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### Course: Electrical engineering > Unit 2

Lesson 5: AC circuit analysis- AC analysis intro 1
- AC analysis intro 2
- Trigonometry review
- Sine and cosine come from circles
- Sine of time
- Sine and cosine from rotating vector
- Lead Lag
- Complex numbers
- Multiplying by j is rotation
- Complex rotation
- Euler's formula
- Complex exponential magnitude
- Complex exponentials spin
- Euler's sine wave
- Euler's cosine wave
- Negative frequency
- AC analysis superposition
- Impedance
- Impedance vs frequency
- ELI the ICE man
- Impedance of simple networks
- KVL in the frequency domain

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# Multiplying by j is rotation

Successive powers of the imaginary unit j correspond to a rotation. Created by Willy McAllister.

## Want to join the conversation?

- This is the most valuable explanation of "j" essence I've ever met. Thank you Willy!(6 votes)
- Hello!

Why do we use real part for resistors and imaginary part for capacitors and coils?

Thank you(0 votes)- Resistors, capacitors, and inductors each have a specific real-world behaviors. We create mathematical models that come as close as possible to what we see happening in real life.

For a resistor by itself the mathematical model is called Ohm's Law. The capacitor and inductor have similar i-v laws.

When we make combinations of R, L, and C,**AND**we limit ourselves to sine wave shaped signals, the mathematical model that seems most useful describes the i-v behavior using a complex number, with R in the real part and L or C in the imaginary part.(4 votes)

- Thanks for this video.

at0:55you say "there is no real part."

Should it not have been "there is no imaginary part."(1 vote) - Likewise, I just noticed he says "there's no REAL part", for j to the zero, when he must mean there's no IMAGINARY part. These little mistakes make it VERY confusing for the student, and shouldn't take but a moment to correct.(1 vote)
- You are correct. I had a brain cramp while recording the video, and only found it much later. The corrected video is on my site

spinningnumbers dot org.

The video is listed in the AC Analysis section.(1 vote)

- The 1 on the complex plane is also 0.(1 vote)
- Luckily I had just studied this in Algebra 2 so this made this lesson much easier to understand.(1 vote)

## Video transcript

- [Voiceover] Okay,
there's one more feature of complex numbers that
I want to share with you and we'll do that down here. So, our definition of
j is j squared equals minus one, and now what I want to do is a sequence of multiplications by j. This is a really important
property of this imaginary unit. We're gonna do powers of j and I'm going to plot them,
while we're doing that I'm gonna plot them on the imaginary, or on the complex plane. So, this is the real axis and
this is the imaginary axis. So, I'm gonna take powers of j, so first one j to the zero,
and anything to the zero is one, so j to the zero is one and if we plot that on the imaginary axis, here's one, and there's no real part. So, it's just right on the real axis. Okay, let's do j to the one, that's j times itself one time, so that is equal to j. If I plot that number, that's up here, that's up on the imaginary
axis right there, there's j. Alright, this has no real
part, it's all imaginary. Let's keep going. Let's do j squared and
what is that equal to? Well, I wrote it down up here, j squared is equal to minus one. So, j squared is equal to minus one, where's that in the complex plane? That's over here at minus one, on the real axis, no imaginary part. Let's do the next one. Let's do j cubed. What is that equal to? J to the third power is
equal to j squared times j and we have those two right here, j squared is minus one
and j to the one is j, so that equals minus j, and where do we plot that? We plot that down on the imaginary axis in the complex plane, right here, minus j. Okay, now we've got four answers. Let's go one or two more. Okay, j to the fourth is equal to what? It's equal to j squared times j squared. Let's look that up. It's minus one times minus one, and what does that equal to? That equals one. So, let's go plot that. Well, we've already plotted it, so that's this answer right here. Alright, so we already have that, let's do one more. J to the fifth, what is that equal to? That equals j to the fourth times j, and j to the fourth is
right here, it's one times j equals j, let's go plot that one. Well, we've already plotted it, it's already right here. Okay, so, you can see
there's a pattern here, one j minus one minus j. One j, it's gonna by
minus one and minus j, it keeps repeating. Here's what's interesting
about this, okay. If we draw this as
vectors, if I draw these imaginary numbers as
vectors, when I multiply by j, when I multiplied one by j, it rotated it 90 degrees and then that was the
first step right here when I multiplied one times j, I got j. Now, when we went to j squared, we ended up at minus one. So, multiplying by j again, caused the vector to
go down here, like that and that was another 90 degrees and if I take it again, the next one went this way and the final one went this way. So, this is the property of j, this is the key property
of the imaginary unit, multiplying by the imaginary unit, the nature of it is
this 90 degree rotation. There's this idea of a number that causes other numbers to rotate
and that's the feature of j that makes it super important and it's the reason we use imaginary numbers in
electrical engineering. So, the key idea here is that j rotates. That's the point. That's what we love about j. So, the last thing I want to mention is the negative powers of j. What happens if we have
j to the minus one? Let's figure out what that means. So, that, of course, is one over j and if I multiply this by j over j and anything over itself is one, so I haven't changed the value of this and that equals j on top
over j times j or j squared and what is j squared equal to? We wrote it down right here,
j squared is minus one, so this equals j over
minus 1 or equals minus j. So, whenever we see a j in a fraction, j to the minus one,
basically it introduces a minus sign and the j comes up to the top out of the fraction. So, j to the minus one equals minus j and we'll use that occasionally
to help us in our math. So, this was a really quick review of complex numbers and if any of this was new to you, I really encourage you to go back and watch Sal's
videos on complex numbers and if this is something
you've seen before, I hope it knocked off some of the rust and we're ready to go
and use these numbers.