chapter 5-2.pptVIP

Chapter 5 Frequency Response Methods Stability in the Frequency Domain Introduction Introduction Introduction Mapping contours in the s-plane Mapping contours in the s-plane Mapping contours in the s-plane Mapping contours in the s-plane Cauchy’s theorem (principle of the argument) The Nyquist Criterion The Nyquist Criterion The Nyquist Criterion The Nyquist Criterion The Nyquist Criterion The Nyquist Criterion The Nyquist Criterion Example 1 Example 3 Example 4 Consider a single-loop control system, whose open-loop transfer function is Two problems Two conclusions from the above example The Nyquist Criterion Example 5 Example 5 * * * * * In previous chapters, we discussed stability and developed various tools to determine stability. For a control system, it is necessary to determine whether the system is stable. To determine the stability of a closed-loop system, we must investigate the characteristic equation of the system. For multi-loop system For single-loop system Characteristic Equation To ensure stability we must ascertain that all the zeros of F(s) lie in the left-hand s-plane. A frequency domain stability criterion was developed by H.Nyquist in 1932. The Nyquist stability criterion is based on a theorem in the theory of the function of a complex variable due to Cauchy. Cauchy’s theorem is concerned with mapping contours in the complex s-plane. H. Nyquist A contour map is a contour or trajectory in one plane mapped into another plane by a relation F(s). s-plane F(s)-plane a b a b c c d d If a contour in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and traversal is in the clockwise direction along the contour, the corresponding contour in the F(s)-plane encircles the origin of the F(s)-plane N times in the counterclockwise direction, where N=P-Z Suppose: then The poles of the open-loop transfer function G(s)H(s) are the pol