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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.
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]
Examples of canonical forms are the realisation of a driving-point impedance by Cauer's canonical ladder network or Foster's canonical form or Brune's realisation of an immittance from his positive-real functions. Topological methods, on the other hand, do not start from a given canonical form.
The impact of this transformation goes beyond Joy’s personal development. The video has struck a chord with audiences, with comments pouring in expressing gratitude for the fosters’ dedication ...
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 himself had found a necessary condition but had failed to prove it to be sufficient. The goal for researchers then was "to remove the restrictions implicit in the Foster-Cauer realisations and find conditions on Z equivalent to realisability by a network composed of arbitrary interconnections of positive-valued R, C and L." [9]