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Concept
Equivalent impedance transforms
Summary
An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.
There are a number of very well known and often used equivalent circuits in linear network analysis. These include resistors in series, resistors in parallel and the extension to series and parallel circuits for capacitors, inductors and general impedances. Also well known are the Norton and Thévenin equivalent current generator and voltage generator circuits respectively, as is the Y-Δ transform. None of these are discussed in detail here; the individual linked articles should be consulted.
The number of equivalent circuits that a linear network can be transformed into is unbounded. Even in the most trivial cases this can be seen to be true, for instance, by asking how many different combinations of resistors in parallel are equivalent to a given combined resistor. The number of series and parallel combinations that can be formed grows exponentially with the number of resistors, n. For large n the size of the set has been found by numerical techniques to be approximately 2.53n and analytically strict bounds are given by a Farey sequence of Fibonacci numbers. This article could never hope to be comprehensive, but there are some generalisations possible. Wilhelm Cauer found a transformation that could generate all possible equivalents of a given rational, passive, linear one-port, or in other words, any given two-terminal impedance. Transformations of 4-terminal, especially 2-port, networks are also commonly found and transformations of yet more complex networks are possible.
The vast scale of the topic of equivalent circuits is underscored in a story told by Sidney Darlington. According to Darlington, a large number of equivalent circuits were found by Ronald M.
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