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.

It is fair to resume that many \ingredients" of a sys-

Loaded lines, often called \Pupin-lines", b ecame the

tematic synthesis theory were in the air when Cauer

rst prototyp es for low-pass lters.

started his scienti c career. Esp ecially Foster's reac-

The tremendous success of Heaviside's network tance theorem was a rst conscious step towards a

mo del for understanding the complicated dynamical thorough mathematical description of classes of cir-

b ehaviour of transmission lines brought in new con- cuits. Cauer immediately recognized the p otential-

cepts such as propagation constants, attenuation and ities of Foster's ideas and started a private corre-

phase, imp edance matching, re ections and mismatch. sp ondence with Foster. Already in his 1926 do c-

[ ]

This renewed appreciation of Heaviside's work marked toral thesis Cauer, 1926 ,hesketched a complete pro-

the b eginning of an increasingly systematic design of gram for network synthesis as a solution to the in-

communication systems by an increasing number of verse problem of circuit analysis: Given the external

as an absolute invariant and that it was p ossible to b ehavior of a linear passive one-p ort in terms of a

pro ceed in a continuous manner from one network to driving-p oint imp edance as a prescrib ed function of

its equivalent network by a linear transformation of frequency,how do es one nd internally passive real-

the instantaneous mesh currents and charges of the izations for this 'black-b ox' ? He then shows that the

network." synthesis problem requires the systematic solution of

On the basis of this fundamentally new concept of three main issues concerning the realizability, approx-

external equivalence of passive linear networks under imation, and realization of a given imp edance volt-

transformations of internal variables, Cauer was able age/current transfer function Z  as a function of

to state the problem of linear circuit synthesis as fol- the complex frequency parameter  =  +j! . At

[ ] [ ]

lows Cauer, 1941, p. 13 , Cauer, 1958, p. 49 : \The the time `passive realization' was clearly a synonym

previous discussions have shown that it is less imp or- for recipro cal circuits consisting of resistors, inductors

tant for the electrical engineer to solve given di eren- p ossibly magnetically coupled, and capacitors with

tial equations than to search for systems of di eren- the p ositive elementvalues R, L and C .

tial equations circuits whose solutions have a desired Cauer had a rm education in mathematical

prop erty. With the realization of circuits with pre- physics, and was well acquainted with Lagrangian-

scrib ed frequency characteristics in mind and in the style analytical mechanics, particularly in terms of the

interest of a systematic pro cedure, the tasks of linear treatises of E. J. Routh and E. T. Whittaker on the

network theory are formulated as follows: dynamics of rigid b o dies. With a view to investigating

realizability conditions, he starts with the symmetric

n  n lo op matrix of a generic passive n-mesh circuit

1 Which classes of functions of  can berealizedas

cf. Fig. 2

frequency characteristics ?

2

A =  L + R + D ; 3

2 Which circuits areequivalent to each other, i.e.

where R, L, and D are p ositive de nite matrices of re-

have the same frequency characteristics ?

sistance, inductance, and elastance recipro cal capac-

itance, resp ectively. The p ertaining quadratic forms

3 How are the interpolation and approximation prob-

corresp ond to the rate of dissipation of energy in heat

lems which constitute the mathematical expression

and the stored electromagnetic and electrostatic ener-

of the circuit problems solved using functions ad-

gies. Assuming the external accessible p ort to b elong

mitted under question 1?"

to the rst mesh, the input imp edance is calculated

by elimination of internal variables as

Later on, this way of studying di erential equations

det A

indirectly through transfer functions of black-b oxes

; 4 Z =

and their input-output pairs b ecame characteristic of

a

11

mo dern linear system theory.

where a is the complement of the element A in

11 11

det A. Cauer emphasizes the analogy b etween realiz-

3.1 Two-Element Kind Networks

able functions and the stability theory of small os-

In his dissertation Cauer complemented Fosters reac-

cillations in classical mechanics by comparing elec-

tance theorem with a more concise pro of of the ana-

trical quantities to their mechanical counterparts,

lytical prop erties of the reactance function Z  and

the Lagrange multipliers, and the classical triple of

by means of his celebrated canonical ladder realiza-

quadratic forms kinetic, p otential and dissipated en-

tions obtained via Stieltjes' continued fraction ex-

ergy. Moreover, he clearly p oints out that on the

pansions. Most notably, he adapted the results for

level of the generic n-mesh circuit there are abso-

purely reactive networks to all two-element kind net-

lutely no additional realizability constraints b eyond

works showing an isomorphism b etween LC , RC and

p ositive de niteness of the three quadratic forms when

RL circuits.

