The root locus
Magnitude and angle conditions, sketching rules, asymptotes and breakaway points, a complete three-pole example, choosing the gain and the effect of an added zero.
As the gain of a proportional controller is increased, the closed-loop poles move in the complex plane: they start from the plant poles and, beyond a certain value, may cross the imaginary axis. The root locus is the drawing of these paths. It is sketched with a few rules, without solving high-degree equations, and shows at a glance which behaviours a simple gain can achieve and which need a more elaborate controller.
The definition
Let 
, with 
of degree 
(zeros 
) and 
of degree 
(poles 
). The closed-loop poles are the roots of
![\[ D(s) + K\,N(s) = 0 \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-2d58b18a9a05ff5ce0062226fd282e76_l3.png "Rendered by QuickLaTeX.com")
The root locus is the set of these roots as 
goes from 0 to 
.
The magnitude and angle conditions. A point 
belongs to the locus if and only if 
, that is
![\[ \sum_i \arg(s - z_i) - \sum_i \arg(s - p_i) = 180° + k\cdot 360°,\qquad K = \frac{\prod|s - p_i|}{\prod|s - z_i|} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-ca1eb6c80024483e9d5bd657bbefa794_l3.png "Rendered by QuickLaTeX.com")
The angle condition decides where the locus lies, the magnitude condition which gain corresponds to each point.
Sketching rules
- The locus has
branches, symmetric about the real axis. - The branches start at the poles of
(
) and end at the zeros (
); 
branches go to infinity. - On the real axis, the points belonging to the locus are those with an odd number of real poles and zeros to their right (from the angle condition: each pole or zero to the right contributes 180°, those to the left 0°, complex pairs cancel).
- Branches going to infinity follow asymptotes with angles
, starting from the centroid
![\[ \sigma_a = \frac{\sum p_i - \sum z_i}{n - m} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-c313b246e18e88f8bb4887aad8c1698e_l3.png "Rendered by QuickLaTeX.com")
- Two branches meeting on the real axis break away at the points where
(breakaway points). - Crossings of the imaginary axis are found with the Routh criterion.
A complete example
![\[ L(s) = \frac{K}{s(s + 2)(s + 4)} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-a5acb42c5a761086d1b1d5035b511915_l3.png "Rendered by QuickLaTeX.com")
Three poles (0, −2, −4), no zeros.
- Real axis: the segment between 0 and −2 (one pole to the right) and everything to the left of −4 (three poles) belong to the locus.
- Asymptotes:
, angles 60°, 180°, 300°, centroid 
. - Breakaway:
, 
gives 
(the other root, −3.15, is not on the locus). There 
. - Imaginary axis: from the lesson on stability,
and 
.
Reading the plot. For 
the poles are real: a response without oscillation but slow. Between 3.08 and 48 the dominant pair is complex and less and less damped. For damping 
(16% overshoot) the poles must lie on the line at 60° from the negative real axis: the locus meets it at 
, with poles at 
; the third pole is at 
, seven times further away, and the pair is truly dominant. The settling time is then about 
s, and it cannot be reduced by changing the gain alone: the real part of the complex poles never goes further left than −2, the centroid of the asymptotes.
Reshaping the locus
This is where the method is useful in design. Adding a zero to the controller attracts the branches and moves them to the left: with a zero at −2.5, for example, the asymptotes become two at ±90° with centroid 
, and the loop stays stable for any gain. This is the effect of derivative action and of lead networks. Adding a pole (like the integrator) pushes the branches to the right and reduces stability. The root locus makes this trade-off visible, which is why, even in the age of simulation software, it remains the tool for reasoning about design.
