--- title: "Root locus of K(s+1)/[(s+2)(s+4)(s+6)] for ζ = 0.5" subject: "Control Systems" chapter: 5 key: "semester_3/control_systems:rl-k-s1-s2-s4-s6" source: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s1-s2-s4-s6 site: BCT Engineering --- # Root locus of K(s+1)/[(s+2)(s+4)(s+6)] for ζ = 0.5 Control Systems · Chapter 5: Root Locus Technique - Asked in: 2078 Bhadra, 8+2 marks - Video: [Graphical Method of Finding Gain K for given Damping Ratio from Root Locus : with GATE 2014 Problem](https://www.youtube.com/watch?v=4GsOFEiy5F0) — Learning Electronics - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s1-s2-s4-s6 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5 ## Question 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. ## 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. ## Answer $$ G(s) = \frac{K(s+1)}{(s+2)(s+4)(s+6)} $$ Characteristic equation: $s^3 + 12s^2 + 44s + 48 + K(s+1) = 0$. #### Root locus data - Poles: $-2, -4, -6$; zero: $-1$. So $n = 3$, $m = 1$, with 2 branches going to infinity. - Real-axis locus: $[-2, -1]$ and $[-6, -4]$. - Asymptotes: $$ \sigma_A = \frac{(-2-4-6) - (-1)}{2} = -5.5, \qquad \theta_A = \pm 90^\circ $$ - Breakaway: $\dfrac{dK}{ds} = 0 \Rightarrow 2s^3 + 15s^2 + 24s - 4 = 0$. The root on the locus is $s = -5.04$, where $K = 0.751$. - The branch from $-2$ moves to the zero at $-1$. The branches from $-4$ and $-6$ meet at $-5.04$, break away, and go vertically along the asymptote $\sigma = -5.5$. #### Locating the ζ = 0.5 point Constant-ζ line: $\theta = \cos^{-1}(0.5) = 60^\circ$ from the negative real axis. Points on it are $s = \omega_n(-0.5 + j0.866)$. Search along this line for $\angle G(s) = -180^\circ$: | Try $\omega_n$ | Point $s$ | $\angle G(s)$ | |---|---|---| | 10.86 | $-5.43 + j9.41$ | $-180^\circ$ | Angle check at $s = -5.43 + j9.41$: | From | Angle | Distance | |---|---|---| | zero $-1$ | $115.22^\circ$ | 10.397 | | pole $-2$ | $110.04^\circ$ | 10.012 | | pole $-4$ | $98.65^\circ$ | 9.514 | | pole $-6$ | $86.54^\circ$ | 9.423 | $$ 115.22^\circ - (110.04^\circ + 98.65^\circ + 86.54^\circ) = -180.0^\circ \;\checkmark $$ #### Gain by magnitude criterion $$ K = \frac{10.012 \times 9.514 \times 9.423}{10.397} = 86.3 $$ #### Check With $K = 86.3$: $s^3 + 12s^2 + 130.3s + 134.3 = 0$. Its roots are $-5.43 \pm j9.41$ and $-1.14$. The third pole at $-1.14$ lies very close to the zero at $-1$, so the two nearly cancel. The complex pair therefore governs the response (second-order approximation valid). ```text . | . | *. | . | . | . | . | .. | . | ---------------x.........x--------x....o----+----- . | .. | . | . | . | . | *. | . | . | -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** ($\omega_n = 10.86$ rad/s, ζ = 0.5). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a dominant pole pair with damping ratio 0.5, on which line do the poles lie? A. A vertical line at sigma = -0.5 B. A line at 30° from the negative real axis C. A line at 60° from the negative real axis D. A circle of radius 0.5 about the origin Answer: C. theta = cos^-1(zeta) = cos^-1(0.5) = 60° from the negative real axis, so s = wn(-0.5 + j0.866). (Key point: Constant-damping line) 2. How is the zeta = 0.5 closed-loop pole found on the root locus? A. Find the point on the 60° line where the angle of G(s) is -180° B. Find the point where the locus crosses the jw axis C. Find the breakaway point at s = -5.04 D. Set K = 1 and solve the characteristic equation Answer: A. The point must lie on both the constant-zeta line and the locus, so the angle criterion (-180°) is searched along the 60° line. (Key point: Locating the point on the locus) 3. At s = -5.43 + j9.41, how is K computed? A. Product of zero distances divided by product of pole distances B. Product of pole distances divided by product of zero distances C. Sum of pole angles minus sum of zero angles D. Sum of pole distances minus sum of zero distances Answer: B. K = (product of distances from poles)/(product of distances from zeros), giving 86.3 here. (Key point: Gain by magnitude criterion) 4. Why is the pole pair -5.43 ± j9.41 accepted as dominant at K = 86.3? A. The third pole is far in the right half-plane B. The third pole at -1.14 nearly cancels the zero at -1 C. The third pole is at the origin D. The complex pair is closer to the jw axis than the third pole Answer: B. The third closed-loop pole at -1.14 almost cancels the zero at -1, so the complex pair governs the response. 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