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All right.
Let's get moving.

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Good morning.

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Today, if everything works out,
we have some fun for you guys.

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I hope it works out.
We'll see.

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What I am going to do today is
a very major application of the

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frequency response and the
frequency domain analysis of

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circuits.
And this application area is

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called filters.
The area of filters often times

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demands a full course or a
couple of full courses all by

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itself.
And filters are incredibly

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useful.
They're used in virtually every

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electronic device in some form
or another.

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They're used in radio tuners.
We will show you a demo of that

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today.
They're also used in your cell

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phones.
Every single cell phone has a

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set of filters.
So, for example,

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how do you pick a conversation?
You pick a conversation by

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picking a certain frequency and
grabbing data from there.

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They are also in wide area
network wireless transmitters.

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Do we have an access point
here?

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I don't see one,
but you've seen wireless access

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points.
Again, there they have filters

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in them.
So, virtually every single

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electronic device contains a
filter at some point or another.

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And so, today we will look at
this major, major application of

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frequency domain analysis.
Before we get into that,

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I'd like to do a bit of review.
The readings for today

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correspond to Chapter 14.4.2,
14.5 and 15.2 in the course

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notes.
All right.

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Let's start with the review.
We looked at this circuit last

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Friday --

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-- where I said that for our
analysis, we are going to focus

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on this small,
small region of the playground.

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And what's special about this
region of our playground is that

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I am going to focus on
sinusoidal inputs.

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And, second,
I am going to focus on the

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steady state response.
How does the response look like

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if I wait a long,
long time?

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And then we said that the full
blown time domain analysis was

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hard.
This was, remember,

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the agonizing approach?
And then I taught you the

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impedance approach in the last
lecture, which was blindingly

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simple.
And, in that impedance

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approach, what we said we would
do is --

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I will apply the approach right
now and in seconds derive the

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result for you.
But the basic idea was we said

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what we are going to do is
assume that we are going to

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apply inputs of the form Vi e to
the j omega t.

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Wherever you see a capital and
a small, there is an implicate e

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to the j omega t next to it.
I'm not showing you that.

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And what I showed last time,
and the class before that was

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once you find out the amplitude
--

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Once you find out the
multiplier that multiplies e to

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the j omega t,
it's a complex number,

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you have all the information
you need.

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And once you have this,
you can find out the time

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domain response by simply taking
the modulus of that,

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or the amplitude and the phase
of that to get the angle.

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And that gives you the time
domain response.

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So, our focus has been on these
quantities.

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The impedance method says what
I am going to do is replace each

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of these by impedances.
And then the corresponding

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impedance model looks like this.

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Instead of R,
I replace that with ZR.

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And instead of the capacitor,
I am going to replace that with

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ZC.
And this is my Vc.

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ZR is simply R and ZC was going
to be one divided by sC where s

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was simply a shorthand notation
for j omega.

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Based on this,
once I converted all my

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elements into impedances,
I can go ahead and apply all

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the good-old linear analysis
techniques.

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I will discuss a bunch of them
today.

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As an example,
I could analyze this using my

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simple voltage divider
relationship.

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Vc is simply ZC divided by ZC
plus ZR times Vi.

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And that, in turn,
is, well, let's say I divide

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this by Vi so I can get the
response relation,

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is ZC divided by ZC plus ZR.
And ZC I know to be one by j

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omega C, plus R.
And multiplying throughout by j

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omega C, I get one divided by
one plus j omega CR.

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It's incredibly simple.
This is simply called the

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frequency response.
And it's a transfer function

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representing the relationship
between the output complex

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amplitude with the input.
We can also plot this.

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Notice that in our entire
analysis we have not only

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assumed sinusoidal input,
but we're also saying that let

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us look at this only in the
steady state.

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So, we will wait for time to be
really, really large,

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and then look at the response.
And so, therefore,

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we will plot the response not
as a function of time,

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but rather we are going to plot
the response as a function of

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omega.
What we are going to say is I

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am going to input a sinusoid and
my output is going to be some

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other sinusoid.
And since I'm waiting for a

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long time to look at the output,
time doesn't make sense

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anymore.
Rather, my free variable is

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going to be my frequency,
so I am going to change the

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frequency of the input that I
apply.

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And so, I am going to plot this
as a function of omega.

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This represents a completely
complimentary view of circuits,

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the time domain view and then
there is a frequency domain

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view.
The frequency domain view says

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how did this circuit behave as I
apply sinusoids of differing

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frequencies?
I can plot that relationship in

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a graph like this,
and this relationship is simply

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given by a parameter edge the
transfer function,

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it's a function of omega.
And I can also plot the

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absolute value of that.
And let's take a look at what

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it looks like.
So, I can look at functions

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like this and very quickly plot
the response.

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I am going to do a whole bunch
of plots just by staring at

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circuits and staring at
expressions like this.

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And you will see a number of
them today.

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First of all,
the way you plot these is look

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for the values where omega is
very small and when omega is

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very large.
When omega is very,

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very small this term goes away.
And so, for very small values

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of omega the output is simply
one.

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Vc by Vi is simply one.
This part goes away.

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What happens when omega is
very, very large?

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When omega is really large,
this part dominates,

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is much greater than one.
If I ignore one in relation to

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this guy and take the absolute
value of that then I simply get

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one divided by omega CR when
omega is very large.

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So, when omega is very large,
I get a decay of the form one

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over omega CR.
I know the value for small

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omega, and it looks like this
for very large omega.

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And, if you plot it out,
this is how it's going to look

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like.
Let's stare at this form for a

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little while longer.
And let's plot some properties

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off it.
First of all,

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you notice something else.
When omega CR equals one then,

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in other words,
when omega equals one by RC,

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notice that the output is given
by one plus j.

