1
00:00:00,000 --> 00:00:04,000
-- will try it again at the end
of this lecture and you show you

2
00:00:04,000 --> 00:00:09,000
that stuff hopefully next time.
For today we are going to start

3
00:00:09,000 --> 00:00:13,000
with nonlinear analysis.
Before we do that I wanted to

4
00:00:13,000 --> 00:00:16,000
do a little bit of review.

5
00:00:29,000 --> 00:00:34,000
I wanted to give you the past
three weeks in perspective and

6
00:00:34,000 --> 00:00:39,000
show you how all of these things
fit into the grand scheme of

7
00:00:39,000 --> 00:00:43,000
things.
We began by building a great

8
00:00:43,000 --> 00:00:47,000
little playground,
and within that playground we

9
00:00:47,000 --> 00:00:52,000
said that by enforcing upon
ourselves the lumped matter

10
00:00:52,000 --> 00:00:58,000
discipline we created the lumped
circuit abstraction.

11
00:00:58,000 --> 00:01:03,000
So within that playfield we
assumed that we had dq by dt and

12
00:01:03,000 --> 00:01:08,000
d phi by dt to be 0 so that gave
us as the lumped circuit

13
00:01:08,000 --> 00:01:11,000
abstraction.
And within that lumped circuit

14
00:01:11,000 --> 00:01:16,000
abstraction, within this
playground we looked at several

15
00:01:16,000 --> 00:01:21,000
methods of analyzing circuits,
including the KVL,

16
00:01:21,000 --> 00:01:24,000
KCL method.
We also learned the method

17
00:01:24,000 --> 00:01:29,000
involving composing resistors,
the voltage dividers and so on

18
00:01:29,000 --> 00:01:34,000
and solving circuits
intuitively.

19
00:01:34,000 --> 00:01:38,000
And we also looked at the node
method, which is kind of the

20
00:01:38,000 --> 00:01:41,000
workhorse of the circuits
industry.

21
00:01:41,000 --> 00:01:46,000
So when in doubt apply the node
method and it will get you where

22
00:01:46,000 --> 00:01:50,000
you want to go.
Now, we also said that this is

23
00:01:50,000 --> 00:01:53,000
good, here is our playground.
We said hey,

24
00:01:53,000 --> 00:01:58,000
if we focus on those circuits
that are linear we come to the

25
00:01:58,000 --> 00:02:04,000
left part of our playground.
And we said that for linear

26
00:02:04,000 --> 00:02:08,000
circuits in this part of the
playground we can further use a

27
00:02:08,000 --> 00:02:12,000
couple of techniques,
a few techniques,

28
00:02:12,000 --> 00:02:15,000
superposition,
Thevenin, Norton and so on.

29
00:02:15,000 --> 00:02:19,000
So these techniques allow you
to very quickly analyze

30
00:02:19,000 --> 00:02:23,000
complicated circuits,
especially when you're looking

31
00:02:23,000 --> 00:02:28,000
to find a single current,
or voltage or some parameter of

32
00:02:28,000 --> 00:02:31,000
interest.
Whenever you see,

33
00:02:31,000 --> 00:02:36,000
if you see a circuit containing
multiple voltage sources or two

34
00:02:36,000 --> 00:02:40,000
or more voltage sources or
current sources,

35
00:02:40,000 --> 00:02:43,000
as a first step think
superposition.

36
00:02:43,000 --> 00:02:48,000
And so these are very powerful
techniques that let you analyze

37
00:02:48,000 --> 00:02:52,000
very complicated circuits very
effectively.

38
00:02:52,000 --> 00:02:56,000
After we did this we said,
oh, let me draw another

39
00:02:56,000 --> 00:03:01,000
playground here.
This is another piece of our

40
00:03:01,000 --> 00:03:05,000
playground.
And if these are linear circuit

41
00:03:05,000 --> 00:03:10,000
then this half of the playground
is nonlinear circuits.

42
00:03:10,000 --> 00:03:16,000
And we said that if you further
focus on discretized values,

43
00:03:16,000 --> 00:03:21,000
if you discretized values and
focused only on circuits that

44
00:03:21,000 --> 00:03:25,000
dealt with binary signals,
highs and lows,

45
00:03:25,000 --> 00:03:32,000
then we came into this small
regime of the playground.

46
00:03:32,000 --> 00:03:36,000
And notice that digital
circuits are,

47
00:03:36,000 --> 00:03:40,000
by their very nature,
nonlinear.

48
00:03:40,000 --> 00:03:46,000
Remember the circuit,
A, B, this was one of our NOR

49
00:03:46,000 --> 00:03:51,000
gate circuits?
And if you look at transfer

50
00:03:51,000 --> 00:03:58,000
functions, that is if I plot,
let's say for example,

51
00:03:58,000 --> 00:04:04,000
for some combination of input
values.

52
00:04:04,000 --> 00:04:06,000
Let's say I plot v in verses v
out.

53
00:04:06,000 --> 00:04:10,000
Let's say, for example,
I turned this guy off by

54
00:04:10,000 --> 00:04:15,000
setting B to 0 and then I simply
apply a low to high transition

55
00:04:15,000 --> 00:04:19,000
at v in, then what I would see
at the output is a transfer

56
00:04:19,000 --> 00:04:24,000
function of the following sort
where as v in changes the output

57
00:04:24,000 --> 00:04:29,000
switches at some point and then
stays at a low value.

58
00:04:29,000 --> 00:04:33,000
So when v in is low v out is
high and v in and high v out is

59
00:04:33,000 --> 00:04:36,000
low.
So that's kind of the v out

60
00:04:36,000 --> 00:04:41,000
versus v in when B is set at 0.
So notice that this is a

61
00:04:41,000 --> 00:04:44,000
nonlinear curve.
This is not a straight line.

62
00:04:44,000 --> 00:04:49,000
It's a nonlinear curve.
And so therefore in the digital

63
00:04:49,000 --> 00:04:54,000
domain we see highly nonlinear
functions that look like this

64
00:04:54,000 --> 00:04:57,000
and so on.
However, take a look at this

65
00:04:57,000 --> 00:05:01,000
circuit.
Suppose I focus on the circuit

66
00:05:01,000 --> 00:05:04,000
for a given set of switch
settings.

67
00:05:04,000 --> 00:05:08,000
Let's say, for example,
I focus on the circuit when A

68
00:05:08,000 --> 00:05:12,000
and B are both 1s.
For a given set of switch

69
00:05:12,000 --> 00:05:17,000
settings, notice that I'm going
to be either in this region or

70
00:05:17,000 --> 00:05:21,000
in this region.
Notice that this region is a

71
00:05:21,000 --> 00:05:24,000
straight line.
So if I focus on let's say both

72
00:05:24,000 --> 00:05:30,000
A and B at once then I get
something like this.

