Hi, it's electronics Fundamentals time again. And in this video, we're going to finish off a basic Dc circuit theory or steady state Dc circuit theory. because we haven't covered transients yet. That's got to be next.

Anyway, we're going to have a look at Dc power efficiency and maximum power transfer. and this is basically the end of the Dc circuit theory, which we covered in voltage and current sources. We did a mesh analysis, nodal analysis, superposition theorem. We did thevenin equivalent circuits Norton equivalent circuits.

So basic Dc circuit power and efficiency will finish all that off nicely. Let's take a look. So in any linear resistor at all here. Yeah, I've used the square box symbol.

You can I know. for all you fanboys, here you go, I'll use the zigzag symbol. There you go Anyway, you've got a resistance. and that resistance can be anything.

As we explained in previous videos, it can be inside a battery, inside a power supply. Any voltage source. the internal resistance of a voltage source. It could be a resistor in your circuit.

It could be a Pcb trace, It could be a wiring. It could be transformer resistance. It could be inductive resistance. It can be almost anything.

And when you pass current I through that resistance, you will get a voltage drop based on Ohm's law, and it will also dissipate power in watts, which is also joules per second. Not to be confused with the rate of energy consumption which I've covered in another video. Not to confuse power and energy, so I'll link that one in up here and down below. So once again, we've got some basic formulas which you'll absolutely have to remember to do Any basic electronics at all.

If you're not remembering these formulas in these red boxes here, then, well, you don't know electronics. They're just like you'll use them absolutely everywhere. So briefly. Ohm's Law Of course, you can think of it as the Ohm's Law Triangle as it's called.

It's just a nice visual representation of voltage. Current which is I and R, which is resistance. and you can just if you visualize that. And we've got an equivalent one for power too.

If you visualize that, then you can. You don't actually have to remember the formulas. If you visualize that, you can create the formulas: Voltage equals Current times resistance because they're next to each other, so it's like dot. It's times and current.

Here is voltage over resistance. Like that that little bar in there. You can think of that as like, divided by so I equals V on R And likewise for resistance. it's voltage on current.

So they're your basic Ohms Law formulas. Now we have a similar one for I've called it the Power Law but that's not really what it is, it's just it's just power. Anyway, we'll run with that similar power triangle. We'll call this P is over I times V.

So power in Watts is current times voltage. And likewise, Current is power divided by voltage and voltage is power divided by current. That's it. Memorize them.

You'll need to know those. And of course, any time you dissipate in power in Watts in any sort of resistance at all, it's going to heat up. It's going to generate power. And that can lead to non-linearity and things like that, which we don't really want to get into because this is all Dc step like linear Dc circuit theorem here.

But basically you know, like a light bulb, for example, is just a resistive wire in it. As it heats up, it goes up in resistance. and we won't get into temperature coefficients and all that sort of stuff. but you just need to know that Ah can and almost always changes with temperature.

There are very low temperature coefficient materials. Oh god, I'm getting into it. No, don't worry. p Learn these now.

p equals I Times V. But there's some instances where you may not know the current because you haven't measured it, or you may not know the voltage because you haven't measured it. So these are this. actually.

Uh derives out into two other formulas which are very common and you must know these as well. Arguably just as important as these because you'll use them everywhere. So the way it works is we can assume. Well, we don't know what voltage is, but Ohm's law says voltage is current times resistance.

so we just substitute that in here. So it's current times, current times, resistance and that works out to I squared r or current times current times resistance. but that's how it's written. I squared r and that formula is used everywhere you want to impress someone at a job interview or something like that.

Just say oh yeah, that's due to I squared our losses. You'll get that everywhere because there's I squared r losses in everything. As I said, Pcb traces, wires, inductors. Have I squared r losses Like why is that inductor heating up I squared our losses.

Why does the transformer heat up I squared are losses? Well, there's magnetic reasons as well, but I squared are losses. and then because there's series resistance in practically every active component out there, it may dissipate a little bit of power. It may have some I squared r losses in it. Even a capacitor, for example, is got an equivalent series resistance lead resistance on the physical capacitor, even if it's a surface mount r capacitor, those little end caps in there, they've all got resistances, Your solder joints have resistances, Your Pcb traces, everything has resistance, and they will have I squared losses.

When you pass any current through it at all, you'll get. I squared our losses, so that's a really cool term to throw around. I squared our losses. Nobody really goes around saying oh V squared on our losses, even though it's the same thing because we can once again how we substituted before when we didn't know the voltage if we don't know the current, then well, current equals voltage on resistance.

so we can just substitute. Oh, here it is. substitute I here with V on R. So it's V on r times voltage.

