11. For the system shown in the figure, with a damping ratio ξ of 0.7. and undamped natural frequency ω_n of 4 rad / sec, the values of K and a are
    
    

1. 1. K = 4, a = 0.35

   
   
   

   2. K = 8, a = 0.455

   
   
   

   3. K = 16, a = 0.225

   
   
   

   4. K = 64, a = 0.9

   
   
   

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1. View Hint View Answer Discuss in Forum

   M(s) =	G(s)	=	K / s ( + 2)
   	1 + G(s) H(s)		1 + (1 + as) K / s (s + 2)

   
   The characteristic equation is
   s (s + 2) + K (1 + as) = 0
   ⇒ s² + s (2 + aK) + K = 0
   Compare with equation s ² + 2 ξ ω_n s + ω_n² = 0, we get
   K = ω_n ² = 4² = 16
   ∴ 2 ξ ω_n = (2 + aK)
   

   ⇒ a =	2 × 0.7 × 4 - 2	=	3.6	= 0.225
   	16		16

   
   

   Correct Option: C

   M(s) =	G(s)	=	K / s ( + 2)
   	1 + G(s) H(s)		1 + (1 + as) K / s (s + 2)

   
   The characteristic equation is
   s (s + 2) + K (1 + as) = 0
   ⇒ s² + s (2 + aK) + K = 0
   Compare with equation s ² + 2 ξ ω_n s + ω_n² = 0, we get
   K = ω_n ² = 4² = 16
   ∴ 2 ξ ω_n = (2 + aK)
   

   ⇒ a =	2 × 0.7 × 4 - 2	=	3.6	= 0.225
   	16		16

   
   

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12. The Nyquist plot of a loop transfer funct ion G(jω) H(jω) of a system encloses the (– 1, j0) point. The gain margin of the system is
    

1. 1. less than zero

   
   
   

   2. zero

   
   
   

   3. greater than zero

   
   
   

   4. infinity

   
   
   

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1. View Hint View Answer Discuss in Forum

   Gain margin of the system for which Nyquist plot of a loop tr ansfer function G(jω) H(jω) encloses (– 1, + j0) point is less than zero.

   Correct Option: A

   Gain margin of the system for which Nyquist plot of a loop tr ansfer function G(jω) H(jω) encloses (– 1, + j0) point is less than zero.

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13. The open-loop transfer function of a unity feedback control system is
    

    G(s) =	K(s + 2)
    	(s + 1)(s - 7)

    
    For K > 6, the stability characteristic of the openloop and closed-loop configurations of the system are respectively
    

1. 1. stable and unstable

   
   
   

   2. stable and stable

   
   
   

   3. unstable and stable

   
   
   

   4. unstable and unstable

   
   
   

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1. View Hint View Answer Discuss in Forum

   In open-loop system, there is a pole in RHP. So system is unstable.
   In closed loop system
   

   T(s) =	K(s + 2)
   	s2 + (K - 6)s + 2K - 7

   
   For stability K > 6, 2K – 7 > 0
   ⇒ K > 6
   So for K > 6, system is stable.

   Correct Option: C

   In open-loop system, there is a pole in RHP. So system is unstable.
   In closed loop system
   

   T(s) =	K(s + 2)
   	s2 + (K - 6)s + 2K - 7

   
   For stability K > 6, 2K – 7 > 0
   ⇒ K > 6
   So for K > 6, system is stable.

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14. The forward-path transfer function of a unity feedback system is
    

    G(s) =	K(s2 - 4)
    	s3 + 3

    
    For the system to be stable the range of K is
    

1. 1. K > -1
      

   
   
   

   2. K <	3
      	4

      
      

   
   
   

   3. -1 < K <	3
      	4

      
      

   
   
   

   4. marginal stable

   
   
   

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1. View Hint View Answer Discuss in Forum

   Closed loop transfer function
   

   T(s) =	K(s2 - 4)
   	(K + 1)s2 + (3 - 4K)

   
   The system can be only marginally stable.

   Correct Option: D

   Closed loop transfer function
   

   T(s) =	K(s2 - 4)
   	(K + 1)s2 + (3 - 4K)

   
   The system can be only marginally stable.

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15. The feedback system with characteristic equation
    s⁴ + 20 Ks³ + 5s² + 10s + 15 = 0 is
    

1. 1. stable for all value of K

   
   
   

   2. stable for positive value of K

   
   
   

   3. stable for ∞ > K > 7.0

   
   
   

   4. unstable for all values of K

   
   
   

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1. View Hint View Answer Discuss in Forum

   By Routh criterion
   
   For this, any value at K the system is unstable.

   Correct Option: D

   By Routh criterion
   
   For this, any value at K the system is unstable.

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