### Black Diagram Reduction, Signal Flow Graph, Mason's Gain formula, Final value theorem - Topicwise Questions in Control Systems (1987 -2015)

2003
1. The Laplace transform of i(t) is given by I(s) = 2 / [s(1 + s)]. As t → ∞, the value of i(t) tends to
a) 0
b) 1
c) 2
d) ∞

2. The signal flow graph of a system is shown in figure.
The transfer function C(s)/R(s) of the system is

2004

1. Consider the signal flow graph shown in figure. The gain X5/X1 is

2005
1. Despite the percentage of negative feedback, control systems still have problems of Instability because the
a) components used have non-linearity
b) dynamic equations of the sub-systems are not know exactly
c) mathematical analysis involves approximations
d) system has large negative phase angle at high frequencies

2006
1. In the system shown below, x(t) = (sint).u(t).
In steady state, the response y(t) will be

2. The unit impulse response of a system is h(t) = e–t, t ≥ 0. For this system, the steady state value of the output for unit step input is equal to
a) –1
b) 0
c) 1
d) ∞

2007
1. If the Laplace Transform of a signal y(t) is Y(s) = 1/[s(s – 1)], then its final value is:
a) – 1
b) 0
c) 1
d) unbounded

2010
1. Given f(t) = L–1[(3s + 1) / (s3 + 4s2 + (k-3)s)]. If Limt -> ∞ f(t) = 1, then the value of k is
a) 1
b) 2
c) 3
d) 4

2011
1. If F(s) = L[f(t)] = 2(s+1)/(s2+4s+7), then the initial and final values of f(t) are respectively
a) 0,2
b) 2,0
c) 0,2/7
d) 2/7,0

Common Data Questions 2 & 3:
The input – output transfer function of a plant H(s) = 100/s(s+10)2. The plant is placed in a unity negative feedback configuration as shown in figure below.
2. The signal flow graph that DOES NOT model the plant transfer function H(s) is

3. The gain margin of the system under closed loop unity negative feedback is
a) 0 dB
b) 20 dB
c) 26 dB
d) 46 dB

2012
1. The unilateral Laplace transform of f(t) is 1/(s2+s+1). The unilateral transform of t.f(t) is

2. With initial condition x(1) = 0.5, the solution of the differential equation, t.(dx/dt)+x = t is
a) x = t – 1/2
b) x = t2 – 1/2
c) x = t2/2
d) x = t/2

2013
1. A system is described by the differential equation d2y/dt2+5dy/dt+6y(t) = x(t). Let x(t) be a rectangular pulse given as x(t) = 1 for 0<t<2 and zero otherwise. Assuming that y(0) = 0 and dy/dt = 0 at t=0, then the Laplace transform of y(t) is

2014
1. For the system shown,
when the X1(s) = 0, the transfer function Y(s)/X2(s) is

2. Consider the state space system expressed by the signal flow diagram shown in the figure.

The corresponding system is
a) always controllable
b) always observable
c) always stable
d) always unstable

3. Consider the following block diagram in the figure.
The transfer function C(s)/R(s) is

4. The input -3e2tu(t), where u(t) is the unit step function, is applied to a system with transfer function (S-2)/(S+3). If the initial value of the output is -2, then the value of the output at steady state is _____________

2015
1. For the signal flow graph shown in the figure, the value of C(s)/R(s) is

2. By performing cascading and/or summing/differencing operations using transfer function blocks G1(s) and G2(s), one CANNOT realize a transfer function of the form

3. The transfer function of a mass-spring-damper system is given by
the frequency response data for the system are given in the following table.
The unit step response of the system approaches a steady state value of _____________

4. Consider the differential equation dx/dt = 10 – 0.2x with initial condition x(0) = 1. The response x(t) for t > 0 is
a) 2 – e-0.2t
b) 2 – e0.2t
c) 50 – 49e-0.2t
d) 50 – 49e0.2t

5. The position control of a DC servo-motor is given in the figure. The values of the parameters are KT = 1 N-mA, Ra = 1 ohm, La = 0.1 H, J = 5 kg-m2, B = 1 N-m/(rad/sec) and Kb = 1 volt/(rad/sec). d is _______.
The steady state position response(in radians) due to unit impulse disturbance torque T

### P, PI, PD, PID Controllers and Lag, Lead, Lag-Lead Compensators - Topicwise Questions in Control Systems (1987 -2015)

2003
1. A PD controller is used to compensate a system. Compared to the uncompensated system, the compensated system has
a) a higher type number
b) reduced damping
c) higher noise amplification
d) larger transient overshoot

2005
1. Which one of the following polar diagrams corresponds to a lag network?