And of course, V times V is V squared. And then don't forget your R divided by R. So there you go. There's your two formulas for power as well as these, but you'll use those all the time as well because you don't often know both of those parameters because you haven't measured them, and because you may want to convert from one specific voltage, current, resistance, or power to another, uh particular thing.

There are actually more formulas uh which you derive from these like uh, Current for example, is square root of P on R and voltage is equal to square root of P times R. And there's actually what's called an Ohm's Law uh, circle or Ohm's Law pie chart which actually, uh, like lists all of these formulas and there's like you don't have to remember them all once you know you can actually just derive them when you need them. like it's not hard to derive these from here. I mean, you know P equals I squared r For example, Well, you want to get I out.

Well, you got P on R. So you take out the r P on R and then I squared. Well, you have to get the square root of that and that's how you get the equation. Yeah, Anyway, yeah, there's a few more.

So this leads us into the second thing we're going to talk about today, which is efficiency. One of three things now. anytime. of course.

Uh, you dissipate in power in a resistor. When you involve power, you're talking about efficiency. How efficient is it to convert that electrical energy in Watts into whatever it is you're trying to do whatever work you're trying to do. Now that is actually such a thing as a hundred percent perfect efficiency.

It's called a resistor because when you actually dissipate one watt in a resistor, that is one watt that's generated as heat. It is 100 efficient. But the resistor is pretty much the only thing that's 100 efficient. Now, unfortunately, when you want to convert electrical energy into another form of energy, be it mechanical energy through a motor light, for example, through an Led, you might want to convert it into some sort of magnetic, uh, form as well.

You might want to convert it into sound. You might want to convert it into a whole host of different types of things. Well, unfortunately, you're not going to be 100 efficient. so you're going to get some losses.

So this is where we have another equation. which you must know, because you'll use it absolutely everywhere in engineering. And that's the efficiency equation for power. Now, if we talk in terms of, say, a Dc to Dc converter which converts one voltage into another voltage like a 240 volt mains.

Ac for example, converts it into Dc. It doesn't have to be Dc to Dc. It can be Ac to Dc or Ac to Ac. or Dc to Ac or whatever it is.

Right when you convert in one form of voltage to another, you're going to have an input power in Watts. and you're going to have an output power in Watts. And of course, because it's not 100 efficient, the output power will always be less than the input power for anything you've got any system at all. because, well, if your output power is greater than your input power, yeah, we're talking about Over Unity Whack Job Central.

and I've done a few videos on that. And because someone will mention there is a thing called coefficient of performance and I've done that in other videos like air conditioners. For example, a one kilowatt air conditioner might have say three kilowatts of cooling power. and there's tricky business there.

There's other things, other outside elements involved in that sort of thing, but any contained system like this, you can't get more output power than you put into it because that's called over Unity. Ain't happening. Laws of physics, captain. We'll use another example where you're lighting an Led.

For example, you've got an input power coming from your power source, your power supply, your battery, whatever it is, and then you might have a series dropper resistor here. I won't go into why, but a series dropper resistor that's going to dissipate some power there, so you already lost some power there. But the Led isn't 100 efficient. In turn, in turning the current in there, or the power that you're putting into that Led into Lumens light output, it's going to have what's called a Lumens per Watt figure.

It's going to have an efficiency figure for just that component. So you're going to lose power there, and you're going to lose power there as well. So that doesn't quite relate to our Power in Power Out Efficiency equation here. Because we're not Power Out, we're Lumens Out other form of energy output.

But if you're talking about a pure electrical circuit with electrical power in and electrical power out, this equation applies. Learn it. The Efficient: You probably already know it. The efficiency and percentage is the output power divided by the input power times 100 and that gives you your percentage figure.

And that's you. Probably learned that in school and because we're effectively talking about losses here, or the useful power that's actually being used here. The efficiency and percentage is also equal to Power Out over Power Out plus losses times a hundred. It's just like it depends which way you want to think about it.

So it's incredibly simple. Efficiency in power can be 100 if you're just talking about a resistor, or it's going to be less than that in terms of like, a useful output. So if you want useful light output lumens. if you want useful mechanical, you know, force in newtons or whatever it is, you've got some actuator or something.

