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Inverse Laplace Transforms There is no integral definition for finding an inverse Laplace transform. Inverse Laplace transforms are found as follows: 1)For simple functions: Use tables of Laplace transform pairs. 2)For complex functions: Decompose the complex function into two or more simple functions using Partial Fraction Expansion (PFE) and then find the inverse transform of each function from a table of Laplace transform pairs. 1 Chapter 12 EGR 272 – Circuit Theory II Read : Ch. 12 in Electric Circuits, 10 th Edition by Nilsson Handouts : Laplace Transform Properties and Common Laplace Transforms Partial Fraction Expansion (using various calculators)

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Chapter 12 EGR 272 – Circuit Theory II. 1. Read : Ch. 12 in Electric Circuits, 9 th Edition by Nilsson Handouts : Laplace Transform Properties and Common Laplace Transforms Partial Fraction Expansion (using various calculators). Inverse Laplace Transforms - PowerPoint PPT Presentation

### Text of Inverse Laplace Transforms

EGR 277 Digital Logic

Inverse Laplace Transforms There is no integral definition for finding an inverse Laplace transform. Inverse Laplace transforms are found as follows:1)For simple functions: Use tables of Laplace transform pairs.2)For complex functions: Decompose the complex function into two or more simple functions using Partial Fraction Expansion (PFE) and then find the inverse transform of each function from a table of Laplace transform pairs.1Chapter 12 EGR 272 Circuit Theory IIRead: Ch. 12 in Electric Circuits, 10th Edition by NilssonHandouts: Laplace Transform Properties and Common Laplace TransformsPartial Fraction Expansion (using various calculators)

2Chapter 12 EGR 272 Circuit Theory II

Table of Laplace Transforms (will be provided on tests)Table of Laplace Transform Properties (will be provided on tests)

3Chapter 12 EGR 272 Circuit Theory IIExample: Find f(t) for F(s) = 16s/(s2 + 4s + 29)(refer to the table of Laplace transforms on slide 5)

4Chapter 12 EGR 272 Circuit Theory IIExample: Find f(t) for F(s) = 16/(s+8)(refer to the table of Laplace transforms on slide 5)Partial Fraction Expansion (or Partial Fraction Decomposition)Partial Fraction Expansion (PFE) is used for functions whose inverse Laplace transforms are not available in tables of Laplace transform.

PFE involves decomposing a given F(s) intoF(s) = A1F1(s) + A2F2(s) + + ANFN(s)

Where F1(s), F2(s), , FN(s) are the Laplace transforms of known functions.

Then by applying the linearity and superposition properties:f(t) = A1f1(t) + A2f2(t) + + ANfN(t)

In most engineering applications,

5Chapter 12 EGR 272 Circuit Theory II

Finding roots of the polynomials yields:

where zi = zeros of F(s)and pi are the poles of F(s)

Note that:

6Chapter 12 EGR 272 Circuit Theory IIPoles and zeros in F(s)Poles and zeros are sometimes plotted on the s-plane. This is referred to as a pole-zero diagram and is used heavily in later courses such as Control Theory for investigating system stability and performance. Poles and zeros are represented on the pole-zero diagram as follows:x - represents a poleo - represents a zeroExampleSketch the pole-zero diagram for the following function:jws-plane

7Chapter 12 EGR 272 Circuit Theory II

Surface plots used to illustrate |F(s)|The names poles and zeros come from the idea of using a surface plot to graph the magnitude of F(s). If the surface, which represents |F(s)|, is something like a circus tent, then the zeros of F(s) are like tent stakes where the height of the tent is zero and the poles of F(s) are like tent poles with infinite height.Example A surface plot is shown to the right.Note: Pole-zero diagrams and surface plots for |F(s)| are not key topics for this course and will not be covered on tests. They are mentioned here as a brief introduction to future topics in electrical engineering.8Chapter 12 EGR 272 Circuit Theory IIAn important requirement for using Partial Fractions Expansion

