Root Locus Breakaway Points

When multiple branches of the root locus travel along the real axis and meet at a point before moving off into the complex plane, that meeting point is called a breakaway point. Conversely, when complex branches return to the real axis and merge before continuing as real-axis branches, the meeting point is called a break-in point. These points represent the gain values at which repeated closed-loop poles occur, which is physically significant for system damping.

For GATE aspirants, the condition to find breakaway and break-in points is one of the most commonly tested calculation rules in root locus. The condition is straightforward and can be applied algorithmically once the characteristic equation is expressed in terms of K.

Core Concept Explanation

A breakaway point occurs between two adjacent poles on the real-axis locus segment. As K increases from zero, two branches travel toward each other along the real axis. At some value of K, they collide at the breakaway point and the system has two identical (repeated) closed-loop poles. For any further increase in K, the poles move off the real axis into the complex plane as a conjugate pair.

A break-in point is the reverse: it occurs between two adjacent zeros (or between a zero and infinity) on the real-axis locus. Complex branches approach the real axis, meet at the break-in point, and then continue as separate real-axis branches.

The physical significance is direct: at a breakaway point, the system transitions from overdamped (two real poles) to underdamped (complex conjugate poles). The damping ratio drops to a critical value at that K. A designer might deliberately operate at a gain slightly below the breakaway to keep both poles real and maintain an overdamped response.

Mathematical Expression

The condition for a breakaway or break-in point is dK/ds = 0, where K is expressed as a function of s from the characteristic equation. Starting from 1 + KG(s)H(s) = 0, we write K = -1/G(s)H(s) = -D(s)/N(s) where D(s) and N(s) are the denominator and numerator polynomials of G(s)H(s). Differentiating this expression with respect to s and setting the derivative to zero gives the candidate points.

An equivalent and often faster formula is: for G(s) = K N(s)/D(s), the breakaway condition becomes N(s) D'(s) - D(s) N'(s) = 0 where primes denote derivatives with respect to s. All solutions to this polynomial that lie on the real-axis locus segment (confirmed by the real-axis rule) are valid breakaway or break-in points.

Practical Understanding

Not every root of dK/ds = 0 is a valid breakaway point. A candidate is valid only if it lies on an actual real-axis locus segment, confirmed by the odd-singularity rule. Also, the value of K at the candidate point must be positive for the standard root locus (K greater than or equal to 0). If the computed K is negative, the candidate belongs to the complementary (positive feedback) locus, not the standard root locus.

Solved Numerical Example

For G(s) = K / [s(s+4)], find the breakaway point. The characteristic equation is s^2 + 4s + K = 0, so K = -(s^2 + 4s) = -s^2 - 4s. Differentiating K with respect to s and setting equal to zero finds the breakaway location.

Given:
G(s) = K / [s(s+4)], poles at 0 and -4, no finite zeros

Why this formula applies:
Breakaway condition is dK/ds = 0 where K = -D(s)/N(s)

Formula:
K = -[s(s+4)] / 1 = -s^2 - 4s
dK/ds = -2s - 4

Substitution:
-2s - 4 = 0

Calculation:
s = -4/2 = -2
Verify: s = -2 lies between poles 0 and -4, on real-axis locus (odd count to right = 1 pole at 0). Valid.
K at breakaway: K = -(-2)^2 - 4(-2) = -4 + 8 = 4

Final Answer:
Breakaway point at s = -2 with K = 4

Exam Tip: After solving dK/ds = 0, always verify the candidate lies on the real-axis locus using the odd-singularity rule AND that K is positive. A common GATE trap is accepting an s value that gives negative K, which does not belong to the standard root locus.

Mechanism: Step-by-Step Procedure

• Express K from the characteristic equation: K = -D(s)/N(s) where G(s)H(s) = KN(s)/D(s).
• Differentiate K with respect to s and set dK/ds = 0. Solve the resulting polynomial for candidate values of s.
• Reject any candidate that does not lie on the real-axis locus segment (check odd-singularity rule).
• Reject any candidate that yields a negative K value when substituted back.
• Breakaway points lie between two adjacent open-loop poles; break-in points lie between two adjacent open-loop zeros.

Quick Revision

• Breakaway point: two real-axis branches meet and depart into complex plane. Occurs between two adjacent OL poles.
• Break-in point: two complex branches merge back onto real axis. Occurs between two adjacent OL zeros.
• Condition: dK/ds = 0 where K = -D(s)/N(s) from characteristic equation 1 + KG(s)H(s) = 0.
• Candidate point is valid only if it lies on real-axis locus AND gives positive K.
• Physically: breakaway corresponds to transition from overdamped to underdamped closed-loop response.
• Exam trap: solving dK/ds = 0 may give multiple roots. Must verify each separately using real-axis rule and K sign.