- [Voiceover] So in the last video
we talked about Euler's formula and then we showed
the expressions for how to extract a cosine and a sine
from Euler's formula and we have a powerful
set of expressions there for relating
exponentials to sine waves.
So now I want to show you an example just to preview
of when we get to the formal AC analyses
how we are gonna exploit these expressions.
How we are going to exploit these formulas.
This is a real world signal.
This cosine turn, let's suppose we build something
that has a cosine to it.
That can be something like a microphone
that's hearing sounds that look like sine waves
and we would model those sine waves as a cosine wave
and they come in to some electronic system.
So let me draw a sketch of the clever approach
that we're gonna use.
So what I'm gonna do is I'm gonna build,
here's a circuit I'm imagining.
There's resistors and there's capacitors
and there's inductors.
Linear elements and we can have sources and stuff like that.
So there's something in here
and we have something going in and something coming out.
Voltage and current coming out.
Voltage and current coming in.
So I'm gonna drive my circuit
with some sort of sinusoid.
I'll call that sinusoid, I'll give it an amplitude
and I'll call it cosine of omega T.
Omega is the frequency, T is time, A is the amplitude
of the signal coming in here.
Now, that signal's gonna go into this circuit here.
Something's gonna happen and there's gonna be
a voltage coming out of here.
So we'll get V out over here of some sort
and it's gonna be some amplitude and it's gonna be
some sort of a cosine wave of omega T
plus some phase angle, some angle.
That's our job to discover this.
That's the circuit analyzes problem for AC analyzes.
We put in the AC signal, we're gonna get out
another AC signal, this is a forced response, remember?
It's gonna look like the input,
it's gonna be at the same frequency but its gonna be
at some different phase angle.
So in order to do this, there's a fair amount
of hard trigonometry we have to do.
There's gonna be a lot of cosines and sines and angles
and things inside this.
So that's pretty challenging analyses.
Now what we do with Euler's formula is we turn it
into exponentials and we already know
how to solve exponentials.
So we take the same circuit, it has the same stuff inside,
as resistors and capacitors, the same exact circuit.
Let's draw the same exact circuit.
There, that's identical.
It has the same outputs.
Output port right here.
And this time what we're gonna do is we're gonna
basically take this cosine and we're gonna,
make up in our head, we're gonna cast this
into an exponential and the way we do that
is we use that formula, we use Euler's formula,
and we basically create two sources, two separate sources
and their exponential sources.
This gonna be A over two,
E to the J
omega T,
that's this source here and this source here
is A over two,
E to the minus J omega T.
That's this wave form and if I add those together
like that.
Now remember, the equation we just looked at,
Euler's formula says that Cosine equals that.
The voltage here is exactly the same
and all we've done is described
the same exact cosine wave form
as these two imaginary exponentials.
Now I can't actually on my work bench
build one of these things.
These don't exists in real life
but they can exists mathematically.
They can exists in my head and I know that if I add
these two voltages together that I do get a cosine.
So, in our heads and on paper,
we can actually draw circuits with these things.
We can't actually build it but on paper, we can do it.
Now that I have two sources,
I can use the principal of superposition.
This is another use
of the very powerful idea of superposition.
So using the idea of superposition, I'm gonna apply
each of these two inputs one at a time
and then add the results together.
Over here, I'm gonna get two outputs.
I'm gonna get a V out one.
Let's call it a V out plus, which is what happens
when I put in this plus source and I suppress this one
which means I short it out.
I'm gonna get a V out plus and how do you solve
a differential equation when you have
a exponential going in?
Well we know this.
It's gonna be an exponential answer.
It's gonna be some constant
times E to the J omega T
plus some angle.
Then I'm gonna solve it again, I'm gonna add to that.
V out
minus
and I do that by suppressing this input
and turning this one back on using superposition
and V out for the plus,
sorry, V out for this source here.
It's gonna equal some other K.
Let's call that K plus and K minus.
E to the minus J omega T
plus V.
So I'm gonna put in two exponentials.
I'm sure I'm gonna get out two exponentials
and now using Euler's formula we know how to combine these.
We can use that same expression
and we can recover our cosine.
We can recover the real signal
and this will have some magnitude B.
We don't know the magnitudes yet
but we know what the shape of the wave form is
and if we look at this, this is the same thing as this here
and we did it by decomposing our cosine into exponentials,
putting each exponential through this and then recombining
to get cosine and we do all those steps
because solving differential equations with exponentials
is the easiest way to do it.
So just that you know, it's not twice as much work,
what's gonna happen it's gonna turn out
that this all solution down here using the negative,
using the negative exponential, the answer is going to be
exactly the same as the positive exponentials
except for there's just this conjugate,
this complex conjugate in here.
So for real signals going in, the answer goes through
and its always, these answers are always
complex conjugates of each other
as long as we start with a real input.
So that was a review of Euler's equation
and a little preview, a little sneak preview
of how we're gonna do AC Analyzes
using this really powerful tool.