Network synthesis

Foster's theorem provided a method of synthesising LC circuits with arbitrary number of elements by a partial fraction expansion of the impedance function.

Other important advances before World War II are due to Otto Brune and Sidney Darlington.

In the 1940s Raoul Bott and Richard Duffin published a synthesis technique that did not require transformers in the general case (the elimination of which had been troubling researchers for some time).

In the 1950s, a great deal of effort was put into the question of minimising the number of elements required in a synthesis, but with only limited success.

In the 2000s, network synthesis began to be applied to mechanical systems as well as electrical, notably in Formula One racing.

[7] A common choice is the Chebyshev approximation in which the designer specifies how much the passband response can deviate from the ideal in exchange for improvements in other parameters.

Modern active components have made this limitation less relevant in many applications,[11] but at the higher radio frequencies passive networks are still the technology of choice.

[15] A major area of research in network synthesis has been to find the realisation which uses the minimum number of elements.

Meanwhile, progress had been made in the United States based on Cauer's pre-war publications and material captured during the war.

South African Otto Brune (1901–1982) later coined the term positive-real function (PRF) for this condition.

Cauer postulated that PRF was a necessary and sufficient condition but could not prove it, and suggested it as a research project to Brune, who was his grad student in the United States at the time.

[22] In general, a PRF will represent an RLC network; with all three kinds of element present the realisation is trickier.

[23] A method of realisation that did not require transformers was provided in 1949 by Hungarian-American mathematician Raoul Bott (1923–2005) and American physicist Richard Duffin (1909–1996).

[29] In 1939, American electrical engineer Sidney Darlington showed that any PRF can be realised as a two-port network consisting only of L and C elements and terminated at its output with a resistor.

[31] Another unsolved problem is finding a proof of Darlington's conjecture (1955) that any RC 2-port with a common terminal can be realised as a series-parallel network.

[40] The unsolved problem of minimisation is much more important in the mechanical domain than the electrical due to the size and cost of components.

Impedance matching over a wide band, however, requires a more complex network, even in the case that the source and load resistances do not vary with frequency.

The only essential difference between a standard filter and a matching network is that the source and load impedances are not equal.

With a given number of elements in the network, narrowing the design bandwidth improves the matching and vice versa.

[53] Which, and how many kinds of, elements are required can be determined by examining the poles and zeroes (collectively called critical frequencies) of the function.

For instance, the partial fraction terms of an RC network in Foster I form will each represent an R and C element in parallel.

It is only necessary to declare whether the function represents an impedance or an admittance at the point that an actual circuit needs to be realised.

Using the same example as used for the Foster I form, or, in more compact notation, The terms of this expansion can be directly implemented as the component values of a ladder network as shown in the figure.

[64] This particular PRF, therefore, cannot be realised in passive components as a Cauer II form without the inclusion of transformers or mutual inductances.

In step two of the cycle it was mentioned that a negative element value must be extracted in order to guarantee a PRF remainder.

Although the Bott-Duffin method avoids the use of transformers and can be applied to any expression capable of realisation as a passive network, it has limited practical use due to the high component count required.

Darlington showed that any PRF can be realised as a two-port network using only L and C elements with a single resistor terminating the output port.

More generally applicable designs of this kind include the Sallen–Key topology due to R. P. Sallen and E. L. Key in 1955 at MIT Lincoln Laboratory, and the biquadratic filter.

A major practical advantage of active implementation is that it can avoid the use of wound components (transformers and inductors) altogether.

[98] Digital realisations, like active circuits, are not limited to PRFs and can implement any rational function simply by programming it in.