ideal transformers are admitted as circuit elements. In

Based on the observation that the p oles and zeros

other words, additional constraints are exclusively im-

of the p ertaining Z  alternate on the real or imagi-

p osed by the top ology of the circuit.

nary -axis, he established a fundamental relationship

[ ] [

In subsequent pap ers Cauer, 1929a , Cauer,

between p olynomial stability tests and realization al-

] [ ]

1929b , Cauer, 1931b Cauer simultaneously sub jects

[

gorithms for two-element kind circuits. In Cauer,

the triple of quadratic forms to a group of real ane

]

1929a , he emphasizes the role of a reversal of these re-

transformations

alization algorithms: They generate parametrizations

 

of the p ertaining sub classes of p ositive-real functions

1 0

T 1;n1

5 T AT ; T =

Z  in an algebraically trivial manner without any

T T

21 22

reference to network graphs, quadratic forms or ana-

and shows that external b ehavior in terms of Z in lytic function theory.

[ ] [ ] [ ]

4 is invariant! In his detailed account of Cauer's In Cauer, 1929b , Cauer, 1931a , Cauer, 1931b ,

[ ]

Cauer, 1934 , Cauer completely solves the questions approach, N. Howitt can barely restrain his enthusi-

[ ]

of minimal realization and equivalence for lossless re- asm Howitt, 1931 : \Considerable has b een written

cipro cal or more generally: two-element kind mul- on electrical networks and the imp edance function,

tip orts with prescrib ed input/output b ehavior in two but it has hardly b een susp ected that electrical net-

works formed a group with the imp edance function steps:

Realization of a given n-p ort by partial fraction

 A

expansion of the reactance matrix Z . Due to

the uniqueness of the partial fraction decomp osi- B B

tion, this realization is canonical minimal.

 Determination of any other externally equivalent

y equivalence transforma-

minimal realization b A

tion 5 of internal variables.

Akey problem is the simultaneous principle axis trans-

Figure 1: Equivalence of a symmetric recipro cal two-

formation of two quadratic semi -de nite forms as op-

[ ]

p ort and a lattice section Cauer, 1927 .

p osed to the standard case where at least one form is

non-degenerate. One should note that in the case

an arbitrary, piecewise linear non-negative function of

of LC multip orts R = 0 itis not dicult to

n

frequency. The p oint is, given a preassigned real part,

write down a Kalman state space realization of the

the imaginary part cannot be chosen at wil l if Z has

imp edance matrix Z  and to show that the equiva-

to b e realizable by a 2-p ole circuit."

lence transformation 5 contains state space equiva-

[ ]

In essence, long b efore algebraic realization the-

lence as a subgroup Belevitch, 1968 .

ory for rational imp edances was pro ducing satisfac-

3.2 General Passive Multip orts

tory results of sucient generality, Cauer obtained a

complete characterization of the realizability class by

As long ago as his 1926 dissertation, Cauer uses

means of analytic function theory later, for this class

a now standard passivity argument to prove that

the term positive-real abbrev. PR was coined in O.

b oundedness of transients in an electrical circuit im-

[ ]

Brune's thesis Brune, 1931 . This included i the

p oses the fundamental necessary realizability condi-

Hilb ert integral relation b etween real and imaginary

tion

parts of characteristic network functions as a limiting

ReZ  +j!  > 0; 8 >0:

case of 6, ii results on their rational interp olation

for any imp edance function Z  at `complex frequen-

and approximation, and iii the relationship to func-

cies'  =  +j! . With regard to rational imp edance

tion theoretical fundamentals suchasSchwarz' lemma,

functions, he studies the prop erties of elementary two-

cross ratios, non-Euclidian metrics and their contrac-

mesh circuits under the additional constraint that no

tion or invariance under mappings induced by passive

ideal transformers are admitted. In an engineering ex-

[ ] [ ]

or lossless circuits cf. Cauer, 1929a , Cauer, 1932b {

p osition of his main ideas Cauer emphasizes the role

[ ]

Cauer, 1933 

of nonrational imp edances: \If we do not restrict our-

In the design of so called image parameter lters,

selves to a xed nite number n of indep endent meshes

Cauer achieved a tremendous progress as compared

but admit in nite networks, one arrives at a very sim-

to previous design metho ds byintro ducing latticere-

ple and complete answer to the question: What is

alizations for symmetric two-p orts as shown in Fig. 1

the general analytic character of a function that may

as well as a new approximation of prescrib ed lter

b e approximated with any requested precision by the

characteristics.

driving p oint imp edance of a nite 2-p ole circuit?

