Hi. This is part two in the Ac Fundamentals tutorial series. In part one, we took a look at what an Ac waveform is, how it's generated, the pros and cons, and we looked at Rms and peak and average and all that. sort of fun stuff.

But now in part two, we're going to take a look at phases and complex numbers. Now, you might have heard of complex numbers before, but trust me, they're not that complex. They're mathematically not complicated at all. And we're going to find out why.

Every calculator, even a very basic one, has this Rdp and P2r button on it. Stay tuned right? so you're familiar with our Ac waveform. This is voltage here. Zero volts here and it goes.

Or it could be current as well. Voltage or current doesn't matter. interchangeable really. And it follows a sinusoidal shape like this.

and it actually it goes negative voltage, positive and then negative voltage. Like this. all positive and negative current. And in the previous video, we saw how this was generated from a moving coil inside a magnetic field.

We've got north and south magnets here and as the rotor turns in there, there's a phase angle theta in there and it actually draw. It starts at zero here and it draws as it sweeps through like that. It actually ends up drawing that waveform as it goes up. There would be the peak waveform and then down here it'd be back to zero and down here it'd be over here and so on.

And that's how you get your sinusoidal shape. It's a natural consequence of a rotating coil in a magnetic field. and in the previous video we took a look at the other button on your calculator here. Degrees, Radians and gradients button.

and we'll forget about gradients because that's a weird thing that like civil engineers and people use. Anyway, Degrees and radians are the two big things in electronics here when you're analyzing Um, Ac and other waveforms like this and radians of course is Uh, Two, Pi and or you can go from zero to 360 in degrees so you know, choose your poison. Of course we looked at Theta. The phase angle here is Omega T because this is time axes like this.

As this thing moves around, it increases in time. and Theta the phase angle is equal to Omega T and that is not a W That's actually Omega which is actually uh, the lowercase version of also Omega which is the Ohms symbol. That's just the uppercase version. So if you want to look at the equation for Ac voltage, it actually looks like this and it's the same for current as well, you just change our V in here for I it makes uh, no difference.

and uh V T here because it's got a time component is equal to V P or V peak. Uh, the peak voltage times Sine which is the sonic because it's a sinusoidal uh wave shape and Omega T plus Phi in here Phi is different to Theta. I've got that here. Theta is there are different Greek symbols and they actually mean different things and it's rather subtle.

In this particular case, when we're looking at and analyzing just the one waveform here, then we use Uh Theta, which is the phase angle. But when we start talking about as we'll do in a minute looking at Uh two waveforms, then uh, we have to start using Phi which is the phase difference between two waveforms or between a waveform and a reference. So equations like this, they start getting a bit ugly when you get all this trig stuff in here. like this gives you the heebie-jeebies so engineers what we've done is we've uh, we use phasers instead of these sorts of equations.

So we're going to have a quick talk about and look at phases first. Now Ohm's law of course V equals I times r Okay, he did that in the Dc Fundamentals series. Well, the Ac version of this is actually V with a little arrow on top which means a vector like this. And then you've got I which is a vector multiplied by.

We should have a little dot in there to signify multiply. and instead of R because we're talking about Ac, it's not resistance anymore. it's impedance. And impedance is designated by Z and Z has a vector component as well.

Because when we start talking and analyzing Ac waveforms, we have vectors and you can see in here this is a vector, It's an arrow like that. They're called vectors and this is where the word phaser actually comes from. It's a combination of vector at the end and phase angle. So we're using a vector and we're rotating it and getting a phase angle hence the name Phasor.

So we work with phases and when we've got our phases which is a vector like this, a vector, a vector. It has a magnitude like this and a phase angle. So now we'll get rid of our magnet thing because we don't want to deal with like we don't care how this sinusoidal wave shape is generated. Now all we care about is how we look at and analyze Ac circuits using phases.

Now we have to introduce a second sinusoidal waveform because this is where all the advantage of complex numbers and phases and everything comes in. So let's just draw another waveform in there. Now you can see that it's lower amplitude like this. Its peak voltage is not as much, but it crosses the zero point at exactly the same time and it reaches the same peak at exactly the same time and didn't have time to build it to scale or to paint it.

But you get the idea right. it's exactly the same waveform except it's a different amplitude. Now, when you have two waveforms like this that cross at exactly the same point and also reach their minimum and maximum at the same time, these are called in-phase signals. And if you wanted to represent this as a phaser like this in our diagram over here.

