Definite integral of sine and cosine product

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.303
- [Voiceover] We're in our quest

00:00:01.303 --> 00:00:02.790
to give ourselves a little bit

00:00:02.790 --> 00:00:05.668
of a mathematical underpinning
of definite integrals

00:00:05.668 --> 00:00:08.157
of various combinations of trig functions.

00:00:08.157 --> 00:00:10.104
So, it'll be hopefully straightforward

00:00:10.104 --> 00:00:12.354
for us to actually find the coefficients,

00:00:12.354 --> 00:00:14.187
our Fourier coefficients,
which we're going to do

00:00:14.187 --> 00:00:15.653
a few videos from now.

00:00:15.653 --> 00:00:17.690
And we've already started
going down this path.

00:00:17.690 --> 00:00:20.012
We've established that
the definite integral

00:00:20.012 --> 00:00:23.169
from zero to two pi of sine
of mt dt is equal to zero,

00:00:23.169 --> 00:00:26.599
and that the definite
integral of cosine mt dt

00:00:26.599 --> 00:00:30.543
is equal to zero for
any non-zero integer m.

00:00:30.543 --> 00:00:32.617
And actually, we can
generalize that a little bit.

00:00:32.617 --> 00:00:36.477
For sine of mt, it could
be for any m actually.

00:00:36.477 --> 00:00:38.594
And if you don't believe
me, I encourage you,

00:00:38.594 --> 00:00:39.427
so let me write this,

00:00:39.427 --> 00:00:43.762
for any, for any integer
m, this top interval's

00:00:43.762 --> 00:00:46.211
going to be zero and this second interval

00:00:46.211 --> 00:00:50.378
for any for non-zero,
non-zero integer, integer m.

00:00:55.615 --> 00:00:58.283
And you can see, if you had
zero in this second case,

00:00:58.283 --> 00:00:59.908
it would be cosine of zero t,

00:00:59.908 --> 00:01:01.701
so this would just evaluate to one,

00:01:01.701 --> 00:01:03.435
so you'd just be integrating

00:01:03.435 --> 00:01:05.846
the value one from zero to two pi,

00:01:05.846 --> 00:01:08.690
and so that's going to
have a non-zero value.

00:01:08.690 --> 00:01:09.949
So, with those two out of the way,

00:01:09.949 --> 00:01:11.492
let's go a little bit deeper,

00:01:11.492 --> 00:01:13.444
get a little bit more foundations.

00:01:13.444 --> 00:01:16.294
So, I now want to establish

00:01:16.294 --> 00:01:19.873
that the definite integral
from zero to two pi

00:01:19.873 --> 00:01:22.873
of sine of mt times cosine of nt dt,

00:01:32.560 --> 00:01:36.727
that this equals zero for any
integers, integers m and n.

00:01:42.014 --> 00:01:43.668
And they could even be the same.

00:01:43.668 --> 00:01:45.129
They don't have to
necessarily be different,

00:01:45.129 --> 00:01:46.715
but they could be different.

00:01:46.715 --> 00:01:48.424
How do we do this?

00:01:48.424 --> 00:01:53.061
Well, let's just rewrite
this part right over here,

00:01:53.061 --> 00:01:55.786
leveraging some trig identities.

00:01:55.786 --> 00:01:57.737
And if this is completely
unfamiliar to you,

00:01:57.737 --> 00:01:59.581
I encourage you to review
your trig identities

00:01:59.581 --> 00:02:00.827
on Khan Academy.

00:02:00.827 --> 00:02:01.725
So, this is the same thing,

00:02:01.725 --> 00:02:04.492
has a definite integral
from zero to two pi.

00:02:04.492 --> 00:02:07.748
This sine of mt times cosine nt,

00:02:07.748 --> 00:02:11.770
we can rewrite it using the
product to sum formulas.

00:02:11.770 --> 00:02:14.215
So, that is, let me use
a different color here.

00:02:14.215 --> 00:02:15.842
So, this thing right over here

00:02:15.842 --> 00:02:17.631
that I've underlined in magenta,

00:02:17.631 --> 00:02:19.905
or that I'm squaring off in magenta,

00:02:19.905 --> 00:02:22.426
that can be rewritten as

00:02:22.426 --> 00:02:25.259
one half times sine of m plus n t,

00:02:30.015 --> 00:02:31.598
sine of m plus n t,

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plus sine, sine of m minus n, m minus n t.

00:02:44.867 --> 00:02:48.700
And then let me just
close that with a dt, dt.

00:02:50.483 --> 00:02:51.728
Now, if we were to just rewrite this

00:02:51.728 --> 00:02:53.354
using some of our integral properties,

00:02:53.354 --> 00:02:55.348
we could rewrite it as,

00:02:55.348 --> 00:02:58.098
so this part over here, we could,

00:02:59.572 --> 00:03:02.585
and let's assume we
distribute the one half,

00:03:02.585 --> 00:03:04.576
so we're gonna distribute the one half

00:03:04.576 --> 00:03:06.611
and use some of our integral properties.

00:03:06.611 --> 00:03:08.849
And so what are we going to get?

00:03:08.849 --> 00:03:11.655
So, this part roughly, right over here,

00:03:11.655 --> 00:03:15.586
we could rewrite as one half
times the definite integral

00:03:15.586 --> 00:03:19.253
from zero to two pi of
sine of m plus n t dt

00:03:25.926 --> 00:03:28.982
and then this part, once
you distribute the one half

00:03:28.982 --> 00:03:31.016
and you use some integration properties,

00:03:31.016 --> 00:03:34.358
this could be plus one half
times the definite integral

00:03:34.358 --> 00:03:38.525
from zero to two pi of sine
of m minus n, m minus n t dt.

00:03:44.732 --> 00:03:47.989
Now, what are each of these
things going to be equal to?

00:03:47.989 --> 00:03:50.072
Well, isn't this right over here,

00:03:50.072 --> 00:03:52.210
isn't that just some integer?

00:03:52.210 --> 00:03:54.490
If I have to take the sum
of two arbitrary integers,

00:03:54.490 --> 00:03:56.173
that's going to be some integer.

00:03:56.173 --> 00:03:58.313
So, that's going to be some integer.

00:03:58.313 --> 00:04:03.042
And this, too, is going to be
some integer right over here.

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And we've already established
that the definite integral

00:04:04.876 --> 00:04:08.585
of sine of some integer
times t dt is zero.

00:04:08.585 --> 00:04:11.189
So, by this first thing
that we already showed,

00:04:11.189 --> 00:04:13.344
this is going to be equal to zero,

00:04:13.344 --> 00:04:14.400
that's going to be equal to zero.

00:04:14.400 --> 00:04:16.193
It doesn't matter that you're
multiplying by one half.

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One half times zero is zero.

00:04:17.700 --> 00:04:19.082
One half times zero is zero.

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This whole thing is going
to evaluate to zero.

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So, there you go; we've
proven that as well.


Formula for first term in Fourier Series

WEBVTT
Kind: captions
Language: en

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- Several videos ago,
we introduced the idea

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of a Fourier series.

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That I could take a periodic function,

00:00:05.348 --> 00:00:07.135
we started with the example of

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this square wave, and that I could

00:00:09.210 --> 00:00:12.793
represent it as the sum of weighted

00:00:12.793 --> 00:00:14.293
sines and cosines.

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And then we took a little bit of

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an interlude of building up

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some of our mathematical foundations,

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just establishing a bunch of properties

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of taking the definite integral

00:00:24.658 --> 00:00:28.032
over the period of that periodic function

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of sine and cosines and we established

00:00:30.511 --> 00:00:32.055
all of these properties.

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And now we're going to get the benefit

00:00:33.638 --> 00:00:35.265
from establishing all those because

00:00:35.265 --> 00:00:37.174
we're going to start actually finding

00:00:37.174 --> 00:00:39.732
at least a formula for
Fourier coefficients

00:00:39.732 --> 00:00:40.871
and then we can apply it to our

00:00:40.871 --> 00:00:42.334
original square wave to see that, hey,

00:00:42.334 --> 00:00:43.877
this could actually be a pretty

00:00:43.877 --> 00:00:45.499
straightforward thing.

00:00:45.499 --> 00:00:48.097
So right over here, I have rewritten

00:00:48.097 --> 00:00:50.905
a Fourier series expression,

00:00:50.905 --> 00:00:52.323
or I've rewritten a Fourier series

00:00:52.323 --> 00:00:54.636
for a periodic function f(t),

00:00:54.636 --> 00:00:57.278
let's say its period is 2π,

00:00:57.278 --> 00:00:58.496
and I'm going to use this in some

00:00:58.496 --> 00:01:00.769
of the properties that we have established

00:01:00.769 --> 00:01:05.236
to start solving for
these actual coefficients.

00:01:05.236 --> 00:01:06.904
And what I'm gonna do in this video,

00:01:06.904 --> 00:01:08.126
I'm gonna first try to solve for

00:01:08.126 --> 00:01:09.508
a-sub-0 and then in the next video,

00:01:09.508 --> 00:01:12.631
we're gonna solve for
an arbitrary a-sub-n.

00:01:12.631 --> 00:01:13.813
And either in that one or the next one,

00:01:13.813 --> 00:01:17.594
we'll also solve for an arbitrary b-sub-n.

00:01:17.594 --> 00:01:19.417
So to solve for a-sub-0,

00:01:19.417 --> 00:01:21.695
what we're going to do is take

00:01:21.695 --> 00:01:23.524
the definite integral of both sides

00:01:23.524 --> 00:01:25.349
from 0 to 2π.

00:01:25.349 --> 00:01:28.726
So 0 to 2π, dt of f(t),

00:01:28.726 --> 00:01:30.106
well that's gonna be the same thing

00:01:30.106 --> 00:01:31.856
as going from 0 to 2π

00:01:33.073 --> 00:01:35.141
of all of this stuff.

00:01:35.141 --> 00:01:36.525
And remember, this is an infinite

00:01:36.525 --> 00:01:37.579
series right over here,

00:01:37.579 --> 00:01:39.123
we have an infinite number of terms.

00:01:39.123 --> 00:01:41.679
And then we would have a dt out there.

00:01:41.679 --> 00:01:44.031
But we know from our
integration properties,

00:01:44.031 --> 00:01:45.735
taking the definite integral of

00:01:45.735 --> 00:01:47.685
a sum, even an infinite sum,

00:01:47.685 --> 00:01:49.189
is the same thing as the sum of

00:01:49.189 --> 00:01:50.487
the definite integrals.

00:01:50.487 --> 00:01:52.032
So that's going to be the same thing

00:01:52.032 --> 00:01:54.032
as taking this integral,

00:01:55.371 --> 00:01:57.204
dt plus this integral.

00:01:58.333 --> 00:02:00.567
And I could take the scaler out.

00:02:00.567 --> 00:02:01.788
Actually, let me just not do that,

00:02:01.788 --> 00:02:03.415
let me just write it like this.

00:02:03.415 --> 00:02:04.248
0 to 2π,

00:02:05.385 --> 00:02:06.943
dt.

00:02:06.943 --> 00:02:08.440
0 to 2π,

00:02:08.440 --> 00:02:09.273
dt.

00:02:10.389 --> 00:02:11.222
0 to 2π,

00:02:12.255 --> 00:02:13.435
dt.

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This is getting a little monotonous,

00:02:14.572 --> 00:02:15.405
but it'll be worth it.

00:02:15.405 --> 00:02:16.638
0 to 2π,

00:02:16.638 --> 00:02:18.181
dt.

00:02:18.181 --> 00:02:19.014
0 to 2π,

00:02:20.172 --> 00:02:21.005
dt.

00:02:22.081 --> 00:02:22.914
0 to 2π,

00:02:23.946 --> 00:02:24.840
dt.

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And we'll do it for every single one

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of the terms.

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And now what's nice is, we can look

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at our integration properties.

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This right over here,

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we could take these coefficients out.

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We could take this a-sub-1,

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put it in front of the integral sign.

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The a-sub-2, put it in front

00:02:39.722 --> 00:02:40.698
of the integral sign.

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The b-sub-1, put it in front

00:02:42.203 --> 00:02:43.665
of the integral sign.

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And then all you're left with is

00:02:44.761 --> 00:02:46.300
an integral from 0 to 2π

00:02:46.300 --> 00:02:47.741
of cosine of

00:02:47.741 --> 00:02:50.324
some integer multiple of t, dt.

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Well, we established
a couple of videos ago

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well, that's always
going to be equal to 0.

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The integral from 0 to 2π of cosine

00:02:59.125 --> 00:03:02.375
of some non-zero integer multiple of t,

00:03:03.263 --> 00:03:05.381
dt, that is equal to 0.

00:03:05.381 --> 00:03:06.642
And then the same thing is true

00:03:06.642 --> 00:03:08.873
for a sine of mt.

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So, this is gonna be fun.

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This is gonna be 0,

00:03:12.668 --> 00:03:13.806
based on what we just saw.

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If you just take that factor out of

00:03:15.310 --> 00:03:16.735
that integral, take that a-sub-1

00:03:16.735 --> 00:03:17.626
out of the integral, it's gonna be

00:03:17.626 --> 00:03:18.883
a-sub-1 times 0.

00:03:18.883 --> 00:03:20.181
This is gonna be a-sub-2 times 0,

00:03:20.181 --> 00:03:21.115
that's gonna be 0.

00:03:21.115 --> 00:03:22.251
That's gonna be 0, that's gonna be 0,

00:03:22.251 --> 00:03:23.265
that's gonna be 0.

