The Use of Zeros and Poles for Frequency Response or Transient Response

Abstract

Abstract The characteristic equation of a control system is often of the form 1/KG(s) + 1 = 0, in which G(s) is (s − q1) (s − q2)… /(s − p1) (s − p2)… Each factor of G(s), such as (s − p1), is a complex variable which can be plotted as a vector in the s-plane from the pole p1 to the point s. The complex product G(s) is itself a vector whose magnitude is the product of magnitudes and whose angle is the sum of the angles of its factors. Frequency response is computed directly for s points along the jω-axis. In order to find the roots of 1/KG(s) = −1, the locus of s is sketched, where for each point on this locus the total angle of G(s) is equal to 180 deg; a point on this locus is then a root if K = 1/|G(s)|. For given initial conditions the Laplace transform permits the transient response of the system to be written directly.