Learning how to draw a root locus is an essential skill for control systems engineers who need to visualize how the closed‑loop poles of a feedback system move as a gain parameter varies. This graphical technique provides insight into stability, transient response, and the effect of controller tuning, making it a cornerstone of classical control theory. By mastering the step‑by‑step procedure and the underlying theory, you can quickly assess design trade‑offs and predict system behavior without solving complex characteristic equations for every gain value Worth knowing..
Introduction
The root locus method, introduced by W. R. Evans in 1948, plots the trajectories of the poles of the closed‑loop transfer function
[ T(s)=\frac{G(s)H(s)}{1+K,G(s)H(s)} ]
as the scalar gain (K) sweeps from zero to infinity. Each point on the locus satisfies the angle condition
[ \angle G(s)H(s) = (2q+1)180^\circ ,\qquad q=0,1,2,\dots ]
and the magnitude condition
[ |K,G(s)H(s)| = 1 . ]
Understanding these conditions lets you sketch the locus by hand, which is invaluable for quick design checks and for teaching fundamental concepts It's one of those things that adds up. Took long enough..
Steps to Draw a Root Locus
Below is a practical, numbered procedure that you can follow for any open‑loop transfer function (G(s)H(s)). Each step builds on the previous one, ensuring a clear and accurate sketch.
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Identify poles and zeros
- Factor the numerator and denominator of (G(s)H(s)).
- Mark poles with an × and zeros with a ○ on the real‑imaginary plane.
- Example: For (G(s)H(s)=\frac{K(s+2)}{(s+1)(s+3)(s+5)}), poles are at (-1,-3,-5) and a zero at (-2).
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Determine the number of branches
- The number of separate loci equals the number of poles (n).
- If there are (m) zeros, (n-m) branches will go to infinity.
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Locate the real‑axis segments
- A point on the real axis belongs to the root locus if the total number of poles and zeros to its right is odd.
- Shade these segments; they indicate where the locus can exist on the real axis.
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Compute asymptotes (for branches heading to infinity)
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The asymptote angles are given by
[ \theta_a = \frac{(2q+1)180^\circ}{n-m},\quad q=0,1,\dots,n-m-1 . ]
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The centroid (intersection of asymptotes with the real axis) is
[ \sigma_a = \frac{\sum \text{poles} - \sum \text{zeros}}{n-m}. ]
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Draw straight lines emanating from the centroid at the calculated angles.
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Find breakaway and break‑in points
- Solve (\frac{d}{ds}[G(s)H(s)] = 0) for real (s) that lie on the root locus.
- These points indicate where multiple branches meet and then diverge (breakaway) or where separate branches converge and then split (break‑in).
- Verify that the gain (K) at these points is positive.
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Calculate angles of departure and arrival
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For a complex pole, the angle of departure (\phi_d) satisfies
[ \phi_d = 180^\circ - \sum \angle(\text{pole to other poles}) + \sum \angle(\text{pole to zeros}) . ]
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For a complex zero, compute the angle of arrival (\phi_a) analogously.
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These angles tell you how the locus leaves a pole or approaches a zero That's the part that actually makes a difference..
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Determine the imaginary‑axis crossings (optional but useful)
- Use the Routh‑Hurwitz criterion on the characteristic equation (1+K,G(s)H(s)=0) to find the gain (K_{cr}) at which the locus crosses the imaginary axis.
- The crossing frequency (\omega_{cr}) gives insight into marginal stability.
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Sketch the locus
- Combine all the information: plot poles/zeros, shade real‑axis segments, draw asymptotes, mark breakaway/break‑in points, indicate departure/arrival angles, and, if needed, mark imaginary‑axis crossings.
- Draw smooth curves that follow the established rules, ensuring symmetry about the real axis.
Following these steps will produce an accurate root locus sketch for most rational transfer functions encountered in undergraduate control courses.
Scientific Explanation
The root locus is rooted in the argument principle of complex analysis. The characteristic equation
[ 1+K,G(s)H(s)=0 ]
can be rewritten as
[ G(s)H(s) = -\frac{1}{K}. ]
Since (-\frac{1}{K}) is a negative real number, its angle is always an odd multiple of (180^\circ). So hence, any point (s) that satisfies the angle condition will make the left‑hand side a negative real number for some positive gain (K). The magnitude condition then determines the exact value of (K) that places the closed‑loop pole at that location Surprisingly effective..
As (K) increases from zero, the poles start at the open‑loop pole locations (when (K=0), the characteristic equation reduces to the denominator of (G(s)H(s)=0)). As (K\to\infty), the poles approach the open‑loop zeros (or go to infinity if there are fewer zeros than poles). The locus thus provides a continuous map of pole migration
