All right, we're talking now about resistance capacitor time constants or RC time constants. And I've drawn a very simple circuit here. And what we have is we've got a battery. We've got a resistor, and we've got a capacitor Okay. Now, when I close the switch, we will complete the circuit and current will flow. So what we have here is, when I when I close the switch, we will have electrons flowing up on this side of the plate of the capacitor is plate Right.

And from our previous discussions, where did those electrons come from, they come from the other side of the plate, and they flow through that resistor. And they go, they go to the positive side of the battery. And and that process will continue until the voltage on the capacitor equals my battery voltage, which is right there. Now, from our previous slides and our previous discussions, we know that if there was no resistor there, that capacitor would charge up instantaneously. Now, I've just cleaned off the slide, it would again that if they was no resistor there, that capacitor would charge up to the battery voltage in this example, instantaneously. But guess what?

There is a resistor and that's what we're talking about RC time constants. So the bullet here and let's just go through this RC time, our time C is the time of one time constant. Okay? It takes five time constants to approximately charged to the supply voltage, in this case 10 volts, each time constant charges to 60% of the supply voltage. Okay, so let's look at this here. So I'm going to calculate the time of one time constant and I've got a one Going We Sr. And I've got a 10 micro farad capacitor, I just do the math.

So I have one megohm times one micro farad arm, and I do the math and that comes out to 10 seconds. So we look at we look at this. And what we're saying is, in 10 seconds, this capacitor will charge up to 63% of my voltage. I'm going to call it voltage supply rebate supply. All right, so in the next slide, we'll see that now we have a curve and the voltage across the capacitor Has that shape, okay? And if you look where it says one time constant right here, it's 66 point 63% of vs. Two is if we get a little chart here, too is 86.53.

Cut three time constants, it's 95% of vs 490 8.2 and five is 99.3%. Ideally after five time constants, this capacitor will be charged. Alright, so I just want to I just want to do one thing here. So before we go on to the next slide, I'm going to wipe this out and do a calculations and show you will show you what I mean by six. The 3% every time constant is 63% of the battery voltage, even though if you look here, it's it's, well, it's not it's 63 and then it's 86.5 and it goes up to 95%, and so forth. So let's see what I mean by that.

Okay, this chart shows you that at one time constant right there, we charged the capacitor up to 63.2. I said 63% previously, it's point two, but we all round off. And that's pretty much the norm they say 63. But anyways, so what's left? What's left, if I charge the capacitor up to 63% And I still have 37% of my voltage left. In other words, when I close the switch here, on the first time constant, the voltage across the capacitor is 63%, in this case 6.3 volts.

All right, what's left 3.7 volts. In other words, to get to the full battery voltage or the, the full source voltage, I've got to go. I've got to charge the capacitor up to 3.7 volts or an additional 3.7 volts, that's fine, but I can only charge 63% each time constant So, number one. All right, let's have that 10 volts 63 Let's go 63% because it's easier. So 6.3 volts. All right.

All right now. I've got 3.7 volts, or let's do it this way. Well, let's, let's continue. Okay, so now, I want to check I've got there I've got 30 in this case, 3.7 volts, right? times another 63%. All right.

And if I do my math, which I'm going to do right now, so if I do my math, all right. I've got 3.7 volts left, because on the first time con constant, I charged my battery voltage up to 6.3. I've got 3.7 volts left to charge the capacitor up to but I can only charge up to 63%. I do my math. And it's 2.33 volts, that 63% of 3.7 volts, and I add them and I get 8.63 volts. All right, 8.63 volts.

All right, so now, if I've got 8.63 volts, that's a second, let's do the third. So the third time constant would be what? Well, what do I have Left I've got 10 minus 8.63. Let's see what that was. So I take the difference, this is the voltage now after the second time con time constant on the capacitor, my supply voltage is 10 volts DC, I subtract the difference and the difference is 1.37 volts, I get 63% of that, and I had get this, then I'm going to add it again to 8.63. And what do you think I'm gonna get?