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And the absolute value of that
is simply one divided the square

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root of two.
So, in other words,

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when omega is one by RC --
When omega is one by CR then

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the output is one by square root
two times its value when omega

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is very, very small.
So, that is one little piece of

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information.
If you look at the form of

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this, I would like you to stare
at it for a few minutes and try

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to understand what this
represents.

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This says that for very low
frequencies the response is

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virtually the same as the input
in amplitude.

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In other words,
if I apply some very low

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frequency sinusoid of some
amplitude then the output

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amplitude is going to be same as
that amplitude.

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And that's a one.
Now, it also says when I apply

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a very high frequency,
at very high frequencies it

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decays.
So, this graph which says I am

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going to pass low frequencies
without any attenuation,

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without hammering it,
but I am going to clobber high

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frequencies and give you a very
low amplitude signal at the

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output but pass through,
almost without attenuation,

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the input at low frequencies.
And so this is an example of

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what is called a low pass filter
or LPF.

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What this is saying is that
this little circuit here acts

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like a low pass filter.
It's a low pass filter because

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it passes low frequencies
without attenuation but kills

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high frequencies.
If I take some music,

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and you will do experiments
with this in lab.

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When is lab three?
People are doing lab three

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right now, right?
Lab three is going on right now

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and early next week as well.
And, in lab three,

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you will play with looking at
the response to music of

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different types of filters.
If apply some music here,

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you will see that the output
will pass low frequencies but

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really attenuate high
frequencies.

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You will hear a lot of the low
sounding base and so on but

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attenuate a lot of the high
frequencies.

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All right.
The other thing that I

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encourage you to do is Websim
has built in pages for a large

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number of such circuits.
You can go in there and play

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with the values of RC,
or L for that matter,

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for a variety of circuits.
And, if you click on frequency

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response, you actually get both
the amplitude response and the

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phase as well.
You can play with various

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values of RLC and see how the
frequency response looks like

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for each of the circuits.
As a next step,

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what I would like to do is just
give you a sense of how

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impedances combine.
This won't be very surprising

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given that they behave just like
resistors, but it's good to go

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through it nonetheless.
Suppose, just to build some

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insight, suppose I had two
resistors in series.

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All right.
R1 and R2.

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And this was my A and B
terminals respectively.

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And let's say the complex
amplitude of the voltage was Vab

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across this.
Then I could relate,

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let's say Iab was the current,
I can relate these resistances.

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Or, I could relate Vab and Iab
as follows.

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Simply Vab divided by Iab
equals R1 plus R2.

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I know that.
And the same thing applies to R

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viewed as an impedance.
It's still impedance R,

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and so this one still goes
ahead and applies.

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The second thing I can try is
the circuit of this form.

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A, B, and I have an R1 and an L
in this case.

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And what I can do is,
in the impedance model,

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I can view this as an impedance
of value j omega L.

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And I can also combine them to
get the impedance between A and

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B.
Much as I got a resistance

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between A and B,
I can get an impedance between

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A and B as Vab divided by Iab.
And that will be given by ZR1

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plus ZL, and that is simply R1
plus j omega L.

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Similarly, I can do an even
more complicated circuit.

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So, resistance.
And here I have a capacitor in

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series with the resistance,
and then I apply inductor to

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it.
This is A, B,

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Iab and plus,
minus Vab.

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And let me call this R1 and let
me call this R2 and this is C

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and L.
I can go about combining these

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in much the same manner that I
combine my resistances in the

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series parallel simplifications.
I can define an impedance Zab

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00:16:13,000 --> 00:16:19,000
between the A and B terminals as
ZR1 plus Z of this combination,

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impedance of this combination,
which is simply impedance of C

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00:16:25,000 --> 00:16:30,000
and that of R2 in parallel with
each other.

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00:16:30,000 --> 00:16:34,000
I get Zc in parallel with ZR2.
Notice that this notation

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00:16:34,000 --> 00:16:38,000
simply says that look at the
impedance of the capacitor in

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parallel with a resistor.
And then, finally,

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I add to that the series
impedance of the inductor ZL.

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00:16:45,000 --> 00:16:49,000
Exactly as you would have done
for resistances,

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if all of these resistances you
would have said R of this piece

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plus the R of the parallel
combination plus the R of

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whatever was here.
This time around we have

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impedances.
And replacing this with the

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values, this is R1.
I know for ZL it's j omega L.

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And so, for ZL,
parallel ZR2 it is given by

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ZCZR2 divided by ZC plus ZR2,
which is simply R1 here and j

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omega L.
And let me just substitute the

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values here.
I know that ZR2 is simply R2,

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00:17:40,000 --> 00:17:46,000
ZC is one by j omega C,
and then one by j omega C plus

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00:17:46,000 --> 00:17:50,000
R2.
And I can go ahead and simplify

232
00:17:50,000 --> 00:17:54,000
that further and get my
impedance Zab.

233
00:17:54,000 --> 00:18:00,000
Notice how simple analysis has
become.

234
00:18:00,000 --> 00:18:03,000
Using this technique,
using the impedance method

235
00:18:03,000 --> 00:18:07,000
we've managed to convert our
analysis from solving

236
00:18:07,000 --> 00:18:11,000
differential equations to going
back to algebra.

237
00:18:11,000 --> 00:18:15,000
A large part of what we do in
circuits is see how we can get

238
00:18:15,000 --> 00:18:20,000
back to really simple algebra
and try to be clever about how

239
00:18:20,000 --> 00:18:23,000
we do things.
So, this is as far as analysis

240
00:18:23,000 --> 00:18:26,000
is concerned.
In the next five minutes,

241
00:18:26,000 --> 00:18:31,000
I want to give you some insight
into how you can build different

242
00:18:31,000 --> 00:18:34,000
kinds of impedances.

