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Test: Laplace Analysis of Networks - Electrical Engineering (EE) MCQ


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15 Questions MCQ Test - Test: Laplace Analysis of Networks

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Test: Laplace Analysis of Networks - Question 1

For the RC parallel circuit, determine the voltage across the capacitor using Laplace transform, assume capacitor is initially relaxed.

Detailed Solution for Test: Laplace Analysis of Networks - Question 1

Concept:

If V0 is the initial voltage across capacitor

Calculation:

Circuit diagram at laplace domain

As capacitor was initially relaxed: V0 = 0 (Voltage source will be replaced by a short circuit)

Given;

C = 1μF

R= 5 Ω 

Use nodal analysis at node VC

⇒ VC/R + VC/(1/sC) = 10/s

⇒ VC( 1/R + sC) = 10/s

⇒ VC( 1+ sRC)/R = 10/s

⇒ VC= 10R/s( 1+ sRC)

⇒ VC= 10R/sRC( 1/RC+ s)

⇒ VC= (10/C)/s( s+ 1/RC)

⇒ VC = A/s + B/(s + 1/RC)

Determine the values of A and B using partial fraction

As=0 = (10/C)/(s + 1/RC) = (10/C)( 0 +1/RC) = 10R

Bs= -1/RC = (10/C)/s = -10R

⇒ VC = (10R)/s + (-10R)/(s + 1/RC)

Using inverse laplace transform:

⇒ VC = 10R -10Re-t/RC

⇒ VC = 10R(1 -e-t/RC)

∴  VC = 50(1 - e-t/5)

Test: Laplace Analysis of Networks - Question 2

Calculate the transfer function of a system if the response is c(t) = e-6t for t > 0. 

Given the input signal is an impulse signal.

Detailed Solution for Test: Laplace Analysis of Networks - Question 2

The ratio of Laplace transforms of output to the Laplace transform of the input assuming all inertial conditions are zero.

L[0] is Laplace transform of output

L[I] is Laplace transform of the input

Solution:

Output = e-6t

L[0] = L[e-6t]

⇒ L[0] = 1 / s+6

We have, input signal is an impulse signal.

⇒ Input = δ(t)

L[I] = 1

From above concept,

Hence, 

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Test: Laplace Analysis of Networks - Question 3

What will be the transfer function of the system shown below?

Detailed Solution for Test: Laplace Analysis of Networks - Question 3

Concept:

Transfer function:

The transfer function is defined as the ratio of the Laplace transform of the output variable to the input variable with all initial conditions zero.

TF = L[output] / L[input]| Initial conditions = 0

TF = C(s) / R(s)

The transfer function of a linear time-invariant system can also be defined as the Laplace transform of the impulse response, with all the initial conditions set to zero. 

Application:

The given circuit can be drawn as in Laplace domain as shown,

For the circuit above,

Transfer Function = 

We have,

Vi (S) = I(S) (R + LS + 1/Cs)

And, Vo (S) = I(S) (1Cs)

From equation (1),

⇒ Transfer Function = 

Test: Laplace Analysis of Networks - Question 4

The voltage transfer function of the network shown in the figure below is

Detailed Solution for Test: Laplace Analysis of Networks - Question 4

Concept:

The Laplace transform resistance, inductor, and capacitance are given by:

  1. Resistance: R
  2. Inductor: sL
  3. Capacitor: 1 / sC

​Calculation:

The circuit diagram in the Laplace domain is given below:

Applying voltage division rule across capacitor:

Test: Laplace Analysis of Networks - Question 5

Which of the following is NOT one of the properties of transfer function?

Detailed Solution for Test: Laplace Analysis of Networks - Question 5

Transfer function:

The transfer function is defined as the ratio of the Laplace transform of the output variable to the input variable with all initial conditions zero.

TF = L[output] / L[input]| Initial conditions = 0

TF = C(s) / R(s)

The transfer function of a linear time-invariant system can also be defined as the Laplace transform of the impulse response, with all the initial conditions set to zero.

Properties of transfer function:

  • The transfer function is defined only for a linear time-invariant system. It is not defined for nonlinear systems.
  • The transfer function is independent of the input and output.
  • Because the transfer function of the system depends on the governing dynamic equation of the system only.
  • If the transfer function is dependent on the input means the system will come under non-linear systems, but actually, the transfer function is defined for linear systems only.
  • Transfer function analysis is not valid for the system that contains variables having initial values.
Test: Laplace Analysis of Networks - Question 6

If the step response to the input step amplitude of 1 V is given by Vo(t) = (1 - e-t / RC), the network can be represented by:

Detailed Solution for Test: Laplace Analysis of Networks - Question 6

Concept:

A transfer function is defined as the ratio of the Laplace transform of output to the Laplace transform of input when initial conditions are zero.

