A root locus plot traces how the closed-loop poles of a transfer function migrate through the complex plane as a system parameter (typically gain K) varies from 0 to infinity. It is a fundamental tool in classical control theory for analyzing system stability and designing controllers. The plot reveals critical information about pole trajectories, stability boundaries, and gain margins.
// anyplot.ai
// root-locus-basic: Root Locus Plot for Control Systems
// Library: echarts 5.5.1 | JavaScript 22.22.3
// Quality: 84/100 | Created: 2026-06-18
//# anyplot-orientation: square
// anyplot.ai
// root-locus-basic: Root Locus Plot for Control Systems
// Library: echarts 5.5.1 | JavaScript 22.22.3
// Quality: 78/100 | Created: 2026-06-18
const t = window.ANYPLOT_TOKENS;
// Open-loop: G(s) = K / [s(s+1)(s+3)] — poles {0, −1, −3}, no zeros
// Char eq: s³ + 4s² + 3s + K = 0
// Real-axis locus: (−∞, −3] ∪ [−1, 0] | Breakaway: s ≈ −0.4514 at K ≈ 0.631
// jω crossing: s = ±j√3 at K = 12 | Asymptotes: 60°, 180°, 300° from (−4/3, 0)
// Parametric formulas for complex conjugate pair given real root σ_C (Vieta):
// σ_AB = (−4 − σ_C)/2, ω² = 3 − σ_AB² − 2·σ_C·σ_AB
function cplxPair(sigC) {
const sig = (-4 - sigC) / 2;
const omegaSq = 3 - sig * sig - 2 * sigC * sig;
return { sig, omega: Math.sqrt(Math.max(0, omegaSq)) };
}
const BKWY = -0.4514; // breakaway real coordinate
const SIG_C0 = -3.0972; // real root (branch C) at start of complex phase
// Locus path arrays: [real, imag] pairs
const bA = [], bB = [], bC = [];
// Phase 1: real-axis convergence toward breakaway (K: 0 → 0.631)
const N1 = 60;
for (let i = 0; i <= N1; i++) {
const f = i / N1;
bA.push([BKWY * f, 0]); // pole 0 → breakaway
bB.push([-1 + 0.5486 * f, 0]); // pole −1 → breakaway
bC.push([-3 + (SIG_C0 + 3) * f, 0]); // pole −3 → SIG_C0
}
// Phase 2: complex locus (K > 0.631), σ_C from SIG_C0 to −6
const N2 = 200;
for (let i = 1; i <= N2; i++) {
const f = i / N2;
const sigC = SIG_C0 + (-6 - SIG_C0) * f;
const { sig, omega } = cplxPair(sigC);
bA.push([sig, omega]);
bB.push([sig, -omega]);
bC.push([sigC, 0]);
}
// Constant damping-ratio reference lines: ζ = 0.5 (angle 60° from −Re axis)
// Direction: (cos 120°, ±sin 120°) = (−0.5, ±√3/2)
// Exits at y = ±4 when r = 4/(√3/2) ≈ 4.619, x = −4.619×0.5 ≈ −2.309
const zeta05up = [[0, 0], [-2.309, 4]];
const zeta05dn = [[0, 0], [-2.309, -4]];
// Natural frequency reference circle: ωn = 2 (full circle, r = 2)
const wnCircle = [];
for (let i = 0; i <= 360; i++) {
const theta = (i / 180) * Math.PI;
wnCircle.push([2 * Math.cos(theta), 2 * Math.sin(theta)]);
}
// Key markers
const poles = [[0, 0], [-1, 0], [-3, 0]];
const jwCross = [[0, Math.sqrt(3)], [0, -Math.sqrt(3)]]; // ±j√3, K=12
// Gain-direction arrows: symbolRotate = atan2(dx, dy) maps chart tangent to
// ECharts clockwise-from-up convention (right→90°, up→0°, left→-90°, down→180°)
function arrowAt(path, idx) {
const i = Math.max(1, Math.min(path.length - 2, idx));
const dx = path[i + 1][0] - path[i - 1][0];
const dy = path[i + 1][1] - path[i - 1][1];
return { value: path[i], symbolRotate: Math.atan2(dx, dy) * 180 / Math.PI };
}
const gainArrows = [
arrowAt(bA, Math.floor(N1 * 0.70)), // phase 1, bA: leftward (K↑ toward breakaway)
arrowAt(bB, Math.floor(N1 * 0.70)), // phase 1, bB: rightward (K↑ toward breakaway)
arrowAt(bA, N1 + Math.floor(N2 * 0.45)), // phase 2, bA: upper-right (K↑ along complex branch)
