page_title: Network synthesis
section_title: History
hierarchical_section_title: Network synthesis / History
caption_attribution_description: English: Raoul Bott in 1986
context_page_description: Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response of a given circuit is calculated. Network synthesis was a great leap forward in circuit design. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits.
  The field was founded by Wilhelm Cauer after reading Ronald M. Foster's 1924 paper A reactance theorem. Foster's theorem provided a method of synthesising LC circuits with arbitrary number of elements by a partial fraction expansion of the impedance function.
context_section_description: The field of network synthesis was founded by German mathematician and scientist Wilhelm Cauer (1900–1945).  The first hint towards a theory came from American mathematician Ronald M. Foster (1896–1998) when he published A reactance theorem in 1924.  Cauer immediately recognised the importance of this work and set about generalising and extending it. His thesis in 1926 was on "The realisation of impedances of specified frequency dependence" and is the beginning of the field.  Cauer's most detailed work was done during World War II, but he was killed shortly before the end of the war.  His work could not be widely published during the war, and it was not until 1958 that his family collected his papers and published them for the wider world.  Meanwhile, progress had been made in the United States based on Cauer's pre-war publications and material captured during the war.
  English self-taught mathematician and scientist Oliver Heaviside (1850–1925) was the first to show that the impedance of an RLC network was always a rational function of a frequency operator, but provided no method of realising a network from a rational function.  Cauer found a necessary condition for a rational function to be realisable as a passive network.  South African Otto Brune (1901–1982) later coined the term positive-real function (PRF) for this condition.  Cauer postulated that PRF was a necessary and sufficient condition but could not prove it, and suggested it as a research project to Brune, who was his grad student in the United States at the time.  Brune published the missing proof in his 1931 doctoral thesis.
  Foster's realisation was limited to LC networks and was in one of two forms; either a number of series LC circuits in parallel, or a number of parallel LC circuits in series.  Foster's method was to expand  into partial fractions.  Cauer showed that Foster's method could be extended to RL and RC networks.  However, Cauer found another method; expanding  as a continued fraction which results in a ladder network, again in two possible forms.  In general, a PRF will represent an RLC network; with all three kinds of element present the realisation is trickier.  Both Cauer and Brune used ideal transformers in their realisations of RLC networks.  Having to include transformers is undesirable in a practical implementation of a circuit.
  A method of realisation that did not require transformers was provided in 1949 by Hungarian-American mathematician Raoul Bott (1923–2005) and American physicist Richard Duffin (1909–1996).  The Bott and Duffin method provides an expansion by repeated application of Richards' theorem, a 1947 result due to American physicist and applied mathematician Paul I. Richards (1923–1978).  The resulting Bott-Duffin networks have limited practical use (at least for rational functionals of high degree) because the number of components required grows exponentially with the degree.  A number of variations of the original Bott-Duffin method all reduce the number of elements in each section from six to five, but still with exponentially growing overall numbers.  Papers achieving this include Pantell (1954), Reza (1954), Storer (1954) and Fialkow & Gest (1955).  As of 2008, there has been no further significant advance in synthesising rational functions.
  In 1939, American electrical engineer Sidney Darlington showed that any PRF can be realised as a two-port network consisting only of L and C elements and terminated at its output with a resistor.  That is, only one resistor is required in any network, the remaining components being lossless.  The theorem was independently discovered by both Cauer and Giovanni Cocci.  The corollary problem, to find a synthesis of PRFs using R and C elements with only one inductor, is an unsolved problem in network theory.  Another unsolved problem is finding a proof of Darlington's conjecture (1955) that any RC 2-port with a common terminal can be realised as a series-parallel network.  An important consideration in practical networks is to minimise the number of components, especially the wound 
caption: Raoul Bott
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