Wilhelm Cauer
Encyclopedia
Wilhelm Cauer was a German mathematician
and scientist
. He is most noted for his work on the analysis and synthesis of electrical filter
s and his work marked the beginning of the field of network synthesis. Prior to his work, electronic filter design used techniques which were essentially approximation methods, as no exact answers were produced for the behaviour of the filter under real conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.
. Cauer became interested in mathematics at the age of thirteen and continued to demonstrate that he was academically inclined as he grew.
Briefly, Cauer served in the German army in the final stages of World War I
. He married Karoline in 1925 and eventually fathered six children.
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
.
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 these subjects;
The relationship of Mix & Genest with Bell gave Cauer an easy path to collaboration with AT&T's engineers at Bell Labs
in the US which must have been of enormous help when Cauer embarked on a study of filter design. Bell were at the forefront of filter design at this time with the likes of George Campbell
in Boston and Otto Zobel in New York making major contributions. However, it was with R. M. Foster
that Cauer had much correspondence and it was his work that Cauer recognised as being of such importance. His paper, A reactance theorem, is a milestone in filter theory and inspired Cauer to generalise this approach into what has now become the field of network synthesis.
In June 1926 Cauer presented his thesis paper, The realisation of impedances of specified frequency dependence, at the Institute of Applied Mathematics and Mechanics of the Technical University of Berlin. This paper is the beginning of modern network synthesis.
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.
Cauer found that he could not support his family during the economic crisis of the 1920s
and in 1930 took his family to the USA where he had obtained a scholarship (a Rockefeller fellowship) to study at MIT and Harvard University
. He worked with Vannevar Bush
who was building machines for the solution of mathematical problems. Essentially, these were what we would now call analogue computers
: Cauer was interested in using them to solve linear systems to aid in filter designs. His work on Filter circuits was completed in 1931 while still in the US.
Cauer met, and had strong contacts with, many of the key researchers in the field of filter design at Bell Labs. These included Hendrik Bode
, George Campbell
, Sidney Darlington
, Foster
and Otto Zobel.
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.
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. This correspondence went beyond his American contacts and included A.C. Bartlett of the General Electric Company in Wembley, Roger Julia of Lignes Telegraph Telephone in Paris, mathematicians Gustav Herglotz
, Georg Pick and Hungarian graph theorist Dénes Kőnig
.
After leaving the Technical Institute for Mix & Genest, Cauer sought to become active in the Verband Deutscher Elektrotechniker (VDE, the German Electrical Engineers Society). He left the VDE, however, in 1942 after a serious falling out with Wagner
, previously his PhD supervisor and ally.
became a major obstacle to Cauer's work from 1933 onwards. The anti-Jewish hysteria of the time forced many academics to leave, including the director of the Mathematics Institute, Richard Courant
. Although Cauer was not Jewish, it became known that he had a Jewish ancestor, Daniel Itzig
, who had been a banker to Frederick II
of Prussia
. Although this was not sufficient to have Cauer removed under the race laws it stifled his future career, he gained the title of professor but was never given a chair.
By 1935 he had three children who he was finding it increasingly difficult to support which prompted him to return to industry. In 1936 he temporarily worked for the aircraft manufacturer Fieseler
at their Fi 156 Storch
works in Kassel
and then became director of the laboratory of Mix & Genest in Berlin
. He did however, continue to lecture at the Technical University in Berlin from 1939.
In 1941, the first volume of his main work, Theory of Linear AC Circuits was published. 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, based on surviving descriptions of the intended contents of volume II.
After taking his children to stay with relatives in Witzenhausen
(in Hesse
) to protect them from the expected fall of Berlin to the Russians, Cauer, against advice, returned to Berlin. His body was located after the end of the war in a mass grave of victims of Russian executions. Cauer had been shot dead in his garden in Berlin-Marienfelde by soviet soldiers as a hostage. Soviet Intelligence was actively looking for scientists they could use in their own researches and Cauer was on their list of people to find, but of course this was not known to all soldiers.
of passive
networks. Indeed, he is considered the founder of the field. He 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.
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.
Initially, the work revolved around one-port impedances. The transfer function between a voltage and a current amounting to the expression for the impedance itself. A useful network can be produced by breaking open a branch of the network and calling that the output.
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 ladder topology and ideal transformers could be dispensed with. From then on network synthesis began to supplant image design as the method of choice.
