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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 .
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'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 (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.
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 transformations. [ 20 ] 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 ...
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.
Human development theory is a theory which uses ideas from different origins, such as ecology, sustainable development, feminism and welfare economics. It wants to avoid normative politics and is focused on how social capital and instructional capital can be deployed to optimize the overall value of human capital in an economy.