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Good morning.

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All right.
Today we are going to take a

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fresh look at some of the stuff
we covered in the last two

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lectures.
And the graph I want you to

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keep in mind as we go through
this lecture in terms of what to

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expect.
This was time.

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And last Tuesday's lecture we
covered some stuff.

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I talked about a method for the
sinusoidal response which was

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agony, I warned you it will be
agony, and then towards the end

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I showed you another method that
was quite a bit easier but still

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pretty hard.
And I promised you that today

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there will be a new method which
is going to be so easy ,

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actually almost trite.
Just imagine.

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I am going to make a statement
right now that I think you will

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all find hard to believe.
What I am going to say is just

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imagine your RLC circuit,
your resistor,

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inductor and capacitor,
a parallel form or series form.

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Imagine that you could write
down the characteristic equation

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for that by observation in 30
seconds or less.

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Just imagine that.
By observation,

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boom, write down the
characteristic equation for

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virtually any RLC circuit or RC
circuit or whatever.

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And we all know that once you
have the characteristic equation

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you could very easily go from
there to the time domain

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response intuitively or to the
sinusoidal steady-state

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response, too.
So just keep that thought in

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mind.
Imagine 30 seconds.

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And that is what you should
expect in today's lecture.

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Students often ask me,
if this stuff is actually so

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easy why do you take us through
this tortuous path?

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Are we just mean?
Do we just want you show you

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how hard things are and then
show the easy way?

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I have argued with myself every
year as to whether to just go

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ahead and give the easy path and
that's it.

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But I think the reason we cover
the basic foundations is that it

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gives you a level of insight
that you would not have

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otherwise gotten if I directly
jumped into the easy method.

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So you need to understand the
foundations and you need to have

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seen that at least once.
And second, once you do

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something the hard way,
you appreciate all the more the

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easy method.
All right.

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Today we cover what is called
"The Impedance Model".

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First let me do a review just
because of the large amount of

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content in the last two
lectures.

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I did them using view graphs.
I usually don't like to do

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that, but even then it was quite
rushed.

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So let me quickly summarize for
you kind of the main points.

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We have been looking at,
on Tuesday, the sinusoidal --

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--looking at the sinusoidal
steady state response.

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Also fondly denoted as SSS.
And the readings for this were

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Chapters 14.1 and 14.2.
what we said was if you took

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this example circuit and we fed
as input cosine of omega t,

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we have an R and a C,
and let's say we cared about

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the output response and we cared
about the capacitor voltage.

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What we talked about was
focused on the sinusoidal

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steady-state response.
And what that meant was first

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of all focus on steady-state.
In other words,

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just to capture the
steady-state behavior when t

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goes to infinity after a long
period of time.

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And for most of the circuits
that we consider,

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because of the R or presence of
any resistance,

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the homogenous response usually
would die out because the

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homogenous response is usually
of the form minus t by tau.

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And as t goes to infinity this
term tends to go to zero.

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We are just looking at the
steady-state.

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And therefore,
because of the circuits we

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looked at, we can ignore the
homogenous response.

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All we are left to do is to
find the particular response to

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sinusoids of this form.
And second was focus on

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sinusoids.
We said the reason for this was

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that, let's say we did not care
particularly

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What happened when I just
turned on my amplifier.

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I just turned on my amplifier,
often times you see some

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distorted sound coming out for a
few seconds and then hear a much

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clearer sound.
And that initial part is due to

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the transient response.
And let's say we don't care

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about that.
We care about the steady state.

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Second we focus on sinusoids
because based on the Fourier

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series experience that you had
previously, we can represent

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repeated signals as a sum of
sines.

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And therefore it is important
to understand the behavior of

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these circuits when the input is
a sinusoid.

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And what was important was this
introduced a new way of looking

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at circuits, and that was the
frequency viewpoint.

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When we looked at transient
responses, we plotted response

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as a function of time.
And when we look at sinusoidal

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steady-state,
it becomes interesting to plot

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the response as a function of
the frequency,

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a function of omega.

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What I will do is draw a little
chart for you to sort of

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visualize the various processes
we have been going through.

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We can liken obtaining the
sinusoidal steady-state response

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to following these steps.
Here is my input.

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What I did as a first step was
fed my input to a usual circuit

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model.
My elements were lumped

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elements, built the circuit and
wrote down the VI relationship

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for the element.
As a second step I set up the

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differential equation.

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This was the first of four
steps, set up a differential

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equation.
And then the path that I took

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first was fraught with real
nightmarish trig.

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By the end of the day it would
still yield an answer.

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It could be a nightmare.
But I would get something

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cosine omega t plus something,
some phase.

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I could grunge through the
trig.

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And I gave up halfway in class
here, but you could grunge

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through it if you would like.
And you would get the answer to

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be some sinusoid with some
amplitude and some phase.

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So Vi cosine omega t would
produce the response that was

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something cosine omega t plus
some phase.

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We said this was too painful so
let's punt this.

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Instead, what we said we would
do is take a detour,

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take an easier path.
And the easier path looked like

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this.
I said let's sneak in --

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-- Vie^(j omega t) drive.
That is just imagine,

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do the math as if you had fed
in not a Vi cosine omega t but a

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Vie^(j omega t).
And from Euler's relation you

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know that the real part is Vi
cosine omega t.

