Presentation on theme: "The Inverse Laplace Transform"— Presentation transcript:
1The Inverse Laplace Transform The University of TennesseeElectrical and Computer Engineering DepartmentKnoxville, Tennesseewlg
2Inverse Laplace Transforms Background:To find the inverse Laplace transform we use transform pairsalong with partial fraction expansion:F(s) can be written as;Where P(s) & Q(s) are polynomials in the Laplace variable, s.We assume the order of Q(s) P(s), in order to be in properform. If F(s) is not in proper form we use long division anddivide Q(s) into P(s) until we get a remaining ratio of polynomialsthat are in proper form.
3Inverse Laplace Transforms Background:There are three cases to consider in doing the partial fraction expansion of F(s).Case 1: F(s) has all non repeated simple roots.Case 2: F(s) has complex poles:(expanded)Case 3: F(s) has repeated poles.(expanded)
4Inverse Laplace Transforms Case 1: Illustration:Given:Find A1, A2, A3 from Heavyside
5Inverse Laplace Transforms Case 3: Repeated roots.When we have repeated roots we find the coefficients of theterms as follows:
6Inverse Laplace Transforms Case 3: Repeated roots.Example???
7Inverse Laplace Transforms Case 2: Complex Roots:F(s) is of the form;K1 is given by,
8Inverse Laplace Transforms Case 2: Complex Roots:
9Inverse Laplace Transforms Case 2: Complex Roots:Therefore:You should put this in your memory:
10Inverse Laplace Transforms Complex Roots: An Example.For the given F(s) find f(t)
11Inverse Laplace Transforms Complex Roots: An Example.(continued)We then have;Recalling the form of the inverse for complex roots;
12Inverse Laplace Transforms Convolution Integral:Consider that we have the following situation.Systemx(t)y(t)h(t)x(t) is the input to the system.h(t) is the impulse response of the system.y(t) is the output of the system.We will look at how the above is related in the time domainand in the Laplace transform.
13Inverse Laplace Transforms Convolution Integral:In the time domain we can write the following:In this case x(t) and h(t) are said to be convolved and theintegral on the right is called the convolution integral.Comment:The convolution integral is extremely important in signals and systems. It can be implemented with a PC in real time and can be very useful in that form. However, if we do the integration by hand, it is tedious, difficult at times. It is much easier to use:Y(s) = X(s)H(s)And the take the inverse to find y(t). Here we simply multiply X(s) and H(s) together, expand in partial fractions and take the inverse to get y(t). Most always this is easier than carrying out the integration.It can be shown that,This is very important* note
14Inverse Laplace Transforms Convolution Integral:Through an example let us see how the convolution integraland the Laplace transform are related.We now think of the following situation:h(t)x(t)y(t)H(s)Y(s)X(s)
15Inverse Laplace Transforms Convolution Integral:From the previous diagram we note the following:h(t) is called the system impulse response for the followingreason.Eq AIf the input x(t) is a unit impulse, (t), the L(x(t)) = X(s) = 1.Since x(t) is an impulse, we say that y(t) is the impulseresponse. From Eq A, if X(s) = 1, then Y(s) = H(s). Since,
16Inverse Laplace Transforms Convolution Integral:A really important thing here is that anytime you are givena system diagram as follows,X(s)Y(s)H(s)the inverse Laplace transform of H(s) is the system’simpulse response.This is important !!
17Inverse Laplace Transforms Convolution Integral:Example using the convolution integral.x(t)y(t)= ?e-4t
18Inverse Laplace Transforms Convolution Integral:Same example but using Laplace.x(t) = u(t)h(t) = e-4tu(t)
19Inverse Laplace Transforms Convolution Integral:Practice problems:1.5[(t) – 2e-2tu(t)][e-6t - 6te-6t]u(t)[(-1/32) + (1/16)t - (1/32)e-4t + (1/16)te-4t]u(t)Answers given on note page
20Inverse Laplace Transforms Circuit theory problem:You are given the circuit shown below.Use Laplace transforms to find v(t) for t > 0.
21Inverse Laplace Transforms Circuit theory problem:We see from the circuit,
22Inverse Laplace Transforms Circuit theory problem:Take the Laplace transformof this equations includingthe initial conditions on vc(t)
23Inverse Laplace Transforms Circuit theory problem: