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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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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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and that of R2 in parallel with
each other.

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I get Zc in parallel with ZR2.
Notice that this notation

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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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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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00:17:15 --> 00:17:21
And so, for ZL,
parallel ZR2 it is given by

231
00:17:21 --> 00:17:29
ZCZR2 divided by ZC plus ZR2,
which is simply R1 here and j

232
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omega L.
And let me just substitute the

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00:17:35 --> 00:17:40
values here.
I know that ZR2 is simply R2,

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ZC is one by j omega C,
and then one by j omega C plus

235
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R2.
And I can go ahead and simplify

236
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that further and get my
impedance Zab.

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Notice how simple analysis has
become.

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Using this technique,
using the impedance method

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

240
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differential equations to going
back to algebra.

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

242
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back to really simple algebra
and try to be clever about how

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we do things.
So, this is as far as analysis

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is concerned.
In the next five minutes,

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

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kinds of impedances.

247
00:18:34 --> 00:18:43


248
00:18:43 --> 00:18:46
And I won't go into too much
detail but give some insight

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into how you can get a sense for
the kind of filters you want to

250
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design.
Or, at the very least,

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

252
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some insight into what kind of
filter it is,

253
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how it performs,
what its frequency response is

254
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and so on.
And, this time around,

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this piece of intuition will be
in honor of Umans.

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

257
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I call this "Unleash it like
Umans".

258
00:19:17 --> 00:19:23
What experts in the field do is
they don't go about sitting

259
00:19:23 --> 00:19:28
around writing differential
equations, but rather use a lot

260
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of insight into how to solve
these things.

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And so in honor of Umans,
I will label this unleash it

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like Umans.
Let's get some insight into how

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the response of various elements
look like.

264
00:19:47 --> 00:19:51
Let's take, for example,
I have some impedance Z.

265
00:19:51 --> 00:19:56
Let's say this could be a
resistor, it could be an

266
00:19:56 --> 00:20:01
inductor or it could be a
capacitor.

267
00:20:01 --> 00:20:05
Let's take a look at what the
frequency response of just these

268
00:20:05 --> 00:20:07
elements look like.
In other words,

269
00:20:07 --> 00:20:10
what are the frequency
dependents of Z itself?

270
00:20:10 --> 00:20:14
Let me just plot the impedance
of each of these elements as a

271
00:20:14 --> 00:20:17
function of frequency.
Let me just take the absolute

272
00:20:17 --> 00:20:21
value of their impedance.
Notice that it's a complex

273
00:20:21 --> 00:20:23
number.
For the inductor it's j omega

274
00:20:23 --> 00:20:25
L.
And let me take the absolute

275
00:20:25 --> 00:20:30
value omega L in that case and
plot it for you.

276
00:20:30 --> 00:20:33
And use that to develop some
insight.

277
00:20:33 --> 00:20:38
Let's do a simple case first.
If Z is a resistance of value R

278
00:20:38 --> 00:20:43
then no matter what the
frequency my value is going to

279
00:20:43 --> 00:20:46
be R.
If I have an inductor of value

280
00:20:46 --> 00:20:51
L then the impedance is going to
look like j omega L,

281
00:20:51 --> 00:20:55
and so I am going to omega L
for that.

282
00:20:55 --> 00:21:00
And the dependence of that
simply says that for low omega

283
00:21:00 --> 00:21:06
the impedance is very small.
For omega zero the impedance is

284
00:21:06 --> 00:21:09
zero and it increases linearly
with omega.

285
00:21:09 --> 00:21:12
So, it's omega L for the
inductor.

286
00:21:12 --> 00:21:16
Impedance increases linerally
as I increase the frequency.

287
00:21:16 --> 00:21:20
What about for the capacitor?
For the capacitor,

288
00:21:20 --> 00:21:23
the impedance is one divided by
j omega C.

289
00:21:23 --> 00:21:27
And so, therefore,
I get the dependence being

290
00:21:27 --> 00:21:33
related to omega C.
Which says that for very high

291
00:21:33 --> 00:21:38
frequencies impedance is very
low, but for very low

292
00:21:38 --> 00:21:45
frequencies the impedance is
very high and I get a behavior

293
00:21:45 --> 00:21:49
pattern that looks something
like this.

