Inverse Laplace Transforms

1. Lecture #3 EGR 261 – Signals & Systems Read: Ch. 12, Sect. 1-9 in Electric Circuits, 9th Edition by Nilsson Ch. 4, Sect. 1-3 and Sect. B.5 in Linear Signals & Systems, 2nd Ed. by Lathi Handout on Bb and webpage: 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. Example: Find f(t) for F(s) = 16/(s+8)

2. Lecture #3 EGR 261 – Signals & Systems Example: Find f(t) for F(s) = 16s/(s2 + 4s + 29) 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 pairs. 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)

3. Lecture #3 EGR 261 – Signals & Systems In most engineering applications, Finding roots of the polynomials yields: where zi = zeros of F(s) and pi are the poles of F(s) Note that:

4. jw s-plane  Lecture #3 EGR 261 – Signals & Systems 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:

5. Lecture #3 EGR 261 – Signals & Systems 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.

6. Lecture #3 EGR 261 – Signals & Systems 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)

7. Lecture #3 EGR 261 – Signals & Systems Methods of performing Partial Fractions Expansion: 1) common denominator method 2) residue method 3) calculators/software Example: (Simple roots) Use PFE to decompose F(s) below and then find f(t). Perform PFE using: 1) common denominator method

8. Lecture #3 EGR 261 – Signals & Systems Example: (continued) 2) residue method 3) calculators (demonstrate with TI-86, TI-89, and MathCAD)

9. Lecture #3 EGR 261 – Signals & Systems 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:

10. Lecture #3 EGR 261 – Signals & Systems Example: (Repeated roots) Find f(t) for F(s) shown below.

11. Lecture #3 EGR 261 – Signals & Systems Example: (Repeated roots) Find f(t) for F(s) shown below.

12. Lecture #3 EGR 261 – Signals & Systems 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:

13. Quadratic factor method Complex linear root method Lecture #3 EGR 261 – Signals & Systems 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.

14. Lecture #3 EGR 261 – Signals & Systems 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

15. Lecture #3 EGR 261 – Signals & Systems Example: (continued) 2) Complex linear root method

16. Lecture #3 EGR 261 – Signals & Systems Example: (Time-delayed function) Find f(t) for Example: (Order of numerator too large) Find f(t) for