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PH2230 - Laplace Transform HW5 1) For the circuit below, the switch has been open for a long time. The switch is then closed at t = 0. Solve for the current into the capacitor as a function of time using the Laplace transform method. The sinusoidal voltage source operates at a frequency of 50 Hz. For the purpose of this exercise, do not use any circuit reduction (e.g. do not create a Thevenin equivalent for the source and resistors). (Remember that all grounds are connected together). Some Laplace Transform Pairs f(t) F(s) 1. 1 1/s 2. t 1/s2 3. e-at 1/(s+a) 4. te-at 1/(s+a)2 5. sin(ωt) ω/(s2+ω2) 6. cos(ωt) s/(s2+ω2) 7. e-at sin(ωt) ω/[(s+a)2+ω2] 8. e-at cos(ωt) (s+a)/[(s+a)2+ω2] 9. sinh(ωt) ω/(s2-ω2) 10. cosh(ωt) s/(s2-ω2) 11. df dt s F(s) - f(0+) ∫ f (τ ) dτ F(s)/s t 12. 0 13. 14. c1 f1(t) + c2 f2(t) ∫f 1 (τ ) f 2 (t − τ ) dτ c1 F1(s) + c2 F2(s) F1(s) F2(s)