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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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00:18:36,000 --> 00:18:41,000
and then you actually get a
straight line of slope minus one

230
00:18:41,000 --> 00:18:45,000
because the log of this gives
you a line with a constant

231
00:18:45,000 --> 00:18:50,000
slope, it's a slope of negative
one so it becomes a straight

232
00:18:50,000 --> 00:18:54,000
line going down.
The other interesting thing to

233
00:18:54,000 --> 00:18:59,000
realize is that this magnitude
is simply one by one plus omega

234
00:18:59,000 --> 00:19:05,000
squared R squared C squared,
the square root of this.

235
00:19:05,000 --> 00:19:13,000
That's the magnitude here.
And notice when omega equals

236
00:19:13,000 --> 00:19:20,000
1/RC, this thing,
the denominator becomes one by

237
00:19:20,000 --> 00:19:27,000
square root of 2.
Somewhere here when omega

238
00:19:27,000 --> 00:19:32,000
equals 1/RC
The output is one by square

239
00:19:32,000 --> 00:19:36,000
root 2 times the input.
It's an interesting point.

240
00:19:36,000 --> 00:19:39,000
And this is called the "break
frequency".

241
00:19:39,000 --> 00:19:44,000
You can view it as a frequency
where I am getting this

242
00:19:44,000 --> 00:19:49,000
transition from one to a lower
value, and it is where the

243
00:19:49,000 --> 00:19:54,000
output is one by square root two
times the value of the input.

244
00:19:54,000 --> 00:20:00,000
Now you can think back on the
demo we showed you earlier.

245
00:20:00,000 --> 00:20:04,000
And in the demo remember that
as I increased the frequency of

246
00:20:04,000 --> 00:20:08,000
my input sinusoid my output kept
becoming smaller and smaller and

247
00:20:08,000 --> 00:20:11,000
smaller.
And you notice that you can see

248
00:20:11,000 --> 00:20:15,000
this dying out or decaying of
the amplitude as I increase my

249
00:20:15,000 --> 00:20:16,000
omega.
Let me go back.

250
00:20:16,000 --> 00:20:20,000
What you have done is that,
we're going to apply a bunch of

251
00:20:20,000 --> 00:20:24,000
sinusoids to the same circuit
and plot the frequency response,

252
00:20:24,000 --> 00:20:28,000
the ratio of the output versus
input as a function of

253
00:20:28,000 --> 00:20:32,000
frequency.
And kept applying a variety of

254
00:20:32,000 --> 00:20:35,000
frequencies.
So you can listen to the

255
00:20:35,000 --> 00:20:39,000
frequencies as they go by,
and we will plot the amplitude

256
00:20:39,000 --> 00:20:43,000
up on the screen for you.
Just for fun we are going to

257
00:20:43,000 --> 00:20:48,000
play frequencies between,
say, 10 hertz and 20 kilohertz.

258
00:20:48,000 --> 00:20:52,000
It will be fun for you to
figure out at what point you

259
00:20:52,000 --> 00:20:57,000
stop hearing the frequencies.
We are going to play from 10

260
00:20:57,000 --> 00:21:01,000
hertz to 20 kilohertz.
And figure out where your ears

261
00:21:01,000 --> 00:21:03,000
cut out.
That will tell you what the

262
00:21:03,000 --> 00:21:06,000
break frequency of your ear is.

263
00:21:11,000 --> 00:21:16,000
You can see the amplitude being
articulated.

264
00:21:16,000 --> 00:21:23,000
The bottom figure is the phase.
This is the frequency axis.

265
00:21:23,000 --> 00:21:28,000
This is the amplitude,
log-log scales.

266
00:22:00,000 --> 00:22:04,000
I am not sure about you but I
cannot hear anymore.

267
00:22:04,000 --> 00:22:09,000
If you bring your canine
friends to class it is quite

268
00:22:09,000 --> 00:22:14,000
possible that they would go
berserk somewhere here.

269
00:22:14,000 --> 00:22:18,000
As I promised you,
when I plot this on a log-log

270
00:22:18,000 --> 00:22:24,000
scale I get a straight line here
and a straight line out there as

271
00:22:24,000 --> 00:22:30,000
well and the bottom line gives
you the phase.

272
00:22:30,000 --> 00:22:34,000
Now, what you can also do is
you can also go to Websim.

