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# Frequency Response

**1. Scientist Bode have contribution in :****a) Asymptotic plots**

b) Polar plots

c) Root locus technique

d) Constant M and n circle**Answer: a****Explanation: Asymptotic plots are the bode plots that are drawn to find the relative stability of the system by finding the phase and gain margin and this was invented by Scientist Bode.****2. Scientist Evans have contribution in :**

a) Asymptotic plots

b) Polar plots**c) Root locus technique**

d) Constant M and n circle**Answer: c****Explanation: Root locus technique is used to find the transient and steady state response characteristics by finding the locus of the gain of the system and this was made Scientist Evans .****3. Scientist Nyquist have contribution in:**

a) Asymptotic plots**b) Polar plots**

c) Root locus technique

d) Constant M and n circle**Answer: b****Explanation: Nyquist plot is used to find the stability of the system by open loop poles and zeroes and the encirclements of the poles and zeroes and satisfying the equation N=P-Z and this is named under the name of scientist Nyquist.****4. For a stable closed loop system, the gain at phase crossover frequency should always be:**

a) < 20 dB

b) < 6 dB

c) > 6 dB**d) > 0 dB****Answer: d****Explanation: Phase crossover frequency is the frequency at which the gain of the system must be 1 and for a stable system the gain is decibels must be 0 db.****5. Which one of the following methods can determine the closed loop system resonance frequency operation?**

a) Root locus method

b) Nyquist method

c) Bode plot**d) M and N circle****Answer: d****Explanation: Closed loop system resonance frequency is the frequency at which maximum peak occurs and this frequency of operation can best be determined with the help of M and N circle.****6. If the gain of the open loop system is doubled, the gain of the system is :****a) Not affected**

b) Doubled

c) Halved

d) One fourth of the original value**Answer: a****Explanation: Gain of the open loop system is doubled then the gain of the system is not affected as the gain of the system is not dependent on the overall gain of the system.****7. Which one of the following statements is correct?****Nichol’s chart is useful for the detailed study of:****a) Closed loop frequency response**

b) Open loop frequency response

c) Close loop and open loop frequency responses

d) None of the mentioned**Answer: a****Explanation: Nichol’s chart has constant M and N circles that are used to find the stability of the system and the detailed study of their can be determined with the help of the closed loop frequency response.****8. Constant M- loci:**

a) Constant gain and constant phase shift loci of the closed-loop system.

b) Plot of loop gain with the variation in frequency

c) Circles of constant gain for the closed loop transfer function**d) Circles of constant phase shift for the closed loop transfer function****Answer: d****Explanation: By definition, Constant M loci are Circles of constant phase shift for the closed loop transfer function.****9. Constant N-loci:**

a) Constant gain and constant phase shift loci of the closed-loop system.

b) Plot of loop gain with the variation in frequency**c) Circles of constant gain for the closed loop transfer function**

d) Circles of constant phase shift for the closed loop transfer function**Answer: c****Explanation: Constant N loci are the circles of constant gain for the closed loop transfer function and the intersection point of the M and N is always the point (-1,0).****10. Nichol’s chart:**

a) Constant gain and constant phase shift loci of the closed-loop system.**b) Plot of loop gain with the variation in frequency**

c) Circles of constant gain for the closed loop transfer function

d) Circles of constant phase shift for the closed loop transfer function**Answer: b****Explanation: Nichol’s chart are plot of loop gain with the variation in frequency and this is used to determine the stability of the system with the variation in the frequency.**

**Correlation between Time and Frequency Response**

**1. The forward path transfer function of a unity feedback system is given by G(s) = 100/(s ^{2}+10s+100). The frequency response of this system will exhibit the resonance peak at:**

a) 10 rad/sec

b) 8.66 rad/sec

**c) 7.07 rad/sec**

d) 5rad/sec

**Answer: c**

**Explanation: G(s) = 100/(s**

^{2}+10s+100)**wn = 10rad/sec**

**G = 10/2wn =0.5**

**wr = 0.707 rad/s.**

**2. Assertion (A): All the systems which exhibit overshoot in transient response will also exhibit resonance peak in frequency response.**