294
00:21:49 --> 00:21:54
It goes as one by omega C.
As omega is very large,

295
00:21:54 --> 00:22:00
my impedance is very small.
If omega is very small,

296
00:22:00 --> 00:22:03
my impedance goes towards that
of an open circuit.

297
00:22:03 --> 00:22:06
This is not surprising.
You've known this before,

298
00:22:06 --> 00:22:09
right?
That a capacitor behaves like

299
00:22:09 --> 00:22:12
an open circuit for DC.
An inductor behaves like a

300
00:22:12 --> 00:22:15
short circuit for DC.
Notice that zero frequency here

301
00:22:15 --> 00:22:18
corresponds to DC.
The capacitor looks like an

302
00:22:18 --> 00:22:21
open circuit for DC,
very high impedance.

303
00:22:21 --> 00:22:24
The inductor looks like a short
circuit for DC,

304
00:22:24 --> 00:22:27
very low impedance.
And the opposite is true at

305
00:22:27 --> 00:22:31
very high frequencies.
While R is a constant

306
00:22:31 --> 00:22:34
throughout.
Let's use this to build some

307
00:22:34 --> 00:22:37
insight into how our circuits
might look.

308
00:22:37 --> 00:22:40
Let me do this example.

309
00:22:40 --> 00:22:45


310
00:22:45 --> 00:22:51
Let's say I have a Vi and I
measure the response across the

311
00:22:51 --> 00:22:53
resistor.

312
00:22:53 --> 00:23:00


313
00:23:00 --> 00:23:05
So, I measure Vr divided by Vi
and take the absolute value and

314
00:23:05 --> 00:23:08
take a look at how it's going to
look like.

315
00:23:08 --> 00:23:13
I want you to stare at this for
me and help me with what the

316
00:23:13 --> 00:23:18
response is going to look like.
Let's take incredibly high

317
00:23:18 --> 00:23:21
frequencies.
At very high frequencies,

318
00:23:21 --> 00:23:26
this has a very high frequency,
what do the capacitor look like

319
00:23:26 --> 00:23:32
to very high frequencies?
Is it an open or is it a short?

320
00:23:32 --> 00:23:35
A short circuit.
At very high frequencies the

321
00:23:35 --> 00:23:38
capacitor looks like a short
circuit.

322
00:23:38 --> 00:23:41
Then Vi simply appears across
the resistor,

323
00:23:41 --> 00:23:45
which means that at very high
frequencies the output is very

324
00:23:45 --> 00:23:49
close to the input.
At very low frequencies what

325
00:23:49 --> 00:23:51
happens?
At very low frequencies the

326
00:23:51 --> 00:23:54
capacitor looks like an open
circuit.

327
00:23:54 --> 00:23:58
If this looks like an open
circuit then very little voltage

328
00:23:58 --> 00:24:03
will drop across this resistor
here because most of it is going

329
00:24:03 --> 00:24:09
to drop across the capacitor.
What is going to happen is,

330
00:24:09 --> 00:24:13
for very low values,
I am going to be looking at

331
00:24:13 --> 00:24:17
something out here.
And, because of that,

332
00:24:17 --> 00:24:23
my response looks like this.
And this is of a different form

333
00:24:23 --> 00:24:27
than the one you saw earlier.
In this case,

334
00:24:27 --> 00:24:32
I pass high frequencies but
attenuate low frequencies.

335
00:24:32 --> 00:24:36
Not surprisingly,
this is called a high pass

336
00:24:36 --> 00:24:38
filter.

337
00:24:38 --> 00:24:44


338
00:24:44 --> 00:24:47
You need to begin to be able to
think about capacitors and

339
00:24:47 --> 00:24:51
inductors in terms of their high
and low frequency properties.