73
00:05:35,000 --> 00:05:39,000
And in this situation,
for a given set of switch

74
00:05:39,000 --> 00:05:45,000
settings, notice that my digital
circuit now can be analyzed

75
00:05:45,000 --> 00:05:50,000
using linear techniques.
So therefore my digital gets

76
00:05:50,000 --> 00:05:57,000
moved into the linear domain for
a given set of switch settings.

77
00:06:02,000 --> 00:06:06,000
So if I fix my switch settings
and look at the circuit then

78
00:06:06,000 --> 00:06:08,000
each circuit,
for a given set of switch

79
00:06:08,000 --> 00:06:12,000
settings, is comprised of
voltage sources and some

80
00:06:12,000 --> 00:06:14,000
resistors and it's a linear
circuit.

81
00:06:14,000 --> 00:06:19,000
Again, I can go back and apply
all my linear techniques to

82
00:06:19,000 --> 00:06:22,000
virtually all the digital
circuits that you will be

83
00:06:22,000 --> 00:06:26,000
dealing with in 6.002.
Again, remember if I fix my

84
00:06:26,000 --> 00:06:29,000
switch settings,
if I fix the inputs then the

85
00:06:29,000 --> 00:06:34,000
output can be determined using
linear techniques.

86
00:06:34,000 --> 00:06:37,000
Because the digital circuits
we're showing you in 6.002

87
00:06:37,000 --> 00:06:41,000
simply comprise linear elements
like voltage sources and

88
00:06:41,000 --> 00:06:44,000
resistors and so on.
You'll see some more later.

89
00:06:44,000 --> 00:06:48,000
But you can apply your linear
techniques and analyze them.

90
00:06:48,000 --> 00:06:51,000
The cool thing here is that
with just two weeks of stuff

91
00:06:51,000 --> 00:06:55,000
that you've learned in 6.002,
you are well on our way to

92
00:06:55,000 --> 00:06:59,000
being able to analyze certain
classes of digital circuits for

93
00:06:59,000 --> 00:07:02,000
a given set of switch settings
and many, many,

94
00:07:02,000 --> 00:07:08,000
many linear circuits.
What we will do today is focus

95
00:07:08,000 --> 00:07:13,000
on nonlinear circuits.
So we look at this space.

96
00:07:13,000 --> 00:07:19,000
Notice again that up until now
we've dealt with these three

97
00:07:19,000 --> 00:07:25,000
methods, which apply to all
circuits within this playground,

98
00:07:25,000 --> 00:07:31,000
the lumped circuit playground.
And the subset of that is the

99
00:07:31,000 --> 00:07:34,000
linear domain.
And we can analyze linear

100
00:07:34,000 --> 00:07:37,000
circuits in this way.
And digital circuits,

101
00:07:37,000 --> 00:07:42,000
for a given set of switch
settings, also fall within this

102
00:07:42,000 --> 00:07:45,000
category.
So notice that you can go ahead

103
00:07:45,000 --> 00:07:50,000
and analyze the digital circuits
using superposition or other

104
00:07:50,000 --> 00:07:54,000
techniques like that.
The next big step for us is to

105
00:07:54,000 --> 00:07:59,000
begin our analysis of nonlinear
circuits today.

106
00:07:59,000 --> 00:08:03,000
The important thing to remember
is that nonlinear circuits are

107
00:08:03,000 --> 00:08:07,000
also within this big playground
in which we are going under the

108
00:08:07,000 --> 00:08:11,000
lumped matter discipline.
So nonlinear circuits are also

109
00:08:11,000 --> 00:08:15,000
lumped circuits.
And therefore because we are in

110
00:08:15,000 --> 00:08:19,000
that playground we can use any
one of our techniques,

111
00:08:19,000 --> 00:08:23,000
KVL, KCL or the node method to
analyze nonlinear circuits.

112
00:08:23,000 --> 00:08:28,000
So if you see a nonlinear
circuit, don't get daunted.

113
00:08:28,000 --> 00:08:31,000
Just remember this is meant to
be simple stuff.

114
00:08:31,000 --> 00:08:37,000
So let me simply write down the
node equation and analyze it.

115
00:08:37,000 --> 00:08:41,000
There is really nothing new in
today's lecture.

116
00:08:41,000 --> 00:08:46,000
I'm just going to show you a
nonlinear circuit and analyzing

117
00:08:46,000 --> 00:08:49,000
using techniques that you
already know.

118
00:08:49,000 --> 00:08:54,000
Today nonlinear circuits.
And we look at several methods

119
00:08:54,000 --> 00:09:01,000
of analyzing nonlinear circuits.
We look at the "Analytic

120
00:09:01,000 --> 00:09:05,000
Method".
We look at a "Graphical

121
00:09:05,000 --> 00:09:11,000
Method".
You will look at a "Piecewise

122
00:09:11,000 --> 00:09:16,000
Linear Method" in the book.

123
00:09:27,000 --> 00:09:29,000
I won't be covering this in
lecture.

124
00:09:29,000 --> 00:09:33,000
You can read Section 4.4 for
the piecewise linear method.

125
00:09:33,000 --> 00:09:38,000
In this method you take your
curves and you approximate them

126
00:09:38,000 --> 00:09:44,000
with a bunch of straight line
segments, kind of like the v

127
00:09:44,000 --> 00:09:49,000
out, v in curve I've shown you
there, and analyze the circuit

128
00:09:49,000 --> 00:09:55,000
using linear techniques within
any given straight line segment.

129
00:09:55,000 --> 00:09:59,000
We will also do incremental
analysis.

130
00:09:59,000 --> 00:10:04,000
This is also called small
signal analysis.

131
00:10:04,000 --> 00:10:08,000
So I will cover these two
today, I will introduce this one

132
00:10:08,000 --> 00:10:13,000
today, and wrap that up during
the next lecture.

133
00:10:25,000 --> 00:10:27,000
Let's start with a simple
example.

134
00:10:36,000 --> 00:10:40,000
So I have some voltage,
V, some voltage source V.

135
00:10:40,000 --> 00:10:43,000
And I have some resistor,
R.

136
00:10:43,000 --> 00:10:49,000
And I have a fictitious device
here that I labeled D.

137
00:10:49,000 --> 00:10:54,000
Let's call this fictitious
device the "Expo Dweeb".

138
00:10:54,000 --> 00:11:00,000
I purposely chose a funky name
because this is a fictitious

139
00:11:00,000 --> 00:11:04,000
device.
Let's call it the Expo Dweeb.

140
00:11:04,000 --> 00:11:09,000
And let me write down the
associated variables for this

141
00:11:09,000 --> 00:11:13,000
device as follows.
iD is the current flowing into

142
00:11:13,000 --> 00:11:17,000
this terminal and vD is the
voltage across this device.

143
00:11:17,000 --> 00:11:21,000
So this is a nonlinear device.

