# laplace transform

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## Presentation Description

Laplace transform for El;ectrical engg applications

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## Presentation Transcript

### Slide 1:

Laplace Transforms And Its Applications By Dr Indrani Pramod Kelkar Vignan’s Institute of Information Technology Visakhaptnam

### Slide 2:

A Laplace transform is a type of integral transform. Plug one function in Get another function out The new function is in a different domain.

### Slide 3:

Properties of Laplace Transforms that may not be readily apparent. First, Laplace transforms, and inverse transforms, are linear :

### Slide 4:

there is a very simple relationship between the Laplace transform of a given function and the Laplace transform of that function’s derivative. Second,

### Slide 5:

First shift theorem L{e−atf(t)} = F(s + a) Second shift thorem L{f(t − d) u(t − d)} = e−dsF(s) Time scaling Other important properties of Laplace Transforms

### Slide 6:

We can use Laplace transforms to turn an initial value problem into an algebraic problem Solve for y(t) Solve for Y(s)

### Slide 7:

1 1 A saw tooth function t Laplace transforms are particularly effective on differential equations with forcing functions that are piecewise, like the Heaviside function, and other functions that turn on and off.

### Slide 8:

I.V.P. Algebraic Equation

### Slide 9:

Algebraic Expression Solution to IVP

### A Calculation :

A Calculation Let This is called the unit step function or the Heaviside function. It’s handy for describing functions that turn on and off.

### Slide 11:

c 1 t The Heaviside Function

### Slide 12:

Calculating the Laplace transform of the Heaviside function is almost trivial.

### The convolution theorem :

The convolution theorem where is called as the convolution of f(t) and g(t), Convolution property: Therefore, Sometimes, denoted as or simply defined by

### Geometrical Understanding of Convolution :

Geometrical Understanding of Convolution

### Impulse? :

Impulse? An impulse is the effect of a force that acts over a very short time interval. Engineers and physicists use the Dirac’s delta function to model impulses. A lightning strike creates an electrical impulse. The force of a major leaguer’s bat striking a baseball creates a mechanical impulse.

### The Dirac’s Delta Function :

The Dirac’s Delta Function This so-called quasi-function was created by P.A.M. Dirac, the inventor of quantum mechanics. The Laplace Transform of theDirac Delta Function is

### Circuit applications :

Circuit applications 1. Transfer functions 2. Convolution integrals 3. RLC circuit with initial conditions

### Laplace transform is another method to transform a signal from time domain to frequency domain. t-domain  s-domain :

Laplace transform is another method to transform a signal from time domain to frequency domain. t-domain  s-domain In frequency domain, In time domain,

### Slide 19:

For the following circuit, find H(s)=Vo(s)/Vi(s). Assume zero initial conditions.

### Solution :

Solution Transform the circuit into s-domain with zero i.c.:

### Slide 21:

Using voltage divider

### Slide 22:

Obtain the transfer function H(s)=Vo(s)/Vi(s), for the following circuit. Example 2 :

### Solution :

Solution Transform the circuit into s-domain (We can assume zero initial condition unless stated in the question)

We found that

### Example 3 : :

Example 3 : Use convolution to find vo(t) in the circuit of Fig.(a) when the excitation (input) is the signal shown in Fig.(b).

### Solution :

Step 1: Transform the circuit into s-domain (assume zero initial condition .) Step 2: Find the TF Solution

### Slide 27:

Step 4: Find vo(t) For t < 0 For t > 0

### Circuit element models :

Circuit element models Apart from the transformations we must model the s-domain equivalents of the circuit elements when they involve initial condition Unlike resistor, both inductor and capacitor are able to store energy

### Slide 29:

Therefore, it is important to consider the initial current of an inductor and the initial voltage of a capacitor For an inductor Taking the Laplace transform on both sides of equation gives or

### Slide 31:

For a capacitor Taking the Laplace transform on both sides of equation gives or

### Typical Scenario :

Typical Scenario The charge on a capacitor in an LRC circuit is given by the following I.V.P. where the emf, f(t) has the following graph.

### Example :

Example Consider the parallel RLC circuit of the following. Find v(t) and i(t) given that v(0) = 5 V and i(0) = −2 A.

### Solution :

Solution Transform the circuit into s-domain (use the given i.c. to get the equivalents of L and C)

### Slide 37:

Then, using nodal analysis

### Slide 38:

Since the denominator cannot be factorized, we may write it as a completion of square: Finding i(t),

### Slide 39:

Using partial fractions, It can be shown that Hence,

### Example :

Example The switch in the following circuit moves from position a to position b at t = 0 second. Compute io(t) for t > 0.

### Solution :

Solution The initial conditions are not given directly. Hence, at first we need to find the initial condition by analyzing the circuit when t ≤ 0:

### Slide 42:

Then, we can analyze the circuit for t > 0 by considering the i.c. Let

### Slide 43:

Using current divider rule, we find that Using partial fraction we have

### Slide 44:

Laplace transforms have limited appeal. You cannot use them to find general solutions to differential equations. Initial conditions at a point other than zero will not do. You cannot use them on initial value problems with initial conditions different from