Hi. This is part three in the Ac Basics tutorial series. We're going to cover complex numbers and they're not that complicated at all. They're really incredibly easy to understand.

and it's how we do uh, Maths operations on Ac waveforms because anytime you introduce, as we saw in the previous video on phases, anytime we introduce a reactive element, a capacitor, or an inductor in a circuit, it causes a current and voltage phase shifts in a circuit. So this is from the previous video. Uh, part Two linked up here down below. Check it out.

First of all, it's important to learn phases before we move into complex numbers. about how waveforms can lag and lead. And as we saw in the previous video, you can actually do mathematical operations on Uh phases and vectors and get a result using graphical methods. But nobody really does that anymore.

We use what are called complex numbers to analyze our Ac circuits when you start including a phase component in them. Because in Dc Fundamentals, you don't have any Uh phase components. There's no difference in Uh phase between waveforms. It's just Dc, It's direct current, It's direct voltage, whereas in Ac it's alternating current, alternating voltage.

and you can get phase differences. And this is where we need to get a bit more clever in our mathematics to actually solve these sorts of things. And you're going to find out Now why every basic scientific a calculator like this has this mysterious Rdp and P2r button on here? What does it do? Well, let's find out with Complex Numbers. So we saw previously once we start dealing with Ac circuits, alternating current, and alternating voltages, we end up with sine wave or other Uh waveforms.

But we'll just stick with uh, sine waves. For all of this that not only have a magnitude like this, they have a peak voltage or an Rms voltage. However you want to look at it, they also have a phase component in them designated by phi here. and then you can have a phase difference between a reference waveform.

You always have a reference waveform as we'll see, and the other waveform has a phase difference either positive or negative phase difference from that waveform. So once you start talking about Ac, you can't just apply regular Ohm's law anymore. It's more complex. I get it.

I'm here all week, so V equals I times r We have to now start talking about vectors. As we said, it's the same formula, but resistance now actually becomes impedance and that impedance will actually have A when you put a show it as a vector over the top. It means that vector has a phase component as we saw before. So if you've got a number line like this, uh, you're familiar with your Re.

This is your real number line from zero to whatever it could be positive, negative, or whatever. Then once you start introducing phase angles, you have to introduce what's called the complex plane which is in this direction like this. And as we saw previously, the magnitude of this Uh vector determines the amplitude, the peak amplitude of the signal. Um, But then you've got that phase angle as well.

In here, you've got that phi angle or the difference between your reference waveform and the other signal. So we have to start using what's called an Argand diagram which shows complex numbers and this is a complex number plane on the X-axis Here we have real numbers and on the Y-axis like this, this is what's called an imaginary number plane. Now, there's nothing imaginary about these numbers at all. They're very real and in fact, they represent real phases.

But when we start talking about Ac and phases, we have to introduce another number method to actually do this. And we do that using the complex plane in this direction. So we saw this equation before, which is the equation for Ac voltage over here or Ac current. You just replace V with I and it's kind of complicated, which is why we don't deal with this sort of stuff because it's got all these trig functions like sine in there and it's got uh, time and frequency and stuff.

and we don't want to deal with that sort of stuff when we're doing our Ac, our circuit, uh, calculations. So we use our complex plane now as it's called with the real component and the imaginary component with the phase angle in there and that takes into account our trig functions like sine and Cos. so we don't have to muck around with that sort of stuff. We can just do all of our basic mathematical operations in our to analyze our circuits using the complex number plane without having to use these trig functions.

It's really neat. So with the complex plane in mathematics, not in engineering mathematics, what they use is the letter I here because it's the imaginary axes in this uh, complex plane. So in this particular case this phasor here this vector has a the equation x plus I y with X being the assume it's like a you know, 45 degrees here or whatever. Anyway, you drop that down to the real axis and this first part of it is that x value when you drop it down and then you can take your line across there like that and this becomes the imaginary complex part of the number.

Remember these are complex numbers so this is what the form of a complex number looks like. You've got the real component and then the imaginary component which is designated with I but that's only in mathematics in engineering we don't use I sorry we use J like that Y We want to be different mathematicians hate it. but yeah this. so this is called the J axis but it's the imaginary axis.