273
00:22:34,000 --> 00:22:37,000
Websim is now linked on your
course homepage.

274
00:22:37,000 --> 00:22:41,000
You can go to Websim and you
can play with various L and C

275
00:22:41,000 --> 00:22:44,000
and R values.
And if you plot frequency

276
00:22:44,000 --> 00:22:48,000
response, if you click on the
frequency response button,

277
00:22:48,000 --> 00:22:52,000
boom, it will give you
frequency responses for your

278
00:22:52,000 --> 00:22:55,000
circuit that look exactly like
that.

279
00:22:55,000 --> 00:22:58,000
You can go and play around with
that.

280
00:22:58,000 --> 00:23:01,000
Thank you.
All right.

281
00:23:01,000 --> 00:23:08,000
As the next step I promised to
show you an easier path.

282
00:23:08,000 --> 00:23:12,000
And let's build some insight.

283
00:23:18,000 --> 00:23:23,000
Is there a simpler way to get
where we would like to get?

284
00:23:23,000 --> 00:23:27,000
In particular,
is there a simpler way to get

285
00:23:27,000 --> 00:23:30,000
Vp?
Let's focus on Vp.

286
00:23:30,000 --> 00:23:33,000
Why Vp?
Because remember Vp was the

287
00:23:33,000 --> 00:23:37,000
complex amplitude of e to the j
omega t.

288
00:23:37,000 --> 00:23:43,000
And once I know Vp then I know
this expression here.

289
00:23:43,000 --> 00:23:49,000
Also notice that this here,
the denominator is simply the

290
00:23:49,000 --> 00:23:55,000
characteristic equation for,
I wonder how many of you

291
00:23:55,000 --> 00:24:01,000
noticed it, is simply the
characteristic equation for the

292
00:24:01,000 --> 00:24:05,000
RC circuit.
If I can write down Vp,

293
00:24:05,000 --> 00:24:08,000
I can write down the
characteristic equation,

294
00:24:08,000 --> 00:24:12,000
it will be in the denominator.
I can also write down the

295
00:24:12,000 --> 00:24:16,000
frequency response very easily
by taking the magnitude and

296
00:24:16,000 --> 00:24:18,000
phase of Vp.
So Vp has all the information

297
00:24:18,000 --> 00:24:21,000
humankind needs for those
circuits.

298
00:24:21,000 --> 00:24:23,000
Is there a simpler way to get
Vp?

299
00:24:23,000 --> 00:24:26,000
To bring some insight,
let's go ahead and write down

300
00:24:26,000 --> 00:24:28,000
--

301
00:24:33,000 --> 00:24:40,000
Let's stare at this for a while
longer and see if light bulbs go

302
00:24:40,000 --> 00:24:46,000
off in our minds.
Of course, I could write this

303
00:24:46,000 --> 00:24:51,000
as Vi/(1+sRC).
I just replaced the shorthand

304
00:24:51,000 --> 00:24:57,000
notation for a j omega.
And I simply divide by SC

305
00:24:57,000 --> 00:25:02,000
throughout.
So I get Vi times,

306
00:25:02,000 --> 00:25:07,000
I simply divide by SC
throughout.

307
00:25:07,000 --> 00:25:11,000
Here is Vi.
I have one by SC,

308
00:25:11,000 --> 00:25:17,000
one by SC plus R.
Light bulbs beginning to go

309
00:25:17,000 --> 00:25:19,000
off?

310
00:25:25,000 --> 00:25:31,000
The form we have here is 1/SC,
some function of my capacitance

311
00:25:31,000 --> 00:25:37,000
divided by something connected
to my capacitance plus R.

312
00:25:37,000 --> 00:25:41,000
This is Vi multiplied by
something connected to

313
00:25:41,000 --> 00:25:48,000
capacitance divided by something
connected to capacitance plus R.

314
00:25:48,000 --> 00:25:52,000
And remember your circuit.

315
00:26:00,000 --> 00:26:04,000
What is that reminiscent of?
What does that remind you of?

316
00:26:04,000 --> 00:26:06,000
Voltage divider?
Hmm.

317
00:26:06,000 --> 00:26:11,000
There is some voltage divider
thing going on here.

318
00:26:11,000 --> 00:26:15,000
I just cannot quite pin it.
It is something about the

319
00:26:15,000 --> 00:26:20,000
capacitor, capacitor plus
booster, some voltage divider

320
00:26:20,000 --> 00:26:25,000
thingamajig happening here.
We will try to figure that out.

