The topic of this problem is The Complete Response of RL Switched Circuits. The problem is to find IL and t at t equals zero minus, IL of t at t is equal to infinity, and define IL of t for t greater than or equal to zero. So IL of t is a combination of the steady state response and the transit response of our circuit. Because we have a switch in our circuit and the switch is going to be closed at time t is equal to 0. So before that it's open and there's no current flowing through the 5 mH inductor and after the switch is closed we have a change in that condition. So the current goes from some value which is 0 to some final value which is a non-zero value for the current. And it goes through this transient period and wish this current is changing from its value before the switch was thrown, to a long time after the switch has been closed. So the complete response IL of t Is that response after long time plus a transit response as it's moving from its condition before the switch is thrown to after the switch is thrown. So we know that we have this general equation which allows us to find IL of t. That is, IL of t is equal to IL of t is equal to infinity plus inside of brackets, IL of t is equal to zero minus, minus IL of t is equal to infinity and that's multiplied by the exponential of t divided by thaw, where thaw is our time constant for this natural response of our circuit. And thaw is equal to l divided by r thevanence. So in order to solve this problem we need to find IL(t) is equal to infinite, IL(t) is equal to zero minus and our thaw for this problem. So in order to do that we have to evaluate it at those timeframes, so the IL at t is equal to 0 minus can be done by redrawing the circuit for that condition. We have an open circuit, Where 4 times t is equals 0. We have a inductor. We know that the inductor acts as a short circuit in Steady State, in a DC circuit. And so this is what our circuit looks like with a 1 kilo Ohm resistor and a 4 milli Amp source. So if we're looking for IL(t)=0 minus We can see that's going to be equal to 0. There's no current flowing in time 2 is equal to 0 minus. What about for t is equal to infinity? So for 2 is equal to infinity let's draw our circuit. We have the 4 milliamp source in it, we have the 1k resistor, our switch is thrown so it's closed now. We have an inductor which in a DC circuit in steady state acts as a short circuit. And so this is our IL, and t is equal to infinity. So if we're looking for this current at time t is equal to infinity, we can see that from our circuit that a four milliamp source is going to flow through the short circuit that's in parallel with the resistor. And so all of the current Is going to be flowing through the inductor in steady state at time t is equal to infinity. So IL at t is equal to infinity is equal to 4 miliamps. So the current goes from 0 miliamps to 4 miliamps As the switch is thrown and the circuit transitions from these two conditions. So our steady state response, which is our response after a long period of time, is 4 milliamps. And our transient response is a combination of those two times an exponential factor which allows it get from its initial state to its final state. So, our iL of t, Is going to be equal to 4 milliamps plus 0 minus 4 milliamps Times e to the minus t over thaw. And our thaw is l divided by r thevanence. So to find r thevanence, we do it the same way that we've always done. And this r thevanence is condition is taking that t is equal to infinity because that's what we're looking for is that equation for IL t for t greater than or equal to zero. So we'll take this and we'll use our knowledge of how to find r thevanence from previous lectures to find, The value for this circuit. So we take our load out which is the inductor and we're looking for RTh back into our network. And so the current source is replaced by an open circuit. We have a resistor which is 1 kiloohm and in fact that is our Rth for this problem. So we have an inductance of 5 millihenries, we have a R Thevenance of 1 kiloohm. So our tau value comes out to be 5. So, overall, our current the complete response for our current to the inductor is going to equal four mili amps minus four E the minus T over five. So lets evaluate T equal 0 to whether or not we get the correct answer. And also at t is equal to infinity. We know at t is equal 0 the current through the conductor is going to be the same as it was at t is equal 0 minus because the current through an inductor can't change instantaneously. So at t is equal to 0 minus our current was 0 so at t is equal to 0 our current will be 0. Does that fit into our equation that we have derived from this problem? So if we put in a time t = 0 this exponential factor goes to 1 so we have 4mA- 4mA and we'll get 0 so indeed we are correct if regarded to that. First condition. If you look at our second condition for t = infinity, our current il(t) is 4mA. So if we put t = infinity into our complete response equation, the exponential factor goes to 0 so this entire second factor of il(t) goes to 0 and we end up with 4mA. And at D that's where we'd expect that should be our current at t is equal to infinity. So this is our correct equation for Il of t. It goes from zero to four miliamps and this time constant tells us how fast the transient occurs between those two points.