Second derivatives (vector-valued functions)

WEBVTT
Kind: captions
Language: en

00:00:00.184 --> 00:00:02.802
- [Voiceover] So I have a
vector valued function h here,

00:00:02.802 --> 00:00:03.861
and when I say vector value,

00:00:03.861 --> 00:00:06.162
it means you give me a
t, it's a function of t,

00:00:06.162 --> 00:00:07.258
and so you give me a t,

00:00:07.258 --> 00:00:09.215
I'm not just going to give you a number,

00:00:09.215 --> 00:00:10.668
I'm going to give you a vector.

00:00:10.668 --> 00:00:11.501
And as we'll see,

00:00:11.501 --> 00:00:13.270
you're going to get a
two dimensional vector.

00:00:13.270 --> 00:00:15.882
You could view this as the
x component of the vector

00:00:15.882 --> 00:00:18.513
and the y component of the vector.

00:00:18.513 --> 00:00:20.902
And you are probably familiar by now,

00:00:20.902 --> 00:00:22.303
that there is multiple notations for

00:00:22.303 --> 00:00:24.179
even a two dimensional vector.

00:00:24.179 --> 00:00:26.938
For example, you could
use what's often viewed

00:00:26.938 --> 00:00:29.106
as engineering notation here,

00:00:29.106 --> 00:00:32.032
where the x component is being multiplied

00:00:32.032 --> 00:00:34.532
by the horizontal unit vector.

00:00:36.355 --> 00:00:37.868
So you might see something like that,

00:00:37.868 --> 00:00:42.831
where that's the unit
vector, plus the y component.

00:00:42.831 --> 00:00:46.483
Four t to the fourth,
plus two t, plus one,

00:00:46.483 --> 00:00:50.303
is multiplied by the vertical unit vector.

00:00:50.303 --> 00:00:52.889
So these are both
representing the same thing,

00:00:52.889 --> 00:00:55.582
it just has a different notation.

00:00:55.582 --> 00:00:57.453
And sometimes you'll see
vector valued functions

00:00:57.453 --> 00:00:59.226
with an arrow on top to make it explicit

00:00:59.226 --> 00:01:00.912
that this is a vector valued function.

00:01:00.912 --> 00:01:02.884
Sometimes you'll just hear people say,

00:01:02.884 --> 00:01:05.227
well, let h be a vector valued function,

00:01:05.227 --> 00:01:07.565
and they might not
write that arrow on top.

00:01:07.565 --> 00:01:08.588
So now that we have that out of the way,

00:01:08.588 --> 00:01:09.812
what we're interested in is,

00:01:09.812 --> 00:01:12.774
well let's find the first
and second derivatives

00:01:12.774 --> 00:01:14.691
of h with respect to t.

00:01:15.837 --> 00:01:18.968
So, let's first take, let's
take the first derivative,

00:01:18.968 --> 00:01:22.313
h prime of t, was as you'll see,

00:01:22.313 --> 00:01:24.130
it's actually quite straight forward.

00:01:24.130 --> 00:01:26.377
You're just going to take
the respective components,

00:01:26.377 --> 00:01:28.205
with respect, take the
derivative of the respective

00:01:28.205 --> 00:01:30.481
components with respect to t.

00:01:30.481 --> 00:01:32.727
So the x component, with respect to t,

00:01:32.727 --> 00:01:35.695
if you were to take the derivative,

00:01:35.695 --> 00:01:39.014
with respect to t, what
are you going to get?

00:01:39.014 --> 00:01:41.513
Well we're going to use the
power rule right over here,

00:01:41.513 --> 00:01:44.005
five times the negative
one or times the negative.

00:01:44.005 --> 00:01:46.966
You're going to get negative
five times t to the five,

00:01:46.966 --> 00:01:49.700
minus one power, so t to the fourth power.

00:01:49.700 --> 00:01:51.633
The derivative with respect
to t of negative six,

00:01:51.633 --> 00:01:53.073
well that's just zero,

00:01:53.073 --> 00:01:56.844
so that's the rate of
change of the x component,

00:01:56.844 --> 00:01:58.850
with respect to t.

00:01:58.850 --> 00:02:01.126
And now we go to the y component.

00:02:01.126 --> 00:02:02.743
So we're going to do the same thing,

00:02:02.743 --> 00:02:05.266
the derivative with respect to t,

00:02:05.266 --> 00:02:06.717
is going to be, and once again,

00:02:06.717 --> 00:02:08.263
we just use the power rule,

00:02:08.263 --> 00:02:11.763
four times four is 16t to the third power.

00:02:13.151 --> 00:02:15.811
Derivative of two t is just two,

00:02:15.811 --> 00:02:16.985
and then derivative of a constant,

00:02:16.985 --> 00:02:19.275
well that's zero, we've already seen that.

00:02:19.275 --> 00:02:21.519
So there you have it.

00:02:21.519 --> 00:02:23.380
So this is the rate of
change of the x component

00:02:23.380 --> 00:02:24.571
with respect to t,

00:02:24.571 --> 00:02:26.139
this is the rate of
change of the y component,

00:02:26.139 --> 00:02:28.012
with respect to t.

00:02:28.012 --> 00:02:29.555
And one way to do it,

00:02:29.555 --> 00:02:32.047
vectors can represent many,
many, many different things,

00:02:32.047 --> 00:02:35.137
but the type of, a two
dimensional vector like this,

00:02:35.137 --> 00:02:37.094
you can imagine this being,

00:02:37.094 --> 00:02:40.881
h of t being a position
vector in two dimensions.

00:02:40.881 --> 00:02:42.298
And then if you're looking
at the rate of change

00:02:42.298 --> 00:02:44.217
of position with respect to time,

00:02:44.217 --> 00:02:46.890
well then this would
be the velocity vector.

00:02:46.890 --> 00:02:48.303
And then if we were to take the derivative

00:02:48.303 --> 00:02:49.816
of this with respect to time,

00:02:49.816 --> 00:02:52.237
well we're going to get
the acceleration vector.

00:02:52.237 --> 00:02:54.987
So if we say h prime, prime of t,

00:02:55.983 --> 00:03:00.883
what is that going to be
equal to, h prime, prime of t?

00:03:00.883 --> 00:03:03.338
Well, we just apply the power rule again.

00:03:03.338 --> 00:03:07.505
So four times negative five
is equal to negative 20t

00:03:09.672 --> 00:03:12.865
to the four minus one,
so t to third power.

00:03:12.865 --> 00:03:16.782
And then we have three
times 16 is 48t squared,

00:03:19.926 --> 00:03:22.789
and then the derivative
of two is just zero.

00:03:22.789 --> 00:03:24.051
And so there you have it,

00:03:24.051 --> 00:03:26.109
for any, if you view t as time,

00:03:26.109 --> 00:03:28.558
for any time, if you view
this one as position,

00:03:28.558 --> 00:03:30.513
this one as velocity and
this is acceleration,

00:03:30.513 --> 00:03:31.992
you could, this would now give you the

00:03:31.992 --> 00:03:34.191
position, velocity and acceleration.

00:03:34.191 --> 00:03:35.956
It's important to realize
that these vectors

00:03:35.956 --> 00:03:40.123
could represent anything of
a two dimensional nature.


Second derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.434 --> 00:00:02.283
- [Voiceover] Let's say that y

00:00:02.283 --> 00:00:04.866
is equal to six over x-squared.

00:00:06.149 --> 00:00:09.171
What I wanna do in this
video is figure out

00:00:09.171 --> 00:00:13.338
what is the second derivative
of y with respect to x.

00:00:16.976 --> 00:00:19.559
And if you're wondering where
this notation comes from

00:00:19.559 --> 00:00:22.587
for a second derivative,
imagine if you started

00:00:22.587 --> 00:00:26.264
with your y, and you
first take a derivative,

00:00:26.264 --> 00:00:28.593
and we've seen this notation before.

00:00:28.593 --> 00:00:30.733
So that would be the first derivative.

00:00:30.733 --> 00:00:33.261
Then we wanna take the derivative of that.

00:00:33.261 --> 00:00:35.883
So we then wanna take
the derivative of that

00:00:35.883 --> 00:00:39.028
to get us our second derivative.

00:00:39.028 --> 00:00:41.695
And so that's where that
notation comes from.

00:00:41.695 --> 00:00:44.879
It likes you have a d-squared, d times d,

00:00:44.879 --> 00:00:46.427
although you're not
really multiplying them.

00:00:46.427 --> 00:00:48.394
You're applying the
derivative operator twice.

00:00:48.394 --> 00:00:50.576
It looks like you have a dx-squared.

00:00:50.576 --> 00:00:51.949
Once again, you're not multiplying 'em,

00:00:51.949 --> 00:00:53.859
you're just applying the operator twice.

00:00:53.859 --> 00:00:56.562
But that's where that
notation actually comes from.

00:00:56.562 --> 00:00:58.841
Well, let's first take
the first derivative

00:00:58.841 --> 00:01:00.473
of y with respect to x.

00:01:00.473 --> 00:01:02.217
And to do that, let's
just remind ourselves

00:01:02.217 --> 00:01:05.338
that we just have to
apply the power rule here,

00:01:05.338 --> 00:01:07.998
and we can just remind
ourselves, based on the fact

00:01:07.998 --> 00:01:12.081
that y is equal to six
x to the negative two.

00:01:13.027 --> 00:01:15.419
So let's take the derivative
of both sides of this

00:01:15.419 --> 00:01:19.586
with respect to x, so with
respect to x, gonna do that,

00:01:23.119 --> 00:01:25.097
and so on the left-hand
side, I'm gonna have

00:01:25.097 --> 00:01:28.930
dy dx is equal to, now
on the right-hand side,

00:01:30.442 --> 00:01:33.670
take our negative two,
multiply it times the six,

00:01:33.670 --> 00:01:37.328
it's gonna get negative 12 x
to the negative two minus one

00:01:37.328 --> 00:01:39.408
is x to the negative three.

00:01:39.408 --> 00:01:41.564
Actually, let me give myself
a little bit more space here.

00:01:41.564 --> 00:01:46.206
So this negative 12 x
to the negative three.

00:01:46.206 --> 00:01:47.938
And now, let's take the derivative of that

00:01:47.938 --> 00:01:49.438
with respect to x.

00:01:50.459 --> 00:01:54.542
So I'm gonna apply the
derivative operator again,

00:01:55.542 --> 00:01:58.715
so the derivative with respect to x.

00:01:58.715 --> 00:02:03.142
Now the left-hand side gets
the second derivative of y

00:02:03.142 --> 00:02:07.402
with respect to to x,
is going to be equal to,

00:02:07.402 --> 00:02:09.447
well, we just use the power rule again,

00:02:09.447 --> 00:02:13.447
negative three times
negative 12 is positive 36,

00:02:14.387 --> 00:02:17.324
times x to the, well,
negative three minus one

00:02:17.324 --> 00:02:19.147
is negative four power,

00:02:19.147 --> 00:02:23.314
which we could also write as
36 over x to the fourth power.

00:02:24.537 --> 00:02:25.787
And we're done.


Inverting and unity-gain op-amp with virtual ground

WEBVTT
Kind: captions
Language: en

00:00:00.674 --> 00:00:02.612
- [Voiceover] Alright, so
now I'm gonna do the analysis

00:00:02.612 --> 00:00:05.278
of this op-amp configuration again,

00:00:05.278 --> 00:00:09.445
and I'm gonna do it using
the idea of virtual ground.

00:00:15.619 --> 00:00:17.744
And the idea of a virtual ground actually,

00:00:17.744 --> 00:00:22.442
it makes really short work of
analyzing a circuit like this.

00:00:22.442 --> 00:00:24.036
To review the virtual ground idea,

00:00:24.036 --> 00:00:26.679
it says if this voltage here, V out,

00:00:26.679 --> 00:00:29.131
is in a reasonable range
between the power supplies

00:00:29.131 --> 00:00:33.364
of its op-amp, and remember
the power supplies are

00:00:33.364 --> 00:00:36.173
not shown here but they're
connected up to the op-amp,

00:00:36.173 --> 00:00:39.389
so this is a number that's
sort of like one hand of volts,

00:00:39.389 --> 00:00:43.441
or two hands of volts,
between zero and plus or minus

00:00:43.441 --> 00:00:45.343
10 volts, something like that.

00:00:45.343 --> 00:00:49.900
And the gain of this op-amp
is huge, it's just gigantic.

00:00:49.900 --> 00:00:51.843
It's up in the 100s of
thousands or millions.

00:00:51.843 --> 00:00:56.600
And so, whenever there's a
normal, finite size voltage here,

00:00:56.600 --> 00:00:59.001
it means the voltage here is about zero,

00:00:59.001 --> 00:01:00.171
it's very close to zero,

00:01:00.171 --> 00:01:02.936
it's down in the very small microvolts.