321
00:26:25,000 --> 00:26:32,000
What I will do is replace those
terms with something called Zc.

322
00:26:32,000 --> 00:26:37,000
Zc plus Zr.
If I can find out the Zr and Zc

323
00:26:37,000 --> 00:26:44,000
somehow, I can write down the Vp
by inspection by the voltage

324
00:26:44,000 --> 00:26:49,000
divider action,
by some generalization of the

325
00:26:49,000 --> 00:26:53,000
good old Ohm's law that I know
about.

326
00:26:53,000 --> 00:27:00,000
Let's proceed further and see
if we can make some kind of a

327
00:27:00,000 --> 00:27:06,000
connection between this and
this.

328
00:27:06,000 --> 00:27:09,000
If I can make the connection
then boom, I'm done.

329
00:27:09,000 --> 00:27:13,000
I will just use voltage
dividers and I am home.

330
00:27:26,000 --> 00:27:28,000
OK, so let's play around and
see.

331
00:27:28,000 --> 00:27:31,000
There is something in there.
By now you should know that we

332
00:27:31,000 --> 00:27:34,000
are very close.
There is something going on in

333
00:27:34,000 --> 00:27:36,000
there.
I just need to get that spark.

334
00:27:36,000 --> 00:27:39,000
I just need to make that spark
so I can bridge the gap between

335
00:27:39,000 --> 00:27:42,000
something that is really easy
versus where I am.

336
00:27:42,000 --> 00:27:45,000
Let's take a look at the
resistor.

337
00:27:55,000 --> 00:28:00,000
I have my resistor with the
voltage vR across it and a

338
00:28:00,000 --> 00:28:05,000
current iR.
Remember to get to any sort of

339
00:28:05,000 --> 00:28:11,000
steady state you are going to be
dealing with the drives of the

340
00:28:11,000 --> 00:28:15,000
form vI e to the j omega t,
exponential drives.

341
00:28:15,000 --> 00:28:20,000
And by taking the real part,
I know I get the input,

342
00:28:20,000 --> 00:28:26,000
and the real part of the output
gives me the actual output.

343
00:28:26,000 --> 00:28:33,000
Let's say my iR is simply
Ire^st and my vR is Vre^st.

344
00:28:33,000 --> 00:28:37,000
The S is, again,
a shorthand notation for j

345
00:28:37,000 --> 00:28:40,000
omega.
If my current Ire^st of the

346
00:28:40,000 --> 00:28:47,000
exponential form shown there and
here is Vr, I need to find out

347
00:28:47,000 --> 00:28:53,000
what relates Vr and Ir for the
element relationship for the

348
00:28:53,000 --> 00:28:58,000
resistor to hold.
In general, Ir and Vr are

349
00:28:58,000 --> 00:29:01,000
complex numbers.
For the resistor,

350
00:29:01,000 --> 00:29:08,000
I know that Vr=RIr.
And I substitute using my

351
00:29:08,000 --> 00:29:15,000
complex drives here.
So it is Vre^st=RIre^st.

352
00:29:15,000 --> 00:29:21,000
I am just substituting for
these drives,

353
00:29:21,000 --> 00:29:30,000
Ohm's law should apply,
and I cancel off e^st.

354
00:29:30,000 --> 00:29:33,000
And so I get Vr=RIr.
Interesting.

355
00:29:33,000 --> 00:29:39,000
For the resistor I find that,
based on the fundamental

356
00:29:39,000 --> 00:29:45,000
principles of resistor action,
the complex amplitude of the

357
00:29:45,000 --> 00:29:51,000
voltage simply relates to the
complex amplitude of the input

358
00:29:51,000 --> 00:29:56,000
by the proportionality factor R.
In other words,

359
00:29:56,000 --> 00:30:02,000
for the resistor --
Just as the time domain V and I

360
00:30:02,000 --> 00:30:06,000
were related by the
proportionality constant R,

361
00:30:06,000 --> 00:30:11,000
the complex amplitudes Vr and
Ir are also related in the same

362
00:30:11,000 --> 00:30:13,000
way.
That's interesting.

363
00:30:13,000 --> 00:30:17,000
Now let's look at the
capacitor.

364
00:30:29,000 --> 00:30:36,000
Some current ic flowing through
it and a voltage vc.

