GATE 2024 EE – Question 57
Consider the closed-loop system shown in the figure with
$$G(s)=\frac{K(s^2-2s+2)}{(s^2+2s+5)}.$$
The root locus for the closed-loop system is to be drawn for $0\le K<\infty$. The angle of departure (between $0^\circ$ and $360^\circ$) of the root locus branch drawn from the pole $(-1+j2)$, in degrees, is __________________ (rounded off to the nearest integer).
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Correct answer: 7
Explanation
The zeros are at $1\pm j$ and the poles at $-1\pm j2$. Angle of departure $=180^\circ+\sum\angle(\text{pole}-\text{zeros})-\sum\angle(\text{pole}-\text{other poles})$. From the pole $-1+j2$: to zero $1+j$: vector $-2+j$, angle $153.4^\circ$; to zero $1-j$: vector $-2+j3$, angle $123.7^\circ$; to the other pole $-1-j2$: vector $j4$, angle $90^\circ$. So $\phi=180+153.4+123.7-90=367.1^\circ\equiv7.1^\circ$, i.e. 7°.