Okay, we can do that. This one here is that length there? Okay, that's shown because the length of the vector in here represents the amplitude there. So during time when this vector goes around like that, that will equal the amplitude over here. So the length of the vector equals the peak amplitude, but the red waveform? Because there's no phase difference between these that phase angle we had before Theta, that's actually 0 or phi.

Actually, because we're talking about two waveforms now. Okay, it's exactly the same, but it's this amplitude here and we can draw another circle in there. and if we dot that across, we'll find that these amplitudes are exactly the same. We've got two different phases here with two different amplitudes, but they're represented pointing in the same direction because there's zero phase difference between them.

They're in phase. Now, if your other waveform, the red one here is the exact opposite of blue. Blue is going to be our reference waveform here. You always when we're talking about all this sort of stuff, you always have a reference waveform and that's where you get the phi from.

You have to have a reference in order to figure out whether you're leading or lagging, or we'll go into that in a minute. But anyway, so you've got a another waveform here that crosses at exactly the same points here, but it goes in the opposite direction. It reaches a peak that's opposite. Like this.

this is called anti-phase or 180 degree. Often, 180 degrees out of phase or simply out of phase. So the common terminology there is in phase. If they cross at the same point and also reach peaks at Uh with the same Uh polarity, then they're in phase.

If they cross at the same time, but they reach opposite Uh peaks at the same time, then they're called antiphase or outer phase and the phaser over here. You guessed it, it's a hundred and eighty degrees like that because the amplitude's a little bit smaller. It's and you can draw your circle in there and dot that across. And it gives you a good physical representation.

So these phases are actually useful for graphically understanding and illustrating and as we'll see in a second. Actually, I'm adding up these waveforms as well. You can actually do maths using Uh phases like this. You can do uh, all your arithmetic graphically.

Now let's draw another sine wave in here at a different phase angle like this. or it has a phase shift. and we have to start talking in terms of Phi here because the difference in the phase angle between where they cross like this. you can take any point, you can take a peak or any other point.

It's just convenient to do it at the zero crossing point, This difference in phase angle between here and here. that is our Phi. That is the phase difference between the two waveforms. As I said, the blue one is going to be our reference.

You have to pick a reference. Technically, it might not matter which one it is, you pick a reference. That's the whole idea of this is you have a reference waveform and then Phi is the phase difference between uh, the second waveform from the first one. How many degrees or radians depending on which system you're using, how many degrees or radians difference? Now, an interesting thing to note.

when you have two sine waves, it doesn't matter what the amplitudes are, It doesn't matter what their phase angle or phase difference is. makes no difference whatsoever. When you add the two of these up or you subtract them, you end up with a sine wave. You always end up with a sine wave of some amplitude and some phase.

So on, our phasor drawing over here. How can we draw in the red? Uh. waveform. Well, we've drawn in the blue waveform.

our reference waveform here. at Time zero. We're drawing it at time zero and then it rotates. It has an angular momentum or an angular frequency as time goes on.

So that would be your angular frequency over here. In fact, that's what Theta is. Um, omega T is your angular frequency over there. I forgot to put it in.

Remember that omega is actually 2 Pi F. There's a frequency component in there, so we have to take that zero reference. and where. what is our value at zero here? Well, we can dot that across like that.

And at that point there, where it crosses our, uh, our amplitude reference circle Like that, I'll call it. I don't think that's I don't think it hasn't Does it have a name? Yeah, Amplitude reference circle. That'll do anyway. Um, we draw that over and then at that point that's the point at time zero.

So you draw a vector in like that and you've got the exact magnitude. This is if you actually did this on graph paper, right? or in a Cad package or whatever. Um, you would this would. These would all perfectly line up.

It's a bit how you're doing here on the whiteboard, but if you drew it up, this is exact. You know you can actually take measurements off this and as we'll see in a minute, you can actually add these waveforms up so that phase angle in there, or phase difference. that's going to be Phi like that. Okay, so let's actually, uh, drop this down onto what's called an Argand diagram and this is what you analyze and you know, sort of like visually represent complex numbers In and we're eventually getting to complex numbers.

We're still, you know, looking at how we're going to add phases. Anyway, we can add these two waveforms together because this is a standard operation in electronics You want to do is you want it. You've got two Ac waveforms and you want to add them Well, we draw them in as our the blue is not very good is it? Anyway, now on our Argan diagram. this one going uh right like this on the X-axis This is actually what's called our reference plane or our uh, you know it's equivalent to our reference waveform down here.

and that's why we draw our reference waveform actually on there. because there's no phase difference between the reference waveform and the reference. The reference waveform is the reference. so that's why it gets put down here as V1 Voltage One.