00:01:02.936 --> 00:01:05.389
And, the idea of a virtual ground,

00:01:05.389 --> 00:01:08.952
says that if this node
here is at zero volts,

00:01:08.952 --> 00:01:11.724
which we've drawn it grounded,
so it's at zero volts,

00:01:11.724 --> 00:01:14.062
that means that this
minus terminal is gonna be

00:01:14.062 --> 00:01:17.053
also held at zero volts,
very close to zero volts.

00:01:17.053 --> 00:01:19.292
And that makes this virtually a ground,

00:01:19.292 --> 00:01:20.959
or a virtual ground.

00:01:22.351 --> 00:01:25.129
Another way to write this
is to say that the V plus

00:01:25.129 --> 00:01:28.568
is approximately equal to V minus.

00:01:28.568 --> 00:01:31.461
Now, my professor who taught me this,

00:01:31.461 --> 00:01:33.624
had a little symbol for
it, what he did was he

00:01:33.624 --> 00:01:37.276
drew it like this, so you
can sketch that on your

00:01:37.276 --> 00:01:39.370
schematics, just a little symbol there,

00:01:39.370 --> 00:01:42.122
and that symbol will help
remind you that those two nodes

00:01:42.122 --> 00:01:44.122
are at the same voltage.

00:01:45.512 --> 00:01:48.242
Okay, so let's do the analysis again,

00:01:48.242 --> 00:01:51.242
we had, this was R1, and this is R2,

00:01:52.114 --> 00:01:55.864
and we have a current
flowing here, called i.

00:01:57.088 --> 00:01:58.756
And we can write an expression for i,

00:01:58.756 --> 00:02:01.256
so the voltage on this side...

00:02:02.569 --> 00:02:05.410
The voltage on this side
of the resistor is V in,

00:02:05.410 --> 00:02:07.389
and the voltage on this
side of the resistor is,

00:02:07.389 --> 00:02:10.139
it's zero volts, it's zero volts,

00:02:11.858 --> 00:02:14.071
because we have a virtual ground here,

00:02:14.071 --> 00:02:15.914
so we know the voltage on both sides.

00:02:15.914 --> 00:02:20.081
So I can say right away, i
equals the voltage difference

00:02:20.952 --> 00:02:24.702
V in minus zero, so just
V in, divided by R1.

00:02:28.702 --> 00:02:31.051
So we have an expression for i.

00:02:31.051 --> 00:02:35.201
Now, as you remember, the thing
we also know about op-amps

00:02:35.201 --> 00:02:38.571
is that this current is zero here,

00:02:38.571 --> 00:02:41.840
there's no current going into
the input of an ideal op-amp.

00:02:41.840 --> 00:02:44.682
A real op-amp there'll be a
really tiny current there.

00:02:44.682 --> 00:02:48.329
For our purposes right now
we can treat this as zero

00:02:48.329 --> 00:02:49.667
current going in here,

00:02:49.667 --> 00:02:51.008
so what does that mean?

00:02:51.008 --> 00:02:53.841
That means that i goes through R2.

00:02:56.753 --> 00:02:59.429
So let's write an expression
for i going through R2,

00:02:59.429 --> 00:03:02.562
and we need to know the
voltage on each side of R2,

00:03:02.562 --> 00:03:06.145
well this side, this
is V naught, or V out.

00:03:07.742 --> 00:03:10.421
This side is V out, and this side, again,

00:03:10.421 --> 00:03:11.981
we get to use the zero,

00:03:11.981 --> 00:03:16.064
so i through R2 equals,
let's get the sign right,

00:03:18.943 --> 00:03:20.275
so the current's going in this way,

00:03:20.275 --> 00:03:23.449
so this is the positive, this
is the relatively positive

00:03:23.449 --> 00:03:25.571
side and this is the negative side,

00:03:25.571 --> 00:03:28.904
so i is zero minus V out, divided by R2.

00:03:37.655 --> 00:03:40.977
Now, let's set these two
currents equal to each other,

00:03:40.977 --> 00:03:45.144
so that means that V in over
R1 equals minus V out over R2,

00:03:50.131 --> 00:03:52.874
and what we want is an expression
that tells us what V out

00:03:52.874 --> 00:03:56.874
is in terms of V in, so
V out equals V in times,

00:04:03.206 --> 00:04:07.289
let's see, R2 goes on top,
R1 goes on the bottom,

00:04:10.213 --> 00:04:12.296
and there's a minus sign.

00:04:14.113 --> 00:04:18.162
And we did it, and that's
the expression for V out

00:04:18.162 --> 00:04:20.803
in terms of V in, for the
inverting op-amp configuration.

00:04:20.803 --> 00:04:23.219
Now, by using the virtual ground idea,

00:04:23.219 --> 00:04:24.842
the analysis became really simple,

00:04:24.842 --> 00:04:28.036
it's basically one two three
steps to get to the answer.

00:04:28.036 --> 00:04:30.477
Compare that to the algebra
that we did in the previous

00:04:30.477 --> 00:04:34.843
video when we did this
from first principles.

00:04:34.843 --> 00:04:38.506
So that's an example of
applying the virtual ground idea

00:04:38.506 --> 00:04:40.923
to analyze an op-amp circuit.

00:04:42.947 --> 00:04:45.280
I'll do one more real quick.

00:04:49.794 --> 00:04:52.528
So here's a different
op-amp configuration,

00:04:52.528 --> 00:04:54.206
we haven't seen this one before.

00:04:54.206 --> 00:04:57.428
First thing to notice,
first thing to notice is

00:04:57.428 --> 00:04:59.305
now plus is on top.

00:04:59.305 --> 00:05:01.359
Always take a peek at your
op-amp to see which terminal

00:05:01.359 --> 00:05:02.617
is on top.

00:05:02.617 --> 00:05:06.450
We have V in here, as
usual, and here's V out,

00:05:08.251 --> 00:05:11.636
and again, what we wanna
do is find out what's V out

00:05:11.636 --> 00:05:13.053
in terms of V in.

00:05:14.374 --> 00:05:17.705
Now this circuit has no resistors in it,

00:05:17.705 --> 00:05:21.014
okay, that's alright, we'll
figure out what's going on here.

00:05:21.014 --> 00:05:24.078
Okay, let's apply the
idea of a virtual ground,

00:05:24.078 --> 00:05:27.509
so I'm gonna make my little
virtual ground symbol here,

00:05:27.509 --> 00:05:29.640
like that, that reminds me
that these two terminals

00:05:29.640 --> 00:05:32.737
are the same voltage if
we're doing the right thing,

00:05:32.737 --> 00:05:35.570
so let's figure out what V out is.

00:05:36.544 --> 00:05:39.922
V out equals this terminal
here, and this one

00:05:39.922 --> 00:05:42.754
is pretty much equal to this one,

00:05:42.754 --> 00:05:44.092
and what's this one?

00:05:44.092 --> 00:05:46.298
This is V out, sorry V in.

00:05:46.298 --> 00:05:47.131
So, boom.

00:05:52.566 --> 00:05:56.825
We just did it, in one
step, that's the answer.

00:05:56.825 --> 00:06:00.158
This configuration is called the buffer.

00:06:01.271 --> 00:06:04.604
Or it's also called a unity-gain buffer.

00:06:14.265 --> 00:06:18.033
This is just a long way
to say the gain is one.

00:06:18.033 --> 00:06:20.176
So that's an example
of a unity-gain buffer,

00:06:20.176 --> 00:06:24.453
and we use the virtual
ground idea to analyze it

00:06:24.453 --> 00:06:25.953
almost instantly.


Summing op-amp circuit

WEBVTT
Kind: captions
Language: en

00:00:00.724 --> 00:00:02.849
- [Voiceover] Another form
of an op-amp circuit is

00:00:02.849 --> 00:00:04.615
called the summing op-amp.

00:00:04.615 --> 00:00:07.310
We're gonna work through
how this one works.

00:00:07.310 --> 00:00:09.017
Let's draw in here now is an inverting

00:00:09.017 --> 00:00:12.103
op-amp circuit with a single input.

00:00:12.103 --> 00:00:13.669
We're gonna call this

00:00:13.669 --> 00:00:14.674
this will be V.

00:00:14.674 --> 00:00:16.757
We'll call the VA for now.

00:00:16.757 --> 00:00:18.687
And we have V out.

00:00:18.687 --> 00:00:21.213
And we worked out what
how this worked before

00:00:21.213 --> 00:00:23.630
where V out was a function of

00:00:24.529 --> 00:00:28.029
VA times the ratio of these two resistors.

00:00:29.332 --> 00:00:31.072
And now I'm gonna add
a little twist to this

00:00:31.072 --> 00:00:33.070
and we're gonna analyze a
different kind of circuit.

00:00:33.070 --> 00:00:36.195
So I'm gonna add another
resistor right here.

00:00:36.195 --> 00:00:37.028
Like this.

00:00:40.411 --> 00:00:41.582
And a second input.

00:00:41.582 --> 00:00:43.900
We'll call this VB.

00:00:43.900 --> 00:00:46.900
And so I have RB, or sorry, RA here.

00:00:47.743 --> 00:00:49.076
This will be RB.

00:00:49.944 --> 00:00:52.527
And we'll call this R feedback.

00:00:54.941 --> 00:00:56.419
And the question again is now,

00:00:56.419 --> 00:00:59.112
what is V out in terms of

00:00:59.112 --> 00:01:00.945
the inputs, VA and VB?

00:01:02.765 --> 00:01:06.515
We're gonna use the idea
of a virtual ground.

00:01:10.572 --> 00:01:11.994
Virtual ground.

00:01:11.994 --> 00:01:13.859
The idea of a virtual
ground applies to most

00:01:13.859 --> 00:01:16.031
op-amp circuits, and
it's a really useful way

00:01:16.031 --> 00:01:18.444
to simplify the analysis.

00:01:18.444 --> 00:01:20.187
And when we use a virtual ground

00:01:20.187 --> 00:01:21.846
one of the things I like to do is just

00:01:21.846 --> 00:01:24.406
draw a little symbol here

00:01:24.406 --> 00:01:26.425
like that, for my virtual ground.

00:01:26.425 --> 00:01:29.465
The virtual ground, as a review,

00:01:29.465 --> 00:01:31.297
if the voltage coming out of this op-amp

00:01:31.297 --> 00:01:33.906
is in a reasonable range,
sort of a plus or minus

00:01:33.906 --> 00:01:35.918
10 volts, or something like that.

00:01:35.918 --> 00:01:38.815
And because the gain of
this op-amp is so enormous

00:01:38.815 --> 00:01:41.898
on the order of 100,000, or a million

00:01:42.929 --> 00:01:45.599
that means that, when
this is working properly

00:01:45.599 --> 00:01:49.430
that these two voltages will
be really close together.

00:01:49.430 --> 00:01:52.388
They'll be only microvolts apart, at most.

00:01:52.388 --> 00:01:54.433
And, because they're so close

00:01:54.433 --> 00:01:56.735
we can just say, let's
just say they're the same

00:01:56.735 --> 00:01:59.241
for the purposes of
analyzing this circuit.

00:01:59.241 --> 00:02:02.819
And, since this input is at zero volts

00:02:02.819 --> 00:02:04.312
this is at zero volts.

00:02:04.312 --> 00:02:07.856
That means that this input
is very close to zero volts.

00:02:07.856 --> 00:02:09.421
So this node here

00:02:09.421 --> 00:02:11.040
this node here, we say

00:02:11.040 --> 00:02:12.957
is at a virtual ground.

00:02:13.990 --> 00:02:15.851
So let's move ahead,
let's analyze this circuit

00:02:15.851 --> 00:02:18.126
and what we want to
find is we want to find

00:02:18.126 --> 00:02:19.959
V out as a function of

00:02:21.887 --> 00:02:22.720
VA and VB.

00:02:25.414 --> 00:02:27.987
Let's see if we can do that.

00:02:27.987 --> 00:02:29.542
So looking at my circuit the other thing

00:02:29.542 --> 00:02:31.528
I know about op-amps
that's really important

00:02:31.528 --> 00:02:33.778
always is, that the current

00:02:34.867 --> 00:02:36.972
going into an op-amp input

00:02:36.972 --> 00:02:38.864
is zero, or practically zero.

00:02:38.864 --> 00:02:42.003
And for the purposes of
what we're doing here

00:02:42.003 --> 00:02:44.596
we can treat it as zero.

00:02:44.596 --> 00:02:45.516
All right.

00:02:45.516 --> 00:02:47.983
So, we have zero volts on this node.

00:02:47.983 --> 00:02:50.001
We have zero current going this way.

00:02:50.001 --> 00:02:51.733
So let's go after

00:02:51.733 --> 00:02:53.932
let's go after this current right here.

00:02:53.932 --> 00:02:55.152
Let's figure out what that is.

00:02:55.152 --> 00:02:56.902
So we'll call that I.

