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 $k=1$

As used in Bott-Duffin synthesis,

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.