---
title: "Criteria for a point to lie on root locus"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-point-criteria"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-point-criteria
site: BCT Engineering
---

# Criteria for a point to lie on root locus

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2078 Bhadra, 4 marks
- Video: [Root Locus | Point lies on root locus | CS | Control Systems | Lec-70](https://www.youtube.com/watch?v=udyoaGUO9Hk) — Education 4u
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-point-criteria
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

Discuss criteria for a point on s-plane that may lie on a root locus.

## Short answer

A point lies on the root locus if it satisfies the angle criterion: the sum of angles from zeros minus the sum from poles equals an odd multiple of 180°. Once on the locus, the magnitude criterion gives K = product of distances to poles divided by distances to zeros.

## Answer

A point $s_1$ in the s-plane lies on the root locus only if it is a root of the characteristic equation $1 + KG(s)H(s) = 0$ for some $K \ge 0$. Writing $KG(s_1)H(s_1) = -1 = 1\angle \pm 180^\circ$ gives two criteria.

#### 1. Angle criterion (decides whether the point is on the locus)
$$
\angle G(s_1)H(s_1) = \sum_{j} \angle(s_1 - z_j) - \sum_{i} \angle(s_1 - p_i) = \pm(2q+1)\,180^\circ
$$

- Draw vectors from every open-loop pole and zero to $s_1$ and measure their angles from the positive real axis.
- If (sum of zero angles) − (sum of pole angles) is an odd multiple of $180^\circ$, the point is on the locus. Otherwise it is not.
- $K$ does not appear in this condition, so it alone fixes the shape of the locus.

#### 2. Magnitude criterion (gives K at that point)
$$
|K\,G(s_1)H(s_1)| = 1 \;\Rightarrow\; K = \frac{\prod_i |s_1 - p_i|}{\prod_j |s_1 - z_j|}
$$

That is, K is the product of the distances from the poles divided by the product of the distances from the zeros (1 if there are no zeros). This is used only after the angle criterion is satisfied.

#### Example
$G(s) = \dfrac{K}{s(s+4)}$, test point $s_1 = -2 + j2$:
- $\angle s_1 = 135^\circ$ and $\angle(s_1 + 4) = 45^\circ$.
- Sum $= -(135^\circ + 45^\circ) = -180^\circ$, so the point is on the locus.
- $K = |s_1|\,|s_1 + 4| = 2.83 \times 2.83 = 8$.

#### Related real-axis rule
A real-axis point is on the locus if the number of real poles plus zeros to its right is odd. This rule follows directly from the angle criterion.

## Check yourself

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

1. A point s1 is on the root locus if the angle of G(s1)H(s1) equals:
   A. 0° or 360°
   B. Any odd multiple of 180°
   C. Any even multiple of 180°
   D. Exactly 90°
   Answer: B. KG(s1)H(s1) = -1 requires the angle to be ±(2q+1)180°. (Key point: Angle criterion)

2. What is true about K in the angle criterion?
   A. K must equal 1
   B. K appears and fixes the gain at the point
   C. K does not appear, so the angle condition alone fixes the shape of the locus
   D. K must be negative
   Answer: C. Since K is a positive real number it contributes no angle, so the angle criterion determines the locus shape. (Key point: Role of K in angle criterion)

3. When is the magnitude criterion used and what does it give?
   A. Before the angle criterion, to test whether the point is on the locus
   B. After the angle criterion is satisfied, to find K at that point
   C. To find the asymptote angles
   D. To find the number of branches
   Answer: B. Once the point passes the angle test, |KG(s1)H(s1)| = 1 gives K = product of pole distances/product of zero distances. (Key point: Magnitude criterion)

4. For G = K/[s(s+4)] at s1 = -2 + j2, what is K?
   A. 4
   B. 2.83
   C. 8
   D. 16
   Answer: C. K = |s1||s1+4| = 2.83 x 2.83 = 8. (Key point: Example calculation)

## Related in Control Systems

- [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
- [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
- [Angle criterion for root locus](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-angle-criterion) — Control Systems, chapter 5
- [Root locus: poles −2±j3, zero at origin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-2j3-zero-0-critical) — Control Systems, chapter 5
- [Root locus with poles 0, −6, −2 ± j3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-0-6-2j3) — 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×

## Related in other subjects

- [Root-locus design: 8 samples per cycle, T = 0.2 s](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aroot-locus-8-samples-t02) — Digital Control System, chapter 4
- [Root-locus design: ζ = 0.5, ts = 2 s, plant 1/(s(s+2))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aroot-locus-design-plant-s2) — Digital Control System, chapter 4, asked 5×
- [Root-locus design: ζ = 0.5, ts = 2 s, plant 1/(s(s+1))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aroot-locus-design-plant-s1) — Digital Control System, chapter 4, asked 5×
- [Root locus of a sampled system (T = 0.5 s)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adraw-root-locus-t-0-5) — Digital Control System, chapter 4