# 11 the principle of the argument complex function

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stability The area to the right of the path Γ is the area enclosed by Γ . 11 The Principle of the Argument Complex Function Mapping • S-plane → F(s) complex plane mapping. jω s2 Γ s3 Im{F} unique s1 F(s1) σ Re{F} non unique F(s2) F(s3) • If F(s) is analytic along the path Γ (no poles of F(s) on Γ) and s starts at s = s1 and traces a closed path terminating at s1 , then F(s) will trace a closed path in the F plane starting at F(s1) and terminating at F(s1) . 12/13/2008 HSK - EPM 381 - Stability 12 The Principle of the Argument • The principle of the argument states that for an The arbitrary closed path Γ in the s-plane, the corresponding closed path in the F plane will encircle the origin as many times as the difference between the number of zeroes of F(s) and poles of F(s) located in the area enclosed by the path Γ . • The direction of the encirclements of origin is the The direction same as the path Γ if the number of zeroes of F(s) is same greater than the poles. 12/13/2008 HSK - EPM 381 - Stability 13 The Nyquist Path • Define the Nyquist Define path Γ such that it encloses the right hand side of the splane, but does not go through any poles of F(s). 12/13/2008 jω j∞ × Poles of F(s) HSK - EPM 381 - Stability ∞ → R Γ σ × × -j∞ 14 The Nyquist Criterion • Map F(s) for the Nyquist path enclosing the RHS. • Then the number of clockwise encirclements of the origin of the F(s) plane is N = Z – P , where Z is the number of zeroes of F(s) in the RHS, P is the number of poles of F(s) in the RHS. Z, corresponds to the number of pol...
View Full Document

## This note was uploaded on 03/03/2010 for the course AUTOMATIC 335 taught by Professor ? during the Winter '10 term at Ain Shams University.

Ask a homework question - tutors are online