00:02:58.433 --> 00:03:01.096
And we can express I in terms of VA

00:03:01.096 --> 00:03:03.852
and VB in these two resistor values.

00:03:03.852 --> 00:03:05.269
So we can write I

00:03:06.289 --> 00:03:07.758
equals, now what is it?

00:03:07.758 --> 00:03:09.591
Well, we have IA here.

00:03:11.164 --> 00:03:13.497
And we have IB flowing here.

00:03:15.463 --> 00:03:18.257
That means that I equals

00:03:18.257 --> 00:03:19.174
IA plus IB.

00:03:22.824 --> 00:03:23.657
All right.

00:03:29.526 --> 00:03:32.480
And let me write IA and IB in terms of

00:03:32.480 --> 00:03:33.553
Ohms Law here.

00:03:33.553 --> 00:03:36.136
So we get I equals, what is IA?

00:03:37.064 --> 00:03:38.564
IA is this voltage

00:03:41.767 --> 00:03:43.055
divided by RA.

00:03:43.055 --> 00:03:45.159
And this voltage is VA

00:03:45.159 --> 00:03:46.576
minus zero volts.

00:03:47.894 --> 00:03:48.727
So it's VA

00:03:50.436 --> 00:03:51.269
over RA.

00:03:52.906 --> 00:03:54.936
And now let's write down what's IB?

00:03:54.936 --> 00:03:55.769
IB

00:03:56.905 --> 00:03:59.322
we look at this voltage here.

00:04:00.316 --> 00:04:02.316
This voltage is VB minus

00:04:03.590 --> 00:04:04.459
zero.

00:04:04.459 --> 00:04:05.542
Or just plain

00:04:07.327 --> 00:04:08.244
VB over RB.

00:04:11.630 --> 00:04:13.600
All right, so that's one step.

00:04:13.600 --> 00:04:16.150
We've written this current in terms of

00:04:16.150 --> 00:04:18.858
what's going on on the
input side over here.

00:04:18.858 --> 00:04:21.604
Now one thing we know,
because this current is zero,

00:04:21.604 --> 00:04:22.477
we know, what?

00:04:22.477 --> 00:04:23.771
We know

00:04:23.771 --> 00:04:25.271
this current is I.

00:04:27.076 --> 00:04:29.832
And, if I's going into RF

00:04:29.832 --> 00:04:33.758
we'll label the voltage on RF like that.

00:04:33.758 --> 00:04:35.710
So now let's write an expression for I

00:04:35.710 --> 00:04:38.793
in terms of RF and this voltage here.

00:04:40.019 --> 00:04:41.200
What is this voltage here?

00:04:41.200 --> 00:04:42.939
Well, let's look at the two ends.

00:04:42.939 --> 00:04:45.494
This is zero volts on this node here.

00:04:45.494 --> 00:04:48.040
And on this side it's

00:04:48.040 --> 00:04:49.562
V out.

00:04:49.562 --> 00:04:51.364
So, the current there, the current

00:04:51.364 --> 00:04:52.364
on that side

00:04:53.461 --> 00:04:54.294
equals

00:04:55.744 --> 00:04:57.577
it's zero minus V out.

00:04:58.414 --> 00:04:59.581
So minus V out

00:05:00.834 --> 00:05:02.001
divided by RF.

00:05:03.874 --> 00:05:07.624
That's Ohms Law for this
resistor right here.

00:05:10.160 --> 00:05:12.694
Now we know that this
current equals this current.

00:05:12.694 --> 00:05:14.841
So let's just set these
two expressions together.

00:05:14.841 --> 00:05:17.008
So I'll do that over here.

00:05:21.131 --> 00:05:23.683
I can write minus V out

00:05:23.683 --> 00:05:24.516
over RF

00:05:26.490 --> 00:05:27.740
equals this one

00:05:30.503 --> 00:05:31.336
VA over RA

00:05:33.523 --> 00:05:34.856
plus VB over RB.

00:05:39.334 --> 00:05:40.722
Now we can get the final function

00:05:40.722 --> 00:05:43.823
I'm just gonna multiply
through by RF on both sides.

00:05:43.823 --> 00:05:45.186
And we get

00:05:45.186 --> 00:05:46.186
V out equals

00:05:47.312 --> 00:05:48.624
take the minus sign with us

00:05:48.624 --> 00:05:49.457
minus

00:05:51.564 --> 00:05:52.397
RF

00:05:53.465 --> 00:05:54.298
over RA

00:05:56.164 --> 00:05:56.997
times VA

00:05:58.413 --> 00:06:00.069
plus

00:06:00.069 --> 00:06:00.902
RF

00:06:02.131 --> 00:06:03.481
over

00:06:03.481 --> 00:06:04.829
RB

00:06:04.829 --> 00:06:06.066
times

00:06:06.066 --> 00:06:06.899
VB.

00:06:09.218 --> 00:06:10.338
So that's our answer.

00:06:10.338 --> 00:06:11.433
That's V out

00:06:11.433 --> 00:06:12.749
as a function of

00:06:12.749 --> 00:06:15.396
VA and VB, the two inputs.

00:06:15.396 --> 00:06:17.034
And you can see that the

00:06:17.034 --> 00:06:19.880
resistor ratio, there's
two resistor ratios here,

00:06:19.880 --> 00:06:22.654
that are participating in the answer.

00:06:22.654 --> 00:06:25.033
So it helps to do a little
special case of this.

00:06:25.033 --> 00:06:27.067
Let's let all the resistors be the same.

00:06:27.067 --> 00:06:28.373
So we'll say that

00:06:28.373 --> 00:06:29.373
RA equals RB

00:06:31.501 --> 00:06:32.334
equals RF.

00:06:33.178 --> 00:06:35.521
And just to pick a real
number we'll just say

00:06:35.521 --> 00:06:38.104
they're all equal to 10 K ohms.

00:06:39.076 --> 00:06:40.956
And let's see what this becomes now.

00:06:40.956 --> 00:06:42.373
Now we have V out

00:06:43.625 --> 00:06:46.449
equals, we still have the negative sign.

00:06:46.449 --> 00:06:48.698
So this is some sort of
inversion going on here.

00:06:48.698 --> 00:06:50.996
RF over RA is one.

00:06:50.996 --> 00:06:53.124
So it's just VA.

00:06:53.124 --> 00:06:54.957
And RF over RB is one.

00:06:56.399 --> 00:06:57.399
So this says

00:06:58.778 --> 00:06:59.611
VB.

00:07:01.885 --> 00:07:04.157
So this is what gives us the nickname

00:07:04.157 --> 00:07:05.189
for this expression

00:07:05.189 --> 00:07:06.606
which is called a

00:07:08.331 --> 00:07:09.164
a summing

00:07:12.125 --> 00:07:12.958
op-amp.

00:07:20.258 --> 00:07:22.263
Let's say I want to use my summing op-amp

00:07:22.263 --> 00:07:25.671
in an application where what I want is

00:07:25.671 --> 00:07:27.754
I want V out to equal say

00:07:29.305 --> 00:07:30.805
minus two times VA

00:07:33.332 --> 00:07:34.999
plus three times VB.

00:07:37.930 --> 00:07:39.331
Okay, this is what I want.

00:07:39.331 --> 00:07:41.150
How do I do that?

00:07:41.150 --> 00:07:43.336
So, these two coefficients here are

00:07:43.336 --> 00:07:45.674
functions of the resistor values.

00:07:45.674 --> 00:07:48.565
That was our original expression up here.

00:07:48.565 --> 00:07:50.787
So what I want is

00:07:50.787 --> 00:07:51.620
RF over RA

00:07:53.905 --> 00:07:55.551
to equal two.

00:07:55.551 --> 00:07:56.634
And I want RF

00:07:58.787 --> 00:07:59.620
over RB

00:08:00.614 --> 00:08:01.864
to equal three.

00:08:03.950 --> 00:08:05.173
So I can pick values

00:08:05.173 --> 00:08:06.691
it's the same RF

00:08:06.691 --> 00:08:09.695
and so I get to adjust RA and RB in here.

00:08:09.695 --> 00:08:12.031
So I can pick component values.

00:08:12.031 --> 00:08:13.031
So if I pick

00:08:13.896 --> 00:08:15.563
RF equals 12 K ohms.

00:08:19.919 --> 00:08:21.932
Then I can pick

00:08:21.932 --> 00:08:23.265
RA equals six K.

00:08:26.287 --> 00:08:27.120
And RB

00:08:29.063 --> 00:08:30.230
equals four K.

00:08:33.097 --> 00:08:34.879
And that would give me the ratios I want

00:08:34.879 --> 00:08:36.496
and that would implement

00:08:36.496 --> 00:08:37.615
this function here.

00:08:37.615 --> 00:08:39.087
So I'm gonna go fill those out

00:08:39.087 --> 00:08:40.491
on the top schematic.

00:08:40.491 --> 00:08:43.156
We'll just write those in.

00:08:43.156 --> 00:08:44.589
And we'll go fill these in here.

00:08:44.589 --> 00:08:47.256
So what we said was RF was 12 K.

00:08:50.104 --> 00:08:51.604
RA was six K ohms.

00:08:53.894 --> 00:08:55.894
RB would be four K ohms.

00:08:58.794 --> 00:09:00.257
And we've designed a circuit

00:09:00.257 --> 00:09:02.785
that implements our summing function.

00:09:02.785 --> 00:09:06.618
So, so this is a pretty
useful op-amp circuit.


Virtual ground

WEBVTT
Kind: captions
Language: en

00:00:00.298 --> 00:00:02.885
- [Voiceover] I wanna take a
look at our two op amp circuits

00:00:02.885 --> 00:00:05.015
and make an interesting observation about

00:00:05.015 --> 00:00:08.245
how this things are behaving
when they are working properly.

00:00:08.245 --> 00:00:09.525
When they're hooked up right,

00:00:09.525 --> 00:00:11.279
there's something these things do

00:00:11.279 --> 00:00:15.446
that is really helpful and
makes life simple for us.

00:00:17.020 --> 00:00:20.570
Let's let the gain of our
op amp be 10 to the third,

00:00:20.570 --> 00:00:21.892
or 10 to the sixth.

00:00:21.892 --> 00:00:24.729
Really high gain, a million.

00:00:24.729 --> 00:00:26.660
And we're gonna let the
output voltage here,

00:00:26.660 --> 00:00:27.493
V out,

00:00:29.205 --> 00:00:30.872
let's say six volts.

00:00:31.950 --> 00:00:34.336
And you remember, what's not
shown here in this circuit

00:00:34.336 --> 00:00:35.919
is the power supply

00:00:37.617 --> 00:00:39.809
is going to both of these op amps.

00:00:39.809 --> 00:00:42.951
Plus or minus, let's say
it's plus or minus 12 volts.

00:00:42.951 --> 00:00:44.639
Those power supplies are implicit.

00:00:44.639 --> 00:00:48.500
They're not shown in the diagram
but we know they're there.

00:00:48.500 --> 00:00:50.888
Alright, now, if V out is six volts

00:00:50.888 --> 00:00:52.704
and A is 10 to the sixth,

00:00:52.704 --> 00:00:53.658
then what's V in?

00:00:53.658 --> 00:00:57.944
V in is the difference between
these two voltages here.

00:00:57.944 --> 00:00:59.756
Let's call this the usual thing.

00:00:59.756 --> 00:01:01.385
We'll call this V plus.

00:01:01.385 --> 00:01:04.267
I'm gonna call this V minus.

00:01:04.267 --> 00:01:06.100
And we know that V in,

00:01:07.276 --> 00:01:08.693
we know that V in

00:01:10.685 --> 00:01:13.018
equals V plus minus V minus.

00:01:14.458 --> 00:01:15.825
And now what the question is

00:01:15.825 --> 00:01:17.453
what is V in in terms of V out?

00:01:17.453 --> 00:01:18.286
Well, V in

00:01:20.693 --> 00:01:22.860
equals V out divided by A.

00:01:25.744 --> 00:01:27.302
If we fill in the values we had,

00:01:27.302 --> 00:01:28.469
it's six volts

00:01:29.814 --> 00:01:31.981
divided by 10 to the sixth,

00:01:31.981 --> 00:01:33.481
or six microvolts.

00:01:35.302 --> 00:01:38.782
So this is six microvolts
between here and here, okay?

00:01:38.782 --> 00:01:40.485
So with six volts here,

00:01:40.485 --> 00:01:42.249
there's six microvolts over here.

00:01:42.249 --> 00:01:45.332
Okay, this is a really small voltage.

00:01:46.706 --> 00:01:48.862
In order for this op amp

00:01:48.862 --> 00:01:50.494
to have an output voltage

00:01:50.494 --> 00:01:52.905
that stays between plus or minus 12 volts,

00:01:52.905 --> 00:01:55.759
this voltage over here
has to be really small.

00:01:55.759 --> 00:01:59.023
It has to be down to the microvolts level.