365
00:30:36,000 --> 00:30:44,000
Let's say the current is Ice^st
and the voltage is Vce^st.

366
00:30:44,000 --> 00:30:51,000
Let's plug these into the
element law for the capacitor

367
00:30:51,000 --> 00:30:59,000
and see if we can find out a way
of relating vc and ic.

368
00:30:59,000 --> 00:31:05,000
I know that ic is simply
Cdvc/dt.

369
00:31:05,000 --> 00:31:18,000
So I replace this with
Ice^st=Cd/dt(vce^st),

370
00:31:18,000 --> 00:31:30,000
which is simply
Ice^st=CsVce^st.

371
00:31:30,000 --> 00:31:34,000
So I can cancel this out again.
Interesting.

372
00:31:34,000 --> 00:31:36,000
Ic=CsVc.
Very interesting.

373
00:31:36,000 --> 00:31:42,000
What is interesting here?
Notice that in the time domain

374
00:31:42,000 --> 00:31:46,000
Ic=Cdvc/dt, the element law for
the capacitor.

375
00:31:46,000 --> 00:31:51,000
So I said let's use exponential
drives, Ice^st,

376
00:31:51,000 --> 00:31:57,000
Vce^st, that's an exponential
drive, and try to find out what

377
00:31:57,000 --> 00:32:04,000
the relationship between the
complex amplitudes are.

378
00:32:04,000 --> 00:32:11,000
I plug them and what do I find?
I find that if my input is

379
00:32:11,000 --> 00:32:16,000
Vce^st, and Vc is the amplitude
of the input,

380
00:32:16,000 --> 00:32:23,000
then the current is simply
given by something multiplied

381
00:32:23,000 --> 00:32:27,000
Vc.
It's very similar in form to

382
00:32:27,000 --> 00:32:31,000
what I saw here.
The resistor,

383
00:32:31,000 --> 00:32:34,000
Vr=RIr.
For the capacitor,

384
00:32:34,000 --> 00:32:39,000
Vc=Ic/sc.
1/sc kind of plays the role of

385
00:32:39,000 --> 00:32:41,000
R.
In other words,

386
00:32:41,000 --> 00:32:48,000
the complex amplitudes around
the capacitor are related by Vc

387
00:32:48,000 --> 00:32:56,000
equals some constant times Ic.
Almost like a funny Ohm's law

388
00:32:56,000 --> 00:33:05,000
kind of relationship where Vc
and IC are complex amplitudes.

389
00:33:05,000 --> 00:33:13,000
For the inductor it is the same
way, iL, vL and L.

390
00:33:13,000 --> 00:33:18,000
Let's say iL=Ile^st and
vL=Vle^st.

391
00:33:18,000 --> 00:33:27,000
Substitute the values for the
inductor into its element

392
00:33:27,000 --> 00:33:34,000
relationship as well.
I know that vL=LdiL/dt.

393
00:33:34,000 --> 00:33:40,000
Therefore, substituting the
complex amplitudes is L.

394
00:33:40,000 --> 00:33:44,000
And diL/dt will simply be
Ilse^st.

395
00:33:44,000 --> 00:33:48,000
So I cancel out the
exponentials.

396
00:33:48,000 --> 00:33:55,000
The reason we're able to do all
of this is simply the remarkable

397
00:33:55,000 --> 00:34:01,000
beauty of exponentials.
Exponentials are absolutely

398
00:34:01,000 --> 00:34:05,000
stunningly beautiful.
The reason is that when I

399
00:34:05,000 --> 00:34:10,000
differentiate them what I get
back is the exponential times

400
00:34:10,000 --> 00:34:14,000
some constant,
and the constant was in its

401
00:34:14,000 --> 00:34:19,000
numerator multiplying t.
And that's the beauty of

402
00:34:19,000 --> 00:34:23,000
exponentials.
If this was a sine then I would

403
00:34:23,000 --> 00:34:27,000
get cosine and a sine.
With exponentials these cancel

404
00:34:27,000 --> 00:34:34,000
out and what I am left with is
something that is LsIl.

405
00:34:34,000 --> 00:34:41,000
Again, for the inductor,
the voltage across the inductor

406
00:34:41,000 --> 00:34:46,000
relates to some constant Ls here
times Il.

