Calculator · Dynamics & Control
Root Locus Calculator
Enter G(s) = N(s)/D(s) in descending powers of s. The locus shows where the closed-loop poles of the unity negative-feedback system travel as the scalar gain K grows from zero, and the table reports the closed-loop poles at the gain you enter.
Input
Result
| Closed-loop verdict at this K | Asymptotically stable |
|---|---|
| Closed-loop poles | -1, -1 |
| Stable range of K | 0 ≤ K (and above, within the range examined) |
| Imaginary-axis crossings | None found |
Root locus plot
- ✕ open-loop poles (2)
- ■ closed-loop poles at K = 1
- — locus branches
- ▬ real-axis locus
- ┄ asymptotes
- ✛ centroid σ_a = -1
- ◌ landmarks too close to separate on screen — the ring lists them all
- ┆ stability boundary Re(s) = 0; shaded side is unstable
Analytical properties
| Branches (n) | 2 |
|---|---|
| Finite zeros (m) | 0 |
| Branches to infinity (n − m) | 2 |
| Asymptote centroid σ_a | -1 |
| Asymptote angles | 90°, 270° |
Open-loop poles and zeros
- × pole s = 0
- × pole s = -2
- No finite zeros.
Real-axis segments
- -2 ≤ Re(s) ≤ 0
Breakaway / break-in points (1)
- ◇ s = -1 · breakaway at K = 1
Technical reference
1 + K·G(s) = 0 ⇔ D(s) + K·N(s) = 0
Angle condition: ∠G(s) = (2l + 1)·180° · Magnitude condition: K = 1/|G(s)|
- G(s)
- open-loop transfer function N(s)/D(s)
- K
- scalar loop gain[–]
- n, m
- degree of D(s) and number of finite zeros of G(s)
A point s lies on the locus exactly when the angle condition holds; the gain that places a closed-loop pole there then follows from the magnitude condition. Each branch starts at an open-loop pole (K = 0) and ends at a finite open-loop zero or, for n − m branches, at infinity along an asymptote as K grows.
An imaginary-axis crossing marks a stability boundary. Determine stability on each gain interval between crossings; as K varies, the system can lose or regain stability. The stable intervals reported above come from evaluating the closed-loop poles over those intervals.
Continue
- · Continuous-time, single-input single-output, linear time-invariant system with real coefficients.
- · Unity negative feedback and a positive scalar gain: K ≥ 0. The complementary K < 0 locus is not drawn.
- · Branches are traced from numerically computed roots of D(s) + K·N(s); very high gains and near-repeated roots can be sensitive.
- · No approximate pole–zero cancellation is performed: a pole and a zero that nearly coincide are reported, not removed.