1
00:00:00 --> 00:00:05
Good morning.
Today we move in the direction

2
00:00:05 --> 00:00:12
that takes a big turn from the
direction we have been going in

3
00:00:12 --> 00:00:16
so far.
All the devices we have had up

4
00:00:16 --> 00:00:24
until now, resistors and voltage
sources, and even your digital

5
00:00:24 --> 00:00:31
devices like the AND gate or the
inverter and so on had a very

6
00:00:31 --> 00:00:36
specific property.
We didn't dwell on that

7
00:00:36 --> 00:00:41
property, but that property was
that these were not what are

8
00:00:41 --> 00:00:43
called memory devices.
In other words,

9
00:00:43 --> 00:00:48
the outputs at any given time
are a function of the inputs

10
00:00:48 --> 00:00:49
alone.
In other words,

11
00:00:49 --> 00:00:54
if you took your inverter or
your NAND gate for that matter

12
00:00:54 --> 00:00:57
and you build a circuit
comprising 50 NAND gates

13
00:00:57 --> 00:01:01
connected in structures that we
have talked about,

14
00:01:01 --> 00:01:06
you apply an input and boom you
get an output.

15
00:01:06 --> 00:01:09
And your output is a function
of the inputs alone,

16
00:01:09 --> 00:01:11
right?
The same thing with your

17
00:01:11 --> 00:01:16
resistors and voltage sources.
At any given point in time your

18
00:01:16 --> 00:01:20
output VO of T was some function
of the input VI of T.

19
00:01:20 --> 00:01:24
What we are going to do today
is discuss a new element which

20
00:01:24 --> 00:01:29
will introduce a whole new class
of fun stuff for all of us to

21
00:01:29 --> 00:01:33
deal with.
And that is called storage.

22
00:01:33 --> 00:01:36
In other words,
the output of a circuit is now

23
00:01:36 --> 00:01:41
going to depend not just on the
inputs but it is going to depend

24
00:01:41 --> 00:01:46
on the background or it is going
to depend on where the circuit

25
00:01:46 --> 00:01:49
has been in the past.
So past is going to matter.

26
00:01:49 --> 00:01:52
It is a very fundamental
difference.

27
00:01:52 --> 00:01:57
And what I would like to do is
start by giving you folks a

28
00:01:57 --> 00:02:02
little bit of a surprise.
I am going to do a little demo

29
00:02:02 --> 00:02:05
taking two of your inverter
circuits.

30
00:02:05 --> 00:02:18


31
00:02:18 --> 00:02:23
I am going to start by taking a
couple of inverters.

32
00:02:23 --> 00:02:30
Remember, I am using this
structure here as an inverter.

33
00:02:30 --> 00:02:37
And I am going to couple this
to another inverter and take an

34
00:02:37 --> 00:02:43
output C, some VS,
some load resistance RL,

35
00:02:43 --> 00:02:47
my B terminal and my A
terminal.

36
00:02:47 --> 00:02:54
So I'm going to apply some
input between ground and my A

37
00:02:54 --> 00:02:59
terminal.
And for fun I want to apply a

38
00:02:59 --> 00:03:06
square wave at the input.
A square wave between zero and

39
00:03:06 --> 00:03:09
5 volts.
And this is how my time goes.

40
00:03:09 --> 00:03:11
Let's assume that VS is 5
volts.

41
00:03:11 --> 00:03:15
So what I am going to do is
plot for you the behavior of

42
00:03:15 --> 00:03:19
this inverter.
I am going to plot for you A,

43
00:03:19 --> 00:03:23
which would look like this.
I am going to plot for you B,

44
00:03:23 --> 00:03:26
which would be the inverted
wave form.

45
00:03:26 --> 00:03:30
And then plot C,
which would be a wave form that

46
00:03:30 --> 00:03:36
looks like this again.
Let me do a plot here.

47
00:03:36 --> 00:03:39
So this is A.

48
00:03:39 --> 00:03:46


49
00:03:46 --> 00:03:49
-- and so on.
Time goes this way.

50
00:03:49 --> 00:03:53
And let's say this is between
zero and 5 volts.

51
00:03:53 --> 00:03:58
And B should be an inverted
wave form that should look like

52
00:03:58 --> 00:04:00
this.

53
00:04:00 --> 00:04:11


54
00:04:11 --> 00:04:15
If all that we believe of the
world so far is true then this

55
00:04:15 --> 00:04:20
is how things should behave,
so C should look like this.

56
00:04:20 --> 00:04:30


57
00:04:30 --> 00:04:33
This is what the world should
look like and if everything that

58
00:04:33 --> 00:04:36
you learned about is true and
correct and all of the good

59
00:04:36 --> 00:04:38
stuff.
Let me show you a little demo

60
00:04:38 --> 00:04:41
and see if I can try to pull the
rug out from under all that you

61
00:04:41 --> 00:04:45
have learned so far and show you
some surprising stuff.

62
00:04:45 --> 00:04:51


63
00:04:51 --> 00:04:58
Here are the three wave forms
that I showed you up here.

64
00:04:58 --> 00:05:02
This is my A.
This is my A wave form.

65
00:05:02 --> 00:05:05
This is the B wave form.
Notice that B,

66
00:05:05 --> 00:05:09
as you expect,
is an inverted form of A.

67
00:05:09 --> 00:05:12
And this is C.
We all expect this,

68
00:05:12 --> 00:05:15
correct?
But what I am going to do is

69
00:05:15 --> 00:05:21
let me expand the time scale on
this so that I can look at these

70
00:05:21 --> 00:05:25
transitions a little bit more
carefully.

71
00:05:25 --> 00:05:30
I am just going to expand the
time scale.

72
00:05:30 --> 00:05:33
There you go.
All I have done is expanded the

73
00:05:33 --> 00:05:37
time scale and spread that out a
little bit.

74
00:05:37 --> 00:05:42
And what you see there is quite
different from what you expect.

75
00:05:42 --> 00:05:48
A is a square wave as expected,
but B is stunningly different.

76
00:05:48 --> 00:05:51
It is a zero as expected
because this is a one.

77
00:05:51 --> 00:05:55
But here I get some really
strange behavior,

78
00:05:55 --> 00:06:00
behavior that is like nothing
on earth.

79
00:06:00 --> 00:06:02
Like nothing you have seen
before.

80
00:06:02 --> 00:06:05
And then, of course,
it becomes a one eventually,

81
00:06:05 --> 00:06:09
but there's some really,
really shady stuff going on

82
00:06:09 --> 00:06:11
here.
And so far you are not prepared

83
00:06:11 --> 00:06:14
to deal with this.
We have not given you the

84
00:06:14 --> 00:06:18
facility to deal with his issue.
What is the problem with this?

85
00:06:18 --> 00:06:22
We could say who cares?
What is the problem with this?

