laplace transform

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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)

Slide 24: 

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

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