340
00:24:51 --> 00:24:55
And, if you develop that
intuition, once you develop the

341
00:24:55 --> 00:24:59
intuition about capacitors and
inductors and their frequency

342
00:24:59 --> 00:25:01
relationship,
that will be a big step forward

343
00:25:01 --> 00:25:04
in 002.
If you get that insight,

344
00:25:04 --> 00:25:07
you will go a long way in terms
of knowing how to tackle

345
00:25:07 --> 00:25:10
problems and being able to
quickly sketch responses.

346
00:25:10 --> 00:25:11
Yes.

347
00:25:11 --> 00:25:22


348
00:25:22 --> 00:25:25
In the case of,
if we get something like j

349
00:25:25 --> 00:25:30
omega L, what you can do is take
the limit as omega goes to zero.

350
00:25:30 --> 00:25:33
If it is omega L then notice
that it is going to start

351
00:25:33 --> 00:25:36
linear.
And, on the other hand,

352
00:25:36 --> 00:25:39
if when you get very high
frequencies, for example,

353
00:25:39 --> 00:25:43
if you get one by something
omega C then this is a

354
00:25:43 --> 00:25:47
hyperbolic relationship,
so it is going to go ahead

355
00:25:47 --> 00:25:50
looking like this.
So, you can take a look at a

356
00:25:50 --> 00:25:55
lot of these functions at their
very low values and see how they

357
00:25:55 --> 00:25:59
look like at that point.
All right.

358
00:25:59 --> 00:26:02
The next one I would like to
draw for you is something that

359
00:26:02 --> 00:26:03
looks like this.

360
00:26:03 --> 00:26:08


361
00:26:08 --> 00:26:12
Let's say, for example,
I have an inductor L and a

362
00:26:12 --> 00:26:16
resistor R and I want to see
what that looks like.

363
00:26:16 --> 00:26:20
In this particular example,
I have H, take the absolute

364
00:26:20 --> 00:26:23
value.
So, what is this going to look

365
00:26:23 --> 00:26:27
like?
I am going to look at the value

366
00:26:27 --> 00:26:32
across the resistor here.
Here what I am going to find is

367
00:26:32 --> 00:26:36
that at very low frequencies
this guy is a short circuit.

368
00:26:36 --> 00:26:40
Since this guy is a short
circuit, all the voltage drops

369
00:26:40 --> 00:26:44
across the resistor so it's
going to look like this.

370
00:26:44 --> 00:26:49
And, at very high frequencies,
what I am going to find is that

371
00:26:49 --> 00:26:52
the inductor is going to appear
like an open circuit.

372
00:26:52 --> 00:26:56
And so, therefore,
all the voltage is going to

373
00:26:56 --> 00:27:00
pretty much drop across the
inductor.

374
00:27:00 --> 00:27:03
It will be R divided by
something plus omega L.

375
00:27:03 --> 00:27:08
So, at high frequencies this
guy is going to taper off to

376
00:27:08 --> 00:27:11
zero and is going to look like
this.

377
00:27:11 --> 00:27:14
And this is back to my low pass
filter.

378
00:27:14 --> 00:27:17
Just to go back to a question
asked earlier,

379
00:27:17 --> 00:27:20
how do you know what this looks
like?

380
00:27:20 --> 00:27:25
I can very quickly write down
the expression for H of j omega.

381
00:27:25 --> 00:27:32
This is simply going to be R
divided by R plus if this is VR.

382
00:27:32 --> 00:27:36
VR is simply R divided by one
by j omega C.

383
00:27:36 --> 00:27:41
I multiply it out by j omega C
in the numerator and the

384
00:27:41 --> 00:27:45
denominator.
I'm going to find j omega C

385
00:27:45 --> 00:27:49
here and I am going to get one
by j omega C here.

386
00:27:49 --> 00:27:55
And what is going to happen
with something like this is that

387
00:27:55 --> 00:28:02
as omega becomes very small then
I am going to ignore this.

