## Frequency Response

### GATE-CH-1995-1-q-ctrl-1mark

1995-1-q-ctrl

Bode diagrams are generated from output response of the system subjected to which of the following input

• impulse

• step

• ramp

• sinusoidal

### GATE-CH-1995-1-t-ctrl-1mark

1995-1-t-ctrl

According to Bode stability criterion, a system is unstable if the open loop frequency response exhibits an amplitude ratio exceeding unity at frequency for which phase lag is

• $$0^\circ$$

• $$45^\circ$$

• $$90^\circ$$

• $$180^\circ$$

### GATE-CH-1998-1-24-ctrl-1mark

1998-1-24-ctrl

A first order system with a time constant of 1 min is subjected to frequency response analysis. At an input frequency of 1 radian/min, the phase shift is

• $$45^{\circ }$$

• $$-90^{\circ }$$

• $$-180^{\circ }$$

• $$-45^{\circ }$$

### GATE-CH-1999-1-27-ctrl-1mark

1999-1-27-ctrl

A sinusoidal variation in the input passing through a linear first-order system

• becomes more oscillatory (frequency increases)

• becomes less oscillatory (frequency decreases)

• gets amplified (magnitude increases)

• gets attenuated (magnitude decrease)

### GATE-CH-2000-1-27-ctrl-1mark

2000-1-27-ctrl

Select the correct statement from the following:

• The frequency response of a pure capacity process is unbounded
• The phase lag of a pure time delay system decreases with increasing frequency
• The amplitude ratio of a pure capacity process is inversely proportional to the frequency
• The amplitude ratio of a pure time delay system increases with frequency

[Index]

### GATE-CH-2000-1-28-ctrl-1mark

2000-1-28-ctrl

For a feedback control system to be stable, the

• roots of the characteristics equation should be real
• poles of the closed loop transfer function should lie in the left half of the complex plane
• Bode plots of the corresponding open loop transfer function should monotonically decrease
• poles of the closed loop transfer function should lie in the right half of the complex plane

### GATE-CH-2002-1-15-ctrl-1mark

2002-1-15-ctrl

A first order system with unity gain and time constant $$\tau$$ is subjected to a sinusoidal input of frequency $$\omega=1/\tau$$. The amplitude ratio for this system is

• 1

• 0.5

• $$1/\sqrt{2}$$
• 0.25

### GATE-CH-2002-2-16-ctrl-2mark

2002-2-16-ctrl

The frequency response of a first order system, has a phase shift with lower and upper bounds given by

• $$[-\infty,\pi/2]$$

• $$[-\pi/2,\pi/2]$$

• $$[-\pi/2,0]$$

• $$[0,\pi/2]$$

### GATE-CH-2010-24-ctrl-1mark

2010-24-ctrl

The transfer function, $$G(s)$$, whose asymptotic Bode diagram is shown below, is

• $$10s+1$$

• $$s-10$$

• $$s+10$$

• $$10s-1$$

### GATE-CH-2012-25-ctrl-1mark

2012-25-ctrl

The Bode stability criterion is applicable when

• Gain and phase curves decreases continuously with frequency
• Gain curve increases and phase curve decreases with frequency
• Gain curve and phase curve both increases with frequency
• Gain curve decreases and phase curve increases with frequency

[Index]

### GATE-CH-1992-19-b-ctrl-6mark

1992-19-b-ctrl

For the loop above, determine:

(i) The maximum gain for stable operation.
{#1}

(ii) The corresponding frequency of oscillation (rad/min).
{#2}

### GATE-CH-1997-26-ctrl-2mark

1997-26-ctrl

The open loop transfer function for a process is $$\dfrac {1}{4(3s+1)^4}$$, where the time constant is in minutes.
Determine:

{#1}

(ii) the ultimate gain.
{#2}

### GATE-CH-2000-14-ctrl-5mark

2000-14-ctrl

A feedback control loop with a proportional controller has an open loop transfer function $$G_L(s) = \dfrac {K_c}{s(5s+1)^2}$$ where time is in minutes.

