is a positive-real function (PRF) then
is a PRF for all real, positive values of
[1] The theorem has applications in electrical network synthesis.
The PRF property of an impedance function determines whether or not a passive network can be realised having that impedance.
Richards' theorem led to a new method of realising such networks in the 1940s.
is a positive real constant, and
is the complex frequency variable, can be written as, where, Since
The zeroes of this function are the poles of
Since a PRF can have no zeroes in the right-half s-plane, then
is given by, Since the PRF condition requires that
is analytic in the right-half s-plane.
[2] Richards' theorem can also be derived from Schwarz's lemma.
[3] The theorem was introduced by Paul I. Richards as part of his investigation into the properties of PRFs.
The term PRF was coined by Otto Brune who proved that the PRF property was a necessary and sufficient condition for a function to be realisable as a passive electrical network, an important result in network synthesis.
[4] Richards gave the theorem in his 1947 paper in the reduced form,[5] that is, the special case where
The theorem (with the more general casse of
being able to take on any value) formed the basis of the network synthesis technique presented by Raoul Bott and Richard Duffin in 1949.
represents the electrical network to be synthesised and
is another (unknown) network incorporated within it (
is merely a positive real number,
in parallel with a capacitor all in series with a network proportional to the inverse of
By a suitable choice for the value of
, a resonant circuit can be extracted from
The whole process can then be applied iteratively to
until the degree of the function is reduced to something that can be realised directly.
[7] The advantage of the Bott-Duffin synthesis is that, unlike other methods, it is able to synthesise any PRF.
Other methods have limitations such as only being able to deal with two kinds of element in any single network.
Its major disadvantage is that it does not result in the minimal number of elements in a network.
The number of elements grows exponentially with each iteration.
[8] Hubbard notes that Bott and Duffin appeared not to know the relationship of Richards' theorem to Schwarz's lemma and offers it as his own discovery,[9] but it was certainly known to Richards who used it in his own proof of the theorem.