What is the meaning of current at infinite time for a series RLC circuit formula

I am reading this doc.

At one point, the author says the current equation form a critically damped series RLC circuit when a step function is applied is:

$I_L(t) = I_{\infty} + (B_1 + B_2t)e^{-\alpha t}$

As far as I know, on a critically damped RLC circuit current will oscillate and be like this $I_{\infty}$ will be an equation, not a value, right?

What is the the meaning of this $I_{\infty}$ on the current equation and how do I find it?

• The ringing is decreasing with time. A step function has only one change, at t=0, after than it is constant. The current will approach a single value over time. – Spehro Pefhany Aug 30 '18 at 13:44
• thanks and how do I find it from the equations? – SpaceDog Aug 30 '18 at 13:49
• If you have the solution to the differential equation, then it's simple, allow t to go to infinity and the exponential term goes to zero. If you don't have the solution, well you know that a capacitor acts as an open circuit after a long time, and an inductor acts as a short. – Spehro Pefhany Aug 30 '18 at 13:57
• so, for a circuit where the capacitor/inductor are both initially discharged current at infinity will stabilize to zero, given the capacitor will be an open circuit, right? Please convert your comment to an answer, so I can accept. THANKS – SpaceDog Aug 30 '18 at 13:59
• Critically damped is $\zeta = 1$, or no ringing. – Chu Aug 30 '18 at 16:26