00:03:23.265 --> 00:03:24.889
Every term's gonna be 0,

00:03:24.889 --> 00:03:28.868
except for this one involving a-sub-0.

00:03:28.868 --> 00:03:31.670
And so, what is this going to be equal to?

00:03:31.670 --> 00:03:33.785
Well, let me write it this way.

00:03:33.785 --> 00:03:36.143
Let's take the integral of,

00:03:36.143 --> 00:03:38.341
the definite integral.

00:03:38.341 --> 00:03:40.613
Let me see where I have some space.

00:03:40.613 --> 00:03:42.563
So we're gonna take the definite integral

00:03:42.563 --> 00:03:45.405
from 0 to 2π of a-sub-0 dt,

00:03:45.405 --> 00:03:46.623
well that's the same thing.

00:03:46.623 --> 00:03:49.985
Once again, we can take
the coefficient out.

00:03:49.985 --> 00:03:52.068
And I can just put the dt

00:03:54.262 --> 00:03:57.757
like this and so that's going to be

00:03:57.757 --> 00:03:59.840
equal to a-sub-0 times t.

00:04:02.321 --> 00:04:04.488
Let me do that in magenta.

00:04:05.690 --> 00:04:08.122
Times t, evaluated at 2π

00:04:08.122 --> 00:04:10.606
and 0, which is going to be equal

00:04:10.606 --> 00:04:12.939
to a-sub-0 times 2π minus 0.

00:04:16.095 --> 00:04:18.248
Times 2π minus 0.

00:04:18.248 --> 00:04:20.972
Well, that's just 2π a-sub-0.

00:04:20.972 --> 00:04:23.324
So I could just write this a-sub-0

00:04:23.324 --> 00:04:24.157
times 2π.

00:04:25.600 --> 00:04:26.900
So this expression right here is

00:04:26.900 --> 00:04:28.317
a-sub-0 times 2π.

00:04:29.461 --> 00:04:31.901
So let me scroll down a little bit.

00:04:31.901 --> 00:04:34.138
So I can rewrite this thing up here,

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the integral from 0 to 2π

00:04:37.635 --> 00:04:39.885
of f(t) dt

00:04:43.324 --> 00:04:45.598
which is equal to this integral.

00:04:45.598 --> 00:04:47.099
Well, we've just figured out that

00:04:47.099 --> 00:04:49.171
the integral from 0 to 2π of a-sub-0 dt

00:04:49.171 --> 00:04:52.947
is the same thing as a-sub-0 times 2π.

00:04:52.947 --> 00:04:55.364
Is equal to a-sub-0 times 2π.

00:04:57.134 --> 00:04:57.967
Times 2π.

00:04:59.167 --> 00:05:00.506
And so now, it's actually pretty

00:05:00.506 --> 00:05:02.702
straightforward to solve for a-sub-0.

00:05:02.702 --> 00:05:05.285
A-sub-0 is going to be equal to

00:05:06.433 --> 00:05:08.626
a-sub-0 is going to be equal to

00:05:08.626 --> 00:05:09.459
1 over 2π,

00:05:11.750 --> 00:05:13.863
1 over 2π times the definite integral

00:05:13.863 --> 00:05:14.946
from 0 to 2π,

00:05:15.816 --> 00:05:17.887
I'll just write the dt,

00:05:17.887 --> 00:05:21.380
of, let me write it a little bit,

00:05:21.380 --> 00:05:23.372
dt of f(t),

00:05:23.372 --> 00:05:24.794
oh I'll just write it like this.

00:05:24.794 --> 00:05:26.299
f(t) dt.

00:05:26.299 --> 00:05:27.271
And this is pretty cool because

00:05:27.271 --> 00:05:28.981
think about what this is.

00:05:28.981 --> 00:05:30.116
This over here,

00:05:30.116 --> 00:05:33.569
this is the average value of our function,

00:05:33.569 --> 00:05:35.683
this is the average value of f

00:05:35.683 --> 00:05:37.850
over the interval 0 to 2π.

00:05:39.013 --> 00:05:40.096
Average value

00:05:42.713 --> 00:05:43.546
of f over,

00:05:46.615 --> 00:05:48.562
over the interval,

00:05:48.562 --> 00:05:52.015
over we could say the interval from 0,

00:05:52.015 --> 00:05:54.291
the interval from 0 to 2π.

00:05:54.291 --> 00:05:55.467
And hopefully that actually makes

00:05:55.467 --> 00:05:57.010
intuitive sense.

00:05:57.010 --> 00:05:58.307
Because if I am,

00:05:58.307 --> 00:05:59.564
if you just think of
it from an engineering

00:05:59.564 --> 00:06:00.497
point of view, if you were just trying

00:06:00.497 --> 00:06:01.956
to engineer this, trying to just

00:06:01.956 --> 00:06:03.262
play around with these numbers,

00:06:03.262 --> 00:06:05.091
you know all of these cosines and sines,

00:06:05.091 --> 00:06:07.895
they oscillate between
positive 1 and negative 1.

00:06:07.895 --> 00:06:09.196
So in order to actually represent

00:06:09.196 --> 00:06:10.293
this function, you're gonna have to

00:06:10.293 --> 00:06:12.730
shift that oscillation.

00:06:12.730 --> 00:06:13.829
Sum of a bunch of oscillation is

00:06:13.829 --> 00:06:15.046
still gonna be an oscillation that's

00:06:15.046 --> 00:06:17.527
going to vary between

00:06:17.527 --> 00:06:20.027
positive 1 and our negative 1.

00:06:21.389 --> 00:06:22.607
And in order to shift it, well,

00:06:22.607 --> 00:06:24.963
that's what our a-sub-0 is going to do.

00:06:24.963 --> 00:06:25.979
And so what you,

00:06:25.979 --> 00:06:27.034
it makes sense that you would want

00:06:27.034 --> 00:06:28.416
to shift the oscillation so it

00:06:28.416 --> 00:06:30.649
oscillates around the average value

00:06:30.649 --> 00:06:32.560
of the function.

00:06:32.560 --> 00:06:33.577
Or you could say the average value

00:06:33.577 --> 00:06:35.204
of the function over,

00:06:35.204 --> 00:06:37.434
over an interval that's representative

00:06:37.434 --> 00:06:39.830
of a period of that function.

00:06:39.830 --> 00:06:42.314
And so that is what a-sub-0 is doing.

00:06:42.314 --> 00:06:44.882
A-sub-0 is just going to be that

00:06:44.882 --> 00:06:47.382
average value of the function.


Fourier coefficients for sine terms

WEBVTT
Kind: captions
Language: en

00:00:00.472 --> 00:00:02.993
- [Voiceover] Many videos ago,
we first looked at the idea

00:00:02.993 --> 00:00:05.141
of representing a periodic function

00:00:05.141 --> 00:00:08.391
as a set of weighted cosines and sines,

00:00:09.558 --> 00:00:10.464
as a sum,

00:00:10.464 --> 00:00:14.922
as the infinite sum of
weighted cosines and sines,

00:00:14.922 --> 00:00:17.847
and then we did some work in order to

00:00:17.847 --> 00:00:21.691
get some basics in terms
of some of these integrals

00:00:21.691 --> 00:00:25.261
which we then started to
use to derive formulas

00:00:25.261 --> 00:00:27.246
for the various coefficients,

00:00:27.246 --> 00:00:28.587
and we are almost there.

00:00:28.587 --> 00:00:31.135
We have figured out a
formula for a sub zero.

00:00:31.135 --> 00:00:34.601
We've figured out a coefficient
in general for any of the

00:00:34.601 --> 00:00:36.534
coefficients on the cosines,

00:00:36.534 --> 00:00:39.657
and now, let's figure
out a way to figure out

00:00:39.657 --> 00:00:42.737
the coefficients on the sines.

00:00:42.737 --> 00:00:44.835
So, how can we do that?

00:00:44.835 --> 00:00:46.675
Well, we're going to use
a very similar technique

00:00:46.675 --> 00:00:49.936
that we did to figure out
the formula for a sub n.

00:00:49.936 --> 00:00:53.148
What we're going to do is multiply

00:00:53.148 --> 00:00:57.128
both sides of this
equation right over here

00:00:57.128 --> 00:00:58.295
by sine of nt.

00:00:59.690 --> 00:01:00.996
So let's do that.

00:01:00.996 --> 00:01:02.246
So, sine of nt,

00:01:04.681 --> 00:01:07.190
and so, if we multiply the
right-hand side by sine of nt,

00:01:07.190 --> 00:01:08.525
that means we're going
to multiply every one of

00:01:08.525 --> 00:01:10.894
the infinite terms by sine of nt,

00:01:10.894 --> 00:01:12.061
so sine of nt,

00:01:14.946 --> 00:01:15.863
sine of nt,

00:01:18.429 --> 00:01:19.262
I think you,

00:01:19.262 --> 00:01:21.557
well, this is always a
little time consuming,

00:01:21.557 --> 00:01:22.390
but I'll get there,

00:01:22.390 --> 00:01:23.307
sine of nt,

00:01:24.510 --> 00:01:25.427
sine of nt,

00:01:28.042 --> 00:01:28.959
sine of nt,

00:01:31.101 --> 00:01:32.018
sine of nt,

00:01:34.953 --> 00:01:35.870
sine of nt,

00:01:36.725 --> 00:01:38.159
and of course there is
these dot, dot, dot's,

00:01:38.159 --> 00:01:39.551
so we're just showing in general,

00:01:39.551 --> 00:01:40.865
we're taking each of the,

00:01:40.865 --> 00:01:43.117
each of these terms and
multiplying them by sine of nt.

00:01:43.117 --> 00:01:44.284
We'll do it for all of the terms.

00:01:44.284 --> 00:01:46.560
We'll keep on doing
this, on and on forever,

00:01:46.560 --> 00:01:48.746
for an infinite number of terms.

00:01:48.746 --> 00:01:51.071
Well, now let's take the
integral of both sides

00:01:51.071 --> 00:01:51.914
of this equation,

00:01:51.914 --> 00:01:54.438
the definite integral from zero to 2 pi,

00:01:54.438 --> 00:01:57.247
and once again, I'm picking
that interval because 2 pi

00:01:57.247 --> 00:02:00.592
is the period of the periodic
function that we care about,

00:02:00.592 --> 00:02:02.009
and everything else that I've been doing

00:02:02.009 --> 00:02:03.797
has been over that interval of 2 pi,

00:02:03.797 --> 00:02:05.881
so I'll just keep focusing in on that,

00:02:05.881 --> 00:02:07.881
so from zero to 2 pi dt,

00:02:08.782 --> 00:02:11.785
and I could say the integral
of this whole thing,

00:02:11.785 --> 00:02:14.171
but we know that the integral
of a sum of a bunch of things

00:02:14.171 --> 00:02:16.446
is also the same thing as
the sum of the integrals

00:02:16.446 --> 00:02:19.738
and we also know that we can
take these constant multipliers

00:02:19.738 --> 00:02:21.442
of expressions outside of integrals

00:02:21.442 --> 00:02:22.980
and so this is going to be equal to,

00:02:22.980 --> 00:02:25.879
if I take the integral
of the right-hand side,

00:02:25.879 --> 00:02:29.740
the definite integral is
going to be from zero to 2 pi,

00:02:29.740 --> 00:02:30.837
dt.

00:02:30.837 --> 00:02:32.254
Zero to 2 pi, dt.

00:02:33.954 --> 00:02:35.371
Zero to 2 pi, dt.

00:02:38.459 --> 00:02:39.876
Zero to 2 pi, dt.

00:02:41.748 --> 00:02:42.689
We're almost there.

00:02:42.689 --> 00:02:44.106
Zero to 2 pi, dt.

00:02:46.371 --> 00:02:47.788
Zero to 2 pi, dt.

00:02:50.185 --> 00:02:53.894
And zero to 2 pi, dt,
and I would be doing this

00:02:53.894 --> 00:02:57.311
for every term in this Fourier expansion.

00:02:58.590 --> 00:03:00.587
Now, this is where some
of that integration work

00:03:00.587 --> 00:03:01.911
is going to be valuable.

00:03:01.911 --> 00:03:05.671
We've already shown that
sine of the definite integral

00:03:05.671 --> 00:03:08.737
from zero to 2 pi of sine of nt, dt

00:03:08.737 --> 00:03:12.232
is going to be equal to zero for n being

00:03:12.232 --> 00:03:13.514
any integer.

00:03:13.514 --> 00:03:15.220
So we saw that,

00:03:15.220 --> 00:03:17.358
we saw that right over there,

00:03:17.358 --> 00:03:18.691
so that tells us

00:03:20.226 --> 00:03:22.867
that that term is going to be zero,

00:03:22.867 --> 00:03:25.437
so that's going to be equal to zero

00:03:25.437 --> 00:03:29.961
and we also know that when
we take a cosine of something

00:03:29.961 --> 00:03:30.802
times t,

00:03:30.802 --> 00:03:32.989
times the sine of something times t,

00:03:32.989 --> 00:03:34.558
where these are integers,

00:03:34.558 --> 00:03:35.621
and we take that definite integral,

00:03:35.621 --> 00:03:37.766
we know that that's going to be zero,

00:03:37.766 --> 00:03:39.903
so all of those are going to be zero,

00:03:39.903 --> 00:03:42.378
and we also know that
when you take the sine

00:03:42.378 --> 00:03:43.962
of two different non,

00:03:43.962 --> 00:03:45.587
two different coefficients right over,

00:03:45.587 --> 00:03:48.343
or sine of some integer
coefficient times t

00:03:48.343 --> 00:03:51.787
times sine of some other
integer coefficient times t,

00:03:51.787 --> 00:03:55.131
that these are also going
to be equal to zero.