Once again, you're going to have some losses in the system due to the conversion process of the actual device you're using to convert any internal resistances of any sort of all the components, all the wiring, everything else inside the thing. It's all got losses internal. But power is never wasted. It's always conserved.

So even if you're you're not getting all your lumens out, the rest of the power will be dissipated as heat, both in this dropper resistor and the heat er from the Led. This is why Leds are mounted on heatsinks. You're almost certainly familiar with this. With Led lighting, they heat up.

Why do they heat up? It's because, like leads aren't that efficient, like they piss away like 70 percent of their power in heat, just in the diode itself. Last thing we're going to look at and a very important principle in electronics is maximum power transfer or the maximum power transfer theorem as it's called. And it's incredibly simple when you've got a voltage source like this. and as we learned about in voltage and current sources, there's no such thing as an ideal voltage source, so it must have an internal resistance.

So we'll call that Rs here for source the source resistance. I'll put the two little dots there. This will power a load resistance. Now this raises the simple question at what value of resistance here and series resistance in your source.

Will you be able to deliver the maximum power into your load? And you might think, well, that's easy, Dave, If Rs is zero, then, uh, you've always got maximum power. But yeah, okay, smart Alec, there's no such thing as an ideal voltage source. Rs must be greater than zero. So how can you actually deliver maximum power into there? Because if you're delivering lots of current in here, this internal resistance of your Uh voltage source here, or your generator with a battery, whatever it is, it's going to start to heat up because it's dissipating power.

And so at what point are we going to deliver max being able to deliver maximum power into our load resistor. Well, the maximum power theorem states maximum power that should be delivered to the load is when the load resistance is equal to the source resistance. And at first thought, that doesn't make sense because you might be thinking, well, if I just make Rs smaller and smaller and smaller and smaller. Surely, I'm able to deliver more power into the load, but that's not actually the case.

and we'll take a look at actually the graph and data for this in a second. So it's not intuitive that that's the case. at least I don't think so anyway. and this also has applications in terms of load matching.

You know transmission lines how you have to match your 50 ohm transmission line with your 50 ohm source to prevent reflections and things like that. Anyway, that's way outside the scope of this, but you know it's a similar sort of thing happening. You have to match your load with your source, and that's when you can deliver the maximum power. Let's move our little troll here and let's have a graph of resistance r here versus p in power like this.

Now, if our load resistance is is incredibly high, it's infinite. It's open. Okay, it's right up here. Then we're going to dissipate no power in that load, right? Because well, dude.

Ohm's law. Everything else we've learnt, right? So let's say it's out here like this, right? That's our little infinite dot there. Now, if our resistance is a short circuit, it's got zero Ohms. And of course, the voltage drop across the resistance is going to be zero.

Because well, any current times zero is going to be zero. and well, a power dissipated in your load. Here, for a current times voltage, Well, you can have as much current as you want. If your voltage is zero, you're going to get zero.

So right over here, we're going to have another dot that's 0. and at some point, the curve is actually going to look something like this. And at this point here, this is our maximum power that we can actually deliver into the load and the theorem states. and it's true.

That will be the case when Rl at matches Rs. And if you don't believe me, let's run some numbers. First of all, we need the equation for the power in the load here and through Ohm's law, power laws. Everything we saw before uh, power in the load equals I squared times R.

But we don't know what I is, so we have to calculate it. Current is just voltage divided by resistance. So it's the voltage source divided by the total resistance here, which is Rs plus Rl like that and then squared times rl. So that's we've just derived our equation using our cool formulas that we had before.

It's really easy to calculate the power in the load. So now let's assume that our voltage source is 1 volt and our source resistance is 10 ohms. Here, that doesn't change. Let's assume it's completely fixed and let's change our load resistance.

So I've got different value load resistors here: 1 Ohms, 5 Ohms, 10, 50 and 100 Ohms here. what is the power in the load? Well get your confuser out and you can calculate for a one Ohm load. the power and the load is eight milliwatts. there.

for five Ohms, it's 22 milliwatts. It's going up for 10 Ohms, it's 25 milliwatts. But ah, when you go above 10, it starts going back down again. 50 Ohms here works out to 14 milliwatts and 100 Ohms drops back to 8 milliwatts.

There is a point here. there's going to be a maximum power point on your curve here where you can deliver the maximum amount of power in the load and it works out as we saw here. Well, if you run enough numbers here, you'll find that it is precisely 10 when the load resistor is 10 Ohms. when it matches the source resistance, That's when you can deliver the maximum amount of power.

