How to find the intersection point in rootlocus diagram?

Question:

Sketch the root locus for the open loop transfer function of a unity feedback control sytem given below and determine the value of K for $\xi=0.5$

$G(s)=\frac{K}{s(s+1)(s+3)}$

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I have drawn the root locus diagram like this

Now i tried all possible way to get the intersection point of rootlocus diagram with the 60 degree line.If i get that point i can easily find the value of K for $\xi=0.5$.

Is is possible to find that point without any software?

The characteristic equation is:

$$s(s+1)(s+3)=-K$$

Hence, plug in any point on the locus, and that will give the corresponding value of K.

It appears that your selected point on the locus is approximately: $s=-0.4+j0.7$, which will give $K\approx 1.8$

For an accurate answer to this particular problem ($\small \zeta=0.5$), write the CE as: $$s(s+1)(s+3)+K= (s+\alpha)(s^2+\omega _n s+ \omega_n^2)$$

solve for $\alpha$ and $\omega_n$, and hence find K. Thus:

$$s^3+(\alpha +\omega_n)s^2+(\alpha\omega_n+\omega_n^2)s+\alpha\omega_n^2=s^3 +4s^2+3s+K$$ Giving $\small K=1.83$.

Systems higher than 3rd order will need a root solver.

But this approach begs the question, why bother to sketch the root locus in the first place?

• I want that $"point"$ to plug into that equation.I think you did't get what i am asking...i am asking that point coordinates where $\xi$ becomes 0.5 – Rohit Oct 8 '17 at 8:48
• how did you get that point coordinates please tell me?thats my doubt only – Rohit Oct 8 '17 at 8:53
• From the graph, assuming the axes are to the same scale. – Chu Oct 8 '17 at 8:55
• Really i am not getting what you are trying to say sir – Rohit Oct 8 '17 at 8:56
• The point on the graph is at (-0.4, 0.7) if the 'y-axis' scale is the same as the 'x-axis' scale. – Chu Oct 8 '17 at 8:58

We search a given line for a point that yields a summation of angles (for both zero and pole angles) equal to an odd multiple of $180^\circ$. Take a look at the following picture

The line in which a point that yields a summation of angles equal to an odd multiple of 180 deg is shown as the Green line in the above picture. The angle of this line is $\beta = \cos^{-1} (\zeta)$ and the radius that satisfies the aforementioned condition is searched by a software that we code. In this example, this is Matlab code that searches for the target point.

r=.75;
while r < .76
zeta=.5;
a=acos(zeta);
x=r*cos(a);
y=r*sin(a);

%poles' angles
p1=0;
Th1=pi-atan( y/x );

p2=-1;
Th2=atan(y/(abs(p2)-x));

p3=-3;
Th3=atan(y/(abs(p3)-x));

angle=-(Th1+Th2+Th3);

Th1=120.000 : Th2=46.102 : Th3=13.898  :  angle=-180.000  : r=0.750

The aforementioned condition is satisfied if $\theta_1 = 120^\circ, \theta_2=46.102^\circ$ and $\theta_3 = 13.898^\circ$ which yields a radius $r=0.75$. The point $P$ is then $(r\cos(\beta), r\sin(\beta))$, or $P=-0.3750+j0.6495$. Now, to compute the gain K, we substitute P in the following formula:
\begin{align} K &= \frac{1}{|G(s)||H(s)|} \\ &= |s(s+1)(s+3)| \Big|_{s=P} \\ &= 1.8281. \end{align}