243
00:18:43,000 --> 00:18:46,000
And I won't go into too much
detail but give some insight

244
00:18:46,000 --> 00:18:50,000
into how you can get a sense for
the kind of filters you want to

245
00:18:50,000 --> 00:18:52,000
design.
Or, at the very least,

246
00:18:52,000 --> 00:18:55,000
given a filter,
how can you very quickly get

247
00:18:55,000 --> 00:18:58,000
some insight into what kind of
filter it is,

248
00:18:58,000 --> 00:19:01,000
how it performs,
what its frequency response is

249
00:19:01,000 --> 00:19:05,000
and so on.
And, this time around,

250
00:19:05,000 --> 00:19:09,000
this piece of intuition will be
in honor of Umans.

251
00:19:09,000 --> 00:19:14,000
And back to our Bend it Like
Beckham series,

252
00:19:14,000 --> 00:19:17,000
I call this "Unleash it like
Umans".

253
00:19:17,000 --> 00:19:23,000
What experts in the field do is
they don't go about sitting

254
00:19:23,000 --> 00:19:28,000
around writing differential
equations, but rather use a lot

255
00:19:28,000 --> 00:19:34,000
of insight into how to solve
these things.

256
00:19:34,000 --> 00:19:39,000
And so in honor of Umans,
I will label this unleash it

257
00:19:39,000 --> 00:19:43,000
like Umans.
Let's get some insight into how

258
00:19:43,000 --> 00:19:47,000
the response of various elements
look like.

259
00:19:47,000 --> 00:19:51,000
Let's take, for example,
I have some impedance Z.

260
00:19:51,000 --> 00:19:56,000
Let's say this could be a
resistor, it could be an

261
00:19:56,000 --> 00:20:01,000
inductor or it could be a
capacitor.

262
00:20:01,000 --> 00:20:05,000
Let's take a look at what the
frequency response of just these

263
00:20:05,000 --> 00:20:07,000
elements look like.
In other words,

264
00:20:07,000 --> 00:20:10,000
what are the frequency
dependents of Z itself?

265
00:20:10,000 --> 00:20:14,000
Let me just plot the impedance
of each of these elements as a

266
00:20:14,000 --> 00:20:17,000
function of frequency.
Let me just take the absolute

267
00:20:17,000 --> 00:20:21,000
value of their impedance.
Notice that it's a complex

268
00:20:21,000 --> 00:20:23,000
number.
For the inductor it's j omega

269
00:20:23,000 --> 00:20:25,000
L.
And let me take the absolute

270
00:20:25,000 --> 00:20:30,000
value omega L in that case and
plot it for you.

271
00:20:30,000 --> 00:20:33,000
And use that to develop some
insight.

272
00:20:33,000 --> 00:20:38,000
Let's do a simple case first.
If Z is a resistance of value R

273
00:20:38,000 --> 00:20:43,000
then no matter what the
frequency my value is going to

274
00:20:43,000 --> 00:20:46,000
be R.
If I have an inductor of value

275
00:20:46,000 --> 00:20:51,000
L then the impedance is going to
look like j omega L,

276
00:20:51,000 --> 00:20:55,000
and so I am going to omega L
for that.

277
00:20:55,000 --> 00:21:00,000
And the dependence of that
simply says that for low omega

278
00:21:00,000 --> 00:21:06,000
the impedance is very small.
For omega zero the impedance is

279
00:21:06,000 --> 00:21:09,000
zero and it increases linearly
with omega.

280
00:21:09,000 --> 00:21:12,000
So, it's omega L for the
inductor.

281
00:21:12,000 --> 00:21:16,000
Impedance increases linerally
as I increase the frequency.

282
00:21:16,000 --> 00:21:20,000
What about for the capacitor?
For the capacitor,

283
00:21:20,000 --> 00:21:23,000
the impedance is one divided by
j omega C.

284
00:21:23,000 --> 00:21:27,000
And so, therefore,
I get the dependence being

285
00:21:27,000 --> 00:21:33,000
related to omega C.
Which says that for very high

286
00:21:33,000 --> 00:21:38,000
frequencies impedance is very
low, but for very low

287
00:21:38,000 --> 00:21:45,000
frequencies the impedance is
very high and I get a behavior

288
00:21:45,000 --> 00:21:49,000
pattern that looks something
like this.

289
00:21:49,000 --> 00:21:54,000
It goes as one by omega C.
As omega is very large,

290
00:21:54,000 --> 00:22:00,000
my impedance is very small.
If omega is very small,

291
00:22:00,000 --> 00:22:03,000
my impedance goes towards that
of an open circuit.

292
00:22:03,000 --> 00:22:06,000
This is not surprising.
You've known this before,

293
00:22:06,000 --> 00:22:09,000
right?
That a capacitor behaves like

294
00:22:09,000 --> 00:22:12,000
an open circuit for DC.
An inductor behaves like a

295
00:22:12,000 --> 00:22:15,000
short circuit for DC.
Notice that zero frequency here

296
00:22:15,000 --> 00:22:18,000
corresponds to DC.
The capacitor looks like an

297
00:22:18,000 --> 00:22:21,000
open circuit for DC,
very high impedance.

298
00:22:21,000 --> 00:22:24,000
The inductor looks like a short
circuit for DC,

299
00:22:24,000 --> 00:22:27,000
very low impedance.
And the opposite is true at

300
00:22:27,000 --> 00:22:31,000
very high frequencies.
While R is a constant

301
00:22:31,000 --> 00:22:34,000
throughout.
Let's use this to build some

302
00:22:34,000 --> 00:22:37,000
insight into how our circuits
might look.

303
00:22:37,000 --> 00:22:40,000
Let me do this example.

304
00:22:45,000 --> 00:22:51,000
Let's say I have a Vi and I
measure the response across the

305
00:22:51,000 --> 00:22:53,000
resistor.