144
00:11:29,000 --> 00:11:33,000
And this device is
characterized by the following

145
00:11:33,000 --> 00:11:35,000
equation.
Much like resistors were

146
00:11:35,000 --> 00:11:40,000
characterized by an iV relation,
V is equal to iR,

147
00:11:40,000 --> 00:11:43,000
or i is equal to V/R.
This device is also

148
00:11:43,000 --> 00:11:47,000
characterized by the following
element relationship.

149
00:11:47,000 --> 00:11:52,000
It's a e raised to bvD.
So there is an exponentiation

150
00:11:52,000 --> 00:11:55,000
here.
Again, this is a fictitious

151
00:11:55,000 --> 00:11:58,000
device.
And I'll show some funky things

152
00:11:58,000 --> 00:12:05,000
that it does in a second.
It's a very simple relation.

153
00:12:05,000 --> 00:12:13,000
It's an exponential relation
where the current relates to the

154
00:12:13,000 --> 00:12:21,000
exponentiated value of the
voltage vD across the element.

155
00:12:21,000 --> 00:12:29,000
So I can plot iD versus vD for
this element as follows.

156
00:12:29,000 --> 00:12:34,000
Notice that when vD is 0 iD is
a, so I have a here,

157
00:12:34,000 --> 00:12:39,000
and it looks like this.
It's a funny device,

158
00:12:39,000 --> 00:12:42,000
a fictitious device.
So when vD is 0,

159
00:12:42,000 --> 00:12:49,000
I have some current flowing the
device, and as vD increases I

160
00:12:49,000 --> 00:12:57,000
get an exponential increase in
the current through that device.

161
00:12:57,000 --> 00:13:00,000
This device is funny in the
sense that it is not a passive

162
00:13:00,000 --> 00:13:04,000
device in that notice that when
vD and iD are positive the

163
00:13:04,000 --> 00:13:06,000
product is positive,
which is fine,

164
00:13:06,000 --> 00:13:08,000
which says that it is consuming
power.

165
00:13:08,000 --> 00:13:11,000
On the other hand,
on the left-hand side notice

166
00:13:11,000 --> 00:13:15,000
that the vI relation is
negative, which means that when

167
00:13:15,000 --> 00:13:19,000
I put a negative voltage on it,
it can still sustain a positive

168
00:13:19,000 --> 00:13:21,000
current.
This must imply that the device

169
00:13:21,000 --> 00:13:24,000
is producing power.
But for the purpose of a

170
00:13:24,000 --> 00:13:29,000
nonlinear analysis we don't have
to worry about that.

171
00:13:29,000 --> 00:13:33,000
Let's just do it mathematically
and find out what it looks like.

172
00:13:33,000 --> 00:13:36,000
So back to this again.
I have a voltage source,

173
00:13:36,000 --> 00:13:40,000
a resistor and my Expo Dweeb
connected in that manner.

174
00:13:40,000 --> 00:13:43,000
Now, again, reflect on this
pattern.

175
00:13:43,000 --> 00:13:47,000
A voltage source or a current
source, a resistor and some

176
00:13:47,000 --> 00:13:50,000
device.
This is a very standard pattern

177
00:13:50,000 --> 00:13:52,000
you will see again and again and
again.

178
00:13:52,000 --> 00:13:55,000
In particular,
if you look at this device,

179
00:13:55,000 --> 00:14:01,000
it's a nonlinear device here.
And facing the nonlinear device

180
00:14:01,000 --> 00:14:05,000
is a voltage source in series
with a resistor.

181
00:14:05,000 --> 00:14:09,000
And the reason I say that this
is an incredibly important

182
00:14:09,000 --> 00:14:14,000
pairing is the following.
Notice that if on the left-hand

183
00:14:14,000 --> 00:14:19,000
side I had any linear circuit
and I had a single nonlinear

184
00:14:19,000 --> 00:14:23,000
element in that circuit.
Notice that by a Thevenin

185
00:14:23,000 --> 00:14:29,000
reduction that you've learned
you can take this entire mess.

186
00:14:29,000 --> 00:14:32,000
If all you care about is the
behavior of the nonlinear

187
00:14:32,000 --> 00:14:36,000
device, for the purpose of
analyzing this nonlinear device,

188
00:14:36,000 --> 00:14:39,000
you can take this entire linear
circuit, no matter how

189
00:14:39,000 --> 00:14:41,000
complicated it is,
voltage sources,

190
00:14:41,000 --> 00:14:44,000
current sources,
resistors and a bunch of other

191
00:14:44,000 --> 00:14:47,000
funky stuff, you can boil all of
that down to a Thevenin

192
00:14:47,000 --> 00:14:50,000
equivalent, a voltage and a
resistor in series.

193
00:14:50,000 --> 00:14:54,000
So we can trick you.
We can give you a complicated

194
00:14:54,000 --> 00:14:57,000
circuit and say ah-ha,
tell me what the current is

195
00:14:57,000 --> 00:14:59,000
through this device if I apply
some voltage,

196
00:14:59,000 --> 00:15:03,000
3 volts there.
What you can do is you can say

197
00:15:03,000 --> 00:15:07,000
ah-ha, I don't care what happens
here so I'm just going to

198
00:15:07,000 --> 00:15:09,000
replace the whole thing with a
Thevenin equivalent.

199
00:15:09,000 --> 00:15:13,000
And you've done your homework
now and you can calculate

200
00:15:13,000 --> 00:15:15,000
Thevenin equivalents for
circuits.

201
00:15:15,000 --> 00:15:18,000
And simply replace this and
then go ahead and solve the

202
00:15:18,000 --> 00:15:19,000
circuit.
Again, remember we are

203
00:15:19,000 --> 00:15:22,000
engineers.
We are looking for answers.

204
00:15:22,000 --> 00:15:24,000
We are looking to build
interesting systems.

205
00:15:24,000 --> 00:15:27,000
And, in general,
we like to take the simplest

206
00:15:27,000 --> 00:15:31,000
path possible to the solution.
So simplify your lives and

207
00:15:31,000 --> 00:15:35,000
create a simple Thevenin coupled
to a nonlinear device and then

208
00:15:35,000 --> 00:15:39,000
you will be rolling.
When we talk about a variety of

209
00:15:39,000 --> 00:15:41,000
other circuits,
nonlinear circuits,

210
00:15:41,000 --> 00:15:45,000
time-varying circuits and so on
in the rest of this course,

211
00:15:45,000 --> 00:15:48,000
we will look at this pattern
again and again and again and

212
00:15:48,000 --> 00:15:52,000
again until we are blue in the
face.

