How to use the Root Locus Plotter
- 1. Enter N(s) and D(s)For example 1 and s(s+1)(s+5), without K.
- 2. Add K or ζ (optional)Check a gain, or find the K that gives a damping ratio.
- 3. PlotSee the locus, all the rule results and the stable range of K.
Root locus rules
The root locus shows how the closed-loop poles, the roots of D(s) + K·N(s) = 0, move as K goes from 0 to ∞. The branches start at the open-loop poles and end at the zeros or at infinity. Points on the real axis belong to the locus if an odd number of real poles and zeros lie to their right. There are n − m asymptotes at angles (2q + 1)180°/(n − m), meeting at the centroid σa = (Σ poles − Σ zeros)/(n − m). Break points solve dK/ds = 0. The jω-axis crossings come from the Routh array or from putting s = jω.
Worked example: G(s)H(s) = K/(s(s+1)(s+5)). The poles are 0, −1 and −5, with three asymptotes at ±60° and 180° from σa = −2. Solving 3s² + 12s + 5 = 0 gives the breakaway point s = −0.472 (K = 1.13). The locus crosses the jω axis at ±j2.236 when K = 30, so the closed loop is stable for 0 < K < 30.
Exam tips
- Only real break-point roots that lie on a real-axis segment (K > 0) are valid.
- The angle of departure from a complex pole is 180° − Σ(angles from the other poles) + Σ(angles from the zeros).
- To meet a damping ratio, draw the line θ = cos⁻¹ζ and find where it cuts the locus. The magnitude condition then gives K.
- Sketch the asymptotes lightly first. They guide the shape of the branches.
Frequently asked questions
How do I enter G(s)H(s)?
Type the numerator and denominator without K, as factors like s(s+1)(s+5) or as coefficients such as 1 6 5 0. The tool multiplies the numerator by K for you.
Is the plot exact or a sketch?
It is exact. The tool solves for the closed-loop roots at many values of K and joins them, and it also lists the hand-sketch rules so you can check your drawing.
Can it find K for a given damping ratio?
Yes. Enter ζ, for example 0.5, and it finds where the constant-ζ line meets the locus and the K needed there.
Is this tool free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. All the maths runs in your browser.