306
00:23:00,000 --> 00:23:05,000
So, I measure Vr divided by Vi
and take the absolute value and

307
00:23:05,000 --> 00:23:08,000
take a look at how it's going to
look like.

308
00:23:08,000 --> 00:23:13,000
I want you to stare at this for
me and help me with what the

309
00:23:13,000 --> 00:23:18,000
response is going to look like.
Let's take incredibly high

310
00:23:18,000 --> 00:23:21,000
frequencies.
At very high frequencies,

311
00:23:21,000 --> 00:23:26,000
this has a very high frequency,
what do the capacitor look like

312
00:23:26,000 --> 00:23:32,000
to very high frequencies?
Is it an open or is it a short?

313
00:23:32,000 --> 00:23:35,000
A short circuit.
At very high frequencies the

314
00:23:35,000 --> 00:23:38,000
capacitor looks like a short
circuit.

315
00:23:38,000 --> 00:23:41,000
Then Vi simply appears across
the resistor,

316
00:23:41,000 --> 00:23:45,000
which means that at very high
frequencies the output is very

317
00:23:45,000 --> 00:23:49,000
close to the input.
At very low frequencies what

318
00:23:49,000 --> 00:23:51,000
happens?
At very low frequencies the

319
00:23:51,000 --> 00:23:54,000
capacitor looks like an open
circuit.

320
00:23:54,000 --> 00:23:58,000
If this looks like an open
circuit then very little voltage

321
00:23:58,000 --> 00:24:03,000
will drop across this resistor
here because most of it is going

322
00:24:03,000 --> 00:24:09,000
to drop across the capacitor.
What is going to happen is,

323
00:24:09,000 --> 00:24:13,000
for very low values,
I am going to be looking at

324
00:24:13,000 --> 00:24:17,000
something out here.
And, because of that,

325
00:24:17,000 --> 00:24:23,000
my response looks like this.
And this is of a different form

326
00:24:23,000 --> 00:24:27,000
than the one you saw earlier.
In this case,

327
00:24:27,000 --> 00:24:32,000
I pass high frequencies but
attenuate low frequencies.

328
00:24:32,000 --> 00:24:36,000
Not surprisingly,
this is called a high pass

329
00:24:36,000 --> 00:24:38,000
filter.

330
00:24:44,000 --> 00:24:47,000
You need to begin to be able to
think about capacitors and

331
00:24:47,000 --> 00:24:51,000
inductors in terms of their high
and low frequency properties.

332
00:24:51,000 --> 00:24:55,000
And, if you develop that
intuition, once you develop the

333
00:24:55,000 --> 00:24:59,000
intuition about capacitors and
inductors and their frequency

334
00:24:59,000 --> 00:25:01,000
relationship,
that will be a big step forward

335
00:25:01,000 --> 00:25:04,000
in 002.
If you get that insight,

336
00:25:04,000 --> 00:25:07,000
you will go a long way in terms
of knowing how to tackle

337
00:25:07,000 --> 00:25:10,000
problems and being able to
quickly sketch responses.

338
00:25:10,000 --> 00:25:11,000
Yes.

339
00:25:22,000 --> 00:25:25,000
In the case of,
if we get something like j

340
00:25:25,000 --> 00:25:30,000
omega L, what you can do is take
the limit as omega goes to zero.

341
00:25:30,000 --> 00:25:33,000
If it is omega L then notice
that it is going to start

342
00:25:33,000 --> 00:25:36,000
linear.
And, on the other hand,

343
00:25:36,000 --> 00:25:39,000
if when you get very high
frequencies, for example,

344
00:25:39,000 --> 00:25:43,000
if you get one by something
omega C then this is a

345
00:25:43,000 --> 00:25:47,000
hyperbolic relationship,
so it is going to go ahead

346
00:25:47,000 --> 00:25:50,000
looking like this.
So, you can take a look at a

347
00:25:50,000 --> 00:25:55,000
lot of these functions at their
very low values and see how they

348
00:25:55,000 --> 00:25:59,000
look like at that point.
All right.

349
00:25:59,000 --> 00:26:02,000
The next one I would like to
draw for you is something that

350
00:26:02,000 --> 00:26:03,000
looks like this.

351
00:26:08,000 --> 00:26:12,000
Let's say, for example,
I have an inductor L and a

352
00:26:12,000 --> 00:26:16,000
resistor R and I want to see
what that looks like.

353
00:26:16,000 --> 00:26:20,000
In this particular example,
I have H, take the absolute

354
00:26:20,000 --> 00:26:23,000
value.
So, what is this going to look

355
00:26:23,000 --> 00:26:27,000
like?
I am going to look at the value

356
00:26:27,000 --> 00:26:32,000
across the resistor here.
Here what I am going to find is

357
00:26:32,000 --> 00:26:36,000
that at very low frequencies
this guy is a short circuit.

358
00:26:36,000 --> 00:26:40,000
Since this guy is a short
circuit, all the voltage drops

359
00:26:40,000 --> 00:26:44,000
across the resistor so it's
going to look like this.

360
00:26:44,000 --> 00:26:49,000
And, at very high frequencies,
what I am going to find is that

361
00:26:49,000 --> 00:26:52,000
the inductor is going to appear
like an open circuit.

362
00:26:52,000 --> 00:26:56,000
And so, therefore,
all the voltage is going to

363
00:26:56,000 --> 00:27:00,000
pretty much drop across the
inductor.

364
00:27:00,000 --> 00:27:03,000
It will be R divided by
something plus omega L.

365
00:27:03,000 --> 00:27:08,000
So, at high frequencies this
guy is going to taper off to

366
00:27:08,000 --> 00:27:11,000
zero and is going to look like
this.

367
00:27:11,000 --> 00:27:14,000
And this is back to my low pass
filter.

