---
title: "Root locus with poles 0, −6, −2 ± j3"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-poles-0-6-2j3"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-poles-0-6-2j3
site: BCT Engineering
---

# Root locus with poles 0, −6, −2 ± j3

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2080 Bhadra, 10 marks
- Video: [root locus in control system](https://www.youtube.com/watch?v=MIVkzOiFvhI) — Smart Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-poles-0-6-2j3
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

For a system having open loop pole-zero plot as in figure, sketch Root locus. [Figure: s-plane with open-loop poles at s = 0, s = −6 and s = −2 ± j3; no zeros]

## Short answer

Open-loop TF: K/[s(s+6)(s²+4s+13)]. Real-axis breakaway at s=-4.20 with K≈104.6. Departure angles from complex poles ∓70.6°. Stable for 0<K<227.8; at marginal K=227.8 it oscillates at 2.79 rad/s.

## Answer

Open-loop poles at $0, -6, -2 \pm j3$ and no zeros give

$$
G(s)H(s) = \frac{K}{s(s+6)(s^2+4s+13)}
$$

Characteristic equation: $s^4 + 10s^3 + 37s^2 + 78s + K = 0$.

#### 1. Branches
$n = 4$, $m = 0$: 4 branches, all ending at infinity.

#### 2. Real-axis locus
Between $0$ and $-6$ (one real pole to the right is an odd count). There is no locus left of −6.

#### 3. Asymptotes
$$
\begin{aligned}
\sigma_A &= \frac{0 - 6 - 2 - 2}{4} = -2.5 \\
\theta_A &= \pm 45^\circ,\ \pm 135^\circ
\end{aligned}
$$

#### 4. Breakaway point
$K = -(s^4 + 10s^3 + 37s^2 + 78s)$

$$
\frac{dK}{ds} = -(4s^3 + 30s^2 + 74s + 78) = 0
$$

Roots: $s = -4.20$ and $s = -1.65 \pm j1.39$. Only $s = -4.20$ is on the real-axis locus. Gain there:

$$
K = 4.20 \times 1.80 \times 13.84 \approx 104.6
$$

#### 5. Angle of departure from $-2 + j3$
Angles to $-2+j3$: from $0$: $123.69^\circ$; from $-6$: $36.87^\circ$; from $-2-j3$: $90^\circ$.

$$
\phi_d = 180^\circ - (123.69^\circ + 36.87^\circ + 90^\circ) = -70.56^\circ
$$

At $-2-j3$, it is $+70.56^\circ$.

#### 6. Imaginary-axis crossing

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | 37 | $K$ |
| $s^3$ | 10 | 78 | |
| $s^2$ | 29.2 | $K$ | |
| $s^1$ | $(2277.6 - 10K)/29.2$ | | |
| $s^0$ | $K$ | | |

$K_{mar} = 227.76$. Auxiliary equation: $29.2s^2 + 227.76 = 0 \Rightarrow \omega = \sqrt{7.8} = 2.79$ rad/s.

#### Sketch
```text
                                       |
    ...                                |      ....
      ....                             |   ....
         ....                x.        .....
            ...               .........X
              ...                      |
                ...                    |
                  ..                   |
----------x........*...................x----------
                  ..                   |
                ...                    |
              ...                      |
            ...               .........X
         ....                x.        .....
      ....                             |   ....
    ...                                |      ....
                                       |
         -6      -4.2       -2         0
x pole   * breakaway (-4.2)
X jw crossing (+-j2.79, K = 227.8)
```
The branches from $0$ and $-6$ meet at $-4.2$ and break away at ±90°, then bend to the ±135° asymptotes. The complex poles leave at $\mp 70.6^\circ$, curve right, cross the $j\omega$-axis at $\pm j2.79$ and follow the ±45° asymptotes.

**Answer:** Breakaway at **s = −4.20 (K ≈ 104.6)**. The departure angle is **∓70.6°**. The system is stable for **0 < K < 227.8** and oscillates at **2.79 rad/s** when K = 227.8.

## Check yourself

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

1. Which segment of the real axis is on the locus?
   A. Left of -6
   B. Between 0 and +infinity
   C. The whole negative axis
   D. Between 0 and -6
   Answer: D. Only between 0 and -6 is one real pole to the right (odd). (Key point: Real-axis locus)

2. What is the asymptote centroid?
   A. -2
   B. -4
   C. -1.5
   D. -2.5
   Answer: D. (0-6-2-2)/4 = -2.5, angles ±45°, ±135°. (Key point: Asymptotes)

3. Which root of dK/ds=0 is the breakaway and what is its gain?
   A. s = -1.65, K ≈ 50
   B. s = -4.20, K ≈ 104.6
   C. s = -2.5, K = 0
   D. s = -6, K = 0
   Answer: B. Only -4.20 lies on the real-axis locus; the complex roots are off it. (Key point: Breakaway)

4. What is the departure angle at -2+j3?
   A. +70.56°
   B. -70.56°
   C. -90°
   D. 123.69°
   Answer: B. 180° - (123.69°+36.87°+90°) = -70.56°. (Key point: Departure angle)

5. What gain limits stability and at what frequency?
   A. K = 104.6 at 2.79 rad/s
   B. K = 227.8 at 2.79 rad/s
   C. K = 227.8 at 4.2 rad/s
   D. K = 78 at 2.5 rad/s
   Answer: B. The s^1 row gives K=227.76 and 29.2s^2+227.76=0 gives ω=2.79 rad/s. (Key point: Stability)

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