Final answer:
The question involves sketching a root locus for a given characteristic equation in a control systems context. A root locus displays how poles of the closed-loop transfer function move in the complex plane as gain K varies. The plot starts at open-loop poles and ends at open-loop zeros while following certain rules.
Step-by-step explanation:
The student has asked for a sketch of the root locus for a closed loop feedback system based on the given characteristic equation. The characteristic equation is s³+2s²+(20K+7)s+100K= 0. The root locus technique is a graphical method used in control systems to determine the stability of the system as the gain, K, varies from 0 to infinity. Unfortunately, without the ability to provide a graphical sketch here, I can instead explain that the root locus plot will show how the roots of the characteristic equation, or the poles of the closed-loop transfer function, move in the complex plane as K increases. To construct a root locus, one would typically use rules that consider angles of departure, the number of poles and zeros, and asymptotes as K approaches infinity. The root locus would begin at the poles of the open-loop transfer function (given by the characteristic equation when K=0) and end at the zeros of the open-loop transfer function as K approaches infinity.