368
00:27:14,000 --> 00:27:17,000
Just to go back to a question
asked earlier,

369
00:27:17,000 --> 00:27:20,000
how do you know what this looks
like?

370
00:27:20,000 --> 00:27:25,000
I can very quickly write down
the expression for H of j omega.

371
00:27:25,000 --> 00:27:32,000
This is simply going to be R
divided by R plus if this is VR.

372
00:27:32,000 --> 00:27:36,000
VR is simply R divided by one
by j omega C.

373
00:27:36,000 --> 00:27:41,000
I multiply it out by j omega C
in the numerator and the

374
00:27:41,000 --> 00:27:45,000
denominator.
I'm going to find j omega C

375
00:27:45,000 --> 00:27:49,000
here and I am going to get one
by j omega C here.

376
00:27:49,000 --> 00:27:55,000
And what is going to happen
with something like this is that

377
00:27:55,000 --> 00:28:02,000
as omega becomes very small then
I am going to ignore this.

378
00:28:02,000 --> 00:28:07,000
When omega becomes very small,
I can ignore this with respect

379
00:28:07,000 --> 00:28:12,000
to one, and I get R j omega C.
Given that, is what I've drawn

380
00:28:12,000 --> 00:28:16,000
here correct or wrong?
This goes away with respect to

381
00:28:16,000 --> 00:28:19,000
one.
I am left with R j omega C,

382
00:28:19,000 --> 00:28:21,000
right?
For very low frequencies.

383
00:28:21,000 --> 00:28:26,000
Given what I have drawn here,
is that correct or is that

384
00:28:26,000 --> 00:28:30,000
wrong?
Well, it's hard to say.

385
00:28:30,000 --> 00:28:36,000
For very, very low frequencies
it starts out being linear

386
00:28:36,000 --> 00:28:40,000
because it's an omega
relationship,

387
00:28:40,000 --> 00:28:47,000
and then it goes up like this
and then goes out there.

388
00:28:47,000 --> 00:28:54,000
Let me go onto another example.
Let me do another example here

389
00:28:54,000 --> 00:29:01,000
which is something like --
I need to make sure I don't

390
00:29:01,000 --> 00:29:05,000
make a mistake here.
If I get R j omega C by R j

391
00:29:05,000 --> 00:29:09,000
omega C, you know what,
this ends up being a first

392
00:29:09,000 --> 00:29:12,000
order system,
and so is going to look like

393
00:29:12,000 --> 00:29:14,000
this.
I blew it there.

394
00:29:14,000 --> 00:29:19,000
Back to this system here.
If I have an L and an R and I

395
00:29:19,000 --> 00:29:23,000
look at this equation to look at
what happens across L,

396
00:29:23,000 --> 00:29:29,000
you can plot that again.
And for very low frequencies it

397
00:29:29,000 --> 00:29:34,000
is going to be zero amplitude
here and for very high

398
00:29:34,000 --> 00:29:38,000
frequencies this is going to be
an open circuit,

399
00:29:38,000 --> 00:29:43,000
and so the response is going to
look something like this.

400
00:29:43,000 --> 00:29:48,000
That's going to end up being
your high pass filter.

401
00:29:48,000 --> 00:29:53,000
As another example,
I would like to do a series RLC

402
00:29:53,000 --> 00:29:55,000
circuit --

403
00:30:10,000 --> 00:30:14,000
-- and try to get you some
sense of what that output looks

404
00:30:14,000 --> 00:30:17,000
like.
Let's use our intuition and

405
00:30:17,000 --> 00:30:22,000
first write down what this looks
like and then go and do some

406
00:30:22,000 --> 00:30:26,000
math and see if the math
corresponds to what our

407
00:30:26,000 --> 00:30:31,000
intuition tells us.
I want to plot Vr with respect

408
00:30:31,000 --> 00:30:34,000
to Vi.
I want to plot it there.

409
00:30:34,000 --> 00:30:38,000
For something like this,
what happens at very low

410
00:30:38,000 --> 00:30:41,000
frequencies?
We are just looking to get

411
00:30:41,000 --> 00:30:46,000
very, very crudely what this
graph is going to look like.

412
00:30:46,000 --> 00:30:51,000
Very, very crudely what this
graph is going to look like.

413
00:30:51,000 --> 00:30:55,000
Given that I am taking the
voltage across VR,

414
00:30:55,000 --> 00:31:00,000
what happens at very low
frequencies?

415
00:31:00,000 --> 00:31:05,000
At incredibly low frequencies,
the inductor looks like a short

416
00:31:05,000 --> 00:31:09,000
circuit, but the capacitor looks
like open circuit.

417
00:31:09,000 --> 00:31:14,000
An open circuit in series with
a short circuit that ends up

418
00:31:14,000 --> 00:31:18,000
looking like an open circuit.
And so, therefore,

419
00:31:18,000 --> 00:31:23,000
all my voltage falls across VR.
Now, what happens at very high

420
00:31:23,000 --> 00:31:26,000
frequencies?
At very high frequencies the

421
00:31:26,000 --> 00:31:32,000
capacitor looks like a short.
But the inductor looks like an

422
00:31:32,000 --> 00:31:36,000
open circuit now for very high
frequencies, correct?

423
00:31:36,000 --> 00:31:39,000
Just remember,
capacitor is short for high

424
00:31:39,000 --> 00:31:42,000
frequencies inductor open for
high frequencies.

425
00:31:42,000 --> 00:31:46,000
So, this ends up having a very
high impedance.

426
00:31:46,000 --> 00:31:50,000
At very high frequencies this
guy has a very high impedance.