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So we said that I am going to
sneak in this thing,

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find the response and just take
the real part of that because

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the real part of the input gives
me this.

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So this is my "sneaky path".
And what I did there,

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as soon as we fed in the e^(j
omega t), because of the

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property of exponentials,
the e^(j omega t) cancelled out

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in my equation.
And what was left was some

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fairly simple complex algebra.
And at the end of the day,

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after I grunged through some
fairly simple complex algebra,

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I ended up with some response
that looked like this.

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Vpe^(j omega t).
What I would find is that for

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the input Vie^(j omega t),
I would get a response Vpe^(j

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omega t).
And then what I said we would

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do is take the real part.
Why take the real part?

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Because this is a fake,
a sneaky input.

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The input I really care about
is the real part of the sneaky

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input.
So this is my sneaky output.

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And what I care about is the
real part of the sneaky output.

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That is sort of the inverse
superposition argument that I

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made on Tuesday that if what I
care about is the real part of

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this input, then I just take the
real part and get the output

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that I care about.
So I take the real part.

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Notice that Vp here,
in the examples we did,

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we did an RC example.
The Vp here was a complex

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number.
So I could represent that

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complex number as,
in many ways.

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This is e^(j omega t).
I could represent Vp in an

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amplitude, as a phasor,
actually polar coordinates.

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I can say that the equivalent
to Vpe to the j angle Vp.

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Vp is a complex number.
If you look at the complex

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appendix in your course notes,
I can represent a complex

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number as an amplitude
multiplied by e raised to j

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times some phase.
It's simple complex algebra.

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And then what I could do here
is take the real part of that.

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And when I took the real part
of that what came about was that

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this was simply Vp.
Notice that the angle Vp goes

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in here so it becomes j times
omega t plus angle Vp.

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It is Vp amplitude times e
raised to j omega t plus j angle

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Vp.
And the real part of that is

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simply Vp cosine of that stuff.
What I end up getting here is

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Vp cosine omega t plus Vp.
The cool thing to notice was

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that once I found out this
response here,

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I could immediately write down
the output based on Vp.

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In other words,
once I had Vp,

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I could stop right there in my
math.

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I got Vp very quickly here.
This step produced Vp very

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quickly, after two algebraic
steps.

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And then from here I could
directly write down the answer

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as homogenous of Vp cosine omega
t plus angle Vp.

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Boom, right there.
So this was a much shorter

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path.
And here I just described to

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you how this yields an
expression for Vp and angle Vp.

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And for our example Vp was
1/(1+j omega RC).

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And we often times write a
shorthand notation 1+sRC,

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where S is simply j omega.
We commonly jump back and forth

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between the shorthand notation S
and j omega.

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S has some other fundamental,
has another fundamental

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significance you will learn
about in future courses,

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but for now S is simply a short
form for j omega.

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This was the path that we took.
There is a hard path and an

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easier path.
Today I am going to claim that

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even this was too hard.
There is an even easier path.

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And today what I am going to
show you is that from here we

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are going to take one step and
get here.

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I am going to show you today
that we won't do this,

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we won't do this,
not this, not this,

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none of this.
One step and then we are going

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to get the answer.
So let's do that.

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Before we jump into the
impedance method and get into

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doing that, I just would like to
plot for you this function here

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just so we can understand a
little bit better exactly what

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is going on.
As I mentioned to you,

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the output vO for our circuit
there was simply Vp cosine of

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omega T plus angle Vp.
Oh, that's Vp so this one

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should be Vi here.
I am showing you Vp so there is

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a Vi in there.
Vp/Vi=1/(1+j omega RC).

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This is a complex number,
and it is simply a number that

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when multiplied with Vi gives me
the output.

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This is also called a transfer
function and represented as H(j

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omega).
This guy is a transfer

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function, much like the gain of
my amplifier.

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Which when multiplied by the
input to get me the output.

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This guy is a complex
multiplier which when multiplied

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by Vi gives me Vp.
And as such we call it a

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transfer function H(j omega).
And we can plot this function.

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Notice that this a function of
omega.

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Remember we are taking the
frequency domain view,

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so where has time vanished?
Remember that we are taking the

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steady state view.
So we are saying in the steady

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state, if I wait long enough
this is how my circuit is going

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to behave, this is how a circuit
is going to behave.

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And the transient responses
have died away and I have time

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in my output here so my output
is a cosine.

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But that in itself is not very
interesting.

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It is a cosine of some
amplitude and has some phase.

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What we will plot is we are
going to plot this property

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here, Vp as a function of the
frequency.

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Vp is frequency dependent.
As an example,

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I could plot the absolute value
of Vp/Vi, the modulus of that

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versus omega.
And notice that when omega is

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zero again intuitive ways of
plotting this is to look at the

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value at zero and look at the
value at large omega.

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For small omega,
omega goes to zero this is one,

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so it starts off here.
And when omega is very large

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then it is much bigger than one
here, so this goes down.

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Far away this one looks like
1/omega RC.