213
00:15:52,000 --> 00:15:56,000
And, just remember,
the reason we keep looking at

214
00:15:56,000 --> 00:16:01,000
this pattern is that whenever
you have some big linear mess

215
00:16:01,000 --> 00:16:06,000
connected to some interesting
device what you can do is if all

216
00:16:06,000 --> 00:16:11,000
you care about is analyzing the
behavior of that device,

217
00:16:11,000 --> 00:16:16,000
you can take this linear mess
and simply figure out the

218
00:16:16,000 --> 00:16:21,000
Thevenin equivalent,
or the Norton equivalent if you

219
00:16:21,000 --> 00:16:26,000
like and replace this whole
thing with its equivalent and

220
00:16:26,000 --> 00:16:32,000
then go ahead and analyze it.
So boil an arbitrarily circuit

221
00:16:32,000 --> 00:16:35,000
down to a very simple pattern of
this sort.

222
00:16:35,000 --> 00:16:38,000
What this means is because of
this brilliant Thevenin

223
00:16:38,000 --> 00:16:41,000
simplification,
going forward through the rest

224
00:16:41,000 --> 00:16:45,000
of this course we will mostly
deal with very simple circuits

225
00:16:45,000 --> 00:16:49,000
like this, voltage source,
resistor and the device.

226
00:16:49,000 --> 00:16:52,000
That's it.
Very, very, very rarely will

227
00:16:52,000 --> 00:16:56,000
you see multiple sources and
lots of resistors in a circuit.

228
00:16:56,000 --> 00:17:00,000
It's usually going to be simple
stuff.

229
00:17:00,000 --> 00:17:03,000
And remember how we got here,
by making a Thevenin

230
00:17:03,000 --> 00:17:07,000
simplification of a linear mess.
All right.

231
00:17:07,000 --> 00:17:10,000
If in homeworks or quizzes or
in real life,

232
00:17:10,000 --> 00:17:15,000
or in many examples of real
life, if you find that you have

233
00:17:15,000 --> 00:17:19,000
to deal with a lot of grunge and
a lot of mess,

234
00:17:19,000 --> 00:17:21,000
step back and think a little
bit.

235
00:17:21,000 --> 00:17:26,000
Try to use intuition and see if
you can simplify things using

236
00:17:26,000 --> 00:17:32,000
some clever trick or method.
Method 1 of analysis.

237
00:17:32,000 --> 00:17:36,000
Let's go ahead and analyze this
pattern here,

238
00:17:36,000 --> 00:17:40,000
this template circuit,
if you will,

239
00:17:40,000 --> 00:17:46,000
a voltage source a resistor and
a nonlinear device.

240
00:17:46,000 --> 00:17:52,000
This is the analytical method.
And remember the node method

241
00:17:52,000 --> 00:17:59,000
applies, so let me go ahead and
apply the node method.

242
00:17:59,000 --> 00:18:01,000
To apply the node method,
what do I do?

243
00:18:01,000 --> 00:18:03,000
I first have to select a ground
node.

244
00:18:03,000 --> 00:18:06,000
Let me insulate this as my
ground node.

245
00:18:06,000 --> 00:18:09,000
Let me label all the nodes with
their voltages.

246
00:18:09,000 --> 00:18:13,000
So this node has voltage V and
this node has label the capital

247
00:18:13,000 --> 00:18:15,000
D.
So let me go ahead and analyze

248
00:18:15,000 --> 00:18:19,000
this using the node method.
So the node method says for

249
00:18:19,000 --> 00:18:23,000
each of the nodes in the circuit
whose voltage is not known go

250
00:18:23,000 --> 00:18:27,000
ahead and write down KCL
implicitly applying the element

251
00:18:27,000 --> 00:18:31,000
relationships to replace the
current values with the voltage

252
00:18:31,000 --> 00:18:35,000
values.
Let's start with the current

253
00:18:35,000 --> 00:18:40,000
going in that direction.
Current going from the vD node

254
00:18:40,000 --> 00:18:44,000
through resistor R,
which looks as follows,

255
00:18:44,000 --> 00:18:48,000
vD - V divided by R.
That's a current going that

256
00:18:48,000 --> 00:18:51,000
way.
And the current going down is

257
00:18:51,000 --> 00:18:54,000
iD.
In general, when I apply the

258
00:18:54,000 --> 00:18:59,000
node method, I don't write iD
here but I go ahead and write

259
00:18:59,000 --> 00:19:04,000
the element relation ae to the
bvD here.

260
00:19:04,000 --> 00:19:08,000
Then I get an equation in vD
and I just solve the mode

261
00:19:08,000 --> 00:19:12,000
voltage.
However, just to make a couple

262
00:19:12,000 --> 00:19:16,000
of extra points later,
let me go ahead and do that in

263
00:19:16,000 --> 00:19:22,000
two steps, write down this and
then go ahead and write down iD

264
00:19:22,000 --> 00:19:25,000
separately as ae to the bvD.
Again, remember,

265
00:19:25,000 --> 00:19:30,000
don't get confused here.
In a node method,

266
00:19:30,000 --> 00:19:32,000
I don't write down a second
step.

267
00:19:32,000 --> 00:19:35,000
I directly write down ae to bvD
in place of iD.

268
00:19:35,000 --> 00:19:38,000
I get one equation in vD,
I go solve it.

269
00:19:38,000 --> 00:19:41,000
Just for fun today,
I'm taking two steps here,

270
00:19:41,000 --> 00:19:45,000
writing iD and explicitly
putting down iD as ae to the

271
00:19:45,000 --> 00:19:46,000
bvD.
Now, that's it.

272
00:19:46,000 --> 00:19:48,000
I mean this is all there is to
it.

273
00:19:48,000 --> 00:19:52,000
You guys can now go ahead and
analyze nonlinear circuits.

274
00:19:52,000 --> 00:19:56,000
You get a bunch of equations,
a bunch of unknowns,

275
00:19:56,000 --> 00:20:00,000
go solve.
I have two equations here.

276
00:20:00,000 --> 00:20:04,000
vD and iD are my unknowns and I
can just go ahead and solve for

277
00:20:04,000 --> 00:20:06,000
them.
Now, in general with nonlinear

278
00:20:06,000 --> 00:20:10,000
circuits, often times it's hard
to get a closed form solution so

279
00:20:10,000 --> 00:20:12,000
you may have to use a bunch of
methods.

280
00:20:12,000 --> 00:20:16,000
You can try a closed form
solution or you can try

281
00:20:16,000 --> 00:20:19,000
numerical solutions or you can
do trial and error.

282
00:20:19,000 --> 00:20:21,000
In this case,
I'll just go ahead and tell

283
00:20:21,000 --> 00:20:27,000
you.
Suppose I choose V as 1 volt,

284
00:20:27,000 --> 00:20:36,000
R is 1 ohm and b is 1 over volt
and a is º amps for those

285
00:20:36,000 --> 00:20:45,000
values, approximately vD is
roughly 0.5 volts and iD is

286
00:20:45,000 --> 00:20:53,000
roughly 0.4 volts.
You can do this by using trial

287
00:20:53,000 --> 00:21:00,000
and error or other methods.
In 6.002 we don't dwell on

288
00:21:00,000 --> 00:21:03,000
working too hard to solve
equations of this sort.

