LC natural response intuition 2

WEBVTT
Kind: captions
Language: en

00:00:01.280 --> 00:00:02.113
- [Voiceover] We've been working on

00:00:02.113 --> 00:00:04.344
an intuitive description
of the natural response

00:00:04.344 --> 00:00:06.209
of an LC circuit,

00:00:06.209 --> 00:00:07.583
and in the last video,

00:00:07.583 --> 00:00:08.842
we got everything set up,

00:00:08.842 --> 00:00:12.356
and now, we're ready to close the switch.

00:00:12.356 --> 00:00:14.273
Let's close our switch.

00:00:19.077 --> 00:00:21.994
And now our switch is closed again,

00:00:24.998 --> 00:00:26.614
and what happens?

00:00:26.614 --> 00:00:28.307
Well, we had V naught.

00:00:28.307 --> 00:00:30.424
We had some voltage on our capacitor,

00:00:30.424 --> 00:00:32.417
and that means now we have some voltage

00:00:32.417 --> 00:00:34.042
on our inductor.

00:00:34.042 --> 00:00:35.720
That's what the switch closer did.

00:00:35.720 --> 00:00:37.310
So all of the sudden,

00:00:37.310 --> 00:00:40.833
our inductor finds itself
with some voltage across it,

00:00:40.833 --> 00:00:42.678
so let's look at this equation here,

00:00:42.678 --> 00:00:44.543
the inductor equation.

00:00:44.543 --> 00:00:46.935
And it says that V naught,

00:00:46.935 --> 00:00:48.293
that's the starting voltage

00:00:48.293 --> 00:00:50.772
from the capacitor that we borrowed,

00:00:50.772 --> 00:00:53.772
is L times the slope of the current.

00:00:57.123 --> 00:00:59.709
Alright, so now we have a finite voltage,

00:00:59.709 --> 00:01:02.724
and then we have a finite value on L,

00:01:02.724 --> 00:01:06.578
and that means that DI, DT is some number,

00:01:06.578 --> 00:01:08.930
and if we put positive charge up here,

00:01:08.930 --> 00:01:11.286
we know that V naught is positive.

00:01:11.286 --> 00:01:12.784
So let's sketch that.

00:01:12.784 --> 00:01:16.534
Let's start a sketch of
what's going on here.

00:01:28.000 --> 00:01:29.843
We're gonna plot together on the same plot

00:01:29.843 --> 00:01:30.973
the current in blue,

00:01:30.973 --> 00:01:33.679
and we'll put the voltage in orange,

00:01:33.679 --> 00:01:36.816
and this is the time axis here.

00:01:36.816 --> 00:01:37.948
So, let's go back and check.

00:01:37.948 --> 00:01:39.373
What was our voltage when we started?

00:01:39.373 --> 00:01:43.206
Our voltage was V naught
when the switch closed.

00:01:44.148 --> 00:01:45.981
So I'll mark that out,

00:01:47.834 --> 00:01:49.582
and what was the current when we started?

00:01:49.582 --> 00:01:51.337
The current was zero,

00:01:51.337 --> 00:01:54.254
so here's a zero current down here.

00:01:57.194 --> 00:01:59.039
And now we're gonna sketch
in what we think happens

00:01:59.039 --> 00:02:00.984
right after the switch closes.

00:02:00.984 --> 00:02:03.144
So we decided that there was gonna be

00:02:03.144 --> 00:02:04.977
some kind of a DI, DT.

00:02:06.851 --> 00:02:09.184
DI, DT equals V naught over L.

00:02:13.720 --> 00:02:15.398
I'm just gonna sketch this in.

00:02:15.398 --> 00:02:18.294
There's gonna be some rising slope.

00:02:18.294 --> 00:02:21.975
There's gonna be some
positive slope on the current.

00:02:21.975 --> 00:02:24.272
Alright, so we took our first step.

00:02:24.272 --> 00:02:26.869
Now, the other thing that's
happening here is that Q,

00:02:26.869 --> 00:02:27.908
what is Q doing?

00:02:27.908 --> 00:02:30.617
Let's track our charge.

00:02:30.617 --> 00:02:33.383
Q is actually gonna start
flowing out of this guy.

00:02:33.383 --> 00:02:36.184
This is the current that
we're talking about.

00:02:36.184 --> 00:02:38.436
DI, DT, this is a finite current,

00:02:38.436 --> 00:02:40.074
and that current is made of the charge

00:02:40.074 --> 00:02:41.788
that's in the capacitor,

00:02:41.788 --> 00:02:43.936
so that current starts flowing around

00:02:43.936 --> 00:02:45.649
and running around to the bottom side

00:02:45.649 --> 00:02:48.452
of the capacitor, right?

00:02:48.452 --> 00:02:50.763
That means the amount of charge on the top

00:02:50.763 --> 00:02:52.504
is going down,

00:02:52.504 --> 00:02:54.519
which means that the
voltage is going down.

00:02:54.519 --> 00:02:56.032
So let's sketch in some voltage.

00:02:56.032 --> 00:02:58.454
The voltage is gonna start going down

00:02:58.454 --> 00:03:02.121
because the charge is
leaving the capacitor.

00:03:03.226 --> 00:03:05.970
Alright, so far, so good.

00:03:05.970 --> 00:03:07.218
So let's keep going back and forth

00:03:07.218 --> 00:03:09.453
between the inductor and capacitor,

00:03:09.453 --> 00:03:12.984
and see how the voltage
and current changes,

00:03:12.984 --> 00:03:14.488
and the way we do that is we just look

00:03:14.488 --> 00:03:16.136
at the equation we had.

00:03:16.136 --> 00:03:20.772
Now, the voltage on the
inductor is not V naught anymore.

00:03:20.772 --> 00:03:22.605
It's changed to DI, DT

00:03:25.900 --> 00:03:28.233
equals whatever V is over L,

00:03:30.300 --> 00:03:32.326
and we notice that V
went down a little bit.

00:03:32.326 --> 00:03:33.248
So that means what?

00:03:33.248 --> 00:03:35.026
That means the slope of the current

00:03:35.026 --> 00:03:36.835
went down a little bit.

00:03:36.835 --> 00:03:38.879
So let me make a little shallower version

00:03:38.879 --> 00:03:40.379
of the slope here.

00:03:42.130 --> 00:03:46.015
Alright, now what happened
next to the voltage?

00:03:46.015 --> 00:03:50.095
A little more charge
has left the capacitor,

00:03:50.095 --> 00:03:52.102
headed over to the opposite plate,

00:03:52.102 --> 00:03:55.769
so the voltage is gonna
continue to go down.

00:03:57.404 --> 00:03:59.806
So now we're just gonna edge forward

00:03:59.806 --> 00:04:01.779
moment to moment and see what happens.

00:04:01.779 --> 00:04:03.470
Every time the voltage goes down,

00:04:03.470 --> 00:04:04.596
the slope of the current

00:04:04.596 --> 00:04:07.505
gets a little shallower, right?

00:04:07.505 --> 00:04:09.083
And then the voltage continues to drop

00:04:09.083 --> 00:04:10.877
because the current's leaving,

00:04:10.877 --> 00:04:15.520
and eventually, that
voltage is gonna reach zero,

00:04:15.520 --> 00:04:17.865
and at the point the voltage reached zero,

00:04:17.865 --> 00:04:21.400
the slope of the current is zero.

00:04:21.400 --> 00:04:23.229
The slope of the current is zero.

00:04:23.229 --> 00:04:25.187
So right when this happens,

00:04:25.187 --> 00:04:27.937
the slope of the current is zero.

00:04:30.027 --> 00:04:33.845
Now, the value of the
current is some number.

00:04:33.845 --> 00:04:36.466
Alright, so now we have
some value of current.

00:04:36.466 --> 00:04:38.195
We have a zero voltage.

00:04:38.195 --> 00:04:39.454
Now, when we have zero voltage,

00:04:39.454 --> 00:04:40.865
let's go back up here.

00:04:40.865 --> 00:04:43.164
When this value of voltage is zero here,

00:04:43.164 --> 00:04:44.506
when this guy is zero,

00:04:44.506 --> 00:04:45.864
that means that Q is zero,

00:04:45.864 --> 00:04:47.066
and what that means is that

00:04:47.066 --> 00:04:50.419
the charge on either side of the capacitor

00:04:50.419 --> 00:04:51.703
is equalized.

00:04:51.703 --> 00:04:53.131
The same charge exists on the top

00:04:53.131 --> 00:04:56.542
and the bottom of the capacitor.

00:04:56.542 --> 00:04:59.003
That's what this means here.

00:04:59.003 --> 00:05:00.181
So in our next moment,

00:05:00.181 --> 00:05:02.503
what's gonna happen next?

00:05:02.503 --> 00:05:03.659
Does the voltage go flat?

00:05:03.659 --> 00:05:05.388
Does it go flat sideways?

00:05:05.388 --> 00:05:06.264
Does it go up?

00:05:06.264 --> 00:05:07.162
Does it go down?

00:05:07.162 --> 00:05:08.203
What does it do?

00:05:08.203 --> 00:05:09.927
What does the current do?

00:05:09.927 --> 00:05:12.006
Let's figure it out.

00:05:12.006 --> 00:05:14.137
So I see we have a finite current.

00:05:14.137 --> 00:05:15.984
Okay, what that means is

00:05:15.984 --> 00:05:18.034
the charge is continuing to flow

00:05:18.034 --> 00:05:18.883
in what direction?

00:05:18.883 --> 00:05:20.439
In this direction here,

00:05:20.439 --> 00:05:24.299
charge is continuing to flow like this,

00:05:24.299 --> 00:05:25.603
and what's happening now

00:05:25.603 --> 00:05:27.474
is we're getting excess plus charges

00:05:27.474 --> 00:05:30.641
flowing on this side of the capacitor.

00:05:33.128 --> 00:05:34.864
So we're actually building
up positive charge

00:05:34.864 --> 00:05:37.465
on the bottom plate of the capacitor,

00:05:37.465 --> 00:05:38.683
and what that means is,

00:05:38.683 --> 00:05:39.665
the charge is reversed,

00:05:39.665 --> 00:05:41.934
so the voltage is reversed,

00:05:41.934 --> 00:05:44.796
and that tells us that we're gonna have

00:05:44.796 --> 00:05:46.160
a negative voltage.

00:05:46.160 --> 00:05:48.778
It's gonna start to grow here.

00:05:48.778 --> 00:05:49.880
I have a negative voltage.

00:05:49.880 --> 00:05:52.296
I go back to my inductor equation.

00:05:52.296 --> 00:05:54.321
This value of V right here,

00:05:54.321 --> 00:05:55.854
this is now negative,

00:05:55.854 --> 00:05:57.515
so that means the DI, DT

00:05:57.515 --> 00:05:58.988
has a negative slope,

00:05:58.988 --> 00:06:03.030
and negative slope looks like
it's sloping down like this.

00:06:03.030 --> 00:06:05.946
It starts to roll over and slope down,

00:06:05.946 --> 00:06:08.029
and a little less current

00:06:09.566 --> 00:06:12.983
means a little less slope on the voltage.

00:06:15.065 --> 00:06:16.871
Now this positive current's gonna continue

00:06:16.871 --> 00:06:17.704
to keep flowing.

00:06:17.704 --> 00:06:19.982
It's gonna continue to deliver charge

00:06:19.982 --> 00:06:21.328
to the bottom edge,

00:06:21.328 --> 00:06:24.411
to the bottom plate of the capacitor.

00:06:25.578 --> 00:06:27.143
At some point, that current will actually

00:06:27.143 --> 00:06:29.054
go through zero again,

00:06:29.054 --> 00:06:30.976
and if we look at this situation,

00:06:30.976 --> 00:06:32.991
this is where we started from.

00:06:32.991 --> 00:06:35.127
This is almost the
situation we started in.

00:06:35.127 --> 00:06:37.327
There's zero current,

00:06:37.327 --> 00:06:38.577
and there's a--

00:06:40.102 --> 00:06:41.614
This time it's a negative voltage.

00:06:41.614 --> 00:06:43.245
We started at a high voltage,

00:06:43.245 --> 00:06:44.453
and we went to a negative voltage.

00:06:44.453 --> 00:06:46.902
This voltage here happens to be

00:06:46.902 --> 00:06:47.985
minus V naught,

00:06:49.492 --> 00:06:50.389
and the reason is

00:06:50.389 --> 00:06:52.992
is that all the charge
that was on the top now

00:06:52.992 --> 00:06:55.558
has made a trip all the
way around to the bottom.

00:06:55.558 --> 00:06:57.759
So the situation is perfectly reversed

00:06:57.759 --> 00:06:59.737
from when we started.

00:06:59.737 --> 00:07:00.952
Now, because of that symmetry,

00:07:00.952 --> 00:07:02.650
we're not gonna make
this video any longer.

00:07:02.650 --> 00:07:05.445
I'm just gonna say that this situation

00:07:05.445 --> 00:07:06.609
will repeat itself.

00:07:06.609 --> 00:07:11.197
Basically, the voltage will
curl back down like that,

00:07:11.197 --> 00:07:13.614
and the current will do this.

00:07:15.012 --> 00:07:18.012
Woops, let's just keep sketching in.

00:07:19.323 --> 00:07:21.298
So we've repeated the same story

00:07:21.298 --> 00:07:22.900
that we did at the beginning part

00:07:22.900 --> 00:07:24.107
of the curve over here,

00:07:24.107 --> 00:07:26.474
just with the voltage sign reversed.

00:07:26.474 --> 00:07:28.363
And now we reach another state.