arrowAt(bB, N1 + Math.floor(N2 * 0.45)), // phase 2, bB: lower-right (K↑ along complex branch)
arrowAt(bC, N1 + Math.floor(N2 * 0.55)), // phase 2, bC: leftward (K↑ along real branch)
];
// ── Chart ─────────────────────────────────────────────────────────────────────
const chart = echarts.init(document.getElementById("container"));
chart.setOption({
animation: false,
color: t.palette,
backgroundColor: "transparent",
title: {
text: "root-locus-basic · javascript · echarts · anyplot.ai",
left: "center",
top: 18,
textStyle: { color: t.ink, fontSize: 22, fontWeight: "bold" },
},
legend: {
bottom: 14,
itemGap: 32,
textStyle: { color: t.inkSoft, fontSize: 13 },
},
// Square grid (2200×2200 within 2400×2400 canvas) for equal axis scaling:
// x range 8 units and y range 8 units → 275 px/unit on each axis
grid: { left: 120, right: 80, top: 80, bottom: 120 },
xAxis: {
type: "value",
name: "Real Axis",
nameLocation: "middle",
nameGap: 36,
nameTextStyle: { color: t.inkSoft, fontSize: 14 },
min: -6,
max: 2,
interval: 1,
axisLabel: { color: t.inkSoft, fontSize: 12 },
axisLine: { show: true, lineStyle: { color: t.inkSoft } },
axisTick: { show: true, lineStyle: { color: t.inkSoft } },
splitLine: { lineStyle: { color: t.grid, width: 1 } },
},
yAxis: {
type: "value",
name: "Imaginary Axis",
nameLocation: "middle",
nameGap: 56,
nameTextStyle: { color: t.inkSoft, fontSize: 14 },
min: -4,
max: 4,
interval: 1,
axisLabel: { color: t.inkSoft, fontSize: 12 },
axisLine: { show: true, lineStyle: { color: t.inkSoft } },
axisTick: { show: true, lineStyle: { color: t.inkSoft } },
splitLine: { lineStyle: { color: t.grid, width: 1 } },
},
series: [
// ζ = 0.5 damping-ratio reference lines (subtle dashed, excluded from legend)
{
type: "line",
data: zeta05up,
showSymbol: false,
lineStyle: { color: t.grid, width: 1.2, type: "dashed" },
silent: true,
legendHoverLink: false,
},
{
type: "line",
data: zeta05dn,
showSymbol: false,
lineStyle: { color: t.grid, width: 1.2, type: "dashed" },
silent: true,
legendHoverLink: false,
},
// ωn = 2 natural frequency reference circle (now renders as true circle)
{
type: "line",
data: wnCircle,
showSymbol: false,
lineStyle: { color: t.grid, width: 1.2, type: "dashed" },
silent: true,
legendHoverLink: false,
},
// Root locus branches (A upper, B lower, C real — share one legend entry)
{
name: "Root Locus",
type: "line",
data: bA,
showSymbol: false,
lineStyle: { color: t.palette[0], width: 2.5 },
},
{
name: "Root Locus",
type: "line",
data: bB,
showSymbol: false,
lineStyle: { color: t.palette[0], width: 2.5 },
},
{
name: "Root Locus",
type: "line",
data: bC,
showSymbol: false,
lineStyle: { color: t.palette[0], width: 2.5 },
},
// Gain-direction arrows (excluded from legend — decorative overlay)
{
type: "scatter",
data: gainArrows,
symbol: "arrow",
symbolSize: 14,
itemStyle: { color: t.palette[0] },
silent: true,
legendHoverLink: false,
},
// Open-loop poles (× markers via rotated cross shape)
{
name: "Open-Loop Poles",
type: "scatter",
data: poles,
symbol:
"path://M-1,-4 L1,-4 L1,-1 L4,-1 L4,1 L1,1 L1,4 L-1,4 L-1,1 L-4,1 L-4,-1 L-1,-1 Z",
symbolSize: 20,
symbolRotate: 45,
itemStyle: { color: t.palette[4] },
},
// Stability boundary crossings ±j√3 at K = 12
{
name: "jω Crossings (K=12)",
type: "scatter",
data: jwCross,
symbol: "diamond",
symbolSize: 18,
itemStyle: { color: t.palette[1] },
},
],
});
Part of Root Locus Plot for Control Systems on anyplot.ai.