00:01:59.023 --> 00:02:00.731
So because I'm a practical engineer,

00:02:00.731 --> 00:02:04.921
I'm just gonna say this
is pretty much zero volts.

00:02:04.921 --> 00:02:05.954
And if I say this is zero,

00:02:05.954 --> 00:02:07.929
that's pretty much the
same thing as saying

00:02:07.929 --> 00:02:11.346
that V plus approximately equals V minus.

00:02:14.815 --> 00:02:18.982
So that's a little observation
we're gonna make right there.

00:02:20.325 --> 00:02:22.984
So in this circuit,
when it's working right,

00:02:22.984 --> 00:02:27.210
these two voltages are
pretty much the same.

00:02:27.210 --> 00:02:30.832
So let's take this idea of V
plus pretty much equals V minus

00:02:30.832 --> 00:02:32.799
and apply it to this circuit over here.

00:02:32.799 --> 00:02:36.693
Now this is our inverting
configuration for an op amp.

00:02:36.693 --> 00:02:38.193
So this is V plus.

00:02:39.126 --> 00:02:42.126
And this is V minus in this circuit.

00:02:43.178 --> 00:02:45.526
And let's do the same
analysis that we did before.

00:02:45.526 --> 00:02:49.769
If this is V out, and
if V out is six volts,

00:02:49.769 --> 00:02:52.769
that means that V plus minus V minus

00:02:56.044 --> 00:02:58.687
divided by 10 to the sixth

00:02:58.687 --> 00:03:01.011
equals six microvolts,

00:03:01.011 --> 00:03:02.928
and that says that this

00:03:03.978 --> 00:03:05.395
is six microvolts

00:03:06.572 --> 00:03:08.116
in this direction.

00:03:08.116 --> 00:03:10.653
When we did this over here,

00:03:10.653 --> 00:03:13.008
because the signs of
the inputs are flipped,

00:03:13.008 --> 00:03:15.758
this was six microvolts this way.

00:03:18.425 --> 00:03:22.121
So again, because of the
enormous gain of this amplifier,

00:03:22.121 --> 00:03:25.244
this is always gonna
be a tiny, tiny number,

00:03:25.244 --> 00:03:28.161
and so, heck, why not make it zero?

00:03:29.934 --> 00:03:31.973
So if I treat this as
zero, what it means is,

00:03:31.973 --> 00:03:35.046
I'm gonna go right in here,

00:03:35.046 --> 00:03:37.379
and I'm gonna change this to

00:03:40.406 --> 00:03:41.323
zero volts.

00:03:43.734 --> 00:03:45.739
So let's make a couple
more observations, okay?

00:03:45.739 --> 00:03:47.471
Right now, it says right here,

00:03:47.471 --> 00:03:48.971
V plus equals zero

00:03:49.897 --> 00:03:51.236
because it's grounded.

00:03:51.236 --> 00:03:54.069
So what does that mean V minus is?

00:03:55.007 --> 00:03:57.370
Well, V minus is also zero.

00:03:57.370 --> 00:03:59.083
V minus is zero.

00:03:59.083 --> 00:04:01.166
So that point right there

00:04:02.028 --> 00:04:03.445
is at zero volts.

00:04:05.728 --> 00:04:07.200
Okay, so that's pretty cool.

00:04:07.200 --> 00:04:09.911
So, that point's at zero volts.

00:04:09.911 --> 00:04:12.230
Now, is it connected to ground?

00:04:12.230 --> 00:04:14.107
It's not connected to ground.

00:04:14.107 --> 00:04:15.269
But at zero volts,

00:04:15.269 --> 00:04:18.167
because of what this op
amp is doing it for us,

00:04:18.167 --> 00:04:22.134
this op amp is making sure,
by this feedback path,

00:04:22.134 --> 00:04:25.568
that this node is always next to this node

00:04:25.568 --> 00:04:27.829
and that means it's always zero.

00:04:27.829 --> 00:04:29.823
There's a really cool
word that we use for this

00:04:29.823 --> 00:04:31.823
and the word is virtual.

00:04:37.224 --> 00:04:38.718
And what does the word virtual mean?

00:04:38.718 --> 00:04:42.635
Well, virtual means that
something is not there

00:04:45.315 --> 00:04:47.315
but it seems like it is.

00:04:59.510 --> 00:05:03.226
So in this case, this node
is not connected to ground

00:05:03.226 --> 00:05:04.758
but it seems like it is.

00:05:04.758 --> 00:05:07.091
So this is referred to as a,

00:05:11.089 --> 00:05:12.506
a virtual ground.

00:05:22.327 --> 00:05:25.009
So these two ideas say the same thing.

00:05:25.009 --> 00:05:27.918
V plus equals V minus
is always the situation

00:05:27.918 --> 00:05:29.928
around the input to an op amp,

00:05:29.928 --> 00:05:31.611
when it's running properly,

00:05:31.611 --> 00:05:35.271
and in the case particularly
of this op amp configuration,

00:05:35.271 --> 00:05:38.454
where the plus terminal
is connected to ground,

00:05:38.454 --> 00:05:40.797
we say that the other terminal, V minus,

00:05:40.797 --> 00:05:44.714
is at a virtual ground,
or is a virtual ground.

00:05:45.593 --> 00:05:47.459
In the next video, I'm gonna go back

00:05:47.459 --> 00:05:50.095
and do this inverting
configuration of the op amp.

00:05:50.095 --> 00:05:51.928
I'm gonna do the analysis again

00:05:51.928 --> 00:05:54.075
with this idea of a virtual ground

00:05:54.075 --> 00:05:55.625
and it's gonna be really easy

00:05:55.625 --> 00:05:57.792
compared to doing all that algebra.


Inverting op-amp circuit

WEBVTT
Kind: captions
Language: en

00:00:01.109 --> 00:00:02.874
- [Voiceover] Now I come
to another configuration

00:00:02.874 --> 00:00:03.822
for an op-amp.

00:00:03.822 --> 00:00:05.793
And it's partially drawn here.

00:00:05.793 --> 00:00:07.407
And I'm gonna talk about this

00:00:07.407 --> 00:00:08.916
as I draw the rest of this circuit in.

00:00:08.916 --> 00:00:11.958
So this is gonna be made
from a resistor configuration

00:00:11.958 --> 00:00:14.059
that looks like this.

00:00:14.059 --> 00:00:16.809
We'll have a resistor on the top.

00:00:20.433 --> 00:00:22.813
And this will be v-out,

00:00:22.813 --> 00:00:24.346
as we did before.

00:00:24.346 --> 00:00:26.888
And now we have a connection like this

00:00:26.888 --> 00:00:29.721
and the connection here to ground.

00:00:31.358 --> 00:00:33.413
This terminal is the minus terminal;

00:00:33.413 --> 00:00:35.758
and this terminal is
the positive terminal.

00:00:35.758 --> 00:00:39.241
This is upside down from
what we've done so far.

00:00:39.241 --> 00:00:41.087
But now pay particular attention here,

00:00:41.087 --> 00:00:43.433
this one has the minus on top.

00:00:43.433 --> 00:00:45.516
So now we have R1 and R2.

00:00:47.763 --> 00:00:49.180
And this is v-in.

00:00:50.875 --> 00:00:53.928
In particular, what we wanna
do is find an expression

00:00:53.928 --> 00:00:54.761
for v-out

00:00:55.903 --> 00:00:57.736
as a function of v-in.

00:01:02.694 --> 00:01:03.527
All right.

00:01:03.527 --> 00:01:07.527
And in this video, I'm
gonna do it the hard way.

00:01:11.842 --> 00:01:13.381
And what the hard way
means is we're gonna do

00:01:13.381 --> 00:01:15.517
all the algebra to do this.

00:01:15.517 --> 00:01:19.999
And then in the next video I'm
gonna show you the easy way.

00:01:19.999 --> 00:01:21.845
The easy way is really fun to learn,

00:01:21.845 --> 00:01:24.434
and it really helps to see
it the hard way one time

00:01:24.434 --> 00:01:28.021
just so you appreciate the easy way.

00:01:28.021 --> 00:01:29.914
The other thing we get to do in this video

00:01:29.914 --> 00:01:32.456
is we'll do the algebra and
we'll see how this gain,

00:01:32.456 --> 00:01:36.623
we take advantage of this
gain to make some assumptions.

00:01:37.472 --> 00:01:38.795
OK, so let's go after this.

00:01:38.795 --> 00:01:42.987
Let's develop an expression
for v-out in terms of v-in.

00:01:42.987 --> 00:01:47.154
First, let's write some things
we know about v-out, OK?

00:01:48.118 --> 00:01:51.799
We know that v-out equals A times...

00:01:51.799 --> 00:01:54.922
Now, it's usually v-plus minus v-minus.

00:01:54.922 --> 00:01:56.172
This is v-plus.

00:01:57.104 --> 00:01:58.437
This is v-minus.

00:02:00.796 --> 00:02:03.792
Usually the expression here
is v-plus minus v-minus.

00:02:03.792 --> 00:02:06.148
And since v-plus is zero,

00:02:06.148 --> 00:02:09.315
we're just gonna put in minus v-minus.

00:02:11.605 --> 00:02:13.938
This is equivalent to saying

00:02:15.001 --> 00:02:16.584
that v-minus equals

00:02:19.491 --> 00:02:21.074
minus v-out over A.

00:02:28.140 --> 00:02:29.940
So what else can we
write for this circuit?

00:02:29.940 --> 00:02:31.670
OK, let's look at these resistors.

00:02:31.670 --> 00:02:34.587
Let's call this plus and minus vR1;

00:02:37.579 --> 00:02:40.829
and we'll call this one plus minus vR2.

00:02:43.175 --> 00:02:47.342
So there's a current flowing
here, and that we'll call I.

00:02:48.492 --> 00:02:51.163
So I'm just gonna use Ohm's Law on R1 here

00:02:51.163 --> 00:02:53.740
and write an expression for I.

00:02:53.740 --> 00:02:54.573
I

00:02:55.748 --> 00:02:56.581
equals

00:02:59.081 --> 00:02:59.914
vR1

00:03:01.117 --> 00:03:01.950
over

00:03:02.847 --> 00:03:03.680
R1.

00:03:04.867 --> 00:03:05.700
Right?

00:03:05.700 --> 00:03:07.085
Another way I can write that.

00:03:07.085 --> 00:03:07.979
What's this voltage here?

00:03:07.979 --> 00:03:09.312
This is v-minus.

00:03:11.473 --> 00:03:15.640
So I can write this in terms
of v-minus, and that equals

00:03:16.651 --> 00:03:17.484
v-in

00:03:18.506 --> 00:03:19.589
minus v-minus

00:03:23.278 --> 00:03:24.111
over R1.

00:03:25.333 --> 00:03:29.663
That's the current going
through this guy here.

00:03:29.663 --> 00:03:31.428
Now I'm gonna use something special.

00:03:31.428 --> 00:03:32.449
I'm gonna use something special

00:03:32.449 --> 00:03:34.911
that I know about this amplifier.

00:03:34.911 --> 00:03:38.440
What I know about an op-amp
is that this current here

00:03:38.440 --> 00:03:39.857
is equal to zero.

00:03:41.157 --> 00:03:44.652
There's no current that
flows into an ideal op-amp.

00:03:44.652 --> 00:03:46.056
So I could take advantage of that.

00:03:46.056 --> 00:03:49.223
What that means is that I flows in R2.

00:03:51.269 --> 00:03:53.359
So let me write and expression for I

00:03:53.359 --> 00:03:56.621
based on what I find
over here, based on R2.

00:03:56.621 --> 00:03:58.371
I can write I equals,

00:03:59.393 --> 00:04:01.143
let's do it, it's vR2

00:04:04.385 --> 00:04:05.218
over R2.

00:04:07.462 --> 00:04:09.379
And I can write vR2 as:

00:04:10.713 --> 00:04:11.546
v-minus

00:04:12.410 --> 00:04:13.327
minus v-out

00:04:19.643 --> 00:04:20.476
over R2.

00:04:23.057 --> 00:04:25.971
All right, so I took
advantage of the zero current

00:04:25.971 --> 00:04:28.444
flowing in here to write
an expression for current

00:04:28.444 --> 00:04:30.882
going all the way through.

00:04:30.882 --> 00:04:31.715
All right.

00:04:31.715 --> 00:04:34.446
So now we're gonna set these
two equal to each other.

00:04:34.446 --> 00:04:38.068
Now we're gonna make these
two equal to each other.

00:04:38.068 --> 00:04:41.528
Let me go over here and do that.

00:04:41.528 --> 00:04:42.361
V-in

00:04:43.873 --> 00:04:44.956
minus v-minus

00:04:47.484 --> 00:04:48.317
over R1.

