Chapter 4 Laplace Transforms Notes Proofread by Yunting Gao and corrections made on 03/30/2021 4 Introduction 4.1 Definition and the Laplace transform of simple functions Given f, a function of time, with value f(t) at time t, the Laplace transform of fwhich is denoted by L(f) (or F) is defined by L(f)(s) = F(s) = Z 1 0 e stf(t)dt s>0: (1 A Laplace transform can be decomposed through partial fraction expansions into terms that can be readily inverse Laplace transformed using Laplace transform primitives. Laplace transforms lead to transfer function models. A transfer function is an algebraic construct that represents the output/input relation in the s-domain. K. Webb MAE 3401 7 Laplace Transforms -Motivation We'll use Laplace transforms to solve differential equations Differential equations in the time domain difficult to solve Apply the Laplace transform Transform to the s‐domain Differential equations becomealgebraic equations easy to solve Transform the s‐domain solution back to the time domain Integral Transform - Laplace Transform -Definition ³ E D F (s) k (s,t) f (t)dt • Tool for solving linear diff. eq. -Integral transform k(s,t) - The kernel of the transformation n D, E( D f ; E f ) f F Transform • Laplace transform whenever this improper integral converges ^ ` ³ f 0 L f (t) F (s) e st f (t)dt l k(s,t) e st LaPlace Transform in Circuit Analysis Recipe for Laplace transform circuit analysis: 1. Redraw the circuit (nothing about the Laplace transform changes the types of elements or their interconnections). 2. Any voltages or currents with values given are Laplace-transformed using the functional and operational tables. 3. 1.1 Laplace Transformation Laplace transformation belongs to a class of analysis methods called integral transformation which are studied in the eld of operational calculus. These methods include the Fourier transform, the Mellin transform, etc. In each method, the idea is to transform a di cult problem into an easy problem. For The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform is defined by Table 1: Properties of Laplace Transforms Number Time Function Laplace Transform Property 1 αf1(t)+βf2(t) αF1(s)+βF2(s) Superposition 2 f(t− T)us(t− T) F(s)e−sT; T ≥ 0 Time delay 3 f(at) 1 a F( s a); a>0 Time scaling 4 e−atf(t) F(s+a) Shift in frequency 5 df (t) dt sF(s)− f(0−) First-order differentiation 6 d2f(t) dt2 s2F(s)− sf(0−)− f(1)(0−) Second-order The Laplace transform can be interpreted as a transforma- tion from the time domain where inputs and outputs are functions of time to the frequency domain where inputs and outputs are functions of complex angular frequency. In order for any function of time f(t) to be Laplace transformable, it must satisfy the following Dirichlet con- ditions [1]: The Laplace Transform is Linear If a is a constant and f and gare functions, then For example, by the above property (1) As an another example, by property (2) L(e5t+cos(3t)) = L(e5t)+L(cos(3t)) = 1 s−5 + s s2+9 ,s>5. L(3t5)=3L(t5)=3 5! s6 = 360 s6 ,s>0. L(af)=aL(f) (1) L(f +g)=L(f)+L(g) (2) 6 An example where both (1) and (2) are used, ME375 Laplace - 4 Definition • Laplace Transform - One Sided Laplace Transform where s is a complex variable that can b