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Root locus of K(s+1)/[(s+2)(s+4)(s+6)] for ζ = 0.5

Short answer

For G(s)=K(s+1)/[(s+2)(s+4)(s+6)], locate the point on the ζ=0.5 line (60° from negative real axis) satisfying the angle criterion. The dominant closed-loop poles are s=-5.43±j9.41. By magnitude criterion, K ≈ 86.3.

Consider a unity feedback system having forward transfer function G(s) = K(s+1)/[(s+2)(s+4)(s+6)]. Determine the value of 'K' such that the damping ratio ξ of the dominant closed loop poles is 0.5 using root locus technique.

G(s)=K(s+1)(s+2)(s+4)(s+6)G(s) = \frac{K(s+1)}{(s+2)(s+4)(s+6)}

Characteristic equation: s3+12s2+44s+48+K(s+1)=0s^3 + 12s^2 + 44s + 48 + K(s+1) = 0.

Root locus data

  • Poles: −2,−4,−6-2, -4, -6; zero: −1-1. So n=3n = 3, m=1m = 1, with 2 branches going to infinity.
  • Real-axis locus: [−2,−1][-2, -1] and [−6,−4][-6, -4].
  • Asymptotes:
σA=(−2−4−6)−(−1)2=−5.5,θA=±90∘\sigma_A = \frac{(-2-4-6) - (-1)}{2} = -5.5, \qquad \theta_A = \pm 90^\circ
  • Breakaway: dKds=0⇒2s3+15s2+24s−4=0\dfrac{dK}{ds} = 0 \Rightarrow 2s^3 + 15s^2 + 24s - 4 = 0. The root on the locus is s=−5.04s = -5.04, where K=0.751K = 0.751.
  • The branch from −2-2 moves to the zero at −1-1. The branches from −4-4 and −6-6 meet at −5.04-5.04, break away, and go vertically along the asymptote σ=−5.5\sigma = -5.5.

Locating the ζ = 0.5 point

Constant-ζ line: θ=cos⁡−1(0.5)=60∘\theta = \cos^{-1}(0.5) = 60^\circ from the negative real axis. Points on it are s=ωn(−0.5+j0.866)s = \omega_n(-0.5 + j0.866). Search along this line for ∠G(s)=−180∘\angle G(s) = -180^\circ:

Try ωn\omega_nPoint ss∠G(s)\angle G(s)
10.86−5.43+j9.41-5.43 + j9.41−180∘-180^\circ

Angle check at s=−5.43+j9.41s = -5.43 + j9.41:

FromAngleDistance
zero −1-1115.22∘115.22^\circ10.397
pole −2-2110.04∘110.04^\circ10.012
pole −4-498.65∘98.65^\circ9.514
pole −6-686.54∘86.54^\circ9.423
115.22∘−(110.04∘+98.65∘+86.54∘)=−180.0∘  ✓115.22^\circ - (110.04^\circ + 98.65^\circ + 86.54^\circ) = -180.0^\circ \;\checkmark

Gain by magnitude criterion

K=10.012×9.514×9.42310.397=86.3K = \frac{10.012 \times 9.514 \times 9.423}{10.397} = 86.3

Check

With K=86.3K = 86.3: s3+12s2+130.3s+134.3=0s^3 + 12s^2 + 130.3s + 134.3 = 0. Its roots are −5.43±j9.41-5.43 \pm j9.41 and −1.14-1.14. The third pole at −1.14-1.14 lies very close to the zero at −1-1, so the two nearly cancel. The complex pair therefore governs the response (second-order approximation valid).

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---------------x.........x--------x....o----+-----
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              -6       -4        -2         0
x pole   o zero (-1)
* zeta = 0.5 poles (-5.43 +- j9.41, K = 86.3)

Answer: K ≈ 86.3. Dominant poles: s = −5.43 ± j9.41 (ωn=10.86\omega_n = 10.86 rad/s, ζ = 0.5).

✓ Answer checked by Claude Opus 5.5 (high effort)

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