= 0°, 120°, and 240° The breakway point is determined by
18.1 that is, or with the solutions
For a system with positive gain, as K increases from K=0, the root at wherTi, H 6 ^ ^ the ^ Untn * reaches ** brekaway point approaches the VZS? ^^ BOm int<> the SeCOnd ^adrant and approaches the 60 /240 asymptote as oo. Similarly, the root at * = (-3,0)
moves to the right along the real axis and into the third quadrant at s2. The real root at j = (-4,0) remains real, moving to the left along the axis (0°/180° asymptote). For a negative gain system, as K decreases from /l = 0, the real root at s = ( — 2,0) remains real, moving to the right along the real axis. The roots at j=(-4,0) and 5=(-3,0) move to the breakaway point sB = sv where they become complex and approach the 120c/300° asymptote as K—> — oo.
The maximum value of K for which the system remains stable corresponds to a pole of the closed-loop transfer function which lies at s=iu on the imaginary axis and is located by inspection from the root locus diagram. From Rule 5, Eq. (18-34), the gain is
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