Wilhelm Cauer
Wilhelm Cauer | |
---|---|
Berlin-Marienfelde, Germany | |
Nationality | German |
Alma mater | Technical University of Berlin |
Scientific career | |
Fields | Mathematics |
Doctoral advisor | Georg Hamel |
Doctoral students | Vitold Belevitch |
Wilhelm Cauer (24 June 1900 – 22 April 1945
Cauer initially specialised in
The manuscripts for some of Cauer's most important unpublished works were destroyed during the war. However, his family succeeded in reconstructing much of this from his notes and volume II of Theorie der linearen Wechselstromschaltungen was published after his death. Cauer's legacy continues today, with network synthesis being the method of choice for network design.
Life and career
Early life and family
Wilhelm Adolf Eduard Cauer was born in
Briefly, Cauer served in the German army in the final stages of World War I. He married Karoline Cauer (a relation)[6] in 1925 and eventually fathered six children.[5][7]
Career
Cauer started off in a field completely unrelated to filters; from 1922 he worked with Max von Laue on general relativity, and his first publication (1923) was in this field. For reasons that are not clear, he changed his field after this to electrical engineering. He graduated in applied physics in 1924 from the Technical University of Berlin.[5]
He then spent a period working for Mix & Genest, a branch of the Bell Telephone Company, applying probability theory to telephone switching. He also worked on timer relays. He had two telecommunications-related publications during this period on "Telephone switching systems" and "Losses of real inductors".[5]
The relationship of Mix & Genest with Bell gave Cauer an easy path to collaboration with
In June 1926 Cauer presented his thesis paper, The realisation of impedances of specified frequency dependence[a], at the Institute of Applied Mathematics and Mechanics of the Technical University of Berlin.[5] This paper is the beginning of modern network synthesis.[10]
In 1927 Cauer went to work as a research assistant at Richard Courant's Institute of Mathematics at the University of Göttingen. In 1928 he obtained his habilitation and became an external university lecturer.[5]
Cauer found that he could not support his family during the
Cauer met, and had strong contacts with, many of the key researchers in the field of filter design at Bell Labs. These included
For a short while, Cauer worked for the Wired Radio Company in Newark, New Jersey but then returned to Göttingen with the intention of building a fast analogue computer there. However, he was unable to obtain funding due to the depression.[5]
Cauer seems to have got on very poorly with his German colleagues. According to Rainer Pauli, his correspondence with them was usually brief and business-like, rarely, if ever, discussing issues in depth. By contrast, his correspondence with his American and European acquaintances was warm, technically deep and often included personal family news and greetings.
After leaving the Technical Institute for Mix & Genest, Cauer sought to become active in the
Nazi era
In November 1933 Cauer signed the Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State.
The rising force of
By 1935 Cauer had three children whom he was finding increasingly difficult to support, which prompted him to return to industry. In 1936 he temporarily worked for the aircraft manufacturer
In 1941, the first volume of his main work, Theory of Linear AC Circuits was published.[e] The original manuscript to the second volume was destroyed as a result of the war. Although Cauer was able to reproduce this work, he was not able to publish it and it too was lost during the war. Some time after his death, however, his family arranged for the publication of some of his papers as the second volume,[f] based on surviving descriptions of the intended contents of volume II.[7]
After taking his children to stay with relatives in
Network synthesis
The major part of Cauer's legacy is his contribution to the network synthesis of passive networks. He is considered the founder of the field and the publication of his principal work in English was enthusiastically greeted, even though this did not happen until seventeen years later (in 1958).[14][15] Prior to network synthesis, networks, especially filters, were designed using the image impedance method. The accuracy of predictions of response from such designs depended on accurate impedance matching between sections. This could be achieved with sections entirely internal to the filter but it was not possible to perfectly match to the end terminations. For this reason, image filter designers incorporated end sections in their designs of a different form optimised for an improved match rather than filtering response. The choice of form of such sections was more a matter of designer experience than design calculation. Network synthesis entirely did away with the need for this. It directly predicted the response of the filter and included the terminations in the synthesis.[16]
