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# Bode - ∠ G jω at several frequencies jω-10 1 θ 2 θ 1...

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Slide 7: Straight line approximation to Bode phase plots: the Bode’s phase-gain relationship For minimum-phase systems, the Bode’s phase-gain relationship holds: G ( ) σ 20 π 2 radians σ 20 90 degrees where σ is the slope of the gain (magnitude) Bode plot in dB/decade. The approximation is most accurate if the slope of the magnitude Bode plot is constant over a large region around the frequency ω ; e.g, if the Bode magnitude plot has a constant slope σ over a two-decade interval centred at ω . Where the slope is not constant over this two-decade interval, an average slope can be used, instead of σ . Example: G ( s ) = 5( s + 1)(0 . 1 s + 1) (10 s + 1)(0 . 01 s + 1) . 10 -2 10 -1 10 0 10 1 10 2 10 3 -20 -10 0 10 20 20log|W(j ω )| (dB) 10 -2 10 -1 10 0 10 1 10 2 10 3 -90 -45 0 45 90 arg(W(j )) (degrees) ω , rad/sec

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Slide 8: Straight line approximation to Bode phase plots: Graphical method Works for non-minimum phase systems. Example: : G ( s ) = 10( s - 1) s + 10 . G ( ) = ( - 1) - ( + 10) = θ 1 - θ 2 . Use the Fgure shown below to measure θ 1 , θ 2 and approximate

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Unformatted text preview: ∠ G ( jω ) at several frequencies. jω-10 1 θ 2 θ 1 ω , rad/sec θ 1 θ 2 phase 180 ◦ ◦ 180 ◦ 0.1 ≈ 174 ◦ ≈ ◦ ≈ 174 ◦ 1 135 ◦ ≈ 6 ◦ 129 ◦ 10 ≈ 96 ◦ 45 ◦ 51 ◦ 100 ≈ 90 ◦ ≈ 84 ◦ ≈ 6 ◦ ∞ 90 ◦ 90 ◦ ◦ Slide 9: Straight line approximation to Bode phase plots: Graphical method (cont.) 10-1 10 10 1 10 2-10 10 20 30 20log|W(j ω )| (dB) 10-1 10 10 1 10 2 45 90 Slide 10: Nyquist polar plots • The Nyquist plot is a plot of the open loop frequency response in the complex plane. • It contains the same information as the Bode plot. • The Matlab command is >> nyquist(num,den,w); For example, >> nyquist([2 5 1],[1 2 3]); >> grid; Real Axis Imaginary Axis Nyquist Diagrams-1-0.5 0.5 1 1.5 2 2.5 3-2-1.5-1-0.5 0.5 1 1.5 2 The Nyquist plot of G ( s ) = 2 s 2 + 5 s + 1 s 2 + 2 s + 3 ....
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Bode - ∠ G jω at several frequencies jω-10 1 θ 2 θ 1...

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