407
00:34:46,000 --> 00:34:54,000
This is absolutely stunning and
almost looks like a form of

408
00:34:54,000 --> 00:35:00,000
Ohm's law here.
What I am going to do is let's

409
00:35:00,000 --> 00:35:07,000
give this the name Zr.
Let's give this 1/sC the name

410
00:35:07,000 --> 00:35:11,000
Zc.
And let's give this the name

411
00:35:11,000 --> 00:35:15,000
ZL.
It kind of behaves like a

412
00:35:15,000 --> 00:35:20,000
resistor, so the resistor simply
becomes Zr.

413
00:35:20,000 --> 00:35:26,000
And 1/sC behaved like a
resistor so I called it Zc.

414
00:35:26,000 --> 00:35:31,000
And this is a ZL.
These are called "impedances".

415
00:35:42,000 --> 00:35:45,000
In other words,
for a capacitor,

416
00:35:45,000 --> 00:35:52,000
as far as complex inputs and
outputs are concerned,

417
00:35:52,000 --> 00:35:59,000
if Vc and Ic is fed to it,
the capacitor can be replaced

418
00:35:59,000 --> 00:36:06,000
by an impedance Zc where I can
write the relationship between

419
00:36:06,000 --> 00:36:13,000
Vc and Ic as Vc=ZcIc.
Where Zc is simply one by sc.

420
00:36:13,000 --> 00:36:18,000
Similarly, for an inductor --

421
00:36:28,000 --> 00:36:37,000
-- I can write its impedance ZL
as sL and I get Vl=ZLIl.

422
00:36:37,000 --> 00:36:46,000
And finally for a resistor it
is pretty simple.

423
00:37:05,000 --> 00:37:08,000
What I am saying is that if I
am in the region of the

424
00:37:08,000 --> 00:37:11,000
playground, if I constrain
myself in the region of the

425
00:37:11,000 --> 00:37:16,000
playground where my inputs are
something Vi e to the j omega t

426
00:37:16,000 --> 00:37:18,000
or exponentials,
in that little region of the

427
00:37:18,000 --> 00:37:21,000
playground now,
I am focusing more and more on

428
00:37:21,000 --> 00:37:26,000
small parts of the playground so
I am kind of boxed in right now.

429
00:37:26,000 --> 00:37:30,000
In that region of the
playground this applies.

430
00:37:30,000 --> 00:37:34,000
In that region of the
playground, I can replace

431
00:37:34,000 --> 00:37:39,000
resistors by impedances,
capacitors with impedances of

432
00:37:39,000 --> 00:37:42,000
value 1/sC.
And within that playground the

433
00:37:42,000 --> 00:37:48,000
beauty of analysis there is that
in that region of the playground

434
00:37:48,000 --> 00:37:54,000
where the inputs are of the form
Vie^st, it turns out that the

435
00:37:54,000 --> 00:38:00,000
element laws are simply
generalizations of Ohm's law.

436
00:38:00,000 --> 00:38:03,000
That is absolutely stunning.
It is one of the biggest

437
00:38:03,000 --> 00:38:06,000
hallelujah moments in learning
circuits.

438
00:38:06,000 --> 00:38:10,000
This is really big.
And I think this is almost as

439
00:38:10,000 --> 00:38:14,000
big as the realization that you
can take a nonlinear circuit,

440
00:38:14,000 --> 00:38:19,000
operate it at a given operating
point, and you can sit around

441
00:38:19,000 --> 00:38:22,000
doing Zen things,
looking at small perturbations

442
00:38:22,000 --> 00:38:26,000
in there, those are going to be
linearly related.

443
00:38:26,000 --> 00:38:31,000
This is one of the big
hallelujah moments in 6.002.

444
00:38:31,000 --> 00:38:34,000
And this is of the same
magnitude as the small signal

445
00:38:34,000 --> 00:38:37,000
response being linear.
It is something that is

446
00:38:37,000 --> 00:38:40,000
completely non-intuitive.
It is something that you just

447
00:38:40,000 --> 00:38:43,000
would not have known until you
had seen it happen.

448
00:38:43,000 --> 00:38:46,000
The same way here.
This is very important so I

449
00:38:46,000 --> 00:38:49,000
will repeat it again.
I have boxed myself into this

450
00:38:49,000 --> 00:38:53,000
small region of the playground
where all I care about are

451
00:38:53,000 --> 00:38:57,000
sinusoidal inputs and
steady-state responses.

