Now, I'd like to introduce the operational amplifier, which is going to be the building block for the guitar amplifier that we're going to build later on. Now, the model for a operational amplifier, or op-amp for short, is a simple dependent voltage source model. So here's our model for an op-amp. Now it's beyond the scope of this course to really talk about what's inside an op-amp. There's some number of you know, 25 to to a few hundred transistors on a integrated circuit inside a real op-amp. But we're going to just deal with an idealized model for how the op-amp operates, so that we can then build it into circuits and analyze them and predict the behavior. Now, the op-amp has, there's an input side and there's an output side. And, the input inside there are two terminals, there is a plus terminal and a minus terminal. And there's going to be a voltage at the plus terminal, VP; VN at the minus terminal. And then, there's going to be a current, iP and iN, flowing into those two terminals. now, internal to the op-amp there is some internal s- input resistance, RI, that connects VP and VN. Now, the output side of the op-amp is something here that represented this diamond that is a dependent voltage that is some number A, which is the game of the loop end of the op-amp times the voltage difference between those two terminals. So, this is just the definition of what this op-amp does. It produces a voltage and it's output that is proportional to the voltage difference at the 2 input terminals. And there is an internal output resistance in the op-amp as well, and then, the output voltage I'll represent with vO, and there's some output current of the op-amp. So, the op-amp, the key element is this dependent voltage sources. The voltage of this source depends upon this voltage difference. Now, a real op-amp typical values are the input resistance may range between a megaohm to teraohm for certain specialized types of op-amps. So this is typically a very large input resistance, so there's going to be very small amounts of current flowing into the input. The output resistance on the other hand is typically fairly small maybe between 10 and 100 ohms, and the gain, the open loop gain is typically very large. maybe from 100,000 to 100 million. Now, the key to the op-amp is that it has this very high input impedance and it has a very high gain. And so, it's going to draw very little current at it's input, but it's going to be able to provide a lot of current at its output. And so, that's how you're going to get gain out of these sorts of devices. and don't worry, this gain seems like a very large number, but the whole key to using op-amps is employ feedback. And the gain of the overall circuit is going to depend upon the components you use in the feedback network, and we're going to talk quite a bit about some of those different configurations. Okay now, the operational amplifier is an active device, which means that you have to provide power to it to make it work, and the power supply that you connect to the op-amp is typically going to be limited to a few volts. And in the case of the amplifier that we're building, it's a 12 volt DC power supply. Now, the output of the op-amp can't exceed the voltage of the power supply. So the fact that the op-amp has a very large. Internal gain, open loop gain, and the, that the output is limited to the power supply, means that here we have this dependent voltage source. So the output voltage is going to be the open loop gain times the difference of the voltage at the positive and negative terminals. This is a big number this is limited to the power supply. That means that VP and VN have to be very close together in voltage. So that's the first observation about this. Now, we're going to introduce an idealized op-amp model, which it seems so simple, it's hard to see how it's going to be useful. But we'll see that it's perfectly adequate for analyzing the kinds of surfaces that we'll build later on. So in the idealized op-amp model we're going to say well, the game is so big. That we're just going to assume it goes to infinity, and then that implies that VP has to approach VN and so we're going to assume that VP equals VN. And then the other part of the idealized op-amp assumption is that the input resistance is so large that there's no current going into either input terminal. So, iP and iN are both 0. And it turns out that this very simple model is perfectly fine and adequate for analyzing a lot of the op-amp circuits and, in particular, the two that we are going to take a look at now.