00:07:28.363 --> 00:07:30.778
Now we have zero volts

00:07:30.778 --> 00:07:33.930
and we have a negative current here,

00:07:33.930 --> 00:07:38.399
and what we look at is we
duplicated this situation here,

00:07:38.399 --> 00:07:40.414
just with the opposite
sign of the current.

00:07:40.414 --> 00:07:42.719
There's positive current,

00:07:42.719 --> 00:07:45.081
and here's zero voltage.

00:07:45.081 --> 00:07:47.578
Now we have zero voltage
and a negative current,

00:07:47.578 --> 00:07:50.213
so the whole story I told over here

00:07:50.213 --> 00:07:51.653
is gonna be in reverse.

00:07:51.653 --> 00:07:55.236
This charge now is
gonna reverse direction.

00:07:56.987 --> 00:07:59.487
Let's take out all our arrows,

00:08:04.284 --> 00:08:05.235
and all the charge we had

00:08:05.235 --> 00:08:09.358
is gonna start flowing back
around this way like that.

00:08:09.358 --> 00:08:10.780
This positive charge that we had

00:08:10.780 --> 00:08:12.080
is gonna flow around

00:08:12.080 --> 00:08:13.567
and meet up with the negative charge

00:08:13.567 --> 00:08:16.243
that we accumulated up here,

00:08:16.243 --> 00:08:18.208
and that's what's going on

00:08:18.208 --> 00:08:20.856
in this part of the cycle,

00:08:20.856 --> 00:08:21.971
and we can continue on.

00:08:21.971 --> 00:08:23.464
Basically, this is gonna go

00:08:23.464 --> 00:08:26.047
the same exact story like that,

00:08:28.050 --> 00:08:32.107
and this, the current, will do the same,

00:08:32.107 --> 00:08:32.940
and we began.

00:08:32.940 --> 00:08:35.463
We've come back to the same exact point

00:08:35.463 --> 00:08:36.746
as the starting point.

00:08:36.746 --> 00:08:40.618
Now we've replicated our starting point.

00:08:40.618 --> 00:08:42.109
So what's gonna happen next?

00:08:42.109 --> 00:08:43.599
Well, what happens next is the same thing

00:08:43.599 --> 00:08:45.027
that happened in the beginning,

00:08:45.027 --> 00:08:48.025
and this curve actually repeats itself

00:08:48.025 --> 00:08:50.121
as long as we let this circuit

00:08:50.121 --> 00:08:51.704
sit here like this.

00:08:52.785 --> 00:08:54.370
So I don't have to
continue with the story.

00:08:54.370 --> 00:08:55.995
It's basically a repetition of everything

00:08:55.995 --> 00:08:57.377
that happened here,

00:08:57.377 --> 00:08:59.112
and what we're gonna
get is a rocking motion

00:08:59.112 --> 00:09:01.057
on the voltage, back and forth,

00:09:01.057 --> 00:09:02.999
and we're gonna get a rocking motion

00:09:02.999 --> 00:09:04.946
on the current, back and forth,

00:09:04.946 --> 00:09:07.613
as these energy storage elements

00:09:08.482 --> 00:09:12.649
change stored charge for
energy in the magnetic field,

00:09:13.549 --> 00:09:14.809
and back to stored charge,

00:09:14.809 --> 00:09:18.849
and then it rocks back
and forth like that.

00:09:18.849 --> 00:09:20.486
So this is our intuitive explanation

00:09:20.486 --> 00:09:22.446
of what happens in an LC circuit

00:09:22.446 --> 00:09:24.933
when we let it do its natural response.

00:09:24.933 --> 00:09:26.019
There'll be this rocking motion

00:09:26.019 --> 00:09:27.319
back and forth,

00:09:27.319 --> 00:09:28.293
and now in the next video,

00:09:28.293 --> 00:09:29.147
we're gonna go through

00:09:29.147 --> 00:09:32.438
and see how this comes out mathematically.


LC natural response derivation 1

WEBVTT
Kind: captions
Language: en

00:00:01.686 --> 00:00:02.961
- [Voiceover] In this
video, we're gonna begin

00:00:02.961 --> 00:00:05.870
the derivation of the LC Natural Response,

00:00:05.870 --> 00:00:09.267
the response of an
inductor capacitor circuit.

00:00:09.267 --> 00:00:11.274
This is a difficult derivation,

00:00:11.274 --> 00:00:13.904
but it really pays off in the end.

00:00:13.904 --> 00:00:15.498
There's a really fun surprise at the end,

00:00:15.498 --> 00:00:19.415
and that is, this is
where sine waves are born.

00:00:21.680 --> 00:00:24.901
We're gonna end up with sine
waves at the end of this.

00:00:24.901 --> 00:00:26.936
And that's a really nice
result because these are

00:00:26.936 --> 00:00:31.103
everywhere in electronics,
and in the natural world.

00:00:31.969 --> 00:00:33.334
We're gonna start out by saying,

00:00:33.334 --> 00:00:37.098
we're gonna put some charge
on this capacitor here.

00:00:37.098 --> 00:00:40.274
And we're gonna have the
current in this circuit, i,

00:00:40.274 --> 00:00:43.274
we're gonna have that start at zero.

00:00:44.781 --> 00:00:46.991
So i zero is zero, but this is the current

00:00:46.991 --> 00:00:50.096
through the circuit, and
that means that current

00:00:50.096 --> 00:00:52.179
exists over here as well.

00:00:53.237 --> 00:00:56.154
Let's put a switch in this circuit.

00:01:02.417 --> 00:01:06.584
And we're gonna close that
switch at t equals zero.

00:01:07.759 --> 00:01:10.602
So we have one variable as
i, and the other variable

00:01:10.602 --> 00:01:13.672
we're gonna use is v, and
v is this voltage here,

00:01:13.672 --> 00:01:15.672
after the switch closes.

00:01:17.474 --> 00:01:19.557
So we wanna find i and v.

00:01:25.112 --> 00:01:28.921
And for this we're gonna
focus on finding i.

00:01:28.921 --> 00:01:31.778
Once we find i it's
straightforward to find e.

00:01:31.778 --> 00:01:33.754
So this is gonna be our
independent variable,

00:01:33.754 --> 00:01:35.004
is the current.

00:01:35.981 --> 00:01:37.541
So let's begin the analysis.

00:01:37.541 --> 00:01:41.514
At the moment, t equals zero,
the charge that exists here

00:01:41.514 --> 00:01:43.593
is gonna start flowing in the circuit,

00:01:43.593 --> 00:01:47.654
and both voltage and current
are gonna start to change.

00:01:47.654 --> 00:01:48.910
So let's write some expressions,

00:01:48.910 --> 00:01:50.936
let's start out by writing
some things that we know

00:01:50.936 --> 00:01:54.572
are true about our two components.

00:01:54.572 --> 00:01:56.572
And we'll start with the capacitor.

00:01:56.572 --> 00:01:59.011
So we'll start by writing an
expression for our capacitor,

00:01:59.011 --> 00:02:01.081
and there's a little sign
flip that happens here,

00:02:01.081 --> 00:02:03.097
so we gotta be a little bit careful.

00:02:03.097 --> 00:02:07.264
So if I have v on a capacitor,
and i in a capacitor,

00:02:09.833 --> 00:02:13.666
I would say that i
capacitor equals c, dv, dt.

00:02:21.735 --> 00:02:24.973
Now if we look here, vc
is the same as v here,

00:02:24.973 --> 00:02:28.568
positive sign at the top,
positive sign at the top,

00:02:28.568 --> 00:02:32.462
negative sign at the bottom,
so vc and v are the same.

00:02:32.462 --> 00:02:35.414
But i, we have to be careful,
i is in the opposite direction

00:02:35.414 --> 00:02:39.581
of the current that I picked
for independent variable,

00:02:41.220 --> 00:02:43.260
so that's upside down.

00:02:43.260 --> 00:02:47.346
So i, there's a negative
sign we get to flip here,

00:02:47.346 --> 00:02:49.513
equals negative c, dv, dt.

00:02:53.014 --> 00:02:57.821
So that's the IV equation
for our particular circuit.

00:02:57.821 --> 00:03:01.879
We have that little extra negative sign.

00:03:01.879 --> 00:03:06.686
And now I wanna write the
integral form of the capacitor

00:03:06.686 --> 00:03:10.103
IV equation, which is v equals one over c

00:03:13.792 --> 00:03:15.959
times the integral of idt.

00:03:20.790 --> 00:03:22.012
And remember our minus sign.

00:03:22.012 --> 00:03:24.892
This is the important
minus sign to bring along.

00:03:24.892 --> 00:03:28.286
Okay so let's do the
voltage across the inductor,

00:03:28.286 --> 00:03:32.453
the inductor voltage is same
variable v, is L, di, dt.

00:03:40.338 --> 00:03:41.986
So there's no extra minus sign here.

00:03:41.986 --> 00:03:45.048
This is assigned according
to the sign convention

00:03:45.048 --> 00:03:46.965
for passive components.

00:03:50.269 --> 00:03:53.244
So now we have an expression
for v on the capacitor,

00:03:53.244 --> 00:03:56.284
and we have an expression
for v on the inductor,

00:03:56.284 --> 00:03:58.768
and we know that that's the same voltage.

00:03:58.768 --> 00:04:00.328
So let's set these two things together,

00:04:00.328 --> 00:04:03.328
so we can say that L, di, dt, equals

00:04:08.919 --> 00:04:12.586
minus one over c, times
the integral of idt.

00:04:16.578 --> 00:04:18.228
All we did was set the
two identical voltages

00:04:18.228 --> 00:04:20.015
equal to each other.

00:04:20.015 --> 00:04:21.648
And now I'm gonna manipulate this a bit,

00:04:21.648 --> 00:04:24.981
it says that L, di, dt, plus one over c,

00:04:28.984 --> 00:04:31.651
the integral of idt equals zero.

00:04:35.225 --> 00:04:36.756
We just gathered terms on one side,

00:04:36.756 --> 00:04:40.006
and we said the other side became zero.

00:04:44.488 --> 00:04:46.074
Now since I'm not used to having integrals

00:04:46.074 --> 00:04:47.953
inside equations, I'd
rather have derivatives,

00:04:47.953 --> 00:04:50.621
since I have some experience
with differential equations,

00:04:50.621 --> 00:04:54.708
let's, if we take a derivative
of this whole equation,

00:04:54.708 --> 00:04:56.913
so here's what we're gonna do.

00:04:56.913 --> 00:04:58.774
We're gonna take a derivative
with respect to time,

00:04:58.774 --> 00:05:01.136
of this whole thing here,

00:05:01.136 --> 00:05:05.053
and that gives us, we
get the second derivative

00:05:05.993 --> 00:05:08.576
of the first term, so we get L,

00:05:13.571 --> 00:05:17.238
second derivative of i
with respect to time,

00:05:19.150 --> 00:05:21.317
plus one over c, and then,

00:05:23.328 --> 00:05:26.067
the derivative of the integral of idt,

00:05:26.067 --> 00:05:30.553
turns into just i, and
the derivative of zero,

00:05:30.553 --> 00:05:32.386
on this side, is zero.

00:05:35.001 --> 00:05:39.168
So this is now the differential
equation for the LC circuit.

00:05:40.928 --> 00:05:42.391
And it's called, it has a name,

00:05:42.391 --> 00:05:45.558
it's called a second order homogeneous

00:05:55.347 --> 00:05:57.930
ordinary differential equation.

00:05:59.289 --> 00:06:00.799
It's a differential
equation because it has

00:06:00.799 --> 00:06:03.749
derivatives in it, it's
homogeneous because it only

00:06:03.749 --> 00:06:08.650
has derivatives of i with
respect to t and nothing else.

00:06:08.650 --> 00:06:11.270
The indicator is that this
side is equal to zero,

00:06:11.270 --> 00:06:13.676
there's no forcing term
over here on this side.

00:06:13.676 --> 00:06:16.013
So when you can write
the equation this way,

00:06:16.013 --> 00:06:18.424
we say it's homogeneous.

00:06:18.424 --> 00:06:20.213
And it's called a second order equation,

00:06:20.213 --> 00:06:24.275
because it has this second
derivative right here.

00:06:24.275 --> 00:06:27.571
So now we've set up our
second order homogeneous ODE,

00:06:27.571 --> 00:06:31.008
and in the next video,
we'll go about solving this.

00:06:31.008 --> 00:06:34.258
We're gonna go through it step by step.


LC natural response intuition 1

WEBVTT
Kind: captions
Language: en

00:00:00.398 --> 00:00:02.250
- [Voiceover] We're gonna talk
about the natural response

00:00:02.250 --> 00:00:06.507
of an LC circuit, inductor
capacitor circuit.

00:00:06.507 --> 00:00:07.808
And this is an interesting one,

00:00:07.808 --> 00:00:11.183
this is a circuit that has
two energy storage elements.

00:00:11.183 --> 00:00:14.543
In the past videos we've done
one energy storage element,

00:00:14.543 --> 00:00:16.931
either a C or an L,

00:00:16.931 --> 00:00:18.689
and this time we're gonna put 'em together

00:00:18.689 --> 00:00:21.127
and see what they do as a pair.

00:00:21.127 --> 00:00:23.841
And there's no resistor in this circuit.

00:00:23.841 --> 00:00:26.058
So this is interesting because we

00:00:26.058 --> 00:00:28.808
have two energy storage elements.

00:00:36.287 --> 00:00:37.120
Elements.

00:00:39.984 --> 00:00:41.039
Well, what does that mean?