86
00:06:22 --> 00:06:25
Let's look at the result.
I am looking at this,

87
00:06:25 --> 00:06:29
I am focusing on this piece
here.

88
00:06:29 --> 00:06:34
And notice that instead of
being a sharp rise it looks like

89
00:06:34 --> 00:06:36
this.
It is going up a little bit

90
00:06:36 --> 00:06:40
more slowly.
What kind of problem would that

91
00:06:40 --> 00:06:44
create?
The problem that it creates is

92
00:06:44 --> 00:06:47
the following.
Let me play around with this

93
00:06:47 --> 00:06:52
graph a little bit more.
What I am going to do is just

94
00:06:52 --> 00:06:56
take this output here,
the C output and line it up

95
00:06:56 --> 00:07:02
against the A output.
And so I am going to line up

96
00:07:02 --> 00:07:05
the C wave form on top of the A
wave form.

97
00:07:05 --> 00:07:09
So you can see for yourself if
something really,

98
00:07:09 --> 00:07:13
really strange and nasty is
happening, I am just going to

99
00:07:13 --> 00:07:17
move up the C wave form and line
it up.

100
00:07:17 --> 00:07:22


101
00:07:22 --> 00:07:28
What is happening out there?
If you look carefully,

102
00:07:28 --> 00:07:33
what you observe is that the C
wave form transitions just ever

103
00:07:33 --> 00:07:37
so slightly later than the A
wave form.

104
00:07:37 --> 00:07:41
Look here.
And I claim that it is because

105
00:07:41 --> 00:07:43
of this.
Because of this,

106
00:07:43 --> 00:07:47
the C wave form falls just a
little bit later,

107
00:07:47 --> 00:07:52
and that little thing we see
out there is a delay.

108
00:07:52 --> 00:07:59
So nothing you have learned so
far prepares you for this.

109
00:07:59 --> 00:08:03
Suddenly, instead of the output
exactly following the input,

110
00:08:03 --> 00:08:07
my output is following the
input but a little bit later.

111
00:08:07 --> 00:08:12
And it is this fact of life
that things happen a little bit

112
00:08:12 --> 00:08:17
later, is really the reason why
each of you and all of us needs

113
00:08:17 --> 00:08:20
to buy new computers every
couple of years.

114
00:08:20 --> 00:08:24
This simple basic fact.
If this fact of life didn't

115
00:08:24 --> 00:08:28
exist, you would buy one
computer and be done with it for

116
00:08:28 --> 00:08:31
life.
Intel would make gobs of money

117
00:08:31 --> 00:08:35
one year, and so would Dell and
Gateway and so on,

118
00:08:35 --> 00:08:36
and then no more.
That's it.

119
00:08:36 --> 00:08:39
This is it.
But because of this a little

120
00:08:39 --> 00:08:43
itty-bitty difference here the
entire semiconductor technology

121
00:08:43 --> 00:08:46
is charging along trying to do
something about that.

122
00:08:46 --> 00:08:49
You buy newer and newer
computers each year.

123
00:08:49 --> 00:08:52
It turns out this little
itty-bitty thing here,

124
00:08:52 --> 00:08:54
that is called the delay,
the inverter delay.

125
00:08:54 --> 00:08:58
And it happens because of a
specific element that has been

126
00:08:58 --> 00:09:03
introduced here that we have not
shown you so far.

127
00:09:03 --> 00:09:06
And a large part of the
semiconductor industry and

128
00:09:06 --> 00:09:11
follow-on courses and design and
so on focuses on how could I

129
00:09:11 --> 00:09:15
make my delay smaller,
how can I get to be faster and

130
00:09:15 --> 00:09:18
faster and faster?
This relates to how fast we can

131
00:09:18 --> 00:09:22
clock your Pentium IV.
Remember it came all the way to

132
00:09:22 --> 00:09:26
1.3 gigahertz?
What's the fasted Pentium money

133
00:09:26 --> 00:09:30
can buy today?
What is the fastest P4?

134
00:09:30 --> 00:09:32
Oh, 3.2 have come out?
I don't know.

135
00:09:32 --> 00:09:35
Ken claims 3.2.
But, yeah, there you go,

136
00:09:35 --> 00:09:38
3.2 gigahertz.
It all has to do with this

137
00:09:38 --> 00:09:42
little itty-bitty thing.
You saw it for the first time

138
00:09:42 --> 00:09:45
here.
When some of you become CTOs at

139
00:09:45 --> 00:09:49
Intel and so on,
just remember that it all began

140
00:09:49 --> 00:09:53
on October 16th with this little
rinky-dink thing here.

141
00:09:53 --> 00:09:58
What you are going to learn now
is some really cool stuff that

142
00:09:58 --> 00:10:04
has huge implications for life.
So why does that happen?

143
00:10:04 --> 00:10:09
Why did this transition happen
just a little bit later?

144
00:10:09 --> 00:10:15
The reason is that remember
when this wave form reaches VT,

145
00:10:15 --> 00:10:21
the threshold voltage of this
MOSFET, this guy is going to

146
00:10:21 --> 00:10:25
switch, right?
So because of the slower rise

147
00:10:25 --> 00:10:30
of the voltage,
the VT is going to be reached a

148
00:10:30 --> 00:10:35
small amount of time later.
So I am going to hit VT

149
00:10:35 --> 00:10:39
slightly later.
And because of that this guy is

150
00:10:39 --> 00:10:43
going to transition just a bit
later because this intermediate

151
00:10:43 --> 00:10:47
wave form B is slower.
It hits VT just a little bit

152
00:10:47 --> 00:10:51
later than if it would have made
an instantaneous transition.

153
00:10:51 --> 00:10:56
And therefore my output falls
just a little bit later and this

154
00:10:56 --> 00:11:00
gives rise to my delay in the
inverter.

155
00:11:00 --> 00:11:06
We can call that d if you would
like, some delay.

156
00:11:06 --> 00:11:13
In your course notes,
this material is covered in

157
00:11:13 --> 00:11:20
Chapters 9 and 10.
That was to kind of motivate

158
00:11:20 --> 00:11:30
why we are going to be doing all
that you we will be doing.

159
00:11:30 --> 00:11:35
Don't anybody come within a
foot of this even by mistake.

160
00:11:35 --> 00:11:39
I mean it.
It is pretty deadly stuff.

161
00:11:39 --> 00:11:43
Today we will talk about the
capacitor.

162
00:11:43 --> 00:11:49
And in the next couple of
lectures I am going to tie it

163
00:11:49 --> 00:11:55
all together and show you how
this relates to that.

164
00:11:55 --> 00:12:01
I will show you exactly how the
delay happens.

165
00:12:01 --> 00:12:04
You can compute it based on
some simple principles that you

166
00:12:04 --> 00:12:07
will learn about in the next
couple of lectures.