388
00:28:02 --> 00:28:07
When omega becomes very small,
I can ignore this with respect

389
00:28:07 --> 00:28:12
to one, and I get R j omega C.
Given that, is what I've drawn

390
00:28:12 --> 00:28:16
here correct or wrong?
This goes away with respect to

391
00:28:16 --> 00:28:19
one.
I am left with R j omega C,

392
00:28:19 --> 00:28:21
right?
For very low frequencies.

393
00:28:21 --> 00:28:26
Given what I have drawn here,
is that correct or is that

394
00:28:26 --> 00:28:30
wrong?
Well, it's hard to say.

395
00:28:30 --> 00:28:36
For very, very low frequencies
it starts out being linear

396
00:28:36 --> 00:28:40
because it's an omega
relationship,

397
00:28:40 --> 00:28:47
and then it goes up like this
and then goes out there.

398
00:28:47 --> 00:28:54
Let me go onto another example.
Let me do another example here

399
00:28:54 --> 00:29:01
which is something like --
I need to make sure I don't

400
00:29:01 --> 00:29:05
make a mistake here.
If I get R j omega C by R j

401
00:29:05 --> 00:29:09
omega C, you know what,
this ends up being a first

402
00:29:09 --> 00:29:12
order system,
and so is going to look like

403
00:29:12 --> 00:29:14
this.
I blew it there.

404
00:29:14 --> 00:29:19
Back to this system here.
If I have an L and an R and I

405
00:29:19 --> 00:29:23
look at this equation to look at
what happens across L,

406
00:29:23 --> 00:29:29
you can plot that again.
And for very low frequencies it

407
00:29:29 --> 00:29:34
is going to be zero amplitude
here and for very high

408
00:29:34 --> 00:29:38
frequencies this is going to be
an open circuit,

409
00:29:38 --> 00:29:43
and so the response is going to
look something like this.

410
00:29:43 --> 00:29:48
That's going to end up being
your high pass filter.

411
00:29:48 --> 00:29:53
As another example,
I would like to do a series RLC

412
00:29:53 --> 00:29:55
circuit --

413
00:29:55 --> 00:30:10


414
00:30:10 --> 00:30:14
-- and try to get you some
sense of what that output looks

415
00:30:14 --> 00:30:17
like.
Let's use our intuition and

416
00:30:17 --> 00:30:22
first write down what this looks
like and then go and do some

417
00:30:22 --> 00:30:26
math and see if the math
corresponds to what our

418
00:30:26 --> 00:30:31
intuition tells us.
I want to plot Vr with respect

419
00:30:31 --> 00:30:34
to Vi.
I want to plot it there.

420
00:30:34 --> 00:30:38
For something like this,
what happens at very low

421
00:30:38 --> 00:30:41
frequencies?
We are just looking to get

422
00:30:41 --> 00:30:46
very, very crudely what this
graph is going to look like.

423
00:30:46 --> 00:30:51
Very, very crudely what this
graph is going to look like.

424
00:30:51 --> 00:30:55
Given that I am taking the
voltage across VR,

425
00:30:55 --> 00:31:00
what happens at very low
frequencies?

426
00:31:00 --> 00:31:05
At incredibly low frequencies,
the inductor looks like a short

427
00:31:05 --> 00:31:09
circuit, but the capacitor looks
like open circuit.

428
00:31:09 --> 00:31:14
An open circuit in series with
a short circuit that ends up

429
00:31:14 --> 00:31:18
looking like an open circuit.
And so, therefore,

430
00:31:18 --> 00:31:23
all my voltage falls across VR.
Now, what happens at very high

431
00:31:23 --> 00:31:26
frequencies?
At very high frequencies the

432
00:31:26 --> 00:31:32
capacitor looks like a short.
But the inductor looks like an

433
00:31:32 --> 00:31:36
open circuit now for very high
frequencies, correct?

434
00:31:36 --> 00:31:39
Just remember,
capacitor is short for high

435
00:31:39 --> 00:31:42
frequencies inductor open for
high frequencies.

436
00:31:42 --> 00:31:46
So, this ends up having a very
high impedance.

437
00:31:46 --> 00:31:50
At very high frequencies this
guy has a very high impedance.

