2.6 Complex numbers: numerical experiment (quadratic roots)

There is a version of the quadratic equation that will arise over and over again in your study of electrical and mechanical systems:

For reasons that can only become clear as you continue your study of engineering, the parameter ${\omega }_0$ is called a resonant frequency , and the parameter $\xi \ge 0$ is called a damping factor . In this experiment, you will begin by

1. finding the “underdamped” range of values $\xi \ge 0$ for which the roots s ₁ and s ₂ are complex;
2. finding the “critically damped” value of $\xi \ge 0$ that makes the roots s ₁ and s ₂ equal; and
3. finding the “overdamped” range of values $\xi \ge 0$ for which s ₁ and s ₂ are real.
4. For each of these ranges, find the analytical solution for $s_{1,2}$ as a function of ω ₀ and ξ ; write your solutions in Cartesian and polar forms and present your resultsas
   $$s_{1,2}=,0\le \xi \le {\xi }_c,\xi ={\xi }_c,\xi \ge {\xi }_c$$
   where ξ _c is the critically damped value of ξ . Write a MATLAB program that computes and plots $s_{1,2}$ for ω ₀ fixed at ${\omega }_0=1$ and ξ variable between 0.0 and 2.0 insteps of 0.1. Interpret all of your findings.

Now organize the coefficients of the polynomial $s^2+2\xi s+1$ into the array $\left[12\xi 1\right]$ . Imbed the MATLAB instructions

in aloop to compute and plot the roots of $s^2+2\xi s+1$ as $\xi$ ranges from 0.0 to 2.0. Note that $r$ is a $1\times 2$ array of complex numbers. You should observe [link] . We call this “half circle and line” the locus of roots for the quadratic equation or the “root locus” in shorthand. for