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And this function,
assuming I have linear scales

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on my X and Y axes looks like
this.

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We also commonly plot this
using log-log scales.

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And when you do log-log scales
you get a straight line here,

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and then you actually get a
straight line of slope minus one

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because the log of this gives
you a line with a constant

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slope, it's a slope of negative
one so it becomes a straight

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line going down.
The other interesting thing to

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realize is that this magnitude
is simply one by one plus omega

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squared R squared C squared,
the square root of this.

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That's the magnitude here.
And notice when omega equals

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1/RC, this thing,
the denominator becomes one by

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square root of 2.
Somewhere here when omega

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equals 1/RC
The output is one by square

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root 2 times the input.
It's an interesting point.

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And this is called the "break
frequency".

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You can view it as a frequency
where I am getting this

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transition from one to a lower
value, and it is where the

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output is one by square root two
times the value of the input.

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Now you can think back on the
demo we showed you earlier.

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And in the demo remember that
as I increased the frequency of

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my input sinusoid my output kept
becoming smaller and smaller and

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smaller.
And you notice that you can see

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this dying out or decaying of
the amplitude as I increase my

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omega.
Let me go back.

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What you have done is that,
we're going to apply a bunch of

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sinusoids to the same circuit
and plot the frequency response,

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the ratio of the output versus
input as a function of

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frequency.
And kept applying a variety of

263
00:20:32 --> 00:20:35
frequencies.
So you can listen to the

264
00:20:35 --> 00:20:39
frequencies as they go by,
and we will plot the amplitude

265
00:20:39 --> 00:20:43
up on the screen for you.
Just for fun we are going to

266
00:20:43 --> 00:20:48
play frequencies between,
say, 10 hertz and 20 kilohertz.

267
00:20:48 --> 00:20:52
It will be fun for you to
figure out at what point you

268
00:20:52 --> 00:20:57
stop hearing the frequencies.
We are going to play from 10

269
00:20:57 --> 00:21:01
hertz to 20 kilohertz.
And figure out where your ears

270
00:21:01 --> 00:21:03
cut out.
That will tell you what the

271
00:21:03 --> 00:21:06
break frequency of your ear is.

272
00:21:06 --> 00:21:11


273
00:21:11 --> 00:21:16
You can see the amplitude being
articulated.

274
00:21:16 --> 00:21:23
The bottom figure is the phase.
This is the frequency axis.

275
00:21:23 --> 00:21:28
This is the amplitude,
log-log scales.

276
00:21:28 --> 00:22:00


277
00:22:00 --> 00:22:04
I am not sure about you but I
cannot hear anymore.

278
00:22:04 --> 00:22:09
If you bring your canine
friends to class it is quite

279
00:22:09 --> 00:22:14
possible that they would go
berserk somewhere here.

280
00:22:14 --> 00:22:18
As I promised you,
when I plot this on a log-log

281
00:22:18 --> 00:22:24
scale I get a straight line here
and a straight line out there as

282
00:22:24 --> 00:22:30
well and the bottom line gives
you the phase.

283
00:22:30 --> 00:22:34
Now, what you can also do is
you can also go to Websim.

284
00:22:34 --> 00:22:37
Websim is now linked on your
course homepage.

285
00:22:37 --> 00:22:41
You can go to Websim and you
can play with various L and C

286
00:22:41 --> 00:22:44
and R values.
And if you plot frequency

287
00:22:44 --> 00:22:48
response, if you click on the
frequency response button,

288
00:22:48 --> 00:22:52
boom, it will give you
frequency responses for your

289
00:22:52 --> 00:22:55
circuit that look exactly like
that.

290
00:22:55 --> 00:22:58
You can go and play around with
that.

291
00:22:58 --> 00:23:01
Thank you.
All right.

292
00:23:01 --> 00:23:08
As the next step I promised to
show you an easier path.

293
00:23:08 --> 00:23:12
And let's build some insight.

294
00:23:12 --> 00:23:18


295
00:23:18 --> 00:23:23
Is there a simpler way to get
where we would like to get?

296
00:23:23 --> 00:23:27
In particular,
is there a simpler way to get

297
00:23:27 --> 00:23:30
Vp?
Let's focus on Vp.

298
00:23:30 --> 00:23:33
Why Vp?
Because remember Vp was the

299
00:23:33 --> 00:23:37
complex amplitude of e to the j
omega t.

300
00:23:37 --> 00:23:43
And once I know Vp then I know
this expression here.

301
00:23:43 --> 00:23:49
Also notice that this here,
the denominator is simply the

302
00:23:49 --> 00:23:55
characteristic equation for,
I wonder how many of you

303
00:23:55 --> 00:24:01
noticed it, is simply the
characteristic equation for the

304
00:24:01 --> 00:24:05
RC circuit.
If I can write down Vp,

305
00:24:05 --> 00:24:08
I can write down the
characteristic equation,

306
00:24:08 --> 00:24:12
it will be in the denominator.
I can also write down the

307
00:24:12 --> 00:24:16
frequency response very easily
by taking the magnitude and

308
00:24:16 --> 00:24:18
phase of Vp.
So Vp has all the information

309
00:24:18 --> 00:24:21
humankind needs for those
circuits.