438
00:31:50 --> 00:31:54
And, because of that,
for a high value of frequency,

439
00:31:54 --> 00:31:59
I end up going in that manner.
This behavior has the effect of

440
00:31:59 --> 00:32:04
the capacitor here.
And for very high frequencies I

441
00:32:04 --> 00:32:10
get the effect of the inductor.
And so this means that I have

442
00:32:10 --> 00:32:14
very low values for low
frequencies, very low values for

443
00:32:14 --> 00:32:18
high frequencies.
And, as the frequency

444
00:32:18 --> 00:32:21
increases, I do something like
this.

445
00:32:21 --> 00:32:25
I keep building up,
then the inductor begins to

446
00:32:25 --> 00:32:30
play a role, and then I taper
off again.

447
00:32:30 --> 00:32:36
This kind of a filter where I
kill low and high frequencies

448
00:32:36 --> 00:32:42
and pass intermediate
frequencies is called a band

449
00:32:42 --> 00:32:46
pass filter, BPF.
This means that it passes

450
00:32:46 --> 00:32:53
frequencies in some band.
Let's get some more insight on

451
00:32:53 --> 00:32:57
this by writing down the
equations.

452
00:32:57 --> 00:33:02
So, Vr divided by Vi is simply
R.

453
00:33:02 --> 00:33:10
Using the impedance relation it
is R divided by j omega L plus

454
00:33:10 --> 00:33:18
one divided by j omega C plus R.
I am going to use this equation

455
00:33:18 --> 00:33:26
later, so let me stash it away
on my stack and put a little

456
00:33:26 --> 00:33:31
notation there.
I am going to multiply

457
00:33:31 --> 00:33:38
throughout by j omega C.
And what I end up getting is j

458
00:33:38 --> 00:33:44
omega RC divided by one plus R j
omega RC, and then here,

459
00:33:44 --> 00:33:51
I get j times j is minus one,
so I get minus omega squared.

460
00:33:51 --> 00:33:59
Let me rewrite it this way.
I get minus omega squared.

461
00:33:59 --> 00:34:04
So, j j is minus one,
omega times omega is omega

462
00:34:04 --> 00:34:10
squared, and then I get an LC.
That's what I end up getting.

463
00:34:10 --> 00:34:16
And if I take the absolute
value here, I end up getting,

464
00:34:16 --> 00:34:23
back to your complex algebra,
the square root of this real

465
00:34:23 --> 00:34:29
value squared plus imaginary
value squared.

466
00:34:29 --> 00:34:34
So, one minus omega squared LC
plus omega RC squared.

467
00:34:34 --> 00:34:38
This is from,
you can look it up in your

468
00:34:38 --> 00:34:42
complex algebra appendix in the
course notes.

469
00:34:42 --> 00:34:47
It's simply omega RC here,
then square of the real value

470
00:34:47 --> 00:34:54
plus the square of the imaginary
value, and take the square root

471
00:34:54 --> 00:34:56
of that.
By staring at this,

472
00:34:56 --> 00:35:04
you can notice that you realize
a really important property.

473
00:35:04 --> 00:35:08
When omega equals LC.
I'm sorry.

474
00:35:08 --> 00:35:14
When omega equals one divided
by LC, what happens?

475
00:35:14 --> 00:35:21
Sorry, square root of LC.
When omega is one divided by

476
00:35:21 --> 00:35:30
square root of LC then omega
squared times LC becomes one.

477
00:35:30 --> 00:35:34
When this is true then this
becomes one, and one and one

478
00:35:34 --> 00:35:37
cancel out.
And, not only that,

479
00:35:37 --> 00:35:41
when these cancel out,
these two cancel out at that

480
00:35:41 --> 00:35:47
point, so I end up getting a
one, which means that when omega

481
00:35:47 --> 00:35:52
equals omega nought equals one
by square root of LC and I end

482
00:35:52 --> 00:35:58
up getting a value that is one.
It's pretty amazing.