167
00:12:07 --> 00:12:11
What I am going to do is first
of all show you,

168
00:12:11 --> 00:12:14
I claim that that delay happens
because of the presence of a

169
00:12:14 --> 00:12:18
capacitor somewhere in there.
What I will do now is take you

170
00:12:18 --> 00:12:21
into a closer look,
take a closer look at the

171
00:12:21 --> 00:12:24
MOSFET and show you were the
capacitor is.

172
00:12:24 --> 00:12:27
This is the MOSFET that you
have seen so far,

173
00:12:27 --> 00:12:34
drain, gate and source.
This is called an n-channel

174
00:12:34 --> 00:12:39
MOSFET.
And what I am going to do is

175
00:12:39 --> 00:12:47
dissect this and show you what
is actually happening,

176
00:12:47 --> 00:12:51
what this looks like on
silicon.

177
00:12:51 --> 00:13:00
So here is my slab of silicon.
It is very thin.

178
00:13:00 --> 00:13:03
And let's say this is,
I won't go into details here.

179
00:13:03 --> 00:13:08
You will learn a lot more about
this in future device classes

180
00:13:08 --> 00:13:12
like 301 and so on,
but suffice it to say I will

181
00:13:12 --> 00:13:16
just introduce it here to give
you a sense of where the

182
00:13:16 --> 00:13:19
capacitor is.
This is p-type silicon.

183
00:13:19 --> 00:13:24
And the way you build a MOSFET
is you create a couple of tubs

184
00:13:24 --> 00:13:28
in which you dope to be n-type.
The basic silicon is dope

185
00:13:28 --> 00:13:33
p-type.
And this guy here is n-type.

186
00:13:33 --> 00:13:39
And what you do is a thin oxide
layer is placed on top of that

187
00:13:39 --> 00:13:44
and then on top of that a thin
metal layer.

188
00:13:44 --> 00:13:50
This is a metal layer.
This is a thin piece of oxide,

189
00:13:50 --> 00:13:56
silicon dioxide.
And this is my P substrate.

190
00:13:56 --> 00:14:00
Now this is a little metal
layer that is really a wire on

191
00:14:00 --> 00:14:03
top of the silicone.
This metal layer could be some

192
00:14:03 --> 00:14:07
sort of a wire that meanders
around on the surface of

193
00:14:07 --> 00:14:10
silicone.
And this is a wire that

194
00:14:10 --> 00:14:13
connects to the gate.
This is the gate of my MOSFET.

195
00:14:13 --> 00:14:17
And this guy here is the drain.
And this guy here is the

196
00:14:17 --> 00:14:19
source.
And this is my gate.

197
00:14:19 --> 00:14:22
So there is a little piece of
metal here.

198
00:14:22 --> 00:14:25
This is this piece of metal
here.

199
00:14:25 --> 00:14:31
And there is a piece of oxide
and then my silicone substrate.

200
00:14:31 --> 00:14:37
Notice that this is my oxide.
When I apply a positive voltage

201
00:14:37 --> 00:14:41
to the gate here with respect to
the substrate,

202
00:14:41 --> 00:14:46
what happens is that I draw up
negative charges.

203
00:14:46 --> 00:14:52
I draw up electrons here into
this channel region and I have

204
00:14:52 --> 00:14:58
corresponding plus type out here
so that I get a view here that

205
00:14:58 --> 00:15:05
looks like a couple of plates.
And I end up with an oxide in

206
00:15:05 --> 00:15:08
the middle.
There is no connection.

207
00:15:08 --> 00:15:14
Two plates separated by a small
distance with plus q and minus q

208
00:15:14 --> 00:15:17
on the plates.
And, because of that,

209
00:15:17 --> 00:15:22
what ends up happening here is
that this piece behaves like a

210
00:15:22 --> 00:15:26
capacitor.
So a capacitor has two plates

211
00:15:26 --> 00:15:30
with a thin insulating material
in the middle with some

212
00:15:30 --> 00:15:35
permittivity epsilon.
And so I get a little piece of

213
00:15:35 --> 00:15:38
a capacitor here.
That is the capacitor that is

214
00:15:38 --> 00:15:40
forming.
I did not set out to build that

215
00:15:40 --> 00:15:43
capacitor, but there is a
capacitor nonetheless.

216
00:15:43 --> 00:15:46
So when I apply a positive
voltage at the gate,

217
00:15:46 --> 00:15:49
negative electrons are pulled
up here which forms a channel,

218
00:15:49 --> 00:15:51
and then a current can then
flow.

219
00:15:51 --> 00:15:53
And that is how the MOSFET
turns on.

220
00:15:53 --> 00:15:57
So n-type electrons back to
n-type, and I get electron flow

221
00:15:57 --> 00:16:00
here and that gives me my
channel.

222
00:16:00 --> 00:16:04
This is just kind of devices in
four minutes or less.

223
00:16:04 --> 00:16:08
You will do an entire course on
this, if you like,

224
00:16:08 --> 00:16:11
if you take 301.
What we do is to be able to

225
00:16:11 --> 00:16:15
capture the behavior that we
just saw, the funny delayed

226
00:16:15 --> 00:16:18
behavior, we have to augment our
model.

227
00:16:18 --> 00:16:21
We have to introduce a new
element.

228
00:16:21 --> 00:16:24
So what we do is here is a
MOSFET, gate,

229
00:16:24 --> 00:16:28
drain and source.
And notice here we model this

230
00:16:28 --> 00:16:33
by putting a little capacitor,
CGS between our gate and the

231
00:16:33 --> 00:16:37
source.
So this becomes a simple model

232
00:16:37 --> 00:16:43
for our MOSFET device which is
the good old gate drain source

233
00:16:43 --> 00:16:48
device from the past with a
little capacitor CGS having some

234
00:16:48 --> 00:16:53
value for CGS in maybe ten to
the minus 14 or thereabouts

235
00:16:53 --> 00:16:56
farads.
So that is a little capacitor

236
00:16:56 --> 00:17:03
that has come about in this
device that we fabricated here.

237
00:17:03 --> 00:17:07
It is that capacitor that is at
between node B and ground

238
00:17:07 --> 00:17:12
because it is between the gate
and the source of the second

239
00:17:12 --> 00:17:15
inverter.
And it is that capacitor that

240
00:17:15 --> 00:17:20
is playing the games that we saw
out there.

241
00:17:20 --> 00:17:27


242
00:17:27 --> 00:17:31
So let's look at some of the
behavior of an ideal linear

243
00:17:31 --> 00:17:34
capacitor.
A capacitor,

244
00:17:34 --> 00:17:38
as I said, has a couple of
plates.

