problem_lab_08_sol_s08 - ECE 220 Spring 2008 Problem Lab#08...

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ECE 220 Spring 2008 Problem Lab #08 Solutions * Note, all figures are at the end of the document. ————————————————————————————– I. Laplace transforms, use of MATLAB laplace() function Read the help file for the MATLAB function laplace(). Question 1. Use laplace() to evaluate the Laplace transform of the following signals: h ( t ) = e - at cos( ωt ) (1) g ( t ) = te - at (2) f ( t ) = A cos( ωt ) + B sin( ωt ) (3) Remember that symbols used in laplace() must be defined by the ’syms’ statement in MAT- LAB. The TAs will help you with that statement. Solution The command help laplace returns the following information: 1. --- help for sym/laplace.m --- LAPLACE Laplace transform. L = LAPLACE(F) is the Laplace transform of the scalar sym F with default independent variable t. The default return is a function of s. If F = F(s), then LAPLACE returns a function of t: L = L(t). By definition L(s) = int(F(t)*exp(-s*t),0,inf), where integration occurs with respect to t. L = LAPLACE(F,t) makes L a function of t instead of the default s: LAPLACE(F,t) <=> L(t) = int(F(x)*exp(-t*x),0,inf). L = LAPLACE(F,w,z) makes L a function of z instead of the default s (integration with respect to w). LAPLACE(F,w,z) <=> L(z) = int(F(w)*exp(-z*w),0,inf). 1
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Examples: syms a s t w x laplace(t^5) returns 120/s^6 laplace(exp(a*s)) returns 1/(t-a) laplace(sin(w*x),t) returns w/(t^2+w^2) laplace(cos(x*w),w,t) returns t/(t^2+x^2) laplace(x^sym(3/2),t) returns 3/4*pi^(1/2)/t^(5/2) laplace(diff(sym(’F(t)’))) returns laplace(F(t),t,s)*s-F(0) See also sym/ilaplace, sym/fourier, sym/ztrans. Reference page in Help browser doc laplace 2. The Matlab code for evaluating the Laplace transforms of the signals in equations 1 through 3 is shown below: syms a w t A B a = laplace(exp(-a*t)*cos(w*t)) b =laplace(t*exp(-a*t)) c = laplace(A*cos(w*t)+B*sin(w*t)) As a result, Matlab returns: a = (s+a)/((s+a)^2+w^2) b = 1/(s+(s+a)/((s+a)^2+w^2))^2 c = (A*s+B*w)/(s^2+w^2) ————————————————————————————– II. Laplace transforms, use of MATLAB laplace() function Read the help file for the MATLAB function ilaplace(). Question 2. Use ilaplace to evaluate the time signals with given Laplace transforms: 2
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A ( s ) = s + 1 ( s + 2)( s + 3) (4) B ( s ) = 8 ( s + 2) 2 (5) C ( s ) = 5 + 4 s s 2 + 4 (6) D ( s ) = 4 s 2 + 3 s + 2 (7) Solution The command help ilaplace returns the following information: 1. --- help for sym/ilaplace.m --- ILAPLACE Inverse Laplace transform. F = ILAPLACE(L) is the inverse Laplace transform of the scalar sym L with default independent variable s. The default return is a function of t. If L = L(t), then ILAPLACE returns a function of x: F = F(x). By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf) where c is a real number selected so that all singularities of L(s) are to the left of the line s = c, i = sqrt(-1), and the integration is taken with respect to s.
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