The lattice equivalence is a global one i.e., inde-

p endent of frequency, where the lattice imp edances

The only rational functions of  that arerealizable

A and B are related to certain op en and short circuit

as impedances of 2-pole networks have to be analytic

imp edances Z and Z measured at the external p orts

o s

in the right halfplane Re > 0, have a positive real

of the 2-p ort in the following way:

part in Re > 0, and take on real values on the real

p p

axis.

Z Z Z ; B = Z Z Z Z : A = Z +

o o s o o o s o

These characteristics are an immediate consequence

Cauer proved that despite the irrational character of

of the representation of these functions as the Poisson

the ab ove formulas the imp edances A and B are ra-

integral

tional functions of frequency and can b e realized bya

1

Z

i h

d x

nite continued fraction resulting in a passive RLC -

; 6 Z =  C +

2

circuit whenever the two-p ort is passive. The de-

 + x

0

cisive role of the lattice realization stems from p os-

sibility to replace the investigation of an imp ortant where C  0,  is a monotonically increasing function,

class of rational 2  2-matrices by the treatment and the integral has to b e taken in Stieltjes' sense.

of two scalar rational functions. In Decemb er 1927 When approximating the integral by a nite sum, we

Cauer got the opp ortunity to present his lattice equiv- get a rational function in  that has an immediate re-

alence theorem for symmetric two-p orts in a session alization as an electric circuit. [. . . ] In this way,we

[ ]

of the Prussian Acadamy of Sciences Cauer, 1927 . obtain an arbitrary close approximation for Z not only

It may b e instructive to illustrate the imp ortance of in the interior of the right half plane, but also on the

this privilege by lo oking through the list of sp eakers b oundary, i.e., for purely imaginary  =j! . When a

in that year Einstein, Hahn, Hasse, Hopf, Landau, complex imp edance Z is represented graphically as a

von Neumann, No ether, Planck, P olya, Schur . . . . function of the frequency ! ,  x can only b e deter-

From a purely mathematical viewp oint, rational ma- mined on the basis of the real part ReZ j! ; it can b e

function to haveaphysical realization, i.e.

 in the form of a nite network with p ositivevalues

of network elements R, L, C or a p ositive de nite

L-matrix in case of coupled coils

 without ideal transformers.

4 Conclusion

Cauer's program marked a milestone in the develop-

ment of network synthesis towards a mathematical de-

scription of classes of circuits | and hence towards

what wenowadays call mathematical systems theory.

It is to Cauer's merit that he turned network and sys-

Figure 2: Canonical realization of an arbitrary nite

tem synthesis into a truly systematic business. He

passivemultip ort by three mesh-connected purely in-

provided it with the appropriate mathematical frame-

ductive, resistive and capacitive n-p orts L, R and D,

[ ]

work just as the discipline was on the rise starting

resp ectively Cauer, 1931b . Their internal structure

from Foster's reactance theorem. In contrast to the

is shown b elow for the resistive R -b ox, where mini-

purely utilitarian or engineering approach, Cauer em-

mality of the numb er of parameters is achieved bya

phasized the scienti c asp ects and provided the rst

sp ecial arrayofmulti-winding transformers obtained

precise statement of the network synthesis problem,

by reduction of the turns-ratio matrix to triangular

one that has b ecome a valid paradigm for all synthe-

form. External p orts can b e intro duced by op ening

[ ]

sis problems Bro ckett, 1977, p. 38 .

one or more of the external meshes.

Beyond the role of p ositive real functions,

Nevanlinna-Pick-interp olation, passivity and lossless-

trices were a rather exotic sub ject: Recall that the

ness in mo dern systems and control theory see

absolutely elementary notion of the McMillan de-

[ ]

Dewilde and et.al. editors, 1995  there remains

greeofarational matrix has b een given a rigorous

amuch more imp ortant connection to classical net-

[

de nition only 25 years later by McMillan McMil-

work synthesis. Instead of solving systems of ordi-

]

lan, 1952 based on a deceptively network-theoretic

nary di erential equations i.e., in the tradition of

reasoning using structural results of Cauer.

classical mechanics, Foster and Cauer started solv-

On the algebraic realization side, Cauer investigated

ing problems indirectly by analysing transfer functions

the internal structure and equivalence of multip orts

of black-b oxes and the admissible external b ehavior

[ ] [

on the basis of 3{5 cf. Cauer, 1931b , Cauer,

of their input-output pairs. This way they studied

] [ ]