245
00:17:38 --> 00:17:44
There are a couple of plates.
Between the plates is some

246
00:17:44 --> 00:17:51
dieletric, permittivity epsilon.
Let's say the area of the

247
00:17:51 --> 00:17:57
plates is A, and let's say the
plates are separated by a

248
00:17:57 --> 00:18:02
distance D.
I get some charge here,

249
00:18:02 --> 00:18:06
let's say q.
So q and minus q on the

250
00:18:06 --> 00:18:10
capacitor.
And the capacitance C is given

251
00:18:10 --> 00:18:15
by epsilon A divided by D.
Epsilon, as I said,

252
00:18:15 --> 00:18:19
is the productivity of the
dielectric.

253
00:18:19 --> 00:18:25
So if it is free space then it
would be epsilon zero which is

254
00:18:25 --> 00:18:32
the permittivity of free space.
That is the capacitance in

255
00:18:32 --> 00:18:35
farads.
And the symbol looks like this.

256
00:18:35 --> 00:18:38
Capacitor C.
Voltage v.

257
00:18:38 --> 00:18:40
Current i.
So this, much like the

258
00:18:40 --> 00:18:46
resistor, voltage source and so
on, this now becomes a primitive

259
00:18:46 --> 00:18:52
element in your tool chest of
elements like the voltage source

260
00:18:52 --> 00:18:56
and so onn.
Capacitance with the voltage v

261
00:18:56 --> 00:19:01
across it and a current i.
And I have assigned the

262
00:19:01 --> 00:19:05
associated variables here
according to the associated

263
00:19:05 --> 00:19:08
variable discipline.
A question to ask ourselves is

264
00:19:08 --> 00:19:13
remember we said we are all now
in a playground from all of

265
00:19:13 --> 00:19:17
nature, in this playground where
the lumped matter discipline

266
00:19:17 --> 00:19:20
holds?
And also remember that we said

267
00:19:20 --> 00:19:23
that for the lumped matter
discipline to hold we have to

268
00:19:23 --> 00:19:29
make a couple of assumptions.
One of those assumptions was

269
00:19:29 --> 00:19:33
that dq/dt, for all their
elements should be zero for all

270
00:19:33 --> 00:19:36
time.
So right now what about the

271
00:19:36 --> 00:19:39
capacitor?
It has got some charge q.

272
00:19:39 --> 00:19:42
So charge must have built up
somehow.

273
00:19:42 --> 00:19:47
Does that mean that I lied all
along, that we are no longer in

274
00:19:47 --> 00:19:51
this playground,
that we have been ejected from

275
00:19:51 --> 00:19:56
the playground because of the
capacitor, or are we still in

276
00:19:56 --> 00:20:01
the circuits playground in which
the lumped matter discipline

277
00:20:01 --> 00:20:06
holds and all good things happen
and so on?

278
00:20:06 --> 00:20:09
It seems like a contradiction,
doesn't it?

279
00:20:09 --> 00:20:12
I took you from Maxwell's
playgrounds to the EECS

280
00:20:12 --> 00:20:17
playground where I said the
lumped matter discipline holds.

281
00:20:17 --> 00:20:21
And one of the foundations of
the LMD was that dq/dt should be

282
00:20:21 --> 00:20:26
zero for all time inside the
elements that we are going to

283
00:20:26 --> 00:20:28
deal with.
And right now boom,

284
00:20:28 --> 00:20:32
it's not four weeks into the
course and Agarwal introduces an

285
00:20:32 --> 00:20:38
element and it has q in it.
It turns out that the capacitor

286
00:20:38 --> 00:20:41
also adheres to the lumped
matter discipline.

287
00:20:41 --> 00:20:45
Remember the discipline says
that dq/dt is zero for all time

288
00:20:45 --> 00:20:48
within elements.
So I am going to be clever.

289
00:20:48 --> 00:20:52
What I am going to do is I want
to choose element boundaries in

290
00:20:52 --> 00:20:55
a very cleaver way.
Notice that if I have q here on

291
00:20:55 --> 00:21:00
this plate then I get minus q on
the other plate.

292
00:21:00 --> 00:21:04
So if I take the whole element,
the element as a whole,

293
00:21:04 --> 00:21:09
if I am careful in terms of how
I package my boundaries,

294
00:21:09 --> 00:21:14
if I put both my plates inside
my element boundary then I still

295
00:21:14 --> 00:21:17
do get the net charge being
zero.

296
00:21:17 --> 00:21:22
So dq/dt is indeed zero for all
time provided I make sure that

297
00:21:22 --> 00:21:27
my element has both the plates.
Therefore, if you come across

298
00:21:27 --> 00:21:32
somebody else that gives you an
element that says I have an

299
00:21:32 --> 00:21:36
idea.
Let's create a new branch of

300
00:21:36 --> 00:21:40
electrical engineering in which
we model the capacitor not as

301
00:21:40 --> 00:21:44
one element for two plates,
but let's build a capacitor by

302
00:21:44 --> 00:21:48
combining two new elements,
two garbage elements called G1

303
00:21:48 --> 00:21:51
and G2.
G1 is like the top plate.

304
00:21:51 --> 00:21:55
G2 is the bottom plate.
I put them together and I get a

305
00:21:55 --> 00:21:58
capacitor.
But notice if I just pick one

306
00:21:58 --> 00:22:03
plate then the element G1 will
not adhere to the LMD.

307
00:22:03 --> 00:22:08
It adheres to the LMD because I
choose my element boundaries in

308
00:22:08 --> 00:22:11
a way that both plates come
within it.

309
00:22:11 --> 00:22:14
So it is very fundamental and
key.

310
00:22:14 --> 00:22:18
And you can read a lot more
about it in the course notes.

311
00:22:18 --> 00:22:23
I purposely dwelt on that
simple point because I think it

312
00:22:23 --> 00:22:29
is foundational and important.
And you really need to

313
00:22:29 --> 00:22:33
understand that the capacitor
does satisfy LMD.

314
00:22:33 --> 00:22:37
We are still in the good old
playground.

315
00:22:37 --> 00:22:41
A few simple facts here.
These are in the notes.

316
00:22:41 --> 00:22:45
And you have also seen this
before, I am sure.

317
00:22:45 --> 00:22:51
I can relate the charge to the
capacitance and the voltage as q

318
00:22:51 --> 00:22:55
is equal to Cv.
And q is in coulombs,

319
00:22:55 --> 00:23:00
this is in farads and this is
in volts.

320
00:23:00 --> 00:23:06
So there is some charge q
stored on the capacitor and it

321
00:23:06 --> 00:23:10
is in coulombs and q is equal to
Cv.

322
00:23:10 --> 00:23:17
So I can differentiate this
with respect to time to get the

323
00:23:17 --> 00:23:21
current, and that becomes
i=dq/dt.

324
00:23:21 --> 00:23:28
So the current at any given
time is dq/dt.

