---
title: "R-H range of k and oscillation frequency"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-s-s2-s1-2"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s-s2-s1-2
site: BCT Engineering
---

# R-H range of k and oscillation frequency

Control Systems · Chapter 4: Stability

- Asked in: 2076 Chaitra, 5+3 marks
- Video: [Numerical 2 : Marginal value of K and Frequency of sustained oscillations](https://www.youtube.com/watch?v=OyDSqsS0eVc) — 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/rh-k-s-s2-s1-2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

The open loop TF of a unity feedback control system is given as G(s) = k/[s(s²+s+1)(s+2) + k]. Determine the range of gain k for the system to be stable. Also determine the value of k which will cause the sustained oscillation and corresponding oscillation frequency.

## Short answer

Form the characteristic equation s⁴+3s³+3s²+2s+k=0 and apply Routh criterion. The system is stable for 0 < k < 1.556, with sustained oscillation at k = 1.556 and frequency ω = 0.816 rad/s.

## Answer

**Reading of the question:** as printed, $\frac{k}{s(s^2+s+1)(s+2)+k}$ is the closed-loop transfer function of a unity feedback system whose open-loop TF is $G(s) = \frac{k}{s(s^2+s+1)(s+2)}$. So the characteristic equation is $s(s^2+s+1)(s+2) + k = 0$.

#### Characteristic equation

$$
\begin{aligned}
s(s^2+s+1)(s+2) + k &= 0 \\
s(s^3 + 3s^2 + 3s + 2) + k &= 0 \\
s^4 + 3s^3 + 3s^2 + 2s + k &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | 3 | $k$ |
| $s^3$ | 3 | 2 | 0 |
| $s^2$ | $\frac{3(3)-1(2)}{3}=\frac{7}{3}$ | $k$ | |
| $s^1$ | $\frac{\frac73(2)-3k}{\frac73}=\frac{14-9k}{7}$ | 0 | |
| $s^0$ | $k$ | | |

#### Range of k for stability

- $s^1$ row: $14 - 9k > 0 \Rightarrow k < \frac{14}{9} = 1.556$
- $s^0$ row: $k > 0$

$$
0 < k < 1.556
$$

#### Sustained oscillation

The system oscillates (is marginally stable) when the $s^1$ row is zero:

$$
k_{mar} = \frac{14}{9} = 1.556
$$

Auxiliary equation from the $s^2$ row:

$$
\begin{aligned}
\frac{7}{3}s^2 + \frac{14}{9} &= 0 \\
s^2 &= -\frac{14}{9}\times\frac{3}{7} = -\frac{2}{3} \\
\omega &= \sqrt{2/3} = 0.816\ \text{rad/s}
\end{aligned}
$$

**Check:** at $k = 14/9$ the roots are $\pm j0.816$ and $-1.5 \pm j0.289$.

**Answer:** stable for $0<k<1.556$; sustained oscillation at $k = 1.556$ with frequency $\omega = 0.816$ rad/s.

**If $G(s)$ is taken literally** as the open-loop TF with unity feedback, the characteristic equation becomes $s(s^2+s+1)(s+2) + 2k = 0$. Every $k$ above is then halved ($0<k<0.778$, $k_{mar}=0.778$), and the frequency stays the same, 0.816 rad/s.

## Check yourself

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

1. What is the expanded characteristic polynomial?
   A. s^4+2s^3+3s^2+3s+k
   B. s^4+3s^3+2s^2+3s+k
   C. s^4+s^3+3s^2+2s+k
   D. s^4+3s^3+3s^2+2s+k
   Answer: D. (s^2+s+1)(s+2)=s^3+3s^2+3s+2; multiply by s and add k. (Key point: Characteristic equation)

2. What range of k keeps the system stable?
   A. 0 < k < 14/9 = 1.556
   B. 0 < k < 7/3
   C. k > 1.556
   D. 0 < k < 0.778
   Answer: A. The s^1 element (14-9k)/7 > 0 gives k < 1.556, and s^0 needs k > 0. (Key point: Stable range)

3. What are the sustained-oscillation k and frequency?
   A. k = 1.556, ω = 1.5 rad/s
   B. k = 0.778, ω = 0.816 rad/s
   C. k = 2.33, ω = 0.667 rad/s
   D. k = 1.556, ω = 0.816 rad/s
   Answer: D. At k=14/9, (7/3)s^2+14/9=0 gives s^2=-2/3, so ω=0.816 rad/s. (Key point: Oscillation)

## Close variants set in other exams

- [R-H range of K for K/[(s+2)(s+4)(s²+6s+25)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-k-s2-s4-s2-6s) — Control Systems, chapter 4, asked 2×

## Related in Control Systems

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- [R-H range of K and oscillation for K(s+13)/[s(s+3)(s+7)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-k-s13-s3-s7) — Control Systems, chapter 4, asked 2×
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