Laplace Transform MCQ Quiz in मराठी - Objective Question with Answer for Laplace Transform - मोफत PDF डाउनलोड करा

Concept:

L [f(t)] = F(s)

L [sin(at) u(t)] ↔ 

Application:

With v(t) = sin (10t) u(t), the Laplace transform will be:

Important Points 

Some common Laplace transforms are:

Concept:

- a\) 

- a\)

Calculation:

Given:

After Partial fraction we’ll get;

Now, Taking Laplace inverse of F(s);

Concept:

Let the Laplace transform of function of f(t) is L [f(t)] = F (s)

By using first shifting rule

 If L [f(t)] = F (s), then

L [e^at f(t)] = F (s – a)

Calculation:

Laplace transform of y(t) = e^-3t cos 5t

By applying the property of shifting,

Solution:

The given periodic function can be expressed as the summation of shifted unit step functions as shown:

Similarly x₃(t), x₄(t), … will be shifted unit step functions.

x₁(t) = u(t) - u(t - 1)

x₂(t) = u(t - 2) - u(t - 3)

x(t) can be expressed as:

x(t) = x₁(t) + x₂(t) + x₃(t) + ….

x(t) = u(t) - u(t - 1) + u(t - 2) - u(t - 3)+ …      ---(1)

Time-shifting affects the frequency as:

The Laplace transform of x(t) from equation (1), can be written as:

Simplifying the geometric series, we can write:

         ---(2)

Given  

Comparing this with equation (2) we get:

k = -1

Given the differential equation,

Taking the Laplace transform,

s²X(s) – sX(0) – X^’(0) = -9X(s)

or s²X(s) – s(1) – (1) = - 9X(s)

or X(s) (s²+9) = s + 1

 (Taking  inverse Laplace.)

Laplace Transform:

The Laplace transform of 

1.)  

2.) 

3.) 

4.) 

Hence, option 2 is correct.

The Fourier transform of a function is equal to its two-sided Laplace transform evaluated on the imaginary axis of the s-plane.

Explanation:

The Fourier transform of x(t) is, denoted by X(jω), is defined as:

The Laplace transform of x(t), denoted by X(s), is defined as:

Where s is a continuous complex variable.

We can also express s as: s = σ + jω

Where σ and ω are the real and imaginary parts of s, respectively

The Laplace transform can be written as:

By comparing the above Laplace and Fourier transform equations, it is clear that Laplace transform of x(t) is equal to the Fourier transform of .

When σ = 0 or s = jω, both are identical.

That is, Laplace transform generalizes Fourier transform.

• In the partial fraction method of Laplace transforms, the function F(s) is generally represented as a rational function, which is the ratio of two polynomials:
   
• The goal of partial fraction decomposition is to express $F(s)$ as a sum of simpler fractions to facilitate the inverse Laplace transform.
• These simpler fractions correspond to known Laplace transform pairs, making it easier to find f(t)" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t)" id="MathJax-Element-15-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t)" id="MathJax-Element-13-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t)" id="MathJax-Element-327-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t)" id="MathJax-Element-10-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t)" id="MathJax-Element-102-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t) $f(t)$ $f(t)$ $f(t)$ $f(t)$ $f(t)$ $f(t)$ f(t) from $F(s)$.

Concept:

Transfer function:

It is the ratio of the Laplace transform of the output variable to the input variable with initial conditions zero.

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Initial conditions = 0

TF = C(s) / R(s)

If R(s) = 1 i.e impulse input

TF = C(s)

Impulse response (IR) = c(t) = L-1 [C(s)] = L-1[TF]

TF = L-1[IR]

Hence, TF may be also called the impulse response of the system.

Explanation:

From the property of Laplace transform

Put a = 2, 

Hence, T.F = 1 / (s+2)

Additional Information

Concept:

The Laplace transform F(s) of a function f(t) is defined by:

From the time-shifting property of Laplace transform:

Application:

L{x(t)} = X(s)

L(x(t-t0)) =