1 Lecture #13 EGR 272 – Circuit Theory II Read: Chapters 12 and 13 in Electric Circuits, 6 th Edition by Nilsson Handout: Laplace Transform Properties and Common Laplace Transform Pairs Laplace Transforms – an extremely important topic in EE! Key Uses of Laplace Transforms: Solving differential equations Analyzing circuits in the s-domain Transfer functions Frequency response Applications in many courses Testing: Calculators have become increasingly powerful in recent years and can often be used to find Laplace transforms and inverse Laplace transforms. However, it is also easy to make mistakes with the calculators and if the student is not familiar with the material, the mistakes might easily go undetected. As a result, calculator usage will be restricted on the test covering this material by giving a 2-part test as follows: Part 1: (No calculators allowed) Covers finding Laplace transforms and inverse Laplace transforms Part 2: (Calculators allowed) Circuit applications using Laplace transforms Courses Using Laplace Transforms: Circuit Theory II Electronics Linear Systems Control Theory Discrete Time Systems (z-transforms) Communications

2 Lecture #13 EGR 272 – Circuit Theory II Notation: F(s) = L {f(t)} = the Laplace transform of f(t). f(t) = L -1 {F(s)} = the inverse Laplace transform of f(t). Uniqueness: Every f(t) has a unique F(s) and every F(s) has a unique f(t). L L -1 Note: Transferring to the s-domain when using Laplace transforms is similar to transferring to the phasor domain for AC circuit analysis.

3 Lecture #13 EGR 272 – Circuit Theory II Definition: (one-sided Laplace transform) where s =  + jw = complex frequency  = Re[s] and w = Im[s] sometimes complex frequency values are displayed on the s-plane as follows:  jw s-plane Note: The s-plane is sometimes used to plot the roots of systems, determine system stability, and more. It is used routinely in later courses, such as Control Theory.

4 Lecture #13 EGR 272 – Circuit Theory II Convergence: A negative exponent (real part) is required within the integral definition of the Laplace Transform for it to converge, so Laplace Transforms are often defined over a specific range (such as for  > 0). Convergence will discussed in the first couple of examples in this course to illustrate the point, but will not be stressed afterwards as convergence is not typically a problem in circuits problems. Determining Laplace Transforms - Laplace transforms can be found by: 1)Definition - use the integral definition of the Laplace transform 2)Tables - tables of Laplace transforms are common in engineering and math texts 3)Using properties of Laplace transforms - if the Laplace transforms of a few basic functions are known, properties of Laplace transforms can be used to find the Laplace transforms of more complex functions.

5 Lecture #13 EGR 272 – Circuit Theory II Example: If f(t) = u(t), find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges). Example: If f(t) = e -at u(t), find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges).

6 Lecture #13 EGR 272 – Circuit Theory II Example: Find F(s) if f(t) = cos(w o t)u(t) (Hint: use Euler’s Identity) Example: Find F(s) if f(t) = sin(w o t)u(t)

7 Lecture #13 EGR 272 – Circuit Theory II Laplace Transform Properties Laplace transforms of complicated functions may be found by using known transforms of simple functions and then by applying properties in order to see the effect on the Laplace transform due to some modification to the time function. Ten properties will be discussed as shown below (also see handout). Table of Laplace Transform Properties

8 Lecture #13 EGR 272 – Circuit Theory II Laplace Transform Properties: 1. Linearity: Example: Use the results of the last two examples plus the two properties above to find F(s) if f(t) = 25(1 – e -3t )u(t) Proof: L {f 1 (t) + f 2 (t) } = F 1 (s) + F 2 (s) 2.Superposition: L {af(t)} = aF(s)

9 Lecture #13 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 3. Modulation: Proof: This means that if you know F(s) for any f(t), then the result of multiplying f(t) by e -at is that you replace each s in F(s) by s+a. Example: Find V(s) if v(t) = 10e -2t cos(3t)u(t) Example: Find I(s) if i(t) = 4e -20t sin(7t)u(t) L {e -at f(t)} = F(s + a)

10 Lecture #13 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 4. Time-Shifting: Example: Find L {4e -2(t - 3) u(t - 3)} Note: Be sure that all t’s are in the (t -  ) form when using this property. Example: Find L {10e -2(t - 4) sin(4[t - 4])u(t - 4)} L {f(t -  )u(t -  )} = e -s  F(s)

11 Lecture #13 EGR 272 – Circuit Theory II Example: Find F(s) if f(t) = 4e -3t u(t - 5) using 2 approaches: A) By applying modulation and then time-shifting B) By applying time-shifting and then modulation

12 Lecture #13 EGR 272 – Circuit Theory II Example: Find L {4e -3t cos(4[t - 6])u(t - 6)}

13 Lecture #13 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 5. Scaling: Example: Find F(s) if f(t) = 12cos(3t)u(t) In other words, the result of replacing each (t) in a function with (at) is that each s in the function is replaced by s/a and the function is also divided by a. Note: This is not a commonly used property.