Handout 2: Gain and Phase margins Eric Feron Feb 6, 2004 Nyquist plots and Cauchy’s principle s 2 + s + 1 Let H(s) be a transfer function. eg H (s) = (s + 1)(s + 3) Evaluate H on a contour in the s-plane. (your plots here) 1 s 2 + s + 1 H = (s + 3)(s ? 3) Evaluate H on another contour of the s-plane (your plots here) 2 Cauchy’s Principle: Control application: Given KG(s), we encircle the entire to get the contour evaluation of Closed-loop roots are poles of They are zeros of If there are no RHPs, then 1 + KG encirclement of 0 means With no RHP poles, KG encirclement of -1 means 3 With right half plane open-loop poles A clockwise contour enclosing a zero of 1 + KG(s) will result in A clockwise contour enclosing a pole of 1 + KG(s) will result in Nyquist plot rules 1. Plot KG(s) for s = ?j∞ to +j∞ 2. Count number of 3. Determine number of 4. Nunber of unstable closed-loop roots is 4 1 Example: G(s) = s 2 + 3s + 1 Bode plot Nyquist plot 5 1 Example: G(s) = s(s + 1) 2 Bode plot Nyquist plot 6 Gain and Phase margins Nyquist plot for G(s). Gain Margin is Phase Margin is 7 1 G(s) = s 2 + 3s + 1 1 G(s) = 2 s(s + 1) 8