289
00:21:03,000 --> 00:21:06,000
If you cannot substitute this
in here and solve it directly,

290
00:21:06,000 --> 00:21:10,000
we don't ask you to go and
learn numerical method and the

291
00:21:10,000 --> 00:21:12,000
techniques and so on to solve
it.

292
00:21:12,000 --> 00:21:15,000
But just remember that you can
use trial and error or you can

293
00:21:15,000 --> 00:21:19,000
use back substitution and other
techniques that you will learn

294
00:21:19,000 --> 00:21:22,000
in future numerical methods
classes and apply it here.

295
00:21:22,000 --> 00:21:25,000
But suffice it to say that,
for here we can stick with

296
00:21:25,000 --> 00:21:30,000
trial and error if you like.
And for these values,

297
00:21:30,000 --> 00:21:33,000
vD and iD are 0.5 and
approximately 0.4.

298
00:21:33,000 --> 00:21:37,000
You're done.
It's really that simple.

299
00:21:37,000 --> 00:21:39,000
Yes.
Oh, I'm sorry.

300
00:21:39,000 --> 00:21:42,000
Good catch.
I know there is one person

301
00:21:42,000 --> 00:21:45,000
that's not sleeping here.
Good.

302
00:21:45,000 --> 00:21:49,000
So, as I said,
there's not a whole lot to it.

303
00:21:49,000 --> 00:21:55,000
Whether it's a nonlinear
circuit or a linear circuit and

304
00:21:55,000 --> 00:22:00,000
as long as I am inside this
playground here where the lumped

305
00:22:00,000 --> 00:22:06,000
circuit abstraction holds,
I can apply my node equations

306
00:22:06,000 --> 00:22:11,000
and then go ahead and solve it.

307
00:22:17,000 --> 00:22:23,000
Let me show you a few more
methods so we can articulate

308
00:22:23,000 --> 00:22:30,000
your repertoire of tools for
nonlinear circuits.

309
00:22:30,000 --> 00:22:33,000
And I'd like to show you a
graphical technique.

310
00:22:33,000 --> 00:22:37,000
I personally rarely use a
graphical technique to solve

311
00:22:37,000 --> 00:22:40,000
circuits.
And why am I sharing this with

312
00:22:40,000 --> 00:22:42,000
you?
It turns out that often times

313
00:22:42,000 --> 00:22:47,000
by looking at things graphically
you can get some better insights

314
00:22:47,000 --> 00:22:51,000
into circuit behavior.
You can also show cool demos

315
00:22:51,000 --> 00:22:55,000
when you show graphs of
responses kind of playing with

316
00:22:55,000 --> 00:23:00,000
each other and so on.
So this is fun for getting

317
00:23:00,000 --> 00:23:06,000
intuition and things like that.
Graphically all I'm really

318
00:23:06,000 --> 00:23:11,000
going to do is solve those two
equations graphically.

319
00:23:11,000 --> 00:23:15,000
So I'm going to plot equation
one.

320
00:23:15,000 --> 00:23:19,000
Let me rewrite equation one as
follows.

321
00:23:19,000 --> 00:23:21,000
iD is --

322
00:23:26,000 --> 00:23:30,000
I'm just rewriting equation one
as follows.

323
00:23:30,000 --> 00:23:40,000
V/R - vD/R.
And I can also draw the second

324
00:23:40,000 --> 00:23:42,000
guy --

325
00:23:51,000 --> 00:24:04,000
OK, I can do this as well.
I can do an iD versus vD plot.

326
00:24:04,000 --> 00:24:09,000
And in this particular
situation, you've seen this

327
00:24:09,000 --> 00:24:14,000
already, that's my iD versus vD
curve right there.

328
00:24:14,000 --> 00:24:19,000
And I can do the same for this
one here.

329
00:24:19,000 --> 00:24:25,000
So this equation establishes
the following straight line

330
00:24:25,000 --> 00:24:29,000
relationship.
It says that when vD is 0,

331
00:24:29,000 --> 00:24:33,000
iD is V/R.
So that's here.

332
00:24:33,000 --> 00:24:38,000
And similarly when iD is 0 then
vD is equal to V so I get

333
00:24:38,000 --> 00:24:41,000
something here.
So that's my straight line

334
00:24:41,000 --> 00:24:46,000
relationship corresponding to
this equation here.

335
00:24:46,000 --> 00:24:50,000
So what I can do is I can
simply solve these by

336
00:24:50,000 --> 00:24:55,000
superimposing the two curves on
the same vD, iD template here

337
00:24:55,000 --> 00:25:00,000
and finding the intersection of
the curves.

338
00:25:00,000 --> 00:25:06,000
So I can take this curve
corresponding to two and I can

339
00:25:06,000 --> 00:25:14,000
take this curve corresponding to
one, and this is V/R and this is

340
00:25:14,000 --> 00:25:19,000
V, 0, and I can find the
intersection point.

341
00:25:19,000 --> 00:25:24,000
This curve here,
for reasons that will be

342
00:25:24,000 --> 00:25:33,000
obvious about three weeks from
now, is called the load line.

343
00:25:33,000 --> 00:25:37,000
It's called the load line.
You will understand why that is

344
00:25:37,000 --> 00:25:42,000
so in a later lecture.
So I've given you a template on

345
00:25:42,000 --> 00:25:46,000
Page 6 to boil these two down
into one equation.

346
00:25:46,000 --> 00:25:49,000
So there, again,
you can substitute the values

347
00:25:49,000 --> 00:25:54,000
for V is 1 volt and R is 1 and
so on and so forth and get the

348
00:25:54,000 --> 00:25:59,000
same kind of result as you did
previously.

349
00:26:05,000 --> 00:26:07,000
So there is really nothing new
here.

350
00:26:07,000 --> 00:26:10,000
All I've done in the second
method is combined the two

351
00:26:10,000 --> 00:26:14,000
equations graphically and found
the solution by looking at where

352
00:26:14,000 --> 00:26:17,000
the two curves intersect.

353
00:26:26,000 --> 00:26:30,000
At the start of the lecture I
also told you that you may want

354
00:26:30,000 --> 00:26:35,000
to go and check out the
piecewise linear technique --

355
00:26:42,000 --> 00:26:46,000
-- in Section 4.4 of the course
notes.

356
00:26:53,000 --> 00:26:58,000
All right.
For today let me do a third

357
00:26:58,000 --> 00:27:05,000
method called "Incremental
Analysis".

358
00:27:13,000 --> 00:27:18,000
This technique is also called
the small signal method.

359
00:27:30,000 --> 00:27:33,000
I'm going to show you,
before I go into the method,

360
00:27:33,000 --> 00:27:38,000
in today's lecture what I'll do
is I'll give you a motivating

361
00:27:38,000 --> 00:27:42,000
example for why we need the
small signal approach.

