EEL5173 Fall 2009 Lecture _9

# EEL5173 Fall 2009 Lecture _9 - (a Mapping Contours •...

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Unformatted text preview: (a) Mapping Contours • Nyquist Path (b) Nyquist Diagram • Frequency Response (c) Nyquist Criterion (2) Nyquist Criterion ) ( ) )( ( ) ( ) )( ( ) ( ) ( 1 ) ( 2 1 2 1 n m p s p s p s z s z s z s k s H s G s F − − − − − − = + = L L Poles of the system, i.e. of T ( s ), come from the zeros of F ( s ). Figure 10.20 Closed-loop control system by N. S. Nise ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( s F s G s H s G s G s R s C s T = + = =- a proper rational function with real coefficients . Where (a) Mapping Contours Figure 10.21 Mapping Contour A through function F ( s ) to Contour B by N. S. Nise A contour is a closed path, with a direction, in a complex plane. ) ( Q F Q = ′ Figure 10.22 Examples of contour mapping by N. S. Nise ɵ ∆ɵ = 0 = Δ = ∠ Δ = ∠ θ θ F F ɵ ϕ- ϕ ∆ϕ = 0 = Δ − = ∠ Δ − = ∠ φ φ F F ɵ ɵ π θ θ 2 − = Δ = ∠ Δ = ∠ F F ∆ɵ = -2 π ∆ϕ = -2 π ϕ- ϕ π φ φ 2 = Δ − = ∠ Δ − = ∠ F F ) 2 ( ) 2 ( = − − − = Δ − Δ = ∠ Δ − = ∠ π π φ θ φ θ F F ɵ ϕ Figure 10.23 Figure 10....
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EEL5173 Fall 2009 Lecture _9 - (a Mapping Contours •...

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