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

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 V_C

⇒ V_C/R + V_C/(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)

⇒ V_C = A/s + B/(s + 1/RC)

Determine the values of A and B using partial fraction

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

B_{s= -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)

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, 

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,

V_i (S) = I(S) (R + LS + 1/Cs)

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

From equation (1),

⇒ Transfer Function = 

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:

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.

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, V_o = Output voltage

V_i = input voltage

Calculation:

V_o(t) = (1 - e^{-t / RC})   ---(1)

V_i(t) = 1 V  ---(2)

Applying Laplace transform to the (1) equation:

Now taking Laplace transform of equation (2):

V_i(s) = 1 / s

Now dividing V₀(s) by V_i(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.

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:

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.

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 :

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'.

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

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

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).

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.

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

= 

[x(t)δ(t) = x(0)δ(t)]

= 

= 1