# Routh Hurwitz (RH) Criterion MCQ

## Routh Hurwitz (RH) Criterion MCQ

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### Questions Type Questions

Q.1. A system described by the transfer function:

$G(s)=\frac{1}{s^{3}+\alpha s^{2}+Ks+3}$ is stable

The constraints on α and K are

• $\alpha > 0, \alpha K< 3$
• $\alpha > 0, \alpha K> 3$
• $\alpha < 0, \alpha K> 3$
• $\alpha < 0, \alpha K< 3$

Answer: $\alpha > 0, \alpha K> 3$

Q.2. The feedback control system in the figure below is stable

• for all K≥0
• only K≥0
• only if 0≤ K < 1
• only if 0 ≤ K ≤ 1

Answer: only if 0≤ K < 1

Q.3. The characteristic polynomial of a system is $q(s)=2s^{5}+s^{4}+4s^{3}+2s^{2}+2s+1$. The system is

• Stable
• Marginally stable
• Unstable
• Oscillatory

Q.4. The open-loop transfer function of a unity feedback system

$G(s)=\frac{K}{s(s^{2}+s+2)(s+3)}$

The range of K for which the system is stable is

• $\frac{21}{4}> K> 0$
• $13> K> 0$
• $\frac{21}{4}< K< \infty$
• $-6< K< \infty$

Answer: $\frac{21}{4}> K> 0$

Q.5. For the polynomial:

$P(s)=s^{5}+s^{4}+2s^{3}+2s^{2}+3s+15$, the number of roots which lie in the right half of the s-plane is

• 4
• 2
• 3
• 1

Q. 6. The Positive values of “K” and ‘a’ so that the system shown in the figure below oscillates at a frequency of 2 rad/sec respectively are

• 1, 0.75
• 2, 0.75
• 1, 1
• 2, 2

Q.7. A certain system has transfer function $G(s)=\frac{s+8}{s^{2}+\alpha s-4}$, where α is a parameter. Consider the standard negative unity feedback configuration as shown below:

Which of the following statements is true ?

• The closed-loop system is never stable for any value of α
• For some positive values of α, the closed-loop system is stable, but not for all positive values.
• For all positive values of α, the closed-loop system is stable.
• The closed-loop system is stable for all values of α, both positive and negative.

Answer: For all positive values of α, the closed-loop system is stable.

Q.8. The number of open right half plane poles of

$G(s)=\frac{10}{s^{5}+2s^{4}+3s^{3}+6s^{2}+5s+3}$
• 0
• 1
• 2
• 3

Q.9. The open-loop transfer function of unity feedback control system is $G(s)=\frac{K}{s(s+a)(s+b)}, 0< a\leq b$

The system is stable is

• $0< K< \frac{(a+b)}{ab}$
• $0< K< \frac{ab}{(a+b)}$
• $0< K< ab(a+b)$
• $0< K< \frac{a}{b}(a+b)$

Answer: $0< K< ab(a+b)$

Q.10. The Routh-Hurwitz criterion cannot be applied when the characteristic equation of the system contains any coefficients which is

• negative real and exponential functions of s
• negative real, both exponential and sinusoidal function of.s.
• both exponential and sinusoidal functions of s
• complex, both exponential and sinusoidal functions of s.

Answer: negative real, both exponential and sinusoidal function of.s.

Q.11. The given characteristic polynomial $s^{4}+s^{3}+2s^{2}+2s+3=0$ has

• zero roots in RHS of s-plane
• one root in RHS of s-plane
• two roots in RHS of s-plane
• three roots in RHS of s-plane

Answer: two roots in RHS of s-plane

Q.12. The characteristic equation of a control system is given by $s^{6}+2s^{5}+8s^{4}+12s^{3}+20s^{2}+16s+16=0$. The number of the roots of the equation which lie in the imaginary axis of s-plane is

• zero
• 2
• 4
• 6

Q.13. An open loop system has a transfer function $\frac{1}{s^{3}+1.5s^{2}+s-1}$. it is converted into a closed loop system by providing a negative feedback having 20(s + 1). Which one of the following is correct?

The open loop and closed loop system are, respectively

• stable and stable
• stable and unstable
• unstable and stable
• unstable and unstable

Q.14. The open loop transfer function of a unity negative feedback control system is given by

$G(s)=\frac{k}{(s+2)(s+4)(s^{2}+6s+25)}$

Which is the value of k which causes sustained oscillations in the closed loop system?

• 666.25
• 790
• 990
• 1190

Q.15. The unit step response of a system is $1-e^{-t}(1+t)$. Which is this system?

• Unstable
• Stable
• Critically stable
• Oscillatory

Q.16. The system having characteristic equation:

$s^{4}+2s^{3}+3s^{2}+2s+K=0$ is to be used as an oscillator. What are the values of k and the frequency of oscillation ω?

• k = 1 and ω = 1 r/s
• k = 1 and ω = 2 r/s
• k = 2 and ω = 1 r/s
• k = 2 and ω = 2 r/s

Answer: k = 2 and ω = 1 r/s

Q.17. The characteristic equation of a control system is

$s^{5}+15s^{4}+85s^{3}+225s^{2}+274s+120=0$

What are the number of roots of the equation which lie to the left of the line $s+1=0$ ?

• 2
• 3
• 4
• 5

Q.18. The open loop transfer function of a unity feedback control system is given by

$G(s)=Ke^{-Ts}$

Where K and T are variables and are greater than zero. The stability of the closed loop system depends on

• K only
• Both K and T
• T only
• Neither K nor T

Q. 19. Consider the unity feedback system with $G(s)=\frac{K}{(s^{2}+2s+2)(s+2)}$

The system is marginally stable. What is the radian frequency of oscillation?

• $\sqrt{2}$
• $\sqrt{3}$
• $\sqrt{5}$
• $\sqrt{6}$

Answer: $\sqrt{6}$

Q.20. For what positive value of K does the polynomial

$s^{4}+8s^{3}+24s^{2}+32s+K$

have roots with zero real parts ?

• 10
• 20
• 40
• 80

$s^{4}+8s^{3}+24s^{2}+32s+K=0$