310
00:24:21 --> 00:24:23
Is there a simpler way to get
Vp?

311
00:24:23 --> 00:24:26
To bring some insight,
let's go ahead and write down

312
00:24:26 --> 00:24:28
--

313
00:24:28 --> 00:24:33


314
00:24:33 --> 00:24:40
Let's stare at this for a while
longer and see if light bulbs go

315
00:24:40 --> 00:24:46
off in our minds.
Of course, I could write this

316
00:24:46 --> 00:24:51
as Vi/(1+sRC).
I just replaced the shorthand

317
00:24:51 --> 00:24:57
notation for a j omega.
And I simply divide by SC

318
00:24:57 --> 00:25:02
throughout.
So I get Vi times,

319
00:25:02 --> 00:25:07
I simply divide by SC
throughout.

320
00:25:07 --> 00:25:11
Here is Vi.
I have one by SC,

321
00:25:11 --> 00:25:17
one by SC plus R.
Light bulbs beginning to go

322
00:25:17 --> 00:25:19
off?

323
00:25:19 --> 00:25:25


324
00:25:25 --> 00:25:31
The form we have here is 1/SC,
some function of my capacitance

325
00:25:31 --> 00:25:37
divided by something connected
to my capacitance plus R.

326
00:25:37 --> 00:25:41
This is Vi multiplied by
something connected to

327
00:25:41 --> 00:25:48
capacitance divided by something
connected to capacitance plus R.

328
00:25:48 --> 00:25:52
And remember your circuit.

329
00:25:52 --> 00:26:00


330
00:26:00 --> 00:26:04
What is that reminiscent of?
What does that remind you of?

331
00:26:04 --> 00:26:06
Voltage divider?
Hmm.

332
00:26:06 --> 00:26:11
There is some voltage divider
thing going on here.

333
00:26:11 --> 00:26:15
I just cannot quite pin it.
It is something about the

334
00:26:15 --> 00:26:20
capacitor, capacitor plus
booster, some voltage divider

335
00:26:20 --> 00:26:25
thingamajig happening here.
We will try to figure that out.

336
00:26:25 --> 00:26:32
What I will do is replace those
terms with something called Zc.

337
00:26:32 --> 00:26:37
Zc plus Zr.
If I can find out the Zr and Zc

338
00:26:37 --> 00:26:44
somehow, I can write down the Vp
by inspection by the voltage

339
00:26:44 --> 00:26:49
divider action,
by some generalization of the

340
00:26:49 --> 00:26:53
good old Ohm's law that I know
about.

341
00:26:53 --> 00:27:00
Let's proceed further and see
if we can make some kind of a

342
00:27:00 --> 00:27:06
connection between this and
this.

343
00:27:06 --> 00:27:09
If I can make the connection
then boom, I'm done.

344
00:27:09 --> 00:27:13
I will just use voltage
dividers and I am home.

345
00:27:13 --> 00:27:26


346
00:27:26 --> 00:27:28
OK, so let's play around and
see.

347
00:27:28 --> 00:27:31
There is something in there.
By now you should know that we

348
00:27:31 --> 00:27:34
are very close.
There is something going on in

349
00:27:34 --> 00:27:36
there.
I just need to get that spark.

350
00:27:36 --> 00:27:39
I just need to make that spark
so I can bridge the gap between

351
00:27:39 --> 00:27:42
something that is really easy
versus where I am.

352
00:27:42 --> 00:27:45
Let's take a look at the
resistor.

353
00:27:45 --> 00:27:55


354
00:27:55 --> 00:28:00
I have my resistor with the
voltage vR across it and a

355
00:28:00 --> 00:28:05
current iR.
Remember to get to any sort of

356
00:28:05 --> 00:28:11
steady state you are going to be
dealing with the drives of the

357
00:28:11 --> 00:28:15
form vI e to the j omega t,
exponential drives.

358
00:28:15 --> 00:28:20
And by taking the real part,
I know I get the input,

359
00:28:20 --> 00:28:26
and the real part of the output
gives me the actual output.

360
00:28:26 --> 00:28:33
Let's say my iR is simply
Ire^st and my vR is Vre^st.

361
00:28:33 --> 00:28:37
The S is, again,
a shorthand notation for j

362
00:28:37 --> 00:28:40
omega.
If my current Ire^st of the

363
00:28:40 --> 00:28:47
exponential form shown there and
here is Vr, I need to find out

364
00:28:47 --> 00:28:53
what relates Vr and Ir for the
element relationship for the

365
00:28:53 --> 00:28:58
resistor to hold.
In general, Ir and Vr are

366
00:28:58 --> 00:29:01
complex numbers.
For the resistor,

367
00:29:01 --> 00:29:08
I know that Vr=RIr.
And I substitute using my

368
00:29:08 --> 00:29:15
complex drives here.
So it is Vre^st=RIre^st.

369
00:29:15 --> 00:29:21
I am just substituting for
these drives,

370
00:29:21 --> 00:29:30
Ohm's law should apply,
and I cancel off e^st.

371
00:29:30 --> 00:29:33
And so I get Vr=RIr.
Interesting.