00:04:50.445 --> 00:04:52.581
That equals this term here,

00:04:52.581 --> 00:04:54.998
which is v-minus minus v-not,

00:04:56.865 --> 00:04:57.948
v-out rather,

00:04:58.792 --> 00:04:59.625
over R2.

00:05:03.830 --> 00:05:05.352
How many variables do we have here?

00:05:05.352 --> 00:05:09.229
We have v-out, we have
v-in, and we have v-minus.

00:05:09.229 --> 00:05:12.632
And what I want is just v-out and v-in,

00:05:12.632 --> 00:05:15.078
so I'm gonna try to eliminate v-minus;

00:05:15.078 --> 00:05:17.702
and the way I'm gonna do that
is this expression over here.

00:05:17.702 --> 00:05:20.732
We're gonna take advantage
of this statement right here

00:05:20.732 --> 00:05:23.232
to replace minus v-out over A.

00:05:25.550 --> 00:05:27.640
So I'll do that right here.

00:05:27.640 --> 00:05:28.789
So let me rewrite this.

00:05:28.789 --> 00:05:29.872
It's gonna be

00:05:31.090 --> 00:05:31.923
v-in

00:05:32.809 --> 00:05:33.642
minus

00:05:34.829 --> 00:05:36.303
minus v-out over A,

00:05:36.303 --> 00:05:38.347
so I get to make this a plus,

00:05:38.347 --> 00:05:40.764
and this becomes v-out over A

00:05:43.409 --> 00:05:44.909
all divided by R1.

00:05:46.729 --> 00:05:48.229
And that equals...

00:05:49.562 --> 00:05:51.979
V-minus is minus v-out over A

00:05:56.806 --> 00:05:57.723
minus v-out

00:06:01.590 --> 00:06:02.423
over R2.

00:06:05.142 --> 00:06:07.894
All right, let's roll down a little bit,

00:06:07.894 --> 00:06:09.061
get some room,

00:06:11.992 --> 00:06:14.396
and we'll keep going.

00:06:14.396 --> 00:06:17.102
What am I gonna do next?

00:06:17.102 --> 00:06:21.149
Next, I'm gonna multiply both sides by A,

00:06:21.149 --> 00:06:23.750
just to get A out of the bottom there.

00:06:23.750 --> 00:06:24.750
So now I get

00:06:26.173 --> 00:06:27.173
A times v-in

00:06:29.180 --> 00:06:30.013
plus v-out

00:06:32.060 --> 00:06:32.893
over R1

00:06:35.438 --> 00:06:36.271
equals

00:06:38.190 --> 00:06:39.107
minus v-out

00:06:40.860 --> 00:06:41.860
minus Av-out

00:06:46.200 --> 00:06:47.033
over R2.

00:06:50.888 --> 00:06:52.769
There's a lot of algebra
here, but trust me,

00:06:52.769 --> 00:06:56.020
it's gonna simplify down
here in just a minute.

00:06:56.020 --> 00:06:57.448
All right, so now I'm gonna break this up

00:06:57.448 --> 00:07:02.370
into separate terms so I
can handle them separately.

00:07:02.370 --> 00:07:03.453
Av-in over R1

00:07:06.616 --> 00:07:08.116
plus v-out over R1

00:07:10.738 --> 00:07:12.905
equals minus v-out over R2

00:07:16.624 --> 00:07:18.374
minus Av-out over R2.

00:07:24.809 --> 00:07:28.579
Let's change colors so we don't get bored.

00:07:28.579 --> 00:07:30.669
Next, what I'm gonna do is start to gather

00:07:30.669 --> 00:07:33.131
the v-out terms on one side

00:07:33.131 --> 00:07:35.104
and the v-in terms on the other side.

00:07:35.104 --> 00:07:37.066
So that means that this v-out term here

00:07:37.066 --> 00:07:39.566
is gonna go to the other side.

00:07:44.984 --> 00:07:45.817
Av-in

00:07:47.242 --> 00:07:48.075
over R1

00:07:49.344 --> 00:07:50.609
equals,

00:07:50.609 --> 00:07:52.942
let's do minus v-out over R2

00:07:56.101 --> 00:07:57.851
minus Av-out over R2.

00:08:00.547 --> 00:08:02.979
And this term comes over as

00:08:02.979 --> 00:08:04.646
minus v-out over R1.

00:08:10.677 --> 00:08:13.427
Haven't made any sign errors yet.

00:08:15.908 --> 00:08:17.233
Now let me clear the R1.

00:08:17.233 --> 00:08:19.900
We'll multiply both sides by R1.

00:08:21.436 --> 00:08:22.269
Av-in

00:08:23.735 --> 00:08:24.568
equals

00:08:26.811 --> 00:08:27.644
R1 over R2

00:08:30.143 --> 00:08:30.976
minus

00:08:32.134 --> 00:08:32.967
v-not

00:08:33.849 --> 00:08:34.682
minus

00:08:35.797 --> 00:08:36.630
R1 over R2

00:08:39.523 --> 00:08:40.356
Av-not

00:08:42.472 --> 00:08:43.472
minus v-not.

00:08:45.351 --> 00:08:49.078
Yeah, the R1s cancel on that last term.

00:08:49.078 --> 00:08:49.911
All right.

00:08:50.761 --> 00:08:51.594
Av-in

00:08:52.767 --> 00:08:53.600
equals.

00:08:56.215 --> 00:08:58.215
Out of here I can factor

00:08:59.303 --> 00:09:00.553
this term here.

00:09:02.055 --> 00:09:04.269
Minus R1 over R2 times v-not,

00:09:04.269 --> 00:09:07.694
I can factor that out of here and here.

00:09:07.694 --> 00:09:09.361
So I can do minus R1

00:09:10.265 --> 00:09:11.348
over R2 v-not

00:09:13.679 --> 00:09:15.196
times

00:09:15.196 --> 00:09:16.029
one plus A

00:09:19.399 --> 00:09:20.399
minus v-not.

00:09:25.192 --> 00:09:26.829
So let's take a look at this expression

00:09:26.829 --> 00:09:30.829
and use our judgement to
decide what to do next.

00:09:32.146 --> 00:09:34.500
Now, because A is so huge,

00:09:34.500 --> 00:09:38.238
that means that this first
term is gonna be gigantic

00:09:38.238 --> 00:09:40.944
compared to this v-not term here.

00:09:40.944 --> 00:09:43.521
V-not is some value like five volts

00:09:43.521 --> 00:09:45.866
or minus five volts or
something like that.

00:09:45.866 --> 00:09:49.964
And A times this is something like 100,000

00:09:49.964 --> 00:09:52.182
or 200,000, something like that.

00:09:52.182 --> 00:09:55.955
It dwarfs this v-not, so I'm
gonna ignore this for now.

00:09:55.955 --> 00:09:58.010
I'm just gonna cross that out,

00:09:58.010 --> 00:10:00.681
and we'll move forward
without that little v-not

00:10:00.681 --> 00:10:03.235
on the end of the expression.

00:10:03.235 --> 00:10:05.278
This is after we've left that out.

00:10:05.278 --> 00:10:08.366
Now we have v-in on this side.

00:10:08.366 --> 00:10:12.033
And I'm gonna take A
over to the other side.

00:10:18.931 --> 00:10:20.431
One plus A over A.

00:10:24.164 --> 00:10:26.881
And this is a point where we
get to use our judgement again.

00:10:26.881 --> 00:10:30.236
Again, A is a huge number, like a million;

00:10:30.236 --> 00:10:33.486
and so A plus one is a million and one.

00:10:34.764 --> 00:10:39.048
That fraction is really
really close to one,

00:10:39.048 --> 00:10:41.637
so I'm gonna ignore it; I'm
gonna just say it's one.

00:10:41.637 --> 00:10:44.137
So we'll send this one to one.

00:10:45.666 --> 00:10:47.071
And let me roll up a little bit more,

00:10:47.071 --> 00:10:50.071
just to have a little bit more room.

00:10:53.689 --> 00:10:55.035
Now what we have is what?

00:10:55.035 --> 00:10:55.868
V-in

00:10:57.346 --> 00:10:58.596
equals minus R1

00:10:59.691 --> 00:11:00.524
over R2

00:11:01.374 --> 00:11:02.524
times v-out.

00:11:02.524 --> 00:11:05.693
And I want the expression
just in terms of v-out,

00:11:05.693 --> 00:11:07.655
so I'm gonna spin this around,

00:11:07.655 --> 00:11:10.572
and we'll get v-out equals minus R2

00:11:11.518 --> 00:11:13.101
over R1 times v-in.

00:11:22.002 --> 00:11:23.837
So this is what our
op-amp is doing for us.

00:11:23.837 --> 00:11:26.832
It basically says v-out is the ratio

00:11:26.832 --> 00:11:29.386
of two resistors times v-in.

00:11:29.386 --> 00:11:31.650
The gain of the overall circuit

00:11:31.650 --> 00:11:34.228
is determined by the ratio
of those two resistors.

00:11:34.228 --> 00:11:36.770
And very importantly, there's
a minus sign in front of it.

00:11:36.770 --> 00:11:39.162
The minus sign came all the way through.

00:11:39.162 --> 00:11:42.829
Again, let me sketch
the circuit real quick.

00:11:45.245 --> 00:11:47.662
We had the minus sign on top.

00:11:49.123 --> 00:11:50.290
There's v-out.

00:12:01.012 --> 00:12:03.012
And there was R1 and R2.

00:12:04.852 --> 00:12:07.801
And the positive input,
the non-inverting input,

00:12:07.801 --> 00:12:10.308
was connected to ground.

00:12:10.308 --> 00:12:11.342
OK?

00:12:11.342 --> 00:12:13.269
So this pattern with the resistor going

00:12:13.269 --> 00:12:15.080
over the top to the minus,

00:12:15.080 --> 00:12:17.247
this is called a inverting

00:12:20.304 --> 00:12:21.137
op-amp.

00:12:28.199 --> 00:12:31.694
And this is a really familiar
pattern in op-amp circuits.

00:12:31.694 --> 00:12:33.006
You can see these on schematics,

00:12:33.006 --> 00:12:35.688
and you'll be designing these on your own.

00:12:35.688 --> 00:12:37.696
So this was quite a bit of algebra it took

00:12:37.696 --> 00:12:39.298
to get down to this point,

00:12:39.298 --> 00:12:41.156
and in the next video I'm gonna show you

00:12:41.156 --> 00:12:44.151
a really easy way to shortcircuit all this

00:12:44.151 --> 00:12:47.193
and be able to do this
analysis really quickly;

00:12:47.193 --> 00:12:49.296
and that's called the virtual ground.


Feedback

WEBVTT
Kind: captions
Language: en

00:00:01.802 --> 00:00:03.745
- [Voiceover] So now I wanna
talk a little bit about

00:00:03.745 --> 00:00:05.745
the concept of Feedback.

00:00:06.746 --> 00:00:09.643
This is a, this is a
really important concept.

00:00:09.643 --> 00:00:13.640
It was developed in the 1920s,
the idea of using Feedback,

00:00:13.640 --> 00:00:15.485
and it was done at, at Bell Labs,

00:00:15.485 --> 00:00:17.818
Bell Telephone Laboratories.

00:00:20.352 --> 00:00:22.820
Remember we talked about
this on the op amp,

00:00:22.820 --> 00:00:26.019
this being the, the non-inverting input,

00:00:26.019 --> 00:00:29.282
so if this voltage went
up, this voltage went up,

00:00:29.282 --> 00:00:32.098
and the negative sign
is the inverting input,

00:00:32.098 --> 00:00:34.401
so these, these voltage here and here,

00:00:34.401 --> 00:00:36.601
move in opposite directions.

00:00:36.601 --> 00:00:38.450
And the key to understanding
what Feedback is,

00:00:38.450 --> 00:00:41.354
is to pay attention to
this inversion path,

00:00:41.354 --> 00:00:43.689
and these non-inverting paths.

00:00:43.689 --> 00:00:46.442
We're gonna start over
here with just isolating

00:00:46.442 --> 00:00:48.243
the voltage divider part of our circuit,

00:00:48.243 --> 00:00:50.493
so that's copied over here.

00:00:56.305 --> 00:00:58.729
And when you think about,
let's look at what happens

00:00:58.729 --> 00:01:00.146
if V out goes up.

00:01:00.994 --> 00:01:03.129
And what happens to V minus in this case?

00:01:03.129 --> 00:01:06.545
Well V minus goes in what
direction in a voltage divider?

00:01:06.545 --> 00:01:07.462
It goes up.

00:01:09.170 --> 00:01:12.561
So this is a, this is a
non-inverting structure.

00:01:12.561 --> 00:01:15.753
If V out goes up, then the
thing we care about V minus,

00:01:15.753 --> 00:01:17.161
goes up as well.

00:01:17.161 --> 00:01:21.328
And likewise if V out goes
down, then V minus goes down.

00:01:24.426 --> 00:01:26.201
So that's isolating
just on this part of it,

00:01:26.201 --> 00:01:29.379
we have no inversion
happening around here.