362
00:27:42,000 --> 00:27:47,000
I'll give you a motivating
example and show you a little

363
00:27:47,000 --> 00:27:49,000
demo.
And then I will close with

364
00:27:49,000 --> 00:27:53,000
showing you a problem with
applying a standard approach,

365
00:27:53,000 --> 00:27:58,000
and I'll ask you to see if you
can figure out a way to handle

366
00:27:58,000 --> 00:28:03,000
it in time for next lecture.
So let me give you the

367
00:28:03,000 --> 00:28:04,000
motivation here.

368
00:28:10,000 --> 00:28:14,000
So here is what I want to do.
Many of you have seen one of

369
00:28:14,000 --> 00:28:17,000
those electric eye garage door
openers, right?

370
00:28:17,000 --> 00:28:22,000
You have a receiver at one end
and you have some kind of a

371
00:28:22,000 --> 00:28:26,000
light beam at the other,
and when you walk through it

372
00:28:26,000 --> 00:28:30,000
stops, or rather it cuts the
circuit and stops the door from

373
00:28:30,000 --> 00:28:33,000
closing.
And when no one is going

374
00:28:33,000 --> 00:28:37,000
through it maintains a
connection and lets the door

375
00:28:37,000 --> 00:28:40,000
close.
So what we did is we went to

376
00:28:40,000 --> 00:28:44,000
Home Depot, or one of those
stores, and bought a very

377
00:28:44,000 --> 00:28:49,000
standard device that essentially
produces some response when

378
00:28:49,000 --> 00:28:53,000
light impinges on it.
And my goal will be to see if I

379
00:28:53,000 --> 00:28:57,000
can send music over the light
beam using a simple garage door

380
00:28:57,000 --> 00:29:01,000
opener device.
So here is the little circuit

381
00:29:01,000 --> 00:29:03,000
that I will do.
We actually went there and

382
00:29:03,000 --> 00:29:05,000
built this.
I will also show you a demo.

383
00:29:11,000 --> 00:29:15,000
Here is my time-varying
voltage, vI(t),

384
00:29:15,000 --> 00:29:20,000
and this is some music signal.

385
00:29:28,000 --> 00:29:32,000
And get some music signal.
And I want to connect this to

386
00:29:32,000 --> 00:29:38,000
this device, which is a device
found in garage door openers.

387
00:29:38,000 --> 00:29:43,000
I am going to call it a LED.
If you like,

388
00:29:43,000 --> 00:29:47,000
you can view it as,
this is very similar to our

389
00:29:47,000 --> 00:29:51,000
Expo Dweeb.
This is called a "Light

390
00:29:51,000 --> 00:29:55,000
Emitting Expo Dweeb".
That's why it is LED.

391
00:29:55,000 --> 00:30:00,000
So what the LED does is,
as I apply this voltage across

392
00:30:00,000 --> 00:30:06,000
it, that same voltage appears
across the Light Emitting Expo

393
00:30:06,000 --> 00:30:11,000
Dweeb.
And there is some current that

394
00:30:11,000 --> 00:30:15,000
flows through the device.
And for our analysis we will

395
00:30:15,000 --> 00:30:19,000
assume that this device
virtually has an identical iD

396
00:30:19,000 --> 00:30:24,000
characteristic to the Expo Dweeb
just that it emits light.

397
00:30:24,000 --> 00:30:29,000
So when I pass a current
through it, it emits light.

398
00:30:29,000 --> 00:30:34,000
And the light intensity is
proportional to the current that

399
00:30:34,000 --> 00:30:38,000
flows through.
So it emits light and light

400
00:30:38,000 --> 00:30:42,000
intensity, LD,
is proportional to iD.

401
00:30:47,000 --> 00:30:49,000
Here is my little light
emitting device,

402
00:30:49,000 --> 00:30:52,000
which when current flows
through it, itproduces light

403
00:30:52,000 --> 00:30:56,000
because its intensity is
proportional to the current.

404
00:30:56,000 --> 00:31:00,000
And what I will do is I will
stick in the receiver here.

405
00:31:00,000 --> 00:31:05,000
Think of it as a photo resistor
or some other device where I am

406
00:31:05,000 --> 00:31:08,000
going to connect that in a
circuit.

407
00:31:08,000 --> 00:31:13,000
I am not going to spend too
much time on this side.

408
00:31:13,000 --> 00:31:17,000
I'm going to focus on the
left-hand side here.

409
00:31:17,000 --> 00:31:22,000
And let's say I have some kind
of amplifier and speakers and so

410
00:31:22,000 --> 00:31:27,000
on and so forth.
Suffice it to say that when the

411
00:31:27,000 --> 00:31:33,000
light falls on this device PR
that iR that goes through here

412
00:31:33,000 --> 00:31:37,000
is proportional to the received
light intensity.

413
00:31:37,000 --> 00:31:41,000
So if the current is
proportional to the received

414
00:31:41,000 --> 00:31:47,000
light intensity then I amplify
that signal in my amplifier and

415
00:31:47,000 --> 00:31:50,000
I get the music playing out
here.

416
00:31:50,000 --> 00:31:55,000
And notice that the following
chain of dependences apply.

417
00:31:55,000 --> 00:32:00,000
So I have an input music signal
VI.

418
00:32:00,000 --> 00:32:06,000
That gets converted to some iD.
These are all time-varying

419
00:32:06,000 --> 00:32:13,000
signals, so VI is a time-varying
signal and so is iD.

420
00:32:13,000 --> 00:32:19,000
And iD gets converted to light
of some intensity LD.

421
00:32:19,000 --> 00:32:26,000
This in turn gets attenuated
somewhat and is received at the

422
00:32:26,000 --> 00:32:31,000
photo resistor.
And I get some intensity LR

423
00:32:31,000 --> 00:32:36,000
impinging on that device there.
And that in turn produces a

424
00:32:36,000 --> 00:32:40,000
current iR and then iR is
amplified and goes through a

425
00:32:40,000 --> 00:32:43,000
speaker and so on and produces
sound.

426
00:32:43,000 --> 00:32:48,000
Notice that using this chain
I've taken a music signal here

427
00:32:48,000 --> 00:32:53,000
and I am playing it here.
And just imagine that this is

428
00:32:53,000 --> 00:32:58,000
your garage door opener device
here where the light emitted is

429
00:32:58,000 --> 00:33:03,000
being articulated by the voltage
signal VI.

430
00:33:03,000 --> 00:33:06,000
And received here.
So notice that if I cut this,

431
00:33:06,000 --> 00:33:09,000
if I stick something in here
and block it then I get no

432
00:33:09,000 --> 00:33:12,000
response here,
but if I take my hand away then

433
00:33:12,000 --> 00:33:15,000
I do get some response.
But this is fine.