372
00:29:33 --> 00:29:39
For the resistor I find that,
based on the fundamental

373
00:29:39 --> 00:29:45
principles of resistor action,
the complex amplitude of the

374
00:29:45 --> 00:29:51
voltage simply relates to the
complex amplitude of the input

375
00:29:51 --> 00:29:56
by the proportionality factor R.
In other words,

376
00:29:56 --> 00:30:02
for the resistor --
Just as the time domain V and I

377
00:30:02 --> 00:30:06
were related by the
proportionality constant R,

378
00:30:06 --> 00:30:11
the complex amplitudes Vr and
Ir are also related in the same

379
00:30:11 --> 00:30:13
way.
That's interesting.

380
00:30:13 --> 00:30:17
Now let's look at the
capacitor.

381
00:30:17 --> 00:30:29


382
00:30:29 --> 00:30:36
Some current ic flowing through
it and a voltage vc.

383
00:30:36 --> 00:30:44
Let's say the current is Ice^st
and the voltage is Vce^st.

384
00:30:44 --> 00:30:51
Let's plug these into the
element law for the capacitor

385
00:30:51 --> 00:30:59
and see if we can find out a way
of relating vc and ic.

386
00:30:59 --> 00:31:05
I know that ic is simply
Cdvc/dt.

387
00:31:05 --> 00:31:18
So I replace this with
Ice^st=Cd/dt(vce^st),

388
00:31:18 --> 00:31:30
which is simply
Ice^st=CsVce^st.

389
00:31:30 --> 00:31:34
So I can cancel this out again.
Interesting.

390
00:31:34 --> 00:31:36
Ic=CsVc.
Very interesting.

391
00:31:36 --> 00:31:42
What is interesting here?
Notice that in the time domain

392
00:31:42 --> 00:31:46
Ic=Cdvc/dt, the element law for
the capacitor.

393
00:31:46 --> 00:31:51
So I said let's use exponential
drives, Ice^st,

394
00:31:51 --> 00:31:57
Vce^st, that's an exponential
drive, and try to find out what

395
00:31:57 --> 00:32:04
the relationship between the
complex amplitudes are.

396
00:32:04 --> 00:32:11
I plug them and what do I find?
I find that if my input is

397
00:32:11 --> 00:32:16
Vce^st, and Vc is the amplitude
of the input,

398
00:32:16 --> 00:32:23
then the current is simply
given by something multiplied

399
00:32:23 --> 00:32:27
Vc.
It's very similar in form to

400
00:32:27 --> 00:32:31
what I saw here.
The resistor,

401
00:32:31 --> 00:32:34
Vr=RIr.
For the capacitor,

402
00:32:34 --> 00:32:39
Vc=Ic/sc.
1/sc kind of plays the role of

403
00:32:39 --> 00:32:41
R.
In other words,

404
00:32:41 --> 00:32:48
the complex amplitudes around
the capacitor are related by Vc

405
00:32:48 --> 00:32:56
equals some constant times Ic.
Almost like a funny Ohm's law

406
00:32:56 --> 00:33:05
kind of relationship where Vc
and IC are complex amplitudes.

407
00:33:05 --> 00:33:13
For the inductor it is the same
way, iL, vL and L.

408
00:33:13 --> 00:33:18
Let's say iL=Ile^st and
vL=Vle^st.

409
00:33:18 --> 00:33:27
Substitute the values for the
inductor into its element

410
00:33:27 --> 00:33:34
relationship as well.
I know that vL=LdiL/dt.

411
00:33:34 --> 00:33:40
Therefore, substituting the
complex amplitudes is L.

412
00:33:40 --> 00:33:44
And diL/dt will simply be
Ilse^st.

413
00:33:44 --> 00:33:48
So I cancel out the
exponentials.

414
00:33:48 --> 00:33:55
The reason we're able to do all
of this is simply the remarkable

415
00:33:55 --> 00:34:01
beauty of exponentials.
Exponentials are absolutely

416
00:34:01 --> 00:34:05
stunningly beautiful.
The reason is that when I

417
00:34:05 --> 00:34:10
differentiate them what I get
back is the exponential times

418
00:34:10 --> 00:34:14
some constant,
and the constant was in its

419
00:34:14 --> 00:34:19
numerator multiplying t.
And that's the beauty of

420
00:34:19 --> 00:34:23
exponentials.
If this was a sine then I would

421
00:34:23 --> 00:34:27
get cosine and a sine.
With exponentials these cancel

422
00:34:27 --> 00:34:34
out and what I am left with is
something that is LsIl.

423
00:34:34 --> 00:34:41
Again, for the inductor,
the voltage across the inductor

424
00:34:41 --> 00:34:46
relates to some constant Ls here
times Il.

425
00:34:46 --> 00:34:54
This is absolutely stunning and
almost looks like a form of

426
00:34:54 --> 00:35:00
Ohm's law here.
What I am going to do is let's

427
00:35:00 --> 00:35:07
give this the name Zr.
Let's give this 1/sC the name

428
00:35:07 --> 00:35:11
Zc.
And let's give this the name

429
00:35:11 --> 00:35:15
ZL.
It kind of behaves like a

430
00:35:15 --> 00:35:20
resistor, so the resistor simply
becomes Zr.

