Iv mirror image of ii there are no net rotations

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Unformatted text preview: cally stable. system φm = 180° − θ 12/13/2008 HSK - EPM 381 - Stability 29 The Nyquist Criterion: Special Example j 2× j∞ ρe K Sec. II: GH ( s) = 2 s +2 + , GH(jω) → K/2 s = j 0 GH s = j ∞, GH(jω) → 0 /-180º GH Sec. I: Let s = j 2 + ρe jθ jθ jω II × IV × 1-j∞ 2/13/2008 I then then III σ where ρ → 0 ; θ = −90o → +90o K θ ( j 2 + ρe j ) 2 + 2 K = − 2 + j 2 2 ρe jθ + ρ 2e j 2θ + 2 GH ( s ) = HSK - EPM 381 - Stability 30 The Nyquist Criterion: Special Example Im{GH} For ρ → 0 R→ ∞ j∞ -1 -j∞ Re{GH} K/2 GH ( s) = K 2 s +2 K GH ( s ) ≈ θ+ 2 2 ρe j ( 90 ) → ∞ /− (θ + 90) • Sec. III: no effect • Sec. IV: mirror image of II. There are no net rotations about the “–1” point, therefore, N = 0. Since P = 0 ( no roots of GH in the RHS), Z = 0 and the closed loop system is stable for all values of K. 12/13/2008 HSK - EPM 381 - Stability 31 Bode Plots and Stability Analysis • In the Nyquist analysis, it became clear that In Nyquist analysis, Section II of the plot was the most critical in determining the stability of the closed-loop determining loop system. system. • The Bode plot of the loop transfer function, The Bode of GH(jω) provides the same magnitude and angle GH information as Section II of the Nyquist plot. • Therefore, the Bode plot of GH(jω) can be used Therefore, GH to evaluate the stability of the closed-loop 12/13/2008 HSK - EPM 381 - Stability 32 system. Bode Plots and Stability Analysis • Consider the definitions of the gain and phase margins in relation to the Bode plot of GH(jω) . GH – Gain Margin: the additional gain required to make | GH(jω) | = 1 when /GH(jω) = −180° . On the Bod...
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