---
title: "Checking whether root locus passes through point P"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-point-p-dominant"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-point-p-dominant
site: BCT Engineering
---

# Checking whether root locus passes through point P

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2075 Chaitra, 4 marks
- 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-p-dominant
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

Consider a point P in s-plane which actually indicates dominant closed loop pole of the system. How would you recognize that the root locus passes through the point P?

## Short answer

Verify the angle criterion: the sum of angles from all finite zeros to P minus the sum from all poles to P must equal ±180°(2q+1). If satisfied, P lies on the locus; then apply the magnitude criterion to find the required gain and check that all other closed-loop poles are far left or cancelled so P is dominant.

## Answer

The root locus passes through a point $P$ (the desired dominant closed-loop pole) only if $P$ satisfies the **angle criterion**. The gain needed there is then found from the **magnitude criterion**.

#### Test steps
1. Mark all open-loop poles ($p_i$) and zeros ($z_j$) of $G(s)H(s)$ and the point $P = -\sigma_d + j\omega_d$.
2. Draw vectors from every pole and zero to $P$ and measure their angles from the positive real axis.
3. Compute
$$
\phi = \sum \angle(P - z_j) - \sum \angle(P - p_i)
$$
4. If $\phi = \pm 180^\circ(2q+1)$, then **P lies on the root locus**. If not, P cannot be a closed-loop pole for any $K$. The difference $\phi_c = \pm 180^\circ - \phi$ is the angle a compensator must supply (the starting point of lead/lag design).
5. If P is on the locus, find the gain:
$$
K = \frac{\prod |P - p_i|}{\prod |P - z_j|}
$$
6. Check that the other closed-loop poles at this $K$ lie well to the left (about 5 times farther from the $j\omega$-axis) or are nearly cancelled by zeros. Only then is P truly dominant.

#### Example
$G(s) = \dfrac{K}{s(s+2)}$, $P = -1 + j\sqrt3$ (ζ = 0.5, $\omega_n = 2$):
- $\angle P = 120^\circ$ and $\angle(P + 2) = 60^\circ$.
- $\phi = -(120^\circ + 60^\circ) = -180^\circ$, so P is on the locus.
- $K = 2 \times 2 = 4$.

## Check yourself

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

1. P lies on the root locus if the sum of zero angles minus pole angles equals:
   A. 0°
   B. An odd multiple of 180°
   C. 90°
   D. 360°
   Answer: B. The angle criterion requires ±180°(2q+1). (Key point: Test for being on locus)

2. If the angle test fails, the difference phi_c = ±180° - phi represents:
   A. The gain needed
   B. The angle a compensator must supply
   C. The breakaway angle
   D. The asymptote angle
   Answer: B. It is the angle deficiency a lead/lag compensator must provide so the locus passes through P. (Key point: Compensator angle)

3. For P to be truly dominant, the other closed-loop poles should:
   A. Be on the right of P
   B. Be at the origin
   C. Lie about 5 times farther from the jw axis or be nearly cancelled by zeros
   D. Be complex and closer to the jw axis
   Answer: C. Other poles must be far left or cancelled so the pair at P governs the response. (Key point: Dominance condition)

## Related in Control Systems

- [Root locus of K(s+1)/[(s+2)(s+4)(s+6)] for ζ = 0.5](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s1-s2-s4-s6) — 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: 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 of K/[s(s+1)(s²+4s+5)] with ξ = 0.5](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s-s1-s2-4s) — 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 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

## 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 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
- [Closed loop MBI system](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Aclosed-loop-mbi-define) — Instrumentation II, chapter 1, asked 2×
- [Determinant form of plane through point and line](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adeterminant-form-plane-through-point) — Mathematics, chapter 3