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Chapter 12 EGR 272 – Circuit Theory II Read: Ch. 12 in Electric Circuits, 9th Edition by Nilsson Handout: Laplace Transform Properties and Common Laplace Transforms 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 Courses Using Laplace Transforms: • Circuit Theory II • Electronics • Control Theory • Discrete Time Systems (z-transforms) • Communications • Others Testing: Some calculators 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: No calculators allowed on Test #3 1 Chapter 12 EGR 272 – Circuit Theory II 2 Notation: F(s) = L{f(t)} = the Laplace transform of f(t). f(t) = L-1{F(s)} = the inverse Laplace transform of F(s). Uniqueness: Every f(t) has a unique F(s) and every F(s) has a unique f(t). L f(t) F(s) L -1 Note: Transferring to the s-domain when using Laplace transforms is similar to transferring to the phasor domain for AC circuit analysis. Chapter 12 EGR 272 – Circuit Theory II 3 Definition: F(s) f(t)e dt -st (one-sided Laplace transform) 0 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. Chapter 12 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. The table on the following page will be provided on tests. 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. 4 Chapter 12 EGR 272 – Circuit Theory II 5 Table of Laplace Transforms (to be provided on tests) Chapter 12 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 Chapter 12 EGR 272 – Circuit Theory II Example: Find F(s) if f(t) = cos(wot)u(t) (Hint: use Euler’s Identity) 7 E uler's Identities e jx = cos(x) jsin(x) cos(x) = e + jx + e -jx 2 sin(x) = e + jx -e 2j Example: Find F(s) if f(t) = sin(wot)u(t) -jx Chapter 12 EGR 272 – Circuit Theory II 8 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. Table of Laplace Transform Properties (will be provided on tests) 1 L inearity L {af(t)} = aF (s) 2 S uperposition L {f 1 (t) + f 2 (t) } = F 1 (s) + F 2 (s) 3 M odulation L {e f(t)} = F (s + a) 4 T im e-S hifting L {f(t - )u(t - )} = e F (s) 5 S caling L f(at) 6 R eal D ifferentiation L 7 R eal Integration L 8 C om plex D ifferentiation L 9 10 C om plex Integration C onvolution -at -s L s F a a 1 d f(t) sF (s) - f(0 ) dt t 1 f(t)d t F (s) 0 s d tf(t) F (s) ds f(t) t F (s)d s s L {f(t) * g(t)} = F (s)· G (s) Chapter 12 EGR 272 – Circuit Theory II Laplace Transform Properties: 1. Linearity: L {af(t)} = aF(s) L {f1(t) + f2(t) } = F1(s) + F2(s) 2. Superposition: 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) 9 Chapter 12 EGR 272 – Circuit Theory II 10 Laplace Transform Properties: (continued) 3. Modulation: L {e-atf(t)} = F(s + a) 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) Solution : L 10cos(3t)u (t) L 10e - 2t 10s s 3 2 cos(3t)u(t ) (known tra 2 10 s 2 s 2 2 9 nsform from table) (modulatio n) This solution shows a good way to use Laplace transform properties: 1) Begin with a known transform 2) Apply property 3) Apply property 4) Etc Chapter 12 EGR 272 – Circuit Theory II Example: Find I(s) if i(t) = 4e-20t sin(7t)u(t) 11 Chapter 12 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 4. Time-Shifting: L {f(t - )u(t - )} = e-sF(s) Example: Find L {4e-2(t - 3) u(t - 3)} Example: Find L {10e-2(t - 4)sin(4[t - 4])u(t - 4)} 12 Note: Be sure that all t’s are in the (t - ) form when using this property. Chapter 12 EGR 272 – Circuit Theory II 13 Note: Properties can sometimes be applied in different orders. However, one of the methods may be easier. Practice helps in deciding which method to use. 