- [Voiceover] Now I want to show you
a really useful manual skill you could use
when you have voltages that look like exponentials.
And we're gonna talk about this exponential
curve here that's generated as part of the
natural response of this RC circuit.
And we worked out that the voltage across here,
this voltage V of T on the capacitor,
its natural response is equal to V zero, V knot
times e to the minus t over RC.
This value of V knot is the starting voltage
that our input source provides and then it
immediately steps down to zero and this circuit
then does its natural response.
We got a current coming out of this capacitor
flowing around in a circle like this.
And that's the natural response of this RC circuit.
I wanna look at the properties of this function
right here, it's got some interesting properties.
I'll write it right here.
V knot, e to the minus t over RC.
So first thing we can look at is this V knot value.
That's this value here.
This is V knot.
In this particular chart, V knot was equal to one volt.
That's what it originally charged up to.
And now we wanna look at this point right here.
Right when it goes through time equals zero
and the curve starts to drop.
We know how high it is, it's V knot volts high.
What I wanna look at now is what's the slope
right at this point?
The slope of a curve is the derivative of a curve
evaluated at this time, at time equals zero.
Let's take a derivative of our function right here
so its d, dt,
V knot, e to the minus t over RC.
And that equals V knot
and the RC comes down, minus one over RC comes down.
Minus one over RC, e to the minus t over RC.
And that's the derivative of this exponential
for all time.
So now we evaluate at
t equals zero
and we get V knot over RC
and there's a minus sign.
And it's e to the, I plug in zero for t,
it's e to the zero and e to the zero is one.
So that equals the slope.
At time equals zero, that equals the slope
at time equals zero.
So that's the value right here.
That's what that slope looks like.
That tangent line to the curve.
Okay, now the next thing I wanna do
is actually take this line and extend it all the way down
till it crosses this axis, till it crosses the time axis.
And the next question we're gonna ask
is what is this point right here?
What's that point?
Let me move up a little bit.
So now we have a line, we've defined a line.
That means we have an equation of a line.
So the equation of a line is y equals slope times x
plus b, b is the intercept on the voltage axis,
m is the slope.
I can plug in my values here for our chart,
y is the voltage axis.
We know what the slope is, it's sitting right here.
It's minus V knot over RC and we multiply by time
and we add the y intercept or the voltage intercept
which we know is V knot, plus V knot.
Alright and now what I wanna do is find out
at what time, what time does voltage equal zero
for this orange line?
So we're gonna plug in zero for volts
and work out what time is.
Zero equals V knot times
one over RC and don't lose the minus sign
times t plus one.
If I divide both sides by V knot,
I get zero on this side and this term on this side.
So I can say zero equals minus one over RC
times t plus one and we wanna isolate t
so we'll take the one on the other side
and multiply by RC so one more step.
Minus one equals minus one over RC times t.
And in the end, we come up with t equals RC.
T equals RC is that time right there.
That's how many seconds after the step that this line
hits the time axis.
And notice something here, there's no mention
of V knot in here.
There's no V knot.
There's no V knot, it's not here.
And it divided out in this step back here.
It's one of our steps, it disappeared going from
this step to this step here.
So independent of how high this thing starts,
it can be high or it can be low,
the slope of this line always goes right through
the time equals RC.
And one more thing we wanna work out is
at time equals RC,
what is the value
of the exponential?
What is this voltage here when time equals RC?
So we can use our equation again.
We can plug into our equation and find out how to do that.
If I go back to
V equals V knot, e to the minus t over RC,
and this time we're gonna plug in RC right here
for this value here and find out what V is.
So V equals V knot, e to the minus RC over RC
or equals V knot, e to the minus one.
Now the value of e
is roughly equal to 2.7
and the value of one over e
is roughly equal to 0.37 or
another way I can say that is 37 percent.
So in the end, this voltage right here
is about 37 percent of V knot.
That's that value right there.
So that's two little things we're gonna
tuck away in our head.
The time for that line to hit the time axis is RC.
If I wanna know the value of where the exponential
actually is at time equals RC,
it's roughly 37 percent of where it started from.
Okay that's the basic idea and in the next
video, I'll show you how to use these ideas to
sketch exponentials really quickly.