Question

What is the nature of locus of the equation

Answer 1

Root Locus and Characteristic Equation

Root Locus on Real Axis

Asymptotes for Infinite Zeros

What You Need to Know (and Don't Need to Know) about Root Locus

In the last class we saw a way to generate the root locus for any system. The roots of the characteristic equation of a system are the system's closed-loop poles. They determine the system response. If we let a parameter in the characteristic equation vary (usually K), we can make a table of closed-loop root locations as a function of K. If we let K vary from 0 to infinity, the root locations trace out the root locus. The path these roots take is called the root locus.

Any system's root locus can be generated by this brute-force method. There is a better, faster way that gives you a rough idea of the shape of the root locus with only a few, easy calculations.