I'm gonna get 94 dot nine, nine. And the difference here is, is I didn't do on I 63.2. So there's a little bit of a rounding or a math error, but we're close enough. So what happens is, and again, let me let me clear this slide off so I can explain it Starting here, the first time constant, I chart my chart, my capacitor charges up to 63.2 of my supply voltage. All right, on the next time constant, I take the voltage that is left, which is the difference between my supply voltage and the voltage across my capacitor and take that and the capacitor will continue to charge up 60 63% of that voltage. And then on the third I do the same thing.

I take the difference between my supply voltage and the voltage on my capacitor and I multiply that by 63%. And what happens is we get a exponent curve here right Right there. And that curve will flatten out after five time constants, all right, and here is the percentage of my supply bolt voltage on my capacitor. After each time In fact, this chart here actually goes down to a half time constant, and then point seven and then one and so forth. So that's what we mean you may hear the term, every time constant 63%. You look at the chart, and you say, well, that's the first one.

The second one but basically, we take 63% of the voltage that is leftover if I can use that term, or the voltage when I subtract My supply voltage, or excuse me, when I subtract the voltage on my capacitor from my supply voltage, that voltage, we then again take 63% of that, that charges the capacitor and so forth. All right, it goes and it'll go on up to five time constants, then for all intents and purposes, after five time constants, my capacitor is charged. Ideally, the capacitor will never get charged. It'll never get up to 100%. But after five, it's five plus it's like 99.99999. And we're good.

So that's it. I wanted to point that out. So let's let's go back to the slide and continue. Okay, on this slide here, Gann we know that one time constant Is 10 seconds. And all I did was calculate the time and this is the voltage across the capacitor. So for instance at point five time constants, which is so right there and here it is here on the curve.

We've got 3.93 volts, if my supply voltages 10 volts and so forth one time constant right here and the voltage across the capacitor will be that and I've, I've, I've brought that out to five times five time constants. But the point I want to make here is this right this this is the curve across the capacitor and the curve across the resistor is the opposite. Because what happens when I them the instant I close This switch, I have maximum current flow, if I've got maximum current flow, I've got maximum voltage across the resistor, which I'm showing you right here. And as the capacitor charges, I have less and less voltage across the resistor. So the, the curve on the resistor is the opposite of the curve on the capacitor. All right?

And take a look at this. So you can stop the recording and take it take a look at that. The manipulation of the map is very, very simple, you should be able to handle that. It's just a multiplication and subtraction and maybe some addition. That's it. So take a look at that and we're going to go on to the next slide.

Okay, first of all, you should ask yourself, how did we get how did we get these numbers? voltage right there? Well, we use this fancy equation right here. We're VC equals vs. minus, that's an act that's called the natural log minus t divided by RC. Okay? And RC right there is there is the product of our resistance times capacitance.

And this is called minus T or tau. And tau is the value that I'm interested in. So for instance, I did one here. I'm going to bring the calculator down because up if you look at it says use calculator. In the old days before the calculator, we had charts. So we could go into the book and and you know, I have a table of charts and we could find it.

But right now with the app The calculator we just plugged in, and I'm going to do that for you in a minute. But his is the equation right here. Let me stop here and close slide off. Basically what I do is I say, VC equals v s minus one, I just just put it in. And now I start plugging in the the equation. All right, so one minus E, and RC is one second.

Right there, and I'm interested in one second or one time constant. So now I just do the math. Okay, I'll show you how we got that in a minute. And in one time constant, it's 6.63 volts. So let me Stop here, pull a calculator down, and we'll go over that. I've brought the calculator down and I'm gonna plug this in here.

All right, so I'm gonna click in. Okay, before I go, I'm I'm brought the calculator down from the operating system, I'm using Windows 10. I make sure that I'm in the scientific mode here. All right, and notice I can change my screen by that arrow. So I want to I want to see this natural log right there. Okay, on natural lips, exponent.

So I'm going to come back here and I'm going to type in 10, or clicking 10. And what I usually do when I have something like that, I know it's 10 times what's in the parentheses, but just for the heck of it. I'll hit the X, meaning multiple location. So now I'm going to one parentheses, okay. And I'm going to go one minus, and one over one with a negative sign is a minus one. All right, so I'm going to do one, change the sign.