We just happen to use J instead of I so this is called the J operator but I know you might think it'd start looking like sort of nasty, but it's not. There's even a actually a simpler form to this because this, uh, in this particular form um of our complex number. this is called rectangular. but there's two different forms in complex numbers.

Rectangular form is one of them. and it's that form. There's a real component and an imaginary component, but we can also represent it exactly the same way in what's called the polar form. So let's say the length of this phasor here, the length of this vector.

Let's just call that R. So in this case, Z would be equal to R and then we use a real handy notation which represents angle like that. and we just are angle Theta like that because Theta is our angle in there. so that's exact.

That's the same quantity, the same number, the same vector, the same phaser, the same component. just represented in two different forms. So there are two forms of complex numbers: polar and rectangular. You remember how I mentioned the calculator before the P to R and R to P, P and R.

It's to convert between polar and rectangular form and it's so important that it's included on like real, ancient scientific calculators. That's how important complex numbers are and to be able to deal with them on your calculator. So this J operator here you can think of that like any other operator, addition, subtraction, everything else. But what it does is J represents, uh, turning the real component like this through 90 degrees like that.

So let's say that you've got a vector like that. a real thing, a real value. That's say, I don't know three. Something like that.

If you rotate that, change the phase angle by 90 degrees, three becomes J Three because J represents shifting it 90 degrees. Now, with the J operator, when we go in the anti-clockwise direction like this, it's positive. Okay, so anywhere from Uh here, right over to 180 degrees like that, that is positive. But if you go in the other direction like this, it actually becomes negative like that.

So if we took our example here of our vector of length A 3, our a real number of Uh three could be three volts or whatever. and we rotated it like this, it'd become negative. J Three got it. Or if you're a polar fanboy and there's reasons to use rectangular and polar and why it's so important that it's on buttons on your basic scientific calculator to convert them between the two, as we'll see in a minute.

If you want to do this in polar form, well, it's right. 3 is just three. If you want to write it in the complex form, it's just three. Angle Zero because there's no angle like that.

But if you rotated it like that up 90 degrees like that, you remember it's a positive. It's well, it was positive before it would become three. Angle 90 like that Um, you can call it angle I just happen to use like angle it's probably a better term for it. Or if you rotate it 90 degrees this way it'd be three angle minus 90.

that's the polar form. These are completely equivalent and you can convert between polar rectangular using basic trigonometry you learned in like early high school. and you'll never get more than plus or minus 180 here because once you rotate this around to 180 degrees, well, you're actually you know if you're down here, you're actually closer to back here. And your negative phase angle? Um, so you know it's it's not like 190 degrees, you'd be minus 170.

So we've got two complex number forms, polar and rectangular and it's easy to convert uh between them. And it's just that. basic trigonometry. Yes, it can.

It comes in useful. That trigonometry. Uh, you learnt in basic high school, right? You've got your triangle like that. You've got your angle Theta in here.

We'll just call this side r here. and this side is length A and this one is B. Um, but because we're talking a complex plane, we'll just add the J in there. but you can just ignore that so you can represent it in either form either length R like that with the phase angle in there, Theta, or you can represent it as length A like that, or J B because it has the J because it's in that, direction, the imaginary direction and they're the two different forms it's written in and then basic trigonometry that you learned that you can then do the conversions between polar and rectangular form and that's how you do it.

And that's exactly what your confuser here is doing. When you push the P to R and R to P buttons, it's just doing these calculations internally, saving you the trouble to having to remember these formulas and convert them. Confuse it does it for you. So we can plot points on our complex plane.

Now remember, we have the real reference plane like this. just like your regular number line. You know we've got zero and positive and negative like this. These are real voltages, real currents, real reactances, right.

Everything's like real. And then you've got your complex plane in this direction and you've got positive values this way and negative values this way. these are positive and negative j This is your complex plane, so we can plot points on this and get the rectangular form for your complex number. So if we had a point like that on our uh, complex plane here, then that would be 2 because the real component first plus j and then it'll be J3 because we're in the positive direction like this.

and if we had another point down here like the opposite side like this, this would be once again two is the real component, but it'd be minus j. three like that because it's in the minus J direction. And if we have say this point here, well that's minus Five is the real component, but it would still be plus J three. So it's not minus j because we're only we're only minus in the real.

