# What are the initial conditions used to find the coefficients in the current equations?

Suppose a RLC series circuit. R, L and C are in series with a battery and a switch. The switch is open. L and C are discharged.

At t=0, the switch is closed and the battery (V1) feeds the circuit.

I apply KVL to the circuit and find three equations:

CRITICALLY DAMPED

$$\ V_C(t) = (At + B) \thinspace e^{-\alpha t} \$$

OVERDAMPED

$$\ V_C(t) = Ae^{m_1t} + Be^{m_2t} \$$

UNDERDAMPED

$$\ V_C(t) = e^{- \alpha t}[K_1 \thinspace Cos(\omega_d t) + K_2 \thinspace Sin(\omega_d t) ] \$$

To find the coefficients of those equations I apply the two initial conditions:

1. I solve the equations for t=0 and for the initial voltage across the capacitor.
2. I take the derivative of the equation and solve for t=0.

Now lets talk about the current equations.

I never understood why but apparently the current equations are the same, or

CRITICALLY DAMPED

$$\ i(t) = (At + B) \thinspace e^{-\alpha t} \$$

OVERDAMPED

$$\ i(t) = Ae^{m_1t} + Be^{m_2t} \$$

UNDERDAMPED

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

What are the two conditions I must use for the current equations to find the coefficients?

In the voltage equations I used the initial voltage across the capacitor and the derivative of voltage (current).

Now I have the current equations.

One condition must be to solve the equations for t=0, but what about the second condition? How do I find the coefficients of the current equations?

• @SpaceDog I've made a minor edit to my answer. You're correct that the current in the circuit is still zero at $t = 0^{+}$. However, current will start to flow, and eventually the inductor will look just like a wire, right? And correct, when the capacitor is fully charged, the current stops. But this time, the voltage across the capacitor is different than it was at $t = 0$. – Shamtam Nov 14 '18 at 17:24
The current through the inductor cannot change abruptly. So, iL(0-) = iL(0+). You can use inductor current at time instant 0 i.e. iL(0) as an initial condition.
For the second initial condition you can use the following formula diL(0)/dt = vL(0)/L. To find vL(0), you can apply KVL at your circuit for the time instant 0.