**Reason (R): Large resonance peak in frequency response corresponds to a large overshoot in transient response.**

**a) Both A and R are true and R is the correct explanation of A**

b) Both A and R are true and R is not the correct explanation of A

c) A is true but R is false

d) A is false but R is true

**Answer: a**

**Explanation: For G<1/√2 frequency parameters ex Mr resonant peak and time response parameters eg. Mp peak overshoot are well correlated. For G>1/√2 the resonant peak Mr does not exist and the correlation breaks down. This is not a serious problem as for**

**this range of G, the step response oscillations are well-damped and Mp is hardly perceptible.**

**3. The transfer function of a system is given by Y(s)/X(s) = e**

^{−0.1}s/1+s. If x(t) is 0.5sint, then the phase angle between the output and the input will be:a) -39.27°

b) -45°

**c) -50.73°**

d) -90°

**Answer: c**

**Explanation: Phase angle = -tan**

^{−1}-0.1*180°/π**w =1 rad/sec**

**Phase angle =-50.73°.**

**4. The critical value of gain for the system is 40. The system is operating at a gain of 20. The gain margin of the system is :**

a) 2 dB

b) 3 dB

**c) 6 dB**

d) 4 dB

**Answer: c**

**Explanation: G.M. =Kmarginal/Kdesire**

**Kmarginal =40**

**Kdesire = 20**

**G.M. =2**

**G.M. (dB) = 6dB.**

**5. The phase angle of the system G(s) =s+5/s**

^{2}+4s+9;varies betweena) 0° and 90°

**b) 0° and -90°**

c) 0° and -180°

d) -90° and -180°

**Answer: b**

**Explanation: Phase = tan**

^{−1}−11w−w^{3}/45−^{2}.**6. The open loop transfer function of a system is :**

**G(s) H(s) =K/ (1+s) (1+2s) (1+3s)**

**The phase crossover frequency wpc is:**

a) √2

**b) 1**

c) Zero

d) √3

**Answer: b**

**Explanation: angle =-180°**

**3wpc/1-2wpc**

^{2}=-3wpc**wpc = 1 rad/sec.**

**7. Which one of the following statements is correct for gain margin and phase margin of two closed-loop systems having loop functions G(s) H(s) and exp(-s) G(s) H(s)?**

a) Both gain and phase margins of the two systems will be identical

b) Both gain and phase margins of G(s) H(s) will be more

**c) Gain margins of the two systems are the same but phase margins of G(s) H(s) will be more**

d) Phase margins of the two systems are the same but gain margin of G(s) H(s) will be less

**Answer: c**

**Explanation: The factor exp (-st) is the cause of the term transportation lag (time delay). The effect of e**

^{-st}term is simply to rotate each point of the G(s) H(s) plot by an angle wT rad in the clockwise direction. So the phase margin of the system reduces as T increases. But since |e^{-s}| =1, therefore the gain margins of both the systems are the same.**8. In a feedback control system, phase margin(PM) is**

**1. Directly proportional to G**

**2. Inversely proportional to G**

**3. Independent of G**

**4. Zero when G =0**

**Which of the above statements are correct?**

a) 1 and 2

b) 2 and 3

c) 3 and 4

**d) 1 and 4**

**Answer: d**

**Explanation: For small values of G, PM is directly proportional to G and at G =0. P.M. =0.**

**9. The gain margin in dBs of a unity feedback control system whose open loop transfer function, G(s) H(s) =1/s(s+1) is**

a) 0

b) 1

c) -1

**d) ∞**

**Answer: d**

**Explanation: wpc = ∞**

**Magnitude of the transfer function =0**

**Gain Margin =∞ dB.**

**10. The loop transfer function of a system is given by G(s) H(s) =10e**

^{-Ls}/s. The phase crossover frequency is 5rad/s. The value of the dead time L isa) π/20

