What Do the Roots Mean in a Root Locus?

A root locus is a plot showing how the poles of a feedback system move as a gain parameter changes. The “roots” are the values of the complex variable $s$ that satisfy the system’s closed-loop characteristic equation. Each point on a root-locus branch therefore represents a pole location at one particular gain.

This interpretation matters because pole locations help predict whether a system is stable and how its response changes. The roots are not simply the zeros or poles of the open-loop transfer function, although those open-loop poles and zeros determine where the root-locus branches begin and end.

The characteristic equation

For a standard negative-feedback system, let the forward path be $K G_0(s)$ and the feedback path be $H(s)$. Its closed-loop transfer function is

$$T(s)=\frac{K G_0(s)}{1+K G_0(s)H(s)}.$$

The characteristic equation is the denominator set equal to zero:

$$1+K G_0(s)H(s)=0.$$

The solutions $s_1,s_2,\ldots$ of this equation are the closed-loop poles for that value of $K$. A root locus collects these solutions as $K$ varies, most commonly over all nonnegative values. In the complex plane, the horizontal coordinate is the real part of $s$ and the vertical coordinate is its imaginary part.

For a rational system, the roots are usually found by clearing denominators and solving a polynomial. The roots can be real or occur in complex-conjugate pairs when the system has real coefficients.

What the roots tell you

For a continuous-time linear system, a pole at $s=\sigma+j\omega$ contributes a natural response with a factor like $e^{\sigma t}$ and, when $\omega\ne 0$, an oscillation at angular frequency $|\omega|$. Thus:

• Poles in the left half-plane ($\sigma<0$) correspond to decaying modes.
• A pole in the right half-plane ($\sigma>0$) corresponds to a growing mode and instability.
• Poles near the imaginary axis decay slowly or produce lightly damped oscillations.
• A real pole gives a non-oscillatory exponential mode; a complex pair gives an oscillatory mode.

For a standard minimal system, all closed-loop poles must be in the left half-plane for asymptotic stability. A transfer-function pole plot can fail to show internal modes that cancel from the input-output transfer function, so checking internal stability may require examining the full system model.

How to read a root locus

For the usual negative-feedback convention, define the open-loop product without the adjustable gain as $L_0(s)=G_0(s)H(s)$. The characteristic equation can be written as $1+K L_0(s)=0$. This shows that the locus consists of the possible values of $s$ that satisfy the equation for some allowed gain $K$.

When $K$ starts at zero, the branches start at the poles of $L_0(s)$. As $K$ increases, they move through the complex plane. For large gain, branches approach the zeros of $L_0(s)$; if there are more poles than zeros, the remaining branches go to infinity. The locus is a map of pole motion, not a time plot: it does not show the system’s output directly.

A practical way to use it is to choose a gain, find the closed-loop poles at that gain, and inspect their locations. You can also use the plot in reverse: identify a region with desirable pole locations and determine which gain places the poles there.

Worked example

Consider a unity negative-feedback system with forward-path transfer function

$$G(s)=\frac{K}{s(s+2)}.$$

Here $H(s)=1$, so the characteristic equation is

$$1+\frac{K}{s(s+2)}=0,$$

or, after multiplying through by $s(s+2)$,

$$s^2+2s+K=0.$$

Solving the quadratic gives

$$s=-1\pm\sqrt{1-K}.$$

For $0<K<1$, both roots are real and negative. At $K=1$, they meet at the repeated root $s=-1$. For $K>1$, they become a complex pair:

$$s=-1\pm j\sqrt{K-1}.$$

For example, at $K=4$ the poles are $s=-1\pm j\sqrt{3}$. Their real part, $-1$, means the corresponding oscillation decays with an exponential envelope proportional to $e^{-t}$; the imaginary parts give an oscillation frequency of $\sqrt{3}$ radians per unit time. As the gain increases above 1, the pair moves vertically while staying on the line with real part $-1$.

This example also shows why “the roots” means the closed-loop poles: their values depend on $K$. At the zero-gain endpoint, root-locus convention treats the branches as starting at the open-loop poles $0$ and $-2$, understood as the limiting pole locations as $K$ approaches zero.

Common mistakes

• Confusing open-loop and closed-loop poles: Open-loop poles are branch starting points; the roots at a selected gain are the closed-loop poles.
• Treating the locus as a single set of roots: Each point corresponds to a particular gain. The locus shows how the roots vary across gains.
• Assuming every point is stable: Stability depends on whether the selected gain puts all relevant poles in the left half-plane.
• Forgetting the feedback sign: The equation $1+K G_0H=0$ assumes negative feedback. With positive feedback, the characteristic equation has a different sign.
• Ignoring cancellations: A pole-zero cancellation in an input-output transfer function can hide an internal mode; a plotted transfer function may not reveal every stability issue.