434
00:33:15,000 --> 00:33:18,000
This should work.
You could try this at home if

435
00:33:18,000 --> 00:33:20,000
you'd like.
If you have a garage door

436
00:33:20,000 --> 00:33:24,000
opener, just stick a little
circuit like this and it should

437
00:33:24,000 --> 00:33:26,000
simply work.
We have a problem,

438
00:33:26,000 --> 00:33:30,000
though.
The problem is that,

439
00:33:30,000 --> 00:33:36,000
as I said, I'm using the Expo
Dweeb here, the light emitting

440
00:33:36,000 --> 00:33:41,000
Expo Dweeb, and its
characteristics are as follows.

441
00:33:41,000 --> 00:33:46,000
iD is exponentially related to
the voltage vD,

442
00:33:46,000 --> 00:33:49,000
so this is nonlinear.

443
00:33:57,000 --> 00:34:02,000
And that's a real problem.
Because this is nonlinear,

444
00:34:02,000 --> 00:34:05,000
I am going to get a distorted
output.

445
00:34:05,000 --> 00:34:10,000
Let me show you a little wave
form, a little graph to show you

446
00:34:10,000 --> 00:34:14,000
how the distortion happens and
then show you a little demo

447
00:34:14,000 --> 00:34:19,000
showing you the distortion.
Let me graphically show you the

448
00:34:19,000 --> 00:34:23,000
kind of distortion that is
happening here,

449
00:34:23,000 --> 00:34:28,000
and I will do it by drawing the
following graph.

450
00:34:36,000 --> 00:34:40,000
So this is the vD,
iD curve for our device.

451
00:34:40,000 --> 00:34:46,000
And what I'm going to plot for
you is if I have a time-varying

452
00:34:46,000 --> 00:34:52,000
vD voltage, I just want to see
what the time-varying iD current

453
00:34:52,000 --> 00:34:56,000
looks like.
And a trick to plot that is to

454
00:34:56,000 --> 00:35:02,000
take your input voltage like so.
And let's say I apply a

455
00:35:02,000 --> 00:35:04,000
sinusoid.
So I am just taking a

456
00:35:04,000 --> 00:35:10,000
time-varying sinusoidal voltage
and rotating the plot 90 degrees

457
00:35:10,000 --> 00:35:15,000
like so, so I can see where
these points correspond to on

458
00:35:15,000 --> 00:35:19,000
that curve.
So what this says is that at

459
00:35:19,000 --> 00:35:21,000
some point here,
for example,

460
00:35:21,000 --> 00:35:25,000
where vI, at this point and
time, vI is here.

461
00:35:25,000 --> 00:35:30,000
Notice vI and vD are the same
thing because vI is applied

462
00:35:30,000 --> 00:35:35,000
across vD.
vI directly applies across the

463
00:35:35,000 --> 00:35:39,000
device, and so vI equals vD at
all time.

464
00:35:39,000 --> 00:35:43,000
So this voltage here
corresponds to this voltage,

465
00:35:43,000 --> 00:35:49,000
it corresponds to this current
and then I can find out what the

466
00:35:49,000 --> 00:35:54,000
current is for that voltage.
By using the same artifice I

467
00:35:54,000 --> 00:35:59,000
can plot the output current iD
like so.

468
00:35:59,000 --> 00:36:02,000
So for this value I get some
current here.

469
00:36:02,000 --> 00:36:07,000
And so at time T0 I start here.
And notice that as this signal

470
00:36:07,000 --> 00:36:09,000
moves up here,
I can find out the

471
00:36:09,000 --> 00:36:14,000
corresponding values of iD by
looking at where a straight line

472
00:36:14,000 --> 00:36:18,000
intersects here and plotting the
values here.

473
00:36:18,000 --> 00:36:22,000
I have a nice little graphical
animation to show you this.

474
00:36:22,000 --> 00:36:29,000
Hopefully, the laptop will work
tomorrow and we can check that.

475
00:36:29,000 --> 00:36:33,000
I am doing nothing new here.
Just showing you a trick to be

476
00:36:33,000 --> 00:36:37,000
able to plot vI versus v out
relationships,

477
00:36:37,000 --> 00:36:41,000
or vI or versus other
relationships based on some kind

478
00:36:41,000 --> 00:36:46,000
of a transfer function.
So what you end up getting is

479
00:36:46,000 --> 00:36:49,000
something that looks like this.
Why is that?

480
00:36:49,000 --> 00:36:54,000
Notice that this curve here
corresponds to the signal.

481
00:36:54,000 --> 00:36:59,000
As this signal moves from here
to here, this point moves from

482
00:36:59,000 --> 00:37:04,000
here to here and that
corresponds to this iD.

483
00:37:04,000 --> 00:37:08,000
When this moves from here to
here that corresponds to a point

484
00:37:08,000 --> 00:37:11,000
moving from this part of the
curve to here,

485
00:37:11,000 --> 00:37:16,000
and that looks like so.
And then for the whole negative

486
00:37:16,000 --> 00:37:20,000
incursion, notice that the whole
negative incursion moves here,

487
00:37:20,000 --> 00:37:25,000
so for that entire negative
incursion I get an output that

488
00:37:25,000 --> 00:37:28,000
looks like this.
Notice that this device has

489
00:37:28,000 --> 00:37:34,000
completely cut off and hammered
negative going signals.

490
00:37:34,000 --> 00:37:37,000
What it's done is that rather
than giving me a nice little

491
00:37:37,000 --> 00:37:41,000
negative spike incursion here,
or excursion here,

492
00:37:41,000 --> 00:37:45,000
what this is doing is that it
is taking this excursion and

493
00:37:45,000 --> 00:37:47,000
simply slamming it down to this
value here.

494
00:37:47,000 --> 00:37:50,000
And then again,
when I go back up,

495
00:37:50,000 --> 00:37:53,000
I get this peak here.
So notice that what was a nice

496
00:37:53,000 --> 00:37:57,000
little sinusoid out there gets
hammered and squished into this

497
00:37:57,000 --> 00:38:01,000
funny curve here.
What this device is doing is

498
00:38:01,000 --> 00:38:06,000
for positive values it tends to
produce exponentially greater

499
00:38:06,000 --> 00:38:09,000
current so I get boom,
high-rising peaks corresponding

500
00:38:09,000 --> 00:38:12,000
to these two,
and for negative going voltages

501
00:38:12,000 --> 00:38:16,000
it simply compresses them to a
low positive value here.

502
00:38:16,000 --> 00:38:19,000
And that's what I see here
corresponding to negative

503
00:38:19,000 --> 00:38:22,000
excursion.
So notice that what this will

504
00:38:22,000 --> 00:38:25,000
do, if I view sound,
if I input sound here,

505
00:38:25,000 --> 00:38:28,000
and sound has negative going
excursions it will simply

506
00:38:28,000 --> 00:38:32,000
scrunch them.
But more or less let the

507
00:38:32,000 --> 00:38:36,000
positive things through.
And that is going to give rise

508
00:38:36,000 --> 00:38:38,000
to a bunch of distortion in my
signal.

