The topic of this problem is the complete response of RL switched circuits. The problem is to find IL at t for t is equal to 0 minus IL of t for t is equal to infinity and the overall IL of t for t greater than 0. IL of t for t greater than 0 is the complete response of the RL switch circuit. So, in order to find this, these values, we're going to look at the conditions for t is equal to 0 minus, or t is equal to infinity, in order to be able to find IL of t for both of those conditions. So, we're going to start with t is equal to 0 minus. For t is equal to 0 minus, we have a switch that is open, it doesn't close until time t is equal to 0, so, if we draw our circuit for that condition, we have our V sub s on the left hand side of the circuit, we have our resistor R1, R2, and R3. We have a switch but the switch is open, it's not closed. We also have an inductor and in this case our inductor is in steady state condition, so it's in a DC circuit in steady state condition and we know that inductors in a DC steady state condition act as a short circuit. So this is our IL of t is equal to 0 minus. So for looking for Il of t is equal to 0 minus, we simply analyze the circuit that we have with a DC voltage source or resistor R1, R2, and R3 resistor which is in parallel with a short circuit. Which is Il of t is equal to 0 minus. So, if we're going to find iL of t is equal to 0 minus, it's very simple for us using Ohm's Law, where we know that iL at t is equal to 0 minus, it's simply going to be the voltage V sub s divided by R sub 1 plus R sub 2. Now if we look at the t is equal to infinity case we can find iL at t is equal to infinity so if t is equal to infinity we still have out voltage source, our V sub S. In this case our resistor R1 has a short circuit, across it, and parallel with it, because we now have thrown the switch and closed the switch in our circuit. We still have our resistor R3 and we're again in steady state. So our inductor acts a short circuit. We have our current. And our current is I sub L at t is equal to infinity, so this is our second steady state condition, a long time after the switch has been thrown. So if we want to find IL at t is equal to infinity, again we can use Ohm's law. And it's going to be equal to V sub s divided by, since R1 has a short, R3 also has a short in parallel with it. Our iL is going to be V sub s divided by R2. So the current goes from a value of V sub s, divided by R1 plus R2 to a higher value of V sub s divided by R2, at t is equal to infinity. Our complete response IL of t describes how this current goes from this initial level to a final level of V sub S divided by R2. The complete response is a sum of the Steady State Response and the Transient Response. The Steady State Response also known as the Forced Response is the response at t is equal to infinity, it's what happens at t is equal to infinity. The transient response is what happens immediately after the switch is thrown. So it has to move from this initial current level to a final current level, and it does that in a transient fashion or also known as a natural response of our circuit. So for the inductor, the general form for IL of t, the complete response is given by the simple equation I sub L, at t is equal to infinity plus I sub L at t is equal to 0 minus, minus I sub L, at t is equal to infinity. E to the minus t over tau. And for the LR circuits or RL circuits, tau is equal to L divided by R Thevanins. So we have R, IL at t is equal to infinity, we have our IL at t is equal to 0 minus, we just need to find our tau value. We have an inductor already, so we have our value for the inductor and we want to find our R Thevenin's. So to find R Thevenin's, we do it just like we've always done. But we're looking at the condition t is equal to infinity. So we take our circuit and we analyse it at t is equal to infinity. And we know that when we're finding R Thevenins, we replace all voltage sources by short circuits. We replace all current sources by open circuits, and this is our circuit once we do just that. And we're looking for R Thevanins equivalent resistance taking our load out, which is our inductor and looking back into our circuit. R1 has a short circuit in parallel with it and so R Thevanin's for this case is going to be R2 in parallel with R3. So if we wanted to write our complete response for the current through the inductor for time t is equal to 0 and greater, we first have our Vs over R2 plus iL at t is equal to 0 minus, which is V sub s divided by R1 plus R2, Minus, R iL at t is equal to infinity which is V sub s divided by R2. E to the minus t over tau and our tau is L, divided by the parallel combination of R2 and R3. So, that's our overall equation for iL of t. Now, let's see, if it makes sense. Let's take the two extreme conditions. First of all, t is equal to 0 minus, a long time before, when the circuit's in steady state before the switch is thrown. And so for t is equal to 0 minus, we know the current is going to be iL, it's going to be V sub s divided by R1 plus R2. We also know that the inductor has the property that the current cannot change instantaneously through it. So as soon as the switch is thrown and it changes from a open state to a closed state, the current through the inductor for that instant is going to remain the same. And then it will work its way toward its final value of V sub s divided by R2. So a t is equal to 0, our current through the inductor should be V sub s divided R1 plus R2. Is this what we get with our equation that we have derived? So if we put t is equal to 0 in here, this term goes to one, the exponential term becomes one. And we end up with V sub s divided by R2 plus what's in the brackets. And so, ultimately, we end up with V sub s, divided by R1 plus R2 for time t is equal to zero. And that is exactly what we'd expect. Now if we look at the time t is equal to infinity in our equation this exponential term goes to 0. So it takes this whole second term of our expression out, and we end up with V sub S divided by R2, which is exactly what we'd expect from our previous analysis. So this is the complete response of the inductor current as a function of time for time t is equal to 0.and greater.