1. For RL series circuit, R = 4Ω and L = 0.5 H. The source voltage V = 12 Vdc is applied. Find the current i(t) using the Laplace transform. Assume all initial conditions are zeros.
a) First, write the voltage equation in terms of a differential equation.
b) Second, apply the Laplace transform on the equation. For example, if you take the Laplace transform on V = 12 V, then the Laplace transform of 12 = 12/s, the Laplace transform of i(t) =I(s) etc.
c) Rearrange the transformed equation for the current I(s).
d) Take the inverse Laplace transform of I(s) to find the current i(t).
e) The answer should be i(t) = 3(1-e^(-8t)).
f) From the answer (e), what is the transient and the final (stable or steady-state) value? Describe i(t) from (e) more specifically.
g) What is the time constant of the circuit? Calculate it.