431
00:35:20 --> 00:35:26
And 1/sC behaved like a
resistor so I called it Zc.

432
00:35:26 --> 00:35:31
And this is a ZL.
These are called "impedances".

433
00:35:31 --> 00:35:42


434
00:35:42 --> 00:35:45
In other words,
for a capacitor,

435
00:35:45 --> 00:35:52
as far as complex inputs and
outputs are concerned,

436
00:35:52 --> 00:35:59
if Vc and Ic is fed to it,
the capacitor can be replaced

437
00:35:59 --> 00:36:06
by an impedance Zc where I can
write the relationship between

438
00:36:06 --> 00:36:13
Vc and Ic as Vc=ZcIc.
Where Zc is simply one by sc.

439
00:36:13 --> 00:36:18
Similarly, for an inductor --

440
00:36:18 --> 00:36:28


441
00:36:28 --> 00:36:37
-- I can write its impedance ZL
as sL and I get Vl=ZLIl.

442
00:36:37 --> 00:36:46
And finally for a resistor it
is pretty simple.

443
00:36:46 --> 00:37:05


444
00:37:05 --> 00:37:08
What I am saying is that if I
am in the region of the

445
00:37:08 --> 00:37:11
playground, if I constrain
myself in the region of the

446
00:37:11 --> 00:37:16
playground where my inputs are
something Vi e to the j omega t

447
00:37:16 --> 00:37:18
or exponentials,
in that little region of the

448
00:37:18 --> 00:37:21
playground now,
I am focusing more and more on

449
00:37:21 --> 00:37:26
small parts of the playground so
I am kind of boxed in right now.

450
00:37:26 --> 00:37:30
In that region of the
playground this applies.

451
00:37:30 --> 00:37:34
In that region of the
playground, I can replace

452
00:37:34 --> 00:37:39
resistors by impedances,
capacitors with impedances of

453
00:37:39 --> 00:37:42
value 1/sC.
And within that playground the

454
00:37:42 --> 00:37:48
beauty of analysis there is that
in that region of the playground

455
00:37:48 --> 00:37:54
where the inputs are of the form
Vie^st, it turns out that the

456
00:37:54 --> 00:38:00
element laws are simply
generalizations of Ohm's law.

457
00:38:00 --> 00:38:03
That is absolutely stunning.
It is one of the biggest

458
00:38:03 --> 00:38:06
hallelujah moments in learning
circuits.

459
00:38:06 --> 00:38:10
This is really big.
And I think this is almost as

460
00:38:10 --> 00:38:14
big as the realization that you
can take a nonlinear circuit,

461
00:38:14 --> 00:38:19
operate it at a given operating
point, and you can sit around

462
00:38:19 --> 00:38:22
doing Zen things,
looking at small perturbations

463
00:38:22 --> 00:38:26
in there, those are going to be
linearly related.

464
00:38:26 --> 00:38:31
This is one of the big
hallelujah moments in 6.002.

465
00:38:31 --> 00:38:34
And this is of the same
magnitude as the small signal

466
00:38:34 --> 00:38:37
response being linear.
It is something that is

467
00:38:37 --> 00:38:40
completely non-intuitive.
It is something that you just

468
00:38:40 --> 00:38:43
would not have known until you
had seen it happen.

469
00:38:43 --> 00:38:46
The same way here.
This is very important so I

470
00:38:46 --> 00:38:49
will repeat it again.
I have boxed myself into this

471
00:38:49 --> 00:38:53
small region of the playground
where all I care about are

472
00:38:53 --> 00:38:57
sinusoidal inputs and
steady-state responses.

473
00:38:57 --> 00:39:01
So there I focus on complex
inputs, Vi e to the j omega t.

474
00:39:01 --> 00:39:05
And I have just shown you that
I can replace inductors,

475
00:39:05 --> 00:39:08
capacitors, resistors with
their impedances.

476
00:39:08 --> 00:39:12
And the amplitudes of the
corresponding signals around

477
00:39:12 --> 00:39:15
them are related by just a
simple Ohm's law like

478
00:39:15 --> 00:39:19
relationship using impedances.
I am sort of boxed into this

479
00:39:19 --> 00:39:22
playground, right?
In my playground it is all

480
00:39:22 --> 00:39:27
about e to the ij omega t.
e to the ij omega t is implicit

481
00:39:27 --> 00:39:30
everywhere.
I just don't show it.

482
00:39:30 --> 00:39:34
If I want to talk to somebody
else outside but within MIT in

483
00:39:34 --> 00:39:38
this small region,
it's all e to the ij omega t in

484
00:39:38 --> 00:39:41
there.
If I want to talk to somebody

485
00:39:41 --> 00:39:45
outside, get out of MIT,
get out of this playground,

486
00:39:45 --> 00:39:49
what else do I have to do?
I have to take the real part.

487
00:39:49 --> 00:39:51
Don't forget that.
Remember that,

488
00:39:51 --> 00:39:54
take for example Vc here,
so Vc is this,

489
00:39:54 --> 00:39:58
so implicit in all of this is
that if I measure Vc at some

490
00:39:58 --> 00:40:04
place it is really going to be
Vce to the j omega t.

