This follows from the fact the impedance of L and C elements are themselves simple rational functions and any algebraic combination of rational functions results in another rational function. This is sometimes referred to as the driving point impedance because it is the impedance at the place in the network at which the external circuit is connected and "drives" it with a signal. In his paper, Foster describes how such a lossless rational function may be realised if it can be realised in two ways. The realisation of the driving point impedance is by no means unique. These are called non-Foster networks.

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This follows from the fact the impedance of L and C elements are themselves simple rational functions and any algebraic combination of rational functions results in another rational function.

This is sometimes referred to as the driving point impedance because it is the impedance at the place in the network at which the external circuit is connected and "drives" it with a signal. In his paper, Foster describes how such a lossless rational function may be realised if it can be realised in two ways. The realisation of the driving point impedance is by no means unique. These are called non-Foster networks. For example, it is possible to create negative capacitance and inductance with negative impedance converter circuits.

For example, the usual passive Foster impedance matching networks can only match the impedance of an antenna with a transmission line at discrete frequencies, which limits the bandwidth of the antenna. A non-Foster network could match an antenna over a continuous band of frequencies.

Practical non-Foster networks are an active area of research. This work was commercially important; large sums of money could be saved by increasing the number of telephone conversations that could be carried on one line.

Cauer was interested in finding the necessary and sufficient condition for realisability of a rational one-port network from its polynomial function, a condition now known to be a positive-real function , and the reverse problem of which networks were equivalent, that is, had the same polynomial function. Both of these were important problems in network theory and filter design. Foster networks are only a subset of realisable networks, [14].

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## FOSTER AND CAUER FORM PDF

Having to include transformers is undesirable in a practical implementation of a circuit. Richards — 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. Despite great efforts being put into minimisation, [18] no general theory of minimisation has ever been discovered as it has for the Boolean algebra of digital circuits. In , British physicist Stephen Butterworth — designed the Butterworth filter , otherwise known as the maximally-flat filter, using Butterworth polynomials.

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## Network synthesis

After passing through a pole the function will be negative and is obliged to pass through zero before reaching the next pole if it is to be monotonically increasing. Minimum positive real functions. An Edo period wood block print of a samurai putting on a fundoshi. The LC ladder development.

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## Foster's reactance theorem

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