step by step process of nyquist plot and its stability
Answers
Step 1 - Check for the poles of G(s) H(s) of jω axis including that at origin.
Step 2 - Select the proper Nyquist contour – a) Include the entire right half of s-plane by drawing a semicircle of radius R with R tends to infinity.
Step 3 - Identify the various segments on the contour with reference to Nyquist path
Step 4 - Perform the mapping segment by segment substituting the equation for respective segment in the mapping function. Basically we have to sketch the polar plots of the respective segment.
Step 5 - Mapping of the segments are usually mirror images of mapping of respective path of +ve imaginary axis.
Step 6 - The semicircular path which covers the right half of s plane generally maps into a point in G(s) H(s) plane.
Step 7- Interconnect all the mapping of different segments to yield the required Nyquist diagram.
Step 8 - Note the number of clockwise encirclement about (-1, 0) and decide stability by N = Z – Pis the Open loop transfer function (O.L.T.F)is the Closed loop transfer function (C.L.T.F)
N(s) = 0 is the open loop zero and D(s) is the open loop pole From stability point of view no closed loop poles should lie in the RH side of s-plane . Characteristics equation 1 + G(s) H(s) = 0 denotes closed loop poles .Now as 1+ G(s) H(s) = 0 hence q(s) should also be zero.Therefore , from the stability point of view zeroes of q(s) should not lie in RHP of s-plane. To define the stability entire RHP (Right Hand Plane) is considered. We assume a semicircle which encloses all points in the RHP by considering the radius of the semicircle R tends to infinity. [R → ∞]. The first step to understand the application of Nyquist criterion in relation to determination of stability of control systems is mapping from s-plane to G(s) H(s) - plane. s is considered as independent complex variable and corresponding value of G(s) H(s) being the dependent variable plotted in another complex plane called G(s) H(s) - plane. Thus for every point in s-plane there exists a corresponding point in G(s) H(s) - plane. During the process of mapping the independent variable s is varied along a specified path in s - plane and the corresponding points in G(s)H(s) plane are joined. This completes the process of mapping from s-plane to G(s)H(s) - plane. Nyquist stability criterion says that N = Z - P. Where, N is the total no. of encirclement about the origin, P is the total no. of poles and Z is the total no. of zeroes. Case 1:- N = 0 (no encirclement), so Z = P = 0 & Z = P If N = 0, P must be zero therefore system is stable. Case 2:- N > 0 (clockwise encirclement), so P = 0 , Z ≠0 & Z > P For both cases system is unstable. Case 3 :- N < 0 (counterclockwise encirclement), so Z = 0, P ≠0 & P > Z System is stable.