483
00:35:58 --> 00:36:01
Which means that if I drive
this at omega nought,

484
00:36:01 --> 00:36:06
if my sinusoid has a frequency
omega nought where omega nought

485
00:36:06 --> 00:36:11
is one by square root of LC,
if I'm sitting here and this is

486
00:36:11 --> 00:36:16
a black box on the right-hand
side, and I drive this at a

487
00:36:16 --> 00:36:20
frequency omega nought equals
one divided by square root of

488
00:36:20 --> 00:36:24
LC, what does this entire
circuit look like to me?

489
00:36:24 --> 00:36:29
I'm sitting there,
the black box here.

490
00:36:29 --> 00:36:33
I'm driving it at omega nought
equals one by square root of LC

491
00:36:33 --> 00:36:36
at that frequency.
What does that circuit look

492
00:36:36 --> 00:36:36
like?
Yes.

493
00:36:36 --> 00:36:40
It looks like a resistor.
It's pretty amazing.

494
00:36:40 --> 00:36:43
It means that even though I
have an L and a C here,

495
00:36:43 --> 00:36:47
if I happen to drive this at
omega nought then the circuit

496
00:36:47 --> 00:36:51
looks purely resistive and it
seems to give me the same input

497
00:36:51 --> 00:36:54
appearing at the output.
In other words,

498
00:36:54 --> 00:36:58
the effect of these two cancels
out.

499
00:36:58 --> 00:37:02
And that aspect is called
driving the circuit at its

500
00:37:02 --> 00:37:05
resonance point.
Resonance is when you're

501
00:37:05 --> 00:37:10
driving the circuit at omega
nought equals one by a square

502
00:37:10 --> 00:37:12
root of LC.

503
00:37:12 --> 00:37:21


504
00:37:21 --> 00:37:27
I will very quickly sketch for
you a couple of other ways of

505
00:37:27 --> 00:37:32
looking at circuits.
Supposing I looked at this

506
00:37:32 --> 00:37:37
value here, Vlc,
I looked at the value across

507
00:37:37 --> 00:37:43
the inductor and the capacitor,
what will the frequency

508
00:37:43 --> 00:37:48
response look like?
I am looking at the voltage

509
00:37:48 --> 00:37:53
across the inductor and the
capacitor in series.

510
00:37:53 --> 00:37:57
Let's see.
Let's go back to our usual

511
00:37:57 --> 00:38:01
mantra.
Think about Steve Umans when

512
00:38:01 --> 00:38:03
you do this.
What would he do?

513
00:38:03 --> 00:38:07
He would say ah-ha,
at very low frequencies the

514
00:38:07 --> 00:38:10
capacitor is going to look like
an open circuit.

515
00:38:10 --> 00:38:14
In my voltage divider,
I am measuring the voltage

516
00:38:14 --> 00:38:18
across an open circuit,
so the entire Vi must drop

517
00:38:18 --> 00:38:20
across the inductor and
capacitor.

518
00:38:20 --> 00:38:25
Similarly, at very high
frequencies the inductor looks

519
00:38:25 --> 00:38:30
like an open circuit now,
so it looks like this.

520
00:38:30 --> 00:38:35
At very high frequencies
inductor is an open circuit.

521
00:38:35 --> 00:38:42
And, again, I'm looking at the
voltage divider across the near

522
00:38:42 --> 00:38:47
infinite resistance,
impedance, so I get a high

523
00:38:47 --> 00:38:52
value here as well.
Well, in the middle the value

524
00:38:52 --> 00:38:56
dips and I get something like
this.

525
00:38:56 --> 00:39:02
So, this thing is called a band
stop filter.

526
00:39:02 --> 00:39:07
Here I can nail any specific
frequency, as long as the

527
00:39:07 --> 00:39:12
frequency falls in roughly that
regime.

528
00:39:12 --> 00:39:15
Yet another example.

529
00:39:15 --> 00:39:20


530
00:39:20 --> 00:39:23
The reason I'm working on so
many examples is that to

531
00:39:23 --> 00:39:27
experts, a large part of what
they do is look at a circuit and

532
00:39:27 --> 00:39:30
boom, give a rough form of how
it looks like.