427
00:31:50,000 --> 00:31:54,000
And, because of that,
for a high value of frequency,

428
00:31:54,000 --> 00:31:59,000
I end up going in that manner.
This behavior has the effect of

429
00:31:59,000 --> 00:32:04,000
the capacitor here.
And for very high frequencies I

430
00:32:04,000 --> 00:32:10,000
get the effect of the inductor.
And so this means that I have

431
00:32:10,000 --> 00:32:14,000
very low values for low
frequencies, very low values for

432
00:32:14,000 --> 00:32:18,000
high frequencies.
And, as the frequency

433
00:32:18,000 --> 00:32:21,000
increases, I do something like
this.

434
00:32:21,000 --> 00:32:25,000
I keep building up,
then the inductor begins to

435
00:32:25,000 --> 00:32:30,000
play a role, and then I taper
off again.

436
00:32:30,000 --> 00:32:36,000
This kind of a filter where I
kill low and high frequencies

437
00:32:36,000 --> 00:32:42,000
and pass intermediate
frequencies is called a band

438
00:32:42,000 --> 00:32:46,000
pass filter, BPF.
This means that it passes

439
00:32:46,000 --> 00:32:53,000
frequencies in some band.
Let's get some more insight on

440
00:32:53,000 --> 00:32:57,000
this by writing down the
equations.

441
00:32:57,000 --> 00:33:02,000
So, Vr divided by Vi is simply
R.

442
00:33:02,000 --> 00:33:10,000
Using the impedance relation it
is R divided by j omega L plus

443
00:33:10,000 --> 00:33:18,000
one divided by j omega C plus R.
I am going to use this equation

444
00:33:18,000 --> 00:33:26,000
later, so let me stash it away
on my stack and put a little

445
00:33:26,000 --> 00:33:31,000
notation there.
I am going to multiply

446
00:33:31,000 --> 00:33:38,000
throughout by j omega C.
And what I end up getting is j

447
00:33:38,000 --> 00:33:44,000
omega RC divided by one plus R j
omega RC, and then here,

448
00:33:44,000 --> 00:33:51,000
I get j times j is minus one,
so I get minus omega squared.

449
00:33:51,000 --> 00:33:59,000
Let me rewrite it this way.
I get minus omega squared.

450
00:33:59,000 --> 00:34:04,000
So, j j is minus one,
omega times omega is omega

451
00:34:04,000 --> 00:34:10,000
squared, and then I get an LC.
That's what I end up getting.

452
00:34:10,000 --> 00:34:16,000
And if I take the absolute
value here, I end up getting,

453
00:34:16,000 --> 00:34:23,000
back to your complex algebra,
the square root of this real

454
00:34:23,000 --> 00:34:29,000
value squared plus imaginary
value squared.

455
00:34:29,000 --> 00:34:34,000
So, one minus omega squared LC
plus omega RC squared.

456
00:34:34,000 --> 00:34:38,000
This is from,
you can look it up in your

457
00:34:38,000 --> 00:34:42,000
complex algebra appendix in the
course notes.

458
00:34:42,000 --> 00:34:47,000
It's simply omega RC here,
then square of the real value

459
00:34:47,000 --> 00:34:54,000
plus the square of the imaginary
value, and take the square root

460
00:34:54,000 --> 00:34:56,000
of that.
By staring at this,

461
00:34:56,000 --> 00:35:04,000
you can notice that you realize
a really important property.

462
00:35:04,000 --> 00:35:08,000
When omega equals LC.
I'm sorry.

463
00:35:08,000 --> 00:35:14,000
When omega equals one divided
by LC, what happens?

464
00:35:14,000 --> 00:35:21,000
Sorry, square root of LC.
When omega is one divided by

465
00:35:21,000 --> 00:35:30,000
square root of LC then omega
squared times LC becomes one.

466
00:35:30,000 --> 00:35:34,000
When this is true then this
becomes one, and one and one

467
00:35:34,000 --> 00:35:37,000
cancel out.
And, not only that,

468
00:35:37,000 --> 00:35:41,000
when these cancel out,
these two cancel out at that

469
00:35:41,000 --> 00:35:47,000
point, so I end up getting a
one, which means that when omega

470
00:35:47,000 --> 00:35:52,000
equals omega nought equals one
by square root of LC and I end

471
00:35:52,000 --> 00:35:58,000
up getting a value that is one.
It's pretty amazing.

472
00:35:58,000 --> 00:36:01,000
Which means that if I drive
this at omega nought,

473
00:36:01,000 --> 00:36:06,000
if my sinusoid has a frequency
omega nought where omega nought

474
00:36:06,000 --> 00:36:11,000
is one by square root of LC,
if I'm sitting here and this is

475
00:36:11,000 --> 00:36:16,000
a black box on the right-hand
side, and I drive this at a

476
00:36:16,000 --> 00:36:20,000
frequency omega nought equals
one divided by square root of

477
00:36:20,000 --> 00:36:24,000
LC, what does this entire
circuit look like to me?

478
00:36:24,000 --> 00:36:29,000
I'm sitting there,
the black box here.

479
00:36:29,000 --> 00:36:33,000
I'm driving it at omega nought
equals one by square root of LC

480
00:36:33,000 --> 00:36:36,000
at that frequency.
What does that circuit look

481
00:36:36,000 --> 00:36:36,000
like?
Yes.
It looks like a resistor.
It's pretty amazing.

482
00:36:40,000 --> 00:36:43,000
It means that even though I
have an L and a C here,

483
00:36:43,000 --> 00:36:47,000
if I happen to drive this at
omega nought then the circuit

484
00:36:47,000 --> 00:36:51,000
looks purely resistive and it
seems to give me the same input

485
00:36:51,000 --> 00:36:54,000
appearing at the output.
In other words,

486
00:36:54,000 --> 00:36:58,000
the effect of these two cancels
out.

