GATE 2020 IN Q.36: solution and answer

GATE 2020 · INQ.36 · 2 marks · NAT

The loop transfer function of a negative feedback system is given by

G (s) H (s) = \frac{K}{s ( s + 2 ) ( s + 6 )},

where $K > 0$ . The value of $K$ at the breakaway point of the root locus for the above system (rounded off to one decimal place) is ____

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The gain at the breakaway point is $K \approx 5.05$ , which rounds to $5.0$ .

Official answer: 5.0 to 5.1Source: Official answer key, GATE 2020 IN (IIT Delhi)

Open-loop poles are $0, - 2, - 6$ , and there are no finite zeros. Real-axis rule: a point on the real axis is on the locus when an odd number of poles and zeros lie to its right. So the locus covers $- 2 < s < 0$ and $s < - 6$ . The breakaway point lies in $(- 2, 0)$ , where the branches starting at $0$ and $- 2$ meet and leave the axis.
Characteristic equation: $1 + G (s) H (s) = 0 \Rightarrow s (s + 2) (s + 6) + K = 0 \Rightarrow s^{3} + 8 s^{2} + 12 s + K = 0$ , so $K = - (s^{3} + 8 s^{2} + 12 s)$ .
Breakaway condition $\frac{d K}{d s} = 0$ : $3 s^{2} + 16 s + 12 = 0 \Rightarrow s = \frac{- 16 \pm 256 - 144}{6} = \frac{- 16 \pm 112}{6}$
The roots are $s = - 0.9028$ and $s = - 4.431$ . The root $s = - 4.431$ lies between $- 2$ and $- 6$ , which is not part of the locus, so reject it. (It would also need $K \approx - 16.9 < 0$ .) The breakaway point is $s_{b} = - 0.9028$ .
Magnitude condition at $s_{b}$ : $K = ∣ s_{b} ∣ ∣ s_{b} + 2∣ ∣ s_{b} + 6∣ = (0.9028) (1.0972) (5.0972)$
$(0.9028) (1.0972) = 0.9906$ , and $0.9906 \times 5.0972 = 5.049$ . As a check, $- (s_{b}^{3} + 8 s_{b}^{2} + 12 s_{b}) = - (- 0.7359 + 6.5208 - 10.8340) = 5.049$ .
So $K \approx 5.049$ , which is $5.0$ to one decimal place.

KEY IDEA

Find the breakaway point from $d K / d s = 0$ , where $K$ comes from the characteristic equation. Keep only the root that lies on a real-axis segment of the locus. Then get the gain there from the magnitude condition $K = \prod ∣ s_{b} - p_{i} ∣$ .

WATCH OUT FOR

Stopping at the breakaway point $s_{b} \approx - 0.90$ and giving that as the answer, when the question asks for $K$ . Another slip is picking the root $s = - 4.43$ , which lies between $- 2$ and $- 6$ . That segment is not on the locus, and that root gives a negative $K$ .

From the GATE 2020 IN paper. Every question of this paper · Download the full paper · All 7 Root locus questions, with notes

GATE 2020 IN Q.36: solution and answer

Answer and solution

More Root locus questions