Remember this, the left side of the plus sign here is the real component and then the complex j component is on the right hand side of the plus. So that would be -5 plus J3 and the same thing down here. You guessed it, it'd be minus 5 Again, minus j 3. And then of course you can draw in your phaser like that.

And then as we looked at before, you can actually convert that into polar form between rectangular and polar. Cool, huh? So I know you're asking, why are we even bothering with all this polar and rectangular form and converting between them and all this sort of rubbish. What benefit does it gain us? Well, it gains us a huge benefit and why it's on every single scientific calculator. It's because once we start talking with Ac components, doesn't matter whether it's voltage, current, impedance, or whatever it is.

When we start talking Ac circuit analysis, they will have an angular phase component and we have to start doing trig functions. If you stayed using that formula that we showed before with the sine function in the trig function, then when you're doing your Ac circuit noise, you'd be solving trig functions all day until the cows come home. So what we do is, we want to actually eliminate trig functions completely from this. We don't want to use a sine cos tan.

and now that Arc rubbish even we don't want to use any trig functions at all. This is what polar and rectangular forms do. Once you've got your complex voltage, current, impedance, whatever you know we're dealing with in our circuits. Once you've got it in complex form with a phase component in it, then we can do simple multiplication, addition, subtraction, and division using just regular mass.

It takes the trig all out of it. and this is the beautiful thing with polar and rectangular form. Let's have a look here. So why both forms? Well, if you want to do addition and subtraction on complex numbers, then you have to do it in rectangular form.

If you want to do a multiplication and division, you have to have it in polar form and it just makes your life so much easier. and why Your calculator allows you to convert easily between those two. Because when we're solving our Ac circuit analysis problems, we want to add voltages together. We want to subtract them, divide them, multiply them, all that sort of stuff, and this is how easy it is to do.

Let's take a look at if you want to multiply two complex voltages together. Okay, V1 and V2 now you. They will be in complex form. So you'll have a real component and a phase component.

Like we showed before. We'll call it R1 here because we had that on the graph before. This is not resistance. Okay, it's not resistor one.

It's just like real. Think of that as the real component. so we'll call it R1. You can label it anything you like.

really like. We've got a ins and B's and C's and D's over here. You can label it anything right? So we've got R1 and phase one like this. This is our complex form of V1 and we want to multiply that by V2.

So we V2 is in its complex form. We'll call that R2 and Phase 2.. So how do you multiply two complex numbers? It's simple. You take the two real components and you multiply them so r1 times r2 and that gives you your resultant real component and then not multiply.

But you'll have to add the phase component so it becomes angle like this. Phase One plus phase Two Simple. You've eliminated all trig functions from your calculations. This is just great.

and it's so simple. Same with division. Well, except you subtract them. So you've got V1 and V2 so your real and your phase component.

so it becomes R1 Phase 1 divided by R2 Phase 2.. Well, you take your two real components like this and you divide them and that gives you your final result. but for the Phase instead of adding them, like for multiplication, we subtract them. So phase One minus phase Two and that gives you your answer.

You've just multiplied and divided two complex numbers. It ain't complex and it's very similar. If you want to add and subtract, well, we need them in polar form like this. So our voltages.

You remember: our voltages can exist in either polar or rectangular form. So if we want to add, add two of two voltage complex voltages together, we put it in its rectangular form, which is a plus J B As I said before, A and B are just generic letters we've chosen and I've chosen C and D here. Uh, not because they represent anything. It's because so we don't confuse them with A and B over here.

But you've got two complex numbers you want to add together. All you do is once again, you add the two real components like this and you simply add the two imaginary components like that. So your answer is a plus C plus J B plus D Easy, and subtraction is just as easy as well. You take your complex voltages here.

You put them in their rectangular form and it's A Instead of a plus C, it's a minus C You take the real components, you subtract them. You take the imaginary components here and here and you subtract them and that gives you answer. Like whoever thought complex numbers were complex is like a fool. Complex numbers are incredibly simple.

These just remember how to do these operations like this and you can perform any maths you want on any complex Ac circuit problem. So how does this tie into real circuit components? Well, I'll have to do a future video on this. I think we might have to cover it. but let's just take an inductor for example.

You remember I mentioned uh, civil before and voltage V represents voltage. I represents Uh current. and for an inductor l Um, the voltage leads the current. Okay, so voltage leads the current.