325
00:23:28 --> 00:23:32
And so I substitute for q in
terms of Cv here.

326
00:23:32 --> 00:23:36
That is what I get.
So the current i=d(Cv)/dt.

327
00:23:36 --> 00:23:41
A 6.002 assumption,
capacitance in general can be

328
00:23:41 --> 00:23:44
time-varying.
I can get time-varying

329
00:23:44 --> 00:23:48
capacitors.
In fact, there are some sensors

330
00:23:48 --> 00:23:51
which are capacitive.
And, as I talk,

331
00:23:51 --> 00:23:57
my sound waves can change the
pressure on the top plate of the

332
00:23:57 --> 00:24:02
capacitor.
And move the top plate of the

333
00:24:02 --> 00:24:08
capacitor, thereby changing the
capacitance by moving the plate.

334
00:24:08 --> 00:24:13
Remember d here,
as the plate moves closer I get

335
00:24:13 --> 00:24:17
a higher capacitance.
So we won't be dealing,

336
00:24:17 --> 00:24:23
unless explicitly said so,
with time-varying capacitances.

337
00:24:23 --> 00:24:29
So what we can do is 6.002
allows us to write Cdv/dt.

338
00:24:29 --> 00:24:33
So my current source capacitor
is Cdv/dt.

339
00:24:33 --> 00:24:39
I can also write down the
energy, capacitors store energy.

340
00:24:39 --> 00:24:42
E=1/2Cv^2.
I am sure you have seen all

341
00:24:42 --> 00:24:46
this before in physics and so
on.

342
00:24:46 --> 00:24:52
That is the amount of energy
stored in the capacitor if it is

343
00:24:52 --> 00:24:56
holding a charge q.
Let me do a little

344
00:24:56 --> 00:25:02
demonstration for you.
They don't make glasses like

345
00:25:02 --> 00:25:07
they used to.
Our friend Lorenzo has charged

346
00:25:07 --> 00:25:11
up this capacitor.
It is a huge capacitor.

347
00:25:11 --> 00:25:15
It is a 250 volt capacitor so
it is nasty.

348
00:25:15 --> 00:25:19
He has charged it up and has
kept it there.

349
00:25:19 --> 00:25:25
And to show you that it does
contain stored charges it has

350
00:25:25 --> 00:25:30
been sitting there holding
charge.

351
00:25:30 --> 00:25:36
Maybe the first row should go
backwards, just step back for a

352
00:25:36 --> 00:25:40
second.
I think you guys would be safe

353
00:25:40 --> 00:25:44
but I just don't want to take
any chances.

354
00:25:44 --> 00:25:48
This is holding a bunch of
charge.

355
00:25:48 --> 00:25:54
It is kind of sitting there.
If I short the terminals it

356
00:25:54 --> 00:25:58
should try to say oh,
I've got a path,

357
00:25:58 --> 00:26:02
let me get my charge out.
All right.

358
00:26:02 --> 00:26:05
Let's do it.
This is always a scary moment

359
00:26:05 --> 00:26:08
for me.
And I say a little prayer

360
00:26:08 --> 00:26:10
before I do this.

361
00:26:10 --> 00:26:19


362
00:26:19 --> 00:26:19
Good?
OK.

363
00:26:19 --> 00:26:21
Gee, you guys would love to see
me getting fried,

364
00:26:21 --> 00:26:22
huh?
All right.

365
00:26:22 --> 00:26:23
Let's see.

366
00:26:23 --> 00:26:45


367
00:26:45 --> 00:26:47
So it did contain charge.

368
00:26:47 --> 00:27:00


369
00:27:00 --> 00:27:04
So there is a reason why
Lorenzo puts one hand inside his

370
00:27:04 --> 00:27:08
pocket when he shorts it,
because there is a natural

371
00:27:08 --> 00:27:12
tendency to hold the wire with
both hands, and la,

372
00:27:12 --> 00:27:16
la, la, la, la and put it
across the capacitor.

373
00:27:16 --> 00:27:21
By doing this you are
guaranteed that you will just be

374
00:27:21 --> 00:27:25
touching it with one hand.
Hopefully you folks will

375
00:27:25 --> 00:27:29
remember for life that a
capacitor can sit around and

376
00:27:29 --> 00:27:33
hold its charge for a while.
All right.

377
00:27:33 --> 00:27:35
That is enough of fun and
games.

378
00:27:35 --> 00:27:39
Let's get on with our business
of building circuits.

379
00:27:39 --> 00:27:41
What I am going to do is,
as I promised you,

380
00:27:41 --> 00:27:46
I am going to close the loop on
that example by halfway through

381
00:27:46 --> 00:27:49
the next lecture.
I'm going take you on a bit of

382
00:27:49 --> 00:27:53
a journey involving capacitors
and resistors and involving some

383
00:27:53 --> 00:27:57
analysis, and then we will close
it all up for you at about the

384
00:27:57 --> 00:28:02
middle of next lecture.
What I would like to do next is

385
00:28:02 --> 00:28:06
here is a new element.
And let's do some fun stuff

386
00:28:06 --> 00:28:10
with elements.
Well, you know about voltage

387
00:28:10 --> 00:28:14
sources, you know about
resistors, let's put them

388
00:28:14 --> 00:28:17
together and see how they
behave.

389
00:28:17 --> 00:28:21
Let's have a capacitor here,
C, vc(t) and some current i.

390
00:28:21 --> 00:28:25
What I am going to do,
in general, whenever I have

391
00:28:25 --> 00:28:30
something new or something
strange, let's say like a

392
00:28:30 --> 00:28:36
capacitor or some other device.
It is interesting to model the

393
00:28:36 --> 00:28:41
rest of the circuit behind it if
it contains only resistors and

394
00:28:41 --> 00:28:46
voltages and linear elements as
a Thevenin equivalent.

395
00:28:46 --> 00:28:50
So let me do that.
This is R and this is vi.

396
00:28:50 --> 00:28:54
This stuff in the back is my
standard pattern,

397
00:28:54 --> 00:28:59
voltage source in series with a
resistor, and I connect that

398
00:28:59 --> 00:29:03
across my capacitor.
But remember,

399
00:29:03 --> 00:29:07
although you saw those funny
wave forms and so on,

400
00:29:07 --> 00:29:10
the capacitor is a linear
device.

401
00:29:10 --> 00:29:15
Because you can see from here
that the current relates to

402
00:29:15 --> 00:29:18
dv/dt.
That is a linear operation.

403
00:29:18 --> 00:29:23
You don't see V squareds and
Vis and things like that in

404
00:29:23 --> 00:29:25
there.
It's is a linear device.

405
00:29:25 --> 00:29:32
Let's go back to our trusty old
method, the node method.

