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 PRFs, a term which he coined. Richards 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

• 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.
• 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.