The best way to have a look at this is let's go to a quick spreadsheet and run the numbers and plot a graph. Let's plot a graph on Excel, shall we? Now, we've got our source voltage here, our source resistance, and we're going to plot the power in the load resistance Rl here. So on the Y-axis we've got the power in the load resistance, in Watts, on X-axis we've got the load resistance in Ohms and over here we've got uh, the voltage which we're going to fix, and our source resistance which we're going to fix. We're going to have 100 Ohms and I won't spoil it yet by typing in a voltage.

And then we've got our load resistance here, which varies in 5 Ohm increments. You can do whatever resolution you want. 5 Ohms is going to be good enough for us to see our graph, and then we're going to calculate the current here, which is the voltage in cell B1 here divided by uh, D2 which is the load resistance plus the fixed source resistance B2 over here. And if you're wondering what these dollars means in a formula in excel like this, it means basically that's a fixed cell don't auto increment.

So when we actually when we put our put our formula into here and then we drag our formula down like that, it's not going to increment. So you'll notice that B1 and B2 there with the dollars next to them are not incrementing, but the other cell does. That's how you put a fixed element a fixed variable into your equation without having to auto increment and then we calculate our load power, which is simply E2 which is the current squared I squared r Remember that so the current result here squared times d2 which is the load resistance and we're going to plot this. So let's put in 100 volts, shall we? Magic There it is.

It starts at zero, it peaks up here, and it will eventually. If you go to infinite, it will eventually go back to zero. It tapers off, but it takes a long time to get this load of resistance up to. you know, Gig Ohms or whatever you know, Mega Ohms Gig Ohms.

That's required to sort of see this drop back down to zero so we can then just muck around with that so we can just muck around with this. We can go to 500 ohms and it shifts like this, but there is still a peak point there where you're going to get that. so it's going to reach a peak value of 25 watts there. And that happens at point 100 which is 100 ohms.

There it is there. and of course that matches that up there. The source equal matches the load resistance and it doesn't matter what value you take, you take it down to ten. Now we're getting towards the resolution of our five Ohm resolution here.

but if we did, we get the exact same smooth graph and if we zoomed in, it would all be exactly the same and we can put a thousand out here and it'll peak at a thousand over here. See magic. Love it. But that's not really intuitive is it? It's remember, we're talking about power in the resistance here, not the current.

If you were talking about the current, then sure enough, when Rl goes to zero, that's when you get maximum current. But we're talking about power, which has this sneaky little squared formula in it. now. Unfortunately, Xl is really stupid and won't let us plot a logarithmic graph on a simple line chart like this.

So you have to do an x y scatter graph in order to get a logarithmic axis. And and sure enough, there's our axis. If I turn off Log, there it is. We're getting the same thing happening here, but I I run out of data at this point.

But we turn on our logarithmic x-axis like this. and bingo, We get a nice smooth bell curve like that. It's beautiful. That's absolutely fantastic, isn't it? And uh, these values aren't correct down here.

Down the bottom because you can't seem to label as X Scattergraph Bloody Excel. Limitations: Unbelievable. So anytime you see a formula with a squared factor in it like that, you know it's probably going to look interesting when you change your axes to logarithmic. And as it turns out, this actually also has implementations in other aspects of electronics like, uh, sharing charge between capacitors and things like that.

That's a real tricky one, which you might do a video on one day. But yeah, it's not obvious because we're not talking about current here. Of course, if you want to deliver absolute maximum current into your load, then of course short out your load. That makes sense, right? You're just going to increase your current and then all of your power is dissipated in your voltage.

uh, source, here, the internal resistance of the voltage source. And of course, if you short out your load, well, you can heat up and blow up your voltage source. whatever it is, unless you got uh, protection. so that's pretty nasty.

But there is a sweet spot in there because we're talking about power. It's a different beast with different equations. the only way you can deliver the maximum power. into that load doesn't matter what it is is to actually match the source resistance.

Not intuitive, but it works out when you actually analyze. it's one of those wow moments. So there you have it, that's Dc power, electrical efficiency, and maximum power Transfer theorem. And that finishes out our basic steady state Dc circuit analysis.

We've pretty much covered everything with all the previous videos. Next thing you probably want to move on to is transients in Dc circuits, so I guess that one will have to be next. So anyway, if you like this series of videos, please give it a big thumbs up. As always, discuss down below and check out all my alternative platforms and not just on the Youtubes.

Catch you next time you.