00:00:41.039 --> 00:00:43.734
That means for a capacitor,

00:00:43.734 --> 00:00:48.213
there's some charge
stored on the capacitor,

00:00:48.213 --> 00:00:49.432
and that typically means that

00:00:49.432 --> 00:00:52.552
there's an excessive charge
on one of the plates.

00:00:52.552 --> 00:00:55.544
So in this case there's an
excessive positive charge

00:00:55.544 --> 00:00:56.696
on the top plate

00:00:56.696 --> 00:00:58.934
or you could say in the same way there's a

00:00:58.934 --> 00:01:00.934
lack of negative charge,

00:01:01.922 --> 00:01:03.905
there's some negative charge
missing from the top plate

00:01:03.905 --> 00:01:05.002
and that centers some extra

00:01:05.002 --> 00:01:07.443
negative charge on the bottom plate.

00:01:07.443 --> 00:01:11.730
So that's what we mean by
capacitor storing charge.

00:01:11.730 --> 00:01:13.579
So how does an inductor store energy?

00:01:13.579 --> 00:01:16.226
Well this stores its
energy in a magnetic field.

00:01:16.226 --> 00:01:18.559
It's out in the space around

00:01:20.120 --> 00:01:21.940
around the inductor.

00:01:21.940 --> 00:01:25.379
So when we have a current
flowing in an inductor,

00:01:25.379 --> 00:01:29.097
its energy is stored in a
magnetic field like that.

00:01:29.097 --> 00:01:33.477
So that's what we mean by
two energy storage elements.

00:01:33.477 --> 00:01:36.281
Now one thing we know
about the q in a capacitor,

00:01:36.281 --> 00:01:39.470
is q equals cv.

00:01:39.470 --> 00:01:41.439
So if there is some q here,

00:01:41.439 --> 00:01:44.356
that means there's some voltage here.

00:01:44.356 --> 00:01:46.724
So this is the voltage we're
gonna track in this circuit,

00:01:46.724 --> 00:01:49.821
that's the voltage between
these two nodes here.

00:01:49.821 --> 00:01:52.779
And because there's an inductor,

00:01:52.779 --> 00:01:54.932
one of the interesting things
is the current in the inductor

00:01:54.932 --> 00:01:58.682
so I'm gonna draw the
current arrow this way.

00:02:00.226 --> 00:02:01.466
And one thing I wanna point out is

00:02:01.466 --> 00:02:04.908
if I define the inductor
current going down

00:02:04.908 --> 00:02:06.961
through the inductor that same current

00:02:06.961 --> 00:02:09.794
is going up through the capacitor.

00:02:11.646 --> 00:02:12.876
So our challenge when we wanna know what

00:02:12.876 --> 00:02:14.928
the natural response of this is,

00:02:14.928 --> 00:02:17.549
is we put in some energy

00:02:17.549 --> 00:02:21.145
and in this case we'll put
in some q on the capacitor

00:02:21.145 --> 00:02:23.645
and we'll let i start at zero.

00:02:24.489 --> 00:02:29.002
And then we step back and we
watch what this circuit does.

00:02:29.002 --> 00:02:31.468
And what that means is we figure out what

00:02:31.468 --> 00:02:33.796
the voltage is as a function of time,

00:02:33.796 --> 00:02:35.368
and the current as a function of time,

00:02:35.368 --> 00:02:37.149
and both of those things together

00:02:37.149 --> 00:02:40.566
is the natural response of an LC circuit.

00:02:41.546 --> 00:02:45.713
So in this video what I wanna
do is predict the shape.

00:02:48.364 --> 00:02:50.614
We're gonna predict the NI,

00:02:52.050 --> 00:02:53.980
we're just gonna do this intuitively.

00:02:53.980 --> 00:02:55.820
And then in the next sequence of videos,

00:02:55.820 --> 00:02:59.295
we'll work it out exactly
with a mathematical precision,

00:02:59.295 --> 00:03:01.477
what this natural response looks like.

00:03:01.477 --> 00:03:02.422
And then we'll look to see if the

00:03:02.422 --> 00:03:05.674
mathematics matches our intuition.

00:03:05.674 --> 00:03:07.584
A good way to make this prediction,

00:03:07.584 --> 00:03:08.687
what we're gonna do,

00:03:08.687 --> 00:03:11.038
is we're gonna follow
and track what happens

00:03:11.038 --> 00:03:12.705
to this charge here

00:03:13.598 --> 00:03:17.966
as this circuit relaxes
in its natural response.

00:03:17.966 --> 00:03:19.906
So, first thing, let's
just write some equations,

00:03:19.906 --> 00:03:22.771
the element equations for the L and the C.

00:03:22.771 --> 00:03:26.938
And we know for an inductor,
v equals L times di dt.

00:03:32.809 --> 00:03:35.756
So voltage is proportional
to the value of the inductor

00:03:35.756 --> 00:03:37.263
times the slope of the current

00:03:37.263 --> 00:03:40.253
or the rate of change of the current.

00:03:40.253 --> 00:03:41.470
For a capacitor, we know that

00:03:41.470 --> 00:03:45.553
i equals C times the slope
of the voltage, dv dt.

00:03:50.066 --> 00:03:53.080
And one thing we know is
that both of these equations

00:03:53.080 --> 00:03:55.174
are true all the time.

00:03:55.174 --> 00:03:56.608
So that's gonna help us out.

00:03:56.608 --> 00:03:58.068
And the way we look at this intuitively

00:03:58.068 --> 00:04:00.155
is we're gonna track the charge,

00:04:00.155 --> 00:04:02.013
and we're gonna look at
what happens in this circuit

00:04:02.013 --> 00:04:06.575
moment to moment as that
charge moves around.

00:04:06.575 --> 00:04:08.648
So what I'm gonna do,
just to get us set up,

00:04:08.648 --> 00:04:09.550
just to get us set up here.

00:04:09.550 --> 00:04:12.587
I'm gonna take out a little
chunk of the circuit here,

00:04:12.587 --> 00:04:15.504
and then put in a switch like that.

00:04:16.957 --> 00:04:21.366
So here's a switch and
that switch is gonna close

00:04:21.366 --> 00:04:23.033
at time equals zero.

00:04:24.809 --> 00:04:26.678
So before the switch closes,

00:04:26.678 --> 00:04:29.746
we're gonna put some
charge on this capacitor,

00:04:29.746 --> 00:04:33.830
there's gonna be a
voltage on the capacitor.

00:04:33.830 --> 00:04:35.710
The capacitor will have
a voltage of v naught,

00:04:35.710 --> 00:04:37.377
so that means that v

00:04:40.920 --> 00:04:44.170
of time less than zero equals v naught,

00:04:45.012 --> 00:04:47.462
we'll just call it v naught.

00:04:47.462 --> 00:04:48.649
And what else do we know?

00:04:48.649 --> 00:04:50.110
Well the switch is open,

00:04:50.110 --> 00:04:53.984
so that means that the
current through this loop,

00:04:53.984 --> 00:04:56.259
the current in our circuit is zero,

00:04:56.259 --> 00:04:59.592
so i, we can write i of t less than zero

00:05:02.684 --> 00:05:03.913
equals zero.

00:05:03.913 --> 00:05:06.464
So there's two things we
know about the circuit.

00:05:06.464 --> 00:05:09.085
So now we're ready to close the switch.

00:05:09.085 --> 00:05:10.699
And we're gonna take a break right now

00:05:10.699 --> 00:05:11.979
and I'll see you in the next video.


Sine and Cosine come from circles

WEBVTT
Kind: captions
Language: en

00:00:00.586 --> 00:00:02.087
- [Voiceover] Now I'm gonna
clear off the screen here

00:00:02.087 --> 00:00:06.125
and we're gonna talk about the
shape of the sine function.

00:00:06.125 --> 00:00:07.292
Let's do that.

00:00:10.086 --> 00:00:11.911
This is a plot of the sine function

00:00:11.911 --> 00:00:15.857
where the angle, theta, this
is the theta axis in this plot,

00:00:15.857 --> 00:00:19.255
where theta has been plotted
out on a straight line

00:00:19.255 --> 00:00:21.660
instead of wrapped around this circle.

00:00:21.660 --> 00:00:23.918
So if we draw a line on here,

00:00:23.918 --> 00:00:26.918
let's make this circle a radius one.

00:00:28.102 --> 00:00:29.663
So if I draw this line up here,

00:00:29.663 --> 00:00:33.317
it's on a unit circle,
the definition of sine

00:00:33.317 --> 00:00:36.150
of theta, this will be theta here,

00:00:37.921 --> 00:00:40.224
is opposite over hypotenuse.

00:00:40.224 --> 00:00:43.179
So this is the opposite side

00:00:43.179 --> 00:00:46.262
and that distance is the opposite leg

00:00:47.361 --> 00:00:50.395
of that triangle, is
this value right here.

00:00:50.395 --> 00:00:53.403
So sine of theta is actually equal to y

00:00:53.403 --> 00:00:56.372
over the hypotenuse and
the hypotenuse is one

00:00:56.372 --> 00:00:58.778
in all cases around this.

00:00:58.778 --> 00:01:01.956
So, if I plot this on a curve,

00:01:01.956 --> 00:01:05.706
this is an angle and I
basically go over here

00:01:07.072 --> 00:01:08.905
and plot it like that.

00:01:10.389 --> 00:01:13.820
And then as theta swings
around the circle,

00:01:13.820 --> 00:01:17.079
I'm gonna plot the different values of y.

00:01:17.079 --> 00:01:20.279
If it comes over this
way, down here like this,

00:01:20.279 --> 00:01:24.029
right, you can see that, that plots

00:01:24.029 --> 00:01:26.106
over there like that.

00:01:26.106 --> 00:01:28.899
Now when the angle gets
back all the way to zero,

00:01:28.899 --> 00:01:32.738
of course, the sine function
comes all the way back to zero

00:01:32.738 --> 00:01:36.310
and then it repeats again
as our vector swings

00:01:36.310 --> 00:01:37.687
around the other way.

00:01:37.687 --> 00:01:41.799
So the sine of two pi is zero,
just like the sine of zero.

00:01:41.799 --> 00:01:44.726
So every two pi, if I go off the screen,

00:01:44.726 --> 00:01:48.393
every two pi comes back
and repeats to zero.

00:01:49.886 --> 00:01:52.045
So now I wanna do the same
thing with the cosine function

00:01:52.045 --> 00:01:56.171
that we did with sine, where
we project the projection

00:01:56.171 --> 00:02:00.338
of this value onto this time
the cosine curve down here.

00:02:01.527 --> 00:02:05.900
This has the cosine curve with
time going down on the page.

00:02:05.900 --> 00:02:08.233
And our definition of cosine

00:02:09.231 --> 00:02:11.648
was adjacent over hypotenuse.

00:02:13.702 --> 00:02:16.455
Hypotenuse is one in our drawing.

00:02:16.455 --> 00:02:17.955
So cosine of theta

00:02:20.263 --> 00:02:24.145
equals adjacent which is x, the x value,

00:02:24.145 --> 00:02:27.107
divided by hypotenuse which is one.

00:02:27.107 --> 00:02:29.825
So in this diagram, the cosine of theta

00:02:29.825 --> 00:02:34.238
is actually the x value
which is this x right here.

00:02:34.238 --> 00:02:37.405
So let me clean this off for a second.

00:02:38.752 --> 00:02:39.880
And we'll start at the beginning.

00:02:39.880 --> 00:02:44.134
Let's start with the radius
pointing straight sideways.

00:02:44.134 --> 00:02:48.301
And we know that cosine of
theta equals zero is one.

00:02:49.819 --> 00:02:51.992
So if I drop that down,
if I project that down

00:02:51.992 --> 00:02:53.659
onto the angle zero,

00:02:56.251 --> 00:02:58.480
that's this point right here on the curve.

00:02:58.480 --> 00:03:02.530
Now as we roll forward,
we go to a higher angle,

00:03:02.530 --> 00:03:06.447
this projection now moves
to here on the curve.

00:03:09.261 --> 00:03:11.761
When the arrow is straight up,

00:03:12.973 --> 00:03:16.129
we are at this point right here,

00:03:16.129 --> 00:03:18.133
we go back to the axis.

00:03:18.133 --> 00:03:21.966
If we go continue on,
this projects down here.

00:03:28.660 --> 00:03:31.243
We're moving this radius vector

00:03:33.550 --> 00:03:35.909
around in a circle like this.

00:03:35.909 --> 00:03:38.127
And actually this one
will be at the same point

00:03:38.127 --> 00:03:41.060
as before, as the one above, but it'll be

00:03:41.060 --> 00:03:43.621
on this part of the curve here.

00:03:43.621 --> 00:03:46.538
And when we get back to zero again,

00:03:48.390 --> 00:03:51.473
the projection is to this point here.

00:03:52.835 --> 00:03:54.793
So that's a way to
visualize the cosine curve

00:03:54.793 --> 00:03:58.960
getting generated by a vector
rotating around this circle.

00:03:59.858 --> 00:04:01.581
The cosine comes out the bottom

00:04:01.581 --> 00:04:03.826
because it's the projection on the x-axis,

00:04:03.826 --> 00:04:05.895
and when we did the sine,
it was the projection

00:04:05.895 --> 00:04:09.867
on the y-axis, produced the
sine wave when we went this way.

00:04:09.867 --> 00:04:13.528
So I like to visualize this
because this rotating vector

00:04:13.528 --> 00:04:16.308
is a really simple and powerful idea,

00:04:16.308 --> 00:04:18.249
and we can see how it actually generates,

00:04:18.249 --> 00:04:21.323
it's a way to generate
sine and cosine waves.

