Root Locus Technique GATE ECE Notes + Formulas + Solved Examples | Control Systems

Introduction

Changing gain changes system behavior. Root locus shows that change as a path of closed-loop pole locations.

It is one of the most visual tools in Control Systems because it links gain, pole movement, stability, and time response.

Why It Matters

It helps select gain for desired damping and speed.
It shows stability range graphically.
It explains how adding poles and zeros changes response.

Prerequisites

Closed-loop characteristic equation.
Poles and zeros.
Complex plane.
Basic stability conditions.

Basic Intuition

Root locus is a map of possible closed-loop behaviors. As gain increases, poles travel along paths, and the system response changes with them.

Read the topic as a physical behavior first, then let the equations describe that behavior.

Root Locus Pole-Zero Plot Diagram Here

Animated Root Locus Gain Slider Visualization

Step-by-Step Visualization

Use this animated view to connect the exam formula with the physical idea behind Root Locus Technique.

Animated concept visual

Closed-Loop Pole Motion as K Changes

Root locus is the path of closed-loop poles while gain K varies.

1
Start at poles
Every branch begins at an open-loop pole.
2
Increase K
Closed-loop poles move along root-locus paths.
3
Watch damping
Pole angle and distance shape overshoot and speed.
4
Choose design point
Use magnitude condition to find the required gain.

Kvariable gain

×open-loop pole

○open-loop zero

σaasymptote centroid

Exam takeaway: Branches start at open-loop poles and end at open-loop zeros or infinity.

Core Theory

Characteristic equation

$$1+KG(s)H(s)=0$$

Root locus is drawn for changing gain K.

Angle condition

$$\angle G(s)H(s)=(2q+1)180^\circ$$

A point lies on root locus if it satisfies the angle condition.

Magnitude condition

$$K|G(s)H(s)|=1$$

After selecting a point, magnitude condition gives gain.

Working Principle

The working method is to move from the physical system to the mathematical model, then use the model to predict or improve behavior.

Plot open-loop poles and zeros.
Find real-axis segments.
Draw asymptotes and centroid.
Locate breakaway or break-in points.
Use angle and magnitude conditions for design points.

Step-by-Step Operation Animation Here

Formula Explanation

Number of branches

$$N=Number\ of\ open-loop\ poles$$

Each branch starts from an open-loop pole.

Asymptote angle

$$\theta=\frac{(2q+1)180^\circ}{P-Z}$$

Angles for branches going to infinity.

Centroid

$$\sigma_a=\frac{\sum poles-\sum zeros}{P-Z}$$

Intersection point of asymptotes.

Diagram Explanation Placeholder

The diagram should show the signal flow, physical interpretation, and the main mathematical variables used in this topic.

Root Locus Pole-Zero Plot Diagram Here

Interactive Framer Motion Visualization Placeholder

Real-World Applications

Gain tuning.
Lead compensator design.
Servo response shaping.
Motor control loops.
Stability range estimation.

Solved Examples

Branches

Open-loop transfer has three poles and one zero.

$$Branches=3,\quad Asymptotes=3-1=2$$

Asymptote angles

If P-Z=2.

$$\theta=90^\circ,270^\circ$$

Common Mistakes

Forgetting branches start at poles and end at zeros.
Drawing root locus on wrong real-axis segments.
Ignoring asymptotes.
Using closed-loop poles as starting points.

Interview Questions

What is root locus?
Why is root locus useful?
State angle and magnitude conditions.
What is breakaway point?
How does adding a zero affect root locus?

Exam Notes

Number of branches equals number of open-loop poles.
Real-axis rule is checked to the right of a test point.
Asymptotes appear when poles exceed zeros.
Root locus helps infer transient response from pole location.

Revision Summary

Root locus shows how closed-loop poles move as system gain changes, making it a visual method for stability and transient-response design.
Number of branches equals number of open-loop poles.
Real-axis rule is checked to the right of a test point.
Asymptotes appear when poles exceed zeros.
Root locus helps infer transient response from pole location.

Root Locus Technique FAQ

Why is Root Locus Technique important for GATE ECE?

Root Locus Technique is important because it supports numerical problem solving in Control Systems and helps connect formulas with practical engineering behavior.

What should I revise first in Root Locus Technique?

Number of branches equals number of open-loop poles.

How should I practice Root Locus Technique for university exams?

Start with the intuition, memorize the core formulas, solve standard examples, and then practice previous-year style questions on root locus rules, asymptotes, breakaway points, angle condition, pole-zero addition..

Practice Questions

Draw real-axis segments for G(s)H(s)=K/[s(s+2)(s+4)].
Find centroid for poles 0,-2,-4 and no zeros.
Find asymptote angles for P-Z=3.
Explain effect of adding a left-half-plane zero.