Cauer extended the work of Bartlett and Brune on geometrically symmetric 2-ports to all symmetric 2-ports, that is 2-ports which are electrically symmetrical but not necessarily topologically symmetrical, finding a number of canonical circuits. He also studied antimetric
2-ports. He also extended Foster's theorem to 2-element LC n-ports (1931) and showed that all equivalent LC networks could be derived from each other by linear transformations.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and scientist
Scientist
A scientist in a broad sense is one engaging in a systematic activity to acquire knowledge. In a more restricted sense, a scientist is an individual who uses the scientific method. The person may be an expert in one or more areas of science. This article focuses on the more restricted use of the word...
. He is most noted for his work on the analysis and synthesis of electrical filter
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
s and his work marked the beginning of the field of network synthesis. Prior to his work, electronic filter design used techniques which were essentially approximation methods, as no exact answers were produced for the behaviour of the filter under real conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.
Early life and family
Wilhelm Cauer was born in Berlin, Germany, on June 24, 1900. He came from a long line of academics. His early grammar school was located in Cauerstrasse, named after his great-grandfather, the founder of the school. He later attended the Mommsen Gymnasium, Berlin. His father, also Wilhelm Cauer, was a Privy Councillor and a professor of railway engineering at the Technical University of BerlinTechnical University of Berlin
The Technische Universität Berlin is a research university located in Berlin, Germany. Translating the name into English is discouraged by the university, however paraphrasing as Berlin Institute of Technology is recommended by the university if necessary .The TU Berlin was founded...
. Cauer became interested in mathematics at the age of thirteen and continued to demonstrate that he was academically inclined as he grew.
Briefly, Cauer served in the German army in the final stages of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
. He married Karoline in 1925 and eventually fathered six children.
Career
Cauer started off in a field completely unrelated to filters; from 1922 he worked with Max von LaueMax von Laue
Max Theodor Felix von Laue was a German physicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals...
on general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, and his first publication (1923) was in this field. For reasons that are not clear, he changed his field after this to electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
. He graduated in applied physics in 1924 from the Technical University of Berlin
Technical University of Berlin
The Technische Universität Berlin is a research university located in Berlin, Germany. Translating the name into English is discouraged by the university, however paraphrasing as Berlin Institute of Technology is recommended by the university if necessary .The TU Berlin was founded...
.
He then spent a period working for Mix & Genest
Mix & Genest
Mix & Genest was a 1879 branch of the ITT Corporation.Mix & Genest was founded on the 1st of October 1879 by the businessman Wilhelm Mix and the engineer Werner Genest in Berlin-Schöneberg. The company was very successful and became one of the pioneers in low voltage devices. Among the products...
, a branch of the Bell Telephone Company
Bell Telephone Company
The Bell Telephone Company, a common law joint stock company, was organized in Boston, Massachusetts on July 9, 1877 by Alexander Graham Bell's father-in-law Gardiner Greene Hubbard, who also helped organize a sister company — the New England Telephone and Telegraph Company...
, applying probability theory to telephone switching. He also worked on timer relays. He had two telecommunications related publications during this period on these subjects;
- Telephone switching systems
- Losses of real inductors
The relationship of Mix & Genest with Bell gave Cauer an easy path to collaboration with AT&T's engineers at Bell Labs
Bell Labs
Bell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its...
in the US which must have been of enormous help when Cauer embarked on a study of filter design. Bell were at the forefront of filter design at this time with the likes of George Campbell
George Ashley Campbell
George Ashley Campbell was a pioneer in developing and applying quantitative mathematical methods to the problems of long-distance telegraphy and telephony. His most important contributions were to the theory and implementation of the use of loading coils and the first wave filters designed to...
in Boston and Otto Zobel in New York making major contributions. However, it was with R. M. Foster
R. M. Foster
Ronald Martin Foster , was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines...
that Cauer had much correspondence and it was his work that Cauer recognised as being of such importance. His paper, A reactance theorem, is a milestone in filter theory and inspired Cauer to generalise this approach into what has now become the field of network synthesis.
In June 1926 Cauer presented his thesis paper, The realisation of impedances of specified frequency dependence, at the Institute of Applied Mathematics and Mechanics of the Technical University of Berlin. This paper is the beginning of modern network synthesis.
In 1927 Cauer went to work as a research assistant at Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
's Institute of Mathematics at the University of Göttingen. In 1928 he obtained his habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
and became an external university lecturer.
