Root Locus Construction Rules

Sketching the root locus by hand requires a set of systematic construction rules that define its key geometric features. These root locus construction rules let you quickly determine the number of branches, their starting and ending points, real-axis segments, asymptote directions, and centroid without solving the full characteristic equation. For GATE and university exams, these rules are among the most frequently tested topics in control systems.

Each rule follows directly from the angle and magnitude conditions of the root locus definition. Understanding why each rule holds, not just memorizing the formula, is what allows you to apply them correctly to unfamiliar transfer functions under exam conditions.

Core Concept Explanation

Rule 1 states that the number of branches of the root locus equals n, the number of open-loop poles. This is because the characteristic equation is of degree n, so there are exactly n roots (closed-loop poles) for every value of K, and each traces out one branch.

Rule 2 establishes start and end points: every branch begins at an open-loop pole (where K = 0) and ends either at a finite open-loop zero (where K approaches infinity) or at infinity. When there are more poles than zeros (n > m), exactly n - m branches go to infinity along asymptotic directions.

Rule 3 identifies which portions of the real axis belong to the root locus. A point on the real axis lies on the locus if and only if the total number of real open-loop poles and zeros to its right is odd. This follows directly from the angle condition: each complex pole-zero pair contributes zero net angle, so only real singularities affect the angle sum.

Rule 4 defines asymptotes: the n - m branches heading to infinity travel along straight lines called asymptotes. The angles of these asymptotes are given by (2q+1) x 180 / (n - m) degrees for q = 0, 1, 2, ..., (n-m-1).

Rule 5 defines the centroid (also called the asymptote intersection point), denoted sigma_a. It is calculated as: sigma_a = (sum of real parts of all OL poles - sum of real parts of all OL zeros) / (n - m). All asymptotes meet at this point on the real axis.

Mathematical Expression

The asymptote angles and centroid are the two most formula-heavy rules. For a system with n poles and m zeros where n > m, the asymptote angles are phi_k = (2k+1) x 180 / (n-m) for k = 0, 1, ..., (n-m-1). The centroid formula is sigma_a = [sum(pole real parts) - sum(zero real parts)] / (n-m). Both formulas appear frequently in GATE numerical questions.

Practical Understanding

In practice, the real-axis rule combined with the asymptote rule gives you the skeleton of the root locus in under a minute. You can determine whether branches will cross into the RHP (indicating possible instability as K increases) just by observing asymptote angles. For example, if asymptote angles include 90 degrees (vertical asymptotes), branches may cross the imaginary axis, warning you of a finite critical gain.

Solved Numerical Example

For G(s) = K / [s(s+2)(s+4)], find the asymptote angles and centroid. Here n = 3 poles (at 0, -2, -4) and m = 0 zeros, so n - m = 3 branches go to infinity. Apply the asymptote angle formula and centroid formula.

Given:
G(s) = K / [s(s+2)(s+4)], n=3, m=0
Poles at: 0, -2, -4
No finite zeros

Why this formula applies:
Asymptote angles because n-m = 3 branches go to infinity
Centroid needed to locate asymptote intersection

Formula:
Angle phi_k = (2k+1) x 180 / (n-m)
Centroid sigma_a = (sum of poles - sum of zeros) / (n-m)

Substitution:
For k=0: phi_0 = (1 x 180) / 3 = 60 degrees
For k=1: phi_1 = (3 x 180) / 3 = 180 degrees
For k=2: phi_2 = (5 x 180) / 3 = 300 degrees
Centroid: sigma_a = (0 + (-2) + (-4) - 0) / 3 = -6/3

Calculation:
Angles: 60 deg, 180 deg, 300 deg
Centroid: -6/3 = -2

Final Answer:
Asymptote angles: 60 deg, 180 deg, 300 deg; Centroid at sigma = -2 on real axis

Exam Tip: For GATE, always check the number of asymptotes first: it is n - m. A common trap is using n instead of n - m. Also remember: all asymptotes radiate from the same centroid point on the real axis, not the origin.

Mechanism: Applying the Rules Step by Step

• Step 1: Count n (poles) and m (zeros). Determine n - m branches going to infinity.
• Step 2: Mark all open-loop poles (branch starts) and zeros (branch ends) on s-plane.
• Step 3: Apply real-axis rule: segments to the left of an odd count of real poles/zeros belong to the locus.
• Step 4: Compute asymptote angles using (2k+1) x 180 / (n-m) and centroid using pole-zero sum formula.
• Step 5: Find breakaway or break-in points on real axis where dK/ds = 0 (covered in detail in next article).
• Step 6: Check imaginary axis crossing using Routh-Hurwitz on the characteristic polynomial to find critical K.

Quick Revision

• Number of branches = n (open-loop poles). Branches start at OL poles (K=0), end at OL zeros or infinity.
• Real-axis rule: point on real axis belongs to locus if odd number of real OL poles and zeros lie to its right.
• Number of asymptotes = n - m. Angles = (2k+1) x 180 / (n-m) degrees for k = 0 to n-m-1.
• Centroid sigma_a = (sum of pole real parts - sum of zero real parts) / (n-m). All asymptotes meet here.
• Locus is symmetric about the real axis. Complex branches always appear in conjugate pairs.
• Exam trap: Centroid formula uses all poles and zeros including complex ones (via their real parts).