452
00:38:57,000 --> 00:39:01,000
So there I focus on complex
inputs, Vi e to the j omega t.

453
00:39:01,000 --> 00:39:05,000
And I have just shown you that
I can replace inductors,

454
00:39:05,000 --> 00:39:08,000
capacitors, resistors with
their impedances.

455
00:39:08,000 --> 00:39:12,000
And the amplitudes of the
corresponding signals around

456
00:39:12,000 --> 00:39:15,000
them are related by just a
simple Ohm's law like

457
00:39:15,000 --> 00:39:19,000
relationship using impedances.
I am sort of boxed into this

458
00:39:19,000 --> 00:39:22,000
playground, right?
In my playground it is all

459
00:39:22,000 --> 00:39:27,000
about e to the ij omega t.
e to the ij omega t is implicit

460
00:39:27,000 --> 00:39:30,000
everywhere.
I just don't show it.

461
00:39:30,000 --> 00:39:34,000
If I want to talk to somebody
else outside but within MIT in

462
00:39:34,000 --> 00:39:38,000
this small region,
it's all e to the ij omega t in

463
00:39:38,000 --> 00:39:41,000
there.
If I want to talk to somebody

464
00:39:41,000 --> 00:39:45,000
outside, get out of MIT,
get out of this playground,

465
00:39:45,000 --> 00:39:49,000
what else do I have to do?
I have to take the real part.

466
00:39:49,000 --> 00:39:51,000
Don't forget that.
Remember that,

467
00:39:51,000 --> 00:39:54,000
take for example Vc here,
so Vc is this,

468
00:39:54,000 --> 00:39:58,000
so implicit in all of this is
that if I measure Vc at some

469
00:39:58,000 --> 00:40:04,000
place it is really going to be
Vce to the j omega t.

470
00:40:04,000 --> 00:40:07,000
And if we the cosine,
the real part,

471
00:40:07,000 --> 00:40:10,000
then I have to take a real part
of this.

472
00:40:10,000 --> 00:40:15,000
And the real part of that would
Vc cosine of omega t angle Vc.

473
00:40:15,000 --> 00:40:18,000
This piece here kind of goes
unsaid.

474
00:40:18,000 --> 00:40:23,000
We will agree that we have to
do it, but we just skip that

475
00:40:23,000 --> 00:40:28,000
step because it is obvious.
We just deal with Vcs and Vls

476
00:40:28,000 --> 00:40:33,000
now.
So a new notation certainly

477
00:40:33,000 --> 00:40:38,000
sneaked by you,
and that notation looks like a

478
00:40:38,000 --> 00:40:45,000
big letter and a small letter.
Remember you have seen vL,

479
00:40:45,000 --> 00:40:50,000
this is the total behavior,
you have seen vl,

480
00:40:50,000 --> 00:40:56,000
that's a small signal behavior,
and now you see this,

481
00:40:56,000 --> 00:41:01,000
Vl, capital V small l.
And we also have DC,

482
00:41:01,000 --> 00:41:05,000
we have labeled operating point
values as VL,

483
00:41:05,000 --> 00:41:09,000
capital V, capital L.
We have one thing left so

484
00:41:09,000 --> 00:41:14,000
nobody go out there inventing
something new because we would

485
00:41:14,000 --> 00:41:17,000
be in trouble.
This is capital V,

486
00:41:17,000 --> 00:41:22,000
small l, and this is simply
"complex amplitude" in the small

487
00:41:22,000 --> 00:41:27,000
boxed region of my playground
where good things happen and

488
00:41:27,000 --> 00:41:32,000
exponentials fly.
Whenever someone gives you a

489
00:41:32,000 --> 00:41:35,000
variable, capital V,
small l, remember it's a

490
00:41:35,000 --> 00:41:38,000
complex amplitude,
a complex number,

491
00:41:38,000 --> 00:41:42,000
and you know how to get to the
time domain from there.

492
00:41:42,000 --> 00:41:45,000
You take that number,
take the real part,

493
00:41:45,000 --> 00:41:49,000
multiple the number by e to the
j omega t and take the real

494
00:41:49,000 --> 00:41:53,000
part, which is tantamount to
magnitude cosine omega t plus

495
00:41:53,000 --> 00:41:57,000
angle of that number.
Actually, you know what?

496
00:41:57,000 --> 00:42:00,000
Let's send this up.