491
00:40:04 --> 00:40:07
And if we the cosine,
the real part,

492
00:40:07 --> 00:40:10
then I have to take a real part
of this.

493
00:40:10 --> 00:40:15
And the real part of that would
Vc cosine of omega t angle Vc.

494
00:40:15 --> 00:40:18
This piece here kind of goes
unsaid.

495
00:40:18 --> 00:40:23
We will agree that we have to
do it, but we just skip that

496
00:40:23 --> 00:40:28
step because it is obvious.
We just deal with Vcs and Vls

497
00:40:28 --> 00:40:33
now.
So a new notation certainly

498
00:40:33 --> 00:40:38
sneaked by you,
and that notation looks like a

499
00:40:38 --> 00:40:45
big letter and a small letter.
Remember you have seen vL,

500
00:40:45 --> 00:40:50
this is the total behavior,
you have seen vl,

501
00:40:50 --> 00:40:56
that's a small signal behavior,
and now you see this,

502
00:40:56 --> 00:41:01
Vl, capital V small l.
And we also have DC,

503
00:41:01 --> 00:41:05
we have labeled operating point
values as VL,

504
00:41:05 --> 00:41:09
capital V, capital L.
We have one thing left so

505
00:41:09 --> 00:41:14
nobody go out there inventing
something new because we would

506
00:41:14 --> 00:41:17
be in trouble.
This is capital V,

507
00:41:17 --> 00:41:22
small l, and this is simply
"complex amplitude" in the small

508
00:41:22 --> 00:41:27
boxed region of my playground
where good things happen and

509
00:41:27 --> 00:41:32
exponentials fly.
Whenever someone gives you a

510
00:41:32 --> 00:41:35
variable, capital V,
small l, remember it's a

511
00:41:35 --> 00:41:38
complex amplitude,
a complex number,

512
00:41:38 --> 00:41:42
and you know how to get to the
time domain from there.

513
00:41:42 --> 00:41:45
You take that number,
take the real part,

514
00:41:45 --> 00:41:49
multiple the number by e to the
j omega t and take the real

515
00:41:49 --> 00:41:53
part, which is tantamount to
magnitude cosine omega t plus

516
00:41:53 --> 00:41:57
angle of that number.
Actually, you know what?

517
00:41:57 --> 00:42:00
Let's send this up.

518
00:42:00 --> 00:42:05


519
00:42:05 --> 00:42:08
Back to an example.

520
00:42:08 --> 00:42:25


521
00:42:25 --> 00:42:26
Oh, I'm sorry.
I'm sorry.

522
00:42:26 --> 00:42:30
This is not good.
This is my time domain circuit.

523
00:42:30 --> 00:42:33
Remember this was my time
domain circuit.

524
00:42:33 --> 00:42:35
A vI input.
A vC output.

525
00:42:35 --> 00:42:40
I wanted to analyze this.
What I am telling you now is

526
00:42:40 --> 00:42:44
let's box ourselves in this
impedance playground.

527
00:42:44 --> 00:42:49
And in the impedance playground
the input becomes the complex

528
00:42:49 --> 00:42:54
amplitude of the input,
my resistance gets replaced by

529
00:42:54 --> 00:43:00
a box Zr, my capacitor gets
replaced by a box Zc.

530
00:43:00 --> 00:43:07
And the voltage I care about
here is Vc.

531
00:43:07 --> 00:43:14
Zr = R and Zc=1/sC.
Now, there we go.

532
00:43:14 --> 00:43:25
I can write down Vc using a
voltage divider action as Vc is

533
00:43:25 --> 00:43:35
simply Zc/(Zc+Zr),
done, times Vi of course.

534
00:43:35 --> 00:43:42
And that gives me 1/sC divided
by 1/sC+R and multiplying

535
00:43:42 --> 00:43:48
throughout by sC I get 1/1+sCR
where S is j omega.

536
00:43:48 --> 00:43:56
Just cannot get any simpler.
How long did I take to do this?

537
00:43:56 --> 00:44:01
30 seconds.
Where I spent a whole lecture

538
00:44:01 --> 00:44:07
on Tuesday grinding through
first trig, giving up halfway

539
00:44:07 --> 00:44:11
and collapsing,
and then showing you the sneaky

540
00:44:11 --> 00:44:16
path which was still pretty
painful, but 30 seconds,

541
00:44:16 --> 00:44:19
boom.
This stuff is spectacularly

542
00:44:19 --> 00:44:23
beautiful.
The really cool thing here is

543
00:44:23 --> 00:44:28
that in this impedance domain
for linear circuits all your

544
00:44:28 --> 00:44:33
good old tricks apply.
Your Thevenin,

545
00:44:33 --> 00:44:35
your Norton,
your superposition,

546
00:44:35 --> 00:44:38
name it and it applies for this
linear circuit.

547
00:44:38 --> 00:44:43
If you close your eyes and make
believe that Zr is like an R and

548
00:44:43 --> 00:44:47
simply apply all the techniques
you have learned so far in this

549
00:44:47 --> 00:44:51
linear playground.
Just a little hack at the end

550
00:44:51 --> 00:44:53
where this is the complex
amplitude.