487
00:36:58,000 --> 00:37:02,000
And that aspect is called
driving the circuit at its

488
00:37:02,000 --> 00:37:05,000
resonance point.
Resonance is when you're

489
00:37:05,000 --> 00:37:10,000
driving the circuit at omega
nought equals one by a square

490
00:37:10,000 --> 00:37:12,000
root of LC.

491
00:37:21,000 --> 00:37:27,000
I will very quickly sketch for
you a couple of other ways of

492
00:37:27,000 --> 00:37:32,000
looking at circuits.
Supposing I looked at this

493
00:37:32,000 --> 00:37:37,000
value here, Vlc,
I looked at the value across

494
00:37:37,000 --> 00:37:43,000
the inductor and the capacitor,
what will the frequency

495
00:37:43,000 --> 00:37:48,000
response look like?
I am looking at the voltage

496
00:37:48,000 --> 00:37:53,000
across the inductor and the
capacitor in series.

497
00:37:53,000 --> 00:37:57,000
Let's see.
Let's go back to our usual

498
00:37:57,000 --> 00:38:01,000
mantra.
Think about Steve Umans when

499
00:38:01,000 --> 00:38:03,000
you do this.
What would he do?

500
00:38:03,000 --> 00:38:07,000
He would say ah-ha,
at very low frequencies the

501
00:38:07,000 --> 00:38:10,000
capacitor is going to look like
an open circuit.

502
00:38:10,000 --> 00:38:14,000
In my voltage divider,
I am measuring the voltage

503
00:38:14,000 --> 00:38:18,000
across an open circuit,
so the entire Vi must drop

504
00:38:18,000 --> 00:38:20,000
across the inductor and
capacitor.

505
00:38:20,000 --> 00:38:25,000
Similarly, at very high
frequencies the inductor looks

506
00:38:25,000 --> 00:38:30,000
like an open circuit now,
so it looks like this.

507
00:38:30,000 --> 00:38:35,000
At very high frequencies
inductor is an open circuit.

508
00:38:35,000 --> 00:38:42,000
And, again, I'm looking at the
voltage divider across the near

509
00:38:42,000 --> 00:38:47,000
infinite resistance,
impedance, so I get a high

510
00:38:47,000 --> 00:38:52,000
value here as well.
Well, in the middle the value

511
00:38:52,000 --> 00:38:56,000
dips and I get something like
this.

512
00:38:56,000 --> 00:39:02,000
So, this thing is called a band
stop filter.

513
00:39:02,000 --> 00:39:07,000
Here I can nail any specific
frequency, as long as the

514
00:39:07,000 --> 00:39:12,000
frequency falls in roughly that
regime.

515
00:39:12,000 --> 00:39:15,000
Yet another example.

516
00:39:20,000 --> 00:39:23,000
The reason I'm working on so
many examples is that to

517
00:39:23,000 --> 00:39:27,000
experts, a large part of what
they do is look at a circuit and

518
00:39:27,000 --> 00:39:30,000
boom, give a rough form of how
it looks like.

519
00:39:30,000 --> 00:39:33,000
That can get you half the way
there in most of what you're

520
00:39:33,000 --> 00:39:37,000
going to do.
How did this look like?

521
00:39:37,000 --> 00:39:43,000
If I take the voltage Vo versus
Vi, let's take a look.

522
00:39:43,000 --> 00:39:49,000
At very low frequencies,
the inductor looks like a short

523
00:39:49,000 --> 00:39:54,000
circuit, correct?
I am talking the voltage across

524
00:39:54,000 --> 00:40:00,000
a short circuit,
so it looks like this.

525
00:40:00,000 --> 00:40:04,000
At very high frequencies,
I am taking a voltage across a

526
00:40:04,000 --> 00:40:09,000
parallel combination,
but the capacitor is now a

527
00:40:09,000 --> 00:40:12,000
short circuit.
So, that looks like a

528
00:40:12,000 --> 00:40:16,000
capacitor.
This looks like an inductor out

529
00:40:16,000 --> 00:40:20,000
here and this is a capacitor
holding sway here.

530
00:40:20,000 --> 00:40:25,000
And so, somewhere in the middle
it goes up and comes down like

531
00:40:25,000 --> 00:40:30,000
that.
So, it's a band pass filter.

532
00:40:30,000 --> 00:40:33,000
What is amazing is that you can
take fairly complicated

533
00:40:33,000 --> 00:40:36,000
circuits, and just by doing a
quick analysis of what happens

534
00:40:36,000 --> 00:40:39,000
at very low frequencies,
what happens at very high

535
00:40:39,000 --> 00:40:42,000
frequencies, you can roughly
sketch the response.

536
00:40:42,000 --> 00:40:45,000
And then what you should do,
in addition to that,

537
00:40:45,000 --> 00:40:48,000
is if it's a second order
circuit, just assume that it's

538
00:40:48,000 --> 00:40:51,000
going to do something
interesting at its resonance

539
00:40:51,000 --> 00:40:54,000
frequency, at omega nought
equals one by square root of LC.

540
00:40:54,000 --> 00:40:57,000
Something interesting is going
to happen.

541
00:40:57,000 --> 00:41:01,000
Check it out.
And for circuits that are first

542
00:41:01,000 --> 00:41:05,000
order, RC or RL,
the important number is the

543
00:41:05,000 --> 00:41:09,000
time constant RC.
Usually, when you're driving it

544
00:41:09,000 --> 00:41:12,000
at one by RC,
omega equals one by RC then

545
00:41:12,000 --> 00:41:17,000
what happens is that you often
times end up getting a value

546
00:41:17,000 --> 00:41:22,000
that is one by square root two
times the input value in the

547
00:41:22,000 --> 00:41:26,000
circuits we looked at here.
Next, what I am going to do is

548
00:41:26,000 --> 00:41:32,000
talk about a major,
major application of filters.

