Richards' Theorem
Richards' theorem is a mathematical result due to Paul I. Richards in 1947. The theorem states that for, :R(s) = \frac if Z(s) is a positive-real function (PRF) then R(s) is a PRF for all real, positive values of k. 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. Proof : R(s) = \frac where Z(s) is a PRF, k is a positive real constant, and s= \sigma + i \omega is the complex frequency variable, can be written as, : R(s) = \dfrac where, : W(s) = \left ( \frac \right ) Since Z(s) is PRF then : 1 + \dfrac is also PRF. The zero (complex analysis), zeroes of this function are the pole (complex analysis), poles of W(s). Since a PRF can have no zeroes in the right-half s-plane, ''s''-plane, then W(s) can have no poles in the right-half ''s''-plane ...
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Paul I
Paul I may refer to: *Paul of Samosata (200–275), Bishop of Antioch *Paul I of Constantinople (died c. 350), Archbishop of Constantinople *Pope Paul I (700–767) *Paul I Šubić of Bribir (c. 1245–1312), Ban of Croatia and Lord of Bosnia *Paul I, Serbian Patriarch, Archbishop of Peć and Serbian Patriarch (c. 1530–1541) *Paul I of Russia (1754–1801), Emperor of Russia *Paul Peter Massad (1806–1890), Maronite Patriarch of Antioch *Paul of Greece (1901–1964), King of Greece * Pavle, Serbian Patriarch (1914–2009), Patriarch of the Serbian Orthodox Church See also

*Patriarch Paul I (other) {{hndis, Paul 01 ...
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