Okay, so now we're going to talk about exactly how that's done. But for now, let's just assume that the capacitor is able to be varied from 0.01 microfarads to 1 microfarad. And so I just generated a set of response curves here. For this circuit basically, it's this voltage divider for these typical values, and different values of the capacitor. Here's the lowest value 0.01 microfarads. And so the center frequency of the response is out here above 2 kHz. If I increase the capacitance to 1 microfarad, then it moves down to about, the response to about 200 Hertz. And the you can see how the width increases as I change the value of the capacitance, as well. And so, by, if I had some way to vary the value of this capacitor, I could make this filter sweep its center frequency over this range. Now, the other thing about this and that's really the, the presence of this R1 It makes this a so called shelving of band-pass filters. So the response does not go to zero at zero frequency like it would if that resistor were not there. Putting that resistor there makes that response go to something finite. So no matter where this filter is, I'm still not killing off the low frequencies. But I'm accentuating some band of frequencies that I can move from maybe 100 Hertz in a typical Wah Pedal. Up to a couple of kilohertz, and so that's functionally what the Wah Pedal does. The trick now is to figure out how in the world do you build this variable capacitor. Okay, now, here is a Op Amp version of the Wah Pedal. That, that, you, there's a, a few ideas for how to do this on, on the internet. But, we, we, put this, this, schematic together, and prototyped it, and, and it works. now, this is kind of a culmination of everything you've learned about in this course. It has a lot of A/C circuit, design and A/C circuit analysis, plus we're using Op Amps. Now, we're introducing this one interesting idea of feedback to the capacitor. Which is, a new idea here and really the essence of what makes this thing work. So I've redrawn the schematic here is the, the two resistors in series, followed by the RLC. And the RNL are grounded. The C is not grounded. It's attached to the output of an amplifier. Now what you do is you take the output, it essentially comes from this point here. And you can put that into a simple inverting Op Amp, and you can set the gain wherever you like. the prototypes that we've built, the gains may be 50 to 100, so that's the ratio of this resistor to that resistor. If you wanted a gain of 50, this resistor is 50 times that resistor. And then, there's the output, and so then what we do is take the output and we go through a potentiometer. And the wiper on the potentiometer then goes to a simple buffer stage, and so this a unity game buffer built with an ambient. And the output of that is connected to the end of the capacitor. So, the capacitor is not grounded, it's attached to this output. Now, what happens when you do that is, you're, instead of grounding the capacitor. You're using feedback on the far side of the capacitor in a way that makes the capacitor look larger or smaller than it really is. Now, this is a little hard to understand but give this a, a little bit of thought. Here's a capacitor, it has a capacitance C and there's a voltage V across that capacitor. And, here's the relationship between the current and the voltage. So, the current is just j omega C times the voltage, or, if you like the voltage. Is I times the impedance and the impedance is 1 over j omega C. Now, to make this thing look like, so if I apply a certain voltage to it, and then I ask, how much current is going to flow through it. That's a reflection of what the capacitance is. If I apply, say 1 volt at, say 1 kHz. I'm going to have a current flowing through this in magnitude, that is proportional to the size of the capacitor. If I could somehow increase that current. Then that's like making the capacitor look like it's bigger, for that given voltage, okay. So that's the key thing, if I have a fixed voltage, but I can find some way to make the current bigger than it would be. With just a grounded capacitor then I can make that capacitor look bigger. And so what you do Is you apply feedback to the other terminal of the capacitor. And then, what happens is, when the voltage on this end is pulled up a little bit, the voltage on the other end of the capacitor is pulled down. And so, notice that this is inverting and this is non-inverting. So if this voltage here goes up a little bit, then this voltage here goes down a little bit. And so I'm pulling the capacitor in two different direction. So, I'm, when I, every time I add a little voltage to it, the current should go up a little bit. but what happens is I'm taking, and I'm taking that voltage, amplifying it, applying it to the other terminal, pulling the other terminal even farther away. So I'm increasing the voltage across this capacitor. And that looks just like I've increased the capacitance. So there, there are two ways to get more current through this capacitor. One of them is to increase the capacitance for a given voltage. The other one is to use this feedback applied to the other terminal of the capacitor. To, every time I, so that every time I increase this voltage a little bit, I pull this one away. And that makes this thing, makes this capacitor, look like it has a larger capacitance. Now, you can kind of think of a mechanical analog of this Imagine you have a spring. And say one end of the spring is attached to a rigid base, and I grab the other end, and I try to pull it up. I'm going to experience the spring constant when I try to pull that. So if I apply a certain force, the spring moves by a certain amount with a certain velocity. Now, if I were to take, and let's say I had some kind of a recover system that monitored how I'm pulling on this spring. So that when I try to pull up, the spring is going to take and move the base and go with me. So it's going to make it easier to pull up. And it's going to make it look like the spring if weaker than it really is and a weaker spring is like having a larger capacitor. If you go back to the, the lecture where I was talking about mechanical analogs to electrical components, a spring and a capacitor are analogous to each other. but you have to be careful, a stiff spring is like a very small capacitor. A large capacitor is like a very weak spring. So, if I want to make the capacitor look larger, what I need to do is pull this voltage the other way and that's going to make C, the effect of C larger. That's the same as with the spring, when I pull on the spring. Moving the other end of the spring along with my, my pole. That makes the spring look weaker, if I turned it around, and, and, every time I try to pull the spring up, I pulled down. Then that would make the spring seem stiffer, and I wouldn't get as much displacement for the same force. So, I hope that this intuitive explanation helps a little bit. But this is a, a nice illustration of how you can use feedback in electronic circuits, to, do some interesting things. So, the bottom line here, and don't worry if you don't understand all of the details of this. But I just wanted to kind of pull everything together. And so you can see how you can use everything you've learned so far to really understand something that looks mighty confusing at first glance, the Wah Pedal schematic. But what you're doing is using feedback to change the apparent size of C. And every time I change the apparent size of this capacitor, I'm shifting the resonant frequency of this guy, and so just by moving the foot switch. So for the foot switch it's all the way down, so it's zero. Then there's no voltage being applied to the capacitor. And this actually looks like a ground because look at this. You know enough about Op Amps to figure this out. If the wiper is down here at ground then this terminal of the Op Amp is grounded. Now, you the know that the ideal Op Amp model always wants to keep these two, inputs, the non-inverting and the inverting input at the same voltage. So, if this one's grounded, then this one looks like a ground. And, so, in that case, when the wah pedal is all the way down here, the potentiometer and the wah pedal is all the way down to ground. Then, this looks like ground. And, the capacitor just has it's nominal value, whatever. If it was a 0.01 microfarad apacitor, it will behave like a 0.01 microfarad capacitor. But if I move the Wah Pedal potentiometer up here. Now, I'm going to be applying a, every time this tries to go positive, I'm going to be applying a negative voltage to this end of the capacitor. And I'm thereby making the capacitor look larger. So the further up this potentiometer the wiper is the larger this capacitor looks and the lower the frequency is going to be. So, that, in a nutshell, is how a wah pedal works. And the last thing is that, assuming that you build the guitar amplifier and you found that to be an enjoyable and rewarding experience. This would be a wonderful project for you to attempt on your own afterward. So and so I hope that this helped you understand this a little bit better. And appreciate how with the simple tools that we've been able to, the analytical tools that we've been able to assemble in just a few weeks. How you can really start to look at some of this electronics that was probably quite mystifying before, and really start to understand how it works.