Cauer treated network synthesis as being the inverse problem of network analysis. Whereas network analysis asks what is the response of a given network, network synthesis on the other hand asks what are the networks that can produce a given desired response. Cauer solved this problem by comparing electrical quantities and functions to their mechanical equivalents. Then, realising that they were completely analogous, applying the known Lagrangian mechanics to the problem.[17]
According to Cauer, there are three major tasks that network synthesis has to address. The first is the ability to determine whether a given transfer function is realisable as an impedance network. The second is to find the canonical (minimal) forms of these functions and the relationships (transforms) between different forms representing the same transfer function. Finally, it is not, in general, possible to find an exact finite-element solution to an ideal transfer function - such as zero attenuation at all frequencies below a given cutoff frequency and infinite attenuation above. The third task is therefore to find approximation techniques for achieving the desired responses.[17]
Initially, the work revolved around
Realisability
- Following on from Foster, Cauer generalised the relationship between the expression for the impedance of a one-port network and its transfer function.[10][18]
- He discovered the necessary and sufficient condition for realisability of a one-port impedance. That is, those impedance expressions that could actually be built as a real circuit.[18] In later papers he made generalisations to multiport networks.[19]
Transformation
- Cauer discovered that all solutions for the realisation of a given impedance expression could be obtained from one given solution by a group of affine transformations.[20]
- He generalised Foster's ladder realisation to filters which included resistors (Foster's were reactance only) and discovered an isomorphism between all two-element kind networks.[18][21]
- He identified the canonical forms of filter realisation. That is, the minimal forms, which includes the ladder networks obtained by Stieltjes's continued fraction expansion.[10][18][21]
Approximation
- He used the Chebyshev approximation to design filters. Cauer's application of Tchebyscheff polynomials resulted in the filters now known as elliptic filters, or sometimes Cauer filters, which have optimally fast passband to stopband transitions for a given maximum attenuation variation. The well known Chebyshev filters can be viewed as a special case of elliptic filters and can be arrived at using the same approximation techniques. So can the Butterworth (maximally flat) filter, although this was an independent discovery by Stephen Butterworth arrived at by a different method.[10][21][22]
Cauer's work was initially ignored because his canonical forms made use of ideal transformers. This made his circuits of less practical use to engineers. However, it was soon realised that Cauer's Tchebyscheff approximation could just as easily be applied to the rather more useful
Further work
Most of the above work is contained in Cauer's first[b] and second[e] monographs and is largely a treatment of one-ports. In his habilitation thesis[c] Cauer begins to extend this work by showing that a global canonical form cannot be found in the general case for three-element kind multiports (that is, networks containing all three R, L and C elements) for the generation of realisation solutions, as it can be for the two-element kind case.[23]
Cauer extended the work of Bartlett and Brune on geometrically symmetric
Publications
- [a]^ Cauer, W, "Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit", Archiv für Elektrotechnik, vol 17, pp355–388, 1926. The realisation of impedances of prescribed frequency dependence (in German)
- Cauer, W, "Über die Variablen eines passiven Vierpols", Sitzungsberichte d. Preuß. Akademie d.Wissenschaften, phys-math Klasse, pp268–274, 1927. On the variables of some passive quadripoles (in German)
- Cauer, W, "Über eine Klasse von Funktionen, die die Stieljesschen Kettenbrüche als Sonderfall enthält", Jahresberichte der Dt. Mathematikervereinigung (DMV), vol 38, pp63–72, 1929. On a class of functions represented by truncated Stieltjes continued fractions (in German)
- Cauer, W, "Vierpole", Elektrische Nachrichtentechnik (ENT), vol 6, pp272–282, 1929. Quadripoles (in German)
- Cauer, W, "Die Siebschaltungen der Fernmeldetechnik", Journal of Applied Mathematics and Mechanics, vol 10, pp425–433, 1930. Telephony filter circuits (in German)
- Cauer, W, "Ein Reaktanztheorem", Sitzungsberichte d. Preuß. Akademie d. Wissenschaften, phys-math. Klasse, pp673–681, 1931. A reactance theorem (in German)
- [b]^ *Cauer, W, Siebschaltungen, VDI-Verlag, Berlin, 1931. Filter circuits (in German)
- [c]^ *Cauer, W, "Untersuchungen über ein Problem, das drei positiv definite quadratische Formen mit Streckenkomplexen in Beziehung setzt", Mathematische Annalen, vol 105, pp86–132, 1931. On a problem where three positive definite quadratic forms are related to one-dimensional complexes (in German)
- Cauer, W, "Ideale Transformatoren und lineare Transformationen", Elektrische Nachrichtentechnik (ENT), vol 9, pp157–174, 1932. Ideal transformers and linear transformations (in German)
- Cauer, W, "The Poisson integral for functions with positive real part", Bull. Amer. Math. Soc., vol 38, pp713–717, 1932.