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 Chapter 12 EGR 272 – Circuit Theory II Example: Find L {4e-3tcos(4[t - 6])u(t - 6)} 14 Chapter 12 EGR 272 – Circuit Theory II 15 Laplace Transform Properties: (continued) In other words, the result of 5. Scaling: 1 s L f(at) F a a Note: This is not a commonly used property. Example: Find F(s) if f(t) = 12cos(3t)u(t) replacing each (t) in a function with (at) is that each s in the transform is replaced by s/a and the transform is also divided by a. Chapter 12 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 6. Real (Time) Differentiation: d L f(t) sF(s) - f(0) dt Example: Find L {f ’(t)} Example: Find L {f ’’(t)} Example: Find L {f ’’(t)} Example: Find L {f n(t)} 16 Chapter 12 EGR 272 – Circuit Theory II Example: Find the Laplace transform of the familiar relationship: i(t) C dv dt 7. Real (Time) Integration: Example: Find t 1 L f(t)dt F(s) 0 s t t L f(t)d td t 0 0 17 Chapter 12 EGR 272 – Circuit Theory II Example: Find the Laplace transform of the familiar relationship: v(t) 1 t i(t)dt C 0 v(0) 18 Chapter 12 EGR 272 – Circuit Theory II 19 Laplace Transform Properties: (continued) 8. Complex Differentiation: d L Example: Find L {tu(t)} Example: Find L {t2 u(t)} Example: Find L {t3 u(t)} Example: Find L {t n u(t)} tf(t) ds F(s) Chapter 12 EGR 272 – Circuit Theory II Example: Find L {3te-2t cos(4t)u(t)} Example: Find L {tu(t - 2)} 20 Chapter 12 EGR 272 – Circuit Theory II Laplace Transform Properties: (continued) 9. Complex Integration: f(t) L t 1 Example: Find L u (t) t F (s)d s s 21 Note: This is not a commonly used property. Multiplying by t is common (such a with repeated roots), but dividing by t is rare. Chapter 12 EGR 272 – Circuit Theory II 22 Laplace Transform Properties: (continued) 10. Convolution: L {f(t) * g(t)} = F(s)·G(s) f(t) * g(t) reads as “f(t) convolved with g(t)” Convolution is defined by the difficult integral relationship shown below. Evaluating this integral is covered in a later course. t f(t) * g(t) g( )f(t - )d 0 Laplace transforms are often used to bypass the convolution integral (illustrated on the following page). Since L {f(t) * g(t)} = F(s) G(s) we can determine f(t)*g(t) using f(t)*g(t) = L -1 {F(s)G(s)} Chapter 12 EGR 272 – Circuit Theory II 23 The method of using Laplace transforms to bypass the convolution integral is illustrated by the diagram below. t f(t) * g(t) Evaluate difficult convolution integral Given: f(t), g(t) g( )f(t - )d 0 f(t)*g(t) = L -1 {F(s)G(s)} Find: f(t)*g(t) = ? Take Laplace Transform F(s), G(s) F(s)·G(s) Multiply Take Inverse Laplace Transform Chapter 12 EGR 272 – Circuit Theory II Example: If f(t) = 2e-2tu(t) and g(t) = 4e-3tu(t), find f(t)*g(t) using Laplace transforms. 24 EGR 272 – Circuit Theory II Chapter 12 25 Impulse Function (t) 0 for t 0 (t) = impulse function The impulse function is defined as: (t)dt 1 (i.e., the area equals 1) - Delta functions are often illustrated as shown below. (t) 2(t-4) Area = 1 (not height) 0 t Area = 2 (not height) 0 4 t f(t) Illustration To illustrate the concept that the area under (t) = 1 (not the height =1), consider the function f(t) shown below: 1 /A (t) lim A0 t -A/2 0 + A/2 f(t) Chapter 12 EGR 272 – Circuit Theory II 26 Example: When do impulse functions occur? Consider the example shown below. Sketch the capacitor current. + v(t) - v(t) i(t) C t i(t) t Chapter 12 EGR 272 – Circuit Theory II 27 Laplace Transform of an impulse function The Laplace transform of an impulse function can be found using the definition of the Laplace transform: L (t) (t)e - st dt 0 Since (t) only exits at t=0, e-st only needs to be evaluated at t = 0 (this is sometimes called the sifting property), so: L (t) (t)e dt (t)dt 1 0 0 so L {(t)} = 1 0 Chapter 12 EGR 272 – Circuit Theory II 28 Impulse functions can occur in real circuits. Constant terms in F(s) will correspond to impulse functions in f(t). We will soon see that if the order of N(s) is not less than the order of D(s), we should begin by using long division before finding an inverse Laplace transform. Example: Find f(t) for F(s) shown below. (Hint: Order of N(s) = 2 and order of D(s) = 2, so begin with long division) 2s 6 2 F(s) s 6s 25 2 Chapter 12 EGR 272 – Circuit Theory II Laplace Transforms of waveforms Piecewise-continuous waveforms can be expressed using unit functions. Laplace transforms of these expressions can then be found. Example: Find F(s) for f(t) shown below. f(t) 6 t 0 2 4 Example: Find F(s) for f(t) shown below. f(t) 12 t 0 4 29