--- title: "R-H range of K for K(s+13)/[s(s+3)(s+7)]" subject: "Control Systems" chapter: 4 key: "semester_3/control_systems:rh-k-s13-s3-s7-range" source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s13-s3-s7-range site: BCT Engineering --- # R-H range of K for K(s+13)/[s(s+3)(s+7)] Control Systems · Chapter 4: Stability - Asked in: 2068 Chaitra, 8 marks - Video: [Q7.b To find the range of values of K, Marginal values of K and frequency of sustained oscillations](https://www.youtube.com/watch?v=vX4tw8FAqSE) — EnggClasses - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s13-s3-s7-range - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4 ## Question Apply RH criteria to determine the range of 'K' for which a unity feedback system with G(s) = K(s+13)/[s(s+3)(s+7)] will be stable. ## Short answer Characteristic equation: s³+10s²+(21+K)s+13K=0. Routh array requires K>0 from s⁰ row and (210−3K)/10>0 from s¹ row, giving K<70. The system is stable for 0 < K < 70. ## Answer For unity feedback, the closed-loop characteristic equation is $1+G(s)=0$. The system is stable when the first column of the Routh array has no sign change. #### Characteristic equation $$ \begin{aligned} 1 + \frac{K(s+13)}{s(s+3)(s+7)} &= 0 \\ s(s^2+10s+21) + Ks + 13K &= 0 \\ s^3 + 10s^2 + (21+K)s + 13K &= 0 \end{aligned} $$ #### Routh array | Row | Col 1 | Col 2 | |---|---|---| | $s^3$ | 1 | $21+K$ | | $s^2$ | 10 | $13K$ | | $s^1$ | $\frac{10(21+K)-1(13K)}{10}=\frac{210-3K}{10}$ | 0 | | $s^0$ | $13K$ | | #### Conditions for stability 1. From the $s^1$ row: $\frac{210-3K}{10} > 0 \Rightarrow K < 70$ 2. From the $s^0$ row: $13K > 0 \Rightarrow K > 0$ $$ 0 < K < 70 $$ #### Marginal stability At $K = 70$ the $s^1$ row vanishes. Auxiliary equation: $$ 10s^2 + 13(70) = 0 \Rightarrow s^2 = -91 \Rightarrow s = \pm j9.54 $$ So at $K=70$ the system oscillates at 9.54 rad/s, and for $K>70$ two poles move into the RHP. **Answer:** the system is stable for $0 < K < 70$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the characteristic equation for G(s)=K(s+13)/[s(s+3)(s+7)] with unity feedback? A. s^3+10s^2+21s+13K=0 B. s^3+10s^2+(21+K)s+13K=0 C. s^3+10s^2+(21+13K)s+K=0 D. s^3+21s^2+(10+K)s+13K=0 Answer: B. s(s+3)(s+7)=s^3+10s^2+21s; adding Ks+13K gives s^3+10s^2+(21+K)s+13K. (Key point: Characteristic equation) 2. Which Routh entry gives the upper limit K < 70? A. First element of the s^1 row, (210-3K)/10 > 0 B. First element of the s^2 row C. The s^0 row, 13K > 0 D. The s^3 row Answer: A. 10(21+K)-13K = 210-3K, which must be positive, giving K < 70. (Key point: Upper limit on K) 3. What is the range of K for stability? A. K < 70 only B. 0 < K < 70 C. K > 70 D. 0 < K < 13 Answer: B. The s^1 row needs K < 70 and the s^0 row needs K > 0. (Key point: Stable range) 4. What happens at K = 70? A. The system becomes absolutely stable B. One pole is at the origin C. Two poles are at +-j9.54, so it oscillates at 9.54 rad/s D. All poles are real and negative Answer: C. At K=70 the s^1 row is zero; the auxiliary equation 10s^2+910=0 gives s=+-j9.54. 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