**b) π/10**

c) -π/20

d) Zero

**Answer: b**

**Explanation: – π/2-180/π*L*5 = -π**

**5L =π/2**

**L =π/10.**

**Polar Plots**

**1. The constant M circle for M=1 is the****a) straight line x=-1/2**

b) critical point (-1j0)

c) circle with r= 0.33

d) circle with r=0.67**Answer: a****Explanation: For M =1 the constant M circle is a straight line at x=-1/2.****2. The polar plot of a transfer function passes through the critical point (-1,0). Gain margin is****a) Zero**

b) -1dB

c) 1dB

d) Infinity**Answer: a****Explanation: Gain margin of a polar plot passing through the critical point is zero.****3. Consider the following statements:****1. The effect of feedback is to reduce the system error****2. Feedback increases the gain of the system in one frequency range but decreases in another****3. Feedback can cause a system that is originally stable to become unstable****Which of these statements are correct.**

a) 1,2 and 3

b) 1 and 2**c) 2 and 3**

d) 1 and 3**Answer: c****Explanation: Feedback can cause the increase in gain and also can cause stable system to become unstable.****4. The open loop transfer function of a system is G(s) H(s)= K / (1+s)(1+2s)(1+3s)****The phase cross over frequency ωc is**

a) V2**b) 1**

c) Zero

d) V3**Answer: b****Explanation: Phase crossover frequency is calculated as by calculating the magnitude of the transfer function and equating it to 1 and the frequency calculated at this magnitude is phase cross over frequency.****5. If the gain of the open-loop system is doubled, the gain margin****a) Is not affected**

b) Gets doubled

c) Becomes half

d) Becomes one-fourth**Answer: a****Explanation: If the gain of the open-loop system is doubled, the gain margin gets doubled.****6. The unit circle of the Nyquist plot transforms into 0dB line of the amplitude plot of the Bode diagram at**

a) 0 frequency

b) Low frequency

c) High frequency**d) Any frequency****Answer: d****Explanation: The unit circle of the Nyquist plot transforms into 0dB line of the amplitude plot of the Bode diagram at any frequency.****7. Consider the following statements:****The gain margin and phase margin of an unstable system may respectively be****1. Positive, positive****2. Positive, negative****3. Negative, positive****4. Negative, negative****Of these statements**

a) 1 and 4 are correct

b) 1 and 2 are correct

c) 1, 2 and 3 are correct**d) 2,3 and 4 are correct****Answer: d****Explanation: For unstable system the signs of gain margin and phase margin are always different or they can both be negative.****8. If a system has an open loop transfer function****1-s / 1+s, then the gain of the system at frequency of 1 rad/s will be**

a) 1

b) 1/2

c) Zero**d) -1****Answer: d****Explanation: The system is all pass system and the gain of the system at frequency of 1 rad/sec.****9. The polar plot of the open loop transfer function of a feedback control system intersects the real axis at -2. The gain margin of the system is**

a) -5dB

b) 0dB**c) -6dB**

d) 40dB**Answer: c****Explanation: Gain margin of the system is inverse of the intersect on the real axis and calculated in decibels.****G(s) = 1+s / s(1+0.5s).****10. The corner frequencies are**

a) 0 and 1

b) 0 and 2

c) 0 and 1**d) 1 and 2****Answer: d****Explanation: Corner frequency can be calculated by time constant form of the transfer function and here the corner frequencies are 1 and 2.****11. For the transfer function****G(s) H(s) = 1 / s(s+1) (s+0.5), the phase cross-over frequency is**

a) 0.5 rad/sec**b) 0.707 rad/sec**

c) 1.732 rad/sec

d) 2 rad/sec**Answer: b****Explanation: Phase cross over frequency is calculated at the point where magnitude of the polar plot is 1.**

Stability In Frequency Domain MCQs

**12. The gain margin (in dB) of a system having the loop transfer function****G(s) H(s) = 2 / s(s+1) is**

a) 0

b) 3

c) 6**d) 8****Answer: d****Explanation: Gain margin of a system is calculated at the phase cross over frequency and expressed in decibels.****13. The gain margin for the system with open loop transfer function****G(s) H(s) = G(s) =2(1+s) / s2 is**

a) 8**b) 0**

c) 1

d) -8**Answer: b****Explanation: Gain margin of a system is calculated at the phase cross over frequency and expressed in decibels.****14. Statement 1: In constant M circles, as M increases from 1 to 8 radius of circle increases from 0 to 8 and Centre shifts from (-1,0) to (-8,0)****Statement 2: The circle intersects real axis at point (-1/2, 0)**

a) Statement 1 is TRUE, 2 is FALSE

b) Statement 1 is FALSE, 2 is TRUE

c) Statement 1 & 2 TRUE**d) Statement 1 & 2 FALSE****Answer: d****Explanation: All the circles pass through the points (0,0) and (-1,0).**

