Root locus of k(s−4)/(s²+6s+18)

$$G(s) = \frac{k(s-4)}{s^2 + 6s + 18}$$

Poles: $-3 \pm j3$. Zero: $+4$ (a right-half-plane, non-minimum-phase zero). Characteristic equation: $s^2 + (6+k)s + (18 - 4k) = 0$.

Root locus construction

1. Branches: $n = 2$, $m = 1$. One branch ends at the zero $+4$; the other goes to $-\infty$ (asymptote at $180^\circ$).
2. Real-axis locus: $(-\infty, +4]$, i.e. everything left of the zero.
3. Angle of departure from $-3 + j3$: angle from zero $+4$: $180^\circ - \tan^{-1}(3/7) = 156.80^\circ$; from pole $-3-j3$: $90^\circ$.

$$\phi_d = 180^\circ + 156.80^\circ - 90^\circ = 246.80^\circ \;(\equiv -113.2^\circ)$$

4. Break-in point:

$$k = -\frac{s^2 + 6s + 18}{s - 4}, \qquad \frac{dk}{ds} = 0 \Rightarrow s^2 - 8s - 42 = 0 \Rightarrow s = -3.616 \text{ or } 11.616$$

$s = 11.616$ gives $k < 0$. Break-in at s = −3.616, where $k = 1.23$. 5. Shape: the complex poles move on a circle centred at the zero $+4$ with radius $\sqrt{7^2 + 3^2} = 7.616$ and meet at $-3.616$. One branch then goes left to $-\infty$. The other moves right along the real axis, passes through the origin, and ends at $+4$.

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    -10              -3.6        0          4
x pole (-3+-j3)  o zero (+4)  * break-in (-3.616)

Range of k for stability

For a quadratic, all coefficients must be positive:

• $6 + k > 0 \Rightarrow k > -6$
• $18 - 4k > 0 \Rightarrow k < 4.5$

At $k = 4.5$ a closed-loop pole is exactly at $s = 0$. For $k > 4.5$ that pole moves into the RHP toward $+4$.

Stable for 0 < k < 4.5 (for the root locus with k ≥ 0; in general −6 < k < 4.5).

Sustained oscillation?

Sustained oscillation needs a pair of poles on the $j\omega$-axis at $\omega \ne 0$. That requires $6 + k = 0$ with $18 - 4k > 0$, i.e. $k = -6$, which lies outside the $k \ge 0$ range. For $k > 0$ the locus crosses the imaginary axis only at the origin ($k = 4.5$), on the real axis. The system then becomes unstable through a real pole (non-oscillatory, exponential growth), not through oscillation.

Answer: The system is stable for 0 < k < 4.5. No, there is no sustained oscillation for any positive k. (Only k = −6 would give oscillation at $\omega = \sqrt{42} = 6.48$ rad/s.)