---
title: "R-H range of K and oscillation for K(s+13)/[s(s+3)(s+7)]"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-s13-s3-s7"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s13-s3-s7
site: BCT Engineering
---

# R-H range of K and oscillation for K(s+13)/[s(s+3)(s+7)]

Control Systems · Chapter 4: Stability

- Asked in: 2081 Bhadra, 6 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

A unity feedback system has the feedforward transfer function G(s) = K(s+13)/[s(s+3)(s+7)]. Using R-H Criterion, find the range of K for the system to be stable. When does the system just oscillate and what would be that frequency during sustained oscillation?

## Short answer

Characteristic equation s^3+10s^2+(21+K)s+13K=0. Routh array gives stability for 0<K<70. At K=70 the s^1 row vanishes, giving marginal stability with sustained oscillation frequency ω=√91≈9.54 rad/s.

## Answer

The closed-loop characteristic equation $1+G(s)=0$ is tested with the Routh array. The system just oscillates (is marginally stable) when the $s^1$ row becomes zero.

#### Characteristic equation

$$
\begin{aligned}
s(s+3)(s+7) + K(s+13) &= 0 \\
s^3 + 10s^2 + (21+K)s + 13K &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 |
|---|---|---|
| $s^3$ | 1 | $21+K$ |
| $s^2$ | 10 | $13K$ |
| $s^1$ | $\frac{10(21+K)-13K}{10}=\frac{210-3K}{10}$ | 0 |
| $s^0$ | $13K$ | |

#### Range of K

- $s^1$ row: $210-3K>0 \Rightarrow K<70$
- $s^0$ row: $13K>0 \Rightarrow K>0$

$$
0 < K < 70
$$

#### Sustained oscillation

At $K = 70$ the $s^1$ row is zero, so the system just oscillates. Auxiliary equation from the $s^2$ row:

$$
\begin{aligned}
10s^2 + 13K &= 0 \\
10s^2 + 910 &= 0 \\
s^2 &= -91 \Rightarrow s = \pm j9.54
\end{aligned}
$$

**Answer:** stable for $0<K<70$; it just oscillates at $K=70$, with frequency $\omega=\sqrt{91}=9.54$ rad/s.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The characteristic equation is
   A. s^3+10s^2+(21+K)s+13K=0
   B. s^3+10s^2+21s+13K=0
   C. s^3+10s^2+(21+13K)s+K=0
   D. s^3+10s^2+(21-K)s+13K=0
   Answer: A. s(s+3)(s+7) = s^3+10s^2+21s; adding K(s+13) gives the stated polynomial. (Key point: Characteristic equation)

2. The stable range of K is
   A. K > 70
   B. 0 < K < 70
   C. 0 < K < 21
   D. K < 0
   Answer: B. s^1: (210-3K)/10>0 gives K<70; s^0: 13K>0 gives K>0. (Key point: Range of K)

3. The system just oscillates at K =
   A. 21
   B. 13
   C. 70
   D. 10
   Answer: C. The s^1 row is zero when 210-3K=0, i.e. K = 70. (Key point: Marginal gain)

4. From 10s^2+13K=0 at K=70, w is
   A. 91 rad/s
   B. 3 rad/s
   C. 7 rad/s
   D. 9.54 rad/s
   Answer: D. s^2 = -91, so w = sqrt(91) = 9.54 rad/s. (Key point: Oscillation frequency)

## Close variants set in other exams

- [R-H range of K for K(s+1)/(s³+bs²+3s+1)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-range-k-ks1-b) — Control Systems, chapter 4, asked 2×

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