📘 Root Locus – Complete Theory & Worked Examples (GATE Focus)
Root Locus is a graphical technique used to study how the poles of a closed-loop system change as system gain (K) varies from 0 to ∞. It is one of the most important topics in GATE Control Systems.
🔹 1. Basic Concept
For unity feedback system:
G(s)H(s) = K * N(s)/D(s)
Characteristic equation:
1 + G(s)H(s) = 0
Root locus shows how roots of this equation move with K.
🔹 2. Angle & Magnitude Condition
Magnitude Condition:
|G(s)H(s)| = 1
Angle Condition:
∠G(s)H(s) = (2n+1)180°
🔹 3. Basic Rules of Root Locus
- Number of branches = Number of open-loop poles
- Root locus starts at open-loop poles
- Ends at open-loop zeros
- If zeros less than poles → remaining branches go to infinity
- Root locus exists on real axis where number of poles + zeros to right is odd
🔹 4. Asymptotes
Number of asymptotes:
N − M
Angles of asymptotes:
θ = (2k+1)180° / (N−M)
Centroid:
(Σ Poles − Σ Zeros) / (N − M)
🔹 5. Worked Example 1
Given:
G(s)H(s) = K / [s(s+2)]
Open-loop poles: 0, -2 No zeros
Number of branches = 2 Asymptotes = 2
Centroid:
(-2 + 0)/2 = -1
Angles:
±90°
🔹 6. Breakaway Point
Breakaway occurs where:
dK/ds = 0
Example:
K = -s(s+2)
Differentiate and solve for s.
🔹 7. Stability Using Root Locus
- If all closed-loop poles lie in left half-plane → Stable
- If any pole crosses imaginary axis → Unstable
🔹 8. Worked Example 2
G(s)H(s) = K / [s(s+1)(s+3)]
Open-loop poles: 0, -1, -3 No zeros Branches = 3 Asymptotes = 3
Angles:
60°, 180°, 300°
🎯 GATE Important Points
- Centroid formula frequently asked
- Real-axis rule very important
- Breakaway calculation common
- Imaginary axis crossing may be tested
Root Locus Visually Explains Stability
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