Plot root locus for control systems. Adjust gain and see pole migration. Identify breakaway points and stability boundaries.
Root locus answers one question: where do the closed-loop poles of a feedback system travel as the loop gain K grows from zero to infinity? Enter the open-loop poles and zeros, and the tool draws those paths across the complex plane, labels gain along each branch, and flags where branches meet and break away and where they cross the imaginary axis, which is the exact gain where the loop goes unstable. The surprise for most users: the plot describes the closed-loop system, so a plant that is stable on its own can still show branches heading into the right half-plane.
A concrete input and expected output from the current implementation.
Input
Open-loop poles: 0, -1, -2. Zeros: none. Transfer function G(s)H(s) = K / (s(s+1)(s+2)).
Expected output
Three branches start at the open-loop poles 0, -1, -2. The real axis carries the locus from -1 to 0 and from -2 to -infinity. Branches meet and break away at s = -0.42 with K = 0.385, then swing into the complex plane along asymptotes from s = -1 at +-60 degrees. The branches cross the imaginary axis at +-j1.41 exactly when K = 6; for any K greater than 6 two poles sit in the right half-plane and the loop is unstable.
With three poles and no zeros the closed-loop characteristic equation is s^3 + 3s^2 + 2s + K = 0. Routh's criterion puts the stability limit at K = 6, and substituting s = j*sqrt(2) verifies the crossing point; the breakaway solves dK/ds = 0 with K = -s(s+1)(s+2), yielding s = -0.42 at K = 0.385.
A point s on the plane lies on the locus when the open-loop transfer function evaluated there has phase exactly 180 degrees (modulo 360), because 1 + G(s)H(s) = 0 requires G(s)H(s) = -1. The tool evaluates this condition across the plane and also computes the gain at each point as K = |D(s)| / |N(s)|, which is why every branch carries a K label.
Breakaway and break-in points solve dK/ds = 0 along the real axis, where two branches merge. Imaginary-axis crossings come from substituting s = j*omega into the closed-loop characteristic equation, and the largest K below the first crossing is the stability limit. Asymptote angles follow from the pole-zero excess: (2q+1)*180 degrees divided by (n - m), anchored at the centroid (sum of poles - sum of zeros)/(n - m).
Only if it never crosses back. Read the locus across the full gain range: a branch that re-enters the left half-plane makes the stable range an interval between two crossings, not a single upper limit. Exactly at the crossing gain the pole sits on the imaginary axis, which gives a sustained oscillation (marginal stability), not a stable response.
Indirectly, through the dominant pole pair. At your chosen K, measure the angle alpha the complex pole makes with the negative real axis, and zeta = cos(alpha); overshoot is exp(-pi*zeta/sqrt(1 - zeta^2)). This is reliable only when the remaining poles and zeros sit well to the left, so they do not shape the response.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
Calculate percentages, percentage change, and what percent one number is of another with step-by-step breakdowns.
Calculate tips and split bills between multiple people with customizable tip percentages and rounding options.
Calculate monthly payments, total interest, and generate amortization schedules for loans and mortgages.
Calculate Body Mass Index from height and weight with category classification and healthy range guidance.
Calculate sale prices, savings amounts, and final costs after single or stacked percentage discounts.
Compute mean, median, mode, standard deviation, variance, and other descriptive statistics from a data set.