1 Lecture #15 EGR 272 – Circuit Theory II Read: Chapter 12 in Electric Circuits, 6 th Edition by Nilsson 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 #15 EGR 272 – Circuit Theory II Partial Fractions Expansion (or Partial Fractions Decomposition) Partial Fractions 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) = A 1 F 1 (s) + A 2 F 2 (s) + … + A N F N (s) Where F 1 (s), F 2 (s), …, F N (s) are the Laplace transforms of known functions. Then by applying the linearity and superposition properties: f(t) = A 1 f 1 (t) + A 2 f 2 (t) + … + A N f N (t) Example: Find f(t) for F(s) = 16s/(s 2 + 4s + 29)

3 Lecture #15 EGR 272 – Circuit Theory II In most engineering applications, Finding roots of the polynomials yields: where z i = zeros of F(s) and p i are the poles of F(s) Note that:

4 Lecture #15 EGR 272 – Circuit Theory II 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:  jw s-plane

5 Lecture #15 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.

6 Lecture #15 EGR 272 – Circuit Theory II 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 #15 EGR 272 – Circuit Theory II 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 #15 EGR 272 – Circuit Theory II Example: (continued) 2)common denominator method 3)calculators (demonstrate with TI-86, TI-89, and MathCAD)

9 Lecture #15 EGR 272 – Circuit Theory II 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 #15 EGR 272 – Circuit Theory II Example: (Repeated roots) Find f(t) for F(s) shown below.

11 Lecture #15 EGR 272 – Circuit Theory II Example: (Repeated roots) Find f(t) for F(s) shown below.

12 Lecture #15 EGR 272 – Circuit Theory II 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: 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 2 nd 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:

13 Lecture #15 EGR 272 – Circuit Theory II 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  t cos(wt +  ) The two methods for handling complex roots are summarized in the table below. Quadratic factor method Complex linear root method

14 Lecture #15 EGR 272 – Circuit Theory II 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 #15 EGR 272 – Circuit Theory II Example: (continued) 2) Complex linear root method

16 Lecture #15 EGR 272 – Circuit Theory II Example: (Time-delayed function) Find f(t) for Example: (Order of numerator too large) Find f(t) for