--- title: "Root locus of K/[s(s²+10s+24)] at ζ = 0.7" subject: "Control Systems" chapter: 5 key: "semester_3/control_systems:rl-k-s-s2-10s-24" source: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s-s2-10s-24 site: BCT Engineering --- # Root locus of K/[s(s²+10s+24)] at ζ = 0.7 Control Systems · Chapter 5: Root Locus Technique - Asked in: 2078 Kartik, 8 marks; 2065 Shrawan (old course), 10 marks (asked 2 times) - Video: [root locus in control system](https://www.youtube.com/watch?v=MIVkzOiFvhI) — Smart Engineer - 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-s-s2-10s-24 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5 ## Question Draw the root locus for the system with open loop transfer function as G(s)H(s) = K/[s(s²+10s+24)] and hence from the root locus, find the gain (K) and corresponding natural frequency of oscillation when the damping ratio is 0.7. ## Short answer For damping ratio 0.7, the closed-loop poles are at -1.39±j1.42, found by solving the angle condition along the ζ=0.7 line. Using the magnitude condition, gain K≈28.4 and natural frequency ωn≈1.98 rad/s. ## Answer **Result:** at $\zeta=0.7$, $K \approx 28.4$, with $\omega_n \approx 1.98$ rad/s and $\omega_d \approx 1.42$ rad/s. The root locus is sketched with the standard rules. The $\zeta = 0.7$ line is then drawn from the origin, and the magnitude condition at its intersection with the locus gives $K$. $$ G(s)H(s) = \frac{K}{s(s^2+10s+24)} = \frac{K}{s(s+4)(s+6)} $$ #### Step 1: Poles and zeros Poles: $0,\ -4,\ -6$ ($n=3$). No zeros ($m=0$), so 3 branches go to infinity. #### Step 2: Real-axis segments Between $0$ and $-4$, and from $-6$ to $-\infty$. #### Step 3: Asymptotes $$ \theta = \frac{(2q+1)180^\circ}{3} = 60^\circ,\ 180^\circ,\ 300^\circ, \qquad \sigma_A = \frac{0-4-6}{3} = -3.33 $$ #### Step 4: Breakaway point $$ \begin{aligned} K &= -(s^3+10s^2+24s) \\ \frac{dK}{ds} &= -(3s^2+20s+24) = 0 \\ s &= \frac{-20\pm\sqrt{400-288}}{6} = -1.57 \ \text{or}\ -5.10 \end{aligned} $$ $s=-5.10$ is not on the locus. So the breakaway point is $s=-1.57$, where $K = |s||s+4||s+6| = 16.9$. #### Step 5: Imaginary-axis crossing Characteristic equation: $s^3+10s^2+24s+K=0$ | Row | Col 1 | Col 2 | |---|---|---| | $s^3$ | 1 | 24 | | $s^2$ | 10 | $K$ | | $s^1$ | $\frac{240-K}{10}$ | | | $s^0$ | $K$ | | $K = 240$ makes the $s^1$ row zero, and $10s^2+240=0$ gives $s = \pm j4.90$. The system is stable for $0 * | -1.39+j1.42 (K=28.4) |. | | . | . = zeta line | .| <====x------x==========o===x----> sigma -6 -4 -1.57 0 (K=16.9) ``` **Answer:** at $\zeta = 0.7$: $K \approx 28.4$, natural frequency $\omega_n \approx 1.98$ rad/s (damped frequency $\omega_d = \omega_n\sqrt{1-\zeta^2} \approx 1.42$ rad/s). Closed-loop dominant poles are at $-1.39 \pm j1.42$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the open-loop poles and how many asymptotes? A. 0, -4, -6 and 3 asymptotes B. 0, -4, -6 and 2 asymptotes C. 0, +4, +6 and 3 asymptotes D. -4, -6 and 2 asymptotes Answer: A. s^2+10s+24=(s+4)(s+6); n=3, m=0 so 3 asymptotes. (Key point: Poles) 2. Which candidate from dK/ds = 0 is the breakaway point? A. s = -1.57, since -5.10 is not on the locus B. s = -5.10, since it lies between 0 and -6 C. Both D. Neither Answer: A. The locus is on 0 to -4 and left of -6; -5.10 is between -4 and -6 so is off the locus. (Key point: Breakaway) 3. At what angle to the negative real axis is the ζ = 0.7 line drawn? A. cos^-1(0.7) = 45.57° B. sin^-1(0.7) = 44.4° C. 70° D. tan^-1(0.7) = 35° Answer: A. The constant-ζ line makes an angle θ = cos^-1 ζ with the negative real axis. (Key point: Damping line) 4. How is K found at the ζ = 0.7 intersection? A. Angle condition giving 45.57 B. Routh array giving 240 C. Breakaway gain 16.9 D. Magnitude condition K = |s||s+4||s+6| at the point, giving 28.4 Answer: D. The point is found by the angle condition and K from the magnitude condition: 1.984x2.971x4.824 = 28.4. (Key point: Finding K) 5. What is ωn at ζ = 0.7? A. 1.42 rad/s B. 1.98 rad/s C. 4.90 rad/s D. 0.7 rad/s Answer: B. ωn = |s| = 1.984 rad/s; 1.42 rad/s is the damped frequency ωd. 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