- [Voiceover] I want to talk about
a new example of an inductor circuit,
and we have one shown here,
where this inductor is now controlled by a switch.
This is a push button switch that we move in and out,
and this metal plate here will touch these two contacts
and complete this circuit here.
This is a lot like the circuit we looked at before
without the switch, where the voltage source
had a pulse going up and down.
This circuit has a real interesting side effect
when this switch opens up.
So, let me give you a couple of examples first,
some practical examples of where
this kind of an inductor will appear,
connected to a voltage source and controlled by a switch.
And there's a couple of examples.
One of them is called a solenoid.
And a solenoid is a coil of wire
that looks like this,
and has a metal
rod going through it.
When you put a pulse of current into this,
what happens is, the rod moves.
The rod will move back and forth,
so this is a way, for instance, a doorbell.
A doorbell has this kind of a actuator.
Another kind of device that has an inductor in it,
or a coil, is called a relay.
And that one, again, has a coil like this.
And,
these are all forms of an electromagnet,
and this one actually has a switch,
some sort of a piece of metal
that will move back and forth.
When the coil is energized,
this piece of metal will actually tip over here,
and when the current's stopped,
it'll move back to its original state.
So, for example, if there is a connection point here,
and a connection point here,
this piece of metal will go
from making a short across there,
and move away,
and open and close this bigger switch.
So, a relay is some sort of a switch.
And you might find this in a car,
where this relay is controlling, like,
the windshield wipers of a car,
where the current for the windshield wiper
goes through this switch,
and there's a smaller current over here
that pulls this switch open and closed
to start and stop the windshield wipers or the motor.
And that's just two examples of where inductors,
or windings of wire can show up in real applications,
and there's some fairly large currents
that flow through these windings.
So, we want to look what happens
when we switch this current on and off,
and this is the example circuit that we'll use.
So, won't assume we have a inductor of 10 milliHenries,
and what we're gonna do is
we're gonna push this push button switch down
and connect it up, and then we're gonna let it go.
And we're gonna look at a couple of different voltages here.
This is gonna be, we'll call this voltage V,
and that's gonna be measured with respect to ground.
And then, there's another voltage here
that's interesting too, which is VL,
and that's the voltage across the inductor.
So V plus VL is equal to three volts, all the time.
And now, we look at what happens
when we push the button down,
and then, after that, we'll look at what happens
when we let go of the button.
Something interesting's gonna happen
when we let go of this button.
So, as we look at our circuit here,
we see there's an open circuit,
so there's no current flowing in here.
That means I equals zero.
We'll call that I zero because it's the initial current
when we close our switch.
So now, let's close our switch.
We'll do that like this.
And we just closed our switch at time, T equals zero.
And let's look at what happens.
All of a sudden now,
our inductor has a voltage across it,
and it's a voltage of,
this node is at three volts,
this node is at zero volts,
so all of a sudden, we have three volts across our inductor.
So, let's use the integral form of the inductor equation
to solve for the current that's gonna happen here.
I equals one over L
times the integral from zero to T
of
V d Tau,
plus I zero.
Let's fill in what we know.
I equals one over 10 milliHenries,
and V is a constant, V is a constant three volts,
times three,
and it's the integral of d Tau,
from zero to T,
and I zero is zero.
And then, we get the final form,
which is I equals three over 10 milliHenries,
times the integral from zero to T of d Tau,
is just T,
and that's the answer,
so what we have here, just like we did before,
when we had the switching power supply,
just like we had before, we're gonna get a current
that has a ramp that looks like that.
The slope of that is three
over 10 milliHenries,
which equals
300 amps per second,
so, oh my goodness.
This current is going up really fast.
That's the slope of that current right there.
This is going up really fast, and that's what it does.
Now, in a real circuit,
there'll be real resistances in here,
but it uh, and so,
there'll be a limit to the amount of current
that will be determined by the resistor,
but for the purposes of showing you
just how this inductor equation works,
that's the kind of slope you would see
at the initial closing of the switch.