497
00:42:05,000 --> 00:42:08,000
Back to an example.

498
00:42:25,000 --> 00:42:26,000
Oh, I'm sorry.
I'm sorry.

499
00:42:26,000 --> 00:42:30,000
This is not good.
This is my time domain circuit.

500
00:42:30,000 --> 00:42:33,000
Remember this was my time
domain circuit.

501
00:42:33,000 --> 00:42:35,000
A vI input.
A vC output.

502
00:42:35,000 --> 00:42:40,000
I wanted to analyze this.
What I am telling you now is

503
00:42:40,000 --> 00:42:44,000
let's box ourselves in this
impedance playground.

504
00:42:44,000 --> 00:42:49,000
And in the impedance playground
the input becomes the complex

505
00:42:49,000 --> 00:42:54,000
amplitude of the input,
my resistance gets replaced by

506
00:42:54,000 --> 00:43:00,000
a box Zr, my capacitor gets
replaced by a box Zc.

507
00:43:00,000 --> 00:43:07,000
And the voltage I care about
here is Vc.

508
00:43:07,000 --> 00:43:14,000
Zr = R and Zc=1/sC.
Now, there we go.

509
00:43:14,000 --> 00:43:25,000
I can write down Vc using a
voltage divider action as Vc is

510
00:43:25,000 --> 00:43:35,000
simply Zc/(Zc+Zr),
done, times Vi of course.

511
00:43:35,000 --> 00:43:42,000
And that gives me 1/sC divided
by 1/sC+R and multiplying

512
00:43:42,000 --> 00:43:48,000
throughout by sC I get 1/1+sCR
where S is j omega.

513
00:43:48,000 --> 00:43:56,000
Just cannot get any simpler.
How long did I take to do this?

514
00:43:56,000 --> 00:44:01,000
30 seconds.
Where I spent a whole lecture

515
00:44:01,000 --> 00:44:07,000
on Tuesday grinding through
first trig, giving up halfway

516
00:44:07,000 --> 00:44:11,000
and collapsing,
and then showing you the sneaky

517
00:44:11,000 --> 00:44:16,000
path which was still pretty
painful, but 30 seconds,

518
00:44:16,000 --> 00:44:19,000
boom.
This stuff is spectacularly

519
00:44:19,000 --> 00:44:23,000
beautiful.
The really cool thing here is

520
00:44:23,000 --> 00:44:28,000
that in this impedance domain
for linear circuits all your

521
00:44:28,000 --> 00:44:33,000
good old tricks apply.
Your Thevenin,

522
00:44:33,000 --> 00:44:35,000
your Norton,
your superposition,

523
00:44:35,000 --> 00:44:38,000
name it and it applies for this
linear circuit.

524
00:44:38,000 --> 00:44:43,000
If you close your eyes and make
believe that Zr is like an R and

525
00:44:43,000 --> 00:44:47,000
simply apply all the techniques
you have learned so far in this

526
00:44:47,000 --> 00:44:51,000
linear playground.
Just a little hack at the end

527
00:44:51,000 --> 00:44:53,000
where this is the complex
amplitude.

528
00:44:53,000 --> 00:44:58,000
And if you want to go to the
time domain part then you do the

529
00:44:58,000 --> 00:45:02,000
usual thing.
Modulus Vc cosine omega t plus

530
00:45:02,000 --> 00:45:04,000
angle Vc.
Just remember that.

531
00:45:04,000 --> 00:45:08,000
That's the jump to get back to
the time domain.

532
00:45:08,000 --> 00:45:13,000
Just to show you that this not
just works for one little

533
00:45:13,000 --> 00:45:17,000
rinky-dink circuit here,
let me take a more complicated

534
00:45:17,000 --> 00:45:20,000
circuit.
If I believe in my own BS,

535
00:45:20,000 --> 00:45:25,000
I should be able to apply this
theory to my series RLC,

536
00:45:25,000 --> 00:45:30,000
the big painful circuit that we
did differential equations for

537
00:45:30,000 --> 00:45:34,000
about a week ago.
Let's do it.

538
00:45:39,000 --> 00:45:44,000
I have an inductor,
a capacitor and a resistor.

539
00:45:44,000 --> 00:45:51,000
What I am going to do is
replace this with the impedance

540
00:45:51,000 --> 00:45:53,000
model.
Input Vi.

