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

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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...
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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.

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