Okay, I'm gonna clean off the screen here,
so I can keep my same circuit.
And now, we're gonna look at what happens here
when we open this switch.
Let me open the switch.
And now, we've opened our switch back up.
It went that way.
All right, so,
we have an initial current,
and it's gonna be some value,
depending on how long we held the switch down,
so it's gonna be some current,
and that's flowing in the inductor.
Now, let's look at what happens
when we open the switch, and all of a sudden,
I goes directly and sharply to zero.
That's what the switch does.
When we open this switch contact,
one moment it's touching,
and the other moment, it's not touching.
So, if we look at, what's delta I, or what's dI.
It's the ending current
minus the starting current, minus I.
And if we look at what's the time involved.
What's the change in time involved in opening the switch?
Well, it's something like zero.
The switch was closed, then it was open,
and that's how much, that took,
let's say that took zero time to happen.
Now, here's something that happens with inductors
that is kind of strange.
Let's calculate the voltage on the inductor
right when this happens.
And we know that V equals
L times dI
dT.
And plug in some numbers here,
so we have V equals L
times what, dI,
is minus I,
over what?
Over zero.
And that equals what?
That equals negative infinity.
What the heck is going on here?
Is that possible?
No, it's not.
Well lets, okay, let's take a second.
Let's say this switch didn't open in zero time.
Let's say this switch,
let's say dT was,
let's say it was one nanosecond.
Let's give it some time to open up.
All right.
Maybe that'll save us.
Maybe that'll make more sense.
Okay, let's do that.
Let's go
V equals L
times dI dT, and dI,
we decided was minus I
over, it took one nanosecond to open the switch.
One nanosecond is one times 10 to the minus nine.
Okay, so what is that equal to?
That equals L,
negative L times I
times 10 to the holy cow.
10 to the plus nine.
Okay, let's plug in some real values
and see what we get here.
Let's say V equals
let's say L was 10 milliHenries,
and let's say I was 10 milliamps
flowing down through the,
so delta I is minus 10 milliamps,
and
the time involved is
10 to the minus nine seconds.
What does that work out to?
10 milliHenries, that's minus three
and minus three,
so it's gonna be 100,
minus three minus three is minus six over minus nine
times 10 to the third.
Okay, that is...
And there's a minus sign here.
This says that V,
this point right here,
or the voltage across the inductor.
Our calculation just said it's gonna be 100,
minus 100,000 volts.
Now, minus 100,000 volts means
that the negative terminal
is 100,000 volts above the positive terminal,
so this voltage V is actually at 100,003 volts.
How can this be?
This is a puzzle about inductors
that we actually have to solve by
actually looking right up close
at what's going on here, right in this switch area here.
So, this is where reality comes in and saves us
from our, our,
the crazy results we're getting from our
ideal equations here.
So, let's pretend, here's the uh,
here's the terminal of the switch, right here.
Let's do a blow up, and here's,
here's that switch plate.
As soon as this switch moves away,
there's an air gap created here.
In our ideal world, this air gap was a perfect insulator,
but what's gonna happen,
because of these extreme numbers here,
because of this is happening,
this air gap is actually not an ideal conductor,
and what's gonna happen
is we're gonna get a bright spark
goes right across here.
This really happens in real life.
There is actually, when you open a switch,
there can be a little spark that goes between,
and that gives this equation here,
this dI dT,
enough time to release that energy
and that current from the inductor
continues to flow and go over into this switch.
That current will flow, and it does it
by breaking down the air molecules,
and when you do that,
that voltage there, for air.
If this gap here,
if this gap is one millimeter,
for air, that's 3,000 volts.
3,000 volts will cause that spark to happen there.
So, this is what actually happens,
and you can build switches
that will take this spark and work for many years,
but it's not always a good idea to let this happen.