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[BLANK_AUDIO].

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So now that we've gotten our feet wet a
little bit with talking about magnetism

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and inductors, let's see how inductors are

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actually going to be behave in real
systems.

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So the aims for today, are to present how

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they're going to work together in these
system settings.

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And just like we did with capacitors,
we're

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going to look at parallel and series
inductors.

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And do some derivations to see what the
equivalent inductance would be.

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And then graphically represent the
relationships, just

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like we did with capacitors, for current,

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voltage, power, and energy.

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So, again, from our previous class, we
started talking about the

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way that currents and voltages relate
within inductors, within single devices.

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and we talked about we meant by
inductance, now we're going

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to go into a little bit more depth on
these things.

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So first we'll analyze the inductors in
series and

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parallel, then analyze DC circuits with
inductors, calculate the energy

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in inductors, doing a duration much like
we did

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with capacitors And then doing the
sketching of the curves.

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Here we place our inductors in series, and
we want

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to do a derivation to find what the
equivalent inductance is.

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But remembering from before, i equals.

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Well, sorry, v equals l di dt.

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We'll do the derivative version instead,
because

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that's what we're going to be using here.

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We know that the current going through
these two

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devices, because they're in series, has to
be equal.

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And we have two voltages here, v1 and v2.

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So to find those voltages, we set up our
equation by saying

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that the total voltage is equal to the sum
of the individual voltages.

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then again, we factor out our two
inductance's

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to find out our equivalent inductance is
right here.

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Sum of the two inductors.

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Remembering from what we learned about
resistors, we place resistors in series.

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It's the same thin as adding the first
instances Okay, so doing inductors,

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we do the exact same thing we did with
resistors for combining them.

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Even though the derivations are different

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in these equations are differential
equations, derived

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equations it still gives us the same
result, we just add them all together.

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Now lets look at how they behave if we
place them in parallel.

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Kirchhoff's Current Law states that this
current has to equal these two currents.

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But we know because they're parallel,
voltages are the same.

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So i equals i1 plus i2.
We're now using the one over L integration

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of v, dt, defined what our currents happen
to be.

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And again combining terms pulling

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things out we discover that our equivalent
inductance is 1

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over L1 plus 1 over L2 and then we invert
that.

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This is the exact same thing you do for
finding

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the the equivalent resistance of resistors
that are placed in parallel.

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And remember the capacitance was the other
way around.

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Capacitors in this configuration you added
them together, and

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if they were in series, you did this
calculation.

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But the nice thing is that they all have
the same basic calculations.

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It's just that when you apply them you
need to be careful, make sure that

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you remember capacitors are the ones that
are

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different, but resisters and conductors
behave very similarly.

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If I put them in a DC circuit.

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Like a capacitor, over a long period of

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time would get to an equalibrium in
voltage.

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An inductor is going to become-, get to an
equalibrium in current.

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And so, we're going to replace this
inductor, with a wire.

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And then do our analysis.

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kind of like where, with a capacitor we
replace it.

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With an open circuit.

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So if I wanted to know, for example, the
current coming through here.

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What I first need to know is what the
current is coming through here.

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So to do that, I have a six ohm here, and
a twelve ohm here, and this is just a nice

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little wire.
So we have 2/12, plus 1/12.

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Invert.
Sets.

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3/12.

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So the equivalent resistance of those two
resistors is four OHMS.

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And then I have one OHM here, which means
overall

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we have one amp of current that's flowing
like this.

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When it gets here.

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We get a voltage divider where 2/3 of it
are going to go through this arm,

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and 1/3 is going to go across this arm.

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If you got lost in that calculation, go
back and

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review the current divider, because that's
essentially what we're doing here.

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So a combination of a kind of a voltage-
divider thing and the

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current divider, and so go back and review
that if you got lost.

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To derive the energy that's stored in an
inductor we're

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going to use the same method we did for a
capacitor.

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P equals I times V.

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W is equal to this integration of power
and now

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we are going to use V equals L di dt.

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So before, where we replaced the i with C
dv/dt,

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now we're going to replace the v with L
di/dt.

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So now we're doing an integration in
current.

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And doing that change of variables

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allows us to get this calculation.
And finally leading us to that energy

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is equal to 1/2 times the inductance, L,
times i squared.

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and if you remember what it was before
capacitance, it was one half CV squared.

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So they have very similar forms.

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And that's one of the nice things, is

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that they've got similarity, makes it
easier to remember.

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To graph these out, we're going to
basically

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follow the same process we did before.
Now our inductance is two henries Alright.

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We were using one, Farad capacitance that
our numbers came out nice.

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Just make sure we are paying attention to
this value and how importance it is.

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To find our voltage, V equals L di dt.
So we're scaling the slope.

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So the slope here is one.

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We're scaling it by two.
Oh, sorry this is one second.

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So the slope here is two.

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Scaling it by two gives us four volts
here.

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This flattens out so this does as well.

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This drops down at twice the rate, this
does drop down here, that negative eight.

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And then it comes back up at the same rate
as this.

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So we're back up to four.

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So this is what the current curve, or the
voltage curve would look like.

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Just like with capacitors you couldn't
instantaneously change the voltage.

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You cannot instantaneously change the
current for an inductor.

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In fact, if you pull something out of the

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wall that's inductive, you see a little
spark jump.

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Because of that behavior.
Because the current has to go, keep

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going.
It can't instantaneously change.

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So if you ever see that happening, that's
what's causing it.

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To find our power, we just multiply the
two functions together.

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So this is going up, and this is a nice
flat, so it comes up like this.

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Drops down here to zero, because of that.

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And drops down here and goes up like this,
as this is going down like that.

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And that's a negative value.

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That basically inverts this line in this
manner and

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this is going to cause it the power to
drop down

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in this way.

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And so we get this kind of this jumpy
spiky power curve.

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That's fine though, power can change
instantaneously, no problem.

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And again because these are lines, just
like with our inductor or capacitor

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example, these lines are going to equate
to these, nice little parabolic curves,

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here, here, and here.
And to calculate the area under this

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curve, we find it to be four.
That's why we have four joules here.

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And we can always check our work.
Just like we did with the capacitors.

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L i squared scaled by 1/2.

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So up here we're at two is our current so
1/2 times two

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times two squared gives us four.

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And so our numbers all match up, it all
works out, we probably did it correctly.

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Now,

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I know we went through that quickly, if
you

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have questions, post to the forums, and
also, we'll

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give you some more examples to practice
this idea

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so you can have some experience doing it
yourself.

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To summarize what we covered today,

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we calculated inductance for inductors in
parallel

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and series configurations, and then we

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identified how inductors behave in DC
circuits.

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Like wires.

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We derived an equation for the energy

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that's stored in an inductor's magnetic
field.

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And then graphically showed these
relationships.

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In the next class we're going to start
talking

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about what happens if things are changing
in our system.

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And, in order to be able to do

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that, we need to know something about
differential equations.

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So we'll present a little bit of a basic.

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intro to first-order differential
equations, and then

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we'll start applying them to some basic

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circuits, to see how these things start

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behaving when you're changing values
inside the circuits.

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Until next time.