1932a , Cauer, 1932c , and Fig. 2. In his habilita-

classes or families of linear dynamical systems via their

tion thesis `On a problem where three p ositive de nite

parametrization in terms of network mo dels. In math-

quadratic forms are related to one-dimensional com-

ematical systems theory the rst systematic studies of

[ ]

plexes' Cauer, 1931b , Cauer concentrated on the an-

families of linear dynamical systems started in 1974

[ ] [ ]

alytic prop erties of RLC multip orts and the algebraic-

Hermann, 1974b , Kalman, 1974 . Although on the

geometric asp ects of their canonical representation i

mathematical side there are some researchers who are

in terms of rational matrices that are generated by

aware of certain antecedents of this developmentin

[ ] [

three p ositive quadratic forms in n variables and ii

classical circuit synthesis Bro ckett, 1977 , Byrnes,

]

in terms of their assignment to a one-dimensional cell

1998 , there has b een alarmingly and amazingly lit-

complex of rst Betti number n in reference to Os-

tle contact and understanding b etween these groups.

vald Veblen's Analysis Situs . As Cauer p oints out,

On the engineering side p eople are increasingly hor-

the main structural distinction b etween general RLC

ri ed by the intense structure of mo dern mathemat-

and two-element kind multip orts is i that it is gen-

ical thought { it certainly cuts o access to p eople

erally not p ossible to simultaneously diagonalize the

outside of mathematics who have not b een trained

three quadratic forms by congruence and ii that the

to understand the style, nor have the intellectual fa-

o ccurrence of additional absolute invariants of 5 im-

naticism to immerse themselves into it cf. Preface

plies the non-existence of global canonical forms for

[ ]

to Hermann, 1974a . It turns out, however, that a

the generation of all realizations. However, he showed

circuit engineer's intuitiveway of lo oking at sets or

among other things that the realization problem can

parametrized families of multip orts in terms of black-

often b e split into 'smaller' ones by simultaneously

box pictures essentially amounts to the coordinate-free

transforming the quadratic forms into a common but

qualitative study of global mo dels for di erentiable

otherwise arbitrary blo ck-diagonal structure.

[ ]

manifolds Pauli, 2000 . It seems worth it to exploit

During his stay at MIT in 1930/31 Cauer suggested these connections more carefully and to re-examine

and sup ervised the do ctoral thesis of Brune. By his recipro cal network mo dels that have b een studied by

famous continued fraction Brune provided the long- Cauer 75 years ago in order to understand their role

unresolved pro of that the PR prop erty is not only a in a \mo dern" context | for instance mo duli spaces

necessary but also sucient condition for a rational for symmetric rational transfer functions.

[ ]

References Cauer, 1933 W. Cauer. Ein Interp olationsproblem

mit Funktionen mit p ositivem Realteil. Mathema-

[ ]

Beauchamp, 2001 K. Beauchamp. History of Teleg-

tische Zeitschrift, 38:1{44, 1933.

raphy. IEE History of Technology. Institution of



Electrical Engineers, London, 2001. [ ]

Cauer, 1934 W. Cauer. Aquivalenz von 2n-Polen

ohne Ohmsche Widerstande. Nachrichten d.

[ ]

Belevitch, 1962 V. Belevitch. Summary of the his-

Gesel lschaft d. Wissenschaften Gottingen, math.-

tory of circuit theory. Proceedings of the IRE,

phys. Kl., 1, N.F.:1{33, 1934.

50:848{855, 1962.

[ ]

Cauer, 1941 W. Cauer. Theorie der linearen

[ ]

Belevitch, 1968 V. Belevitch. Classical Network

Wechselstromschaltung en, Vol.I. Akad. Verlags-

Theory. Holden-Day, 1968.

Gesellschaft Becker und Erler, Leipzig, 1941.

[ ]

Bro ckett, 1977 R. W. Bro ckett. Control theory and

[ ]

Cauer, 1958 W. Cauer. Synthesis of Linear Commu-

analytical mechanics. In C. Martin and R. Her-

nication Networks. McGraw-Hill, New York, 1958.

mann, editors, The 1976 AMES Research Center

Transl. by G. E. Knausenb erger and J. N. War eld.

NASA ConferenceonGeometric Control Theory,

[ ]

pages 1{48. Math Sci Press, Bro okline. MA, 1977.

Darlington, 1984 S. Darlington. A history of net-

work synthesis and lter theory for circuits com-

[ ]

Brune, 1931 O. Brune. Synthesis of a nite two-

p osed of resistors, inductors, and capacitors. IEEE

terminal network whose driving-p oint imp edance is

Trans. Circuits and Systems, 31:3{13, 1984.

a prescrib ed function of frequency. J. of Mathemat-

[ ]

Dewilde and et.al. editors, 1995 P. Dewilde and ical Physics, 10:191{236, 1931.

et.al. editors. Sp ecial issue in memory of wilhelm

[ ]

Byrnes, 1998 C. I. Byrnes. On the global analysis

cauer: The role of losslessness in theory and appli-

of linear systems. In Baillieul and Willems, edi-



cations. AEU Int. J. Electron. Commun., 495/6,

tors, Mathematical Control Theory, pages 99{139.

