Question

Assume we have a series RLC circuit. The model of the RLC circuit can be represented by The circuit is driven by voltage source ean). And the crcuit elements are resistance R 0.4 capacitance C 0.04F, and inductance L 0.002H. At time t 0, the voltage source is stepped from zero to 2V (the circuit elements initially have zero charge and zero current). Determine the solution for charge q(t) stored in the capacitor using Laplace transform methods.

Answer 1

L d^2 q/ d t^2 + R d q/ d t + q/ c = e_i n E (s) = 2/ s 0.002 d^2 q/ d t^2 + 0.4 d q/ d t + q/ 0.04 = e_i n L T of equation 0.002 r^2 Q W + 0.4 s Q (s) + Q (s)/ 0.04 = 2/ s Q (s) = [2/ 1000 s^2 + 4 s/ 10 + 25] = 2/ s Q (s) [2 S^2 + 400 S + 25000/ 1000] = 2/ S Q (S) = 2000/ 2 s (s^2 + 200 s + 12500) Q (s) = 1000/ s (s^2 + 200 s + 12500) Q (s) = 1000/ r (s^2 + 200 s + 12500) s^2 + 200 s + 12500 s = -200 plusminus squareroot 200^2 - 4 (12500)/ 2 s = -100 plusminus 50 j Q (S) = 1000/ s (s + 100 + 50 j) (s + 100 - 50 j) using partial fraction 1000/ s (s + 100 + 50 j) (s + 100 - 50 j) = A? s + B/ s + 100 + 50 j + c/ s +