Today during a lecture out teacher was demonstrating Matlab's rlocus function, but before that he decided to do it on chalkboard.
Below we have a simple transfer function:
$$ \dfrac{1}{(s^2+s+k)} $$ We find roots for it using quadratic formula: $$ s_1 = \dfrac{-1 + \sqrt{1-4k}}{2} = \dfrac{-1}{2} + \dfrac{\sqrt{4k-1}}{2}i $$ $$ s_2 = \dfrac{-1 - \sqrt{1-4k}}{2} = \dfrac{-1}{2} - \dfrac{\sqrt{4k-1}}{2}i $$
Given that $k=[0;\infty)$ we can plot these roots:
As you can see, when $k = 0, s_1 = -1 $ and $ s_2 = 0$. When $ k $ goes to infinity and beyond, $s_1$ goes up and $s_2$ goes down (on imaginary axis).
Now, when we tried to do this on Matlab using rlocus function and transfer function:
$$
\dfrac{1}{(s^2+s)},
$$
it gave us this plot:
Here $s_1$ goes down and $s_2$ goes up.
So, my question is - why? Did we do something wrong with calculations, equations or is it Matlab's fault?