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

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

$Vs=5V,R=3\mathrm{.}092K,C=0\mathrm{.}0208F,L=42\mathrm{.}7mH$

Case I: $>{}_0-$ overdamped case

Solution equation is

$x(t)=K_1{e}^{s_{4}t}+K_2{e}^{s_{2}t}$

Evaluated at $t-0,$ we get

Derivative of the solution is

$\frac{dx}{dt}=K_1{s}_1{e}^{s_1{l}_4}+K_2{s}_2{e}^{s_{2}t}$

Evaluated at $t=0,$ we get

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