00:03:55.131 --> 00:03:56.516
We saw that,

00:03:56.516 --> 00:03:58.195
we saw that right over here.

00:03:58.195 --> 00:04:00.042
That is this property

00:04:00.042 --> 00:04:02.543
and so, really, all you're left with,

00:04:02.543 --> 00:04:05.225
the only one that does not become zero

00:04:05.225 --> 00:04:06.834
is this term right over here,

00:04:06.834 --> 00:04:10.084
where it's sine of nt times sine of nt,

00:04:11.223 --> 00:04:13.179
and what is that going to be equal to?

00:04:13.179 --> 00:04:15.871
Well, this, inside of the integral,

00:04:15.871 --> 00:04:16.726
that's the same thing.

00:04:16.726 --> 00:04:18.157
Let me do this in a different
color, since I'm using orange

00:04:18.157 --> 00:04:21.137
to cross out everything
that ends up becoming zero,

00:04:21.137 --> 00:04:22.925
so that is the same thing
as the definite integral

00:04:22.925 --> 00:04:24.504
from zero to 2 pi of

00:04:24.504 --> 00:04:26.421
sine-squared of nt, dt,

00:04:29.160 --> 00:04:29.993
and I'll do the,

00:04:29.993 --> 00:04:31.898
or the dt is still there
so I'm just replacing

00:04:31.898 --> 00:04:34.811
the blue stuff with
the sine-squared of nt,

00:04:34.811 --> 00:04:37.378
and we know what that's going to be.

00:04:37.378 --> 00:04:41.060
We know that when the coefficient on the t

00:04:41.060 --> 00:04:42.999
is a non-zero integer,

00:04:42.999 --> 00:04:45.639
that this is going to result in pi.

00:04:45.639 --> 00:04:46.889
So we know that

00:04:48.212 --> 00:04:50.238
all of this stuff right over here,

00:04:50.238 --> 00:04:51.869
we'll do that in a different color,

00:04:51.869 --> 00:04:52.702
all of,

00:04:53.651 --> 00:04:58.305
we know that all of this
stuff is going to evaluate,

00:04:58.305 --> 00:05:00.170
or actually we should
say, all of this stuff

00:05:00.170 --> 00:05:02.254
is going to become pi,

00:05:02.254 --> 00:05:06.591
so we know that b sub n times
pi is going to be equal to

00:05:06.591 --> 00:05:08.030
this definite integral.

00:05:08.030 --> 00:05:09.151
Or, we could write,

00:05:09.151 --> 00:05:10.364
let me write it this way,

00:05:10.364 --> 00:05:12.781
b sub n times pi is equal to,

00:05:14.860 --> 00:05:17.034
'cause everything else
ends up becoming zero,

00:05:17.034 --> 00:05:18.954
is going to be equal to zero,

00:05:18.954 --> 00:05:20.819
the definite integral from zero to 2 pi.

00:05:20.819 --> 00:05:22.664
I'll put my dt out here,

00:05:22.664 --> 00:05:23.497
of f of t,

00:05:25.056 --> 00:05:27.056
f of t times sine of nt,

00:05:28.481 --> 00:05:29.398
sine of nt,

00:05:32.034 --> 00:05:36.065
dt, and so we can then
divide both sides by,

00:05:36.065 --> 00:05:38.032
divide both sides by pi

00:05:38.032 --> 00:05:39.937
and we get a little bit of a drum roll,

00:05:39.937 --> 00:05:41.209
we get b sub n,

00:05:41.209 --> 00:05:42.846
actually, I could write that,

00:05:42.846 --> 00:05:44.789
I'll write that in that same blue color,

00:05:44.789 --> 00:05:46.039
we get b sub n,

00:05:46.921 --> 00:05:48.539
I'll do it actually right here,

00:05:48.539 --> 00:05:51.228
b sub n is going to be equal to

00:05:51.228 --> 00:05:54.821
1 over pi times the definite integral

00:05:54.821 --> 00:05:56.689
from zero to 2 pi,

00:05:56.689 --> 00:05:58.507
zero to 2 pi,

00:05:58.507 --> 00:06:00.005
we have our dt here,

00:06:00.005 --> 00:06:02.172
you have your f of t here,

00:06:03.034 --> 00:06:05.350
and then you have your sine,

00:06:05.350 --> 00:06:06.183
sine nt,

00:06:08.689 --> 00:06:10.610
so once again, that was pretty
straight forward now that

00:06:10.610 --> 00:06:11.882
we knew these,

00:06:11.882 --> 00:06:13.292
these properties of integrals

00:06:13.292 --> 00:06:14.847
so we knew how to evaluate these integrals

00:06:14.847 --> 00:06:15.680
that are,

00:06:15.680 --> 00:06:17.548
definite integrals that
are taking a cosine or sine

00:06:17.548 --> 00:06:20.694
or products of cosines and sines.

00:06:20.694 --> 00:06:22.946
And using these three formulas,

00:06:22.946 --> 00:06:27.004
we can now attempt to find
the Fourier expansion,

00:06:27.004 --> 00:06:27.868
the Fourier series,

00:06:27.868 --> 00:06:31.368
find the coefficients for our square wave.


Visualizing Fourier expansion of square wave

WEBVTT
Kind: captions
Language: en

00:00:00.528 --> 00:00:01.369
- [Voiceover] So we started

00:00:01.369 --> 00:00:02.506
with a square wave

00:00:02.506 --> 00:00:04.239
that had a period of two pi,

00:00:04.239 --> 00:00:05.524
then we said, hmm,

00:00:05.524 --> 00:00:07.146
can we represent it

00:00:07.146 --> 00:00:09.545
as an infinite series

00:00:09.545 --> 00:00:11.877
of weighted sines and cosines,

00:00:11.877 --> 00:00:14.487
and then working from that idea,

00:00:14.487 --> 00:00:15.320
we were actually able

00:00:15.320 --> 00:00:16.767
to find expressions

00:00:16.767 --> 00:00:17.908
for the coefficients,

00:00:17.908 --> 00:00:19.795
for a sub zero and a sub n

00:00:19.795 --> 00:00:21.092
when n does not equal zero,

00:00:21.092 --> 00:00:22.472
and the b sub ns.

00:00:22.472 --> 00:00:23.723
And evaluating it for this

00:00:23.723 --> 00:00:25.295
particular square wave,

00:00:25.295 --> 00:00:26.128
we were able to get

00:00:26.128 --> 00:00:27.341
that a sub n is going

00:00:27.341 --> 00:00:28.307
to be equal,

00:00:28.307 --> 00:00:29.467
or a sub zero is going

00:00:29.467 --> 00:00:30.926
to be 3/2,

00:00:30.926 --> 00:00:31.759
that a sub n is going

00:00:31.759 --> 00:00:32.730
to be equal to zero

00:00:32.730 --> 00:00:33.963
for any n

00:00:33.963 --> 00:00:35.664
other than zero,

00:00:35.664 --> 00:00:37.036
and that b sub n

00:00:37.036 --> 00:00:39.105
is going to be equal

00:00:39.105 --> 00:00:41.587
to zero if n is even

00:00:41.587 --> 00:00:43.109
and six over n pi

00:00:43.109 --> 00:00:44.109
if n is odd.

00:00:45.343 --> 00:00:46.306
So one way to think about it,

00:00:46.306 --> 00:00:49.804
you're gonna get your a sub zero,

00:00:49.804 --> 00:00:50.637
you're not gonna have any

00:00:50.637 --> 00:00:52.048
of the cosine terms,

00:00:52.048 --> 00:00:53.189
and you're only going to have

00:00:53.189 --> 00:00:54.772
the odd sine terms.

00:00:55.844 --> 00:00:56.852
And if you think about it

00:00:56.852 --> 00:00:57.885
just visually, if you look

00:00:57.885 --> 00:00:59.039
at the square wave,

00:00:59.039 --> 00:01:00.088
it makes sense

00:01:00.088 --> 00:01:01.213
that you're gonna have

00:01:01.213 --> 00:01:03.205
the sines and not the cosines

00:01:03.205 --> 00:01:04.857
because a sine function

00:01:04.857 --> 00:01:07.573
is gonna look something like this.

00:01:07.573 --> 00:01:10.132
So a sine function is gonna look

00:01:10.132 --> 00:01:11.696
something like this,

00:01:11.696 --> 00:01:12.869
while a cosine function

00:01:12.869 --> 00:01:14.619
looks something like,

00:01:17.052 --> 00:01:19.360
let me make it a little bit neater,

00:01:19.360 --> 00:01:20.731
a cosine function would look

00:01:20.731 --> 00:01:22.398
something like that.

00:01:25.054 --> 00:01:26.317
And so a cosine

00:01:26.317 --> 00:01:28.226
and multiples of cosine

00:01:28.226 --> 00:01:30.910
of two x, cosine of three x,

00:01:30.910 --> 00:01:32.185
is gonna be out of phase,

00:01:32.185 --> 00:01:33.501
while the sine of x,

00:01:33.501 --> 00:01:35.071
or I should say cosine of ts

00:01:35.071 --> 00:01:36.457
and the sines of ts,

00:01:36.457 --> 00:01:38.082
sine two t, sine three t,

00:01:38.082 --> 00:01:39.310
is gonna be more in phase

00:01:39.310 --> 00:01:40.417
with the way this function

00:01:40.417 --> 00:01:41.764
just happened to be.

00:01:41.764 --> 00:01:43.875
So it made sense that our

00:01:43.875 --> 00:01:45.758
a sub ns were all zero

00:01:45.758 --> 00:01:48.066
for n not equaling zero.

00:01:48.066 --> 00:01:50.044
And so based on what we found

00:01:50.044 --> 00:01:51.409
for our a sub zero,

00:01:51.409 --> 00:01:52.242
and our a sub ns,

00:01:52.242 --> 00:01:53.662
and our b sub ns,

00:01:53.662 --> 00:01:54.758
we could expand out

00:01:54.758 --> 00:01:56.445
this actual,

00:01:56.445 --> 00:01:58.018
we did in the previous video,

00:01:58.018 --> 00:01:59.212
what is this Fourier series

00:01:59.212 --> 00:02:00.247
actually look like?

00:02:00.247 --> 00:02:01.779
So 3/2 plus six over pi

00:02:01.779 --> 00:02:04.255
sine of t plus six over three pi

00:02:04.255 --> 00:02:05.127
sine of three t plus

00:02:05.127 --> 00:02:05.960
six over five pi

00:02:05.960 --> 00:02:07.111
sine of five t,

00:02:07.111 --> 00:02:08.532
and so on and so forth.

00:02:08.532 --> 00:02:09.365
And so a lot of you might

00:02:09.365 --> 00:02:10.198
be curious what does this

00:02:10.198 --> 00:02:11.831
actually look like.

00:02:11.831 --> 00:02:13.011
And so I actually just,

00:02:13.011 --> 00:02:14.055
you can type these things

00:02:14.055 --> 00:02:14.964
into Google and it will

00:02:14.964 --> 00:02:16.645
just graph it for you.

00:02:16.645 --> 00:02:17.801
And so this right over here

00:02:17.801 --> 00:02:19.058
is just the first two terms.

00:02:19.058 --> 00:02:22.090
This is 3/2 plus six over pi

00:02:22.090 --> 00:02:22.923
sine of t.

00:02:23.973 --> 00:02:24.806
And notice it's starting

00:02:24.806 --> 00:02:26.356
to look right because our

00:02:26.356 --> 00:02:27.635
square wave looks something

00:02:27.635 --> 00:02:28.802
like, it goes,

00:02:30.612 --> 00:02:33.595
it looks something like this.

00:02:33.595 --> 00:02:35.492
So it's gonna go like that

00:02:35.492 --> 00:02:38.106
and then it's gonna go down

00:02:38.106 --> 00:02:40.606
to zero and then it's gonna go

00:02:42.831 --> 00:02:44.559
up, looks something like that.

00:02:44.559 --> 00:02:45.392
It doesn't have the pis

00:02:45.392 --> 00:02:46.455
and the two pis

00:02:46.455 --> 00:02:47.302
marked off between these

00:02:47.302 --> 00:02:48.135
because it's gonna look

00:02:48.135 --> 00:02:49.731
something like that.

00:02:49.731 --> 00:02:51.128
So even just the two terms,

00:02:51.128 --> 00:02:52.539
it's kind of a decent approximation

00:02:52.539 --> 00:02:53.912
for even two terms,

00:02:53.912 --> 00:02:54.745
but then as soon as you get

00:02:54.745 --> 00:02:55.578
to three terms,

00:02:55.578 --> 00:02:57.861
if you add the six

00:02:57.861 --> 00:02:59.364
over three pi sine of three t

00:02:59.364 --> 00:03:01.085
to the first two terms.

00:03:01.085 --> 00:03:01.918
So if you look at these

00:03:01.918 --> 00:03:03.566
first three terms,

00:03:03.566 --> 00:03:04.399
now it's looking a lot

00:03:04.399 --> 00:03:05.664
more like a square wave.

