A single-loop negative-feedback system has $G(s)=K(s+2)/(s^2+2s+2)$ and H(s)=1, with real K>0. The breakaway-point location on its root locus is ________.
The characteristic equation gives $K=-(s^2+2s+2)/(s+2)$. Stationary gain points satisfy $$\frac{dK}{ds}=0\quad\Longrightarrow\quad s^2+4s+2=0.$$ Thus s=−2±√2. Only s=−2−√2 lies to the left of the real zero −2 on the positive-gain real-axis locus; the other gives negative K.
The required location is **−3.41421**, within −3.45 to −3.35. The complex branches meet there and split along the real axis.
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