Cauer found that he could not support his family during the economic crisis of the 1920s
Inflation in the Weimar Republic
The hyperinflation in the Weimar Republic was a three year period of hyperinflation in Germany between June 1921 and July 1924.- Analysis :...
and in 1930 took his family to the USA where he had obtained a scholarship (a Rockefeller fellowship) to study at MIT and Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
. He worked with Vannevar Bush
Vannevar Bush
Vannevar Bush was an American engineer and science administrator known for his work on analog computing, his political role in the development of the atomic bomb as a primary organizer of the Manhattan Project, the founding of Raytheon, and the idea of the memex, an adjustable microfilm viewer...
who was building machines for the solution of mathematical problems. Essentially, these were what we would now call analogue computers
Analog computer
An analog computer is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved...
: Cauer was interested in using them to solve linear systems to aid in filter designs. His work on Filter circuits was completed in 1931 while still in the US.
Cauer met, and had strong contacts with, many of the key researchers in the field of filter design at Bell Labs. These included Hendrik Bode
Hendrik Wade Bode
Hendrik Wade Bode , was an American engineer, researcher, inventor, author and scientist], of Dutch ancestry. As a pioneer of modern control theory and electronic telecommunications he revolutionized both the content and methodology of his chosen fields of research.He made important contributions...
, George Campbell
George Ashley Campbell
George Ashley Campbell was a pioneer in developing and applying quantitative mathematical methods to the problems of long-distance telegraphy and telephony. His most important contributions were to the theory and implementation of the use of loading coils and the first wave filters designed to...
, Sidney Darlington
Sidney Darlington
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair...
, Foster
R. M. Foster
Ronald Martin Foster , was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines...
and Otto Zobel.
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.
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. This correspondence went beyond his American contacts and included A.C. Bartlett of the General Electric Company in Wembley, Roger Julia of Lignes Telegraph Telephone in Paris, mathematicians Gustav Herglotz
Gustav Herglotz
Gustav Herglotz was a German mathematician. He is best known for his works on the theory of relativity and seismology....
, Georg Pick and Hungarian graph theorist Dénes Kőnig
Dénes König
Dénes Kőnig was a Jewish Hungarian mathematician who worked in and wrote the first textbook on the field of graph theory....
.
After leaving the Technical Institute for Mix & Genest, Cauer sought to become active in the Verband Deutscher Elektrotechniker (VDE, the German Electrical Engineers Society). He left the VDE, however, in 1942 after a serious falling out with Wagner
Karl Willy Wagner
Karl Willy Wagner was a German pioneer in the theory of electronic filters. He is noted by Hendrik Bode as being one of two Germans whose;The other German being referred to is Wilhelm Cauer...
, previously his PhD supervisor and ally.
Nazi era
The rising force of NazismNazism
Nazism, the common short form name of National Socialism was the ideology and practice of the Nazi Party and of Nazi Germany...
became a major obstacle to Cauer's work from 1933 onwards. The anti-Jewish hysteria of the time forced many academics to leave, including the director of the Mathematics Institute, Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
. Although Cauer was not Jewish, it became known that he had a Jewish ancestor, Daniel Itzig
Daniel Itzig
Daniel Itzig was a Court Jew of Kings Frederick II the Great and Frederick William II of Prussia....
, who had been a banker to Frederick II
Frederick II of Prussia
Frederick II was a King in Prussia and a King of Prussia from the Hohenzollern dynasty. In his role as a prince-elector of the Holy Roman Empire, he was also Elector of Brandenburg. He was in personal union the sovereign prince of the Principality of Neuchâtel...
of Prussia
Prussia
Prussia was a German kingdom and historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organized and effective army. Prussia shaped the history...
. Although this was not sufficient to have Cauer removed under the race laws it stifled his future career, he gained the title of professor but was never given a chair.
By 1935 he had three children who he was finding it increasingly difficult to support which prompted him to return to industry. In 1936 he temporarily worked for the aircraft manufacturer Fieseler
Fieseler
The Gerhard Fieseler Werke was a German aircraft manufacturer of the 1930s and 40s. The company is remembered mostly for its military aircraft built for the Luftwaffe during the Second World War.-History:...
at their Fi 156 Storch
Fieseler Fi 156
The Fieseler Fi 156 Storch was a small German liaison aircraft built by Fieseler before and during World War II, and production continued in other countries into the 1950s for the private market...
works in Kassel
Kassel
Kassel is a town located on the Fulda River in northern Hesse, Germany. It is the administrative seat of the Kassel Regierungsbezirk and the Kreis of the same name and has approximately 195,000 inhabitants.- History :...
and then became director of the laboratory of Mix & Genest in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
. He did however, continue to lecture at the Technical University in Berlin from 1939.
