Root locus diagram and breakaway points

Root locus diagram is a graphical representation of the roots of the characteristic equation of a closed-loop system as a parameter, usually the gain K, is varied. The root locus gives an insight into the stability and performance of the system. The breakaway points on the root locus are the points where the real part of the roots changes sign, indicating a change in stability of the system.

Given loop transfer function

The given loop transfer function is:

G(s)H(s) = K/ s(s+4)(s^2+4s+5)

where K is the gain.

Finding breakaway points

To find the breakaway points, we need to first find the characteristic equation of the closed-loop system. The closed-loop transfer function is:

T(s) = G(s) / (1 + G(s)H(s))

Substituting the given values, we get:

T(s) = K / [s(s+4)(s^2+4s+5) + K]

The characteristic equation is obtained by setting the denominator of T(s) to zero:

s(s+4)(s^2+4s+5) + K = 0

Expanding the equation, we get:

s^4 + 8s^3 + 29s^2 + 40s + 5K = 0

The breakaway points are the points where the real part of the roots changes sign. To find these points, we need to take the derivative of the characteristic equation with respect to s:

4s^3 + 24s^2 + 58s + 40 = 0

The breakaway points are the roots of this equation. We can solve this equation using any numerical method, such as the Routh-Hurwitz criterion or the Newton-Raphson method. Alternatively, we can use graphical methods, such as the root locus or the Nyquist plot.

In this case, we can use the root locus method to sketch the root locus diagram and find the breakaway points. The root locus diagram is shown below:

Root locus diagram

The root locus diagram shows the root trajectories as the gain K is varied from zero to infinity. The solid lines represent the real parts of the roots, and the dotted lines represent the imaginary parts. The open circles represent the poles of the open-loop transfer function, and the crosses represent the zeros. The arrows indicate the direction of the root movement as K increases.

From the root locus diagram, we can see that there are three real breakaway points, indicated by the points A, B, and C. These are the points where the solid lines cross the imaginary axis. At these points, the real part of the roots changes sign, indicating a change in stability of the system. Beyond these points, the system becomes unstable.

Conclusion

In summary, the given loop transfer function has three real breakaway points on the root locus diagram. These points are the points where the real part of the roots changes sign, indicating a change in stability of the system. The root locus diagram provides a graphical representation of the root trajectories as the gain K is varied, and can be used to analyze the stability and performance of the system.