Show that expressing F(s) as

leads to an important requirement for performing Partial Fractions Expansion:

If F(s) does not satisfy the condition above, use long division to place it (the remainder) in the proper form (to be demonstrated later).order of N(s) < order of D(s)

9Chapter 12 EGR 272 Circuit Theory IIMethods of performing Partial Fractions Expansion:1)common denominator method2)residue method3)calculators, MATLAB, etcExample: (Simple roots)Use PFE to decompose F(s) below and then find f(t). Perform PFE using:1)common denominator method

10Chapter 12 EGR 272 Circuit Theory IIExample: (continued)2)residue method

11Chapter 12 EGR 272 Circuit Theory IIRepeated rootsA term in the decomposition with a repeated root in the denominator could in general be represented as:

(Note that in general the order of the numerator should be 1 less than the order of the denominator).F(s) above is inconvenient, however, since it is not the transform of any easily recognizable function. An equivalent form for F(s) works better since each part is a known transform:

12Chapter 12 EGR 272 Circuit Theory II

Example: (Repeated roots) Find f(t) for F(s) shown below. 13Chapter 12 EGR 272 Circuit Theory II

Example: (Repeated roots) Find f(t) for F(s) shown below. 14Chapter 12 EGR 272 Circuit Theory II

Complex rootsComplex roots always yield sine and/or cosine terms in the time domain. Complex roots may be handled in one of two ways:

Also note that cosine and sine terms can be represented as a single cosine term with a phase angle using the identity shown below:1)using quadratic factors Leave the portion of F(s) with complex roots as a 2nd order term and manipulate this term into the form of the transform for sine and cosine functions (with or without exponential damping). Keep the transform pairs shown to the right in mind:

15Chapter 12 EGR 272 Circuit Theory II2)using complex roots a complex term can be represented using complex linear roots as follows:

where the two terms with complex roots will yield a single time-domain term that is represented in phasor form as

or in time-domain form as 2Betcos(wt + )

The two methods for handling complex roots are summarized in the table below.

Quadratic factor methodComplex linear root method16Chapter 12 EGR 272 Circuit Theory IIExample: (Complex roots) Find f(t) for F(s) shown below. Use both methods described above and show that the results are equivalent.1) Quadratic factor method 17Chapter 12 EGR 272 Circuit Theory II

Example: (continued) 2) Complex linear root method 18Chapter 12 EGR 272 Circuit Theory II

Example: (Time-delayed function) Find f(t) for Example: (Order of numerator too large) Find f(t) for 19Chapter 12 EGR 272 Circuit Theory II

Hint: Form a new function such that F(s) = F1(s)e-2s.Find f1(t). f(t) is simply a delayed version of f1(t).

1LinearityL{af(t)} = aF(s)

2SuperpositionL {f1(t) + f2(t) } = F1(s) + F2(s)

3ModulationL {e-atf(t)} = F(s + a)

4Time-ShiftingL {f(t - ()u(t - ()} = e-s(F(s)

5Scaling

6Real Differentiation

7Real Integration

8Complex Differentiation

9Complex Integration

10Convolution

{f(t) * g(t)} = F(s)(G(s)

_1128104668.unknown

_1128104692.unknown

_1128104723.unknown

_1128104748.unknown

_1128104680.unknown

_1128104626.unknown

F(t)F(s)

Ke-atu(t)

Kte-atu(t)

?

_1128536282.unknown

_1128536283.unknown

_1128536123.unknown

F(t)F(s)

Kcos(wt)u(t)

Ksin(wt)u(t)

Ke-atcos(wt)u(t)

Ke-atsin(wt)u(t)

_950309100.unknown

_950309170.unknown

_950309206.unknown

_950309069.unknown

F(t)F(s)

Ke-atcos(wt)u(t)

Ke-atsin(wt)u(t)

2Be-atcos(wt + ()u(t)

_950309206.unknown

_1128537388.unknown

_950309170.unknown

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