00:01:29.379 --> 00:01:30.721
So we go around here now, and let's look

00:01:30.721 --> 00:01:33.089
at what happens on this path here,

00:01:33.089 --> 00:01:37.089
if we go up in V in, then
we know V out goes up.

00:01:39.498 --> 00:01:43.165
If we then change colors
to, let's try this.

00:01:44.482 --> 00:01:46.065
If V minus goes up,

00:01:47.948 --> 00:01:49.649
then V out goes down.

00:01:49.649 --> 00:01:51.816
That's the inverting path.

00:01:52.697 --> 00:01:54.881
So there's, so there's one
inversion in this circuit,

00:01:54.881 --> 00:01:57.877
and it happens right here
where V minus goes to V out,

00:01:57.877 --> 00:02:01.009
so that's where the inversion is.

00:02:01.009 --> 00:02:03.849
So now let me set up
just an example circuit.

00:02:03.849 --> 00:02:05.932
We'll set R1 equal to R2.

00:02:10.465 --> 00:02:13.529
And from the last video, we
developed a gain expression,

00:02:13.529 --> 00:02:17.696
and we said that V out
equals R1 plus R2, over R2,

00:02:23.516 --> 00:02:27.683
times V in, and with these,
with these resistor values,

00:02:29.908 --> 00:02:32.241
V out equals two times V in.

00:02:35.733 --> 00:02:39.483
Alright, so this is
equal to, two times V in.

00:02:42.047 --> 00:02:43.736
And what does that make this point here?

00:02:43.736 --> 00:02:46.108
V minus, this is V minus,

00:02:46.108 --> 00:02:49.457
and from our voltage divider,
we know a voltage divider

00:02:49.457 --> 00:02:52.790
says that V minus equals V out times R2,

00:02:57.351 --> 00:02:59.934
which is just R, over R plus R,

00:03:03.914 --> 00:03:06.747
or V minus equals one-half, V out.

00:03:10.205 --> 00:03:12.506
So, we have let's put a,

00:03:12.506 --> 00:03:14.938
let's put a voltage on here.

00:03:14.938 --> 00:03:16.241
Let's put a real voltage on here.

00:03:16.241 --> 00:03:19.491
Let's say this is at one volt, alright?

00:03:20.529 --> 00:03:21.777
And going through our amplifier,

00:03:21.777 --> 00:03:24.777
we know that V out equals two volts,

00:03:25.896 --> 00:03:29.633
and that means that V minus
equals one-half of V out,

00:03:29.633 --> 00:03:31.550
so V minus is one volt.

00:03:32.491 --> 00:03:34.574
So this is one volt here.

00:03:37.233 --> 00:03:39.288
So let's say for the moment,

00:03:39.288 --> 00:03:40.757
that something happens to the circuit,

00:03:40.757 --> 00:03:42.874
like we heated up or something like that,

00:03:42.874 --> 00:03:47.153
and let's say the gain
goes up a little bit.

00:03:47.153 --> 00:03:48.553
Now what that means is,

00:03:48.553 --> 00:03:51.585
that this amplifier, which is amplifying

00:03:51.585 --> 00:03:53.971
this voltage difference right here,

00:03:53.971 --> 00:03:55.213
is gonna be a little higher,

00:03:55.213 --> 00:03:59.193
so the voltage here is
gonna go up a little bit.

00:03:59.193 --> 00:04:00.943
Let's use this color.

00:04:02.001 --> 00:04:04.553
It goes up a little bit,

00:04:04.553 --> 00:04:06.628
and that means that this output voltage

00:04:06.628 --> 00:04:09.046
is gonna go up a little bit.

00:04:09.046 --> 00:04:12.080
And we already decided from
looking at this voltage divider,

00:04:12.080 --> 00:04:16.161
that if this point goes up,
that this point will go up.

00:04:16.161 --> 00:04:17.375
It goes up half as much, but it,

00:04:17.375 --> 00:04:19.797
it goes in the up direction.

00:04:19.797 --> 00:04:22.041
When this voltage goes up, that means

00:04:22.041 --> 00:04:24.833
this voltage goes up, and
now we find ourselves,

00:04:24.833 --> 00:04:26.729
we're at the inverting input.

00:04:26.729 --> 00:04:29.665
We're at the inverting
input to the amplifier,

00:04:29.665 --> 00:04:30.529
and that means what?

00:04:30.529 --> 00:04:34.206
When a change at the
inverting input goes up,

00:04:34.206 --> 00:04:37.486
that means the output goes down.

00:04:37.486 --> 00:04:39.025
And that's in the opposite direction

00:04:39.025 --> 00:04:41.348
of the original change.

00:04:41.348 --> 00:04:43.465
So this is the mechanism of Feedback.

00:04:43.465 --> 00:04:44.986
A went up a little bit.

00:04:44.986 --> 00:04:47.726
We thought that V out would go up a bit,

00:04:47.726 --> 00:04:49.666
which meant this point goes up,

00:04:49.666 --> 00:04:53.657
which meant it gets fed back to the input,

00:04:53.657 --> 00:04:55.041
to the inverting input,

00:04:55.041 --> 00:04:56.499
and then it goes back down,

00:04:56.499 --> 00:04:59.305
and this balancing act
that's going on right here,

00:04:59.305 --> 00:05:03.472
that is the mechanism, that
is what we call Feedback.

00:05:04.321 --> 00:05:06.058
You get this Feedback effect,

00:05:06.058 --> 00:05:08.077
when this connection is made right here,

00:05:08.077 --> 00:05:10.465
back to the inverting
input, to the op amp.

00:05:10.465 --> 00:05:13.105
And in particular, because
it's the inverting input,

00:05:13.105 --> 00:05:15.855
this is called Negative Feedback.

00:05:23.444 --> 00:05:25.275
So this is the mechanism of Feedback,

00:05:25.275 --> 00:05:27.757
in particular, Negative Feedback and,

00:05:27.757 --> 00:05:29.329
what it does for us is,

00:05:29.329 --> 00:05:32.226
it provides us a way
to exploit and to use,

00:05:32.226 --> 00:05:35.033
this enormous gain that
these amplifiers have,

00:05:35.033 --> 00:05:39.200
to create really stable, really
nicely controlled circuits,

00:05:40.369 --> 00:05:42.939
that are controlled by the
values of the components

00:05:42.939 --> 00:05:46.022
we attached to the, to the amplifier.

00:05:47.932 --> 00:05:49.177
So that's the idea of Feedback,

00:05:49.177 --> 00:05:51.153
a really powerful idea,

00:05:51.153 --> 00:05:54.936
and really at the heart
of analog electronics.


Non-inverting op-amp circuit

WEBVTT
Kind: captions
Language: en

00:00:01.724 --> 00:00:02.909
- [Voiceover] Okay, now
we're going to work on our

00:00:02.909 --> 00:00:05.118
first Op-amp circuit.

00:00:05.118 --> 00:00:07.197
Here's what the circuit's
going to look like.

00:00:07.197 --> 00:00:11.364
Watch where it puts the plus
sign is on the top on this one.

00:00:13.825 --> 00:00:17.992
And we're going to have a
voltage source over here.

00:00:23.070 --> 00:00:26.467
This will be plus or minus V in,

00:00:26.467 --> 00:00:28.467
that's our input signal.

00:00:29.904 --> 00:00:31.590
And over on the output,

00:00:31.590 --> 00:00:33.007
we'll have V out,

00:00:34.542 --> 00:00:37.642
and it's hooked up this way.

00:00:37.642 --> 00:00:39.499
The resistor,

00:00:39.499 --> 00:00:40.890
another resistor,

00:00:40.890 --> 00:00:41.723
to ground,

00:00:42.803 --> 00:00:46.303
and this goes back to the inverting input.

00:00:47.982 --> 00:00:49.133
Now we're going to look at this circuit

00:00:49.133 --> 00:00:50.673
and see what it does.

00:00:50.673 --> 00:00:52.158
Now we know that connected up here

00:00:52.158 --> 00:00:53.588
the power supply's hooked up

00:00:53.588 --> 00:00:55.338
to these points here,

00:00:56.577 --> 00:00:59.937
and the ground symbol is zero volts.

00:00:59.937 --> 00:01:02.183
And we want to analyze this circuit.

00:01:02.183 --> 00:01:03.241
And what do we know about this?

00:01:03.241 --> 00:01:06.241
We know that V out equals some gain,

00:01:07.418 --> 00:01:09.978
I'll write the gain right there.

00:01:09.978 --> 00:01:12.894
A big, big number times V minus,

00:01:12.894 --> 00:01:15.144
sorry V plus minus V minus,

00:01:17.268 --> 00:01:18.321
and let's label that.

00:01:18.321 --> 00:01:21.016
V plus is this point right here,

00:01:21.016 --> 00:01:24.378
and V minus is this point right here.

00:01:24.378 --> 00:01:25.976
And we also know that the currents,

00:01:25.976 --> 00:01:28.893
let's call them i plus and i minus,

00:01:31.545 --> 00:01:35.980
equals zero, and that's
the currents going in here.

00:01:35.980 --> 00:01:38.916
This is i minus here, and that's i plus,

00:01:38.916 --> 00:01:41.583
and we know those are both zero.

00:01:43.615 --> 00:01:45.498
So now what I want to do it describe

00:01:45.498 --> 00:01:47.868
what's going on inside this
triangle symbol in more detail

00:01:47.868 --> 00:01:50.946
by building a circuit model.

00:01:50.946 --> 00:01:53.012
Alright, and a circuit
model for an amplifier

00:01:53.012 --> 00:01:54.773
looks like this.

00:01:54.773 --> 00:01:57.606
We have V minus here, V plus here,

00:01:58.965 --> 00:02:00.298
so this is V in,

00:02:02.651 --> 00:02:03.887
and over on this side we have an,

00:02:03.887 --> 00:02:07.594
here's a new symbol that
you haven't seen before.

00:02:07.594 --> 00:02:09.719
It's usually drawn as a diamond shape,

00:02:09.719 --> 00:02:12.136
and this is a voltage source,

00:02:13.874 --> 00:02:17.374
but it's a special kind of voltage source.

00:02:19.055 --> 00:02:21.638
It's called a voltage-dependent

00:02:30.845 --> 00:02:32.095
voltage source.

00:02:36.961 --> 00:02:38.986
And it's the same as a
regular ideal voltage source

00:02:38.986 --> 00:02:40.342
except for one thing,

00:02:40.342 --> 00:02:43.544
it says that the V, in this case V out,

00:02:43.544 --> 00:02:46.794
equals gain times V plus minus V minus.

00:02:49.861 --> 00:02:54.498
So the voltage here depends
on the voltage somewhere else,

00:02:54.498 --> 00:02:56.442
and that's what makes
it a voltage-dependent,

00:02:56.442 --> 00:02:57.719
that's what that means.

00:02:57.719 --> 00:03:00.756
So, we've just taken our
gain expression here,

00:03:00.756 --> 00:03:03.246
added, drawn circuit
diagram that represents

00:03:03.246 --> 00:03:06.496
our voltage expression for our circuit.

00:03:07.378 --> 00:03:09.464
Now, specifically over here

00:03:09.464 --> 00:03:12.157
we've drawn an open circuit on V plus,

00:03:12.157 --> 00:03:15.721
and V minus so we know that
those currents are zero.

00:03:15.721 --> 00:03:18.719
So this model, this circuit sketch

00:03:18.719 --> 00:03:22.386
represents our two
properties of our Op-amp.

00:03:25.098 --> 00:03:26.414
So I'm going to take a second here

00:03:26.414 --> 00:03:29.512
and I'm going to draw
the rest of our circuit

00:03:29.512 --> 00:03:30.930
surrounding this model,

00:03:30.930 --> 00:03:33.847
but I need a little bit more space.

00:03:36.702 --> 00:03:38.868
So let's put in the rest
of our circuit here.

00:03:38.868 --> 00:03:41.035
We had our voltage source,

00:03:44.316 --> 00:03:45.983
connected to V plus,

00:03:47.091 --> 00:03:48.424
and that's V in,

00:03:50.898 --> 00:03:52.660
and over here we had V out.

00:03:52.660 --> 00:03:56.827
Let's check, V out was
connected to two resistors,

00:04:01.678 --> 00:04:04.873
and the bottom is connected to ground,

00:04:04.873 --> 00:04:07.290
and this was connected there.

00:04:09.832 --> 00:04:11.290
So what our goal is right now,

00:04:11.290 --> 00:04:14.957
we want to find V out
as a function of V in.

00:04:19.586 --> 00:04:21.659
That's what we're shooting for.

00:04:21.659 --> 00:04:23.705
So let's see if we can do that.

00:04:23.705 --> 00:04:25.377
Let's give our resistors some names.

