---
title: "R-H relation between K and T for stability"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-t-relation"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-t-relation
site: BCT Engineering
---

# R-H relation between K and T for stability

Control Systems · Chapter 4: Stability

- Asked in: 2081 Baisakh, 6 marks
- Video: [#188 Problem on routh criterion to find K for stability of system || EC Academy](https://www.youtube.com/watch?v=xcssuw3YTZY) — EC Academy
- 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-t-relation
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Using Routh-Hurwitz criterion determine the relation between "K" and "T" so that unity feedback control system whose open loop transfer function given below is stable. G(s) = K/(s[s(s+10) + T])

## Short answer

Characteristic equation s^3+10s^2+Ts+K=0. Routh array yields stability condition 0<K<10T with T>0. The system is marginally stable at K=10T, oscillating at ω=√T rad/s.

## Answer

For unity feedback, the characteristic equation is $1+G(s)=0$. All first-column Routh elements must be positive.

#### Characteristic equation

$$
\begin{aligned}
s[s(s+10)+T] + K &= 0 \\
s^3 + 10s^2 + Ts + K &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 |
|---|---|---|
| $s^3$ | 1 | $T$ |
| $s^2$ | 10 | $K$ |
| $s^1$ | $\frac{10T-K}{10}$ | 0 |
| $s^0$ | $K$ | |

#### Conditions

- $s^0$ row: $K > 0$
- $s^1$ row: $10T - K > 0 \Rightarrow K < 10T$
- From these, $T > 0$ (also needed so that all coefficients are positive).

$$
0 < K < 10T
$$

**Interpretation:** for a given $T$, the gain can be raised up to $10T$. At $K = 10T$ the system is marginally stable and oscillates at $\omega = \sqrt{K/10} = \sqrt{T}$ rad/s (from $10s^2+K=0$). For example, with $T = 5$ the system is stable for $0<K<50$.

**Answer:** relation for stability: $0 < K < 10T$ (with $T>0$).

## Check yourself

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

1. For G = K/(s[s(s+10)+T]), the characteristic equation is
   A. s^3+10s^2+Ts+K=0
   B. s^3+10s^2+K s+T=0
   C. s^2+10s+T+K=0
   D. s^3+10s^2+T+K=0
   Answer: A. s[s^2+10s+T]+K = s^3+10s^2+Ts+K. (Key point: Characteristic equation)

2. The s^1 element of the array is
   A. (10K-T)/10
   B. (10T-K)/10
   C. (T-K)/10
   D. 10T+K
   Answer: B. (10*T - 1*K)/10. (Key point: s^1 element)

3. The stability relation between K and T is
   A. K > 10T
   B. 0 < T < 10K
   C. 0 < K < 10T
   D. K = 10T
   Answer: C. K > 0 from s^0 and 10T-K > 0 from s^1. (Key point: Stability relation)

4. At K = 10T the system oscillates at
   A. w = 10T
   B. w = T
   C. w = sqrt(10)
   D. w = sqrt(T)
   Answer: D. 10s^2 + K = 0 gives w^2 = K/10 = T. (Key point: Marginal case)

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