533
00:39:30 --> 00:39:33
That can get you half the way
there in most of what you're

534
00:39:33 --> 00:39:37
going to do.
How did this look like?

535
00:39:37 --> 00:39:43
If I take the voltage Vo versus
Vi, let's take a look.

536
00:39:43 --> 00:39:49
At very low frequencies,
the inductor looks like a short

537
00:39:49 --> 00:39:54
circuit, correct?
I am talking the voltage across

538
00:39:54 --> 00:40:00
a short circuit,
so it looks like this.

539
00:40:00 --> 00:40:04
At very high frequencies,
I am taking a voltage across a

540
00:40:04 --> 00:40:09
parallel combination,
but the capacitor is now a

541
00:40:09 --> 00:40:12
short circuit.
So, that looks like a

542
00:40:12 --> 00:40:16
capacitor.
This looks like an inductor out

543
00:40:16 --> 00:40:20
here and this is a capacitor
holding sway here.

544
00:40:20 --> 00:40:25
And so, somewhere in the middle
it goes up and comes down like

545
00:40:25 --> 00:40:30
that.
So, it's a band pass filter.

546
00:40:30 --> 00:40:33
What is amazing is that you can
take fairly complicated

547
00:40:33 --> 00:40:36
circuits, and just by doing a
quick analysis of what happens

548
00:40:36 --> 00:40:39
at very low frequencies,
what happens at very high

549
00:40:39 --> 00:40:42
frequencies, you can roughly
sketch the response.

550
00:40:42 --> 00:40:45
And then what you should do,
in addition to that,

551
00:40:45 --> 00:40:48
is if it's a second order
circuit, just assume that it's

552
00:40:48 --> 00:40:51
going to do something
interesting at its resonance

553
00:40:51 --> 00:40:54
frequency, at omega nought
equals one by square root of LC.

554
00:40:54 --> 00:40:57
Something interesting is going
to happen.

555
00:40:57 --> 00:41:01
Check it out.
And for circuits that are first

556
00:41:01 --> 00:41:05
order, RC or RL,
the important number is the

557
00:41:05 --> 00:41:09
time constant RC.
Usually, when you're driving it

558
00:41:09 --> 00:41:12
at one by RC,
omega equals one by RC then

559
00:41:12 --> 00:41:17
what happens is that you often
times end up getting a value

560
00:41:17 --> 00:41:22
that is one by square root two
times the input value in the

561
00:41:22 --> 00:41:26
circuits we looked at here.
Next, what I am going to do is

562
00:41:26 --> 00:41:32
talk about a major,
major application of filters.

563
00:41:32 --> 00:41:41
And that is an AM receiver.
Let me do Radios 101 for 30

564
00:41:41 --> 00:41:48
seconds.
These guys have an antenna.

565
00:41:48 --> 00:41:57
You take a ground here.
You pick up a signal at your

566
00:41:57 --> 00:42:02
antenna.
There is an implied ground as

567
00:42:02 --> 00:42:03
well.
And what you do,

568
00:42:03 --> 00:42:07
as a first step,
is you begin processing the

569
00:42:07 --> 00:42:10
signal now.
What we place right there is a

570
00:42:10 --> 00:42:12
little filter that looks like
this.

571
00:42:12 --> 00:42:16
It is a inductor and a
capacitor in parallel.

572
00:42:16 --> 00:42:20
And this capacitor is really
your tuner that you can tune to

573
00:42:20 --> 00:42:24
radio frequencies.
And then what you have here is

574
00:42:24 --> 00:42:30
a bunch of other processing and
end up with your speaker.

575
00:42:30 --> 00:42:36
And the processing that happens
here is you have a demodulator,

576
00:42:36 --> 00:42:42
you have an amplifier and a
bunch of other things that let's

577
00:42:42 --> 00:42:48
not worry about them for now.
What we do here is the antenna

578
00:42:48 --> 00:42:51
picks up a signal.
So, in some sense,

579
00:42:51 --> 00:42:56
this part of the circuit here
is your source.

580
00:42:56 --> 00:43:03
I could replace it with its
Thevenin equivalent as follows.