2006
1. The transfer function of a phase lead compensator is given by Gc(s) = (1 + 3Ts)/(1 + Ts) where T ≥ 0. The maximum phase shift provided by such a compensator is
a) π/2
b) π/3
c) π/4
d) π/6

2007
1. A control system with a PD controller is shown in the figure.

If the velocity error constant Kv = 1000 and the damping ratio ξ = 0.5, then the values of Kp and KD are
a) Kp = 100, KD = 0.09
b) Kp = 100, KD = 0.9
c) Kp = 10, KD = 0.09
d) Kp = 10, KD = 0.9

2. The open loop transfer function of a plant is given as G(s) = 1/(s2 – 1). If the plant is operated in a unity feedback configuration, then the lead compensator that can stabilize this control system is

2008
1. Group – I gives two possible choices for the impedance Z in the diagram. The circuit elements in Z satisfy the condition R2C2>R1C1. The transfer function Vo/V1 represents a kind of controller. Match the impedance's in Group – I with the types of controllers in Group – II.
a) Q – 1, R – 2
b) Q – 1, R – 3
c) Q – 2, R – 3
d) Q – 3, R – 2

2009
1. The magnitude plot of a rational transfer function G(s) with real coefficients is shown below.
Which of the following compensator has such a magnitude plot?
b) Lag compensator
c) PID compensator
d) Lead – Lag compensator

2. If the transfer function of the following circuit is Vo(s)/Vi(s) = 1/(2+sCR),L is
then the value of the load resistor R
a) R/4
b) R/2
c) R
d) 2R

2010
1. A unity negative feedback closed loop system has a plant with the transfer function G(s) = 1/(s2 + 2s + 2) and a controller Gc(s) in the feed forward path. For a unit step input, the tranfer function of the controller that gives minimum steady state error is

2012
Statement for Linked Answer Questions : 1 and 2

The transfer function of a compensator is given as Gc(s)= (s+a)/(s+b).
1. Gc(s) is a lead compensator if
a) a = 1, b =2
b) a = 3, b =2
c) a = –3, b = –1
d) a = 3, b =1

2. The phase of the above lead compensator is maximum at
Solution (1 & 2) : https://www.youtube.com/watch?v=CSWEFDPoSsA

2015
1. A lead compensator network includes a parallel combination fo R and C in the feed-forward path. If the transfer function of the compensator is GC(s) = (s+2)/(s+4) , the value of RC is ___________

2. The transfer function of a first order controller is given as Gc(s) = K(s+a) / (s+b), where K, a and b are positive real numbers. The condition for this controller to act as a phase lead compensator is
a) a < b
b) a > b
c) K < ab
d) K > ab

### Frequency Response and Bode Plots (Magnitude & Phase) - Topicwise Questions in Control Systems (1987 -2015)

2003
7. The approximate Bode magnitude plot of a minimum phase system is shown in figure.
The transfer function of the system is

2004
3. Consider the Bode magnitude plot shown in figure.
The transfer function H(s) is

6. A system has poles at 0.01 Hz, 1 Hz and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz. The approximate phase of the system response at 20 Hz is
a) - 90o
b) 0o
c) 90o
d) - 180o

2006
3. Consider two transfer functions G1(s) = 1 / (s2 + as + b) and G2(s) = s / (s2 + as + b). The 3 dB bandwidths of their frequency responses are, respectively

Solution :

2007
7. The asymptotic Bode plot of a transfer function is as shown in figure.
The transfer function G(s) corresponding to this Bode plot is

2008
7. The magnitude of frequency response of an under damped second order system is 5 at 0 rad/sec and peaks to 10/√3 at 5√2 rad/sec. The transfer function of the system is

2010
2. A system with transfer function Y(s)/X(s) = s/(s + p) has an output y(t) = cos(2t – π/3) for the input signal x(t) = p cos(2t – π/2). Then, the system parameter 'p' is
a) √3
b) 2/√3
c) 1
d) √3/2

3. For the asymptotic bode magnitude plot shown below,
the system transfer function can be

2013
1. The Bode plot of a transfer function G(s) is shown in the figure below.
The gain (20log|G(s)|) is 32 dB and – 8 dB at 1 rad/sec and 10 rad/sec respectively. The phase is negative for all ω. Then G(s) is
a) 39.8/s
b) 39.8/s2
c) 32/s
d) 32/s2

2014
11. The Bode asymptotic magnitude plot of a minimum phase system is shown in the figure.
If the system is connected in a unity negative feedback configuration, the steady state error of the closed loop system, to a unit ramp input is _____________

18. 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) – 80 dB/decade
b) – 40 dB/decade
c) + 40 dB/decade
d) + 80 dB/decade

2015
14. Consider the Bode plot shown in figure.
Assume that all the poles and zeros are real valued. The value of fH – fL (in Hertz) is ____________________

### GATE 2004 ECE Network Theory - Complete Video Solutions with Answers

2004
1. Consider the network graph shown in figure. Which one of the following is NOT a ‘tree’ of this group?