509
00:38:38,000 --> 00:38:41,000
So I would like to show you a
little demo.

510
00:38:41,000 --> 00:38:45,000
Actually, we've gone ahead and
built a little device like this.

511
00:38:45,000 --> 00:38:48,000
We have an honest to goodness
little device costing,

512
00:38:48,000 --> 00:38:51,000
I don't know,
50 cents or $1 or something,

513
00:38:51,000 --> 00:38:54,000
which is a little voltage,
it's a device that emits light

514
00:38:54,000 --> 00:38:59,000
proportional to the current
flowing through it.

515
00:38:59,000 --> 00:39:01,000
I have a receiver.
And I am going to play some

516
00:39:01,000 --> 00:39:04,000
music, and you will listen to
the output here.

517
00:39:04,000 --> 00:39:07,000
And hopefully you should see a
bunch of distortion because of

518
00:39:07,000 --> 00:39:10,000
that effect that I showed you.

519
00:39:16,000 --> 00:39:19,000
And what I will do is,
before we do that,

520
00:39:19,000 --> 00:39:21,000
you will see two curves up
there.

521
00:39:21,000 --> 00:39:25,000
The yellow, I believe is the
vI, is the input,

522
00:39:25,000 --> 00:39:27,000
and the green,
I believe, is a signal

523
00:39:27,000 --> 00:39:31,000
proportionate to --
The other way around.

524
00:39:31,000 --> 00:39:33,000
Oh, I see.
So green is the input.

525
00:39:33,000 --> 00:39:37,000
So green, the lower one is the
input and the upper one is the

526
00:39:37,000 --> 00:39:40,000
distorted output.
So we are going to play some

527
00:39:40,000 --> 00:39:43,000
sound through it,
music through it and you can

528
00:39:43,000 --> 00:39:47,000
listen, through a little CD
player.

529
00:40:25,000 --> 00:40:30,000
So a couple of things.
The good news is that it works.

530
00:40:30,000 --> 00:40:33,000
However, I doubt that music
artists will come to my studio

531
00:40:33,000 --> 00:40:37,000
to record if this is the quality
of what I produce.

532
00:40:37,000 --> 00:40:41,000
Do notice that there are hardly
any negative going excursions in

533
00:40:41,000 --> 00:40:43,000
that curve up there,
right?

534
00:40:43,000 --> 00:40:47,000
All the negative ones have been
like scrunched up down into a

535
00:40:47,000 --> 00:40:50,000
flat line there,
and that's the reason I get

536
00:40:50,000 --> 00:40:54,000
this distortion.
And just to prove to you that I

537
00:40:54,000 --> 00:40:58,000
am indeed using a garage door
opener device and not faking it

538
00:40:58,000 --> 00:41:02,000
here, I am going to just shut
the signal off by stopping the

539
00:41:02,000 --> 00:41:06,000
light using a piece of paper
here.

540
00:41:06,000 --> 00:41:11,000
So notice that this device here
is the little device that has a

541
00:41:11,000 --> 00:41:16,000
light beam going through the
center, and I am going to take

542
00:41:16,000 --> 00:41:20,000
this piece of paper,
can you turn it up?

543
00:41:34,000 --> 00:41:36,000
So let's have some fun with
this.

544
00:41:36,000 --> 00:41:39,000
If I were to put this piece of
paper halfway down,

545
00:41:39,000 --> 00:41:42,000
I should get half the
intensity, right.

546
00:41:42,000 --> 00:41:46,000
So my sound should diminish in
volume a little bit.

547
00:41:46,000 --> 00:41:49,000
Maybe that will work.
Let's see if it works.

548
00:41:49,000 --> 00:41:52,000
Nothing to do with 002 but it's
just fun.

549
00:41:52,000 --> 00:41:55,000
Louder.
You can make it loud.

550
00:42:12,000 --> 00:42:15,000
Too much coffee.
My hand is shaking.

551
00:42:15,000 --> 00:42:19,000
I guess you did see the
lowering of volume,

552
00:42:19,000 --> 00:42:20,000
right?

553
00:42:30,000 --> 00:42:32,000
OK.
Just way too much coffee,

554
00:42:32,000 --> 00:42:38,000
and so my hand was shaking too
fast imposing its own sine wave

555
00:42:38,000 --> 00:42:41,000
on top of the signal.
What did I show you?

556
00:42:41,000 --> 00:42:43,000
This was garbage,
right?

557
00:42:43,000 --> 00:42:48,000
We had a nice little signal
input, and the output was

558
00:42:48,000 --> 00:42:54,000
completely distorted because I
was playing sound over this and

559
00:42:54,000 --> 00:42:59,000
this is what happened.
Switch to Page 9.

560
00:42:59,000 --> 00:43:03,000
Now, this is what I would have
liked to have happened.

561
00:43:03,000 --> 00:43:08,000
On Page 9 what I would have
liked to see happen is this.

562
00:43:16,000 --> 00:43:22,000
Suppose I had a light emitting
device that looked linear,

563
00:43:22,000 --> 00:43:29,000
a straight line where the
current was linearly related to

564
00:43:29,000 --> 00:43:31,000
vD.
Then what I would see,

565
00:43:31,000 --> 00:43:35,000
if I had a sinusoid here then I
would get a sinusoid here.

566
00:43:35,000 --> 00:43:37,000
No distortion there,
right?

567
00:43:37,000 --> 00:43:41,000
If only things were like I
wanted them, if I had a linear

568
00:43:41,000 --> 00:43:44,000
device, but I don't have a
linear device.

569
00:43:44,000 --> 00:43:48,000
I have an Expo Dweeb.
Now you know why I call it a

570
00:43:48,000 --> 00:43:51,000
dweeb.
Well, I'd like a linear device

571
00:43:51,000 --> 00:43:56,000
and it's exponential.
But this is what I would like.

572
00:44:02,000 --> 00:44:05,000
And if I had this I wouldn't
show it to you today.

573
00:44:05,000 --> 00:44:09,000
If I had this my music would go
through without any distortion

574
00:44:09,000 --> 00:44:12,000
and I wouldn't have to run
cables through my attic.

575
00:44:12,000 --> 00:44:16,000
I could just use my garage door
opener to play signals from my

576
00:44:16,000 --> 00:44:18,000
bedroom and living room and so
on, right?

577
00:44:18,000 --> 00:44:21,000
So the key thing here is how do
I get this?

578
00:44:21,000 --> 00:44:25,000
And what I would like you to do
is think about it yourselves.

579
00:44:25,000 --> 00:44:29,000
What I am given is something
like this.

580
00:44:29,000 --> 00:44:32,000
This about it yourselves,
you know, what would you do?

581
00:44:32,000 --> 00:44:35,000
See if you can come to me
before lecture tomorrow or

582
00:44:35,000 --> 00:44:38,000
Thursday and tell me the answer,
OK?