- [Voiceover] Now we're gonna talk about the two forms
of the Inductor Equation and get
familiar with these things.
I'm gonna do some examples to show you how
the Inductor Equations work.
So, we know that the Inductor Equation is the
voltage across an inductor is
a factor called L, the inductance, times di, dt.
So the voltage is proportional
to the slope or the rate of change of current.
Let me do a quick review of the two letters
that are used as variables for inductors.
When we have a resistor,
we have, we have a,
we call it a R,
and it's in units of ohms,
and that's named after Ohm, himself.
And when we have a capacitor,
the units are c, for capacitance,
and f for faraday, farad,
that's named after Michael Faraday.
For the inductor, we have two
other letters we use, L and H,
and L is the name of the component,
and that's named after Lenz, Heinrich Lenz,
and he did some research in induction,
and the unit is named after Henry,
and so that's the corresponding R, c, and L
are the names of the components,
and ohms, farads, and henries are the units.
Okay, let's get back to our inductor.
There's another form of the inductor equation
where we write it in terms of just i,
in terms of v,
and to get to that point,
we need to get rid of this derivative here.
So, what we, what I'm gonna do
is take the integral of both sides of this equation.
So, we take the integral of v,
with respect to time,
and that equals L times
take the integral of di, dt
with respect to time,
and these two terms go away,
and have the integral of di,
the antiderivative of di is, is
what, what function has a derivative of di?
And that would be just i.
So, let me flip this all around here now.
I'm gonna write i, on the left side
equals one over L
times the integral of v, dt.
Now we always put limits on this integral,
so the limits in this case are
time is goes from minus infinity
till some time now, and this tells us that the, ah,
that the current through an inductor
depends on the voltage across that inductor
for the entire life
all the way back to time equals minus infinity, and
that's not so convenient.
We don't want to track this,
the inductor's voltage back that far in time.
So what we'll do instead is we'll,
we'll say, let's say at t equals zero,
that we know the vol, that we know the current,
that we know some,
we know some i of zero.
And then we can change the limits on our,
our integral here to be zero on the bottom.
One over L, integral from time is zero until now
time t, big T,
of v of T times dt.
In order to account for everything that's happened
before time equals zero,
we'll just say, what was the current
at time equals zero?
So that's i of zero.
And I'll do one more little change.
I, I, I want this variable up here to be
small t, and that means I have to replace this t
inside here with some other symbol as a,
as a dummy variable, and so now I'm gonna get
i of t, little t
one over L, integral from zero
to little t of vd,
let's just call it Tao,
plus i of zero.
So this is the integral form.
There's the integral form of the inductor equation
and here is the derivative form.
And we'll use both these.
What I'm gonna do now,
is we're gonna do an exercise
where we use this one down here.
So, in our circuit we have a voltage source
with a time pattern voltage in it,
and we have a current going through this thing.
And we wanna find out, what's this current?
So, we're connected to a inductor.
It's a 10 millihenry inductor
and the voltage waveform is,
it starts at zero,
at time equals zero,
the, the voltage goes up to two volts,
then after two milliseconds it goes to minus two volts.
After two more milliseconds,
it goes to plus two volts and on and on.
So, we know v of t,
and what we're going to figure out is, what's,
what's i through the, in the inductor?
And to do that,
we're gonna use our, our
inductor equation here, the integral
form of the inductor equation.
So I actually need some room on the screen here, so
let me take a second and clean out some stuff.
Okay, we're ready to go.
So let's look at what's going on here
before time equals zero.
So, in the before state,
we're gonna make an assumption.
We're gonna assume that
i is zero a long time ago,
and if we look at our chart, here,
it says that v is zero, as well.
And if I take this fact
and I look at this equation right here,
if v is zero,
L is some positive number,
so we can say that,
so then we know that
di, dt equals zero.
So this circuit is pretty much doing nothing
before this waveform starts working.
Over here I'm gonna make a little time chart
of what, what we discover.
That's time, and this'll be, ah, current, here.
So before zero, before time is zero,
we can say that the current is zero.
Now, we've plotted in part of our, our curve.
Okay, now something happens.
Now the voltage goes up two volts,
and we're gonna use this equation,
the integral form of the,
the inductor equation to figure out what happens
to i of t now that voltage has changed.
So, we can say that i
equals one over L
times the integral from zero,
times zero is our starting point here,
to t, some time t of v of t.
Now, v, in this little region of time here,
is two volts plus two
times d Tao plus i zero,
and we decided i zero was the current
that it started at right there;
i zero is zero.
Let's keep working on this.
We can say that i
equals one over 10 millihenries
times the two comes out of the integral,
and now it's the integral
from zero to t of d Tao.
And the zero goes away,
that equals two over 10 millihenries times what?
What's this integral evaluate to?
It evaluates to t.
Integral of d Tao between zero and t
is t minus zero or just plain t.
So looking at this, I see that
i equals a constant times t,
and that's the equation of a line.
That's the equation of a line
if I plot it on t and i,
so two over 10 millihenries is the slope,
and to plot that line,
what I need is two points on the line.
I know one of the points already.
I know one of the points is gonna be
right there at zero zero.
What is the current at the end of this pulse here?
So, at two milliseconds,
let's figure out what the current is,
and that equals, at two milliseconds it's
two times two milliseconds
divided by 10 millihenries.
The millis cancel themselves out
and I get four over 10, or
zero point four, what, Amperes.
So, at two milliseconds,
the value of the current is
point four amperes, and I can draw a straight between there.
So now we have the value of the current
during this first excursion,
this first part of the voltage waveform.
Now, we're gonna switch again.
Now we have a new i zero.
Let's, let's go to a new point, here.
Let's go to a new point, at, now we have i,
i zero equals point four,
and we have v equals what?
Now it goes to minus two volts,
minus two volts, so let's use our inductor
equation one more time
with these, with these, ah, initial conditions
that tells us i equals one over L
integral from, what's the new point?
Two milliseconds to t
and v of t after two milliseconds is
minus two volts, minus two d Tao
plus, now we have a starting current.
Our starting current was point four amps.
Zero point four amps.
And let's continue on here.
Let's, ah, maybe can squeeze it in here;
i equals one over 10 millihenries
times minus two times the integral,
the integral from two milliseconds to t of d Tao
plus zero point four;
i equals minus two over 10 millis
times, what is this?
It's again, it's t minus,
t minus two milliseconds plus point four,
and again, we have the equation of a line.
This is the equation of a line, i,
there's a constant times t,
there's a con, a,
followed by a constant.
And this is basically a sloping line.
It has a negative slope,
and the slope is negative two over 10 millis,
which is just the opposite of what we had over here.
We had two over 10 millihenries,
so it's the same looking kind of line.
So let me mark out here.
Here's four milliseconds and it's gonna
be a line that looks like this.
So, the current's gonna ramp up,
it's gonna turn around and ramp straight back down
because the, the slope up
and the time up is the same as
the slope down and the time down,
just with a negative slope.
So there's an example of the
inductor equation in action.