Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response (or other behaviour) of a given circuit is calculated. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits. The field was founded by Wilhelm Cauer after reading Ronald M. Foster's 1924 paper A reactance theorem. Foster's theorem provided a method of synthesising LC circuits with arbitrary number of elements by a partial fraction expansion of the impedance function. Cauer extended Foster's method to RC and RL circuits, found new synthesis methods, and methods that could synthesise a general RLC circuit. 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. Little was done in the field until the 2000s when the issue of minimisation again became an active area of research, but as of 2023, is still an unsolved problem. A primary application of network synthesis is the design of network synthesis filters but this is not its only application. Amongst others are impedance matching networks, time-delay networks, directional couplers, and equalisation. In the 2000s, network synthesis began to be applied to mechanical systems as well as electrical, notably in Formula One racing.
Overview Network synthesis is all about designing an electrical network that behaves in a prescribed way without any preconception of the network form. Typically, an impedance is required to be synthesised using passive components. That is, a network consisting of resistances (R), inductances (L) and capacitances (C). Such networks always have an impedance, denoted Z ( s ) {\displaystyle Z(s)} , in the form of a rational function of the complex frequency variable s. That is, the impedance is the ratio of two polynomials in s. There are three broad areas of study in network synthesis; approximating a requirement with a rational function, synthesising that function into a network, and determining equivalents of the synthesised network.
Approximation The idealised prescribed function will rarely be capable of being exactly described by polynomials. It is therefore not possible to synthesise a network to exactly reproduce it. A simple, and common, example is the brick-wall filter. This is the ideal response of a low-pass filter but its piecewise continuous response is impossible to represent with polynomials because of the discontinuities. To overcome this difficulty, a rational function is found that closely approximates the prescribed function using approximation theory. In general, the closer the approximation is required to be, the higher the degree of the polynomial and the more elements will be required in the network. There are many polynomials and functions used in network synthesis for this purpose. The choice depends on which parameters of the prescribed function the designer wishes to optimise. One of the earliest used was Butterworth polynomials which results in a maximally flat response in the passband. 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. Other approximations are available for optimising time delay, impedance matching, roll-off, and many other requirements.
Realisation Given a rational function, it is usually necessary to determine whether the function is realisable as a discrete passive network. All such networks are described by a rational function, but not all rational functions are realisable as a discrete passive network. Historically, network synthesis was concerned exclusively with such networks. Modern active components have made this limitation less relevant in many applications, but at the higher radio frequencies passive networks are still the technology of choice. There is a simple property of rational functions that predicts whether the function is realisable as a passive network. Once it is determined that a function is realisable, there a number of algorithms available that will synthesise a network from it.
Equivalence A network realisation from a rational function is not unique. The same function may realise many equivalent networks. It is known that affine transformations of the impedance matrix formed in mesh analysis of a network are all impedance matrices of equivalent networks (further information at Analogue filter § Realisability and equivalence). Other impedance transformations are known, but whether there are further equivalence classes that remain to be discovered is an open question. A major area of research in network synthesis has been to find the realisation which uses the minimum number of elements. This question has not been fully solved for the general case, but solutions are available for many networks with practical applications.
History
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