichtigungSchnellerElektrischerSchwingungen, published in 1907. According to the above quoted article on his life, Barkhausen equation first appeared in a book he published in 1917. More on Barkhausen’s life at http://de.wikipedia.org/wiki/Heinrich_Barkhausen. Publications by Barkhausen are listed in the Katalog der DeutschenNationalbibliothek, at https://portal.d-nb.de/opac.htm?query=Woe%3D118657240&method=simpleSearch.