**Bode Plots**

**1. Assertion (A): Relative stability of the system reduces due to the presence of transportation lag.****Reason (R): Transportation lag can be conveniently handled by Bode plot.**

a) Both A and R are true but R is correct explanation of A**b) Both A and R are true but R is correct explanation of A**

c) A is true but R is false

d) A is false but R is true**Answer: b****Explanation: Transportation lag can be conveniently handled on Bode plot as well without the need to make any approximation.****2. Assertion (A): The phase angle plot in Bode diagram is not affected by the variation in the gain of the system.****Reason(R): The variation in the gain of the system has no effect on the phase margin of the system.**

a) Both A and R are true but R is correct explanation of A

b) Both A and R are true but R is correct explanation of A**c) A is true but R is false**

d) A is false but R is true**Answer: c****Explanation: The variation in the gain of the system has effect on the phase margin but phase plot is not affected.****3. A system has poles at 0.01 Hz, 1 Hz and 80Hz, zeroes at 5Hz, 100Hz and 200Hz. The approximate phase of the system response at 20 Hz is :****a) -90°**

b) 0°

c) 90°

d) -180°**Answer: a****Explanation: Pole at 0.01 Hz gives -180° phase. Zero at 5Hz gives 90° phase therefore at 20Hz -90° phase shift is provided.****4. The constant M-circle represented by the equation x ^{^}2+2.25x+y^{^}2=-1.25 has the value of M equal to:**

a) 1

b) 2

**c) 3**

d) 4

**Answer: c**

**Explanation: Comparing with the M circle equation we have the value of M =3.**

**5. What is the value of M for the constant M circle represented by the equation 8x**

^{2}+18x+8y^{2}+9=0?a) 0.5

b) 2

**c) 3**

d) 8

**Answer: c**

**Explanation: Comparing with the M circle equation we have the value of M =3.**

**6. The constant N loci represented by the equation x**

^{^}2+x+y^{^}2=0 is for the value of phase angle equal to:a) -45°

b) 0°

c) 45°

**d) 90°**

**Answer: d**

**Explanation: Centre = (-0.5, 0)**

**Centre of N circle is (-1/2, 1/2N)**

**N =tanα**

**α =90°.**

**7. All the constant N-circles in G planes cross the real axis at the fixed points. Which are these points?**

**a) -1 and origin**

b) Origin and +1

c) -0.5 and 0.5

d) -1 and +1

**Answer: a**

**Explanation: Centre of N circle is (-1/2, 1/2N)**

**N =tanα**

**Constant –N circles always pass through (-1, 0) and (0, 0).**

**8. Consider the following statements:**

**Nichol’s chart gives information about.**

**i. Closed loop frequency response.**

**ii. The value of the peak magnitude of the closed loop frequency response Mp.**

**iii. The frequency at which Mp occurs.**

**Which of the above statements are correct?**

a) 2 and 3

b) 1 and 2

c) 1 and 3

**d) 1,2 and 3**

**Answer: d**

**Explanation: Nichol’s chart gives information about closed loop frequency response, value of the peak magnitude of the closed loop frequency response Mp and the frequency at which Mp occurs.**

**9. Which one of the following statements is correct? Nichol’s chart is useful for the detailed study analysis of:**

**a) Closed loop frequency response**

b) Open loop frequency response

c) Close loop and open loop frequency responses

d) None of the above

**Answer: a**

**Explanation: Nichol’s chart is useful for the detailed study analysis of closed loop frequency response.**

**10. In a bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4th order all-pole system?**

**a) -80dB/decade**

b) -40 dB/decade

c) 40 dB/decade

d) 80 dB/decade

**Answer: a**

**Explanation: A 4th order all pole system means that the system must be having no zero or s-term in numerator and s**

^{4}terms in denominator. One pole exhibits -20dB/decade slope, so 4 pole exhibits a slope of -80 dB /decade.**11. Frequency range of bode magnitude and phases are decided by :**

a) The lowest and higher important frequencies of dominant factors of the OLTF

b) The lowest and highest important frequencies of all the factors of the open loop transfer function