00:04:21.323 --> 00:04:23.283
And you can see how sort of naturally

00:04:23.283 --> 00:04:24.930
they come out at different phases, right.

00:04:24.930 --> 00:04:28.534
The sine starts at zero and
the cosine starts at one.

00:04:28.534 --> 00:04:31.271
With this way of drawing it,
you could see why that happens.

00:04:31.271 --> 00:04:33.291
So this relationship between circles

00:04:33.291 --> 00:04:35.943
and rotating vectors and sines and cosines

00:04:35.943 --> 00:04:37.321
is a very powerful idea.

00:04:37.321 --> 00:04:39.007
We're really gonna take advantage of this.


Lead Lag

WEBVTT
Kind: captions
Language: en

00:00:00.383 --> 00:00:01.491
- [Voiceover] In this
video we're gonna introduce

00:00:01.491 --> 00:00:03.489
a couple of words to help talk about the

00:00:03.489 --> 00:00:06.385
relationship between sines and cosines,

00:00:06.385 --> 00:00:09.615
or different sinusoids that
have the same frequency,

00:00:09.615 --> 00:00:12.072
but a different timing relationship.

00:00:12.072 --> 00:00:13.231
So, what I've shown here,

00:00:13.231 --> 00:00:15.015
is a plot of a cosine,

00:00:15.015 --> 00:00:16.768
and a sine wave.

00:00:16.768 --> 00:00:18.954
And the axis here is in theta,

00:00:18.954 --> 00:00:21.011
in the angle, the radian angle

00:00:21.011 --> 00:00:22.844
of the cosine or sine.

00:00:23.755 --> 00:00:25.380
Now I can label these, the pi over two

00:00:25.380 --> 00:00:27.963
represents a 90 degrees change.

00:00:29.754 --> 00:00:30.587
Pi is 180,

00:00:32.629 --> 00:00:33.765
degrees.

00:00:33.765 --> 00:00:35.432
This is 270 degrees.

00:00:36.648 --> 00:00:39.014
And this is 360 degrees.

00:00:39.014 --> 00:00:41.097
Those are two equivalent,

00:00:41.097 --> 00:00:44.452
scales for the angle, the angle axis,

00:00:44.452 --> 00:00:46.535
in degrees or in radians.

00:00:48.557 --> 00:00:50.704
Now, one thing we notice
here is that sine and cosine,

00:00:50.704 --> 00:00:53.095
they look the same,
but they don't overlap.

00:00:53.095 --> 00:00:55.618
If I change this to a time,

00:00:55.618 --> 00:00:58.368
If I change this to a time, axis,

00:00:59.574 --> 00:01:00.757
what I can say is

00:01:00.757 --> 00:01:04.924
the cosine wave reaches its
peak at time equals zero,

00:01:07.274 --> 00:01:09.835
and the sine peak, the sine wave,

00:01:09.835 --> 00:01:12.676
reaches its peak at a later time,

00:01:12.676 --> 00:01:15.438
this is increasing time going this way.

00:01:15.438 --> 00:01:18.727
So, the sine is delayed
compared to the cosine.

00:01:18.727 --> 00:01:20.624
The peak here is delayed here.

00:01:20.624 --> 00:01:23.015
If I go down and look at these two peaks,

00:01:23.015 --> 00:01:25.230
we see the same relationship.

00:01:25.230 --> 00:01:30.043
This sine peak in orange,
is behind, is delayed, from

00:01:30.043 --> 00:01:30.960
the cosine.

00:01:31.796 --> 00:01:33.381
So when we have this timing relationship

00:01:33.381 --> 00:01:35.577
between two periodic waves,

00:01:35.577 --> 00:01:37.934
what we say is, in this case,

00:01:37.934 --> 00:01:39.851
we say that the cosine,

00:01:43.236 --> 00:01:44.069
leads,

00:01:46.294 --> 00:01:47.923
the sine wave.

00:01:47.923 --> 00:01:51.243
And the amount of lead
is the difference between

00:01:51.243 --> 00:01:52.305
these two points,

00:01:52.305 --> 00:01:54.536
and we can say the lead is

00:01:54.536 --> 00:01:58.536
270 minus 180, in this
case, it'd be 90 degrees.

00:02:00.028 --> 00:02:02.361
So we say that cosine, leads

00:02:03.733 --> 00:02:04.566
sine

00:02:05.856 --> 00:02:07.023
by 90 degrees.

00:02:09.507 --> 00:02:11.627
Now I can take exactly the
opposite point of view.

00:02:11.627 --> 00:02:16.394
If I actually measure where
the sine is relative to cosine,

00:02:16.394 --> 00:02:20.144
I say it's behind, then
we would say it lags.

00:02:22.675 --> 00:02:24.571
So the phrase we hear would be

00:02:24.571 --> 00:02:25.904
sine lags cosine

00:02:29.363 --> 00:02:30.530
by 90 degrees.

00:02:32.416 --> 00:02:35.407
So that's the term lead and
lag, that's what those mean.

00:02:35.407 --> 00:02:37.819
Now these terms apply,
this idea of a delay,

00:02:37.819 --> 00:02:40.644
this only applies when these
frequencies are the same.

00:02:40.644 --> 00:02:42.487
If the frequencies are different,

00:02:42.487 --> 00:02:44.359
the relationship between
the two wave functions

00:02:44.359 --> 00:02:45.624
changes all the time.

00:02:45.624 --> 00:02:47.482
So, we use the word lead and lag when we

00:02:47.482 --> 00:02:49.902
know that the two signals
we are talking about

00:02:49.902 --> 00:02:52.316
are exactly the same frequency.

00:02:52.316 --> 00:02:53.790
One thing I want to be able to do

00:02:53.790 --> 00:02:56.830
is express sines and cosines
in terms of each other.

00:02:56.830 --> 00:02:58.829
So, if I have a sine wave,

00:02:58.829 --> 00:03:01.487
could I actually express this orange curve

00:03:01.487 --> 00:03:03.178
as a cosine wave?

00:03:03.178 --> 00:03:04.649
How would a do that?

00:03:04.649 --> 00:03:05.482
So what I've noticed,

00:03:05.482 --> 00:03:07.989
if I look at the value of sine right here,

00:03:07.989 --> 00:03:12.065
and this is sine at 90 degrees,
or sine at pi over two,

00:03:12.065 --> 00:03:14.525
if I look at this value here,

00:03:14.525 --> 00:03:17.510
what I notice is that,
this has the same value,

00:03:17.510 --> 00:03:19.883
which is the peak of one.

00:03:19.883 --> 00:03:23.051
Cosine has the same
value, the peak of one,

00:03:23.051 --> 00:03:25.999
but 90 degrees earlier, 90 degrees before,

00:03:25.999 --> 00:03:29.052
because it's a leading function.

00:03:29.052 --> 00:03:30.723
So this suggests a conversion factor.

00:03:30.723 --> 00:03:33.503
Any time I pick out a value of the sine,

00:03:33.503 --> 00:03:35.730
if I look back 90 degrees,

00:03:35.730 --> 00:03:38.485
I'll see the same value for cosine.

00:03:38.485 --> 00:03:39.701
So I can write something like this,

00:03:39.701 --> 00:03:42.034
I can say that sine of theta

00:03:44.280 --> 00:03:46.530
equals the cosine of theta,

00:03:50.560 --> 00:03:51.977
minus 90 degrees.

00:03:55.604 --> 00:03:56.930
If I go out to some value,

00:03:56.930 --> 00:03:59.663
let's say there on the sine curve,

00:03:59.663 --> 00:04:01.604
and if I back up 90 degrees,

00:04:01.604 --> 00:04:04.975
like that, I'll read the same
value on the cosine curve,

00:04:04.975 --> 00:04:09.142
so these two functions will
give me the same number.

00:04:12.185 --> 00:04:14.598
So I can write this
identity in reverse also.

00:04:14.598 --> 00:04:16.370
If I have a cosine, if I'm riding along

00:04:16.370 --> 00:04:18.066
on this cosine wave,

00:04:18.066 --> 00:04:21.162
what I notice is, if
let's say I'm right here,

00:04:21.162 --> 00:04:24.018
I'll notice I have my peak value here,

00:04:24.018 --> 00:04:26.668
and if I added, if I went later in time,

00:04:26.668 --> 00:04:29.021
or if I added 90 degrees,

00:04:29.021 --> 00:04:32.351
I would have the same value
on that orange sine curve.

00:04:32.351 --> 00:04:34.578
So if I look here, on cosine,

00:04:34.578 --> 00:04:37.544
if I want to know what that is,
in terms of a sine function,

00:04:37.544 --> 00:04:40.427
if I add 90 degrees to the argument,

00:04:40.427 --> 00:04:43.214
the sine function will
give me the same value.

00:04:43.214 --> 00:04:46.131
So what that says, is cosine theta,

00:04:49.476 --> 00:04:51.976
equals sine of theta, plus 90.

00:04:57.519 --> 00:04:59.122
So these are two
identities, we can use this

00:04:59.122 --> 00:05:02.349
to convert a something
expressed as a sine,

00:05:02.349 --> 00:05:04.766
into a cosine, or vice versa.

00:05:05.660 --> 00:05:07.910
Now I want to show you
two more identities,

00:05:07.910 --> 00:05:10.577
that are actually pretty useful.

00:05:12.597 --> 00:05:16.025
And here what I have is, I
sketched on in dash lines

00:05:16.025 --> 00:05:17.571
the negative of the orange curve,

00:05:17.571 --> 00:05:20.355
so this is a negative sine wave.

00:05:20.355 --> 00:05:21.621
You can see it the opposite

00:05:21.621 --> 00:05:24.760
of the original sine wave we had.

00:05:24.760 --> 00:05:28.408
So now I have the case
here where the cosine

00:05:28.408 --> 00:05:32.480
is trailing, or lagging,
the negative sine.

00:05:32.480 --> 00:05:36.061
Right, it comes later
in time, right there.

00:05:36.061 --> 00:05:38.478
So cosine lags negative sine.

00:05:39.671 --> 00:05:40.517
And what I'll do is, I'll write

00:05:40.517 --> 00:05:42.363
the same sort of identities here,

00:05:42.363 --> 00:05:45.403
but in terms of this negative sine,

00:05:45.403 --> 00:05:47.820
and those come out like this.

00:05:48.877 --> 00:05:50.210
Cosine of theta,

00:05:51.300 --> 00:05:52.800
equals minus sine,

00:05:54.533 --> 00:05:56.616
so that's the dashed one,

00:05:57.480 --> 00:05:59.730
of theta, minus 90 degrees.

00:06:03.472 --> 00:06:07.639
So what that means, if I want
to know the value of cosine,

00:06:09.976 --> 00:06:11.785
and I can flip that around the same way,

00:06:11.785 --> 00:06:13.285
and I can say that

00:06:15.811 --> 00:06:16.894
negative sine

00:06:18.832 --> 00:06:19.665
of theta

00:06:22.089 --> 00:06:23.172
equals cosine

00:06:25.282 --> 00:06:26.699
of theta plus 90.

00:06:30.547 --> 00:06:32.903
That's the same identity but in reverse.

00:06:32.903 --> 00:06:36.269
If I want to know the
value of negative sine,

00:06:36.269 --> 00:06:37.969
I just take that argument,

00:06:37.969 --> 00:06:41.052
I add it, I advance it by 90 degrees,

00:06:42.081 --> 00:06:44.755
and take the cosine,
it'll have the same value.

00:06:44.755 --> 00:06:46.172
So, this identity

00:06:48.158 --> 00:06:48.991
and,

00:06:51.442 --> 00:06:53.176
this identity,

00:06:53.176 --> 00:06:54.537
are pretty useful to have around.

00:06:54.537 --> 00:06:57.932
This one allows us to convert
sines and cosines together.

00:06:57.932 --> 00:07:02.537
This pair here is useful for
moving minus sines around.

00:07:02.537 --> 00:07:04.893
So that's what I wanted
to say about lead and lag.

00:07:04.893 --> 00:07:06.980
These are sort of slang or jargon,

00:07:06.980 --> 00:07:09.513
the nicknames of the relationship between

00:07:09.513 --> 00:07:12.017
two different wave forms
of the same frequency

00:07:12.017 --> 00:07:15.292
but different phase timing,
different phase delay.

00:07:15.292 --> 00:07:17.678
And then we worked out
some, some identities

00:07:17.678 --> 00:07:20.008
that are kind of useful to have around,

00:07:20.008 --> 00:07:23.166
to be able to convert these two wave forms

00:07:23.166 --> 00:07:24.937
from one to the other and back.


AC analysis intro 2

WEBVTT
Kind: captions
Language: en

00:00:01.109 --> 00:00:02.313
- [Voiceover] So in the last video

00:00:02.313 --> 00:00:03.937
we started working on the analysis

00:00:03.937 --> 00:00:07.431
of an RLC circuit that
had a forcing function,

00:00:07.431 --> 00:00:10.075
and the math for doing
that gets really hard.

00:00:10.075 --> 00:00:12.578
And so what we decided to
do was see what happens

00:00:12.578 --> 00:00:16.249
if we limit ourselves to
using just sinusoidal inputs,

00:00:16.249 --> 00:00:19.191
inputs that look like sines and cosines.

00:00:19.191 --> 00:00:20.824
So I wanna continue the introduction

00:00:20.824 --> 00:00:23.346
to the sinusoidal analysis technique

00:00:23.346 --> 00:00:27.513
and just give you a preview of
where we're headed with that.

00:00:30.740 --> 00:00:33.996
So when we make this limitation
to sinusoidal inputs,

00:00:33.996 --> 00:00:36.579
there's a big prize at the end.