00:04:25.377 --> 00:04:27.523
Let's call this R1, and R2,

00:04:27.523 --> 00:04:30.100
our favorite names always,

00:04:30.100 --> 00:04:32.809
and now everything is labeled.

00:04:32.809 --> 00:04:35.547
Now and we can label this point here,

00:04:35.547 --> 00:04:38.310
and this point we can call V minus,

00:04:38.310 --> 00:04:39.143
V minus.

00:04:42.753 --> 00:04:43.826
So that's our two unknowns.

00:04:43.826 --> 00:04:46.563
Our unknowns are V not, V out,

00:04:46.563 --> 00:04:50.553
and V minus, so let's
see if we can find them.

00:04:50.553 --> 00:04:52.052
So what I'm going to do is just start

00:04:52.052 --> 00:04:55.211
writing some expressions for
things that I know are true.

00:04:55.211 --> 00:04:57.711
For example, I know that V out

00:05:00.678 --> 00:05:03.678
equals A times V plus minus V minus.

00:05:07.766 --> 00:05:11.933
Alright, that's what this
Op-amp is telling us is true.

00:05:12.870 --> 00:05:14.146
Now what else do I know?

00:05:14.146 --> 00:05:16.252
Let's look at this resistor chain here.

00:05:16.252 --> 00:05:18.054
This resistor chain actually looks

00:05:18.054 --> 00:05:20.889
a lot like a voltage divider,

00:05:20.889 --> 00:05:22.975
and it's actually a very
good voltage divider.

00:05:22.975 --> 00:05:24.494
Remember we said this current here,

00:05:24.494 --> 00:05:25.952
what is this current here?

00:05:25.952 --> 00:05:26.785
It's zero.

00:05:28.807 --> 00:05:32.169
I can use the voltage divider
expression that I know.

00:05:32.169 --> 00:05:34.175
In that case, I know that V minus,

00:05:34.175 --> 00:05:37.835
this is the voltage divider equation,

00:05:37.835 --> 00:05:38.835
equals V out

00:05:42.073 --> 00:05:42.977
times what?

00:05:42.977 --> 00:05:45.556
Times the bottom resistor

00:05:45.556 --> 00:05:47.181
remember this?

00:05:47.181 --> 00:05:48.764
R2 over R1 plus R2,

00:05:53.657 --> 00:05:55.885
so the voltage divider
expression says that

00:05:55.885 --> 00:05:58.274
when you have a stack
of resistors like this,

00:05:58.274 --> 00:06:01.190
with the voltage on the top
and ground on the bottom,

00:06:01.190 --> 00:06:05.357
this is the expression for
the voltage at the midpoint.

00:06:08.367 --> 00:06:10.303
Kay, so what I'm going to do next

00:06:10.303 --> 00:06:12.572
is I'm going to take this expression

00:06:12.572 --> 00:06:14.905
and stuff it right in there.

00:06:17.002 --> 00:06:18.169
Let's do that.

00:06:19.777 --> 00:06:21.822
See if we got enough room,

00:06:21.822 --> 00:06:23.179
okay now let's go over here.

00:06:23.179 --> 00:06:25.179
Now I can say that V out

00:06:26.622 --> 00:06:29.372
equals A times V plus minus V out

00:06:36.160 --> 00:06:38.243
times R2 over R1 plus R2,

00:06:44.381 --> 00:06:45.940
alright so far so good.

00:06:45.940 --> 00:06:48.310
Let's keep going, let's
keep working on this.

00:06:48.310 --> 00:06:51.810
V not equals A times V plus minus A V not,

00:06:59.777 --> 00:07:01.360
R2 over R1 plus R2.

00:07:05.787 --> 00:07:08.562
Alright, so now I'm going
to gather all the V not

00:07:08.562 --> 00:07:11.559
terms over on the left hand side.

00:07:11.559 --> 00:07:13.118
Let's try that.

00:07:13.118 --> 00:07:15.183
So that gives me,

00:07:15.183 --> 00:07:16.766
V not plus A V not,

00:07:22.251 --> 00:07:25.584
times R2 over R1 plus R2 and that equals

00:07:29.805 --> 00:07:31.055
A times V plus,

00:07:32.744 --> 00:07:34.098
and actually I can change that now

00:07:34.098 --> 00:07:35.840
V plus is what?

00:07:35.840 --> 00:07:37.090
V plus is V in.

00:07:43.113 --> 00:07:47.828
Okay let's keep going I
can factor out the V not.

00:07:47.828 --> 00:07:48.828
V not is one

00:07:50.906 --> 00:07:52.989
plus A R2 over R1 plus R2

00:08:00.790 --> 00:08:02.623
and that equals AV in.

00:08:04.578 --> 00:08:06.885
Alright so we're getting close,

00:08:06.885 --> 00:08:08.181
and our original goal,

00:08:08.181 --> 00:08:11.077
we want to find V out in terms of V in.

00:08:11.077 --> 00:08:14.256
So I'm going to take this
whole expression here

00:08:14.256 --> 00:08:16.160
and divide it over to the other side,

00:08:16.160 --> 00:08:18.631
so then I have just V not on this side,

00:08:18.631 --> 00:08:20.881
and V in on the other side.

00:08:22.519 --> 00:08:24.186
Make some more room.

00:08:25.739 --> 00:08:27.095
I can do that, I can say

00:08:27.095 --> 00:08:28.678
V not equals A V in

00:08:34.750 --> 00:08:37.181
divided by this big old expression,

00:08:37.181 --> 00:08:39.681
one plus A R2 over R1 plus R2.

00:08:47.448 --> 00:08:48.744
Alright so that's our answer.

00:08:48.744 --> 00:08:50.607
That's the answer.

00:08:50.607 --> 00:08:54.107
That's V out equals some function of V in.

00:08:59.153 --> 00:09:01.725
Now I want to make a really
important observation here.

00:09:01.725 --> 00:09:04.864
This is going to be a
real cool simplification.

00:09:04.864 --> 00:09:07.334
Okay, so this is the point where Op-amp

00:09:07.334 --> 00:09:09.198
theory gets really cool.

00:09:09.198 --> 00:09:10.454
Watch what happens here.

00:09:10.454 --> 00:09:12.545
We know that A is a giant number.

00:09:12.545 --> 00:09:14.179
A is something like 10 to the fifth,

00:09:14.179 --> 00:09:15.800
or 10 to the sixth,

00:09:15.800 --> 00:09:17.500
and it's whatever we have here,

00:09:17.500 --> 00:09:20.397
if our resistors are sort
of normal-sized resistors

00:09:20.397 --> 00:09:22.867
we know that a giant number
times a normal number

00:09:22.867 --> 00:09:26.898
is still going to be a very
big number compared to one.

00:09:26.898 --> 00:09:30.482
So this one is almost
insignificant in this expression

00:09:30.482 --> 00:09:32.507
down here, so what I'm going to do,

00:09:32.507 --> 00:09:33.865
bear with me,

00:09:33.865 --> 00:09:34.917
I'm going to cross it out.

00:09:34.917 --> 00:09:37.975
I'm going to say no, I
don't need that anymore.

00:09:37.975 --> 00:09:40.506
So if this, if this number
here, if A is a million,

00:09:40.506 --> 00:09:42.050
10 to the sixth,

00:09:42.050 --> 00:09:45.258
and this expression here
is something like one half

00:09:45.258 --> 00:09:48.571
then this total thing is
one half 10 to the sixth

00:09:48.571 --> 00:09:50.171
or a half a million,

00:09:50.171 --> 00:09:52.037
and that's huge compared to one.

00:09:52.037 --> 00:09:54.221
So I can pretty safely ignore the one,

00:09:54.221 --> 00:09:56.408
it's very, very small.

00:09:56.408 --> 00:09:57.846
Now when I do that,

00:09:57.846 --> 00:09:59.122
well look what happens next,

00:09:59.122 --> 00:10:02.585
now I have A top and
bottom in the expression,

00:10:02.585 --> 00:10:04.266
and I can cancel that too.

00:10:04.266 --> 00:10:06.373
So the A goes away,

00:10:06.373 --> 00:10:07.830
now this is pretty astonishing.

00:10:07.830 --> 00:10:10.017
We have this amplifier circuit

00:10:10.017 --> 00:10:12.205
and all of a sudden I
have an expression here

00:10:12.205 --> 00:10:13.561
where A doesn't appear,

00:10:13.561 --> 00:10:15.060
the gain does not appear,

00:10:15.060 --> 00:10:16.518
and what does this turn into?

00:10:16.518 --> 00:10:18.563
This is called V not

00:10:18.563 --> 00:10:20.563
equals V in, times what?

00:10:22.816 --> 00:10:24.233
Times R1 plus R2,

00:10:27.616 --> 00:10:28.783
divided by R2.

00:10:31.262 --> 00:10:34.643
So our amplifier, our
feedback circuit came down to

00:10:34.643 --> 00:10:39.139
V out is V in multiplied by
the ratio of the resistors

00:10:39.139 --> 00:10:41.690
that we added to the circuit.

00:10:41.690 --> 00:10:44.465
This is one of the really cool
properties of using Op-amps

00:10:44.465 --> 00:10:46.834
in circuits, really high-gain amplifiers.

00:10:46.834 --> 00:10:47.989
What we've done is

00:10:47.989 --> 00:10:50.114
we have chosen the gain of our circuit

00:10:50.114 --> 00:10:52.342
based on the components that we picked

00:10:52.342 --> 00:10:53.962
to add to the amplifier.

00:10:53.962 --> 00:10:56.251
It's not determined by
the gain of the amplifier

00:10:56.251 --> 00:11:00.546
as long as the amplifier
gain is really, really big.

00:11:00.546 --> 00:11:02.974
And for Op-amps, that's a good assumption,

00:11:02.974 --> 00:11:04.291
it is really big.

00:11:04.291 --> 00:11:07.451
So this expression came out
with a positive sign, right?

00:11:07.451 --> 00:11:09.637
All the R's are positive values,

00:11:09.637 --> 00:11:13.054
so this is referred to as a non-inverting

00:11:18.547 --> 00:11:20.630
Op-amp circuit amplifier.

00:11:25.048 --> 00:11:26.587
So just to do a quick example,

00:11:26.587 --> 00:11:29.017
if R1 and R2 are the same,

00:11:29.017 --> 00:11:31.062
then we end up with an
expression that looks like this

00:11:31.062 --> 00:11:35.229
V out equals R1 plus R2, R
plus R over R is equal to two

00:11:38.515 --> 00:11:41.015
so the gain is two times V in.

00:11:45.218 --> 00:11:46.373
So just to do a quick sketch

00:11:46.373 --> 00:11:50.206
just to remind ourselves
what this looks like,

00:11:51.658 --> 00:11:52.825
this was V in,

00:11:54.398 --> 00:11:55.607
and we had what out here?

00:11:55.607 --> 00:11:57.693
We had a resistor,

00:11:57.693 --> 00:12:00.360
we had a resistor to the ground,

00:12:02.493 --> 00:12:03.993
and this is V out.

00:12:08.669 --> 00:12:11.281
So this is the configuration
of a non-inverting amplifier

00:12:11.281 --> 00:12:13.813
built with an Op-amp,

00:12:13.813 --> 00:12:17.057
the two resistors in this
voltage divider string

00:12:17.057 --> 00:12:19.724
connected to the negative input.

00:12:21.346 --> 00:12:23.756
So that's what non-inverting
Op-amp circuit looks like,

00:12:23.756 --> 00:12:25.662
and it's going to be one of the familiar

00:12:25.662 --> 00:12:27.644
patterns that you see over and over again

00:12:27.644 --> 00:12:31.394
as you read schematics and
you design your own circuits.


What is an operational amplifier?

WEBVTT
Kind: captions
Language: en

00:00:00.689 --> 00:00:01.522
- [Voiceover] We're gonna talk about

00:00:01.522 --> 00:00:05.190
the operational amplifier,
or op-amp for short,

00:00:05.190 --> 00:00:09.332
and this is the workhorse
of all analog electronics.

00:00:09.332 --> 00:00:12.554
The operational amplifier,
it's a type of amplifier.

00:00:12.554 --> 00:00:14.554
An amplifier is anything

00:00:15.849 --> 00:00:18.716
that you put an electronic signal in,

00:00:18.716 --> 00:00:21.820
and you get out a larger
version of the signal.

00:00:21.820 --> 00:00:25.680
So this will be an amplifier
with some sort of gain,

00:00:25.680 --> 00:00:27.750
and if I put a signal x in here,

00:00:27.750 --> 00:00:30.128
usually a voltage or a current,

00:00:30.128 --> 00:00:34.350
then the signal that comes
out here is A times x,

00:00:34.350 --> 00:00:36.888
and that's what we mean by amplification.