2. The equivalent inductance measured between the terminals 1 and 2 for the circuit shown in figure, is
a) L1 + L2 + M
b) L1 + L2 - M
c) L1 + L2 + 2M
d) L1 + L2 - 2M

3. The circuit shown in figure, has R = 1/3 Ω, L = 1/4 H, C = 3 F has input voltage v(t) = sin2t. The resulting current i(t) is

4. For the circuit shown, the time constant RC = 1 ms. The input voltage is vi(t) = √2 sin103t. The output voltage vo(t) is equal to

5. For the R-L circuit shown, the input voltage vi(t) = u(t). The current i(t) is

6. For the lattice circuit shown in figure, Za = j2Ω and Zb = 2Ω. The values of the open circuit impedance parameters (Z-parameters) are

7. The circuit shown in figure has initial current iL(0-) = 1 A through the inductor and an initial voltage Vc(0-) = - 1 volt across the capacitor. For input v(t) = u(t), the Laplace transform of the current i(t) for t ≥ 0 is

8. Consider the following statements S1 and S2
S1: at the resonant frequency, the impedance of a series RLC circuit is zero.
S2: In a parallel G-L-C circuit, increasing the conductance G results in increase in its Q-factor.
Which one of the following is correct?
a) S1 is FALSE and S2 is TRUE
b) Both S1 and S2 are TRUE
c) S1 is TRUE and S2 is FALSE
d) Both S1 and S2 are FALSE

### GATE 2005 ECE Network Theory - Complete Video Solutions with Answers

2005

1. The condition on R, L and C such that the step response y(t) in figure has no oscillations, is

2. The ABCD parameters of an ideal n:1 transformer shown in figure are given below. The value of 'x' will be
a) n
b) 1/n
c) n2
d) 1/n2

3. In a series RLC circuit, R = 2 kΩ, L = 1 H, and C = 1/400 µF. The resonant frequency is
a) 2 x 104 Hz
b) (1/π) x 104 Hz
c) 104 Hz
d) 2π x 104 Hz

4. The maximum power that can be transferred to the load resistor RL form the voltage source in figure is
a) 1 W
b) 10 W
c) 0.25 W
d) 0.5 W

5. The first and Last critical frequency of an RC driving point impedance function must respectively be
a) a zero and a pole
b) a zero and a zero
c) a pole and a pole
d) a pole and a zero

6. For the circuit in figure, the instantaneous current i1(t) is

7. Impedance Z as shown in figure is
a) j29 Ω
b) j9 Ω
c) j19 Ω
d) j39 Ω

8. For the circuit shown in figure, Thevenin's voltage and Thevenin's equivalent resistance at terminals a-b is
a) 5 V and 2 Ω
b) 7.5 V and 2.5 Ω
c) 4 V and 2 Ω
d) 3 V and 2.5 Ω

9. If R1 = R2 = R4 = R and R3 = 1.1R in the bridge circuit shown in figure, then the reading in the ideal voltmeter connected between a and b is
a) 0.238 V
b) 0.138 V
c) - 0.238 V
d) 1 V

10. The h-parameters of the circuit shown in figure are

11. A square pulse of 3 volts amplitude is applied to RC circuit shown in figure. The capacitor is initially uncharged. The output voltage Vo at time t = 2 sec is
a) 3 V
b) -3 V
c) 4 V
d) -4 V

### GATE 2006 ECE Network Theory - Complete Video Solutions with Answers

2006

1. A two port network is represented by ABCD parameters as shown below. If port-2 is terminated by RL, the input impedance seen at port-1 is given by

2. In the two port network shown in the figure below, Z12 and Z21 are respectively

3. The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by
a) RL network only
b) RC network only
c) LC network only
d) RC as well as RL networks

4. A 2 mH inductor with some initial current can be represented as shown below, where 's' is the Laplace Transform variable. The value of initial current is :
a) 0.5 A
b) 2.0 A
c) 1.0 A
d) 0.0 A

5. In the figure shown below, assume that all the capacitors are initally uncharged. If Vi(t) = 10 u(t) volts, then Vo(t) is given by

6. A negative resistance Rneg is connected to a passive network N having driving point impedance Z1(s) as shown below. For Z2(s) to be positive real,