The topic of this problem is The Complete Response of RC Switched Circuits. The problem is to find vc of t at t is equal to 0 minus, vc of t at is equal to infinity. And ultimately, the vc of t for t greater than 0 for the circuit shown below. Circuit has a DC voltages source, three resistors, a capacitor, and a switch in it. The switch is at the top of the circuit and at time t is equal to 0, the switch changes from an open state to a closed state. So that the path around R1 or in parallel with R1 is a short circuit for time t is equal to 0 plus so this switch is thrown at t is equal to 0. So in order to find vc of t, we first have to find vc at t for t is equal to zero minus and also vc of t for t is equal to infinity. As two of the variables that are in our overall vc of t equation. We know that, what's happening is the voltage across this capacitor is going to change when the switch is thrown, when it goes from an open state to a close state. So this will be represented by this value before the switch is thrown which is vc of t at t t is equal to zero minus and a long time after the switch is thrown which is vc of t for t is equal to infinity. So let's find those first. So in order to find those, we have to re draw the circuit for the two conditions. We'll draw the circuit first, for the condition t is equal to zero minus and this is before the switch has been thrown. So before the switch has been closed, it's an open circuit and parallel with R1. We still have our R2s and our R3s with our circuit And we have the capacitor which is the part of a DC circuit and we know a capacitor and a DC circuit which is in steady state acts as an open circuit. So we replace the capacitor by an open circuit. So what we're looking for in this case is Vc at t is equal zero minus. Now that we've redrawn our circuit for this condition, it becomes pretty obvious what Vc at t is equal to zero minus is. it's voltage division of our source between R1, R2, and R3. We're interested in the amount of the voltage source that's across R3. So that's going to be V sub s times R sub 3 divided by R sub 1 plus R sub 2 plus R sub 3. Now if we look at the t is equal to infinity case which is a long time after the switch has been thrown, we can draw a circuit for that condition. We still have our v sub s. We have our r sub one, r sub two and r sub threes, And we have a short circuit in parallel with R1 because our switch is now closed. We also have our capacitor out here and it's still a DC circuit. It's still in steady state. So the capacitor acts as an open circuit at t is equal to infinity. So now if we want to find Vc and t is equal to infinity it's going to be another simple voltage division where we're taking our source voltage and we're dropping some of it across R2, and some of it across R3 because R1 has really been taken out of the circuit because of the short circuit which is in parallel with it. So Vc af t is equal to infinity is going to be V sub s times R3 divided by R2 plus R3. Okay, so the voltage level does change. At t is equal to 0 minus, it's one level at t is equal to infinity, it's another level. If we're looking for our total VC of t, we know that that is our complete response with something that is a complete response. Of our circuit so Vc of t is equal to a transit response. Plus a steady state response. And our steady state response is that response at t is equal to infinity. If we want to put this in terms of information that we know, then we can write this as follows, our steady state response, we'll write that first, is Vc of t, at t is equal to infinity. Our transient response is going to be the difference in our voltage levels at t is equal to 0 minus, and t is equal to infinity. Times the exponential of minus t over tau. We have our values for t is equal to infinity and we have our value for t is equal to zero minus. The only thing that we don't have in this equation for the complete response of the capacitor for time t is equal to zero and moving forward is our tau value. And we know that tau in these problems are defined as R Thevenins times the capacitance. So what we do is we look at our circuit at time t is equal to infinity to analyze R Thevenins for our problem. So, let's take a look at that. If we do that, we know that we're analyzing R Thevenins. We take the voltage sources and we replace those by a short circuit, so we're going to take V sub S and remove it and put a short circuit in. We still have R1 and it still has a short circuit, in parallel with it. We have R2, we have R3 and we're looking for the thevanence equivalent resistance with the load taken out. So with the capacitor taken out, so we have R1, R 2 and R3. So R Thevenins in this case is pretty is to see. Looking back into the circuit, it's going to be R3 in parallel with R2. So our total response, Vc of t, is going to be equal to V sub s R3 divided by R2 plus R3 plus Vc at at t is equal to 0 minus, which is V sub s. R3 over R1 plus R2 plus R3, Minus Vc at t is equal to infinity, which is V sub s. R3 over R2 plus R3, e to the minus t over tau, and tau in our case it's going to be R2 parallel with R3, times RC. So this is our complete response for the capacitor, as the capacitor voltage changes from an initial value of V sub S R3 divided by R1 plus R2 plus R3, to a final value of V sub S R3 divided by R2 plus R3. We can check that in our equation for the complete response. So if we put in a value for t, of t is equal to infinity, this exponential goes to zero. And so this second term of our equation goes to zero. And that leaves us with our first term which is correct. So for t is equal to infinity, it should be V sub s, R3 divided by R2 plus R3, and indeed that's what we get. Similarly, if we look at time t is equal to zero, we know what we should have. It should be V sub s, R3 divided by R1 plus R2 plus R3. And the reason is because the capacitor has a property in which the voltage cannot change across it instantaneously. So at t is equal to 0 minus, this is a voltage at t is equal to 0 plus, this is a voltage. So if we go over here, and we look at our equation, we evaluate it for t is equal to 0, this term goes to 1, the exponential does, and what that leaves us with is these three terms and we look at that we would end up with this as our Vc at t is equal to 0 and that's correct because it's the same as what we have for Vc for t is equal to 0 minus. So this indeed is a correct, complete response for the capacitor voltage as a function of time for t greater than or equal to zero.