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. 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.

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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.

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.

They found this result surprising as it showed that the Bott-Duffin method was not quite so non-minimal as previously thought. This is an enumeration of all distinct RLC networks with no more than two reactances and three resistances.

Edward Ladenheim carried out this work in while a student of Foster. The relevance of the catalogue is that all these networks are realised by biquadratic functions. The modern designs of such filters are almost always some form of network synthesis filter. Impedance matching at a single frequency requires only a trivial network—usually one component.

Impedance matching over a wide band, however, requires a more complex network, even in the case that the source and load ressitances do not vary with frequency. Doing this with passive elements and without the use of transformers results in a filter-like design. Furthermore, if the load is not a pure resistance then it is only possible to achieve a perfect match at a number of discrete frequencies; the match over the band as a whole must be approximated. The only essential difference between a standard filter and a matching network is that the source and load impedances are not equal.

Unless the network has a dual function, the designer is not too concerned over the behaviour of the impedance matching network outside the passband. It does not matter if the transition band is not very narrow, or that the stopband has poor attenuation.

In fact, trying to improve the bandwidth beyond what is strictly necessary will detract from the accuracy of the impedance match. With a given number of elements in the network, narrowing the design bandwidth improves the matching and vice versa. The limitations of impedance matching networks were first investigated by American engineer and scientist Hendrik Wade Bode in , and the principle that they must necessarily be filter-like was established by Italian-American computer scientist Robert Fano in For impedance matching networks, a better match can be obtained by also setting a minimum loss.

That is, the gain never rises to unity at any point. It is not possible to design a delay network that has a constant delay at all frequencies in a band. An approximation to this behaviour must be used limited to a prescribed bandwidth. The prescribed delay will occur at most at a finite number of spot frequencies. The Bessel filter has maximally-flat time-delay. It can be applied to systems in any energy domain that can be represented as a network of linear components. In particular, network synthesis has found applications in mechanical networks in the mechanical domain.

Consideration of mechanical network synthesis led Malcolm C. Smith to propose a new mechanical nework element, the inerter , which is analogous to the electrical capacitor. If the function is to be implemented with passive components, the function must also meet the conditions of a positive-real function PRF.

A one-element-kind network is a trivial case.

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## Network synthesis

Tolkis It is possible to construct non-Foster networks using active components such as amplifiers. 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. 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 netwoek positive-real functionand the reverse problem of which networks were equivalent, that is, had the same polynomial function. By using this site, you agree to the Terms of Use and Privacy Policy. Requires Control System Toolbox.

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## Thermische Charakterisierung elektronischer Systeme

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