(i) The crossover frequency in radians/min = _________

{#1}

(ii)The ultimate controller gain = _________

{#2}

### GATE-CH-2006-84-85-ctrl-4mark

2006-84-85-ctrl

For the system shown below, $$G_1(s) = \dfrac {1}{\tau _1s+1}$$, $$G_2(s)=\dfrac {1}{\tau _2s+1}$$ and $$\tau _2=2\tau _1$$.

When the system is excited by the sinusoidal input $$X(t) = \sin \omega t$$, the intermediate response $$Y$$ is given by $$Y = A\sin (\omega t+\phi )$$

(i) If the response $$Y$$ lags behind the input $$X$$ by 45$$^\circ$$ and $$\tau _1=1$$, then the input frequency $$\omega$$ is
{#1}

(ii) For the same input, the amplitude of the output $$Z$$ will be
{#2}

### GATE-CH-1998-2-21-ctrl-2mark

1998-2-21-ctrl

The frequency response of a dynamic element shows a constant magnitude ratio at all frequencies. The element exhibits a negative phase shift at all frequencies. The absolute value of the phase shift increases linearly with frequency. The element has the transfer function

• $$e^{-\tau s}$$

• $$\dfrac {\tau _1s+1}{\tau _2s+1}$$

• $$\dfrac {\tau _1s}{\tau _2s+1}$$

• $$\dfrac {\tau _1s+1}{\tau _2s}$$

[Index]

### GATE-CH-2000-2-23-ctrl-2mark

2000-2-23-ctrl

The time constant of a unity gain, first order plus time delay process is 5 min. If the phase lag at a frequency of 0.2 rad/min is 60o, then the dead time (in minutes) is

• $$5\pi/12$$
• $$\pi/6$$
• $$\pi/12$$
• $$\pi/3$$

### GATE-CH-2001-2-18-ctrl-2mark

2001-2-18-ctrl

An ideal PID controller has the transfer function $$\left[1+\dfrac{1}{0.5s} + 0.2s\right]$$. The frequency at which the magnitude ratio of the controller is 1, is

• 0.5/0.2

• 0.2/0.5

• 0.2$$\times$$0.5

• $$1/\sqrt{0.2\times0.5}$$

### GATE-CH-2003-77-ctrl-2mark

2003-77-ctrl

Find the ultimate gain and frequency for a proportional controller in the case of a process having the following transfer function

$G_p(s) = \frac{1}{(4s+1)(2s+1)(s+1)}$

• $$\displaystyle \omega = \frac{1}{\sqrt{14}}; \ K_c = \frac{45}{7\sqrt{14}}$$

• $$\displaystyle \omega = \sqrt{\frac{7}{6}}; \ K_c = \frac{46}{3}$$

• $$\omega = 1; \ K_c = 13$$

• $$\displaystyle \omega = \sqrt{\frac{7}{8}}; \ K_c = \frac{45}{4}$$

### GATE-CH-2004-80-ctrl-2mark

2004-80-ctrl

Consider a system with open-loop transfer function

$G(s) = \frac{1}{(s+1)(2s+1)(5s+1)}$

Match the range of $$\omega$$ (frequency) in Group I with the slope of the asymptote of the $$\log \text{AR}$$ (amplitude ratio) versus $$\log \omega$$ plot in Group II.

Group I Group II (1) $$\quad -5$$ (2) $$\quad -3$$ (3) $$\quad -2$$ (4) $$\quad -1$$ (5) $$\quad$$zero

• P-5, Q-2

• P-4, Q-2

• P-5, Q-3

• P-4, Q-1

### GATE-CH-2005-50-ctrl-2mak

2005-50-ctrl

The value of ultimate period of oscillation $$P_u$$ is 3 minutes, and that of the ultimate controller gain $$K_{cu}$$ is 2. Select the correct set of tuning parameters (controller gain $$K_c$$, derivative time constant $$\tau_D$$ in minutes, and the integral time constant $$\tau_I$$ in minutes) for a PID controller using Ziegler-Nichols controller settings.

