[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.