# Regh if the 1 point is inside the gh path then gh 1 n2

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: ) = s ( s + a )( s + b) jω j∞ II III σ × I IV -j∞ 12/13/2008 Sec. II: s = j 0+ , GH(jω) → ∞ /-90º GH s = j ∞, GH(jω) → 0 /-270º GH Sec. III: no effect. Sec. IV: mirror image of II. Sec. I: (k = 1) Then GH(j0) rotates (k GH 0) 180º clockwise from GH(j0–) to 180 clockwise GH to GH(j0+) with a magnitude of ∞ . GH with HSK - EPM 381 - Stability 26 The Nyquist Criterion: Example 2 j∞ Im{GH} GH ( s ) = GH(jωc) K s ( s + a )( s + b) R→ ∞ • Where is the “–1” point? Re{GH} • If the “–1” point is inside the GH path, then GH -1 N=2 , P= 0 and Z=2 ∴ the system is unstable. • If the “–1” point is to the left -j∞ of the GH path, then GH Find |GH(jωc)| where ωc is defined N=0 , P= 0 and Z=0 by Im{GH(jωc)} = 0 . ∴ the system is stable. 12/13/2008 HSK - EPM 381 - Stability 27 The Nyquist Criterion: Example 2 K K = GH ( s ) ω ( jω + a )( jω + b) j s ( s + a )( s + b) K = − jω 3 + ab jω − ( a + b)ω 2 GH ( jω ) = For, Im{GH ( jω c )} = 0 − jω c3 + ab jω c = 0 → ω c = ab Check if this is less than –1 . K K = Then, GH ( jω c ) = 2 − ( a + b)ω c − ( a + b)ab 12/13/2008 HSK - EPM 381 - Stability 28 Relative Stability Gain and Phase Margins • Gain and phase margins are a measure of Gain how close the system is to instability. how Gain Margin: the additional gain in db, that will make the system db, critically stable. Im{GH} |GH(jωc)| -1 φm Re{GH} θ 20 log10 1 GH ( jω c ) Phase Margin: the additional phase the lag in degrees, that will make the system criti...
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