00:03:05.664 --> 00:03:08.458
And then if you add the next term,

00:03:08.458 --> 00:03:09.291
well, it looks like even more

00:03:09.291 --> 00:03:10.519
like a square wave,

00:03:10.519 --> 00:03:12.058
and then if you add to that

00:03:12.058 --> 00:03:14.220
what we already wrote down here,

00:03:14.220 --> 00:03:15.646
if you were to add to that

00:03:15.646 --> 00:03:16.988
six over seven pi times

00:03:16.988 --> 00:03:18.156
sine of seven t,

00:03:18.156 --> 00:03:19.044
it looks even more

00:03:19.044 --> 00:03:20.382
like a square wave.

00:03:20.382 --> 00:03:21.675
So this is pretty neat.

00:03:21.675 --> 00:03:22.945
You can visually see

00:03:22.945 --> 00:03:24.405
that we were actually able

00:03:24.405 --> 00:03:25.238
to do it.

00:03:25.238 --> 00:03:26.454
And it all kind of just

00:03:26.454 --> 00:03:28.860
fell out from the mathematics.


Eulers formula

WEBVTT
Kind: captions
Language: en

00:00:00.523 --> 00:00:02.107
- [Voiceover] So in this
video we're goint to talk about

00:00:02.107 --> 00:00:03.347
Euler's formula.

00:00:03.347 --> 00:00:06.252
And one of the things I
want to start out with is

00:00:06.252 --> 00:00:08.638
why do we want to talk about this

00:00:08.638 --> 00:00:10.438
rather odd looking formula?

00:00:10.438 --> 00:00:12.327
What's the big deal about this?

00:00:12.327 --> 00:00:14.095
And there is a big deal.

00:00:14.095 --> 00:00:15.928
And the big deal is e.

00:00:18.422 --> 00:00:19.255
We love e.

00:00:22.100 --> 00:00:23.997
And I'll underline that twice.

00:00:23.997 --> 00:00:25.806
Now the reason is, because

00:00:25.806 --> 00:00:28.407
when we take a derivative of e

00:00:28.407 --> 00:00:29.240
D DT

00:00:31.312 --> 00:00:32.395
of e to the x

00:00:33.803 --> 00:00:35.303
equals e to the x.

00:00:37.765 --> 00:00:38.598
And D DT

00:00:41.483 --> 00:00:42.650
of e to the ax

00:00:43.564 --> 00:00:45.657
so where a is anything

00:00:45.657 --> 00:00:47.324
equals ae to the ax.

00:00:50.118 --> 00:00:52.382
And so the property is that
when you take a derivative

00:00:52.382 --> 00:00:54.404
of the function, the
same function comes out.

00:00:54.404 --> 00:00:57.046
Or, if you take a
derivative of the function

00:00:57.046 --> 00:01:00.558
a scaled version of the
same function comes out.

00:01:00.558 --> 00:01:02.044
And we love this, because why?

00:01:02.044 --> 00:01:05.461
Because when we do differential equations

00:01:07.591 --> 00:01:08.424
e to the x

00:01:09.329 --> 00:01:10.662
is the solution.

00:01:13.998 --> 00:01:15.067
Almost every time.

00:01:15.067 --> 00:01:16.538
Whenever we did a circuit

00:01:16.538 --> 00:01:18.205
e to the x was the answer.

00:01:18.205 --> 00:01:20.359
If you recall from when we were solving

00:01:20.359 --> 00:01:22.910
circuit simple circuits
with differential equations

00:01:22.910 --> 00:01:24.434
that we always said something like

00:01:24.434 --> 00:01:27.185
well we're gonna guess that V of T

00:01:27.185 --> 00:01:29.768
is some constant times e to the

00:01:30.883 --> 00:01:31.716
st.

00:01:33.057 --> 00:01:34.662
That was a proposed solution.

00:01:34.662 --> 00:01:37.303
This turned out to work every time.

00:01:37.303 --> 00:01:38.909
So there's something else we love, too.

00:01:38.909 --> 00:01:39.965
And that is

00:01:39.965 --> 00:01:40.798
sinusoids.

00:01:43.925 --> 00:01:45.758
Or, sines and cosines.

00:01:49.384 --> 00:01:50.237
Okay.

00:01:50.237 --> 00:01:51.404
We love these.

00:01:53.589 --> 00:01:54.676
And that gets two lines.

00:01:54.676 --> 00:01:55.755
Now why do we love these?

00:01:55.755 --> 00:01:58.505
Is because they happen in nature.

00:02:00.201 --> 00:02:01.368
If you whistle

00:02:04.224 --> 00:02:06.807
the air pressure looks like a sine wave.

00:02:06.807 --> 00:02:08.848
If you ring a bell

00:02:08.848 --> 00:02:11.448
the bell moves in a sine wave.

00:02:11.448 --> 00:02:13.115
In any kind of music

00:02:14.833 --> 00:02:16.519
if you look at the notes in music

00:02:16.519 --> 00:02:17.526
the sound they make

00:02:17.526 --> 00:02:19.152
the pressure waves look like

00:02:19.152 --> 00:02:20.899
sine waves.

00:02:20.899 --> 00:02:22.950
And circuits make sine waves, remember?

00:02:22.950 --> 00:02:24.931
We analyzed this circuit in great detail

00:02:24.931 --> 00:02:25.931
it was the

00:02:27.096 --> 00:02:28.283
LC circuit.

00:02:28.283 --> 00:02:30.436
We looked at the natural response of this

00:02:30.436 --> 00:02:32.519
and that was a sine wave.

00:02:33.971 --> 00:02:34.804
Okay.

00:02:34.804 --> 00:02:36.358
So, electric circuits make sine waves.

00:02:36.358 --> 00:02:37.629
All these things make sine waves.

00:02:37.629 --> 00:02:39.570
They occur in nature.

00:02:39.570 --> 00:02:41.420
And we want to be able to analyze

00:02:41.420 --> 00:02:44.101
things that happen when
sine waves are present.

00:02:44.101 --> 00:02:46.244
So, we have two things we love and we want

00:02:46.244 --> 00:02:47.686
to relate these two things.

00:02:47.686 --> 00:02:49.547
And these are going to be related

00:02:49.547 --> 00:02:51.364
through that.

00:02:51.364 --> 00:02:52.655
Euler's Formula.

00:02:52.655 --> 00:02:55.702
That's how we connect
these two separate ideas.

00:02:55.702 --> 00:02:57.785
Let me, let's go do that.

00:03:01.087 --> 00:03:03.037
So Euler's Formula says that

00:03:03.037 --> 00:03:03.954
e to the jx

00:03:07.792 --> 00:03:08.625
equals

00:03:10.424 --> 00:03:11.257
cosine

00:03:12.354 --> 00:03:13.533
X

00:03:13.533 --> 00:03:14.691
plus

00:03:14.691 --> 00:03:16.072
j

00:03:16.072 --> 00:03:17.155
times sine x.

00:03:22.867 --> 00:03:24.878
Sal has a really nice
video where he actually

00:03:24.878 --> 00:03:26.974
proves that this is true.

00:03:26.974 --> 00:03:29.208
And he does it by taking the MacLaurin

00:03:29.208 --> 00:03:31.380
series expansions of e,

00:03:31.380 --> 00:03:33.017
and cosine, and sine

00:03:33.017 --> 00:03:35.577
and showing that this expression is true

00:03:35.577 --> 00:03:38.767
by comparing those series expansions.

00:03:38.767 --> 00:03:40.474
And I'm not going to repeat that here

00:03:40.474 --> 00:03:42.680
we're just going to state that as fact.

00:03:42.680 --> 00:03:45.309
And now we're going to look at
this equation a little bit more.

00:03:45.309 --> 00:03:49.161
So, this is the expression
that relates exponentials

00:03:49.161 --> 00:03:52.330
that we love, to sines
and cosines that we love.

00:03:52.330 --> 00:03:55.488
And part of the price of
doing that is we introduce

00:03:55.488 --> 00:03:57.491
complex numbers into our world.

00:03:57.491 --> 00:03:59.970
Here's two complex numbers.

00:03:59.970 --> 00:04:00.803
Okay.

00:04:00.803 --> 00:04:02.285
This is where complex numbers come into

00:04:02.285 --> 00:04:04.948
electrical engineering.

00:04:04.948 --> 00:04:06.422
So we have to mention the other form of

00:04:06.422 --> 00:04:08.172
this formula which is

00:04:09.653 --> 00:04:11.388
e to the, I put a minus sign in here,

00:04:11.388 --> 00:04:12.888
e to the minus jx.

00:04:14.134 --> 00:04:15.384
And that equals

00:04:16.227 --> 00:04:17.060
cosine X

00:04:18.511 --> 00:04:19.344
minus J

00:04:21.001 --> 00:04:21.834
sine X.

00:04:23.447 --> 00:04:26.069
So these two expressions together

00:04:26.069 --> 00:04:29.666
are Euler's Formula, or Euler's Formulas.

00:04:29.666 --> 00:04:30.997
And we're gonna exploit this by taking

00:04:30.997 --> 00:04:34.209
we'll be able to take
the cosines and sines

00:04:34.209 --> 00:04:35.389
that we find in nature

00:04:35.389 --> 00:04:38.231
we're going to be able to fashion
them into exponentials.

00:04:38.231 --> 00:04:39.928
And these exponentials then go into our

00:04:39.928 --> 00:04:42.561
differential equations
and give us solutions.

00:04:42.561 --> 00:04:44.764
And we're going to come back
and pull out the cosines

00:04:44.764 --> 00:04:45.597
and sines.

00:04:46.633 --> 00:04:49.031
That's the rhythm of how
we're going to use this equation

00:04:49.031 --> 00:04:51.198
to help us solve circuits.

00:04:52.659 --> 00:04:54.751
One small point I want to share,

00:04:54.751 --> 00:04:57.837
notice that in both these
equations the sine comes first?

00:04:57.837 --> 00:04:59.279
And the sine is over here

00:04:59.279 --> 00:05:00.834
on the imaginary side.

00:05:00.834 --> 00:05:02.855
So the cosine is the real side.

00:05:02.855 --> 00:05:05.842
This is the reason that we
have a preference in the future

00:05:05.842 --> 00:05:07.214
we're going to have a preference

00:05:07.214 --> 00:05:09.214
if we're talking about
our real world signals

00:05:09.214 --> 00:05:10.973
in terms of the cosine function.

00:05:10.973 --> 00:05:12.892
It's because in this Euler's Formula

00:05:12.892 --> 00:05:16.059
the cosine comes first, in both cases.

00:05:18.143 --> 00:05:19.272
So what I want to do now is

00:05:19.272 --> 00:05:21.547
take a second and I want to see

00:05:21.547 --> 00:05:25.457
if we had our signal expressed
in these exponentials

00:05:25.457 --> 00:05:29.155
how do we recover the
cosine and the sine term?

00:05:29.155 --> 00:05:31.543
How do we flip these equations around

00:05:31.543 --> 00:05:34.185
so we can solve for the
cosine and the sine?

00:05:34.185 --> 00:05:35.912
It's a simple bit of algebra here.

00:05:35.912 --> 00:05:37.891
It's good to see.

00:05:37.891 --> 00:05:42.058
All right, so if I want to
isolate the cosine term.

00:05:43.063 --> 00:05:44.616
If I want to isolate the cosine term

00:05:44.616 --> 00:05:47.449
let me get rid of these guys here.

00:05:51.330 --> 00:05:52.624
So now to isolate the cosine term

00:05:52.624 --> 00:05:55.335
what I'm going to do is add
these two equations together

00:05:55.335 --> 00:05:57.244
and that plus and minus
are going to cancel out

00:05:57.244 --> 00:05:58.839
this second term here.

00:05:58.839 --> 00:06:00.137
That's what I'm going for.

00:06:00.137 --> 00:06:01.137
So, if I add

00:06:03.645 --> 00:06:05.312
I'll get e to the jx

00:06:08.095 --> 00:06:08.928
plus

00:06:09.943 --> 00:06:11.360
e to the minus jx

00:06:13.345 --> 00:06:15.095
equals, cosine doubles,

00:06:16.364 --> 00:06:17.447
two cosine X.

00:06:20.355 --> 00:06:22.366
And the two sine terms cancel out.

00:06:22.366 --> 00:06:23.727
All right?

00:06:23.727 --> 00:06:26.287
So then I could write

00:06:26.287 --> 00:06:27.628
I'll write it over here.

00:06:27.628 --> 00:06:29.628
I can write, cosine of X

00:06:32.455 --> 00:06:33.288
equals

00:06:34.963 --> 00:06:36.296
e to the plus jx

00:06:37.746 --> 00:06:39.579
plus e to the minus jx

00:06:42.145 --> 00:06:42.978
over two.

00:06:44.756 --> 00:06:45.589
All right?

00:06:45.589 --> 00:06:46.441
So that's the expression

00:06:46.441 --> 00:06:48.384
that's the expression
for cosine in terms of

00:06:48.384 --> 00:06:50.134
complex exponentials.

00:06:51.066 --> 00:06:53.086
Okay, let's go back and
see if we can get sine.

00:06:53.086 --> 00:06:54.872
So what I'm gonna do with sine

00:06:54.872 --> 00:06:56.705
is I'm going subtract.

00:07:02.066 --> 00:07:04.058
And that gives me

00:07:04.058 --> 00:07:05.996
what that'll do is that'll
get the cosine terms

00:07:05.996 --> 00:07:07.722
to fall away.

00:07:07.722 --> 00:07:09.652
And then I get

00:07:09.652 --> 00:07:10.569
e to the jx

00:07:13.180 --> 00:07:14.277
subtracting, right?

