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 , 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.Aatre, p. 259
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.E. Cauer et al., p. 4
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.Storer, p.3 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.Robertson et al., p. 107 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.Robertson et al., p. 108
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.Paarman, p. 15 One of the earliest used was Butterworth polynomials which results in a maximally flat response in the passband.Robertson et al., p. 107 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.Shenoi, pp. 4, 18 Other approximations are available for optimising time delay, impedance matching, roll-off, and many other requirements.Paarman, pp. 15–16
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.Bakshi & Chitode, p. 6-1 Historically, network synthesis was concerned exclusively with such networks. Modern active components have made this limitation less relevant in many applications,Anderson & Vongpanitlerd, p. 509 but at the higher radio frequencies passive networks are still the technology of choice.Wanhammar, p. 10 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.Bakshi & Chitode, p. 6-1
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 ).E. Cauer et al., p.4 Other impedance transformations are known, but whether there are further equivalence classes that remain to be discovered is an open question.Hughes et al., p. 288
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,Hughes et al., p. 288 but solutions are available for many networks with practical applications.Bakshi & Chitode, p. 6-1
History
The field of network synthesis was founded by German mathematician and scientist Wilhelm Cauer (1900–1945). The first hint towards a theory came from American mathematician Ronald M. Foster (1896–1998) when he published A reactance theorem in 1924. Cauer immediately recognised the importance of this work and set about generalising and extending it. His thesis in 1926 was on "The realisation of impedances of prescibed frequency dependence" and is the beginning of the field. Cauer's most detailed work was done during World War II, but he was killed shortly before the end of the war. His work could not be widely published during the war, and it was not until 1958 that his family collected his papers and published them for the wider world. Meanwhile, progress had been made in the United States based on Cauer's pre-war publications and material captured during the war.E. Cauer et al., pp. 3-4
English self-taught mathematician and scientist Oliver Heaviside (1850–1925) was the first to show that the impedance of an RLC network was always a rational function of a frequency operator, but provided no method of realising a network from a rational function.Kalman, p. 4 Cauer found a necessary condition for a rational function to be realisable as a passive network. 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.E. Cauer et al., pp. 6-7 Brune published the missing proof in his 1931 doctoral thesis.Kalman, p. 6
Foster's realisation was limited to LC networks and was in one of two forms; either a number of series LC circuits in parallel, or a number of parallel LC circuits in series. Foster's method was to expand into partial fractions. Cauer showed that Foster's method could be extended to RL and RC networks. Cauer also found another method; expanding as a continued fraction which results in a ladder network, again in two possible forms.Kalman, p. 4 In general, a PRF will represent an RLC network; with all three kinds of element present the realisation is trickier. Both Cauer and Brune used ideal transformers in their realisations of RLC networks. Having to include transformers is undesirable in a practical implementation of a circuit.Hubbard, p. 3
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).Hubbard, p. 3 The Bott and Duffin method provides an expansion by repeated application of Richards' theorem, a 1947 result due to American physicist and applied mathematician Paul I. Richards (1923–1978).Wing, p. 122 The resulting Bott-Duffin networks have limited practical use (at least for rational functionals of high degree) because the number of components required grows exponentially with the degree.Kalman, p. 7 A number of variations of the original Bott-Duffin method all reduce the number of elements in each section from six to five, but still with exponentially growing overall numbers.Chen & Smith, p. 38 Papers achieving this include Pantell (1954), Reza (1954), Storer (1954) and Fialkow & Gest (1955).Chen & Smith, pp. 38, 50 As of 2010, there has been no further significant advance in synthesising rational functions.
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. 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.E. Cauer et al., p. 7 The corollary problem, to find a synthesis of PRFs using R and C elements with only one inductor, is an unsolved problem in network theory.Wing, p. 128 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.Belevitch, p. 854 An important consideration in practical networks is to minimise the number of components, especially the wound components—inductors and transformers. Despite great efforts being put into minimisation,Lee, pp. 756-757 no general theory of minimisation has ever been discovered as it has for the Boolean algebra of digital circuits.Kalman, p. 10
Cauer used elliptic rational functions to produce approximations to ideal filters.E. Cauer et al., p. 5 A special case of elliptic rational functions is the Chebyshev polynomials due to Pafnuty Chebyshev (1821–1894) and is an important part of approximation theory.Swanson, p. 58 Chebyshev polynomials are widely used to design filters. In 1930, British physicist Stephen Butterworth (1885–1958) designed the Butterworth filter, otherwise known as the maximally-flat filter, using Butterworth polynomials.McCarthy, p. 51 Butterworth's work was entirely independent of Cauer, but it was later found that the Butterworth polynomials were a limiting case of the Chebyshev polynomials.Swanson, p. 58 Even earlier (1929) and again independently, American engineer and scientist Edward Lawry Norton (1898–1983) designed a maximally-flat mechanical filter with a response entirely analogous to Butterworth's electrical filter.Darlington, p. 7