c) Resonant frequencies of the second factors

**d) None of the above**

**Answer: d**

**Explanation: T. F. = Kp (1+Tds)**

**There is only one zero which will give slope of +20dB/decade.**

**12. OLTF contains one zero in right half of s-plane then**

a) Open loop system is unstable

b) Close loop system is unstable

**c) Close loop system is unstable for higher gain**

d) Close loop system is stable

**Answer: c**

**Explanation: OLTF contains one zero in right half of s-plane then Close loop system is unstable for higher gain.**

**13. The critical value of gain for a system is 40 and gain margin is 6dB. The system is operating at a gain of:**

**a) 20**

b) 40

c) 80

d) 120

**Answer: a**

**Explanation: Gm (dB) = 20logGM**

**GM =2**

**As we know, GM =K (marginal)/K (desired)**

**K desired =40/2 =20.**

**14. Nichol’s chart is useful for the detailed study and analysis of:**

**a) Closed loop frequency response**

b) Open loop frequency response

c) Close loop and open loop frequency responses

d) open loop and Close loop frequency responses

**Answer: a**

**Explanation: Nichol’s chart is useful for the detailed study and analysis of closed loop frequency response.**

**15. The roots of the characteristic equation of the second order system in which real and imaginary part represents the**:

a) Damped frequency and damping

**b) Damping and damped frequency**

c) Natural frequency and damping ratio

d) Damping ratio and natural frequency

**Answer: b**

**Explanation: Real part represents the damping and imaginary part damped frequency.**

**All-pass and Minimum-phase Systems**

**1. A minimum phase unity feedback system has a bode plot with a constant slope of -20dB/decade for all frequencies. What is the value of the maximum phase margins of the system?**

a) 0°**b) 90°**

c) -90°

d) 180°**Answer: b****Explanation: For given Bode plot, G(s) H(s) =K/jw****As H(s) =1****PM (maximum) =90°.****2. The range of K for the stability of system is 0<K<100. For K =10, The gain Margin of the system****a) 10**

b) 5

c) 0.1

d) 0.5**Answer: a****Explanation: Gain margin of the system is 10.****3. The frequency at which the Nyquist diagram cuts the unit circle is known as:****a) Gain crossover frequency**

b) Phase crossover frequency

c) Damping frequency

d) Corner frequency**Answer: a****Explanation: The frequency at which the Nyquist diagram cuts the unit circle is known as gain cross over frequency.****4. The forward path transfer function of a unity feedback system is given by G(s) = 1/(1+s) ^{^}2.**

**What is the phase margin of the system?**

a) –π rad

b) 0 rad

c) π/2 rad

**d) π rad**

**Answer: d**

**Explanation: Magnitude at gain cross over frequency is 1 and PM is π rad.**

**5. Consider the following statements:**

**1. The delay time is the time required to reach 50% of the final value in the first time.**

**2. The rise time is the time required for the response to rise from 10% to 90% of its final value for underdamped systems.**

**3. The rise time is the time required for the response to rise from 0 to 100% for the underdamped systems.**

**Which of these statements are correct?**

a) 1,2 and 3 only

b) 1 and 2 only

**c) 1 and 3 only**

d) 2 and 3 only negative real axis

**Answer: c**

**Explanation: The rise time is the time required for the response to rise from 10% to 90% of its final value for overdamped systems.**

**6. For minimum phase systems:**

a) Pole must lie on left plane

b) Zeroes must lie on left plane

**c) Poles and zeroes must lie on left plane**

d) Both must lie on right plane

**Answer: c**

**Explanation: For minimum phase systems poles and zeroes must lie on the right s-plane.**

**7. For non-minimum phase systems:**

a) Poles and zeroes must lie on the right s-plane

b) Zeroes must lie on right of s-plane

c) Both lie on left of s-plane

**d) Poles must be on the left and zeroes can be on the right s –plane.**

**Answer: d**

**Explanation: For non-minimum phase systems poles must be on the left and zeroes can be on the right s –plane.**

**8. For all-pass systems:**

a) Poles must lie on right of the s-plane

b) Zeroes must lie on the left of s-plane

**c) Poles must lie on the left and zeroes on the mirror image of the pole can be on the right**

d) None of the mentioned

**Answer: c**

**Explanation: For all-pass systems poles must lie on the left and zeroes on the mirror image of the pole can be on the right.**

