1)The feedback control system has

G(s)=K(s+6)/(s+4)

and an ideal integrator with unity gain in feedback path.Determine the following with reference to root locus technique:

(i)K=0 and K=infinity points on root loci

(ii)number of asymptotes

(iii)breakaway points,if any

2)if root locus of s(s+2)+K(s+4)=0 is a circle.What are the coordinates of the centre of this circle?

3)Also sktech the asymptotes for K>0 and K<0 for

G(H)=K/s(s+2)(s+1+j)(s+1-j)

Community Answer

Honor CodeSolved 1 Answer

See More Answers for FREE

Enhance your learning with StudyX

Receive support from our dedicated community users and experts

See up to 20 answers per week for free

Experience reliable customer service

Get Started

Answer:Given: \rightarrow\begin{aligned}G(s) & =\frac{k(s+6)}{(s+4)} \\H(s) & =\frac{1}{s} \\G(s) H(s) & =\frac{k(s+6)}{s(s+4)}\end{aligned}(i) K=0, represents the location of open leop poles i-e s=-4 and s=0 in the given case.K=\infty, represents the open leop sero i.es=-6 \text { and } s=-\infty \text {. }(ii)\text { Number of asymptotes }=\text { no. of open loop poles }(P)-\text { no: of }\begin{aligned}& \text { open loop zeros (z) } \\= & (2-1)=1\end{aligned}(iii) Characteristic equation:\begin{array}{r}1+\frac{k(s+6)}{s(s+4)}=0 \\k=\frac{-s(s+4)}{(s+6)}\end{array}Fer breakaway points, \frac{d k}{d s}=0\begin{array}{c}\Rightarrow\left[\frac{(2 s+4)(s+6)-\left(s^{2}+4 s\right)}{(s+6)^{2}}\right]=0 \\2 s^{2}+16 s+24-s^{2}-4 s=0 \\s^{2}+12 s+24=0 \\s=-2.536,-9.464 \\\uparrow\end{array}Breakauay Break in Point point\begin{array}{c}s(s+2)+k(s+4)=0 \\1+\frac{k(s+4)}{s(s+2)}=0 \quad \text { (Dividing by } s(s+2) \\\therefore \quad G(s) H(s)=\frac{k(s+4)}{s(s+2)}=\frac{k(s+b)}{s(s+a)} \text { (Comparing) } \\\text { Centre }=(-b, 0)=(-4,0) .\end{array}Therefere, the co-ardinates of the centre of this circle is (-4,0)3)Given,G(H)=\frac{K}{s(S+2)(s+1+j)(s+1-j)}The center of asymptotes is\sigma_{c}=-\frac{(0+2+1+j+1-j)}{4}=-1Fer k>0, the angles of the asymptotes areQ=45^{\circ}, 135^{\circ}, 225^{\circ} \text {, and } 315^{\circ}Fer k<0, the angles of the asymptotes are\theta=0^{\circ}, 90^{\circ}, 180^{\circ} \text {, and } 270^{\circ}The plot's are shoun abore. ...