00:00:37.698 --> 00:00:41.698
And the prize is that the
differential equations

00:00:43.330 --> 00:00:44.830
turn into algebra.

00:00:49.398 --> 00:00:50.942
That's the reason we're doing this.

00:00:50.942 --> 00:00:53.375
It just basically because really simple,

00:00:53.375 --> 00:00:55.626
just like the resistor
circuits that we used to do,

00:00:55.626 --> 00:00:56.693
they were all algebra.

00:00:56.693 --> 00:00:57.982
There was no calculus.

00:00:57.982 --> 00:01:00.246
We're gonna turn these
kinds of differential

00:01:00.246 --> 00:01:03.102
equation circuits into algebra.

00:01:03.102 --> 00:01:05.159
So if we're able to convert this circuit

00:01:05.159 --> 00:01:08.377
into an algebra problem instead
of a differential equation,

00:01:08.377 --> 00:01:11.369
that means we could use
Kirchhoff's Voltage Law.

00:01:11.369 --> 00:01:13.732
We could use Kirchhoff's Current Law.

00:01:13.732 --> 00:01:15.899
We could use node voltage.

00:01:19.541 --> 00:01:23.708
The node voltage method or
the mesh current method.

00:01:24.792 --> 00:01:26.639
Just like we did for resistors.

00:01:26.639 --> 00:01:29.133
And this whole set of techniques then

00:01:29.133 --> 00:01:31.186
automatically gets applied to circuits

00:01:31.186 --> 00:01:33.378
that have inductors
and capacitors in them,

00:01:33.378 --> 00:01:35.506
just like we learned how
to do with resistors.

00:01:35.506 --> 00:01:37.733
It's a major simplification.

00:01:37.733 --> 00:01:39.769
So let me draw this circuit
over here again real quick,

00:01:39.769 --> 00:01:42.006
the one we were looking at earlier.

00:01:42.006 --> 00:01:42.839
We have V.

00:01:44.037 --> 00:01:46.120
We have an inductor here.

00:01:47.098 --> 00:01:50.015
We have a resistor and a capacitor.

00:01:52.692 --> 00:01:55.808
So for V, in, we're gonna limit ourselves

00:01:55.808 --> 00:01:57.938
to just sinusoids.

00:01:57.938 --> 00:01:59.884
So that means that the input
is gonna look something

00:01:59.884 --> 00:02:02.551
like A cosine, omega t plus phi.

00:02:07.065 --> 00:02:09.384
Phi is a phase angle.

00:02:09.384 --> 00:02:11.517
And we're gonna represent this in a way

00:02:11.517 --> 00:02:12.350
that looks like this.

00:02:12.350 --> 00:02:13.989
This is gonna get transformed or changed

00:02:13.989 --> 00:02:16.478
into something that looks like this.

00:02:16.478 --> 00:02:19.740
It's gonna be called
A at the angle of phi.

00:02:19.740 --> 00:02:21.740
This is an angle symbol.

00:02:22.927 --> 00:02:25.442
And this is referred to as a phasor.

00:02:25.442 --> 00:02:28.442
This has a name this way of writing.

00:02:31.004 --> 00:02:34.182
This way of writing down sinusoids.

00:02:34.182 --> 00:02:35.556
If I have a sinusoid that looks like

00:02:35.556 --> 00:02:37.012
this is a function of
time, I could write it

00:02:37.012 --> 00:02:41.179
as a phasor where I say V
equals A at an angle of phi

00:02:43.492 --> 00:02:46.342
and understood is that
there's this omega t term,

00:02:46.342 --> 00:02:49.089
this cosine omega t term nearby.

00:02:49.089 --> 00:02:50.357
So the other thing we're
gonna learn about is

00:02:50.357 --> 00:02:52.607
how to transform a circuit.

00:02:56.917 --> 00:03:00.723
So we can use this sinusoidal
steady state analysis.

00:03:00.723 --> 00:03:03.554
So the inductor, it
gets transformed from L,

00:03:03.554 --> 00:03:06.011
instead we write down SL,

00:03:06.011 --> 00:03:08.820
where S is that same natural frequency.

00:03:08.820 --> 00:03:11.878
Whenever we have a resistor,
we write down just R,

00:03:11.878 --> 00:03:15.550
just like we usually do, and
whenever we have a capacitor,

00:03:15.550 --> 00:03:18.217
we write one over SC, and again,

00:03:19.773 --> 00:03:22.387
this S is the same thing as we had before,

00:03:22.387 --> 00:03:24.220
the natural frequency.

00:03:26.812 --> 00:03:29.421
And in a future video, we'll justify

00:03:29.421 --> 00:03:33.194
why we could make this
transformation and what this means.

00:03:33.194 --> 00:03:35.938
The big payoff here is I'm
gonna write a KVL equation

00:03:35.938 --> 00:03:38.286
around this loop and watch what happens.

00:03:38.286 --> 00:03:41.509
Watch how easy this is, it's amazing.

00:03:41.509 --> 00:03:43.418
So let me real quick I'm gonna just

00:03:43.418 --> 00:03:45.190
indicate signs of these voltages.

00:03:45.190 --> 00:03:46.943
There's the inductor voltage.

00:03:46.943 --> 00:03:48.662
There's the resistor voltage.

00:03:48.662 --> 00:03:50.672
There's the capacitor voltage.

00:03:50.672 --> 00:03:52.998
And here's the voltage on V, in.

00:03:52.998 --> 00:03:55.107
Like that, we'll give it that polarity.

00:03:55.107 --> 00:03:58.088
It's not obvious yet, but I
get to use these quantities,

00:03:58.088 --> 00:04:00.237
SL and one over SC,

00:04:00.237 --> 00:04:03.103
just like they were a
resistance value in Ohm's Law.

00:04:03.103 --> 00:04:04.811
And watch how this happens.

00:04:04.811 --> 00:04:08.795
I'm just gonna write KVL around this loop.

00:04:08.795 --> 00:04:11.128
And what I get is that V, in

00:04:13.928 --> 00:04:17.295
is equal to the voltage
across the inductor

00:04:17.295 --> 00:04:20.173
plus the resistor plus the capacitor,

00:04:20.173 --> 00:04:21.420
and I can write it like this.

00:04:21.420 --> 00:04:24.503
I can write SL times i plus R times i

00:04:29.419 --> 00:04:31.502
plus one over SC times i.

00:04:36.337 --> 00:04:37.861
And if I can write that again,

00:04:37.861 --> 00:04:40.377
I'll write that one more time,

00:04:40.377 --> 00:04:42.210
it's i times SL plus R

00:04:48.266 --> 00:04:49.683
plus one over SC.

00:04:52.620 --> 00:04:54.365
All right, that was a
straightforward application

00:04:54.365 --> 00:04:55.565
of Kirchhoff's Law.

00:04:55.565 --> 00:04:56.866
Now if we look at this expression here,

00:04:56.866 --> 00:04:58.586
look at this right here,

00:04:58.586 --> 00:05:00.533
this is the characteristic equation.

00:05:00.533 --> 00:05:04.450
We just wrote down the
characteristic equation.

00:05:07.925 --> 00:05:09.175
Characteristic.

00:05:14.837 --> 00:05:16.534
We just wrote down the
characteristic equation

00:05:16.534 --> 00:05:20.701
of this circuit using these
transformed components.

00:05:22.448 --> 00:05:23.696
Now what I wanna do next,

00:05:23.696 --> 00:05:26.369
we're gonna actually get a new concept.

00:05:26.369 --> 00:05:27.666
I can write an equation like this.

00:05:27.666 --> 00:05:30.166
I can say V, in, divided by i,

00:05:31.554 --> 00:05:34.821
I'm just gonna take i over
this side of the equation here,

00:05:34.821 --> 00:05:37.654
equals SL plus R plus one over SC.

00:05:43.040 --> 00:05:44.217
This is an interesting idea.

00:05:44.217 --> 00:05:46.592
Here is a ratio, right here.

00:05:46.592 --> 00:05:49.938
This is a ratio of voltage to current.

00:05:49.938 --> 00:05:53.016
Now if this was just a plain resistor,

00:05:53.016 --> 00:05:57.598
V over i for a plain ole
resistor, is what, is R.

00:05:57.598 --> 00:06:00.806
That's an expression of Ohm's Law.

00:06:00.806 --> 00:06:03.986
So now I have another expression over here

00:06:03.986 --> 00:06:05.547
for something that's written in terms

00:06:05.547 --> 00:06:08.735
of my component values and this natural,

00:06:08.735 --> 00:06:12.138
this frequency S that's going on in here,

00:06:12.138 --> 00:06:15.771
and this is gonna lead us to
a general idea of resistance

00:06:15.771 --> 00:06:17.854
that is called impedance.

00:06:23.781 --> 00:06:24.999
So that's what this is right here.

00:06:24.999 --> 00:06:27.662
This is this ratio of voltage to current

00:06:27.662 --> 00:06:31.745
and the symbol usually
used for impedance is a Z.

00:06:34.137 --> 00:06:37.419
So this is where we're headed
over the next several videos.

00:06:37.419 --> 00:06:38.993
To justify what we're doing here,

00:06:38.993 --> 00:06:40.298
we need to go through some steps,

00:06:40.298 --> 00:06:42.503
and so what we're gonna do
in the next couple of videos

00:06:42.503 --> 00:06:45.003
is we're gonna do some review.

00:06:47.872 --> 00:06:49.423
So here are the things we're gonna review.

00:06:49.423 --> 00:06:52.506
We're gonna review some trigonometry.

00:06:53.729 --> 00:06:56.896
So cosine and sine and those functions

00:06:57.905 --> 00:06:59.272
and what they mean, especially

00:06:59.272 --> 00:07:01.505
when they're functions of time.

00:07:01.505 --> 00:07:04.922
We're also gonna review Euler's identity.

00:07:12.067 --> 00:07:13.431
Euler's identity is important,

00:07:13.431 --> 00:07:15.340
because it's the thing that allows us

00:07:15.340 --> 00:07:20.266
to relate e to the jx and we
get some sort of relationship

00:07:20.266 --> 00:07:22.933
to sine of x and cosine of x.

00:07:27.856 --> 00:07:29.457
And if you remember when we were solving

00:07:29.457 --> 00:07:32.697
differential equations,
this was always the form

00:07:32.697 --> 00:07:34.467
that was the easiest solution

00:07:34.467 --> 00:07:36.824
to come up with, e to the something.

00:07:36.824 --> 00:07:39.468
And if we're limiting
ourselves to sines and cosines

00:07:39.468 --> 00:07:42.428
for inputs, we need to have a way to make

00:07:42.428 --> 00:07:44.415
a really easy way to solve equations.

00:07:44.415 --> 00:07:47.141
So Euler's identity is the trigger

00:07:47.141 --> 00:07:49.308
that allows us to do that.

00:07:50.620 --> 00:07:52.492
Now when we use Euler's identity,

00:07:52.492 --> 00:07:54.105
we're gonna get this little complex number

00:07:54.105 --> 00:07:55.293
that keeps coming up.

00:07:55.293 --> 00:07:58.460
So we're gonna review complex numbers.

00:08:06.987 --> 00:08:08.487
That's the three review topics.

00:08:08.487 --> 00:08:12.931
And then we're gonna move
on and we're gonna do

00:08:12.931 --> 00:08:17.098
after that, we'll define
something like these phasors.

00:08:19.322 --> 00:08:22.489
Then we'll look at the transformation.

00:08:28.238 --> 00:08:31.155
So that's SL and R and one over SC.

00:08:34.919 --> 00:08:38.143
Phasor is the idea
where we change a cosine

00:08:38.143 --> 00:08:41.149
into something at a phase angle.

00:08:41.149 --> 00:08:45.316
And then finally what we get
to do is we get to solve.

00:08:48.018 --> 00:08:49.058
So that's the sequence of events.

00:08:49.058 --> 00:08:52.231
That's what's coming up over
the next couple of videos.

00:08:52.231 --> 00:08:53.514
It's a really powerful technique

00:08:53.514 --> 00:08:56.180
for handling some very
complicated circuits

00:08:56.180 --> 00:08:57.677
and getting it to do what we want.


AC analysis intro 1

WEBVTT
Kind: captions
Language: en

00:00:00.543 --> 00:00:01.891
- [Voiceover] We now
begin a whole new area

00:00:01.891 --> 00:00:06.811
of circuit analysis called
sinusoidal steady state analysis.

00:00:06.811 --> 00:00:09.561
You can also call it AC analysis.

00:00:10.478 --> 00:00:12.718
AC stands for alternating current.

00:00:12.718 --> 00:00:15.341
It means it is a voltage or a current that

00:00:15.341 --> 00:00:17.507
where the signal actually changes sign.

00:00:17.507 --> 00:00:18.518
It is positive sometimes.

00:00:18.518 --> 00:00:20.378
It is negative sometimes.

00:00:20.378 --> 00:00:22.696
And the conventional name for that is AC

00:00:22.696 --> 00:00:24.263
or alternating current.

00:00:24.263 --> 00:00:26.591
It could've been called
alternating voltage

00:00:26.591 --> 00:00:27.663
but that's not the name.

00:00:27.663 --> 00:00:28.913
The name is AC.

00:00:29.899 --> 00:00:31.065
What I want to do in this video

00:00:31.065 --> 00:00:32.840
is we will do a quick review

00:00:32.840 --> 00:00:34.881
of what it is like to solve an equation

00:00:34.881 --> 00:00:35.987
like the one showed here.

00:00:35.987 --> 00:00:38.194
This is an RLC circuit.