- Cauer, W, "Über Funktionen mit positivem Realteil", Mathematische Annalen, vol 106, pp369–394, 1932. On positive-real functions (in German)
- Cauer, W, "Ein Interpolationsproblem mit Funktionen mit positivem Realteil", Mathematische Zeitschrift, vol 38, pp1–44, 1933. An interpolation problem of positive-real functions (in German)
- [d]^ Cauer, W, "Äquivalenz von 2n-Polen ohne Ohmsche Widerstände", Nachrichten d. Gesellschaft d. Wissenschaften Göttingen, math-phys. Kl., vol 1, N.F., pp1–33, 1934. Equivalence of 2-poles without resistors (in German)
- Cauer, W, "Vierpole mit vorgeschriebenem Dämpfungsverhalten", Telegraphen-, Fernsprech-, Funk- und Fernsehtechnik, vol 29, pp185–192, 228–235, 1940. Quadripoles with prescribed insertion loss (in German)
- [e]^ Cauer, W, Theorie der linearen Wechselstromschaltungen, Vol.I, Akad. Verlags-Gesellschaft Becker und Erler, Leipzig, 1941. Theory of Linear AC Circuits, Vol I (in German)
- Cauer, W, Synthesis of Linear Communication Networks, McGraw-Hill, New York, 1958. (published posthumously)
- [f]^ Cauer, W, Theorie der linearen Wechselstromschaltungen, Vol. II, Akademie-Verlag, Berlin, 1960. Theory of Linear AC Circuits, Vol II (published posthumously in German)
- [g]^ Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", J. Math. and Phys., vol 10, pp191–236, 1931.
See also
- Black box
- Cauer topology
- Tchebyscheff filter
References
- ^ a b Piloty, Hans (1957), "Cauer, Wilhelm", Neue Deutsche Biographie (in German), vol. 3, Berlin: Duncker & Humblot, pp. 179–180; (full text online)
- ^ Emil Cauer: Wilhelm Cauer: His Life and the Reception of his Work
- ^ "Die Geschichte unserer Schule", Ludwig Cauer Grundschule official site (in German), accessed and archived 29 July 2012.
- ^ "Ludwig-Cauer-Grundschule Berlin", Architektur Bild Archiv (in German), accessed and archived 29 July 2012.
- ^ a b c d e f g h i E. Cauer et al., p2
- ^ O'Connor, John J.; Robertson, Edmund F., "Wilhelm Cauer", MacTutor History of Mathematics Archive, University of St Andrews Accessed and archived 29 July 2012.
- ^ a b c d e E. Cauer et al., p3
- ^ Bray, p62
- ^ Foster, R M, "A reactance theorem", Bell System Technical Journal, Vol. 3, pp259–267, 1924.
- ^ a b c d e f g Belevitch, p850
- ^ a b E. Cauer et al., p8
- ^ a b E. Cauer et al., p9
- ISBN 978-3-89811-587-2.
- ISBN 9814282294.
- ^ K.C. Garner, "Reviews", The Aeronautical Journal, volume 63, page 375, Royal Aeronautical Society 1959.
- ^ Mathis et al., pp.83-84
- ^ a b E. Cauer et al., p4
- ^ a b c d Cauer, 1926
- positive-realor PR.
- ^ Cauer, 1929, 1931
- ^ a b c E. Cauer et al., p5
- ^ Cauer, 1927, 1933
- ^ E. Cauer et al., p6
Bibliography
Referenced works
- E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June 2000. Retrieved online 19 September 2008.
- Belevitch, V, "Summary of the History of Circuit Theory", Proceedings of the IRE, vol 50, pp848–855, May 1962.
- Bray, J, Innovation and the Communications Revolution, Institute of Electrical Engineers, 2002 ISBN 0852962185.
- Matthaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964.
Further reading
- Guillemin, E A, "A recent contribution to the design of electrical filter networks". Journ. Math. Phys., vol 11, pp150–211, 1931–32. A comparison of the methods of Cauer and Zobel
- Julia, R, "Sur la Theorie des Filtres de W. Cauer", Bull. Soc. Franc. Electr., October 1935. Recommended by R. Pauli as the most profound treatise on Cauer's theory (in French).
- Wilhelm Cauer: His Life and the Reception of his Work Archived 3 March 2016 at the Wayback Machine Mathis, W and Cauer, E, University of Hannover, 2002. A PowerPoint presentation.
External links