Roughly sketch the root loci for the pole-zero maps as shown in Fig. 5.51 without the aid of a computer. Show your estimates of the center and angles of the asymptotes, a rough evaluation of arrival and departure angles for complex poles and zeros, and the loci for positive values of the parameter K. Each pole-zero map is from a characteristic equation of the form b(s) 1+K = 0, a(s) where the roots of the numerator b(s) are shown as small circles o and the roots of the denominator a(s) are shown as x's on the s-plane. Note that in Fig. 5.51(c) there are two poles at the origin. (a) (b) (c) (d) (c) (1)