00:07:14.277 --> 00:07:16.194
Minus e to the minus jx

00:07:17.941 --> 00:07:18.774
equals

00:07:20.454 --> 00:07:22.951
cosine terms, they subtract out.

00:07:22.951 --> 00:07:24.534
And I get two times

00:07:26.232 --> 00:07:27.065
j

00:07:28.894 --> 00:07:29.977
times sine X.

00:07:33.403 --> 00:07:35.497
All right, and that means I can write

00:07:35.497 --> 00:07:36.330
sine X

00:07:38.047 --> 00:07:38.880
equals

00:07:39.906 --> 00:07:41.239
e to the plus jx

00:07:42.851 --> 00:07:44.768
minus e to the minus jx

00:07:47.639 --> 00:07:48.556
over two j.

00:07:52.068 --> 00:07:53.729
And it's really easy

00:07:53.729 --> 00:07:55.273
that j term is down there.

00:07:55.273 --> 00:07:57.501
That's easy to forget sometimes.

00:07:57.501 --> 00:07:59.072
So I put a square around these guys

00:07:59.072 --> 00:08:01.838
'cause that's important.

00:08:01.838 --> 00:08:04.926
So there's the two expressions.

00:08:04.926 --> 00:08:06.693
That's the two expressions for

00:08:06.693 --> 00:08:08.737
if you have complex
exponentials and you want

00:08:08.737 --> 00:08:10.310
to extract the cosine,

00:08:10.310 --> 00:08:11.398
this is how you do it.

00:08:11.398 --> 00:08:13.308
And if you want to extract the sine

00:08:13.308 --> 00:08:15.340
this is how you do it.

00:08:15.340 --> 00:08:16.173
Okay?

00:08:16.173 --> 00:08:18.405
And you can either

00:08:18.405 --> 00:08:20.460
you can put these in your head or

00:08:20.460 --> 00:08:21.840
probably easier for me,

00:08:21.840 --> 00:08:23.406
if you just remember

00:08:23.406 --> 00:08:25.830
Euler's Formula, this is
a pretty straightforward

00:08:25.830 --> 00:08:27.247
quick derivation.

00:08:28.106 --> 00:08:31.127
So the other thing we put a square around

00:08:31.127 --> 00:08:32.544
is this guy here.

00:08:34.323 --> 00:08:36.104
So we have Euler's Formula.

00:08:36.104 --> 00:08:39.222
And basically the cosine
and sine extracted

00:08:39.222 --> 00:08:41.029
from Euler's Formula.


Fourier Series introduction

WEBVTT
Kind: captions
Language: en

00:00:00.107 --> 00:00:02.936
- [Voiceover] So I have
the graph of y is equal to

00:00:02.936 --> 00:00:06.381
f of t here, our horizontal
axis is in terms of time,

00:00:06.381 --> 00:00:08.115
in terms of seconds.

00:00:08.115 --> 00:00:10.183
And this type of function
is often described as

00:00:10.183 --> 00:00:14.294
a square wave, and we see that
it is a periodic function,

00:00:14.294 --> 00:00:18.101
that it completes one
cycle every two pi seconds.

00:00:18.101 --> 00:00:22.552
And so we could say its
period is equal to two pi,

00:00:22.552 --> 00:00:25.160
if we wanna put the units
we could say two pi,

00:00:25.160 --> 00:00:29.327
two pi seconds per cycle,
we could write it like that.

00:00:30.956 --> 00:00:32.657
We could also just write s for seconds.

00:00:32.657 --> 00:00:34.849
And its frequency is
gonna be one over that.

00:00:34.849 --> 00:00:39.004
So we could write its
frequency is equal to one over

00:00:39.004 --> 00:00:42.671
two pi cycles per second,
cycles per second,

00:00:45.476 --> 00:00:48.011
it could also be described as hertz.

00:00:48.011 --> 00:00:50.029
And what we're gonna
explore in this video,

00:00:50.029 --> 00:00:53.059
is can we take a periodic
function like this and

00:00:53.059 --> 00:00:56.599
represent it as an infinite
sum of sines and cosines

00:00:56.599 --> 00:01:00.294
of different periods or
different frequencies?

00:01:00.294 --> 00:01:02.175
So to write that out a
little bit more clearly,

00:01:02.175 --> 00:01:05.755
can we take our f of t,
so can we take our f of t,

00:01:05.755 --> 00:01:10.297
and write it as the sum
of sines and cosines?

00:01:10.297 --> 00:01:13.248
So can we write it, so
it's going to be sum,

00:01:13.248 --> 00:01:16.643
let's say baseline constant,
that'll shift it up or down,

00:01:16.643 --> 00:01:18.989
and as we'll see, that's
going to be based on the

00:01:18.989 --> 00:01:23.024
average value of the
function over one period.

00:01:23.024 --> 00:01:26.650
So a sub zero, and
then, let's start adding

00:01:26.650 --> 00:01:28.783
some periodic functions here.

00:01:28.783 --> 00:01:32.616
And so let's take a sub
one times cosine of t.

00:01:36.050 --> 00:01:38.606
Now, why am I starting with cosine of t?

00:01:38.606 --> 00:01:40.095
And I could also add a sine of t,

00:01:40.095 --> 00:01:42.762
so plus b sub one, of sine of t.

00:01:47.890 --> 00:01:50.750
Why am I starting with
cosine of t and sine of t?

00:01:50.750 --> 00:01:53.862
Well, if our original function
has a period of two pi,

00:01:53.862 --> 00:01:55.903
and I just set up this one so it does have

00:01:55.903 --> 00:01:58.251
a period of two pi, well
it would make sense that it

00:01:58.251 --> 00:02:01.271
would involve some functions
that have periods of two pi.

00:02:01.271 --> 00:02:03.935
And these weights will tell
us how much they involve it.

00:02:03.935 --> 00:02:07.307
If a one is much larger
than b one, well it says,

00:02:07.307 --> 00:02:09.557
okay, this has a lot more
of cosine of t in it,

00:02:09.557 --> 00:02:12.628
than it has of sine of t in it.

00:02:12.628 --> 00:02:14.973
And that by itself isn't going
to describe this function,

00:02:14.973 --> 00:02:16.362
because we know what this would look like.

00:02:16.362 --> 00:02:18.895
This would look like
a very clean sinusoid,

00:02:18.895 --> 00:02:20.472
not like a square wave.

00:02:20.472 --> 00:02:23.812
And so what we're gonna do
is we're gonna add sinusoids

00:02:23.812 --> 00:02:26.788
of frequencies that are
multiples of these frequencies.

00:02:26.788 --> 00:02:30.955
So let's add a sub two, so
another waiting coefficient,

00:02:34.708 --> 00:02:36.541
times cosine of two t.

00:02:37.910 --> 00:02:40.159
This has a frequency of one over two pi,

00:02:40.159 --> 00:02:41.954
this has twice the frequency,

00:02:41.954 --> 00:02:44.133
this has a frequency of one over pi.

00:02:44.133 --> 00:02:47.883
And then a sub three
times cosine of three t.

00:02:49.034 --> 00:02:52.349
And I'm gonna keep going
on and on and on forever.

00:02:52.349 --> 00:02:54.211
And I'm gonna do the same
thing with the sines.

00:02:54.211 --> 00:02:58.378
So plus b two sine of two t
plus b three sine of three t.

00:03:05.034 --> 00:03:07.396
And you might be saying,
well, okay, this seems like

00:03:07.396 --> 00:03:09.978
a fun little mathematical exercise,

00:03:09.978 --> 00:03:12.092
but why do folks even do this?

00:03:12.092 --> 00:03:16.064
Well this was first
explored, and they're named,

00:03:16.064 --> 00:03:18.440
series like this, infinite
series where you represent

00:03:18.440 --> 00:03:21.187
something by essentially
weighted sines and cosines,

00:03:21.187 --> 00:03:23.951
this was explored originally by Fourier,

00:03:23.951 --> 00:03:25.562
and they're called Fourier Series.

00:03:25.562 --> 00:03:28.549
And they were interesting
to him in the study of

00:03:28.549 --> 00:03:30.646
differential equations,
because a lot of differential

00:03:30.646 --> 00:03:34.813
equations are easy to solve
when you involve sines

00:03:36.774 --> 00:03:40.212
and cosines, but not as
obvious to solve when you have

00:03:40.212 --> 00:03:43.952
more general functions, like
maybe a square wave here.

00:03:43.952 --> 00:03:45.423
But if you could
represent that square wave

00:03:45.423 --> 00:03:47.551
as sums of sines and cosines,

00:03:47.551 --> 00:03:49.485
then all of a sudden you
might be able to find

00:03:49.485 --> 00:03:53.708
more general solutions to
your differential equations.

00:03:53.708 --> 00:03:56.006
Another really interesting
thing about this,

00:03:56.006 --> 00:03:58.975
and this is really the
foundation of signal processing,

00:03:58.975 --> 00:04:02.692
it's very used, it's heavily
used in electrical engineering,

00:04:02.692 --> 00:04:06.109
is you can view these
coefficients as they are,

00:04:06.109 --> 00:04:08.494
they're weights on
these cosines and sines,

00:04:08.494 --> 00:04:10.384
but another way to think about it is,

00:04:10.384 --> 00:04:12.504
they tell you how much
of different frequencies

00:04:12.504 --> 00:04:14.769
that this function contains.

00:04:14.769 --> 00:04:19.710
So, for example, if a one
is much bigger than a two,

00:04:19.710 --> 00:04:21.922
then that tells you that
that function contains

00:04:21.922 --> 00:04:25.331
a lot more of the one over
two pi hertz frequency

00:04:25.331 --> 00:04:28.098
than the one over pi frequency.

00:04:28.098 --> 00:04:32.015
Or maybe a three is bigger
than a one or a two.

00:04:33.536 --> 00:04:36.156
And so you can start to
say, hey, this helps us

00:04:36.156 --> 00:04:38.781
start to think of a function
not just in terms of the time

00:04:38.781 --> 00:04:42.213
domain which f of t does,
but it can start bringing us

00:04:42.213 --> 00:04:44.354
to saying well, how much do
we have of each frequency?

00:04:44.354 --> 00:04:47.266
And as we'll see with Fourier
Series and eventually,

00:04:47.266 --> 00:04:49.248
Fourier Transforms, that's
going to get us into

00:04:49.248 --> 00:04:52.219
the frequency domain,
where we can start doing

00:04:52.219 --> 00:04:54.272
some signal processing.

00:04:54.272 --> 00:04:56.615
So we're going to explore
all of that in future videos.

00:04:56.615 --> 00:04:58.077
In order to understand how we can actually

00:04:58.077 --> 00:05:02.358
find these coefficients, we're
gonna review a little bit

00:05:02.358 --> 00:05:04.807
of our trigonometry, especially
integrating trig functions,

00:05:04.807 --> 00:05:06.500
and then we're gonna solve for these,

00:05:06.500 --> 00:05:08.545
and we're gonna see how
good we can approximate

00:05:08.545 --> 00:05:09.795
our function f.


Negative frequency

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:01.960
- [Voiceover] I wanna
talk a little bit about

00:00:01.960 --> 00:00:05.213
one of the quirkier ideas
in signal processing,

00:00:05.213 --> 00:00:07.886
and that's the idea of negative frequency.

00:00:07.886 --> 00:00:10.649
This is a phrase that initially
may not make any sense

00:00:10.649 --> 00:00:13.721
at all, what does it mean
to be a negative frequency?

00:00:13.721 --> 00:00:17.101
Could there be a sine
wave that goes up and down

00:00:17.101 --> 00:00:20.931
at a rate of minus 10 cycles per second,

00:00:20.931 --> 00:00:22.906
what on earth does that mean?

00:00:22.906 --> 00:00:27.073
Or could there be a radio
station of minus 680 kilohertz?

00:00:29.362 --> 00:00:32.076
What is that, that doesn't
sound like it means anything.

00:00:32.076 --> 00:00:34.207
There is a sense in
which negative frequency

00:00:34.207 --> 00:00:36.723
is understandable and
we're gonna just quickly

00:00:36.723 --> 00:00:38.384
talk about that here.

00:00:38.384 --> 00:00:42.000
You remember, we can
describe a cosine like this,

00:00:42.000 --> 00:00:46.167
cosine of omega t equals
1/2 e to the plus j omega t,

00:00:53.828 --> 00:00:56.328
plus e to the minus j omega t.

00:00:59.855 --> 00:01:01.503
And each of these components,

00:01:01.503 --> 00:01:03.334
each of these complex exponentials here,

00:01:03.334 --> 00:01:06.751
we can draw as a rotating complex number.

00:01:07.847 --> 00:01:09.861
So for this first one, we
could actually draw it,

00:01:09.861 --> 00:01:11.569
if we wanted to, we
could draw it like this,

00:01:11.569 --> 00:01:14.909
we could draw a complex
number out here in space,

00:01:14.909 --> 00:01:17.743
and think of it as a rotating vector,

00:01:17.743 --> 00:01:20.826
and that would be e to the j omega t.

00:01:23.843 --> 00:01:25.699
This one has a plus sign.

00:01:25.699 --> 00:01:29.116
Now the other thing, this term over here,

00:01:30.298 --> 00:01:33.413
would look like a similar thing,

00:01:33.413 --> 00:01:36.163
it would be sum vector and space,

00:01:37.467 --> 00:01:41.634
there's that number right
there, e to the minus j omega t,

00:01:43.132 --> 00:01:45.435
and this one's rotating
in the negative direction.

00:01:45.435 --> 00:01:48.768
So that means this is rotating this way.

