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Root Locus

Trace closed-loop pole movement and predict stability with simple construction rules.

Control Systems8-10 marks45 min

Overview Concepts Formulas Examples Revision FAQ

Topic Overview

Start here for the big picture before memorizing formulas or steps.

Root locus is a graphical method for seeing how the closed-loop poles of a feedback system move as the loop gain changes. It turns abstract characteristic-equation behavior into a visual stability tool.

Most exam questions focus on the standard construction rules: where the locus begins and ends, which parts of the real axis belong to it, how asymptotes behave, and how gain affects stability.

The most important intuition is this: as the pole locations move, the system response changes. So root locus is not just a plotting topic; it is directly tied to damping, oscillation, and settling behavior.

Subtopics Covered

Basic rulesAsymptotesBreakaway and break-inImaginary-axis crossing

Core Concepts

Read these ideas in plain language and use them as your understanding checklist.

Learning Goals

Interpret root locus as the path of closed-loop poles as gain varies.

Use the basic rules for real-axis segments, asymptotes, and breakaway points.

Connect pole movement to stability and transient-response behavior.

Key Concepts

The locus starts at open-loop poles and ends at open-loop zeros or infinity.

A real-axis point belongs to the root locus if the number of poles and zeros to its right is odd.

Asymptotes show where branches go when there are more poles than zeros.

Pole movement toward the right half-plane indicates reduced stability.

Quick Concept Map

AsymptotesBreakawayAngle criterion

Formulas and Meaning

Keep formulas close to their meaning so they are easier to remember and apply.

Characteristic condition

1 + K G(s)H(s) = 0

Closed-loop poles are the roots of the characteristic equation.

Angle condition

angle G(s)H(s) = (2q + 1)180 deg

A point lies on the root locus when the phase condition is satisfied.

Asymptote centroid

sigma = (sum of poles - sum of zeros) / (n - m)

Used when the number of poles exceeds the number of zeros.

Worked Examples

Use these solved examples to see how the concept is applied step by step.

Interpret gain increase

If a root-locus branch crosses into the right half-plane as gain increases, what does that suggest about the closed-loop system?

Remember that closed-loop poles determine stability.

A pole in the right half-plane means the response grows instead of decays.

So crossing the imaginary axis marks a stability boundary.

Answer

It indicates the system becomes unstable beyond that gain range.

Revision and Exam Focus

Use this block for last-minute revision, common traps, and exam-oriented reading.

Common Mistakes

Trying to memorize the whole sketch without first counting poles and zeros.

Forgetting that branches begin at poles and end at zeros or infinity.

Treating the root locus as pure geometry and not relating it to system stability.

Exam Pointers

Start every root-locus question by counting poles and zeros.

Check the real-axis rule before trying to sketch asymptotes or breakaway points.

Link pole locations back to damping and stability to interpret the result quickly.

Quick Revision

Root locus shows closed-loop pole paths as gain varies.

Real-axis segments are chosen using the odd-number-to-the-right rule.

Crossing into the right half-plane means instability.

Exam Insight

Root locus becomes much easier once you stop treating it like a drawing exercise and start reading it as a stability story.

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Root Locus FAQ

Quick answers for students searching root locus explained, control systems notes, and GATE ECE preparation.

What should I study first in Root Locus?

Interpret root locus as the path of closed-loop poles as gain varies.

How is Root Locus useful for GATE ECE and university exams?

Root Locus is useful for Control Systems notes because it combines concept clarity, formula-based revision, and exam-style worked examples for ECE students.

Which topics should I revise after Root Locus?

After Root Locus, revise Time Response, Laplace Transform.