1995.

Springer, 1998.

[ ]

Foster, 1924 R. M. Foster. A reactance theorem.

[ ]

Campb ell, 1937 G. A. Campb ell. The Col lected Pa-

Bel l System Techn. J., 3:259{267, 1924.

pers of George Ashley Campbel l. American Tele-

[ ]

phone and Telegraph Comp., New York, 1937.

Foster, 1962 R. M. Foster. Academic and theoretical

asp ects of circuit theory. Proceedings of the IRE,

[ ]

Cauer, 1926 W. Cauer. Die Verwirklichung der

50:866{871, 1962.

Wechselstromwiderstande vorgeschrieb ener Fre-

[ ]

quenzabhangigkeit. Archiv fur Elektrotechnik,

Hermann, 1974a R. Hermann. Algebraic Topics in

17:355{388, 1926.

Systems Theory, volume III of Interdisciplinary

Mathematics.MATH SCI PRESS, Bro okline, MA,



[ ]

Cauer, 1927 W. Cauer. Ub er die Variablen eines

1974.

passiven Vierp ols. Sitzungsberichte d. Preu.

[ ]

Hermann, 1974b R. Hermann. Geometric Structure

Akademie d. Wissenschaften, phys.-math. Klasse,

Theory of Systems { Control Theory and Physics,

pages 268{274, 1927.

Part A,volume IX of Interdisciplinary Mathemat-



[ ]

Cauer, 1929a W. Cauer. Ub er eine Klasse von Funk-

ics.MATH SCI PRESS, Bro okline, MA, 1974.

tionen, die die Stieljesschen Kettenbruc  he als Son-

[ ]

Howitt, 1931 N. Howitt. Group theory and the elec-

derfall enthalt. Jahresberichte der Dt. Mathematik-

trical circuit. Physical Review, 37:1583{1595, 1931.

ervereinigung DMV, 38:63{72, 1929.

[ ]

Kalman, 1974 R. E. Kalman. Algebraic geometric

[ ]

Cauer, 1929b W. Cauer. Vierp ole. Elektrische

description of the class of linear systems of constant

Nachrichtentechnik ENT, 6:272{282, 1929.

dimension. In Proc. 8th Annual Princeton Con-

[ ]

Cauer, 1931a W. Cauer. Ein Reaktanztheorem.

ference on Information Sciences & Systems, pages

Sitzungsberichte d. Preu. Akademie d. Wis-

189{191, Princeton, March 1974. Dept. of Electrical

senschaften, phys.-math. Klasse, pages 673{681,

Engineering, Princeton University.

1931.

[ ]

McMillan, 1952 B. McMillan. Intro duction to for-

[ ]

Cauer, 1931b W. Cauer. Untersuchungenub  er ein

mal realizability theory { I, I I. The Bel l System

Problem, das drei p ositiv de nite quadratische For-

Technical J., 31:217{279, 541{600, 1952.

men mit Streckenkomplexen in Beziehung setzt.

[ ]

Pauli, 2000 R. Pauli. Di erentiable manifolds and

Mathematische Annalen, 105:86{132, 1931.

engineering black-b ox mo dels for linear physical

[ ]

Cauer, 1932a W. Cauer. Ideale Transforma-

systems. In Proc. MTNS2000, pages 1{10, Perpig-

toren und lineare Transformationen. Elektrische

nan, France, June 19{23 2000. CD-ROM.

Nachrichtentechnik ENT, 9:157{174, 1932.

[ ]

Rohrer, 1990 R. A. Rohrer. Taking circuits seriously.

[ ]

Cauer, 1932b W. Cauer. The Poisson integral for

IEEE Circuits and Devices Magazine, pages 27{31,

functions with p ositive real part. Bul l. Amer. Math.

July 1990.

Soc., 38:713{717, 1932.

[ ]

Standage, 1998 T. Standage. The Victorian Inter-



[ ]

Cauer, 1932c W. Cauer. Ub er Funktionen mit

net.Walker Publ. Co., New York, 1998.

p ositivem Realteil. Mathematische Annalen,

[ ]

Wunsch, 1985 G. Wunsch. Geschichte der Sys-

106:369{394, 1932. auch Verhandl. d. Intern.

temtheorie.Akademie-Verlag, Berlin, 1985.

Mathematiker-Kongresses, Zuric  h 1932.