00:01:50.798 --> 00:01:52.443
So the idea of a negative frequency,

00:01:52.443 --> 00:01:56.138
when we talk about rotating
vectors, makes good sense.

00:01:56.138 --> 00:02:00.107
If we are rotating in the
positive direction, like this,

00:02:00.107 --> 00:02:02.452
you could say that's a positive frequency.

00:02:02.452 --> 00:02:05.447
And if we're rotating in the
negative direction, like this,

00:02:05.447 --> 00:02:09.166
you could say that's a negative frequency.

00:02:09.166 --> 00:02:12.205
So omega t gives us the speed,

00:02:12.205 --> 00:02:16.152
and this sign right here,
and this sign right here,

00:02:16.152 --> 00:02:17.855
give us the direction.

00:02:17.855 --> 00:02:21.445
The frequency here is plus omega t,

00:02:21.445 --> 00:02:25.283
and the frequency here is minus omega t.

00:02:25.283 --> 00:02:27.368
So in this sense of rotating vectors,

00:02:27.368 --> 00:02:31.060
negative frequency seems
like a pretty simple idea.

00:02:31.060 --> 00:02:33.570
Okay, so let's go where
it's not a simple idea,

00:02:33.570 --> 00:02:35.707
and that's when we do
this thing we did before

00:02:35.707 --> 00:02:39.874
where we projected these
vectors onto a cosine wave,

00:02:41.372 --> 00:02:44.811
it had spread out the time
axis linearly, like this,

00:02:44.811 --> 00:02:47.157
going down the page for
the cosine, remember,

00:02:47.157 --> 00:02:50.706
we did this, we drew a
line here on this side,

00:02:50.706 --> 00:02:53.308
we're gonna do plus omega t.

00:02:53.308 --> 00:02:57.391
So we're gonna plot e to
the plus j omega t here.

00:02:58.227 --> 00:03:03.008
This will be the real axis,
this will be the imaginary axis,

00:03:03.008 --> 00:03:07.073
and down here, this'll
be the voltage axis,

00:03:07.073 --> 00:03:09.490
and this'll be the time axis.

00:03:15.043 --> 00:03:17.779
So when we start out at
time equals zero, we have,

00:03:17.779 --> 00:03:22.335
we project it down here, and
we got that value right there.

00:03:22.335 --> 00:03:25.789
And then as time goes on,
if we tip our arrow up,

00:03:25.789 --> 00:03:29.850
like that, to that point,
then we project down to the

00:03:29.850 --> 00:03:31.850
cosine curve right here.

00:03:32.963 --> 00:03:36.400
If we let our arrow go all
the way to the other side,

00:03:36.400 --> 00:03:38.650
it projects like this down,

00:03:40.351 --> 00:03:42.902
it projects down to this
point on the cosine curve.

00:03:42.902 --> 00:03:44.414
And as we go farther and farther,

00:03:44.414 --> 00:03:46.366
let's go straight down for a second,

00:03:46.366 --> 00:03:49.497
that one projects to right there.

00:03:49.497 --> 00:03:51.522
and as we come over here,

00:03:51.522 --> 00:03:54.627
it projects to that point right there.

00:03:54.627 --> 00:03:57.506
And eventually, when we
get back to home again,

00:03:57.506 --> 00:03:58.957
when we get back to zero,

00:03:58.957 --> 00:04:02.451
the projection is right
to this point here.

00:04:02.451 --> 00:04:05.547
And so with one rotation
we've carved out one cycle

00:04:05.547 --> 00:04:06.714
of the cosine.

00:04:08.473 --> 00:04:09.644
So that seems pretty clear.

00:04:09.644 --> 00:04:12.719
Omega t is there, omega t is down here.

00:04:12.719 --> 00:04:14.507
Okay, so let's do it again,

00:04:14.507 --> 00:04:16.529
but this time we'll go over on this side,

00:04:16.529 --> 00:04:19.862
and we'll plot e to the minus j omega t.

00:04:25.462 --> 00:04:30.223
So this again is the real axis,
and this is the imaginary.

00:04:30.223 --> 00:04:34.390
And this is the voltage axis,
and this is the time axis.

00:04:36.958 --> 00:04:39.283
So let's start out again,
we're going straight sideways

00:04:39.283 --> 00:04:43.319
to time equals zero, so
let's project that down.

00:04:43.319 --> 00:04:45.971
And okay, that's pretty good,
that's the same as before,

00:04:45.971 --> 00:04:48.195
we got the same v here.

00:04:48.195 --> 00:04:49.102
Now let's tip it down,

00:04:49.102 --> 00:04:53.417
let's say we rotate this way a little bit,

00:04:53.417 --> 00:04:55.744
and here's our new position of our vector.

00:04:55.744 --> 00:04:58.079
And that projection, after
a little bit of time,

00:04:58.079 --> 00:04:59.662
goes to right here.

00:05:00.983 --> 00:05:03.868
And then if we go over this way,

00:05:03.868 --> 00:05:06.720
eventually, rotate some more,

00:05:06.720 --> 00:05:09.470
we'll project to this point here.

00:05:10.851 --> 00:05:12.221
And when we're straight sideways,

00:05:12.221 --> 00:05:15.388
it'll project to this peak right here.

00:05:17.891 --> 00:05:19.471
And you can see what's
happening is basically

00:05:19.471 --> 00:05:22.439
the exact same thing is happening
as happened on the left.

00:05:22.439 --> 00:05:26.272
Which is these points
carve out a cosine wave,

00:05:29.657 --> 00:05:31.324
just as we'd expect,

00:05:33.577 --> 00:05:38.502
and when this vector gets all
the way back around to zero,

00:05:38.502 --> 00:05:40.594
we've done one cycle, we're back home,

00:05:40.594 --> 00:05:44.704
and now we're projecting
to this point here.

00:05:44.704 --> 00:05:47.836
What happened here is we
took two different vectors

00:05:47.836 --> 00:05:49.303
rotating in opposite directions,

00:05:49.303 --> 00:05:51.505
this one clearly had a positive frequency

00:05:51.505 --> 00:05:54.699
cause it was going counter-clockwise.

00:05:54.699 --> 00:05:56.409
This one had a negative frequency

00:05:56.409 --> 00:05:58.826
cause it was going clockwise.

00:06:00.511 --> 00:06:02.927
And both of them carved out
the exact same cosine wave,

00:06:02.927 --> 00:06:04.344
when we got done.

00:06:05.484 --> 00:06:08.152
So in the vector world where
we're spinning vectors around,

00:06:08.152 --> 00:06:10.522
it seems very natural to talk
about plus and minus frequency

00:06:10.522 --> 00:06:14.689
but when we cast this back
into say a real-world v of t,

00:06:16.038 --> 00:06:19.258
notice that the idea of
negative frequency just sort of,

00:06:19.258 --> 00:06:23.670
it melts away, it evaporates,
it's not really there anymore.

00:06:23.670 --> 00:06:26.990
And it's been removed by
this process of projection.

00:06:26.990 --> 00:06:28.737
So when the idea of
negative frequency comes up,

00:06:28.737 --> 00:06:30.585
and it seems like it
doesn't make sense in this

00:06:30.585 --> 00:06:33.068
time to main view of the signals,

00:06:33.068 --> 00:06:36.097
but then just remember that
when we go back up here,

00:06:36.097 --> 00:06:37.677
and we look at the rotating vectors,

00:06:37.677 --> 00:06:40.471
that it just means which
way the vector's spinning.


AC analysis superposition

WEBVTT
Kind: captions
Language: en

00:00:00.766 --> 00:00:01.724
- [Voiceover] So in the last video

00:00:01.724 --> 00:00:03.968
we talked about Euler's
formula and then we showed

00:00:03.968 --> 00:00:07.109
the expressions for how to
extract a cosine and a sine

00:00:07.109 --> 00:00:09.787
from Euler's formula
and we have a powerful

00:00:09.787 --> 00:00:12.106
set of expressions there for relating

00:00:12.106 --> 00:00:13.876
exponentials to sine waves.

00:00:13.876 --> 00:00:16.900
So now I want to show you
an example just to preview

00:00:16.900 --> 00:00:19.451
of when we get to the formal AC analyses

00:00:19.451 --> 00:00:21.988
how we are gonna exploit
these expressions.

00:00:21.988 --> 00:00:24.255
How we are going to
exploit these formulas.

00:00:24.255 --> 00:00:25.828
This is a real world signal.

00:00:25.828 --> 00:00:27.812
This cosine turn, let's
suppose we build something

00:00:27.812 --> 00:00:29.812
that has a cosine to it.

00:00:30.659 --> 00:00:32.173
That can be something like a microphone

00:00:32.173 --> 00:00:34.749
that's hearing sounds
that look like sine waves

00:00:34.749 --> 00:00:38.005
and we would model those
sine waves as a cosine wave

00:00:38.005 --> 00:00:39.956
and they come in to
some electronic system.

00:00:39.956 --> 00:00:42.551
So let me draw a sketch
of the clever approach

00:00:42.551 --> 00:00:44.301
that we're gonna use.

00:00:46.660 --> 00:00:48.829
So what I'm gonna do is I'm gonna build,

00:00:48.829 --> 00:00:51.209
here's a circuit I'm imagining.

00:00:51.209 --> 00:00:55.192
There's resistors and there's capacitors

00:00:55.192 --> 00:00:57.548
and there's inductors.

00:00:57.548 --> 00:01:01.327
Linear elements and we can have
sources and stuff like that.

00:01:01.327 --> 00:01:02.728
So there's something in here

00:01:02.728 --> 00:01:06.895
and we have something going
in and something coming out.

00:01:08.174 --> 00:01:10.729
Voltage and current coming out.

00:01:10.729 --> 00:01:13.229
Voltage and current coming in.

00:01:14.288 --> 00:01:16.705
So I'm gonna drive my circuit

00:01:17.846 --> 00:01:19.385
with some sort of sinusoid.

00:01:19.385 --> 00:01:22.606
I'll call that sinusoid,
I'll give it an amplitude

00:01:22.606 --> 00:01:25.523
and I'll call it cosine of omega T.

00:01:26.513 --> 00:01:30.670
Omega is the frequency, T
is time, A is the amplitude

00:01:30.670 --> 00:01:32.735
of the signal coming in here.

00:01:32.735 --> 00:01:36.765
Now, that signal's gonna
go into this circuit here.

00:01:36.765 --> 00:01:38.880
Something's gonna happen
and there's gonna be

00:01:38.880 --> 00:01:41.297
a voltage coming out of here.

00:01:42.201 --> 00:01:45.180
So we'll get V out over here of some sort

00:01:45.180 --> 00:01:49.347
and it's gonna be some
amplitude and it's gonna be

00:01:51.127 --> 00:01:54.210
some sort of a cosine wave of omega T

00:01:55.212 --> 00:01:58.045
plus some phase angle, some angle.

00:02:00.481 --> 00:02:02.236
That's our job to discover this.

00:02:02.236 --> 00:02:05.879
That's the circuit analyzes
problem for AC analyzes.

00:02:05.879 --> 00:02:08.331
We put in the AC signal,
we're gonna get out

00:02:08.331 --> 00:02:12.438
another AC signal, this is
a forced response, remember?

00:02:12.438 --> 00:02:14.131
It's gonna look like the input,

00:02:14.131 --> 00:02:16.285
it's gonna be at the same
frequency but its gonna be

00:02:16.285 --> 00:02:18.785
at some different phase angle.

00:02:19.858 --> 00:02:21.877
So in order to do this,
there's a fair amount

00:02:21.877 --> 00:02:23.700
of hard trigonometry we have to do.

00:02:23.700 --> 00:02:26.647
There's gonna be a lot of
cosines and sines and angles

00:02:26.647 --> 00:02:28.283
and things inside this.

00:02:28.283 --> 00:02:30.510
So that's pretty challenging analyses.

00:02:30.510 --> 00:02:32.624
Now what we do with Euler's
formula is we turn it

00:02:32.624 --> 00:02:34.630
into exponentials and we already know

00:02:34.630 --> 00:02:37.381
how to solve exponentials.

00:02:37.381 --> 00:02:40.888
So we take the same circuit,
it has the same stuff inside,

00:02:40.888 --> 00:02:44.969
as resistors and capacitors,
the same exact circuit.

00:02:44.969 --> 00:02:47.055
Let's draw the same exact circuit.

00:02:47.055 --> 00:02:49.055
There, that's identical.

00:02:50.637 --> 00:02:52.324
It has the same outputs.

00:02:52.324 --> 00:02:54.637
Output port right here.

00:02:54.637 --> 00:02:56.367
And this time what we're
gonna do is we're gonna

00:02:56.367 --> 00:02:59.473
basically take this
cosine and we're gonna,

00:02:59.473 --> 00:03:01.128
make up in our head, we're gonna cast this

00:03:01.128 --> 00:03:03.298
into an exponential and the way we do that

00:03:03.298 --> 00:03:06.882
is we use that formula,
we use Euler's formula,

00:03:06.882 --> 00:03:11.049
and we basically create two
sources, two separate sources

00:03:12.282 --> 00:03:14.903
and their exponential sources.

00:03:14.903 --> 00:03:16.986
This gonna be A over two,

00:03:17.903 --> 00:03:18.736
E to the J

00:03:20.804 --> 00:03:21.637
omega T,

00:03:22.663 --> 00:03:26.330
that's this source here
and this source here

00:03:27.397 --> 00:03:28.564
is A over two,

00:03:30.424 --> 00:03:32.507
E to the minus J omega T.