And then we've got voltage two, which is a vector that has a phase angle. So we can actually add graphically these two waveforms. because as I said, remember, the length of these vectors is actually represents the actual real magnitude, the peak voltage of, uh, the waveform. So what we can do, we can add these graphically.

So what we do is we get our ruler here, recommend a micro ruler and you actually get the exact length of it like that and you keep the angle. so hold my tongue at the right angle and I'll move this across like this and I will draw a vector in like that. And of course you will also find that that length there if you move that up like that should be the exact length over there like that. And this gives us a new point over here.

and what can do so we can draw in a line like that. Bingo. You guessed it. The amplitude of this line.

The length. Oh look, it's almost the length. Oh, it's pretty much precisely the length of my ruler, which is one tenth of a smooth. By the way.

this is the only ruler in existence that actually has a smooth scale. This is one tenth of a smooth. so our vector here is one tenth of a smooth, so the amplitude will actually be up here somewhere. Whoa.

It's gone off into the title. I didn't plan this very well. So this is a graphical technique you can use to add two waveforms together and get an actual proper magnitude result out of it instead of using your confuser over here. Now, you start at time zero Like this.

Okay, so we've got our two waveforms our blue and our red like this. and basically we start at time Zero here. and then we rotate around and we add them at er. So as we rotate this like this, we're going along the time axes and we add them up.

So the blue wave form is zero. At this point, the red wave forms up here. so our green waveform we'll call that our sum waveform is the green one. It's the addition of the two of them.

Obviously, it starts out at this point, and then it's going to be a bit how you're doing here on the whiteboard, so it's not exactly correct. Okay, but if you did this as I said on proper graph paper, and you actually did it properly, you would get a real, uh, proper magnitude response and so you add them together. So at this point here, for example, where they cross, obviously they're the same value. so it's going to be double the height up there.

And you know. So you do this at every point along here and you eventually plot out this green sum response like this. And as I said, any two sinusoidal waveforms you add together or subtract, you will always end up with a sinusoidal result. So it'll be some similar soil result.

It'll be a larger amplitude because you can see them actually add like this. So we get our resultant summed waveform using phaser addition it's called. It's a graphical technique. You can do this, but pretty much nobody really does this anymore because that's what complex numbers are for.

When you start dealing, when you start talking mathematically about analyzing Ac waveforms, you're not doing it graphically with vectors. but you can. And that's where it started. And I'm sure we talked in the previous video about leading and lagging waveforms.

If they're not in phase, then, uh, your V2 here because you always talk in terms of the reference V1. So our voltage waveform V2 here. the red one is either leading or lagging. Uh, depending on the phase difference between from the reference here.

In this particular case, it's a lead in because just just take the zero crossing point as the reference. You can take any point. But it's easy. As I said, to take the zero crossing reference.

It actually crosses the zero point and goes negative before the blue waveform does. So it's leading this waveform. it. it does the business first.

So that's called a lead-in waveform. But if the red waveform was like this and and it actually uh, crossed the zero point and went negative after our blue reference waveform, then it would be lagging. I'm sure in a previous video I've mentioned, uh, the convenient um, acronym civil here. gradients anyone? Anyway, for capacitors and inductors.

in this particular case, okay, V is the voltage, I is the current and for a capacitor like this. So C is the capacitor, the current, I leads the voltage, so the current leads the voltage. and for an inductor, the current lags the voltage because it's after, so you know. Anyway, you get the point and that's why this is not just about generating like voltages with motors and stuff like that.

In circuits you have capacitors and inductors and capacitors and inductors create a phase difference between waveforms, voltage, and current waveform. So hence you know it's important to know this. and for a capacitor, the current leads the voltage. And when you add reactive components like this as they've called and we might see why in a minute, reactive components change the phase angle of the voltage and or current in your circuit.

And this is why we get into complex numbers. Because it does like it gets a bit more complex to analyze it. That's not why it's called complex. Anyway, it's called the complex plane here.

so that's phases. So this is graphical addition using phases. and it's important to learn the concept of where this all comes from because nobody really like uses these graphical methods to actually do calculations anymore. That's what complex numbers are for.

So complex numbers is not that complex. But because we spend a significant amount of time on the background of phases here, we'll leave complex numbers for the next video and link that up here and down below anyway. hope you found phases interesting. Come check out complex numbers in the next video.

Catch you next time you.