00:00:38.194 --> 00:00:39.466
I'm going to show what
it is like to solve this

00:00:39.466 --> 00:00:41.321
in differential equation form,

00:00:41.321 --> 00:00:43.578
which is gonna be a lot of work.

00:00:43.578 --> 00:00:45.371
I want to introduce the idea

00:00:45.371 --> 00:00:49.556
of a new point of view
or a new analysis method

00:00:49.556 --> 00:00:53.258
that we refer to as the
sinusoidal steady state.

00:00:53.258 --> 00:00:55.246
And it is a transformation

00:00:55.246 --> 00:00:57.167
that we are gonna do on this circuit.

00:00:57.167 --> 00:00:59.357
And it is gonna be a
big reward at the end.

00:00:59.357 --> 00:01:01.265
I want to go through what's the reward

00:01:01.265 --> 00:01:03.697
and what's it gonna look like at the end.

00:01:03.697 --> 00:01:04.831
Then we are gonna know some of the math

00:01:04.831 --> 00:01:07.866
that we have to review in
order to fully understand

00:01:07.866 --> 00:01:09.306
this change that we are gonna go through.

00:01:09.306 --> 00:01:12.028
This change of point of view.

00:01:12.028 --> 00:01:13.797
Let's first take a look
at this circuit here.

00:01:13.797 --> 00:01:17.158
This is a circuit that's
now a driven RLC circuit.

00:01:17.158 --> 00:01:18.378
So here's the drive function.

00:01:18.378 --> 00:01:19.656
It is a voltage.

00:01:19.656 --> 00:01:20.979
It is some waveform.

00:01:20.979 --> 00:01:23.025
It is driving a sequence of an inductor,

00:01:23.025 --> 00:01:25.680
a resistor, and a capacitor.

00:01:25.680 --> 00:01:26.918
Now in an earlier video

00:01:26.918 --> 00:01:29.177
we derived the natural response

00:01:29.177 --> 00:01:30.427
of this circuit

00:01:31.273 --> 00:01:33.889
and to do that, we shorted out

00:01:33.889 --> 00:01:35.028
we removed the source

00:01:35.028 --> 00:01:36.054
and shorted it out

00:01:36.054 --> 00:01:38.841
and added a little bit
of energy to this circuit

00:01:38.841 --> 00:01:40.616
and saw what it did on its own.

00:01:40.616 --> 00:01:42.815
What was its natural response.

00:01:42.815 --> 00:01:43.894
Now we've upgraded this.

00:01:43.894 --> 00:01:45.246
We've added a source

00:01:45.246 --> 00:01:46.834
and now we have to solve this again

00:01:46.834 --> 00:01:48.716
including the source.

00:01:48.716 --> 00:01:51.618
If we use the differential
equation technique

00:01:51.618 --> 00:01:53.316
this is how we are gonna go about it.

00:01:53.316 --> 00:01:55.898
So the first step in a
circuit analysis like this

00:01:55.898 --> 00:01:58.148
is to write a KVL equation.

00:01:59.192 --> 00:02:01.704
We are gonna solve for
this current right here.

00:02:01.704 --> 00:02:03.567
That's the one current that's in this.

00:02:03.567 --> 00:02:05.846
So I is our independent variable.

00:02:05.846 --> 00:02:08.745
If I write KVL as you recall

00:02:08.745 --> 00:02:11.696
when we did this for natural response,

00:02:11.696 --> 00:02:13.481
we ended up with a differential equation

00:02:13.481 --> 00:02:15.535
that looked like this.

00:02:15.535 --> 00:02:18.952
We had L times the second derivative of I

00:02:23.974 --> 00:02:27.141
plus R times the first derivative of I

00:02:31.322 --> 00:02:33.322
plus one over C times I.

00:02:38.397 --> 00:02:39.457
So these are the voltages.

00:02:39.457 --> 00:02:41.225
Each of these individual terms

00:02:41.225 --> 00:02:45.058
are the voltages across
these components here.

00:02:46.207 --> 00:02:48.790
So that's the resistor voltage.

00:02:49.821 --> 00:02:52.054
This is the capacitor voltage

00:02:52.054 --> 00:02:53.579
and this here is the inductor voltage

00:02:53.579 --> 00:02:55.445
so it is inductor voltage,

00:02:55.445 --> 00:02:56.323
resistor voltage,

00:02:56.323 --> 00:02:57.943
capacitor voltage.

00:02:57.943 --> 00:02:59.854
And all those if we add those up,

00:02:59.854 --> 00:03:01.771
those have to equal VN.

00:03:04.530 --> 00:03:07.905
So this is now a forced equation,

00:03:07.905 --> 00:03:10.317
which means this is the forcing function

00:03:10.317 --> 00:03:11.654
and we are gonna have to solve this

00:03:11.654 --> 00:03:13.032
and the math for doing this is

00:03:13.032 --> 00:03:14.204
pretty difficult.

00:03:14.204 --> 00:03:16.500
It was hard enough to
do the natural response

00:03:16.500 --> 00:03:18.271
and we add in this

00:03:18.271 --> 00:03:21.021
and it gets to be even more work.

00:03:22.727 --> 00:03:24.175
So as we did before,

00:03:24.175 --> 00:03:26.822
what we do now is we propose a solution

00:03:26.822 --> 00:03:30.445
and the solution we are in
the habit of doing this now

00:03:30.445 --> 00:03:32.930
is gonna be sum constant times E

00:03:32.930 --> 00:03:36.013
to the sum natural frequency times T.

00:03:38.458 --> 00:03:40.520
So AE to the ST

00:03:40.520 --> 00:03:45.114
is our proposed solution
for I as a function of time.

00:03:45.114 --> 00:03:46.823
You remember we called S.

00:03:46.823 --> 00:03:47.991
S is a frequency term

00:03:47.991 --> 00:03:51.322
because S times T has to have no units

00:03:51.322 --> 00:03:55.438
so S has units of one
over time or frequency

00:03:55.438 --> 00:03:58.688
so that's called the natural frequency.

00:04:04.390 --> 00:04:06.052
And when we plug in I

00:04:06.052 --> 00:04:08.828
the way to tell if I is a solution

00:04:08.828 --> 00:04:12.172
is to plug this into this equation here.

00:04:12.172 --> 00:04:13.744
And we got an equation
that looks like this.

00:04:13.744 --> 00:04:16.806
We ended up after factoring our I

00:04:16.806 --> 00:04:18.139
we ended up with

00:04:20.258 --> 00:04:21.758
LS squared plus RS

00:04:24.538 --> 00:04:25.788
plus one over C

00:04:29.093 --> 00:04:31.369
and all that is equal to

00:04:31.369 --> 00:04:33.369
for the natural response

00:04:35.544 --> 00:04:36.377
we put in,

00:04:40.207 --> 00:04:42.691
so we solve this equation by setting

00:04:42.691 --> 00:04:44.788
this term here equal to zero

00:04:44.788 --> 00:04:45.996
and solving for s

00:04:45.996 --> 00:04:48.267
to find out what the natural frequency is

00:04:48.267 --> 00:04:50.571
and then we go back and we find out A

00:04:50.571 --> 00:04:52.874
by looking at the initial
conditions over here.

00:04:52.874 --> 00:04:55.166
Whatever initial energy
was in this circuit

00:04:55.166 --> 00:04:57.452
determines the value of A here.

00:04:57.452 --> 00:04:59.298
The next step in this forced response

00:04:59.298 --> 00:05:01.224
where VN is driving the circuit

00:05:01.224 --> 00:05:03.974
is we have to set this back to VN

00:05:05.249 --> 00:05:07.332
and solve for the forced.

00:05:13.289 --> 00:05:17.404
Now if we let VN be sort of
any forcing function we want,

00:05:17.404 --> 00:05:18.932
any kind of waveform,

00:05:18.932 --> 00:05:22.413
this is gonna be a really
hard piece of mathematics.

00:05:22.413 --> 00:05:24.412
This is gonna be a really
difficult calculation.

00:05:24.412 --> 00:05:26.855
It is gonna take a long time and basically

00:05:26.855 --> 00:05:28.202
I don't want to do it.

00:05:28.202 --> 00:05:29.650
I'm gonna wish there was
some other way to do these

00:05:29.650 --> 00:05:32.212
kinds of equations and there is.

00:05:32.212 --> 00:05:35.409
The way we simplify this
process substantially

00:05:35.409 --> 00:05:37.896
is we make a little
limitation on ourselves

00:05:37.896 --> 00:05:39.396
on what VN can be.

00:05:40.515 --> 00:05:41.549
If we make a rule,

00:05:41.549 --> 00:05:45.219
if we basically volunteer
to limit ourselves

00:05:45.219 --> 00:05:47.052
to VN equals sinusoids

00:05:53.057 --> 00:05:55.867
that means that VN is of the form

00:05:55.867 --> 00:05:58.034
cosine of omega T plus phi

00:06:01.079 --> 00:06:02.771
where phi is some angle

00:06:02.771 --> 00:06:04.354
or it could be sine

00:06:06.211 --> 00:06:08.461
of omega T plus some angle.

00:06:10.795 --> 00:06:13.295
Any waveform of this form here

00:06:14.822 --> 00:06:16.768
is called the sinusoid.

00:06:16.768 --> 00:06:20.682
So sinusoid is the general
name for cosine and sine

00:06:20.682 --> 00:06:23.600
and signals that look like that.

00:06:23.600 --> 00:06:25.972
Because we don't want this
math to blow up on us,

00:06:25.972 --> 00:06:29.184
with the general input over here,

00:06:29.184 --> 00:06:30.780
we are gonna develop a really elegant way

00:06:30.780 --> 00:06:32.238
to solve circuits with this

00:06:32.238 --> 00:06:35.685
we limit ourselves to sinusoidal inputs.

00:06:35.685 --> 00:06:37.044
We will take a pause right here

00:06:37.044 --> 00:06:38.666
and continue on in the next video

00:06:38.666 --> 00:06:42.003
to introduce the idea
of sinusoidal analysis.


Trigonometry review

WEBVTT
Kind: captions
Language: en

00:00:00.651 --> 00:00:02.161
- [Voiceover] I want
to do a quick overview

00:00:02.161 --> 00:00:05.578
of trigonometry, and
aspects of trig functions

00:00:05.578 --> 00:00:08.432
that are important to us
as electrical engineers.

00:00:08.432 --> 00:00:11.429
So this isn't meant to be a
full class on trigonometry.

00:00:11.429 --> 00:00:13.430
If you haven't had this subject before,

00:00:13.430 --> 00:00:15.971
this is something you
can study on Khan Academy,

00:00:15.971 --> 00:00:18.853
and Sal does a lot of good
videos on trig functions

00:00:18.853 --> 00:00:20.886
and how they work.

00:00:20.886 --> 00:00:23.887
So the way I remember my trig functions,

00:00:23.887 --> 00:00:25.387
is with the phrase

00:00:26.716 --> 00:00:27.549
Soh cah

00:00:31.011 --> 00:00:31.844
toa.

00:00:35.264 --> 00:00:37.191
So this is the little
phrase I use to remember

00:00:37.191 --> 00:00:39.170
how to do my trig functions.

00:00:39.170 --> 00:00:41.150
And we draw a triangle, like this,

00:00:41.150 --> 00:00:42.567
a right triangle,

00:00:44.256 --> 00:00:47.364
with an angle here of theta.

00:00:47.364 --> 00:00:50.779
And we label the sides of the triangle as,

00:00:50.779 --> 00:00:53.856
this is the side adjacent to theta,

00:00:53.856 --> 00:00:56.856
this is the side, opposite of theta.

00:00:58.014 --> 00:01:02.676
And this side over here is the
hypotenuse of the triangle.

00:01:02.676 --> 00:01:06.093
So this says that the definition of sine,

00:01:07.187 --> 00:01:08.020
of theta,

00:01:08.905 --> 00:01:12.336
equals opposite over hypotenuse.

00:01:12.336 --> 00:01:14.419
Opposite over hypotenuse.

00:01:15.772 --> 00:01:19.439
This phrase here means
the cosine, of theta,

00:01:20.806 --> 00:01:23.473
equals adjacent over hypotenuse.

00:01:27.758 --> 00:01:30.210
And the last one here is for the tangent.

00:01:30.210 --> 00:01:32.627
It says the tangent of theta,

00:01:35.042 --> 00:01:37.857
equals opposite over adjacent.

00:01:37.857 --> 00:01:40.369
Opposite divided by adjacent.

00:01:40.369 --> 00:01:42.286
Opposite over adjacent.

00:01:43.968 --> 00:01:48.135
So, soh cah toa helps you
remember your trig functions.

00:01:49.115 --> 00:01:51.796
So lets take that idea over here,

00:01:51.796 --> 00:01:55.796
and draw a line out, and
make some calculations.

00:01:58.420 --> 00:02:01.993
We have a graph here of a unit circle.

00:02:01.993 --> 00:02:05.222
That means the radius of
this is one everywhere.

00:02:05.222 --> 00:02:06.634
And what I want to know is,

00:02:06.634 --> 00:02:09.128
here's my angle theta.

00:02:09.128 --> 00:02:13.211
And angles are measured
from the positive X axis.

00:02:14.910 --> 00:02:16.674
Here's the X axis.

00:02:16.674 --> 00:02:18.174
Here's the Y axis.

00:02:19.835 --> 00:02:23.418
Angles are measured
going counterclockwise.

00:02:24.667 --> 00:02:27.665
So let's talk for a second
about how angles are measured.

00:02:27.665 --> 00:02:29.131
Angles are measured in two ways.

00:02:29.131 --> 00:02:31.714
Angles are measured in degrees.

00:02:35.834 --> 00:02:37.251
From zero to 360.