In 1941, the first volume of his main work, Theory of Linear AC Circuits was published. 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, based on surviving descriptions of the intended contents of volume II.
After taking his children to stay with relatives in Witzenhausen
Witzenhausen
Witzenhausen is a small town in the Werra-Meißner-Kreis in northeastern Hesse, Germany.It was granted town rights in 1225, and until 1974, it was a district seat....
(in Hesse
Hesse
Hesse or Hessia is both a cultural region of Germany and the name of an individual German state.* The cultural region of Hesse includes both the State of Hesse and the area known as Rhenish Hesse in the neighbouring Rhineland-Palatinate state...
) to protect them from the expected fall of Berlin to the Russians, Cauer, against advice, returned to Berlin. His body was located after the end of the war in a mass grave of victims of Russian executions. Cauer had been shot dead in his garden in Berlin-Marienfelde by soviet soldiers as a hostage. Soviet Intelligence was actively looking for scientists they could use in their own researches and Cauer was on their list of people to find, but of course this was not known to all soldiers.
Network synthesis
The major part of Cauer's legacy is his contribution to the network synthesisNetwork synthesis filters
Network synthesis is a method of designing signal processing filters. It has produced several important classes of filter including the Butterworth filter, the Chebyshev filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but...
of passive
Passivity (engineering)
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems...
networks. Indeed, he is considered the founder of the field. He treated network synthesis as being the inverse problem of network analysis
Network analysis (electrical circuits)
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this...
. 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
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
to the problem.
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.
Initially, the work revolved around one-port impedances. The transfer function between a voltage and a current amounting to the expression for the impedance itself. A useful network can be produced by breaking open a branch of the network and calling that the output.
Realisability
- Following on from Foster, Cauer generalised the relationship between the expression for the impedance of a one-port network and its transfer functionTransfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
. - 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. In later papers he made generalisations to multiport networks.
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 transformationAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s. - 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.
- He identified the canonical forms of filter realisation. That is, the minimal forms, which includes the ladder networks obtained by StieltjesThomas Joannes StieltjesThomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France. He was a pioneer in the field of moment problems and contributed to the study of continued fractions....
's continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
expansion.
Approximation
- He used the Tchebyscheff approximation to design filters. Cauer's application of Tchebyscheff polynomials resulted in the filters now known as elliptical filters, or sometimes Cauer filters, which have optimally fast passbandPassbandA passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
to stopbandStopbandA stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level...
transitions for a given maximum attenuation variation. The well known Tchebyscheff filters can be viewed as a special case of elliptical filters and can be arrived at using the same approximation techniques. So can the ButterworthButterworth filterThe Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
(maximally flat) filter, although this was an independent discovery by Stephen ButterworthStephen ButterworthStephen Butterworth was a British physicist who invented the Butterworth filter, a class of electrical circuits that are used to separate different frequencies of electrical signals....
arrived at by a different method.
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 ladder topology and ideal transformers could be dispensed with. From then on network synthesis began to supplant image design as the method of choice.
Further work
Most of the above work is contained in Cauer's first and second monographs and is largely a treatment of one-ports. In his habilitation thesis 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.Cauer extended the work of Bartlett and Brune on geometrically symmetric 2-ports to all symmetric 2-ports, that is 2-ports which are electrically symmetrical but not necessarily topologically symmetrical, finding a number of canonical circuits. He also studied antimetric
Antimetric (electrical networks)
An antimetric electrical network is one that exhibits anti-symmetrical electrical properties. The term is often encountered in filter theory, but it applies to general electrical network analysis...
2-ports. He also extended Foster's theorem to 2-element LC n-ports (1931) and showed that all equivalent LC networks could be derived from each other by linear transformations.
See also
- Cauer topology
- Elliptic (Cauer) filterElliptic filterAn elliptic filter is a signal processing filter with equalized ripple behavior in both the passband and the stopband...
- Chebyshev filterChebyshev filterChebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters...