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The Scienti C Work of Wilhelm Cauer and Its Key Position at The Pro c. 16th Eur. Meeting on Cyb ernetics and Systems Research EMCSR, Vienna, April 2 { 5, 2002, vol. 2, pp. 934-939 = Cyb ernetics and Systems, R. Trappl, ed., Austrian So c. for Cyb ernetics Studies. The Scienti c Work of Wilhelm Cauer and its Key Position at the Transition from Electrical Telegraph Techniques to Linear Systems Theory Rainer Pauli Munich UniversityofTechnology D { 80290 Munich Germany email: [email protected] Abstract together with their mutual relations which are de ned by the frequency-indep endent holonomic constraints The pap er sketches a line of historical con- imp osed byinterconnection rules. Therefore, network tinuity ranging from telegraph pioneers and theory ab initio was set up as a purely mathemati- early submarine cable technology of the 19th cal discipline. Most notably, under the standard as- century to some asp ects of mo dern systems sumptions of linearity and time-invariance, the the- and control theory. Sp ecial emphasis is laid ory of passive networks is equivalent to the theory of on the intermediary role of the scienti c work dissipative mechanical systems that p erform small os- of Wilhelm Cauer 1900-1945 on the synthe- cillations around a stable equilibrium. Esp ecially in sis of linear networks. [ ] his early pap ers on network synthesis Cauer, 1926 , [ ] Cauer, 1931b Cauer's mathematical starting p ointis this relation to classical analytical mechanics. Nev- 1 Intro duction ertheless, it is fair to recall that the motivation and While it is almost commonplace that mo dern metho ds detailed structure of the problem was rmly ro oted of data compaction, co ding and encryption in to day's in electrical telegraph techniques of the 19th century, communications all have their ro ots in the techniques esp ecially in submarine cable transmission problems; [ ] of the telegraph pioneers Beauchamp, 2001 , analo- [ ] [ ] [ see Belevitch, 1962 , Darlington, 1984 and Wunsch, gous facts concerning the evolution of classical net- ] 1985 . In order to keep the numb er of citations un- work and systems theory seem to fall into oblivion. der control, we refer the reader to these references for This is partly due to the dichotomy of the historical more details ab out the historical asp ects sketched in ro ots of mo dern system and control theory into ana- the subsequent section. lytical mechanics and into classical network theory in the style of Bo de, Cauer, Darlington, Foster, Youla, 2 Before Network Synthesis and many others combined with a widespread un- 2.1 Electrical Telegraph Techniques in willingness to lo ok b eyond the b orders of one's own the 19th Century eld of research. Additionally, in recent decades there has b een a dramatic decline of knowledge ab out clas- Although network analysis has emerged as a near- sical network theory that has resulted in the expulsion mathematical discipline in the mid of the 19th century, of the basic academic and educational asp ects of net- the development of submarine telegraph cables had a work mo dels from electrical engineering curricula and very imp ortant in uence on the progress of network a progressive fading out of network theory as an inde- theory, the main reasons b eing i the extremely com- [ ] p endent scienti c discipline Rohrer, 1990 . plicated dynamical b ehavior of transmission cables as Long b efore the advent of general electromagnetic compared to the ubiquitous RLC -circuits of mo dest theory rst formulated in 1873 by Maxwell in his complexity and ii the b oiling hot economic interest celebrated Treatise the theory of electrical networks in the advancement of electrical telegraph technology emerged as an indep endent discipline with original | comparable only to the Internet euphoria of to day. concepts and metho ds. These rst achievements were For a vivid p opular account on the remarkable paral- Ohm's law 1827, Kirchhoff's laws 1845 and the lels in the history of the electrical telegraph and the p ertaining top ological rules 1847, as well as the revolution in communications due to the Internet, see [ ] sup erp osition principle and the so-called Thevenin- Standage, 1998 though from a decidedly american equivalent circuits credited to Helmholtz 1853. viewp oint. Typically,aphysical device such as a resistor made of The rst working electrical telegraph had b een con- metallic wire or carb on is considered a black box de- structed in 1833 by Gau and Weber in Gottingen ned by a linear relation v = Ri between the external using de ected needle technology, followed in 1837 by electrical variables v and i . This abstraction turned Steinheil's recording telegraph in Munichaswell as the analysis of realistic physical devices into the study by