00:03:34.493 --> 00:03:39.361
That's this wave form and
if I add those together

00:03:39.361 --> 00:03:40.194
like that.

00:03:44.308 --> 00:03:46.722
Now remember, the equation
we just looked at,

00:03:46.722 --> 00:03:50.472
Euler's formula says
that Cosine equals that.

00:03:52.634 --> 00:03:55.017
The voltage here is exactly the same

00:03:55.017 --> 00:03:56.273
and all we've done is described

00:03:56.273 --> 00:03:58.322
the same exact cosine wave form

00:03:58.322 --> 00:04:01.322
as these two imaginary exponentials.

00:04:02.525 --> 00:04:05.033
Now I can't actually on my work bench

00:04:05.033 --> 00:04:06.235
build one of these things.

00:04:06.235 --> 00:04:08.538
These don't exists in real life

00:04:08.538 --> 00:04:10.188
but they can exists mathematically.

00:04:10.188 --> 00:04:12.940
They can exists in my head
and I know that if I add

00:04:12.940 --> 00:04:16.732
these two voltages together
that I do get a cosine.

00:04:16.732 --> 00:04:18.823
So, in our heads and on paper,

00:04:18.823 --> 00:04:21.315
we can actually draw
circuits with these things.

00:04:21.315 --> 00:04:24.758
We can't actually build it
but on paper, we can do it.

00:04:24.758 --> 00:04:26.446
Now that I have two sources,

00:04:26.446 --> 00:04:29.023
I can use the principal of superposition.

00:04:29.023 --> 00:04:30.013
This is another use

00:04:30.013 --> 00:04:34.291
of the very powerful
idea of superposition.

00:04:34.291 --> 00:04:36.862
So using the idea of
superposition, I'm gonna apply

00:04:36.862 --> 00:04:40.592
each of these two inputs one at a time

00:04:40.592 --> 00:04:43.163
and then add the results together.

00:04:43.163 --> 00:04:45.297
Over here, I'm gonna get two outputs.

00:04:45.297 --> 00:04:47.590
I'm gonna get a V out one.

00:04:47.590 --> 00:04:50.459
Let's call it a V out
plus, which is what happens

00:04:50.459 --> 00:04:54.122
when I put in this plus
source and I suppress this one

00:04:54.122 --> 00:04:55.922
which means I short it out.

00:04:55.922 --> 00:04:59.041
I'm gonna get a V out
plus and how do you solve

00:04:59.041 --> 00:05:01.479
a differential equation when you have

00:05:01.479 --> 00:05:03.388
a exponential going in?

00:05:03.388 --> 00:05:04.531
Well we know this.

00:05:04.531 --> 00:05:06.406
It's gonna be an exponential answer.

00:05:06.406 --> 00:05:08.816
It's gonna be some constant

00:05:08.816 --> 00:05:10.816
times E to the J omega T

00:05:13.227 --> 00:05:14.560
plus some angle.

00:05:17.022 --> 00:05:20.393
Then I'm gonna solve it
again, I'm gonna add to that.

00:05:20.393 --> 00:05:21.226
V out

00:05:22.972 --> 00:05:23.973
minus

00:05:23.973 --> 00:05:26.522
and I do that by suppressing this input

00:05:26.522 --> 00:05:29.602
and turning this one back
on using superposition

00:05:29.602 --> 00:05:31.635
and V out for the plus,

00:05:31.635 --> 00:05:34.468
sorry, V out for this source here.

00:05:36.592 --> 00:05:38.741
It's gonna equal some other K.

00:05:38.741 --> 00:05:41.999
Let's call that K plus and K minus.

00:05:41.999 --> 00:05:43.999
E to the minus J omega T

00:05:45.619 --> 00:05:46.452
plus V.

00:05:49.163 --> 00:05:51.335
So I'm gonna put in two exponentials.

00:05:51.335 --> 00:05:53.622
I'm sure I'm gonna get
out two exponentials

00:05:53.622 --> 00:05:57.872
and now using Euler's formula
we know how to combine these.

00:05:57.872 --> 00:05:59.595
We can use that same expression

00:05:59.595 --> 00:06:02.534
and we can recover our cosine.

00:06:02.534 --> 00:06:05.034
We can recover the real signal

00:06:08.337 --> 00:06:11.600
and this will have some magnitude B.

00:06:11.600 --> 00:06:13.566
We don't know the magnitudes yet

00:06:13.566 --> 00:06:16.083
but we know what the
shape of the wave form is

00:06:16.083 --> 00:06:20.250
and if we look at this, this
is the same thing as this here

00:06:21.226 --> 00:06:25.131
and we did it by decomposing
our cosine into exponentials,

00:06:25.131 --> 00:06:28.307
putting each exponential through
this and then recombining

00:06:28.307 --> 00:06:31.172
to get cosine and we do all those steps

00:06:31.172 --> 00:06:34.437
because solving differential
equations with exponentials

00:06:34.437 --> 00:06:36.770
is the easiest way to do it.

00:06:38.060 --> 00:06:40.251
So just that you know, it's
not twice as much work,

00:06:40.251 --> 00:06:42.646
what's gonna happen it's gonna turn out

00:06:42.646 --> 00:06:46.904
that this all solution down
here using the negative,

00:06:46.904 --> 00:06:49.792
using the negative exponential,
the answer is going to be

00:06:49.792 --> 00:06:52.299
exactly the same as the
positive exponentials

00:06:52.299 --> 00:06:54.419
except for there's just this conjugate,

00:06:54.419 --> 00:06:56.619
this complex conjugate in here.

00:06:56.619 --> 00:07:00.542
So for real signals going
in, the answer goes through

00:07:00.542 --> 00:07:03.385
and its always, these answers are always

00:07:03.385 --> 00:07:05.318
complex conjugates of each other

00:07:05.318 --> 00:07:09.221
as long as we start with a real input.

00:07:09.221 --> 00:07:11.526
So that was a review of Euler's equation

00:07:11.526 --> 00:07:13.592
and a little preview,
a little sneak preview

00:07:13.592 --> 00:07:15.557
of how we're gonna do AC Analyzes

00:07:15.557 --> 00:07:18.224
using this really powerful tool.


Eulers formula magnitude

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:02.212
- [Voiceover] In this video
we're gonna talk a bunch

00:00:02.212 --> 00:00:05.720
about this fantastic
number e to the j omega t.

00:00:05.720 --> 00:00:07.807
And one of the coolest things
that's gonna happen here,

00:00:07.807 --> 00:00:09.060
we're gonna bring together what we know

00:00:09.060 --> 00:00:12.677
about complex numbers
and this exponential form

00:00:12.677 --> 00:00:14.837
of complex numbers

00:00:14.837 --> 00:00:17.870
and sines and cosines
as a function of time.

00:00:17.870 --> 00:00:19.218
And what we're gonna
end up with is the idea

00:00:19.218 --> 00:00:21.135
of a number that spins.

00:00:24.261 --> 00:00:25.784
I think this is really
one of the coolest things

00:00:25.784 --> 00:00:28.746
in electronics and it's
really at the essence

00:00:28.746 --> 00:00:31.413
of all signal processing theory.

00:00:33.006 --> 00:00:34.668
So this number here, e to the j omega t,

00:00:34.668 --> 00:00:37.466
this is based on Euler's formula.

00:00:37.466 --> 00:00:40.460
Just as a reminder, Euler's formula is

00:00:40.460 --> 00:00:44.770
e to the j, we'll use
theta as our variable,

00:00:44.770 --> 00:00:46.353
equals cosine theta

00:00:48.696 --> 00:00:50.946
plus j times sine of theta.

00:00:52.756 --> 00:00:55.253
That's one form of Euler's formula.

00:00:55.253 --> 00:00:57.596
And the other form is with a
negative up in the exponent.

00:00:57.596 --> 00:01:00.013
We say e to the minus j theta

00:01:01.723 --> 00:01:03.306
equals cosine theta

00:01:05.458 --> 00:01:07.041
minus j sine theta.

00:01:11.982 --> 00:01:16.149
Now if I go and plot this,
what it looks like is this.

00:01:20.180 --> 00:01:22.009
So if I plot this on a complex plane,

00:01:22.009 --> 00:01:25.925
and that's a plane that has a real axis

00:01:25.925 --> 00:01:27.758
and an imaginary axis.

00:01:29.135 --> 00:01:32.767
Now we remember that j is
the variable that we use

00:01:32.767 --> 00:01:34.544
for the imaginary unit.

00:01:34.544 --> 00:01:36.978
A j squared is equal to minus one.

00:01:36.978 --> 00:01:41.145
And we use that in electrical
engineering instead of i.

00:01:42.190 --> 00:01:43.553
One way to express this number is

00:01:43.553 --> 00:01:45.888
by plotting it on this complex plane.

00:01:45.888 --> 00:01:50.564
And so if I pick out a location
for some complex number

00:01:50.564 --> 00:01:54.949
and I say, oh, okay,
this is the x-coordinate

00:01:54.949 --> 00:01:56.532
is cosine of theta.

00:01:59.208 --> 00:02:01.875
And what's this coordinate here?

00:02:02.848 --> 00:02:05.598
On the j-axis, that's sine theta.

00:02:08.486 --> 00:02:12.494
And if I draw a line
right through our number,

00:02:12.494 --> 00:02:14.494
this is the angle theta.

00:02:15.673 --> 00:02:18.682
So this is one of the
representations of complex numbers

00:02:18.682 --> 00:02:21.874
is this Euler's formula
or the exponential form.

00:02:21.874 --> 00:02:25.304
And we can represent it this way here.

00:02:25.304 --> 00:02:26.530
This notation is challenging.

00:02:26.530 --> 00:02:29.203
I can't help but every time I look at this

00:02:29.203 --> 00:02:31.894
I start to do e to the
complex to something

00:02:31.894 --> 00:02:35.665
and everything I know
about taking exponents

00:02:35.665 --> 00:02:38.838
and things like that, it kind
of confuses me in my head.

00:02:38.838 --> 00:02:41.260
But what I've done over
time is basically say,

00:02:41.260 --> 00:02:45.460
e to the j anything, that
whole thing is a complex number

00:02:45.460 --> 00:02:49.059
and this is what that complex
number looks like right there.

00:02:49.059 --> 00:02:50.564
So let's take a look at
some of the properties

00:02:50.564 --> 00:02:52.003
of this complex number.

00:02:52.003 --> 00:02:54.380
One of the things we can ask is

00:02:54.380 --> 00:02:57.880
what is the magnitude of e to the j theta?

00:02:59.513 --> 00:03:01.256
If I put magnitude absolute value

00:03:01.256 --> 00:03:03.125
or magnitude bars around that.

00:03:03.125 --> 00:03:06.359
And what that says is what
is this value here for r?

00:03:06.359 --> 00:03:10.526
And we can figure that out
using the Pythagorean Theorem.

00:03:11.739 --> 00:03:14.072
What we know is this squared

00:03:16.006 --> 00:03:19.423
equals x value squared which is cosine

00:03:20.390 --> 00:03:23.850
plus the y value squared
which is sine theta.

00:03:23.850 --> 00:03:26.850
So it equals cosine squared of theta

00:03:28.339 --> 00:03:30.589
plus sine squared of theta.

00:03:34.076 --> 00:03:36.476
Okay, that's just, we just
applied the Pythagorean Theorem

00:03:36.476 --> 00:03:38.939
to this right triangle right here,

00:03:38.939 --> 00:03:40.606
this right triangle.

00:03:42.426 --> 00:03:45.801
Now from trigonometry, from
trigonometry we know what?

00:03:45.801 --> 00:03:47.277
What's the value of this?

00:03:47.277 --> 00:03:51.444
Cosine squared plus sine squared
for any angle equals one.

00:03:53.748 --> 00:03:58.478
So that tells us that e to
the j theta magnitude squared

00:03:58.478 --> 00:03:59.867
is equal to one

00:03:59.867 --> 00:04:04.046
or that e to the j theta
magnitude is also one.

00:04:04.046 --> 00:04:08.845
All right, so we can write
down e to the j theta,

00:04:08.845 --> 00:04:12.659
the magnitude of that is equal to one.

00:04:12.659 --> 00:04:13.696
Let's go back to our friend up here.

00:04:13.696 --> 00:04:16.112
That says that the length of this vector,

00:04:16.112 --> 00:04:17.504
this is a complex number,

00:04:17.504 --> 00:04:20.602
that is distance one away from the origin.

00:04:20.602 --> 00:04:21.535
So we'll tuck that away.

00:04:21.535 --> 00:04:24.918
We know that the magnitude
of e to the j theta is one

00:04:24.918 --> 00:04:26.461
and I can actually go over here

00:04:26.461 --> 00:04:29.711
and draw now a circle on here like this

00:04:31.729 --> 00:04:33.917
and if I put the circle
right through there,

00:04:33.917 --> 00:04:37.236
there's the unit circle
and it has a radius of one.

00:04:37.236 --> 00:04:39.771
So I know that for any value of theta

00:04:39.771 --> 00:04:43.104
that my complex number e to the j theta,

00:04:46.222 --> 00:04:49.546
it's gonna be somewhere
on this yellow circle.

00:04:49.546 --> 00:04:52.110
So e to the j theta is
somewhere on this circle

00:04:52.110 --> 00:04:53.431
and the angle is what?

00:04:53.431 --> 00:04:55.065
The angle is right here.

00:04:55.065 --> 00:04:58.635
It's whatever is multiplied
by j up in that exponent.

