TEAM PROJECT. Laplace Transform of Periodic Functions (a) Theorem. The Laplace

Chapter 6, Problem 6.1.152

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TEAM PROJECT. Laplace Transform of Periodic Functions (a) Theorem. The Laplace trallSform of a piecewise COlltilluOllS fUllctioll f(t) n'ith period p is (I \) I JV ~(f) = 1 _ e-Ps 0 e-stf(t) dt (.I' > 0). Prove this theorem. Hint: Write {o'" = I P + tv + ... o v . Set t = (n - I)p in the nth integral. Take out e -(n-l)p from under the integral sign. Use the sum formula for the geometric series.(b) Half-wave rectifier. Using (11), show that thehalf-wave rectification of sin wt in Fig. 135 has theLaplace transformw(A half-wave rectifier clips the negative portions of thecurve. Afull-wave rectifier converts them to positive;see Fig. 136.)2rrlmv~_ 3rrlmFig. 135. Half-wave rectificationf(t) Ilr",~~Y~o TrIm 2rrlm 3rrlmFig. 136. Full-wave rectification(c) Full-wave rectifier. Show that the Laplacetransform of the full-wave rectification of sin wt isW 7TS2 2 coth -. S + w 2w(d) Saw-tooth wave. Find the Laplace transform ofthe saw-tooth wave in Fig. 137.fit)k Ii/o p 2p 3pFig. 137. Saw-tooth wave(e) Staircase function. Find the Laplace transform ofthe staircase function in Fig. 138 by noting that it isthe difference of ktlp and the function in (d).~':~ O~-----pL------2~p----~3pL------Fig. 138. Staircase function