Richards' theorem

Due to Paul I. Richards, 1947.

Richards' theorem states that for,

$R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}}$

if $Z(s)$ is a positive-real function (PRF) then $R(s)$ is a PRF for all real, positive values of $k$.

<note>Is this an extension of Richards' original statement?</note>

Proof

Let,

$W(s)={1-{\dfrac {Z(s)}{Z(k)}} \over 1+{\dfrac {Z(s)}{Z(k)}}}\left({\frac {k+s}{k-s}}\right)$

where $Z(s)$ is a PRF and k is a positive real constant.

Since $Z(s)$ is PRF then

$1+{\dfrac {Z(s)}{Z(k)}}$

is also PRF. The zeroes of this function are the poles of $W(s)$. Since a PRF can have no zeroes in the right-half s-plane, then $W(s)$ can have no poles in the right-half s-plane and hence is analytic in the right-half s-plane.

Richards' theorem can also be derived from Schwarz's lemma.

Uses

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. Richards gave the theorem in his 1947 paper in the reduced form,

$R(s)={\frac {Z(s)-sZ(1)}{Z(1)-sZ(s)}}$

that is, the special case where $k=1$

The theorem (with the more general casse of $k$ being able to take on any value) formed the basis of the network synthesis technique presented by Raoul Bott and Richard Duffin in 1949. In the Bott-Duffin synthesis, $Z(s)$ represents the electrical network to be synthesised and $R(s)$ is another (unknown) network incorporated within it. Making $Z(s)$ the subject gives

$Z(s)=\left({\frac {F(s)}{Z(k)}}+{\frac {k}{sZ(k)}}\right)^{-1}+\left({\frac {1}{Z(k)F(s)}}+{\frac {s}{kZ(k)}}\right)^{-1}$

Since $Z(k)$ is merely a positive real number, $Z(s)$ can be synthesised as a new network proportional to $R(s)$ in parallel with a capacitor all in series with a network proportional to the inverse of $R(s)$ in parallel with an inductor. By a suitable choice for the value of $k$, a resonant circuit can be extracted from $R(s)$ leaving a function $Z'(s)$ two degrees lower than $Z(s)$. The whole process can then be applied iteratively to $Z'(s)$ until the degree of the function is reduced to something that can be realised directly.

Bibliography

• Cauer, Emil; Mathis, Wolfgang; Pauli, Rainer, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000.
• Richards, Paul I., "A special class of functions with positive real part in a half-plane", Duke Mathematical Journal, vol. 14, no. 3, 777–786, 1947.
• Wing, Omar, Classical Circuit Theory, Springer, 2008 ISBN 0387097406.

Unprocessed sources

• Bott, Raoul; Duffin, Richard, "Impedance synthesis without use of transformers", Journal of Applied Physics, vol. 20, iss. 8, p. 816, August 1949.
• Hubbard, John H., "The Bott-Duffin synthesis of electrical circuits", pp. 33–40 in, Kotiuga, P. Robert (ed), A Celebration of the Mathematical Legacy of Raoul Bott, American Mathematical Society, 2010 ISBN 9780821883815.