---
title: "Angle criterion for root locus"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-angle-criterion"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-angle-criterion
site: BCT Engineering
---

# Angle criterion for root locus

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2076 Chaitra, 2 marks
- Video: [Root Locus Magnitude and Angle condition](https://www.youtube.com/watch?v=RGPcYWnrNxU) — techgurukula
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-angle-criterion
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

Describe angle criteria for a point in s-plane such that the root locus would cross through the point.

## Short answer

The root locus passes through a point if the sum of angles from all open-loop zeros minus the sum from all open-loop poles equals ±(2q+1)180°. Vectors are drawn from each pole and zero to the test point; if the net angle is an odd multiple of 180°, the point is on the locus.

## Answer

The root locus passes through a point $s_1$ only if $s_1$ satisfies the **angle criterion**:

$$
\angle G(s_1)H(s_1) = \sum \angle(s_1 - z_j) - \sum \angle(s_1 - p_i) = \pm(2q+1)\,180^\circ
$$

with $q = 0, 1, 2, \dots$ To apply it, draw vectors from all open-loop zeros and poles to $s_1$. Then take the sum of the zero angles minus the sum of the pole angles. If the result is an odd multiple of $180^\circ$, the point lies on the root locus for $K > 0$; otherwise it does not. Example: for $G = K/[s(s+2)]$, the point $-1 + j1$ gives $-(135^\circ + 45^\circ) = -180^\circ$, so it is on the locus.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The angle criterion is computed as:
   A. Sum of pole angles minus sum of zero angles, equal to 0°
   B. Sum of zero angles minus sum of pole angles, equal to an odd multiple of 180°
   C. Sum of zero angles plus sum of pole angles, equal to 360°
   D. Product of angles equal to 180°
   Answer: B. Angle of GH = sum of zero angles - sum of pole angles = ±(2q+1)180°. (Key point: Angle formula)

2. How are angles obtained in applying the criterion?
   A. Draw vectors from each open-loop pole and zero to s1 and measure their angles
   B. Measure angles of the asymptotes only
   C. Use the closed-loop poles only
   D. Measure angles from the jw axis to the origin
   Answer: A. Vectors from every open-loop pole and zero to the test point give the angles used. (Key point: Applying the test)

3. For G = K/[s(s+2)] at -1 + j1, the angle of G is:
   A. -90°
   B. -135°
   C. -180°
   D. -45°
   Answer: C. -(135° + 45°) = -180°, so the point is on the locus. (Key point: Example)

## Related in Control Systems

- [Checking whether root locus passes through point P](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-point-p-dominant) — Control Systems, chapter 5
- [Criteria for a point to lie on root locus](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-point-criteria) — Control Systems, chapter 5
- [Root locus criteria; poles 0, −1±j2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-criteria-poles-0-1j2) — Control Systems, chapter 5
- [Root locus of k/[s(s+4)(s²+4s+20)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s-s4-s2-4s) — Control Systems, chapter 5, asked 6×
- [Root locus of K(s²−2s+5)/(s²+1.5s−1)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s2-2s5-s2-15s) — Control Systems, chapter 5, asked 6×
- [Root locus of K/[s(s+6)(s²+4s+13)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s-s6-s2-4s) — Control Systems, chapter 5, asked 6×

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