📐 Root Locus Explorer

About this simulation

The root locus is a graphical method, introduced by Walter R. Evans in 1948, for showing how the poles of a closed-loop control system move through the complex plane as a single parameter — usually the loop gain K — is swept from zero to infinity. Because a linear system is stable only when every closed-loop pole has a negative real part (sits in the left half of the complex plane), the root locus is one of the fastest ways to see whether turning up feedback gain will make a system faster — or send it unstable.

How it works

• The open-loop poles (✕) and zeros (○) are fixed by the plant and controller; the characteristic equation is 1 + K·G(s)H(s) = 0.
• As K rises from 0, each branch starts at an open-loop pole. As K → ∞, branches end at a finite open-loop zero or run off to infinity along straight asymptotes.
• The imaginary axis (highlighted) is the stability boundary: the right half-plane is tinted red because any pole crossing into it makes the system unstable.
• Real poles moving toward each other collide and "break away" into a complex-conjugate pair — the classic root-locus signature of an underdamped response appearing as gain increases.

Controls

• System — pick an open-loop pole/zero configuration to explore different qualitative behaviours.
• Loop gain K — slides the closed-loop poles along the pre-computed locus branches live.
• Zero position — (complex-pair preset) repositions the compensator zero and re-traces the locus.
• Show asymptotes — toggles the dashed guide lines that branches approach as K → ∞.
• Auto-sweep gain — animates K from 0 up and back so you can watch the branches move.

Did you know?

Root locus design predates digital computers by decades — engineers originally sketched loci by hand using angle and magnitude conditions, and the method is still taught today because it makes the trade-off between speed and stability directly visible, something a single time-domain simulation cannot show.