**9. All pass system and minimum phase system constitute_________________**

a) Pole must lie on left plane

b) Minimum phase system

**c) Non-minimum phase system**

d) Both must lie on right plane

**Answer: c**

**Explanation: All pass system and minimum phase system constitute non-minimum phase system.**

**10. Minimum phase systems have unique relationship between its phase and magnitude curves**

**a) True**

b) False

**Answer: a**

**Explanation: Minimum phase system has the least phase angle range for a given magnitude curve.**

**Experimentation of Transfer Function**

**1. The impulse response of a LTI system is a unit step function, then the corresponding transfer function is****a) 1/s**

b) 1/s^{2}

c) 1

d) s**Answer: a****Explanation: The impulse response of a LTI system is the transfer function itself and hence for the unit step function . As input then the transfer function will be 1/s.****2. For a type one system, the steady – state error due to step input is equal to**

a) Infinite**b) Zero**

c) 0.25

d) 0.5**Answer: b****Explanation: The steady state error is defined as the error between the final value and the desired response and the difference in the value of both will be the steady state error due to step input for type one system is zero.****3. The equation 2s ^{4}+s^{3}+3s^{2}+5s+10=0 has roots in the left half of s–plane:**

a) One

**b) Two**

c) Three

d) Four

**Answer: b**

**Explanation: The roots of the equation can be calculated using Routh-Hurwitz criterion and hence there are 2 sign changes in the first column of the row and therefore the two roots lie on the right half of s-plane.**

**4. If the Nyquist plot of the loop transfer function G (s)H (s) of a closed-loop system encloses the (1, j0) point in the G (s)H (s) plane, the gain margin of the system is**

a) Zero

b) Greater than zero

**c) Less than zero**

d) Infinity

**Answer: c**

**Explanation: Nyquist plot deals with the open loop poles and zero and equals the encirclements to the open loop poles of the system.**

**5. Consider the function F (s) =5/s (s**

^{2}+s+2) , where F (s) is the Laplace transform f (t). Then the final value theorem is equal toa) 5

**b) 5/2**

c) Zero

d) Infinity

**Answer: b**

**Explanation: Final value theorem is given for the stable system only and this is a type 1 system and for step input the final value can be calculated as 5/2.**

**6. The transfer function of a phase-lead controller is given by**

**a) (1+aTs)/(1+Ts) , a>1 T>0**

b) (1+aTs)/(1+Ts) , a<1 T>0

c) (1-aTs)/(1+Ts) , a>1 T>0

d) (1±Ts)/(1+Ts) , a<1 T>0

**Answer: a**

**Explanation: For the phase lead controller in which the stability and speed of response is more for the system, the magnitude of the pole must be greater than the magnitude of the zero.**

**7. If the system matrix of a linear time invariant continuous system is given by**

**Its characteristic equation is given by:**

**a) s**

^{2}+5s+3=0b) s

^{2}-3s-5=0

c) s

^{2}+3s+5=0

d) s

^{2}+s+2=0

**Answer: a**

**Explanation: The transfer function is calculated by the state variable analysis and hence the transfer function is calculated by state transition matrix and taking the inverse Laplace transform.**

**8. Given a unity feedback control system with G (s) = K/s(s+4), the value of K for which the damping ratio is 0.5.**

a) 1

**b) 16**

c) 64

d) 32

**Answer: b**

**Explanation: The value is found by using the Routh- Hurwitz criteria and equating one of the row of the Routh-Hurwitz criteria equal to zero and hence finding the value of K.**

**9. The LVDT is used in the measurement of:**

**a) Displacement**

b) Acceleration

c) Velocity

d) Humidity

**Answer: a**

**Explanation: The LVDT is the linear variable differential transformer and it is used to calculate the displacement with the inductor process.**

**10. A system with gain margin close to unity and phase margin close to zero is :**

a) Highly stable

b) Oscillatory

**c) Relatively stable**

d) Unstable

**Answer: c**

**Explanation: A system is relative stable not stable if the phase margin is close to zero then the stability is checked by gain margin.**