00:02:39.933 --> 00:02:44.100
And angles are also measured
in something called radians.

00:02:46.566 --> 00:02:49.399
And that goes from zero to two pi.

00:02:50.922 --> 00:02:53.306
So these are two different angle measures.

00:02:53.306 --> 00:02:54.874
And when you're measuring in degrees,

00:02:54.874 --> 00:02:58.303
we put the little degree mark up here.

00:02:58.303 --> 00:02:59.136
That's what that means.

00:02:59.136 --> 00:03:03.004
Radians don't get a degree mark on them.

00:03:03.004 --> 00:03:05.706
So if I mark this out in degrees,

00:03:05.706 --> 00:03:07.695
here's zero degrees.

00:03:07.695 --> 00:03:09.195
Here's 90 degrees.

00:03:11.067 --> 00:03:12.650
Here's 180 degrees.

00:03:14.676 --> 00:03:15.676
This is 270.

00:03:18.709 --> 00:03:22.876
And when I get back to the
beginning, it's 360 degrees.

00:03:25.601 --> 00:03:27.428
If I measure the same angles in radians,

00:03:27.428 --> 00:03:29.833
this'll be zero radians.

00:03:29.833 --> 00:03:34.000
When I get back here it's
gonna be two pi radians.

00:03:36.318 --> 00:03:39.206
Going all the way around the
circle is two pi radians.

00:03:39.206 --> 00:03:40.227
That means that going half way

00:03:40.227 --> 00:03:42.894
around the circle is pi radians.

00:03:44.505 --> 00:03:47.397
That's equivalent to 180 degrees.

00:03:47.397 --> 00:03:49.861
If I do a quarter of a circle,

00:03:49.861 --> 00:03:52.944
that's equal to pi over two, radians.

00:03:54.796 --> 00:03:58.727
And if I do three quarters of a circle,

00:03:58.727 --> 00:04:00.894
that's three halves pi, or

00:04:02.190 --> 00:04:03.690
three pi over two.

00:04:06.115 --> 00:04:08.418
So, we'll use degrees
and radians all the time,

00:04:08.418 --> 00:04:12.062
and we'll flip back
and forth between them.

00:04:12.062 --> 00:04:14.029
So, now let's do some trig functions,

00:04:14.029 --> 00:04:17.244
on our angle theta right in here.

00:04:17.244 --> 00:04:20.347
Let's work out the sine,
cosine, and tangent.

00:04:20.347 --> 00:04:22.739
Now let me give a name to this hypotenuse.

00:04:22.739 --> 00:04:24.239
Let's call that R.

00:04:25.353 --> 00:04:27.247
And R equals one, right?

00:04:27.247 --> 00:04:30.937
I said this was a unit
circle, so R is equal to one.

00:04:30.937 --> 00:04:32.956
And when we calculate our right triangle,

00:04:32.956 --> 00:04:36.445
what we do is we drop a
perpendicular down here,

00:04:36.445 --> 00:04:38.101
to the X axis.

00:04:38.101 --> 00:04:42.268
And we also draw a horizontal
over here from the Y axis.

00:04:44.427 --> 00:04:47.637
This side right here,
of the X axis right here

00:04:47.637 --> 00:04:50.556
is the side adjacent to the angle theta.

00:04:50.556 --> 00:04:54.141
And this distance right here
on this side of our triangle,

00:04:54.141 --> 00:04:55.891
is the side opposite.

00:04:57.645 --> 00:05:02.200
Okay, and basically there's
gonna be a Y intercept here,

00:05:02.200 --> 00:05:05.071
and an X intercept right
here, where those happen.

00:05:05.071 --> 00:05:06.662
These'll be some number,

00:05:06.662 --> 00:05:09.995
depending on the tilt of this line here.

00:05:11.154 --> 00:05:12.237
So, the sine,

00:05:14.069 --> 00:05:16.552
of theta, is equal to what?

00:05:16.552 --> 00:05:17.552
Is equal to,

00:05:18.513 --> 00:05:19.886
let's look at our definition,

00:05:19.886 --> 00:05:22.218
it's equal to opposite over hypotenuse.

00:05:22.218 --> 00:05:23.385
Opposite is Y,

00:05:26.363 --> 00:05:29.622
over the hypotenuse which is R.

00:05:29.622 --> 00:05:32.047
If I look at cosine,

00:05:32.047 --> 00:05:32.880
theta,

00:05:34.938 --> 00:05:36.707
adjacent over hypotenuse.

00:05:36.707 --> 00:05:39.567
Adjacent is the X distance,

00:05:39.567 --> 00:05:42.567
X distance, and the hypotenuse is R.

00:05:45.605 --> 00:05:47.688
And if we do the tangent,

00:05:49.277 --> 00:05:50.110
of theta,

00:05:51.074 --> 00:05:52.170
that equals what?

00:05:52.170 --> 00:05:54.302
It's opposite over adjacent.

00:05:54.302 --> 00:05:57.802
So it's opposite, which is the Y distance,

00:05:59.935 --> 00:06:03.685
over X, the adjacent X,
X is the adjacent, X.

00:06:08.236 --> 00:06:09.814
One thing to notice here about tangent,

00:06:09.814 --> 00:06:10.647
Y over X

00:06:11.626 --> 00:06:12.459
is the

00:06:13.496 --> 00:06:15.962
rise divided by the run,

00:06:15.962 --> 00:06:19.193
going from this point, up to this point.

00:06:19.193 --> 00:06:20.943
So that is the slope.

00:06:25.499 --> 00:06:27.453
So the idea of slope
and the idea of tangent,

00:06:27.453 --> 00:06:29.703
are really closely related.

00:06:31.158 --> 00:06:32.277
Just as one small point,

00:06:32.277 --> 00:06:34.077
let's work out, what is one radian?

00:06:34.077 --> 00:06:37.854
What's an angle of one radian in degrees?

00:06:37.854 --> 00:06:40.587
I can do that conversion
just doing some units.

00:06:40.587 --> 00:06:42.504
If we have 180 degrees,

00:06:46.371 --> 00:06:48.371
that equals pi, radians.

00:06:51.486 --> 00:06:53.986
So that means that one radian,

00:06:55.760 --> 00:06:57.343
equals 180 over pi.

00:07:02.411 --> 00:07:04.335
If you plug that in the calculator,

00:07:04.335 --> 00:07:06.252
it'll come out to 57.3,

00:07:07.819 --> 00:07:09.236
roughly, degrees.

00:07:10.291 --> 00:07:13.865
So, one radian actually is
a little above 45 degrees.

00:07:13.865 --> 00:07:16.608
One radian is 57 degrees.

00:07:16.608 --> 00:07:18.691
It looks about like that.

00:07:19.736 --> 00:07:20.802
We don't use this very often.

00:07:20.802 --> 00:07:23.644
Mostly we talk about radians
in terms of multiples of pi,

00:07:23.644 --> 00:07:25.566
'cause it makes more sense on this circle.

00:07:25.566 --> 00:07:29.372
But, just to let you know,
that's roughly one radian.


Allopatric and sympatric speciation

WEBVTT
Kind: captions
Language: en

00:00:00.391 --> 00:00:02.285
- [Voiceover] In any discussion of biology

00:00:02.285 --> 00:00:03.924
or discussion of evolution,

00:00:03.924 --> 00:00:08.395
the idea of a species will
come up over and over again.

00:00:08.395 --> 00:00:11.399
And we have a whole
separate video on species.

00:00:11.399 --> 00:00:12.859
But the general idea,

00:00:12.859 --> 00:00:14.794
or the mainstream definition of a species

00:00:14.794 --> 00:00:17.963
is a group of organisms
that can interbreed,

00:00:17.963 --> 00:00:21.463
interbreed, and produce fertile offspring,

00:00:22.414 --> 00:00:24.664
fertile, fertile offspring.

00:00:26.251 --> 00:00:28.904
So for example, in this
picture right over here,

00:00:28.904 --> 00:00:30.756
you have a bunch of species

00:00:30.756 --> 00:00:34.227
of both modern elephants and previous,

00:00:34.227 --> 00:00:36.365
or now nonexistent species,

00:00:36.365 --> 00:00:38.122
that are related to modern elephants.

00:00:38.122 --> 00:00:41.111
But today on earth you
have Asian elephants

00:00:41.111 --> 00:00:42.713
and you have African elephants,

00:00:42.713 --> 00:00:44.732
and they are each a species.

00:00:44.732 --> 00:00:46.151
An Asian elephant can interbreed

00:00:46.151 --> 00:00:48.734
and produce fertile offspring
with another Asian elephant,

00:00:48.734 --> 00:00:50.488
and an African elephant can interbreed

00:00:50.488 --> 00:00:52.909
and produce fertile offspring
with another African elephant,

00:00:52.909 --> 00:00:54.595
but they can't do it with each other.

00:00:54.595 --> 00:00:55.909
An Asian elephant and an African elephant

00:00:55.909 --> 00:00:57.693
cannot get together and interbreed

00:00:57.693 --> 00:00:59.393
to produce fertile offspring.

00:00:59.393 --> 00:01:02.616
We know, people have actually tried this.

00:01:02.616 --> 00:01:04.486
But the next question, or
the most obvious question,

00:01:04.486 --> 00:01:07.487
and this is one of the central
questions of evolution,

00:01:07.487 --> 00:01:10.181
is, well, how do you get these species?

00:01:10.181 --> 00:01:12.435
We see drawings like we have on the right,

00:01:12.435 --> 00:01:13.595
on the left here we have actually

00:01:13.595 --> 00:01:15.320
some of Darwin's original drawings

00:01:15.320 --> 00:01:17.495
showing this evolutionary tree,

00:01:17.495 --> 00:01:18.961
showing how over and over again

00:01:18.961 --> 00:01:22.996
we have this branching from
a parent species into two,

00:01:22.996 --> 00:01:24.819
I guess you could say
different child species.

00:01:24.819 --> 00:01:28.289
You see this here with the elephants.

00:01:28.289 --> 00:01:31.582
At some point, the Asian
elephant and the African elephant

00:01:31.582 --> 00:01:33.166
shared a common ancestor,

00:01:33.166 --> 00:01:34.595
and it was also, based on this diagram,

00:01:34.595 --> 00:01:36.012
a common ancestor

00:01:38.311 --> 00:01:39.144
of the mammoth.

00:01:39.144 --> 00:01:41.097
And you go even further back,

00:01:41.097 --> 00:01:43.870
it's the common ancestor of this species

00:01:43.870 --> 00:01:46.059
that I'm not familiar with, the Anancus.

00:01:46.059 --> 00:01:46.930
And you can keep going back.

00:01:46.930 --> 00:01:48.809
But how does this tree branch?

00:01:48.809 --> 00:01:50.798
How do you actually get speciation?

00:01:50.798 --> 00:01:53.916
How does the variation
within a population,

00:01:53.916 --> 00:01:55.674
within a species get so extreme

00:01:55.674 --> 00:01:58.282
and in some ways so
separate from each other

00:01:58.282 --> 00:02:00.236
that they can no longer interbreed

00:02:00.236 --> 00:02:02.740
and produce fertile offspring?

00:02:02.740 --> 00:02:05.186
Well, there's a couple of
ways to think about it.

00:02:05.186 --> 00:02:07.517
The most obvious way that you
could imagine this happens,

00:02:07.517 --> 00:02:09.059
or maybe the most intuitive way

00:02:09.059 --> 00:02:10.336
that you could imagine this happening,

00:02:10.336 --> 00:02:13.086
is through geographic separation.

00:02:15.222 --> 00:02:17.924
And the technical term for speciation,

00:02:17.924 --> 00:02:20.126
which is the formation of new species,

00:02:20.126 --> 00:02:23.450
so speciation, actually, let
me just write it this way,

00:02:23.450 --> 00:02:25.827
the technical term for speciation

00:02:25.827 --> 00:02:29.972
due to geographic
separation is allopatric.

00:02:29.972 --> 00:02:31.805
Allopatric speciation.

00:02:34.856 --> 00:02:36.421
So speciation is just how,

00:02:36.421 --> 00:02:39.093
the formation of new species.

00:02:39.093 --> 00:02:42.193
And "allo" comes from the word "other,"

00:02:42.193 --> 00:02:46.504
and "patric" comes from the
root or the word "homeland."

00:02:46.504 --> 00:02:48.746
So it's really talking
about other geographies,

00:02:48.746 --> 00:02:52.044
or other homelands, or
geographic separation.

00:02:52.044 --> 00:02:54.760
And one commonly cited example here

00:02:54.760 --> 00:02:57.010
are the antelope squirrels.

00:02:57.990 --> 00:02:59.307
So if you go to the,

00:02:59.307 --> 00:03:01.296
if you were to go to
the American Southwest

00:03:01.296 --> 00:03:05.127
a long time ago, before the
Grand Canyon was a canyon,

00:03:05.127 --> 00:03:08.947
when the Colorado River was
just kinda going through

00:03:08.947 --> 00:03:10.716
and wasn't a major barrier,

00:03:10.716 --> 00:03:12.221
there was a parent species,

00:03:12.221 --> 00:03:15.002
an ancestral species to
both of these characters

00:03:15.002 --> 00:03:17.312
that lived on both sides of the river.

00:03:17.312 --> 00:03:19.346
And on different times of the year,

00:03:19.346 --> 00:03:21.749
it was able to get across the river,

00:03:21.749 --> 00:03:24.113
it's able to, the squirrels on the north

00:03:24.113 --> 00:03:25.441
and the squirrels on the south

00:03:25.441 --> 00:03:29.170
were able to interbreed and
produce fertile offspring,

00:03:29.170 --> 00:03:31.303
so they were all one species.

