Root locus of K/[(s²+2s+2)(s²+2s+5)]

$$G(s) = \frac{K}{(s^2+2s+2)(s^2+2s+5)}$$

Characteristic equation: $s^4 + 4s^3 + 11s^2 + 14s + 10 + K = 0$.

1. Poles and zeros

• Poles: $-1 \pm j1$ and $-1 \pm j2$ ($n = 4$). No zeros.
• All 4 branches go to infinity.

2. Real-axis locus

There are no real poles or zeros, so no part of the real axis is on the locus.

3. Asymptotes

$$\sigma_A = \frac{-1-1-1-1}{4} = -1, \qquad \theta_A = \pm 45^\circ,\ \pm 135^\circ$$

4. Angles of departure

• At $-1 + j1$: angles from $-1-j1$: $90^\circ$; from $-1+j2$: $-90^\circ$; from $-1-j2$: $90^\circ$.

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

The branch leaves straight up.

• At $-1 + j2$: the three angles are all $90^\circ$.

$$\phi_d = 180^\circ - 270^\circ = -90^\circ$$

The branch leaves straight down.

So the two branches move toward each other along the line $\sigma = -1$ and meet.

5. Breakaway points

$K = -(s^4 + 4s^3 + 11s^2 + 14s + 10)$

$$\frac{dK}{ds} = -(4s^3 + 12s^2 + 22s + 14) = -2(s+1)(2s^2+4s+7) = 0$$

• $s = -1$: $K = -(1)(4) = -4 < 0$, so it is not on the locus.
• $s = -1 \pm j1.581$: at $s = -1 + jb$, $(s^2+2s+2) = 1 - b^2 = -1.5$ and $(s^2+2s+5) = 4 - b^2 = 1.5$. Hence $K = 2.25 > 0$, a valid complex breakaway point.

At $K = 2.25$ the poles meet at $-1 \pm j1.581$ and then split horizontally (±90° from the vertical) toward the 45° and 135° asymptotes.

6. Imaginary-axis crossing (Routh)

$K_{mar} = 16.25$. Auxiliary equation: $7.5s^2 + 26.25 = 0 \Rightarrow \omega = \sqrt{3.5} = 1.871$ rad/s.

Sketch

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     ....                        |       ....
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             ......     .x     ..X...
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---------------------------------+----------------
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             ......     .x     ..X...
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       -3              -1        0
x pole   * breakaway (-1 +- j1.58, K = 2.25)
X jw crossing (+-j1.87, K = 16.25)

Each pair moves along $\sigma = -1$, meets at $-1 + j1.581$, and turns. One branch heads to the 135° asymptote; the other heads right, crossing the $j\omega$-axis at $j1.871$ and following the 45° asymptote.

Answer: Complex breakaway at −1 ± j1.581 (K = 2.25). Departure angles are +90° from −1+j1 and −90° from −1+j2. The system is stable for 0 < K < 16.25 and oscillates at 1.871 rad/s when K = 16.25.