This set of Network Theory Multiple Choice Questions & Answers (MCQs) focuses on “Laplace Transform of a Periodic Function”.

1. For a network having 1 Ω resistor and 1 F capacitor in series, the impedance Z(s) is ____________

a) \(\frac{s+1}{s}\)

b) \(\frac{s+2}{s}\)

c) \(\frac{s}{s+1}\)

d) \(\frac{s}{s+2}\)

View Answer

Explanation: We know that the impedance, Z(s) is given by,

Z(s) = resistance + 1/capacitor

= 1 + \(\frac{1}{s}\)

= \(\frac{s+1}{s}\).

2. A system function H(s) = \(\frac{25}{s^2+4s+100}\), the resonant frequency in rad/sec and bandwidth in rad/sec is ____________

a) 5, 10

b) 5, 4

c) 10, 4

d) 10, 10

View Answer

Explanation: Given that, H(s) = \(\frac{25}{s^2+4s+100}\)

So, we can infer that, ω

_{n}= (100)

^{0.5}

= 10

And hence, ω

_{r}≃ 10

So, resonant frequency = 10 rad/sec

And Bandwidth = 10 rad/sec.

3. Given a series RLC circuit. The impedance Z(s) of the circuit will be?

a) \(\frac{5(s^2+5)}{s^2+s+1}\)

b) \(\frac{5(s^2-5)}{s^2-s-1}\)

c) \(\frac{s^2-10s-100}{s}\)

d) \(\frac{s^2+10s+100}{s}\)

View Answer

Explanation: The impedance Z(s) = R + sL + \(\frac{1}{sC}\)

= \(\frac{s^2 LC+RCs+1}{sC}\)

This is similar to \(\frac{s^2+10s+100}{s}\).

4. A system function H(s) = \(\frac{V(s)}{I(s)} = \frac{s}{(s+4)}\). For i(t) = u(t) and value of system is 0 for t<0. Then v(t) is ___________

a) 1 – e^{-4t}

b) e^{-4t}

c) e^{4t}

d) 1 + e^{-4t}

View Answer

Explanation: Given that, V(s) = \(\frac{s}{s+4}.\frac{1}{s}\)

= \(\frac{1}{s+4}\)

So, v (t) = e

^{-4t}.

5. A system is at rest for t < 0. It is given by \(\frac{dy}{dt}\) + 2y = u(t)sin(2t+A). If steady state is reached at t = 0, then the value of angle A is ___________

a) 0°

b) 45°

c) -45°

d) ∞

View Answer

Explanation: Given that, \(\frac{dy}{dt}\) + 2y = u(t)sin(2t+A)

Or, y(s).s + 2y(s) = \(\frac{ω}{s^2+4}\)

Or, s + 2 = 0

Or, j ω + 2 = 0

Or, j2 + 2 = 0

Or, 1 + j = 0

Or, tan

^{-1}(1) = 45°.

6. The value of 2 [u (t – 1) – u (1 – 2t)] (u ( t + 1) + u (t)) at t = 3 sec is ____________

a) 0

b) 4

c) ∞

d) 1

View Answer

Explanation: Putting t=3 in the given equation, we get,

2[u (2) – u (1 – 6)] [u (4) + u (3)]

= 2 [1 – 0] [1 + 1]

= 4.

7. Barlett’s Bisection Theorem is applicable to ___________

a) Unsymmetrical networks

b) Symmetrical networks

c) Both unsymmetrical and symmetrical networks

d) Neither to unsymmetrical nor to symmetrical networks

View Answer

Explanation: A symmetrical network can be split into two halves.

So the z parameters of the network are symmetrical as well as reciprocal of each other. Hence Barlett’s Bisection Theorem is applicable to Symmetrical networks.

8. The values of z_{11} and z_{21} for a T circuit having resistances 20 Ω each is _____________

a) 40, 20

b) 40, 60

c) 60, 40

d) 40, 40

View Answer

Explanation: To determine the values of z

_{11}and z

_{21}, we apply a voltage source V

_{1}to the input port and leave the output port open.

Thus, z

_{11}= \(\frac{V_1}{I_1} = \frac{(20+20) I_1}{I_1}\)

= 40 Ω

Now, z

_{21}is the input impedance at port 1.

So, z

_{21}= \(\frac{V_2}{I_1}\) = 20 Ω.