581
00:43:03 --> 00:43:09


582
00:43:09 --> 00:43:13
So, the front end of your radio
looks like a Vi,

583
00:43:13 --> 00:43:17
R, L and a C.
Where have you seen this

584
00:43:17 --> 00:43:19
before?
Right there.

585
00:43:19 --> 00:43:26
That's the front end of radios.
Let me tell you why I need a

586
00:43:26 --> 00:43:31
band pass filter in a radio out
here.

587
00:43:31 --> 00:43:35
The way life works is as
follows.

588
00:43:35 --> 00:43:41
I have my frequency.
Let me do this not in radians

589
00:43:41 --> 00:43:48
but in kilohertz for now,
and let me plot your radio

590
00:43:48 --> 00:43:52
signal strength.
In the Boston area,

591
00:43:52 --> 00:43:59
the signals go between 540
kilohertz and they go all the

592
00:43:59 --> 00:44:06
way to 1600 kilohertz.
In some areas we have begun to

593
00:44:06 --> 00:44:10
use the 1700 extra band as well
for some new stations.

594
00:44:10 --> 00:44:13
This is the frequency range of
interest.

595
00:44:13 --> 00:44:17
If you look at your radio
tuner, you will see 540

596
00:44:17 --> 00:44:22
kilohertz all the way up to 1600
and you can tune your AM radio.

597
00:44:22 --> 00:44:27
The way it works is that each
station is given 10 kilohertz of

598
00:44:27 --> 00:44:31
spectrum here.
And so, this is at 1000

599
00:44:31 --> 00:44:36
kilohertz, 1010 kilohertz and so
on.

600
00:44:36 --> 00:44:43
And each station transmits its
signal in plus or minus 5

601
00:44:43 --> 00:44:50
kilohertz around that point.
And this station transmits it

602
00:44:50 --> 00:44:57
here and this station transmits
it here and so on.

603
00:44:57 --> 00:45:02
This is 1030.
This guy is WBZ News Radio

604
00:45:02 --> 00:45:06
1030, for those of you who
listen to it.

605
00:45:06 --> 00:45:12
What happens is that at 10
kilohertz, each station gets 10

606
00:45:12 --> 00:45:18
kilohertz, and so WBZ transmits
in the 10 kilohertz around 1030.

607
00:45:18 --> 00:45:23
Notice that each of these
signals transmitted by radio

608
00:45:23 --> 00:45:28
stations happen within small
bands.

609
00:45:28 --> 00:45:31
Now, you will learn a lot more
about modulation and how do you

610
00:45:31 --> 00:45:34
get a signal to go in a small
band and all that stuff.

611
00:45:34 --> 6.003.
You will learn about that in

612
6.003. --> 00:45:36


613
00:45:36 --> 00:45:39
For now, don't worry about how
I did all of this.

614
00:45:39 --> 00:45:41
How do you listen to that
station?

615
00:45:41 --> 00:45:44
The way you listen to that
station is you put a low pass

616
00:45:44 --> 00:45:47
filter here.
You put a low pass filter that

617
00:45:47 --> 00:45:49
does the following.
Let's say I want to hear WBZ

618
00:45:49 --> 1030.


619
1030. --> 00:45:51


620
00:45:51 --> 00:45:56


621
00:45:56 --> 00:46:00
If I pass this entire signal
through that filter.

622
00:46:00 --> 00:46:04
And if I arrange to have the
omega nought of my filter at

623
00:46:04 --> 1030.


624
1030. --> 00:46:07
If I can arrange to have the

625
00:46:07 --> 00:46:11
omega nought at 1030 then this
is the response of my filter.

626
00:46:11 --> 00:46:17
And I am going to pick out this
guy and cut out everything else.

627
00:46:17 --> 00:46:20
I am just going to get this.

628
00:46:20 --> 00:46:40


629
00:46:40 --> 00:46:42
Let's listen to the station for
some time.

630
00:46:42 --> 00:47:52


631
00:47:52 --> 47:55
So, you can see I can tune to
the station WBUL.