# Homework Help: Root locus varying H

1. Nov 6, 2013

### weavertri

Hello guys, I need help to sketch the root locus of the system when K is varying from 0 to infinity. (K>=0)
The system:

I'm stuck. I really don't know how proceed.

2. Nov 6, 2013

### TimeToShine

Plot the poles and zeroes (There are no zeroes here). Use the root locus drawing rules.

3. Nov 6, 2013

### weavertri

yeah, that's what I'm doing... the problem is: K is not a constant. to use the root locus drawing rules the characteristic equation of the closed loop system must be written as 1 + GH = 0. All that I got in this case is
(s+2)(s+5)(s+K) + 10K = 0.

4. Nov 7, 2013

### weavertri

I guess I finally did it.
my characteristic equation is (s+2)(s+5)(s+K) + 10K = 0.
we expand to s³ + 7s² + 10s + Ks² 7Ks + 20K = 0
the C.E. must be written as 1 + K*F(s) = 0, then

s³ + 7s² + 10s + K(s² + 7s + 20) = 0

1 + K(s² + 7s + 20)/( s³ + 7s² + 10s) = 0

this is equal to

1 + K(s + 3.5 - 2.8i)(s + 3.5 + 2.8i)/(s(s+2)(s+5)) = 0

now we can sketch the root locus where the zeros are (z1= 3.5 + 2.8i and z2 = 3.5 - 2.8i) and the poles p1=0, p2=-2 and p3=-5.

:)