So Ohm's law V on I would normally equal r but we're actually talking about a reactive component now. So we actually Um designated X. Anyway, let's see how to get down here. So it's actually Omega which is 2 pi F.

So it's omega L Uh, the inductance. So the reactance of the inductor or the effective Ac resistance of the inductor so to speak. Um has to have uh, that phase component in it and it's dependent upon frequency. So because voltage leads the current, its angle 90 degrees like that.

And if we convert that into complex Uh notation, V on I equals j Omega L So we've introduced the imaginary component in there, the complex Uh component. So the reactants or the Ac resistance. Uh, So to speak of our Uh inductor here is j Omega L in Ohms. So when you start talking uh, com reactive components like inductors and capacitors as well, you can go through exactly the same thing for a capacitor and this is where uh, you have to take into account that voltage and phase component.

And so the reactants are components. So J So when you start doing your Ac circuit analysis problems, you'll have like reactance values for inductors and capacitors in complex form. So then you can start doing all your regular circuit calculations, but instead of doing them in Dc, you do them in Ac and they'll have that J component. but that's for a future video.

I just wanted to show you how it's relevant and just for completeness, we'll do the capacitor because now current for capacitance, current leads the voltage like this. So if we do Ohm's law and get our which would normally equal resistance. but because we're talking about Ac, it's now and it's reactive Uh component, we're now talking. We want the value of Xc Ic like the Ac resistance.

So um, it's one on Omega C angle minus 90 Now because it's 180 degrees or antiphase or outer phase compared to the inductor and that's assuming it's a pure capacitor and pure inductor of course. And so you can work that out and the capacitive reactance is now minus J 1 on Omega C instead of plus J Omega L for the inductor got it. So just a simple worked example for multiplying two voltages here. Got them in the complex form: 5 Angle 20.

So 5 volts could be 5 volts Rms with a 20 degree phase angle and you want to add it to a 2 volt Rms signal. You usually use Rms for this with 30 degrees phase angle. Well, you multiply the real components like that so that becomes 10 and then you add the phase components 20 plus 30 which becomes 50.. And say, for example, if this one was minus 30 degrees, it was.

You know down here on the graph. So one was up here. One's down here like this: What would you end up with? Well, you would, uh, it would be 20 plus minus 30 which would be minus 10 degrees. Easy.

What would that look like on our polar diagram here? Well, uh, we've got five. Uh, five volts. Um, at angle 20 Like that. multiplying.

Uh, this one which is uh, two volts so it's shorter at an angle of minus 30 and it gives a result of 10. So that's much longer at an angle of minus 10 degrees. Like that. Beauty and the power of rectangular and polar forms.

Don't just stop at your regular um. four arithmetic functions. You can do powers and roots as well. Like powers are for example in polar form here.

So you've got r Angle uh, Theta, It's just r to the power and then all that to the power of n. So if you want to take your voltage to the power of N. well, it's just the real component to the power of n. Um, Angle, the power of n times the phase angle, that's it.

and roots. You actually, uh, do a similar way. So if you wanted to get a square root of your complex up voltage, it'd just be the real component to the power of a half because that's what a square root is. And if you wanted a cube root, then it would be the power of one on R3 and Angle Theta times a half.

It's just really easy, no trig functions involved. So there you go. hopefully. I've given you a very simple overview of why we use complex numbers and why we put them in polar and rectangular forms, and why even the most basic scientific calculators from way back? This Fx82 actually has P to R and R to P polar to rectangular rectangular polar conversion on it.

because it's so damn useful for engineers. So, and that's why it's got like the engineering, uh, display mode as well. You know these things were made for engineers and engineers invented. Um, these sort of things to do Ac circuit analysis and in future videos I might be able to show you look at um, the Ohms Law for Ac basically and it's exactly the same as Ohm's law for Dc.

and all the stuff we learned in Dc fundamentals is exactly the same for Ac, except your voltages and your currents and your uh, reactances and stuff. They now have phase components to them, so everything's in complex form, but your Ohm's Law remains the same. You're just dealing with complex numbers instead of real numbers and complex numbers ain't complex. It's really easy to add them up and hopefully are giving you a good idea how they represent with phases and everything else, so I hope you found that useful.

If you did, please give it a big thumbs up. As always, discuss down below: Catch you next time you.