Break Away/In Points

Introduction:

In root locus analysis, "breakaway" and "break-in" points are key points at specific locations on the root locus plot where the system's open-loop poles enter or exit the complex plane. These points play a crucial role in understanding the stability and behavior of a control system.

Breakaway Point:

1. A breakaway point is a point on the root locus plot where two branches of the locus approach and depart from each other.
2. It indicates the locations on the complex plane where two complex conjugate poles of the open-loop system cross each other while varying a system parameter (usually the controller gain K) continuously.
3. At the breakaway point, the system is marginally stable, and any change in K will result in the poles moving into or out of the complex plane.
4. The breakaway point is determined by solving the characteristic equation for the open-loop system and finding the values of K where the equation has a double root (repeated pole). These repeated poles signify the breakaway point.

Break-in Point:

1. A break-in point is a point on the root locus plot where a branch of the locus re-enters the complex plane.
2. It indicates where a complex conjugate pole pair of the open-loop system, which initially left the complex plane as the gain K was increased, re-enters the complex plane as K continues to increase.
3. The break-in point typically occurs when a branch of the locus reaches the point where two complex poles become real and separate.
4. The break-in point is also determined by solving the characteristic equation for the open-loop system, but in this case, it's where two real roots become complex conjugate poles.

Steps for Finding the breakaway and break-in points in a root locus plot:

D(s) = 0

• For breakaway points, look for points where two branches approach each other on the real axis and then split apart. This means that two complex poles become a real pole and a complex conjugate pair.
• At the breakaway point, the following equation holds: D'(s) = 0, where D'(s) is the derivative of the characteristic equation with respect to s.
• Solve for the value of s that satisfies this equation. This value of s represents the location of the breakaway point.
• To find the corresponding value of K, substitute the breakaway point s-value back into the characteristic equation

(D(s) = 0).

• For break-in points, identify the locations where a real pole and a complex conjugate pole on the real axis join to form a complex conjugate pair as K is increased.
• At the break-in point, the following equation holds: D(s) = 0, where D(s) is the characteristic equation.
• Solve for the value of s that satisfies this equation. This value of s represents the location of the break-in point.
• To find the corresponding value of K, substitute the break-in point s-value back into the characteristic equation (D(s) = 0).