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