00:04:58.635 --> 00:05:02.406
Anything that multiplies
by j, that's an angle.

00:05:02.406 --> 00:05:03.519
So what if I wanted something

00:05:03.519 --> 00:05:04.965
that wasn't on the unit circle?

00:05:04.965 --> 00:05:06.767
What if I wanted something
that was farther away

00:05:06.767 --> 00:05:08.840
from the origin than that?

00:05:08.840 --> 00:05:11.048
Well what I would do is I would say,

00:05:11.048 --> 00:05:15.131
I would take A, some
amplitude, e to the j theta,

00:05:17.057 --> 00:05:20.777
and this amplitude would expand
the length of that vector.

00:05:20.777 --> 00:05:24.652
So, I would say, if I wanted
to know how far away it is,

00:05:24.652 --> 00:05:26.907
well, the magnitude of
e to the j theta is one,

00:05:26.907 --> 00:05:28.968
and the magnitude of A is A.

00:05:28.968 --> 00:05:30.385
So this equals A.

00:05:33.840 --> 00:05:36.192
And if I was to sketch
in a circle for that,

00:05:36.192 --> 00:05:39.942
let's say A was a little
bit bigger than one,

00:05:41.070 --> 00:05:42.774
it would be a little bit bigger out,

00:05:42.774 --> 00:05:45.524
and this would have a value here.

00:05:47.156 --> 00:05:50.942
That would be a value of
A, the radius would be A.

00:05:50.942 --> 00:05:53.047
Pretty flexible notation.

00:05:53.047 --> 00:05:55.504
So with this notation we
can represent any number

00:05:55.504 --> 00:05:59.855
in the complex plane with
this kind of format here.

00:05:59.855 --> 00:06:01.126
We'll take a little break here

00:06:01.126 --> 00:06:03.419
and in the next video
we'll put in for theta,

00:06:03.419 --> 00:06:05.733
we'll put in an argument
that has to do with time

00:06:05.733 --> 00:06:09.077
and we'll see what happens
to this complex exponential.


Complex exponentials spin

WEBVTT
Kind: captions
Language: en

00:00:00.868 --> 00:00:02.595
- [Voiceover] In the last
video we did a quick review

00:00:02.595 --> 00:00:04.342
of the exponential and what it means.

00:00:04.342 --> 00:00:05.975
And then we looked in and figured out

00:00:05.975 --> 00:00:07.878
what the magnitude exponential is,

00:00:07.878 --> 00:00:10.609
the magnitude is 92 is equal to one.

00:00:10.609 --> 00:00:14.054
Now we're gonna look closely
at this complex exponential

00:00:14.054 --> 00:00:17.887
as it represents a cosine,
a part of a cosine.

00:00:21.702 --> 00:00:23.754
Now we're gonna keep
combining some of our ideas

00:00:23.754 --> 00:00:24.994
from the last couple of videos.

00:00:24.994 --> 00:00:27.696
And you remember if one
of the things we did

00:00:27.696 --> 00:00:29.604
is we used Euler's formula

00:00:29.604 --> 00:00:33.678
and inside out and developed
an expression for cosine.

00:00:33.678 --> 00:00:35.928
So if I say cosine of theta

00:00:37.773 --> 00:00:40.273
I can say that equals one half

00:00:41.146 --> 00:00:42.729
times E to the plus

00:00:43.998 --> 00:00:44.831
J theta

00:00:45.733 --> 00:00:47.981
plus E to the minus

00:00:47.981 --> 00:00:48.814
J theta.

00:00:51.585 --> 00:00:54.300
This is cosine of theta expressed as two

00:00:54.300 --> 00:00:56.133
separate exponentials.

00:00:58.066 --> 00:00:59.734
And now we're gonna take
a really special step.

00:00:59.734 --> 00:01:02.712
I'm gonna put in an
argument right here of time.

00:01:02.712 --> 00:01:04.032
I'm gonna say

00:01:04.032 --> 00:01:05.532
cosine of omega T.

00:01:07.832 --> 00:01:09.262
T is time.

00:01:09.262 --> 00:01:11.029
N is shape here

00:01:11.029 --> 00:01:12.992
this symbol here is the lowercase omega

00:01:12.992 --> 00:01:14.557
from the Greek alphabet.

00:01:14.557 --> 00:01:16.514
And that's the frequency.

00:01:16.514 --> 00:01:19.097
So time is in units of seconds.

00:01:21.412 --> 00:01:22.887
And omega is a frequency

00:01:22.887 --> 00:01:24.350
so it's in

00:01:24.350 --> 00:01:26.605
units of one over seconds.

00:01:26.605 --> 00:01:29.893
That's the units of
frequency are per seconds.

00:01:29.893 --> 00:01:31.661
And when these multiply together

00:01:31.661 --> 00:01:34.314
we get dimensionless number right here.

00:01:34.314 --> 00:01:38.581
And we can take the cosine
of the dimensionless number.

00:01:38.581 --> 00:01:39.698
So what does this equal to?

00:01:39.698 --> 00:01:40.615
This equals

00:01:41.648 --> 00:01:42.798
one half times

00:01:42.798 --> 00:01:43.881
E to the plus

00:01:45.671 --> 00:01:46.504
J omega T

00:01:47.967 --> 00:01:49.050
plus E to the

00:01:51.003 --> 00:01:52.336
minus J omega T.

00:01:55.249 --> 00:01:58.196
Well we make T the argument
of the cosine here.

00:01:58.196 --> 00:02:01.040
T is the stuff that keeps
going up and up and up.

00:02:01.040 --> 00:02:03.663
The number T gets just
gets bigger all the time.

00:02:03.663 --> 00:02:05.310
And so we ended up with

00:02:05.310 --> 00:02:07.024
a cosine wave formula

00:02:07.024 --> 00:02:10.398
I'll just to make a bad
cosine looking thing here.

00:02:10.398 --> 00:02:12.531
That's what a cosine looks like.

00:02:12.531 --> 00:02:15.161
And it keeps going and going and going.

00:02:15.161 --> 00:02:18.400
So we have an idea of what
a cosine wave looks like.

00:02:18.400 --> 00:02:21.195
The frequency determines how
fast this goes up and down

00:02:21.195 --> 00:02:23.563
or how often it goes up and down.

00:02:23.563 --> 00:02:25.087
Now what I want to do is I want to look at

00:02:25.087 --> 00:02:26.713
a really special thing.

00:02:26.713 --> 00:02:28.886
I want to look at what
is this thing right here?

00:02:28.886 --> 00:02:32.734
What is this thing, E
to the plus J omega T?

00:02:32.734 --> 00:02:34.027
And what we see is

00:02:34.027 --> 00:02:37.591
this cosine here is made
of two of these things.

00:02:37.591 --> 00:02:39.034
So whatever these things are

00:02:39.034 --> 00:02:42.528
I can make a cosine out of them.

00:02:42.528 --> 00:02:44.897
So now we're gonna really carefully add

00:02:44.897 --> 00:02:46.480
E to the J omega T.

00:02:48.928 --> 00:02:50.249
What we just reviewed was

00:02:50.249 --> 00:02:51.825
that this is a complex number.

00:02:51.825 --> 00:02:54.408
Let's draw that complex number.

00:03:10.332 --> 00:03:12.679
So we're gonna put a number out here.

00:03:12.679 --> 00:03:15.414
We know it falls on the unit circle.

00:03:15.414 --> 00:03:17.547
We know its angle

00:03:17.547 --> 00:03:19.648
is whatever is multiplying

00:03:19.648 --> 00:03:21.506
the J up in this exponent.

00:03:21.506 --> 00:03:24.443
Whatever's up in the exponent
is the angle of this thing.

00:03:24.443 --> 00:03:26.595
So this angle right here

00:03:26.595 --> 00:03:27.512
is omega T.

00:03:29.238 --> 00:03:30.882
And we know the magnitude of this is

00:03:30.882 --> 00:03:32.927
as we decided before,
the magnitude is one.

00:03:32.927 --> 00:03:35.711
That's why it falls on the unit circle.

00:03:35.711 --> 00:03:36.817
Okay, so now look at this.

00:03:36.817 --> 00:03:38.939
Here's this number T that's
determining the angle.

00:03:38.939 --> 00:03:39.945
And that means, what?

00:03:39.945 --> 00:03:43.472
That means that the angle
is increasing with time.

00:03:43.472 --> 00:03:45.472
If time is equal to zero

00:03:46.630 --> 00:03:48.215
the point is right here.

00:03:48.215 --> 00:03:50.299
At time equals zero.

00:03:50.299 --> 00:03:51.992
Because angle is zero.

00:03:51.992 --> 00:03:53.996
As time proceeds

00:03:53.996 --> 00:03:56.353
the angle keep starts growing and growing

00:03:56.353 --> 00:04:00.520
and it basically keeps
growing and it keeps going.

00:04:02.282 --> 00:04:05.699
It comes back to here after
omega T equals two pi.

00:04:05.699 --> 00:04:06.857
And then what happens?

00:04:06.857 --> 00:04:09.315
It goes, it keeps going around it again.

00:04:09.315 --> 00:04:12.077
And, this basically goes along

00:04:12.077 --> 00:04:14.671
for as long as time goes along.

00:04:14.671 --> 00:04:16.203
So here's this number.

00:04:16.203 --> 00:04:18.206
Here's this complex number

00:04:18.206 --> 00:04:20.665
moving along the unit circle

00:04:20.665 --> 00:04:24.534
in time, over and over and over again.

00:04:24.534 --> 00:04:27.094
So this is a number that is rotating,

00:04:27.094 --> 00:04:29.011
the number is rotating.

00:04:30.487 --> 00:04:32.306
So I can write here,

00:04:32.306 --> 00:04:33.139
E to the J

00:04:34.672 --> 00:04:35.505
omega T.

00:04:36.451 --> 00:04:38.624
And I know that that
because time's up here

00:04:38.624 --> 00:04:41.712
I know it's rotating in time.

00:04:41.712 --> 00:04:45.188
All right, now I'm gonna put
a different number on there.

00:04:45.188 --> 00:04:46.683
Let's put it over here say.

00:04:46.683 --> 00:04:50.377
Actually let's start this number at zero.

00:04:50.377 --> 00:04:52.431
And I'm gonna call this number

00:04:52.431 --> 00:04:53.598
E to the minus

00:04:55.042 --> 00:04:55.875
J omega T.

00:04:56.943 --> 00:04:58.162
What does that number look like?

00:04:58.162 --> 00:04:59.014
That's this guy here.

00:04:59.014 --> 00:05:00.275
That's this one here.

00:05:00.275 --> 00:05:02.108
We'll make him orange.

00:05:02.976 --> 00:05:05.212
So at time equals zero is E to the zero

00:05:05.212 --> 00:05:07.974
or one, just as we would expect.

00:05:07.974 --> 00:05:09.844
Now as time gets bigger

00:05:09.844 --> 00:05:10.677
the angle

00:05:11.661 --> 00:05:13.237
the thing multiplying J

00:05:13.237 --> 00:05:15.025
is minus omega T.

00:05:15.025 --> 00:05:17.980
And so the angle is becoming
more and more negative.

00:05:17.980 --> 00:05:20.397
So after a little bit of time

00:05:21.690 --> 00:05:23.335
it's here.

00:05:23.335 --> 00:05:25.347
And after a little bit more time

00:05:25.347 --> 00:05:26.180
it's here.

00:05:26.180 --> 00:05:27.805
And what we notice is

00:05:27.805 --> 00:05:30.222
it keeps it rotates this way.

00:05:31.754 --> 00:05:33.239
This is what happens when you have

00:05:33.239 --> 00:05:35.270
E to the minus J omega T.

00:05:35.270 --> 00:05:36.603
You rotate in this direction.

00:05:36.603 --> 00:05:39.354
And it keeps going and going and going.

00:05:39.354 --> 00:05:41.975
So these two numbers are
pretty similar in behavior

00:05:41.975 --> 00:05:44.858
except one rotates counterclockwise

00:05:44.858 --> 00:05:46.831
and the other rotates clockwise

00:05:46.831 --> 00:05:50.998
in our coordinate system
which is the complex plane.

00:05:52.247 --> 00:05:54.787
So in summary, if you see
either of these shapes

00:05:54.787 --> 00:05:56.726
E to the minus plus J omega T

00:05:56.726 --> 00:05:59.059
or E to the minus J omega T.

00:06:02.293 --> 00:06:04.201
What pops into your head is

00:06:04.201 --> 00:06:05.868
a number that spins.

00:06:10.257 --> 00:06:12.553
So for me the simple idea is

00:06:12.553 --> 00:06:14.544
I have a number here and
I have a number here.

00:06:14.544 --> 00:06:16.759
They both spin in a complex base.

00:06:16.759 --> 00:06:20.233
And to represent those
in mathematical notation

00:06:20.233 --> 00:06:22.021
I need this kind of notation here

00:06:22.021 --> 00:06:23.658
which is a little bit awkward

00:06:23.658 --> 00:06:25.201
but as I get used to it

00:06:25.201 --> 00:06:27.405
E to the J omega T is a spinning number.

00:06:27.405 --> 00:06:29.460
E to the minus J omega T

00:06:29.460 --> 00:06:31.146
is a spinning number.

00:06:31.146 --> 00:06:33.544
This is a amazingly powerful idea

00:06:33.544 --> 00:06:36.063
and we'll be able to describe every single

00:06:36.063 --> 00:06:39.170
that happens using these kind of terms.