406
00:29:32 --> 00:29:36
If you just blindly apply the
node method and simply grunge

407
00:29:36 --> 00:29:40
through a bunch of math,
you should be able to get to

408
00:29:40 --> 00:29:44
the answer, that is for some
voltage v or some form of

409
00:29:44 --> 00:29:49
voltage vi, I should be able to
figure out what vc looks like.

410
00:29:49 --> 00:29:52
So let's do that.
This is the node that is of

411
00:29:52 --> 00:29:56
interest here with the unknown
node voltage vc.

412
00:29:56 --> 00:29:58
So let me apply the node
method.

413
00:29:58 --> 00:30:03
(vc-vi)/R is the current going
this way.

414
00:30:03 --> 00:30:09
That plus the current through
the capacitor should equal zero.

415
00:30:09 --> 00:30:14
And what is the current through
the capacitor?

416
00:30:14 --> 00:30:20
The node method tells me that,
get the current in terms of the

417
00:30:20 --> 00:30:25
element values.
We know that the current is

418
00:30:25 --> 00:30:31
given by CdvC/dt.=O.
Just shuffling things around a

419
00:30:31 --> 00:30:35
little bit, I can write RC
dvc/dt+vc=vi.

420
00:30:35 --> 00:30:41
We are writing the node
equation and then getting the

421
00:30:41 --> 00:30:47
equation that characterizes this
little circuit.

422
00:30:47 --> 00:30:51
Notice here that this has units
of volts.

423
00:30:51 --> 00:30:58
And since I have time here,
this also must have units of

424
00:30:58 --> 00:31:00
time.

425
00:31:00 --> 00:31:06


426
00:31:06 --> 00:31:12
Let's go about solving this
little circuit and understanding

427
00:31:12 --> 00:31:16
how it behaves.
The specific example that we

428
00:31:16 --> 00:31:22
will look at looks like this.
Let's say the capacitor voltage

429
00:31:22 --> 00:31:25
at time T=0 is V0.
This is given.

430
00:31:25 --> 00:31:30
So at time T=0,
I am telling you that the

431
00:31:30 --> 00:31:36
capacitor contains a charge.
And because of that there is a

432
00:31:36 --> 00:31:40
voltage V0 across it.
That capacitor had a voltage of

433
00:31:40 --> 00:31:45
250 volts across it and most of
the devices we deal with in

434
00:31:45 --> 00:31:48
laptops and so on today,
like the Pentium IV,

435
00:31:48 --> 00:31:53
voltages are on the order of
1.5 volts, very small voltages.

436
00:31:53 --> 00:31:57
So that is the value in the
capacitor, the voltage.

437
00:31:57 --> 00:32:02
That is called a state.
This is called the state,

438
00:32:02 --> 00:32:05
capacitor state.
It is the state of the

439
00:32:05 --> 00:32:08
capacitor.
And I also give you that

440
00:32:08 --> 00:32:10
vi(t)=VI.
So my voltage is VI.

441
00:32:10 --> 00:32:14
And somehow,
I am not telling you how,

442
00:32:14 --> 00:32:19
but some how it arranged to
have the capacitor voltage be V0

443
00:32:19 --> 00:32:22
at time T=0.
Now I want to look to the

444
00:32:22 --> 00:32:28
solution to this for t greater
than or equal to zero.

445
00:32:28 --> 00:32:35
And in that time my voltage vi
is at some capital VI,

446
00:32:35 --> 00:32:41
some DC voltage VI.
So I am going to solve the

447
00:32:41 --> 00:32:48
differential equation RC
dvc/dt+vc=vi given these two

448
00:32:48 --> 00:32:53
values.
Input is DC voltage VI and VC0

449
00:32:53 --> 00:33:00
is V0, the initial charge in the
capacitor.

450
00:33:00 --> 00:33:03
So from now until almost to the
end of the lecture,

451
00:33:03 --> 00:33:08
it is just going to be math by
solving this very simple first

452
00:33:08 --> 00:33:12
order differential equation.
And the key here will be that

453
00:33:12 --> 00:33:16
throughout 6.002 we will be
following one method to solve

454
00:33:16 --> 00:33:19
these.
There are many methods to

455
00:33:19 --> 00:33:23
solving differential equations,
and we will follow one method.

456
00:33:23 --> 00:33:27
That method is called the
method of homogenous and

457
00:33:27 --> 00:33:31
particular solutions.
In 1802, I believe,

458
00:33:31 --> 00:33:36
you would have learned maybe
this, and certainly other

459
00:33:36 --> 00:33:40
methods.
You can use any method to solve

460
00:33:40 --> 00:33:43
it.
We will just stick to one

461
00:33:43 --> 00:33:46
method.
And this is also used in the

462
00:33:46 --> 00:33:50
course notes.
In this method what we do is

463
00:33:50 --> 00:33:55
take the solution VC by finding
two other components.

464
00:33:55 --> 00:34:00
One is called the homogenous
solution.

465
00:34:00 --> 00:34:03
And summing that up with the
particular solution.

466
00:34:03 --> 00:34:08
And that is the total solution.
So total solution is the sum of

467
00:34:08 --> 00:34:11
the homogenous and the
particular solutions.

468
00:34:11 --> 00:34:15
And the method has three steps.
As I said before,

469
00:34:15 --> 00:34:19
we will be using this method
again and again with every

470
00:34:19 --> 00:34:23
differential equation that we
encounter in this course.

471
00:34:23 --> 00:34:26
And you won't encounter a while
lot.

472
00:34:26 --> 00:34:31
The first step we find the
particular solution.

473
00:34:31 --> 00:34:39
The second step,
find the homogenous solution.

474
00:34:39 --> 00:34:47
The total solution is the sum
of the two.

475
00:34:47 --> 00:34:51
And then find ---

476
00:34:51 --> 00:34:56


477
00:34:56 --> 00:34:58
There will be some unknown
constants depending on the

478
00:34:58 --> 00:35:02
equation that you have.
And in the end we simply find

479
00:35:02 --> 00:35:06
the unknown constants by
applying the initial conditions

480
00:35:06 --> 00:35:09
that we have.
Boom, boom, boom.

481
00:35:09 --> 00:35:10
Particular.
Homogenous.

482
00:35:10 --> 00:35:13
Find constants.
Three things.

483
00:35:13 --> 00:35:17
So let's go about solving this
equation and apply those three

484
00:35:17 --> 00:35:19
conditions.
Again, remember,

485
00:35:19 --> 00:35:24
what I am doing now for the
next 10 minutes or 15 minutes is

486
00:35:24 --> 00:35:29
using math that you know about
to simply solve this first order

487
00:35:29 --> 00:35:35
of differential equations.
There is nothing really new

488
00:35:35 --> 00:35:39
that I am going to talk about
here.

489
00:35:39 --> 00:35:43
One is to find the particular
solution vCP,

490
00:35:43 --> 00:35:49
which will then be added into
the vCH to get me the solution.

