And so this whole thing, with the pick up connected to this tone control circuit reduces to this. Now, I can reduce that a little bit more and I realize that the tone control impedance, ZT, is just in parelle with R0. Now there's one little complexity here that's worth a little bit of comment. We're going to assume that whenever I connect something to the output, it's going to have a very high impinges and so no appreciable current is going to flow out the output line. And so I just have a total resistance, R zero to ground from this point. So what I've done is I've put R0 in parallel with Zt. And so this whole thing then is the series combination of the pickup impedance And then the tone control impedance in parallel with our zero. And so this is the entire circuit, reduces to this simple thing. And the output voltage then is this factored g, which tells you the potentiometer setting. Times the voltage at that midpoint there. Now I don't think I said anything about this, over here on the volume control circuit, the total potentiometer resistances are 0. And will represent the position of the slider with g. So if g is zero, then the slider is all the way at the bottom here, and there's no output. And then g can go from zero all the way up to one. And so the resistance of the, if the slider is all the way at the top. Then there's, then there's no resistance from this point to the slider. But it's, it's 1 times R0 from the slider to ground. So this is just the, the potentiometer position G. A number that goes from 0 to 1. Now back to this all I have to do now is calculate Vout. And that's fairly straightforward. It's going to be V1 and then this thing is just a voltage divider and so I want the voltage across, the paralell combination of ZT and R0. And then the then I have the series combination of ZP and this, so remember the voltage divider equation for the voltage across this impedance down here, is just this impedance divided by the sum of both impedance's and I'm putting the G factor in that's just a simple way. I don't want to draw another voltage divider, but it's a simple way to represent this internal voltage divider in here. But this, the net behavior of this entire part of the circuit here is just R0 and parallel with Zt. And so there's V out, that's V1 the pickup voltage times the position of the volumen control pot, times This, the parallel combination of zt and R0 over the sum of zp and the parallel combination of those, z, zt and R0. So there's the output. Okay, so again, take a, a few minutes and pencil and paper and work this out for yourself. There's lots of J omegas floating around and all that, but it's It's good for you to work through that. Now let's take a look at what the frequency response curves really look like. So here again is the, we reduced the whole complex pickup and tone volume control circuit to this. And here is the expression we found for the output. So what I did is, I just put in typical values that you would, say look up for a typical A telecaster of tone and volume control pot resistances and the size of the capacitor one would use and and then wrote a simple program to compute the frequency response or what I really have is just, I've taken this factor which is a function of frequency. If you go back you know, everything, there are j omegas and zt. And z pickup has a bunch of frequency factors in it. And so I can write a, you could write an Excel program or a MATLAB A program to compute this. And then what I plot is the magnitude of this factor. So I'm just going to say G is one. Let's say the volume's turned all the way up. And I'm plotting this factor as a function of the resistance. In the tone control circuit. So that's the variable, it's this resistor. So, it's not very interesting to look at how things change as a function of, of G. It's just the whole, the whole frequency response group moves up and down. there's no change with frequency. But as I change RT, the resistor in the tone control circuit, the frequency response changes. And it changes in this way. So when RT is very small. Relatively small, than the high frequencies are reduced. And so this is like the base setting, so this is the low the low setting on your tone control pot. The high frequencies are cutout and the base frequencies are, are, are still present. So that makes it sound, have a much more base-y. Kind of tone. And then as I increase, and I've increased by factors of two here, as you increase rt, you see that as it gets large, the frequency response approaches something that's relatively flat. Well, given the fact that you're hearing increases as frequency goes up, your hearing acuity increases, and the pickup actually. The output of the pickup is going to go up a little bit as frequency increases. the net effect is that you're going to hear this as a very troubly kind of sound. So so anyway, there you have it. There's we went all the way from. A looking at the, the collection of wires inside your telecaster. To figuring out the equivalent circuit to working out the details of that and this is exactly what the frequency response of such a circuit would do, okay?