00:03:31.303 --> 00:03:33.458
But over time, the Colorado River

00:03:33.458 --> 00:03:36.146
started to erode more
and more soil and rock,

00:03:36.146 --> 00:03:38.694
and so this became what we now consider

00:03:38.694 --> 00:03:40.550
to be the Grand Canyon.

00:03:40.550 --> 00:03:42.422
And so over time,

00:03:42.422 --> 00:03:46.721
this became a very significant
geographic barrier.

00:03:46.721 --> 00:03:49.739
No longer could, before
they could travel across,

00:03:49.739 --> 00:03:51.430
but once it became the Grand Canyon,

00:03:51.430 --> 00:03:53.774
it became very difficult or impossible

00:03:53.774 --> 00:03:56.270
for them to travel across.

00:03:56.270 --> 00:03:58.381
And so now you have these
two different populations.

00:03:58.381 --> 00:03:59.808
They have the same parent species,

00:03:59.808 --> 00:04:02.008
but they're now geographically isolated.

00:04:02.008 --> 00:04:04.375
And since the creation of the,

00:04:04.375 --> 00:04:06.579
or while we have the
creation of the Grand Canyon,

00:04:06.579 --> 00:04:09.619
since it became very hard
or impossible them to cross,

00:04:09.619 --> 00:04:12.338
you've now had enough, both genetic drift,

00:04:12.338 --> 00:04:14.170
also natural selection,

00:04:14.170 --> 00:04:15.804
these are the evolutionary processes

00:04:15.804 --> 00:04:17.159
that we've talked about,

00:04:17.159 --> 00:04:20.110
where the Harris' antelope squirrel,

00:04:20.110 --> 00:04:22.953
which lives on the south side

00:04:22.953 --> 00:04:26.199
and is right over here, it's this picture,

00:04:26.199 --> 00:04:29.449
and the white-tailed antelope squirrel,

00:04:30.560 --> 00:04:32.156
which lives on the north side.

00:04:32.156 --> 00:04:33.718
Even though they look quite similar,

00:04:33.718 --> 00:04:35.349
as you can see from these pictures,

00:04:35.349 --> 00:04:36.760
they have now diverged enough

00:04:36.760 --> 00:04:38.740
that they are different species,

00:04:38.740 --> 00:04:42.573
that they no longer will
be able to interbreed

00:04:43.535 --> 00:04:46.305
and produce fertile offspring.

00:04:46.305 --> 00:04:48.902
So it's fairly intuitive

00:04:48.902 --> 00:04:51.867
how allopatric speciation can work.

00:04:51.867 --> 00:04:54.194
Geographic separation,
no longer can interbreed,

00:04:54.194 --> 00:04:57.239
and over time their genes change

00:04:57.239 --> 00:04:59.808
through natural selection
and genetic drift.

00:04:59.808 --> 00:05:03.109
But what about situations where
they stay in the same place,

00:05:03.109 --> 00:05:06.705
where theoretically,
they could get together,

00:05:06.705 --> 00:05:08.241
they could interact.

00:05:08.241 --> 00:05:10.459
Could you still have speciation?

00:05:10.459 --> 00:05:12.258
And the answer is yes.

00:05:12.258 --> 00:05:13.707
And that form of speciation,

00:05:13.707 --> 00:05:16.303
where you are still in the same geography,

00:05:16.303 --> 00:05:19.824
that is called sympatric speciation.

00:05:19.824 --> 00:05:21.668
Let me write that down.

00:05:21.668 --> 00:05:23.418
Sympatric speciation.

00:05:27.627 --> 00:05:28.544
Speciation.

00:05:29.825 --> 00:05:32.464
And examples of sympatric speciation

00:05:32.464 --> 00:05:34.321
are a little bit less obvious,

00:05:34.321 --> 00:05:36.012
or a little bit less intuitive,

00:05:36.012 --> 00:05:39.218
but there's an example that people believe

00:05:39.218 --> 00:05:43.918
is sympatric speciation
happening before our eyes.

00:05:43.918 --> 00:05:48.085
So this species, the technical
term Rhagoletis pomonella,

00:05:49.269 --> 00:05:51.759
I know I'm mispronouncing
it right over here,

00:05:51.759 --> 00:05:53.635
this is native to North America,

00:05:53.635 --> 00:05:55.545
and before European settlers

00:05:55.545 --> 00:05:57.925
brought apples to North America,

00:05:57.925 --> 00:05:59.815
they hung out and they laid their eggs

00:05:59.815 --> 00:06:02.565
and their maggots were inside of,

00:06:04.396 --> 00:06:07.424
or they leveraged the hawthorn
fruit right over here.

00:06:07.424 --> 00:06:09.291
So they would go to the hawthorn trees,

00:06:09.291 --> 00:06:11.038
and they would lay inside of the haw,

00:06:11.038 --> 00:06:14.161
they would use the hawthorn fruit

00:06:14.161 --> 00:06:17.074
for their, to lay their eggs,

00:06:17.074 --> 00:06:20.170
and for their young to kind of consume.

00:06:20.170 --> 00:06:23.537
But once the European settlers came

00:06:23.537 --> 00:06:26.404
and introduced apples into North America,

00:06:26.404 --> 00:06:29.216
a certain, I guess you
could say a subgroup,

00:06:29.216 --> 00:06:33.967
of Rhagoletis pomonella
started to leverage the apples,

00:06:33.967 --> 00:06:34.981
so started to lay their eggs

00:06:34.981 --> 00:06:37.071
and their eggs and
their maggots would grow

00:06:37.071 --> 00:06:38.821
inside of the apples.

00:06:41.309 --> 00:06:43.546
And they've actually now diverged,

00:06:43.546 --> 00:06:45.811
not into fully different species now.

00:06:45.811 --> 00:06:48.054
In theory, they can still interbreed

00:06:48.054 --> 00:06:50.072
and produce fertile offspring.

00:06:50.072 --> 00:06:52.042
But they don't tend to do it any more.

00:06:52.042 --> 00:06:53.124
That it tends to be,

00:06:53.124 --> 00:06:54.538
even though they're in the same geography,

00:06:54.538 --> 00:06:57.196
and it's not hard to fly
from the hawthorn tree

00:06:57.196 --> 00:06:58.812
to the apple tree,

00:06:58.812 --> 00:07:00.038
they don't tend to do it.

00:07:00.038 --> 00:07:03.524
And because of this behavioral divergence,

00:07:03.524 --> 00:07:05.286
that some decide to go to the apple,

00:07:05.286 --> 00:07:07.175
some decide to stay at the hawthorn,

00:07:07.175 --> 00:07:10.310
they actually are now
developing different traits

00:07:10.310 --> 00:07:14.056
that are selected for depending on that,

00:07:14.056 --> 00:07:15.602
I guess you could say initial preference,

00:07:15.602 --> 00:07:18.091
or that initial bias for
which fruit they want to use

00:07:18.091 --> 00:07:19.321
to lay their eggs in.

00:07:19.321 --> 00:07:21.672
So for example, the ones
that are in the apple tree,

00:07:21.672 --> 00:07:24.794
they now have, their breeding cycle

00:07:24.794 --> 00:07:27.547
is more aligned with the
growing season for apples,

00:07:27.547 --> 00:07:29.099
while the ones in the hawthorn tree,

00:07:29.099 --> 00:07:31.008
their breeding cycle is
more aligned with what,

00:07:31.008 --> 00:07:33.631
for the growing cycle for the hawthorn.

00:07:33.631 --> 00:07:35.975
And so biologists believe
that this is an example

00:07:35.975 --> 00:07:39.836
of sympatric speciation
happening before our eyes,

00:07:39.836 --> 00:07:42.845
that if we were to wait
a few hundred more years,

00:07:42.845 --> 00:07:44.950
possibly a thousand years or more,

00:07:44.950 --> 00:07:48.848
that this will diverge
into two different species

00:07:48.848 --> 00:07:49.967
that will no longer be able

00:07:49.967 --> 00:07:53.546
to interbreed and produce
fertile offspring.

00:07:53.546 --> 00:07:55.985
Another example of sympatric speciation,

00:07:55.985 --> 00:07:59.325
which is a little bit
more wild in some ways,

00:07:59.325 --> 00:08:00.825
it's a little bit more out there,

00:08:00.825 --> 00:08:03.137
but this would be an example with plants.

00:08:03.137 --> 00:08:07.379
So as we learn in other
Khan Academy videos,

00:08:07.379 --> 00:08:09.029
organisms like human beings,

00:08:09.029 --> 00:08:11.793
and in fact many sexually
reproducing organisms,

00:08:11.793 --> 00:08:13.384
they're diploid organisms.

00:08:13.384 --> 00:08:16.217
They have two sets of chromosomes.

00:08:17.851 --> 00:08:22.384
For example, human beings have
two sets of 23 chromosomes,

00:08:22.384 --> 00:08:23.995
for a total of 46 chromosomes,

00:08:23.995 --> 00:08:27.541
23 from your mom, 23 from your dad.

00:08:27.541 --> 00:08:30.017
And so we are diploid organisms.

00:08:30.017 --> 00:08:32.544
And in general, there are errors

00:08:32.544 --> 00:08:33.975
that occur during reproduction

00:08:33.975 --> 00:08:36.678
and errors that occur during meiosis

00:08:36.678 --> 00:08:38.446
that can lead to polyploidy,

00:08:38.446 --> 00:08:42.529
where an organism can have
more than two sets of,

00:08:43.818 --> 00:08:46.250
or can start, or a potential organism

00:08:46.250 --> 00:08:48.449
could have more than
two sets of chromosomes.

00:08:48.449 --> 00:08:49.699
In the animal kingdom,

00:08:49.699 --> 00:08:51.168
that doesn't work out too well.

00:08:51.168 --> 00:08:54.049
Usually, that does not
produce a viable embryo,

00:08:54.049 --> 00:08:55.677
a viable zygote.

00:08:55.677 --> 00:08:56.903
But in the plant kingdom,

00:08:56.903 --> 00:08:59.712
this is, it tolerates
it a little bit more.

00:08:59.712 --> 00:09:00.893
So you could have a situation

00:09:00.893 --> 00:09:02.815
where you have a diploid plant,

00:09:02.815 --> 00:09:06.140
and through meiosis,
through an error in meiosis,

00:09:06.140 --> 00:09:09.179
instead of producing
haploid egg and sperms,

00:09:09.179 --> 00:09:11.500
it produces diploid egg and sperm,

00:09:11.500 --> 00:09:14.152
which then are able to get together

00:09:14.152 --> 00:09:16.139
to form a tetraploid plant,

00:09:16.139 --> 00:09:19.881
so a plant that has
four sets of chromosomes

00:09:19.881 --> 00:09:22.020
instead of two sets of chromosomes.

00:09:22.020 --> 00:09:24.709
And then once that
tetraploid plant exists,

00:09:24.709 --> 00:09:26.647
it might only be able to reproduce

00:09:26.647 --> 00:09:30.985
with other tetraploid plants
versus the diploid plant.

00:09:30.985 --> 00:09:32.675
So you have the diploid plants here,

00:09:32.675 --> 00:09:34.817
when meiosis is working "properly,"

00:09:34.817 --> 00:09:35.650
and I'll put that in quotes,

00:09:35.650 --> 00:09:37.324
'cause maybe, you know,
arguably this error

00:09:37.324 --> 00:09:39.535
is what helps for a speciation sometimes,

00:09:39.535 --> 00:09:41.567
especially in the plant kingdom,

00:09:41.567 --> 00:09:45.117
it is, it can produce
this haploid egg or sperm.

00:09:45.117 --> 00:09:47.069
The tetroid plant would then,

00:09:47.069 --> 00:09:48.208
would, through its meiosis,

00:09:48.208 --> 00:09:50.865
its ploidy, I guess you could say halves,

00:09:50.865 --> 00:09:52.774
when it goes to the egg or the sperm,

00:09:52.774 --> 00:09:54.790
but you're now going to have a nonviable

00:09:54.790 --> 00:09:56.565
or infertile triploid plant,

00:09:56.565 --> 00:09:59.445
because the separation
won't happen properly

00:09:59.445 --> 00:10:01.043
in meiosis for this egg,

00:10:01.043 --> 00:10:02.710
if it's even viable.

00:10:03.772 --> 00:10:05.756
So all of a sudden this tetraploid plant

00:10:05.756 --> 00:10:08.568
is now a, you've had speciation occur.

00:10:08.568 --> 00:10:10.272
It could be viewed as a new species.

00:10:10.272 --> 00:10:13.267
And you could think about
things like this happening

00:10:13.267 --> 00:10:15.318
as a potential, and we
don't understand all of it,

00:10:15.318 --> 00:10:17.183
we don't understand how
all of the speciation

00:10:17.183 --> 00:10:19.945
that we now observe has actually occurred,

00:10:19.945 --> 00:10:22.313
but you can even imagine
as this being a mechanism

00:10:22.313 --> 00:10:25.420
for why you have an increase
in the number of chromosomes

00:10:25.420 --> 00:10:28.473
in certain species versus others.

00:10:28.473 --> 00:10:32.240
So hopefully this is starting
to answer some questions,

00:10:32.240 --> 00:10:34.035
and hopefully even
introduce more questions,

00:10:34.035 --> 00:10:36.017
'cause this is a very exciting topic,

00:10:36.017 --> 00:10:38.996
about how we get species from,

00:10:38.996 --> 00:10:41.251
more species, how do we get the diversity

00:10:41.251 --> 00:10:44.668
from parent species or ancestral species.