9. The values of z_{12} and z_{22} for a T circuit having resistances 20 Ω each is _____________

a) 40, 20

b) 40, 60

c) 20, 40

d) 40, 40

View Answer

Explanation: To find z

_{12}and z

_{22}, we apply a voltage source V

_{2}to the output port and leave the input port open.

Thus, z

_{12}= \(\frac{V_1}{I_2} = \frac{(20) I_2}{I_2}\)

= 20 Ω

Now, z

_{22}is the input impedance at port 1.

So, z

_{22}= \(\frac{V_2}{I_2} = \frac{(20+20) I_2}{I_2}\) = 40 Ω.

10. If y(t) = 120e^{10x(t)}, then the relation is _________

a) Dynamic

b) Static

c) Memory

d) Memoryless but not static

View Answer

Explanation: Given relation, y (t) = 120

^{10x(t)}.

The system represented by the above relation is static because the present output of the system as well as memoryless as its present output does not depend on its past input. It is not a dynamic system since the value of the system increases exponentially.

11. The z parameters form a matrix of the form ___________

a) [z_{11} z_{12}; z_{21} z_{22}]

b) [z_{11} z_{12}; z_{22} z_{21}]

c) [z_{12} z_{11}; z_{21} z_{22}]

d) [z_{11} z_{22}; z_{12} z_{21}]

View Answer

Explanation: Z parameters are also called as the impedance parameters.

There are 4 main types of Z-parameters, z

_{11}, z

_{12}, z

_{21}, z

_{22}

They are arranged in the form of a matrix given by [z

_{11}z

_{12}; z

_{21}z

_{22}].

12. The Laplace transform of the function e^{-2t}cos(3t) + 5e^{-2t}sin(3t) is ____________

a) \(\frac{(s+2)-15}{(s+2)^2-9}\)

b) \(\frac{(s+2)+15}{(s+2)^2-9}\)

c) \(\frac{(s+2)+15}{(s+2)^2+9}\)

d) \(\frac{(s+2)-15}{(s+2)^2+9}\)

View Answer

Explanation: L {e

^{-2t}cos(3t) + 5e

^{-2t}sin(3t)} = \(\frac{(s+2)}{(s+2)^2+9} + \frac{5 3}{(s+2)^2+9}\)

= \(\frac{(s+2)+15}{(s+2)^2+9}\).

13. The Laplace transform of the function 6e^{5t}cos(2t) – e^{7t} is ______________

a) \(\frac{6(s-5)}{(s-5)^2+4} – \frac{1}{s-7}\)

b) \(\frac{6(s-5)}{(s-5)^2+4} + \frac{1}{s-7}\)

c) \(\frac{6(s+5)}{(s+5)^2+4} – \frac{1}{s-7}\)

d) \(\frac{6(s+5)}{(s+5)^2+4} + \frac{1}{s-7}\)

View Answer

Explanation: L {6e

^{5t}cos(2t) – e

^{7t}} = \(\frac{6(s-5)}{(s-5)^2+4} – \frac{1}{s-7}\).

14. The Laplace transform of the function cosh^{2}(t) is ____________

a) \(\frac{s^2+2}{s(s^2+4)}\)

b) \(\frac{s^2-2}{s(s^2-4)}\)

c) \(\frac{s^2-2}{s(s^2+4)}\)

d) \(\frac{s^2+2}{s(s^2-4)}\)

View Answer

Explanation: L ((\(\frac{1}{2}\)(e

^{t}– e

^{-t}))

^{2})

= L \(\left(\frac{e^{2t}}{4} + \frac{1}{2} + \frac{e^{-2t}}{4}\right)\)

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

= \(\frac{s^2-2}{s(s^2-4)}\).

15. Given a system function H(s) = \(\frac{V(s)}{I(s)} = \frac{(s+4)}{(s+3)^2}\). And i(t) is a unit step, then V(t) in the steady state is ___________

a) \(\frac{4}{9}\)

b) \(\frac{4}{3}\)

c) 0

d) ∞

View Answer

Explanation: V(s) = \(\frac{(s+4)}{(s+3)^2}, I(s) = \frac{(s+4)}{s(s+3)^2}\)

At steady state, sV(s) = \(\frac{0+4}{(0+3)^2}\)

= \(\frac{4}{9}\).

**Sanfoundry Global Education & Learning Series – Network Theory.**

To practice all areas of Network Theory, __here is complete set of 1000+ Multiple Choice Questions and Answers__.