541
00:45:53,000 --> 00:46:00,000
Let's say this was vI.
Let's say I cared about vR.

542
00:46:00,000 --> 00:46:05,000
L, C and R.
The impedance model would

543
00:46:05,000 --> 00:46:11,000
simply be Vi.
What's the impedance of an

544
00:46:11,000 --> 00:46:13,000
inductor?
SL.

545
00:46:13,000 --> 00:46:17,000
And for the capacitor it is
1/sC.

546
00:46:17,000 --> 00:46:22,000
And for a resistor it is simply
R.

547
00:46:22,000 --> 00:46:29,000
And just remember,
if I can find out VR then for

548
00:46:29,000 --> 00:46:38,000
an input cosine of the form Vi
cosine omega t the output will

549
00:46:38,000 --> 00:46:48,000
given by |Vr| cosine of omega t
plus angle Vr.

550
00:46:48,000 --> 00:46:53,000
Just remember this last step.
But Vr itself is trivially

551
00:46:53,000 --> 00:46:57,000
determined.
It is the voltage divider

552
00:46:57,000 --> 00:47:03,000
action again times Vi.
And the voltage divider action

553
00:47:03,000 --> 00:47:09,000
is in the denominator I sum
these thingamajigs,

554
00:47:09,000 --> 00:47:13,000
so ZL+ZC+ZR,
ZR in the numerator.

555
00:47:13,000 --> 00:47:17,000
And Zr is simply R.
ZL is sL.

556
00:47:17,000 --> 00:47:20,000
Zc is 1/sC.
And R is R.

557
00:47:20,000 --> 00:47:23,000
Vi.
And I multiply through by,

558
00:47:23,000 --> 00:47:30,000
in this particular situation,
by s/L.

559
00:47:30,000 --> 00:47:38,000
I want to get it into the same
form as you've seen before.

560
00:47:38,000 --> 00:47:46,000
Multiply throughout,
the numerator and denominator

561
00:47:46,000 --> 00:47:53,000
by s/L, what do I get?
I get RS/L and out here I end

562
00:47:53,000 --> 00:48:03,000
up getting S squared plus 1/LC,
and I get plus R/L S.

563
00:48:03,000 --> 00:48:04,000
I am done.
Look at that.

564
00:48:04,000 --> 00:48:07,000
Well, a little more than 30
seconds.

565
00:48:07,000 --> 00:48:09,000
Maybe a minute.
What is this?

566
00:48:09,000 --> 00:48:11,000
Where have you seen this
before?

567
00:48:11,000 --> 00:48:14,000
The denominator of this
expression here?

568
00:48:14,000 --> 00:48:17,000
Ah, characteristic equation for
the RLC.

569
00:48:17,000 --> 00:48:22,000
Remember I promised you in the
beginning that when we come to

570
00:48:22,000 --> 00:48:26,000
the end of the day using a
simple one-minute expression I

571
00:48:26,000 --> 00:48:32,000
am going to write down the
characteristic equation?

572
00:48:32,000 --> 00:48:36,000
Boom, here is what I get.
Did somebody hear an echo in

573
00:48:36,000 --> 00:48:39,000
there?
Notice that just by doing a

574
00:48:39,000 --> 00:48:43,000
simple voltage divider
thingamajig, I got this

575
00:48:43,000 --> 00:48:46,000
expression.
And now I can write down the

576
00:48:46,000 --> 00:48:51,000
frequency response by replacing
s is equal to j omega.

577
00:48:51,000 --> 00:48:56,000
Even more beautiful and what is
even more stunningly pretty here

578
00:48:56,000 --> 00:49:03,000
is that remember the intuitive
method I taught you about?

579
00:49:03,000 --> 00:49:06,000
The characteristic equation
gives you alpha,

580
00:49:06,000 --> 00:49:09,000
omega nought,
omega d and Q.

581
00:49:09,000 --> 00:49:13,000
And based on those we can
sketch even the time domain

582
00:49:13,000 --> 00:49:15,000
response.
Guess what?

583
00:49:15,000 --> 00:49:20,000
RLC circuits are passÈ now.
You can just write this thing

584
00:49:20,000 --> 00:49:23,000
down and you're done,
30 seconds or less.

585
00:49:23,000 --> 00:49:26,000
No DEs, no trig,
no nothing.

586
00:49:26,000 --> 00:49:29,000
OK.