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The example used for the Cauer I form and the Foster forms when expanded as a Cauer II form results in some elements having negative values. [64] This particular PRF, therefore, cannot be realised in passive components as a Cauer II form without the inclusion of transformers or mutual inductances .
Foster's work was an important starting point for the development of network synthesis. It is possible to construct non-Foster networks using active components such as amplifiers. These can generate an impedance equivalent to a negative inductance or capacitance. The negative impedance converter is an example of such a circuit.
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.
The transmission line example is one of a class of practical problems that can be modelled by infinitesimal elements (the distributed-element model). Other examples are launching waves into a continuous medium, fringing field problems, and measurement of resistance between points of a substrate or down a borehole.
Wilhelm Cauer (following on from R. M. Foster [10]) did much of the early work on what mathematical functions could be realised and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer. [11]
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]
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.
Image theory defines quantities in terms of an infinite cascade of two-port sections, and in the case of the filters being discussed, an infinite ladder network of L-sections. Here "L" should not be confused with the inductance L – in electronic filter topology , "L" refers to the specific filter shape which resembles inverted letter "L".