I'm going to use this. And then I'm going to close the parentheses. And then I'm going to multiply it because when I close the parentheses with the calculator, I've gotten the result from here to here, but we know we're going to multiply that by 10. So I'm going to hit the equal sign. Well, let me bring it down. And then we go 6.32 volts.

And I'm sure I got got that 6.63 It's a rounding error. I may have rounded a number, but that That's right. And that's how we that's how we use the calculator. And give it a shot, bring it down, give it a shot, get used to the keystrokes. All right on the calculator. So with that said, let's stop here and go on to the next slide.

All right, on this slide here, I've drawn a circuit and we've got a switch. And we're taking my output there. And that's, that's the waveform right there. Okay. And first of all, All, let's find the RC time constant of that, which is 100 k times one micro farad. And I've done it here.

And when I do my math, I get 0.1 second. Okay, let's notice I've got a wave form a square wave. Some people or some books will call it a step voltage. All right? I'll call it a square wave, because that's what they call it an industry. So we have a square wave.

And the time that it is on is point one second, so it's highs. All right, so let's say somehow I've got a mechanism to switch that on and off. All right. So when it is on I'm charging this capacitor through this resistor and this is 100 volts just like I'm showing you there, okay 100 volts. So, I switch that on was 0.1 seconds. Well, we know from the chart in from previous slides and explanation that in one time constant, the capacitor will charge to 63.2% of the voltage.

So therefore, right here at that instance, I have 63.2 volts across the capacitor, all right so, So now what happens? All right? I go from 100 to zero volts right there. So what happens? Let me clear the slide. Okay, I've cleaned the slide off.

So at this point here, which is one time constant, we have 63 dot two volts across the capacitor. But look at what happens my step voltage now transitions to off. All right. Now, what will happen? Well, let's look over here. So now my switch is closed.

And what happens is, the capacitor will discharge through the resistor because this is there's no connection there. Okay, so now my capacitor will discharge this way. All right. So what happens? We've got 63.2% of the charge on the capacitor which happens to be 63.2 volts. Because we made it easy.

We, we said we've got 100 volts here. Okay. So now we discharge but we discharge 63.2%. So what I've done here is the voltage across the capacitor at this point here, right there is 63.2. All right. It's going to discharge 63 Point 2% of that.

All right, so now 63.2 minus 6.632 times 63 two, I find that some and I subtract, and at the end of two time constants right there. I've got 23 dot to three volts. Now, what does this sign? What does this look like here? Well, let's go back to the previous slide. All right.

So if we look at this chart, okay, here is my charging curve across the capacitor. Alright, and we have 12345 times constants. And now, once I get a charge on the capacitor and discharge it through a resistor or a some load, I get the opposite, which is showing you this curve here. All right. So that's what I wanted to point out. Okay, so when I'm charging it, I get this exponential curve.

And when I discharge the capacitor, I get the exponential curve. flipped around. In other words, if I take this here, and flip it vertically, I get this curve here because they're opposite. All righty. Okay, so now we're here and we go down to 25. 3.23 volts and we do it again.

Okay, so now I want to charge up here. Okay, again, one time constants, what do I charge? What is my charging ratio 63.2%. So we've got what we've got 23.3 volts here, we've got 100 volt source. All right, so now 100 minus 23.3, and I multiply that by 6.632. And I get 71.7.

So at this point here, it's 71.7. I discharged 63%. And now I'm at 26 dot four. So you can See the capacitor charges discharges and it in each time and each time charges discharges charges discharges and each time I either charge to 63% of the voltage or discharge 63% of the voltage and I take into consideration the when my step volt voltage change from here to here to here, I had to take into consideration the existing voltage on the capacitor all right by the way The waveform here. That's called a sawtooth. Okay, it's called a sawtooth right there and there is a W that that's a w, t o th sawtooth.

Alright. And that's what it's called. All right, we're going to stop here. I've got a little bit more to go, but I'm gonna, this one really ran long and I've still got a little bit more to do on this. So I'm going to stop it here, break it up, and then we're going to continue on part two of this in the next section. Alright, talk to you soon.