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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.
Wilhelm Cauer found a transformation that could generate all possible equivalents of a given rational, [note 9] passive, linear one-port, [note 8] or in other words, any given two-terminal impedance. Transformations of 4-terminal, especially 2-port, networks are also commonly found and transformations of yet more complex networks are possible.
Foster published his paper the following year which included his canonical realisation forms. [13] Cauer in Germany grasped the importance of Foster's work and used it as the foundation of network synthesis. Amongst Cauer's many innovations was the extension of Foster's work to all 2-element-kind networks after discovering an isomorphism ...
However, it was with Ronald 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, [9] is a milestone in filter theory and inspired Cauer to generalise this approach into what has now become the field of network synthesis. [5]
Cauer's second form of driving point impedance consists of a ladder of series capacitors and shunt inductors and is most useful for high-pass filters. Foster's first form of driving point impedance consists of parallel connected LC resonators (series LC circuits) and is most useful for band-pass filters.
Ronald Martin Foster (3 October 1896 – 2 February 1998), was an American mathematician at Bell Labs whose work was of significance regarding electronic filters for use on telephone lines.
Wilhelm Cauer expanded on the work of Foster (1926) [47] and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them.
Campbell published in 1922 but had clearly been using the topology for some time before this. Cauer first picked up on ladders (published 1926) inspired by the work of Foster (1924). There are two forms of basic ladder topologies: unbalanced and balanced. Cauer topology is usually thought of as an unbalanced ladder topology.