491
00:35:49 --> 00:35:54
So the way you find the vCP is
you find any solution that

492
00:35:54 --> 00:36:00
satisfies this equation.
This is the equation.

493
00:36:00 --> 00:36:03
You find any solution that
satisfies it.

494
00:36:03 --> 00:36:07
And find the simplest possible
solution that money can buy.

495
00:36:07 --> 00:36:10
Find it.
That's the particular solution.

496
00:36:10 --> 00:36:13
Any solution is fine.
In this case,

497
00:36:13 --> 00:36:16
a really simple one would be
vCP equals VI.

498
00:36:16 --> 00:36:21
Let's see if a constant works.
One thing you will realize in

499
00:36:21 --> 00:36:25
differential equations is that
they are actually much simpler

500
00:36:25 --> 00:36:30
than they seem.
And the reason is that almost

501
00:36:30 --> 00:36:33
every time you have to assume
you know the answer,

502
00:36:33 --> 00:36:37
and then you are checking to
see what you assumed was

503
00:36:37 --> 00:36:40
correct.
Assume the answer is this like

504
00:36:40 --> 00:36:44
you are really smart,
and then check it out and say

505
00:36:44 --> 00:36:46
oh, yeah, that must have been
the answer.

506
00:36:46 --> 00:36:51
So here we assume that I think
VI is going to work so let's try

507
00:36:51 --> 00:36:53
it out.
Substituting in here.

508
00:36:53 --> 00:36:55
RC dvc/dt is 0.
vi is a constant.

509
00:36:55 --> 00:36:59
So I get vi equals vi,
so therefore this is a

510
00:36:59 --> 00:37:02
particular solution.
Done.

511
00:37:02 --> 00:37:05
I substitute vi here.
So dvi/dt=0.

512
00:37:05 --> 00:37:08
This vanishes and vi=VI.
Bingo.

513
00:37:08 --> 00:37:12
Therefore, VI is a solution to
this equation.

514
00:37:12 --> 00:37:15
So I am done with my vCP.

515
00:37:15 --> 00:37:22


516
00:37:22 --> 00:37:23
And in general what you have to
do is use trial and error.

517
00:37:23 --> 00:37:24
By trial and error try out a
bunch of solutions until you get

518
00:37:24 --> 00:37:24
lucky.
In general, again,

519
00:37:24 --> 00:37:25
in all of 6.002 for many of the
excitations a simple constant

520
00:37:25 --> 00:37:26
usually suffices.
Our second step is to find the

521
00:37:26 --> 00:37:27
homogenous solution.
And we can also do that very

522
00:37:27 --> 00:37:27
quickly.
And to do that we have to find

523
00:37:27 --> 00:37:29
a general solution to the
homogenous equation.

524
00:37:29 --> 00:37:52
The homogenous equation is the
same differential equation but

525
00:37:52 --> 00:38:04
with the drive set to zero.

526
00:38:04 --> 00:38:11


527
00:38:11 --> 00:38:15
We want to follow a set pattern
to solve the differential

528
00:38:15 --> 00:38:19
equations here,
and the set pattern is find

529
00:38:19 --> 00:38:23
vCP, vCH, find constants.
And to find vCH we are also

530
00:38:23 --> 00:38:29
going to follow a set pattern to
find the homogenous solution.

531
00:38:29 --> 00:38:34
So we set the drive to zero,
so vi is set to be zero.

532
00:38:34 --> 00:38:38
And I need to find a general
solution to this.

533
00:38:38 --> 00:38:43
As I promised earlier,
diff equations are really,

534
00:38:43 --> 00:38:49
really simple because the way
we are going to solve them is we

535
00:38:49 --> 00:38:55
are going to assume we know the
answer and then go check it.

536
00:38:55 --> 00:39:00
So let's try Ae^st.
Let's try and see if this can

537
00:39:00 --> 00:39:05
solve this particular equation
for some values of A and S.

538
00:39:05 --> 00:39:10
I am telling you that the
solution is going to be of this

539
00:39:10 --> 00:39:12
form.
Assume it.

540
00:39:12 --> 00:39:16
And then simply go ahead and
find me A and S,

541
00:39:16 --> 00:39:21
and do that by substituting it
back into the equation and find

542
00:39:21 --> 00:39:27
out the corresponding As and Ss.
So let's go ahead and do that.

543
00:39:27 --> 00:39:36
I get RC.
I substitute this back up so I

544
00:39:36 --> 00:39:50
get dAe^(st)/dt+Ae^st=0.
And let me plug that in and see

545
00:39:50 --> 00:40:00
what comes.
I get RCAse^st+Ae^st=0.

546
00:40:00 --> 00:40:05
I want to discard the trivial
solution of A being 0.

547
00:40:05 --> 00:40:11
That is a trivial solution so I
will discard that.

548
00:40:11 --> 00:40:16
And what I will do is cancel
out the As from here,

549
00:40:16 --> 00:40:21
assuming A is not zero,
and cancel e^st here.

550
00:40:21 --> 00:40:28
And what is left is RCs+1=0.
What this is saying is that if

551
00:40:28 --> 00:40:34
I can find an S such that this
is true then Aest is a general

552
00:40:34 --> 00:40:40
solution to my homogenous
equation.

553
00:40:40 --> 00:40:44
This is easy enough.
And so S=-1/RC.

554
00:40:44 --> 00:40:51
If I choose my S to be -1/RC
then the simple math that I have

555
00:40:51 --> 00:40:57
gone through shows me that this
must be the solution to the

556
00:40:57 --> 00:41:02
homogenous equation.
Or in other words

557
00:41:02 --> 00:41:07
vCH=Ae^(-t/RC).
All this is saying is that

558
00:41:07 --> 00:41:12
Ae^(-t/RC) is a solution to my
homogenous equation.

559
00:41:12 --> 00:41:16
A is an unknown constant.
A is some constant.

560
00:41:16 --> 00:41:21
I don't know what that is yet.
Notice RC has popped up again.

561
00:41:21 --> 00:41:26
And the cool thing about RC is
that, this is time,

562
00:41:26 --> 00:41:33
this also has units of time.
We commonly represent RC as

563
00:41:33 --> 00:41:38
some time constant tau,
as units of time.

564
00:41:38 --> 00:41:45
Associated with that circuit is
the time constant tau,

565
00:41:45 --> 00:41:50
which is simply RC.
I commonly write this as

566
00:41:50 --> 00:41:53
Ae^(-t/tau).