551
00:44:53 --> 00:44:58
And if you want to go to the
time domain part then you do the

552
00:44:58 --> 00:45:02
usual thing.
Modulus Vc cosine omega t plus

553
00:45:02 --> 00:45:04
angle Vc.
Just remember that.

554
00:45:04 --> 00:45:08
That's the jump to get back to
the time domain.

555
00:45:08 --> 00:45:13
Just to show you that this not
just works for one little

556
00:45:13 --> 00:45:17
rinky-dink circuit here,
let me take a more complicated

557
00:45:17 --> 00:45:20
circuit.
If I believe in my own BS,

558
00:45:20 --> 00:45:25
I should be able to apply this
theory to my series RLC,

559
00:45:25 --> 00:45:30
the big painful circuit that we
did differential equations for

560
00:45:30 --> 00:45:34
about a week ago.
Let's do it.

561
00:45:34 --> 00:45:39


562
00:45:39 --> 00:45:44
I have an inductor,
a capacitor and a resistor.

563
00:45:44 --> 00:45:51
What I am going to do is
replace this with the impedance

564
00:45:51 --> 00:45:53
model.
Input Vi.

565
00:45:53 --> 00:46:00
Let's say this was vI.
Let's say I cared about vR.

566
00:46:00 --> 00:46:05
L, C and R.
The impedance model would

567
00:46:05 --> 00:46:11
simply be Vi.
What's the impedance of an

568
00:46:11 --> 00:46:13
inductor?
SL.

569
00:46:13 --> 00:46:17
And for the capacitor it is
1/sC.

570
00:46:17 --> 00:46:22
And for a resistor it is simply
R.

571
00:46:22 --> 00:46:29
And just remember,
if I can find out VR then for

572
00:46:29 --> 00:46:38
an input cosine of the form Vi
cosine omega t the output will

573
00:46:38 --> 00:46:48
given by |Vr| cosine of omega t
plus angle Vr.

574
00:46:48 --> 00:46:53
Just remember this last step.
But Vr itself is trivially

575
00:46:53 --> 00:46:57
determined.
It is the voltage divider

576
00:46:57 --> 00:47:03
action again times Vi.
And the voltage divider action

577
00:47:03 --> 00:47:09
is in the denominator I sum
these thingamajigs,

578
00:47:09 --> 00:47:13
so ZL+ZC+ZR,
ZR in the numerator.

579
00:47:13 --> 00:47:17
And Zr is simply R.
ZL is sL.

580
00:47:17 --> 00:47:20
Zc is 1/sC.
And R is R.

581
00:47:20 --> 00:47:23
Vi.
And I multiply through by,

582
00:47:23 --> 00:47:30
in this particular situation,
by s/L.

583
00:47:30 --> 00:47:38
I want to get it into the same
form as you've seen before.

584
00:47:38 --> 00:47:46
Multiply throughout,
the numerator and denominator

585
00:47:46 --> 00:47:53
by s/L, what do I get?
I get RS/L and out here I end

586
00:47:53 --> 00:48:03
up getting S squared plus 1/LC,
and I get plus R/L S.

587
00:48:03 --> 00:48:04
I am done.
Look at that.

588
00:48:04 --> 00:48:07
Well, a little more than 30
seconds.

589
00:48:07 --> 00:48:09
Maybe a minute.
What is this?

590
00:48:09 --> 00:48:11
Where have you seen this
before?

591
00:48:11 --> 00:48:14
The denominator of this
expression here?

592
00:48:14 --> 00:48:17
Ah, characteristic equation for
the RLC.

593
00:48:17 --> 00:48:22
Remember I promised you in the
beginning that when we come to

594
00:48:22 --> 00:48:26
the end of the day using a
simple one-minute expression I

595
00:48:26 --> 00:48:32
am going to write down the
characteristic equation?

596
00:48:32 --> 00:48:36
Boom, here is what I get.
Did somebody hear an echo in

597
00:48:36 --> 00:48:39
there?
Notice that just by doing a

598
00:48:39 --> 00:48:43
simple voltage divider
thingamajig, I got this

599
00:48:43 --> 00:48:46
expression.
And now I can write down the

600
00:48:46 --> 00:48:51
frequency response by replacing
s is equal to j omega.

601
00:48:51 --> 00:48:56
Even more beautiful and what is
even more stunningly pretty here

602
00:48:56 --> 00:49:03
is that remember the intuitive
method I taught you about?

603
00:49:03 --> 00:49:06
The characteristic equation
gives you alpha,

604
00:49:06 --> 00:49:09
omega nought,
omega d and Q.

605
00:49:09 --> 00:49:13
And based on those we can
sketch even the time domain

606
00:49:13 --> 00:49:15
response.
Guess what?

607
00:49:15 --> 00:49:20
RLC circuits are passé now.
You can just write this thing

608
00:49:20 --> 00:49:23
down and you're done,
30 seconds or less.

609
00:49:23 --> 00:49:26
No DEs, no trig,
no nothing.

610
00:49:26 --> 49:29
OK.