similar exp eriments in the same year by Wheat- of sets or \systems" of well-de ned idealized ob jects stone in England and Morse in the USA. The rst protagonists Breisig, Campbell, Franke, Haus- cable to cross the English Channel was laid in 1851, rath, Kennelly, Strecker, to mention just a few. so on followed by the rst deep-sea cable b etween Sar- Many new circuits and devices such as arti cial lines, dinia and Tunisia in 1854. When the celebrated attenuators, hybrid coils, balancing networks, T - and transatlantic cable connecting Europ e and America -equivalent circuits for two-p orts, lters, and allpass went on-line in 1858 misconceptions and the lackof lattice sections and equalizers were designed. electrical mo dels for understanding transmission line op eration p osed serious practical diculties due to an At the same time complex calculus vulgarized in excessive spread of transmitted pulses. Telegraph en- electrical p ower engineering by Steinmetz b ecame gineering at that time was crude empiricism. Not even ubiquitous; rational functions and p oles and zeros Ohm's law had found its way to many of the early ex- gradually entered into engineering pap ers and the no- p erimentalists. tion of a multip ort and its various matrix descriptions emerged. LaCour 1903 seems to have b een the Various steps towards network mo delling of trans- rst to write down the equations for a symmetric two- mission lines had b een undertaken by Lord Kelvin p ort in chain matrix form that is also the solution to since 1855, but his mo dels were not well suited to ex- eqn.1 plain most of the practical diculties. After Bell's " invention of the \sp eaking telegraph" in 1876 pre- cosh g Z sinh g ceeded ab out 15 years by Reis these de ciencies b e- U U 1 2 = ; 2 1 sinh g cosh g came even more evident when engineers tried to trans- I I 1 2 Z mit sp eech signals over long aerial or mo dest subma- rine distances. In the early 1880's there were ab out while the general black box concept of a two-p ort was 150 000 kilometers of submarine telegraph cables op er- formulated by Breisig in 1921. Campbell and Fos- ating byhookorby cro ok when Heaviside prop osed ter established the theory of the ideal transformer as his celebrated analysis of transmission lines in terms a new network element and gave a complete enumera- of his telegraphist's equations tion of all realizations of lossless four-p orts comp osed [ ] of ideal transformers Campb ell, 1937, pp. 119-168 . @i @u The rst breakthrough towards a systematic syn- = Ri + L ; @x @t thesis of lters was achieved in 1924 by Foster in his @i @u [ ] pap er Areactance theorem Foster, 1924 . The im- = Gu + C : 1 @x @t p ortance of this celebrated theorem rests essentially in the giving of necessary and sucient conditions on Here, R; L or G; C are the p er unit length series re- the imp edance function of a lossless one-p ort in order sistance and inductance, or cross conductance and ca- that it b e realized by means of capacitors and induc- pacitance, resp ectively, of the well-known lump ed net- tors. Moreover, by use of a partial fraction decomp o- work mo del for an in nitesimal piece of transmission sition Foster presents a circuit con guration realiza- line. Initially, Heaviside's mo del was not much appre- tion that is canonical, in the sense that it can b e used ciated, though in 1893 he predicted on purely theoret- to realize any arbitrary lossless rational imp edance ical grounds that a distortion-free transmission of sig- function. nals is feasible by designing cables such that R; L; G; C For completeness, let us remark that even Foster's are related in a certain manner. 1924 pap er originated from work on submarine cables. \It might b e of some slight historical interest to note 2.2 First Steps Towards Network that the case of networks comp osed of resistors and Synthesis 1900 { 1924 capacitors was the rst for which the general solution At the turn of the century the situation changed al- was obtained, in the very early 1920's, in connection most abruptly when Pupin con rmed by exp eriment with the design of balancing networks for submarine the advantage of Heaviside's theoretical prop osal to in- telephone cables. But, when written up for publica- sert lump ed inductors in the line at uniformly spaced tion in 1924, it was presented in terms of inductors p oints. Miraculously, the increased series inductance [ ] and capacitors." Foster, 1962 did not further deteriorate the transmission prop er- ties; rather it turned out that these loaded lines ex- 3 Cauer's Program for Network hibited quite p erfect transmission characteristics up Synthesis 1926 { 1929 to a certain \cuto frequency" while presenting a re- markable increase of attenuation at higher frequencies.
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