$v_L=L\frac{di}{dt}$

Determine the expression for the inductor voltage $v_2(t)$ analytically using the series RLC circuit theory. To do this, you need to solve the circuit current $i(f),$ then differentiate it as explained in the theory section of this lab handout.

$Vs=5V,R=1\mathrm{.}104k,C=0\mathrm{.}0208F,L=42\mathrm{.}7mH$

Case III: underdamped case

Solution equation is

$x(t)=e^{-at}(K_{1}cos{}_{n}t+K_{2}sin{}_{n}t)$

Fvaluated at $t-0,$ we get

$x(0)=K_1$

Derivative of the solution is

$\frac{dx}{dt}=-e^{-t}(K_{1}cos{}_{n}t+K_{2}sin{}_{n}t)+e^{-i}(-K_1{}_{n}sin{}_{n}t+K_2{}_{n}cos{}_{n}t)$

Evaluated at $f-0,$ we get

$2=\frac{R}{L},($ series $RLC),{}_o^2=\frac{1}{LC}$