- [Voiceover] We now begin a whole new area
of circuit analysis called sinusoidal steady state analysis.
You can also call it AC analysis.
AC stands for alternating current.
It means it is a voltage or a current that
where the signal actually changes sign.
It is positive sometimes.
It is negative sometimes.
And the conventional name for that is AC
or alternating current.
It could've been called alternating voltage
but that's not the name.
The name is AC.
What I want to do in this video
is we will do a quick review
of what it is like to solve an equation
like the one showed here.
This is an RLC circuit.
I'm going to show what it is like to solve this
in differential equation form,
which is gonna be a lot of work.
I want to introduce the idea
of a new point of view or a new analysis method
that we refer to as the sinusoidal steady state.
And it is a transformation
that we are gonna do on this circuit.
And it is gonna be a big reward at the end.
I want to go through what's the reward
and what's it gonna look like at the end.
Then we are gonna know some of the math
that we have to review in order to fully understand
this change that we are gonna go through.
This change of point of view.
Let's first take a look at this circuit here.
This is a circuit that's now a driven RLC circuit.
So here's the drive function.
It is a voltage.
It is some waveform.
It is driving a sequence of an inductor,
a resistor, and a capacitor.
Now in an earlier video
we derived the natural response
of this circuit
and to do that, we shorted out
we removed the source
and shorted it out
and added a little bit of energy to this circuit
and saw what it did on its own.
What was its natural response.
Now we've upgraded this.
We've added a source
and now we have to solve this again
including the source.
If we use the differential equation technique
this is how we are gonna go about it.
So the first step in a circuit analysis like this
is to write a KVL equation.
We are gonna solve for this current right here.
That's the one current that's in this.
So I is our independent variable.
If I write KVL as you recall
when we did this for natural response,
we ended up with a differential equation
that looked like this.
We had L times the second derivative of I
plus R times the first derivative of I
plus one over C times I.
So these are the voltages.
Each of these individual terms
are the voltages across these components here.
So that's the resistor voltage.
This is the capacitor voltage
and this here is the inductor voltage
so it is inductor voltage,
resistor voltage,
capacitor voltage.
And all those if we add those up,
those have to equal VN.
So this is now a forced equation,
which means this is the forcing function
and we are gonna have to solve this
and the math for doing this is
pretty difficult.
It was hard enough to do the natural response
and we add in this
and it gets to be even more work.
So as we did before,
what we do now is we propose a solution
and the solution we are in the habit of doing this now
is gonna be sum constant times E
to the sum natural frequency times T.
So AE to the ST
is our proposed solution for I as a function of time.
You remember we called S.
S is a frequency term
because S times T has to have no units
so S has units of one over time or frequency
so that's called the natural frequency.
And when we plug in I
the way to tell if I is a solution
is to plug this into this equation here.
And we got an equation that looks like this.
We ended up after factoring our I
we ended up with
LS squared plus RS
plus one over C
and all that is equal to
for the natural response
we put in,
so we solve this equation by setting
this term here equal to zero
and solving for s
to find out what the natural frequency is
and then we go back and we find out A
by looking at the initial conditions over here.
Whatever initial energy was in this circuit
determines the value of A here.
The next step in this forced response
where VN is driving the circuit
is we have to set this back to VN
and solve for the forced.
Now if we let VN be sort of any forcing function we want,
any kind of waveform,
this is gonna be a really hard piece of mathematics.
This is gonna be a really difficult calculation.
It is gonna take a long time and basically
I don't want to do it.
I'm gonna wish there was some other way to do these
kinds of equations and there is.
The way we simplify this process substantially
is we make a little limitation on ourselves
on what VN can be.
If we make a rule,
if we basically volunteer to limit ourselves
to VN equals sinusoids
that means that VN is of the form
cosine of omega T plus phi
where phi is some angle
or it could be sine
of omega T plus some angle.
Any waveform of this form here
is called the sinusoid.
So sinusoid is the general name for cosine and sine
and signals that look like that.
Because we don't want this math to blow up on us,
with the general input over here,
we are gonna develop a really elegant way
to solve circuits with this
we limit ourselves to sinusoidal inputs.
We will take a pause right here
and continue on in the next video
to introduce the idea of sinusoidal analysis.