ECE 201 Spring 2010 Homework 25 Solutions Problem 43 a To find the initial conditions at t 0 we can write the following KVL equations 20 40iL 0 60 0 1 iL 0 0 iL 0 iL 0 iC 0 0 14 A vC 0 vC 0 20 0 14 40 14 4 V After t 0 the circuit is a series RLC circuit with the following characteristic equation R 1 s 0 L LC s2 15s 50 0 s2 s1 5 s2 10 vC t c1 e 5t c2 e 10t c1 c2 14 4 5 10 3 5c1 10c2 0 14 c1 23 2 c2 8 8 vC t 23 2e 5t 8 8e 10t V The plot of vC t for 0 t 1 is shown on the next page 1 Plot of vC t part a for 0 t 1s 15 vC t Volts 10 5 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 t seconds b In this case the situation after t 0 can be visualized as a series RLC circuit with a constant input voltage source after source transformation which will have the value 0 5 60 30 V The initial conditions and the characteristic equation do not change However the expression for vC t will now have a term due to the constant input also Thus vC t c1 e 5t c2 e 10t 30 c1 c2 30 14 4 5c1 10c2 28 vC t 36 8e 5t 21 2e 10t 30 V The plot of vC t for 0 t 1 follows 2 Plot of vC t part b for 0 t 1s 30 28 26 vC t Volts 24 22 20 18 16 14 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 t seconds Problem 49 a At t 0 the capacitor is open circuit and the inductor is short circuit At t 0 the current source is off and the voltage source is still operating Thus we can write the following equations vC 0 vC 0 10 V 10 iL 0 0 005 1000 iL 0 iL 0 0 005 A 10 vL 0 vC 0 0 vL 0 0 10 0 005 iC 0 1000 iC 0 0 005 A 3 b For t 0 we can write the following KVL and KCL equations 10 L diL vC 0 dt dvC vC dt R 10 LC iL C 1 dvC vC d2 vC 2 dt RC dt LC Thus the characteristic equation is s2 2000s 1 087 107 0 s1 2 1000 j1000 vC t e 1000t A cos 1000 t B sin 1000 t 10 10 A 0 B Using initial conditions 10 vC t 10 e 1000t sin 1000 t V dvC vC iL t C dt R 1 0 005 2 e 1000t sin 1000 t cos 1000 t A c vC t is computed above in part b d vL t 10 vC t 10 1000t e sin 1000 t V e dvC dt 0 005 1000t e sin 1000 t cos 1000 t A iC t C The plots for the respective quantities follow 4 Plot of iL t for 0 t 5ms 3 12 x 10 11 10 iL t A 9 8 7 6 5 0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 t seconds 5 3 x 10 Plot of vC t for 0 t 5ms 11 10 5 10 vC t V 9 5 9 8 5 8 7 5 0 0 5 1 1 5 2 2 5 t seconds 5 3 3 5 4 4 5 5 3 x 10 Plot of vL t for 0 t 5ms 2 5 2 1 5 vL t V 1 0 5 0 0 5 1 0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 t seconds Plot of iC t for 0 t 5ms 3 3 5 3 x 10 x 10 2 1 iC t A 0 1 2 3 4 5 0 0 5 1 1 5 2 2 5 t seconds 6 3 3 5 4 4 5 5 3 x 10
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