Inverse Laplace Transforms

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

2. Chapter 12 EGR 272 – Circuit Theory II 2 Table of Laplace Transforms(will be provided on tests)

3. Chapter 12 EGR 272 – Circuit Theory II 3 Table of Laplace Transform Properties (will be provided on tests)

4. Chapter 12 EGR 272 – Circuit Theory II 4 Example: Find f(t) for F(s) = 16/(s+8) (refer to the table of Laplace transforms on slide 5) Example: Find f(t) for F(s) = 16s/(s2 + 4s + 29) (refer to the table of Laplace transforms on slide 5)

5. Chapter 12 EGR 272 – Circuit Theory II 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) into F(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,

6. Chapter 12 EGR 272 – Circuit Theory II 6 Finding roots of the polynomials yields: where zi = zeros of F(s) and pi are the poles of F(s) Note that:

7. Chapter 12 EGR 272 – Circuit Theory II 7 jw s-plane  Poles 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 pole o - represents a zero Example Sketch the pole-zero diagram for the following function:

8. Chapter 12 EGR 272 – Circuit Theory II 8 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.

9. Chapter 12 EGR 272 – Circuit Theory II 9 An 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)

10. Chapter 12 EGR 272 – Circuit Theory II 10 Methods of performing Partial Fractions Expansion: 1) common denominator method 2) residue method 3) calculators, MATLAB, etc Example: (Simple roots) Use PFE to decompose F(s) below and then find f(t). Perform PFE using: 1) common denominator method

11. Chapter 12 EGR 272 – Circuit Theory II 11 Example: (continued) 2) residue method

12. Chapter 12 EGR 272 – Circuit Theory II 12 Repeated roots A 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:

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

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

15. Chapter 12 EGR 272 – Circuit Theory II 15 Complex roots Complex roots always yield sine and/or cosine terms in the time domain. Complex roots may be handled in one of two ways: 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: Also note that cosine and sine terms can be represented as a single cosine term with a phase angle using the identity shown below:

16. Chapter 12 EGR 272 – Circuit Theory II 16 Quadratic factor method Complex linear root method 2) 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 2Betcos(wt +  ) The two methods for handling complex roots are summarized in the table below.

17. Chapter 12 EGR 272 – Circuit Theory II 17 Example: (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

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

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