# How do I determine the unknowns for this equation?

Suppose a series RLC circuit in series with a 12 V battery and a switch initially opened. R = 100 Ω, L = 100mH and C = 10uF. L and C are initially discharged.

A t=0 the switch is closed.

I want to know the current equation.

I calculate $\alpha = \frac{R}{2L} = \frac{100}{2 \times 100x10^{-3}} = 500$

then $\omega_0 = \sqrt{\frac{1}{LC}} = = \sqrt{\frac{1}{100 \times 10^{-3} \times 10 \times 10^{-6}}} = 1000$

$\alpha < \omega_0$ , so we are dealing with an underdamped circuit and the current equation has the form

$i(t) = e^{- \alpha t} [K_1 \thinspace Cos(\omega_d t) + K_2\thinspace Sen(\omega_d t)] + I_{\infty}$

I need to determine $K_1$ and $K_2$

I apply the first initial conditions. Current at time 0 is 0

$i(t=0) = 0 = e^{0} [K_1 \thinspace Cos(0) + K_2\thinspace Sen(0)] + 0$

0 = [K_1 + 0] + 0  K_1 = 0$Second condition,$ i'(0) = 0 $The derivate of the i equation gives me$ \frac{V_L}{L} = -\alpha e^{- \alpha t} [K_1 \thinspace Cos(\omega_d t) + K_2\thinspace Sen(\omega_d t)] - \omega_d e^{- \alpha t} \thinspace K_1 \thinspace Sen(\omega_d t) + \omega_d e^{- \alpha t} \thinspace K_2\thinspace Cos(\omega_d t) $when I apply t=0 that gives me$ K_2 = 0$With K1 and K2 equal to zero there is no current equation. How do I get these unknowns? ## 1 Answer If$K_1 = 0$and steady-state current is$i_{ss}=0$:  i(t) = K_2e^{-\alpha t}sin(\omega t) Then,  \frac{di(t)}{dt} = K_2 \left[ -\alpha e^{-\alpha t}sin(\omega t) + \omega cos(\omega t)e^{-\alpha t} \right ]  K_2 = \frac{1}{\omega}\frac{di(0_+)}{dt}   \frac{di(0_+)}{dt} = \frac{v_{L_{(0+)}}}{L} But$ v_{L_{(0+)}} = 12 V $and$L=0.1 H$and K_2 = 0.12 A • I guess you have a typo there on the first uknown. You mean K1 = 0. Anyway, you say that$ i(t) = K_2e^{-\alpha t}sin(wt) $, shouldn't it be$ i(t) = K_1e^{-\alpha t}cos(wt) \\$ ? sin(wt) would be zero, because sin(0) = 0 – MLL Aug 31 '18 at 0:35
• I forgot to delete the third line of original text. – Dirceu Rodrigues Jr Aug 31 '18 at 1:39
• @MLL: The statement in your question: "K2 = 0" is not correct. Have you noticed that VL (0+) = 12 V? – Dirceu Rodrigues Jr Sep 1 '18 at 3:24