# Where are the K = ± ∞ points on the root loci of the characteristic equation of the closed loop control system located at?

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Where are the K = ± ∞ points on the root loci of the characteristic equation of the closed loop control system located at?
1. Poles of G(s) H(s)
2. Zeroes of G(s) H(s)
3. Both Zeroes and Poles of G(s) H(s)
4. Neither at Zeroes nor at Poles of G(s) H(s)

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Correct Answer - Option 2 : Zeroes of G(s) H(s)

Root locus technique:

The root locus is a graphical representation of closed-loop poles in the s-domain.

Angle condition:  $\angle G(s)H(s)=\pm(2q+1)180$

Magnitude condition:  |G(s)H(s)| = 1.

Every branch of a root locus diagram starts at the open-loop pole (K = 0) and terminates at the open-loop zero (K = ∞) of the open-loop transfer function.

For, k = ∞ the closed-loop poles are equal to the open-loop zero (zero of G(s) H(s))

1. Root locus diagram is symmetrical with respect to the real axis.

2. The number of branches of the root locus diagram is:

N = P if P ≥ Z

= Z, if P ≤ Z

3. Number of asymptotes in a root locus diagram = |P – Z|

4. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

$\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}$

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

5. Angle of asymptotes:

l = 0, 1, 2, … |P – Z| – 1

6. On the real axis to the right side of any section, if the sum of the total number of poles and zeros are odd, the root locus diagram exists in that section.

7. Break-in/away points: These exist when there are multiple roots on the root locus diagram.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of $\frac{{dK}}{{ds}}$ are the breakpoints.