567
00:41:53 --> 00:42:02


568
00:42:02 --> 00:42:08
I am very the end here.
I have the particular solution

569
00:42:08 --> 00:42:11
here.
I have got the homogenous

570
00:42:11 --> 00:42:16
solution there.
I need to tell you about

571
00:42:16 --> 00:42:21
something else.
The way I found the homogenous

572
00:42:21 --> 00:42:27
solution was in four steps.
I assumed a solution of the

573
00:42:27 --> 00:42:32
form Ae^st.
I created this equation here in

574
00:42:32 --> 00:42:34
S.
This is called the

575
00:42:34 --> 00:42:38
characteristic equation for that
circuit.

576
00:42:38 --> 00:42:44
We will see this time and time
again for RC and other forms of

577
00:42:44 --> 00:42:47
circuits.
Assume a solution of this form.

578
00:42:47 --> 00:42:51
Construct the characteristic
equation.

579
00:42:51 --> 00:42:55
Find the roots of the
characteristic equation.

580
00:42:55 --> 00:43:00
In this case it is an equation
in S.

581
00:43:00 --> 00:43:05
So this is the root.
And then form the solution

582
00:43:05 --> 00:43:08
based on that root.
Four steps.

583
00:43:08 --> 00:43:15
Ae^st, characteristic equation,
root and then write down the

584
00:43:15 --> 00:43:21
general homogenous solution.
Four steps there.

585
00:43:21 --> 00:43:28
And finally I want to write
down the total solution.

586
00:43:28 --> 00:43:33
And the total solution is
simply vCP+vCH.

587
00:43:33 --> 00:43:38
And vCP was VI and vCH was
Ae^(-t/tau).

588
00:43:38 --> 00:43:43
tau was simply RC.
That is my solution.

589
00:43:43 --> 00:43:50
Now, remember the last step.
The last step was form the

590
00:43:50 --> 00:43:59
total solution and find out the
remaining constants.

591
00:43:59 --> 00:44:05
Find out the remaining
constants by using my initial

592
00:44:05 --> 00:44:10
conditions.
At t=0, I know that vC=V0.

593
00:44:10 --> 00:44:15
I know that.
And so therefore I can

594
00:44:15 --> 00:44:20
substitute t=0 to find the
constant.

595
00:44:20 --> 00:44:26
So I know that VO=VI+A.
t=0, this thing becomes 1,

596
00:44:26 --> 00:44:35
and so I get this equation from
which I get A=V0-Vi.

597
00:44:35 --> 00:44:38
In other words,
my solution vC is simply

598
00:44:38 --> 00:44:43
VI+(VO-VI) e^(-t/tau).
So the last 15 minutes have

599
00:44:43 --> 00:44:47
just been math.
No electrical engineering here,

600
00:44:47 --> 00:44:53
but electrical engineering
stopped at the point where you

601
00:44:53 --> 00:44:58
wrote this differential equation
down, went through a bunch of

602
00:44:58 --> 00:45:03
math and came up with a
solution.

603
00:45:03 --> 00:45:07
Purely mathematically.
So here I simply used math to

604
00:45:07 --> 00:45:11
get you the solution.
And, as I have been promising

605
00:45:11 --> 00:45:16
you throughout this course,
in the next lecture I will give

606
00:45:16 --> 00:45:19
you an intuitive EE method of
doing it.

607
00:45:19 --> 00:45:23
Real electrical engineers,
real EECS folks don't do it

608
00:45:23 --> 00:45:26
this way.
Real EECS folks do it

609
00:45:26 --> 00:45:30
intuitively.
And I will show you how to do

610
00:45:30 --> 00:45:35
it in four easy seconds in the
next lecture.

611
00:45:35 --> 00:45:41
But you need to understand the
foundations of how this comes

612
00:45:41 --> 00:45:44
about, and so this is the
answer.

613
00:45:44 --> 00:45:49
You can also get the current iC
is simply Cdvc/dt.

614
00:45:49 --> 00:45:53
I won't do that for you,
but you can simply

615
00:45:53 --> 00:45:57
differentiate it and get the
current.

616
00:45:57 --> 00:46:02
So I can plot for you vC,
time t, vC.

617
00:46:02 --> 00:46:08
The intuitive way of looking at
this is I have VI which is the

618
00:46:08 --> 00:46:15
final value of the voltage.
When t is infinity this part

619
00:46:15 --> 00:46:19
goes to zero so the vC is simply
VI.

620
00:46:19 --> 00:46:26
And then there is a component
V0-VI which decays according to

621
00:46:26 --> 00:46:32
this starting out at an initial
value of V0.

622
00:46:32 --> 00:46:38
Notice when t is zero vC is V0,
you can see that in the

623
00:46:38 --> 00:46:45
equation, and so it starts out
at V0 and ends up at VI.

624
00:46:45 --> 00:46:49
I start here,
I end up here.

625
00:46:49 --> 00:46:55
And this portion V0-VI decays
out over time like this.

626
00:46:55 --> 00:47:04
And this decay is governed by
the RC time constant or tau.

627
00:47:04 --> 00:47:10
I am going to show you very
quickly a couple of examples of

628
00:47:10 --> 00:47:17
wave forms, one that goes like
this and one that looks like

629
00:47:17 --> 00:47:21
this.
This is when I start with some

630
00:47:21 --> 00:47:28
value V0 and I don't apply any
input, it should decay down to

631
00:47:28 --> 00:47:32
zero, t, t, vC,
vC.

632
00:47:32 --> 00:47:37
If I apply zero for VI then
this should simply decay down to

633
00:47:37 --> 00:47:41
nothing over time.
And if I apply some VI but

634
00:47:41 --> 00:47:45
there is no state in the
capacitor then that same

635
00:47:45 --> 00:47:48
equation is going to look like
this.

636
00:47:48 --> 00:47:53
You can go and confirm for
yourselves that when I apply

637
00:47:53 --> 00:47:58
some input but the capacitor has
zero state, I start at zero,

638
00:47:58 --> 00:48:04
I finish up at VI and my wave
form looks like this.

639
00:48:04 --> 00:48:07
There you go.
That's the first one.

640
00:48:07 --> 00:48:13
The second one where I have 5
volts on the capacitor and no

641
00:48:13 --> 00:48:16
input.
Assume that at time equals zero

642
00:48:16 --> 00:48:21
I take away an input,
short the input voltage to

643
00:48:21 --> 00:48:25
ground for example,
apply zero volts.

644
00:48:25 --> 00:48:31
You will see the decay from 5
volts to 0 volts.

645
00:48:31 --> 00:48:37
And in the first case I start
with zero volts in my capacitor,

646
00:48:37 --> 00:48:42
I apply input of 5 volts,
and notice that at t=0 the

647
00:48:42 --> 00:48:45
capacitor rises up to that
level.

648
00:48:45 --> 00:48:51
So notice that these circuits
with capacitor and resistors are

649
00:48:51 --> 00:48:56
typified by wave forms that are
exponential rises and

650
00:48:56 --> 00:49:01
exponential decays.
We will see more of that next

651
00:49:01 --> 49:04
time.