549
00:41:32,000 --> 00:41:41,000
And that is an AM receiver.
Let me do Radios 101 for 30

550
00:41:41,000 --> 00:41:48,000
seconds.
These guys have an antenna.

551
00:41:48,000 --> 00:41:57,000
You take a ground here.
You pick up a signal at your

552
00:41:57,000 --> 00:42:02,000
antenna.
There is an implied ground as

553
00:42:02,000 --> 00:42:03,000
well.
And what you do,

554
00:42:03,000 --> 00:42:07,000
as a first step,
is you begin processing the

555
00:42:07,000 --> 00:42:10,000
signal now.
What we place right there is a

556
00:42:10,000 --> 00:42:12,000
little filter that looks like
this.

557
00:42:12,000 --> 00:42:16,000
It is a inductor and a
capacitor in parallel.

558
00:42:16,000 --> 00:42:20,000
And this capacitor is really
your tuner that you can tune to

559
00:42:20,000 --> 00:42:24,000
radio frequencies.
And then what you have here is

560
00:42:24,000 --> 00:42:30,000
a bunch of other processing and
end up with your speaker.

561
00:42:30,000 --> 00:42:36,000
And the processing that happens
here is you have a demodulator,

562
00:42:36,000 --> 00:42:42,000
you have an amplifier and a
bunch of other things that let's

563
00:42:42,000 --> 00:42:48,000
not worry about them for now.
What we do here is the antenna

564
00:42:48,000 --> 00:42:51,000
picks up a signal.
So, in some sense,

565
00:42:51,000 --> 00:42:56,000
this part of the circuit here
is your source.

566
00:42:56,000 --> 00:43:03,000
I could replace it with its
Thevenin equivalent as follows.

567
00:43:09,000 --> 00:43:13,000
So, the front end of your radio
looks like a Vi,

568
00:43:13,000 --> 00:43:17,000
R, L and a C.
Where have you seen this

569
00:43:17,000 --> 00:43:19,000
before?
Right there.

570
00:43:19,000 --> 00:43:26,000
That's the front end of radios.
Let me tell you why I need a

571
00:43:26,000 --> 00:43:31,000
band pass filter in a radio out
here.

572
00:43:31,000 --> 00:43:35,000
The way life works is as
follows.

573
00:43:35,000 --> 00:43:41,000
I have my frequency.
Let me do this not in radians

574
00:43:41,000 --> 00:43:48,000
but in kilohertz for now,
and let me plot your radio

575
00:43:48,000 --> 00:43:52,000
signal strength.
In the Boston area,

576
00:43:52,000 --> 00:43:59,000
the signals go between 540
kilohertz and they go all the

577
00:43:59,000 --> 00:44:06,000
way to 1600 kilohertz.
In some areas we have begun to

578
00:44:06,000 --> 00:44:10,000
use the 1700 extra band as well
for some new stations.

579
00:44:10,000 --> 00:44:13,000
This is the frequency range of
interest.

580
00:44:13,000 --> 00:44:17,000
If you look at your radio
tuner, you will see 540

581
00:44:17,000 --> 00:44:22,000
kilohertz all the way up to 1600
and you can tune your AM radio.

582
00:44:22,000 --> 00:44:27,000
The way it works is that each
station is given 10 kilohertz of

583
00:44:27,000 --> 00:44:31,000
spectrum here.
And so, this is at 1000

584
00:44:31,000 --> 00:44:36,000
kilohertz, 1010 kilohertz and so
on.

585
00:44:36,000 --> 00:44:43,000
And each station transmits its
signal in plus or minus 5

586
00:44:43,000 --> 00:44:50,000
kilohertz around that point.
And this station transmits it

587
00:44:50,000 --> 00:44:57,000
here and this station transmits
it here and so on.

588
00:44:57,000 --> 00:45:02,000
This is 1030.
This guy is WBZ News Radio

589
00:45:02,000 --> 00:45:06,000
1030, for those of you who
listen to it.

590
00:45:06,000 --> 00:45:12,000
What happens is that at 10
kilohertz, each station gets 10

591
00:45:12,000 --> 00:45:18,000
kilohertz, and so WBZ transmits
in the 10 kilohertz around 1030.

592
00:45:18,000 --> 00:45:23,000
Notice that each of these
signals transmitted by radio

593
00:45:23,000 --> 00:45:28,000
stations happen within small
bands.

594
00:45:28,000 --> 00:45:31,000
Now, you will learn a lot more
about modulation and how do you

595
00:45:31,000 --> 00:45:34,000
get a signal to go in a small
band and all that stuff.

596
00:45:34,000 --> 00:00:06,003
You will learn about that in

597
00:45:36,000 --> 00:45:39,000
For now, don't worry about how
I did all of this.

598
00:45:39,000 --> 00:45:41,000
How do you listen to that
station?

599
00:45:41,000 --> 00:45:44,000
The way you listen to that
station is you put a low pass

600
00:45:44,000 --> 00:45:47,000
filter here.
You put a low pass filter that

601
00:45:47,000 --> 00:45:49,000
does the following.
Let's say I want to hear WBZ

602
00:17:10,000 --> 00:45:51,000
If I can arrange to have the

603
00:45:56,000 --> 00:46:00,000
If I pass this entire signal
through that filter.

604
00:46:00,000 --> 00:46:04,000
And if I arrange to have the
omega nought of my filter at

605
00:46:07,000 --> 00:46:11,000
omega nought at 1030 then this
is the response of my filter.

606
00:46:11,000 --> 00:46:17,000
And I am going to pick out this
guy and cut out everything else.

607
00:46:17,000 --> 00:46:20,000
I am just going to get this.

608
00:46:40,000 --> 00:46:42,000
Let's listen to the station for
some time.

609
00:47:52,000 --> 00:47:55,000
So, you can see I can tune to
the station WBUL.