---
title: "Find K and a for oscillation at 2 rad/s"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-a-oscillate-2rad"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-a-oscillate-2rad
site: BCT Engineering
---

# Find K and a for oscillation at 2 rad/s

Control Systems · Chapter 4: Stability

- Asked in: 2076 Chaitra, 6 marks
- Video: [GATE 2012 ECE Find the values of K and a for the system to oscillate at 2 rad/sec](https://www.youtube.com/watch?v=IBPC5lyX_X4) — GATE Paper Satish Bojjawar
- 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-a-oscillate-2rad
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

In the following system, determine K (and a) if the system just oscillates at a frequency 2 rad/sec. [Figure: unity negative feedback system with forward path G(s) = k(s+1)/(s³ + as² + 2s + 1)]

## Short answer

Characteristic equation s^3+as^2+(2+K)s+(1+K)=0. For marginal stability, set s^1 row to zero and from auxiliary equation ω^2=2+K. With ω=2 rad/s, K=2 and a=0.75.

## Answer

A system "just oscillates" when it is marginally stable. In the Routh array, the $s^1$ row then becomes zero, and the auxiliary equation from the $s^2$ row gives the frequency of oscillation.

#### Characteristic equation

$$
\begin{aligned}
1 + \frac{K(s+1)}{s^3+as^2+2s+1} &= 0 \\
s^3 + as^2 + (2+K)s + (1+K) &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 |
|---|---|---|
| $s^3$ | 1 | $2+K$ |
| $s^2$ | $a$ | $1+K$ |
| $s^1$ | $\frac{a(2+K)-(1+K)}{a}$ | 0 |
| $s^0$ | $1+K$ | |

#### Condition for sustained oscillation

$s^1$ row $=0$:

$$
a(2+K) = 1+K \qquad \text{...(1)}
$$

Auxiliary equation from the $s^2$ row, with $s = j\omega$:

$$
\begin{aligned}
as^2 + (1+K) &= 0 \\
\omega^2 &= \frac{1+K}{a} \qquad \text{...(2)}
\end{aligned}
$$

From (1), $\frac{1+K}{a} = 2+K$, so (2) becomes $\omega^2 = 2+K$.

#### Substitute $\omega = 2$ rad/s

$$
\begin{aligned}
4 &= 2+K \Rightarrow K = 2 \\
a &= \frac{1+K}{2+K} = \frac{3}{4} = 0.75
\end{aligned}
$$

#### Check

With $K=2$, $a=0.75$: $s^3+0.75s^2+4s+3 = (s^2+4)(s+0.75)$. The roots are $\pm j2$ and $-0.75$, so the system oscillates at 2 rad/s.

**Answer:** $K = 2$, $a = 0.75$.

## Check yourself

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

1. 'Just oscillates' means
   A. The s^1 row is zero, giving imaginary roots
   B. The s^0 element is zero
   C. K = 0
   D. All roots are in the RHP
   Answer: A. Marginal stability occurs with the s^1 row zero and the roots on the jw axis. (Key point: Just oscillates)

2. Combining a(2+K) = 1+K with w^2 = (1+K)/a gives
   A. w^2 = 1+K
   B. w^2 = 2+K
   C. w^2 = a
   D. w^2 = K/a
   Answer: B. (1+K)/a = 2+K from the zero-row condition. (Key point: Frequency relation)

3. For w = 2 rad/s, K is
   A. 4
   B. 1
   C. 2
   D. 3
   Answer: C. 4 = 2+K gives K = 2. (Key point: Value of K)

4. The corresponding a is
   A. 1.5
   B. 0.5
   C. 1
   D. 0.75
   Answer: D. a = (1+K)/(2+K) = 3/4. (Key point: Value of a)

## Related in Control Systems

- [R-H range of k and oscillation frequency](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-k-s-s2-s1-2) — Control Systems, chapter 4, asked 2×
- [Stability of 56/[s(s⁴+7s³+6s²+42s+8)] feedback system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Astability-56-s5) — Control Systems, chapter 4
- [Block diagram reduction: G1–G4, H1, H2, unity feedback](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1-g4-h1h2-unity-2076c) — Control Systems, chapter 3, asked 3×
- [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×
- [Root locus of K/[(s²+2s+2)(s²+2s+5)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s2-2s-2-s2) — Control Systems, chapter 5
- [Tachometer feedback for 16/[s(s+1)] to get ζ = 0.6](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atacho-16-s-s1-zeta06) — Control Systems, chapter 3

## Related in other subjects

- [Numerical: Okumura median path loss (unity gain)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-median-loss-unity-gain) — Wireless Communication, chapter 3, asked 3×
- [Damped LCR oscillation, damped frequency](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-lcr-frequency-significance) — Engineering Physics, chapter 1, asked 2×
- [Short note: Positive and negative feedback](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ashort-note-feedback) — Basic Electronics Engineering, chapter 4
- [Frequency and magnitude scaling to 10⁵ rad/s](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ascaling-ladder-1e5-numerical) — Filter Design, chapter 1