Question

The differential equation for a single closed RL-circuit is, by Kirchoffs Second Law, di where i() is the current in the circuit at time t, L is the inductance, R is the resistance, and E(t) is the impressed voltage. In this lab you will investigate the current under voltages that are nonzero for only a brief period of time. Assuming the values L R 1, solve the LR- circuit initial value problem below using the Laplace transform. Then graph the current using an appropriate viewing window. atEC, i(0) - 0, t>0, where ECO) is a) mpressed vorage Et

I did this normallt but I need to use laplace tranforms.

Answer 1

L d i/d t + R I = E (t) L = R = 1 Rightarrow d i/d t + I = E (t) d i/d t leftrightarrow S I (s) - I (0) Rightarrow Apply Laplace transform, Rightarrow s I (s) - I (0) + I (s) = E (s) Rightarrow I (s) = E (s)/s + 1 Rightarrow I (t) = e^- t u (t) * E (t) E (t) = 4 (t - 1) - 4 (t - 2) Rightarrow E (s) = e^- s/s - e^- 2 s/s Rightarrow I (s) = E (s)/s + 1 = e^- s - e^- 2 s/(s + 1). s Rightarrow I (s) = e^- s - e^- 2 s [A/S + B/S + 1] Rightarrow A (S + D + B S = 1 Rightarrow A + B = 0, A = 1 Rightarrow B = -1 Rightarrow I (s) = e^- s/s - e^- 2 s/s - e^- s/s + 1 + e^- 2 s/s + 1 Rightarrow I (t) = 4 [t - 1] - 4 [t - 2] - e^- (t - 1) u (t - 1) + e^- (t - 2) u (t - 2)