Where, Vo = Output voltage

V= input voltage

Calculation:

Vo(t) = (1 - e-t / RC)   ---(1)

Vi(t) = 1 V  ---(2)

Applying Laplace transform to the (1) equation:

Now taking Laplace transform of equation (2):

Vi(s) = 1 / s

Now dividing V0(s) by Vi(s) we get:

Now consider the diagram given in option(a):

Finding V0 by using voltage division rule:

Hence it gives the same transfer function as calculated in equation (4)

Hence option (a) is the correct answer.

Test: Laplace Analysis of Networks - Question 7

The transfer function of a linear system is the

Detailed Solution for Test: Laplace Analysis of Networks - Question 7

Transfer function:

  • The transfer function of a control system is defined as the ratio of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero.
  • It is also defined as the Laplace transform of the impulse response.

If the input is represented by R(s) and the output is represented by C(s), then the transfer function will be:

 

Test: Laplace Analysis of Networks - Question 8

Transfer functions can be derived for a

Detailed Solution for Test: Laplace Analysis of Networks - Question 8

Concept:

  • Transfer Functions are  particularly useful in LTI systems for two reasons:
  • If your input is made up of a bunch of these little parts, then you can use the transfer function to find the combined output. For instance, if your input was sin(t)+sin(2t), then your output would be:

ysin(t) + sin(2t) = ysin(t) + ysin(2t) = 10sin(t) + 2sin(2t)

So,  the transfer function tells us how the system reacts to all types of inputs. This is not true for non-linear systems, so the transfer function isn't useful there.

If your input is delayed, then the output is also delayed:

ysin(t − 1) = 10sin(t - 1)

So, the transfer function is valid for time-shifted inputs. Again, for time-variant systems, this isn't true, so if your input has phase shifts, then the transfer function is useless.

To sum up: LTI systems have nice properties that let us use a simple-looking transfer function to deal with non-simple inputs. When you take away linearity and time-invariance, the transfer function doesn't give you enough information to be helpful.

Test: Laplace Analysis of Networks - Question 9

The average power consumed by the following circuit is

Vrms = 20∠ 53.13°V 

Detailed Solution for Test: Laplace Analysis of Networks - Question 9

Concept:

  • “Average power observed by Resistor” (P)
  • “Reactive power observed by Inductor OR capacitor” (Q).

Calculation:

Since we know active power OR average power observed by Inductor is zero.

Method 1:

Method 2:

Current through Resistor :

Test: Laplace Analysis of Networks - Question 10

The final value of  will be

Detailed Solution for Test: Laplace Analysis of Networks - Question 10

Concept:

1. Final value theorem:

A final value theorem allows the time domain behavior to be directly calculated by taking a limit of a frequency domain expression

The final value theorem states that the final value of a system can be calculated by

Where F(s) is the Laplace transform of the function.

For the final value theorem to be applicable system should be stable in a steady-state and for that real part of the poles should lie on the left side of s plane.

2. Initial value theorem:

It is applicable only when the number of poles of C(s) is more than the number of zeros of C(s).

Calculation:
Given that, 

Poles lies at s = 0, ±j 1

Roots lies on the imaginary axis, so it is marginally stable.

So the Final value theorem is not applicable as the system is oscillatory in nature. 

∴ The correct answer is option C 'cannot be determined'.

Test: Laplace Analysis of Networks - Question 11

Transfer function may be defined as ______

Detailed Solution for Test: Laplace Analysis of Networks - Question 11

Transfer function may be defined as the ratio of laplace transform of output to input with zero initial conditions.

Test: Laplace Analysis of Networks - Question 12

Any system is said to be stable if and only if ____________

Detailed Solution for Test: Laplace Analysis of Networks - Question 12

Any system is said to be stable if and only if it poles lies at the left of imaginary axis.

Test: Laplace Analysis of Networks - Question 13

The transfer function of a system having the input as X(s) and output as Y(s) is?

Detailed Solution for Test: Laplace Analysis of Networks - Question 13

The transfer function is defined as the s-domain ratio of the laplace transfrom of the output to the laplace transfrom of the input. The transfer function of a system having the input as X(s) and output as Y(s) is H(s) = Y(s)/X(s).

Test: Laplace Analysis of Networks - Question 14

Find the ROC of x(t) = e-2t u(t) + e-3t u(t).

Detailed Solution for Test: Laplace Analysis of Networks - Question 14

Given x(t) = e-2t u(t) + e-3t u(t)

Laplace transform, L{x(t)} = X(s)

ROC is {σ > -2}∩{σ > -3}
Hence, the ROC is σ > -2.

Test: Laplace Analysis of Networks - Question 15

Find the Laplace transform of δ(t).

Detailed Solution for Test: Laplace Analysis of Networks - Question 15

Laplace transform, L{x(t)} = X(s)

[x(t)δ(t) = x(0)δ(t)]

= 1

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