Question (1) Poles at j, –j, j2, –j2; zeros at –2, 2 (2) Poles at j, –j, j2, –j2; zeros at –1, 1 Please write down all calculations. Thanks. 9-12. Construct the root-locus diagram for each of the following control systems for which the poles and zeros of G(s)H(s) are given. The characteristic equation is obtained by equating the numerator of 1 + G(s)H(s) to zero.

Solution:-(i) poles of j,-j, j 2,-j 2; zeros of -2,2 construction Rules of Root locey : -Root locus its symmetrical obout real axilyIf P>Z, the number of branchey terminating at infinity =p-zA point on real axil if faid to be on root locus If, to the right side of thit point, the sum of open loop pores and zeras if an odd number.Angle of Asymptotey!-\text { Number of A-ymptates } \begin{array}{l}=p-2 \\\Rightarrow 4-2=2\end{array}\rightarrow Angle of Asymptotey \phi=\frac{(2 q+1) 180^{\circ}}{p-z} \phi_{1}=\frac{(2 \times 0+1) 180^{\circ}}{2}=90^{\circ}\begin{array}{l}\phi_{1}=\frac{(2 \times 0+1) 180^{\circ}}{2}=90^{\circ} \\\phi_{2}=\frac{(2 \times 1+1) 180}{2}=270^{\circ} \\\phi=90^{\circ}, 270^{\circ}\end{array}centroid!:centroid l's the point of entertection of Asymptotes on the real axi's\begin{array}{l}=\frac{\sum \text { Real part of cpen loop poly }-\sum \text { zeray }}{p-2} \\=\frac{\sum 0-\sum-2+2}{z}=0 \\\text { centroid }=0\end{array}[2] similak for poles \left.j,-j, j_{2},-j_{2}\right\} zerajat -1,1 ...