• $$K_c=1.1; \quad \tau_I=2.1; \quad \tau_D=1.31$$

• $$K_c=1.5; \quad \tau_I=1.8; \quad \tau_D=0.51$$

• $$K_c=1.5; \quad \tau_I=1.8; \quad \tau_D=0.51$$

• $$K_c=1.2; \quad \tau_I=1.5; \quad \tau_D=0.38$$

[Index]

### GATE-CH-2006-58-ctrl-2mark

2006-58-ctrl

A process is perturbed by a sinusoidal input, $$u_t = A \sin \omega t$$. The resulting process output is $$Y(s) = \dfrac{K A\omega}{(\tau s+1)(s^2+\omega^2)}$$. If $$y(0) = 0$$, the differential equation representing the process is

• $$\dfrac{dy(t)}{dt} + \tau y(t) = K u(t)$$

• $$\tau \dfrac{dy(t)}{dt} + y(t) = K Au(t)$$

• $$\tau \dfrac{dy(t)}{dt} + y(t) = K u(t)$$

• $${\displaystyle \tau \left[\dfrac{dy(t)}{dt} + y(t)\right] = K Au(t)}$$

### GATE-EE-2013-A-15-ctrl-1mark

EE-2013-A-15-ctrl

The Bode plot of a transfer function $$G(s)$$ is shown in the figure below.

The gain $$(20\log |G(s)|)$$ is 32 dB and -8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all $$\omega$$. Then $$G(s)$$ is

• $$\dfrac{39.8}{s}$$

• $$\dfrac{39.8}{s^2}$$

• $$\dfrac{32}{s}$$

• $$\dfrac{32}{s^2}$$

### GATE-IN-2014-23-ctrl-1mark

IN-2014-23-ctrl

A plant has an open-loop transfer function, $G_p(s) = \frac{20}{(s+0.1)(s+2)(s+100)}$ The approximate model obtained by retaining only one of the above poles, which is closest to the frequency response of the original transfer function at low frequency is

• $$\dfrac{0.1}{s+0.1}$$
• $$\dfrac{2}{s+2}$$
• $$\dfrac{100}{s+100}$$
• $$\dfrac{20}{s+0.1}$$

### GATE-CH-2017-51-ctrl-2mark

2017-51-ctrl

The open loop transfer function of a process with a proportional controller (gain $$K_c$$) is $G_{OL} = K_c\frac{e^{-2s}}{s}$ Based on Bode criterion for closed-loop stability, the ultimate gain of the controller, rounded to 2 decimal places, is ______

### GATE-CH-2018-44-ctrl-2mark

2018-44-ctrl

Consider the following transfer function: $G(s) = \frac{3}{(5s+1)^2}$ where, the natural period of oscillation is in min. The amplitude ratio at a frequency of 0.5 rad/min is ________ (rounded off to second decimal place).

[Index]

### GATE-CH-2019-49-ctrl-2mark

2019-49-ctrl

For the closed loop system shown in figure, the phase margin (in degrees) is ___________

### GATE-IN-2014-46-ctrl-2mark

IN-2014-46-ctrl

The transfer function of a system is given by $$\displaystyle G(s) = \frac{e^{-s/500}}{s+500}$$. The input to the system is $$x(t)=\sin 100\pi t$$. In periodic steady state the output of the system is found to be $$y(t)=A\sin(100\pi t - \phi)$$. The phase angle ($$\phi$$) in degree is _______

### GATE-CH-1999-2-18-ctrl-2mark

1999-2-18-ctrl

Each item given in the left-hand column is closely associated with a specific characteristic listed in the right-hand column. Match each of the items with the corresponding characteristic.

• I. Transportation lag

• II. Control valve

[Index]