[BLANK_AUDIO].
So now that we've gotten our feet wet a
little bit with talking about magnetism
and inductors, let's see how inductors are
actually going to be behave in real
systems.
So the aims for today, are to present how
they're going to work together in these
system settings.
And just like we did with capacitors,
we're
going to look at parallel and series
inductors.
And do some derivations to see what the
equivalent inductance would be.
And then graphically represent the
relationships, just
like we did with capacitors, for current,
voltage, power, and energy.
So, again, from our previous class, we
started talking about the
way that currents and voltages relate
within inductors, within single devices.
and we talked about we meant by
inductance, now we're going
to go into a little bit more depth on
these things.
So first we'll analyze the inductors in
series and
parallel, then analyze DC circuits with
inductors, calculate the energy
in inductors, doing a duration much like
we did
with capacitors And then doing the
sketching of the curves.
Here we place our inductors in series, and
we want
to do a derivation to find what the
equivalent inductance is.
But remembering from before, i equals.
Well, sorry, v equals l di dt.
We'll do the derivative version instead,
because
that's what we're going to be using here.
We know that the current going through
these two
devices, because they're in series, has to
be equal.
And we have two voltages here, v1 and v2.
So to find those voltages, we set up our
equation by saying
that the total voltage is equal to the sum
of the individual voltages.
then again, we factor out our two
inductance's
to find out our equivalent inductance is
right here.
Sum of the two inductors.
Remembering from what we learned about
resistors, we place resistors in series.
It's the same thin as adding the first
instances Okay, so doing inductors,
we do the exact same thing we did with
resistors for combining them.
Even though the derivations are different
in these equations are differential
equations, derived
equations it still gives us the same
result, we just add them all together.
Now lets look at how they behave if we
place them in parallel.
Kirchhoff's Current Law states that this
current has to equal these two currents.
But we know because they're parallel,
voltages are the same.
So i equals i1 plus i2.
We're now using the one over L integration
of v, dt, defined what our currents happen
to be.
And again combining terms pulling
things out we discover that our equivalent
inductance is 1
over L1 plus 1 over L2 and then we invert
that.
This is the exact same thing you do for
finding
the the equivalent resistance of resistors
that are placed in parallel.
And remember the capacitance was the other
way around.
Capacitors in this configuration you added
them together, and
if they were in series, you did this
calculation.
But the nice thing is that they all have
the same basic calculations.
It's just that when you apply them you
need to be careful, make sure that
you remember capacitors are the ones that
are
different, but resisters and conductors
behave very similarly.
If I put them in a DC circuit.
Like a capacitor, over a long period of
time would get to an equalibrium in
voltage.
An inductor is going to become-, get to an
equalibrium in current.
And so, we're going to replace this
inductor, with a wire.
And then do our analysis.
kind of like where, with a capacitor we
replace it.
With an open circuit.
So if I wanted to know, for example, the
current coming through here.
What I first need to know is what the
current is coming through here.
So to do that, I have a six ohm here, and
a twelve ohm here, and this is just a nice
little wire.
So we have 2/12, plus 1/12.
Invert.
Sets.
3/12.
So the equivalent resistance of those two
resistors is four OHMS.
And then I have one OHM here, which means
overall
we have one amp of current that's flowing
like this.
When it gets here.
We get a voltage divider where 2/3 of it
are going to go through this arm,
and 1/3 is going to go across this arm.
If you got lost in that calculation, go
back and
review the current divider, because that's
essentially what we're doing here.
So a combination of a kind of a voltage-
divider thing and the
current divider, and so go back and review
that if you got lost.
To derive the energy that's stored in an
inductor we're
going to use the same method we did for a
capacitor.
P equals I times V.
W is equal to this integration of power
and now
we are going to use V equals L di dt.
So before, where we replaced the i with C
dv/dt,
now we're going to replace the v with L
di/dt.
So now we're doing an integration in
current.
And doing that change of variables
allows us to get this calculation.
And finally leading us to that energy
is equal to 1/2 times the inductance, L,
times i squared.
and if you remember what it was before
capacitance, it was one half CV squared.
So they have very similar forms.
And that's one of the nice things, is
that they've got similarity, makes it
easier to remember.
To graph these out, we're going to
basically
follow the same process we did before.
Now our inductance is two henries Alright.
We were using one, Farad capacitance that
our numbers came out nice.
Just make sure we are paying attention to
this value and how importance it is.
To find our voltage, V equals L di dt.
So we're scaling the slope.
So the slope here is one.
We're scaling it by two.
Oh, sorry this is one second.
So the slope here is two.
Scaling it by two gives us four volts
here.
This flattens out so this does as well.
This drops down at twice the rate, this
does drop down here, that negative eight.
And then it comes back up at the same rate
as this.
So we're back up to four.
So this is what the current curve, or the
voltage curve would look like.
Just like with capacitors you couldn't
instantaneously change the voltage.
You cannot instantaneously change the
current for an inductor.
In fact, if you pull something out of the
wall that's inductive, you see a little
spark jump.
Because of that behavior.
Because the current has to go, keep
going.
It can't instantaneously change.
So if you ever see that happening, that's
what's causing it.
To find our power, we just multiply the
two functions together.
So this is going up, and this is a nice
flat, so it comes up like this.
Drops down here to zero, because of that.
And drops down here and goes up like this,
as this is going down like that.
And that's a negative value.
That basically inverts this line in this
manner and
this is going to cause it the power to
drop down
in this way.
And so we get this kind of this jumpy
spiky power curve.
That's fine though, power can change
instantaneously, no problem.
And again because these are lines, just
like with our inductor or capacitor
example, these lines are going to equate
to these, nice little parabolic curves,
here, here, and here.
And to calculate the area under this
curve, we find it to be four.
That's why we have four joules here.
And we can always check our work.
Just like we did with the capacitors.
L i squared scaled by 1/2.
So up here we're at two is our current so
1/2 times two
times two squared gives us four.
And so our numbers all match up, it all
works out, we probably did it correctly.
Now,
I know we went through that quickly, if
you
have questions, post to the forums, and
also, we'll
give you some more examples to practice
this idea
so you can have some experience doing it
yourself.
To summarize what we covered today,
we calculated inductance for inductors in
parallel
and series configurations, and then we
identified how inductors behave in DC
circuits.
Like wires.
We derived an equation for the energy
that's stored in an inductor's magnetic
field.
And then graphically showed these
relationships.
In the next class we're going to start
talking
about what happens if things are changing
in our system.
And, in order to be able to do
that, we need to know something about
differential equations.
So we'll present a little bit of a basic.
intro to first-order differential
equations, and then
we'll start applying them to some basic
circuits, to see how these things start
behaving when you're changing values
inside the circuits.
Until next time.