00:00:36.888 --> 00:00:39.476
And a signal, that I've shown here as x,

00:00:39.476 --> 00:00:40.842
is anything that we're interested in,

00:00:40.842 --> 00:00:42.917
it could be a voltage or a current,

00:00:42.917 --> 00:00:44.518
and when we put it through an amplifier,

00:00:44.518 --> 00:00:46.156
we get a larger version of it.

00:00:46.156 --> 00:00:50.067
So this is a really common
activity in electronic design.

00:00:50.067 --> 00:00:51.325
Now, when we talk specifically about

00:00:51.325 --> 00:00:54.592
an operational amplifier,
the symbol for that,

00:00:54.592 --> 00:00:58.858
we use for an operational
amplifier, is a triangle.

00:00:58.858 --> 00:01:00.358
It has two inputs,

00:01:03.406 --> 00:01:06.282
one is the plus input,
one is the minus input,

00:01:06.282 --> 00:01:08.601
and it has an output,

00:01:08.601 --> 00:01:10.935
and it also has two power supplies to it.

00:01:10.935 --> 00:01:15.001
There's some sort of plus
voltage that goes into it,

00:01:15.001 --> 00:01:17.584
and some sort of minus voltage.

00:01:19.068 --> 00:01:22.555
So this is the abstract
symbol for an op-amp.

00:01:22.555 --> 00:01:23.883
So when we say the word op-amp,

00:01:23.883 --> 00:01:26.876
we have some specific properties in mind.

00:01:26.876 --> 00:01:29.376
One is op-amps have high gain.

00:01:32.760 --> 00:01:37.420
So in this case, the gain,
usually with a symbol A,

00:01:37.420 --> 00:01:41.587
is something like 10 to the
fifth to 10 to the sixth.

00:01:42.449 --> 00:01:44.116
Really, really high.

00:01:45.358 --> 00:01:47.994
Another thing we think about
when we talk about op-amps

00:01:47.994 --> 00:01:51.577
is that they're used
for feedback circuits.

00:01:54.196 --> 00:01:56.520
And we'll talk about feedback
in the next couple of videos

00:01:56.520 --> 00:01:57.779
and what that means,

00:01:57.779 --> 00:02:01.946
but that's the application
that we use op-amps for.

00:02:03.101 --> 00:02:05.371
And the third that's
distinctive about op-amps

00:02:05.371 --> 00:02:07.912
is that they have this kind of input.

00:02:07.912 --> 00:02:09.349
This kind of input here is referred to

00:02:09.349 --> 00:02:10.766
as a differential

00:02:15.489 --> 00:02:16.322
input.

00:02:20.647 --> 00:02:22.941
So an op-amp usually
has differential inputs,

00:02:22.941 --> 00:02:25.186
and that's as opposed to something we call

00:02:25.186 --> 00:02:29.353
a single-ended input, which
will be just one wire.

00:02:31.045 --> 00:02:33.900
And what a differential input means,

00:02:33.900 --> 00:02:36.155
it says, we have, we can
label the voltages here.

00:02:36.155 --> 00:02:39.860
We'll call this v out,
we'll call this v plus,

00:02:39.860 --> 00:02:42.448
and we'll call this input v minus.

00:02:42.448 --> 00:02:45.698
And differential input means that v out

00:02:46.641 --> 00:02:48.391
equals the gain times

00:02:50.614 --> 00:02:52.364
v plus minus v minus.

00:02:55.829 --> 00:02:59.996
So the output signal here is
proportional to the difference

00:03:01.000 --> 00:03:04.833
in the voltage between
these two signals here.

00:03:06.520 --> 00:03:08.785
So I wanna make a plot of
this equation right here,

00:03:08.785 --> 00:03:12.868
just so we get a good idea
of what it looks like.

00:03:14.483 --> 00:03:17.566
And the axis here are v in and v out,

00:03:21.377 --> 00:03:25.544
where v in specifically
equals v plus minus v minus.

00:03:29.367 --> 00:03:31.852
v plus minus v minus,
that's the input signal

00:03:31.852 --> 00:03:34.971
to the op-amp, and we're gonna
apply the gain factor to it

00:03:34.971 --> 00:03:36.679
to get v out.

00:03:36.679 --> 00:03:38.975
So that'll look like
this, something like this,

00:03:38.975 --> 00:03:41.927
it's gonna be a very steep line,

00:03:41.927 --> 00:03:44.594
and the slope of that line,

00:03:44.594 --> 00:03:46.927
the slope of this line is A.

00:03:47.990 --> 00:03:49.789
So the slope is going
to be 10 to the fifth

00:03:49.789 --> 00:03:51.249
or 10 to the sixth, something like that,

00:03:51.249 --> 00:03:52.916
very, very vertical.

00:03:54.021 --> 00:03:56.508
Now one of the properties
of this is that v out

00:03:56.508 --> 00:04:01.396
cannot go above or below
its power supply voltages.

00:04:01.396 --> 00:04:04.347
So on this plot here,
that's called saturation.

00:04:04.347 --> 00:04:08.339
If v out gets up to V
plus, we say it saturates.

00:04:08.339 --> 00:04:11.415
It looks like this, it
goes flat basically here

00:04:11.415 --> 00:04:15.120
and here, where this voltage value here

00:04:15.120 --> 00:04:17.287
is minus the power supply,

00:04:18.717 --> 00:04:23.284
and this voltage right here
is the positive power supply.

00:04:23.284 --> 00:04:25.814
But over this range here,
over this range in here

00:04:25.814 --> 00:04:29.882
between those two points, is quite linear.

00:04:29.882 --> 00:04:32.419
It goes through zero,

00:04:32.419 --> 00:04:36.838
and this is where we
use it most of the time.

00:04:36.838 --> 00:04:39.122
So now I wanna talk a
little bit more detail about

00:04:39.122 --> 00:04:42.091
what this symbol means
here and what's inside it,

00:04:42.091 --> 00:04:46.091
and how it's actually
connected up in a circuit.

00:04:58.066 --> 00:05:00.922
So we talked about the
voltage behavior of an op-amp.

00:05:00.922 --> 00:05:03.839
This is v plus, v minus, and v out.

00:05:06.242 --> 00:05:07.729
There's one thing more
that we need to know,

00:05:07.729 --> 00:05:09.016
and that is the current,

00:05:09.016 --> 00:05:10.568
this current right here

00:05:10.568 --> 00:05:12.907
and this current right here,

00:05:12.907 --> 00:05:16.240
for an op-amp, an ideal op-amp, is zero,

00:05:17.122 --> 00:05:18.749
no current flows in here.

00:05:18.749 --> 00:05:21.727
So this op-amp is just sensing
the voltages at these points

00:05:21.727 --> 00:05:23.727
but no current flows in.

00:05:24.916 --> 00:05:27.763
So this is the second key
property of an op-amp.

00:05:27.763 --> 00:05:30.836
The first one is the voltage behavior,

00:05:30.836 --> 00:05:33.086
v out equals the gain times

00:05:35.428 --> 00:05:37.178
v plus minus v minus.

00:05:38.761 --> 00:05:42.928
Another way we can write this
is v out equals A times v in,

00:05:44.677 --> 00:05:48.844
where v in, of course,
equals v plus minus v minus.

00:05:52.524 --> 00:05:55.695
So these are the two electrical properties

00:05:55.695 --> 00:05:58.090
that are gonna allow us
to analyze these circuits.

00:05:58.090 --> 00:06:02.027
Analyzing these circuits is
actually gonna be pretty simple.

00:06:02.027 --> 00:06:04.156
So you're probably wondering,
what is inside here,

00:06:04.156 --> 00:06:06.051
what's going on inside here?

00:06:06.051 --> 00:06:07.448
So what's inside here is,

00:06:07.448 --> 00:06:12.074
somewhere between 20 and 50 or
so transistors and resistors,

00:06:12.074 --> 00:06:13.950
sometimes capacitors.

00:06:13.950 --> 00:06:16.816
These are really complex
designs, and for right now,

00:06:16.816 --> 00:06:18.814
if we just concentrate
on the two properties

00:06:18.814 --> 00:06:22.142
that we have here, we'll be
able to use these circuits

00:06:22.142 --> 00:06:25.512
even without understanding
exactly what's inside.

00:06:25.512 --> 00:06:27.743
Suffice to say it's a
differential amplifier

00:06:27.743 --> 00:06:29.002
with really high gain,

00:06:29.002 --> 00:06:30.158
and with just that knowledge,

00:06:30.158 --> 00:06:32.971
we can work out how these circuits work.

00:06:32.971 --> 00:06:35.853
So let me do a couple of more details on

00:06:35.853 --> 00:06:38.106
how this thing is actually hooked up.

00:06:38.106 --> 00:06:41.933
So we have a plus terminal
and a minus terminal.

00:06:41.933 --> 00:06:42.919
There's more terminals on this,

00:06:42.919 --> 00:06:45.760
there's a power supply like this,

00:06:45.760 --> 00:06:47.260
that's plus big V,

00:06:48.450 --> 00:06:49.652
and there has to be a minus supply,

00:06:49.652 --> 00:06:53.133
typically a minus supply, minus V,

00:06:53.133 --> 00:06:54.558
and there'll be a ground pin,

00:06:54.558 --> 00:06:57.391
there'll be a ground node on here,

00:06:58.836 --> 00:07:00.006
like that.

00:07:00.006 --> 00:07:02.506
And when this is used in a circuit,

00:07:02.506 --> 00:07:04.917
there'll be, over to the side,

00:07:04.917 --> 00:07:07.261
there'll be two power supplies,

00:07:07.261 --> 00:07:08.762
and this will be 12 volts,

00:07:08.762 --> 00:07:12.286
12 volts is a real typical value,

00:07:12.286 --> 00:07:14.619
and there'll be another one,

00:07:17.590 --> 00:07:20.085
and this is a plus 12
volts supply as well,

00:07:20.085 --> 00:07:22.435
and they'll be connected together,

00:07:22.435 --> 00:07:24.432
and this node right between them

00:07:24.432 --> 00:07:26.618
will be the ground node,

00:07:26.618 --> 00:07:28.636
that's the voltage reference,

00:07:28.636 --> 00:07:31.369
and these two guys will
be hooked up like that,

00:07:31.369 --> 00:07:34.231
the two power supplies
will be hooked up this way.

00:07:34.231 --> 00:07:38.398
So, with respect to ground,
this node is at minus 12 volts

00:07:39.413 --> 00:07:43.246
and this node from ground
is at plus 12 volts.

00:07:44.530 --> 00:07:46.446
And ground is right in the middle.

00:07:46.446 --> 00:07:50.450
And when we measure v
out, we'll measure it

00:07:50.450 --> 00:07:53.200
with respect to this ground node.

00:07:56.810 --> 00:07:59.442
So this is the voltage where we measure

00:07:59.442 --> 00:08:02.109
plus or minus v out right there.

00:08:03.056 --> 00:08:04.627
And what we're gonna do
is we're gonna assume

00:08:04.627 --> 00:08:06.846
that all of this stuff
is always hooked up,

00:08:06.846 --> 00:08:10.846
and we're just gonna use
an even simpler symbol,

00:08:11.853 --> 00:08:15.255
just the three terminals like that,

00:08:15.255 --> 00:08:17.277
and you'll know that all of
the rest of the power supply's

00:08:17.277 --> 00:08:19.059
hooked up that way.

00:08:19.059 --> 00:08:20.619
And the thing to keep in mind is

00:08:20.619 --> 00:08:22.461
there's a large minus voltage,

00:08:22.461 --> 00:08:24.134
there's a large plus voltage,

00:08:24.134 --> 00:08:26.692
and the ground level, the ground node,

00:08:26.692 --> 00:08:28.100
is right in between,

00:08:28.100 --> 00:08:30.774
so positive voltage is high on the page,

00:08:30.774 --> 00:08:33.804
negative voltage is low on the page,

00:08:33.804 --> 00:08:38.085
and v out can go both positive
and negative around ground.

00:08:38.085 --> 00:08:42.252
So that's your voltage
framework to keep in your head.

00:08:45.186 --> 00:08:47.459
And the op-amp that we've been looking at

00:08:47.459 --> 00:08:49.376
has a symbol like this,

00:08:53.280 --> 00:08:56.947
and we know that v out
equals some huge gain

00:08:58.291 --> 00:09:00.541
times v plus minus v minus,

00:09:02.434 --> 00:09:04.546
so this is a differential input,

00:09:04.546 --> 00:09:06.177
here's v plus,

00:09:06.177 --> 00:09:08.175
here's v minus.

00:09:08.175 --> 00:09:09.867
One way I think about this

00:09:09.867 --> 00:09:14.573
is to look at the way a
change in voltage on the input

00:09:14.573 --> 00:09:16.244
modifies the output.

00:09:16.244 --> 00:09:18.244
Let me label v out here.

00:09:23.167 --> 00:09:26.410
So, if there's a change on the input,

00:09:26.410 --> 00:09:28.493
say v plus goes this way,

00:09:29.484 --> 00:09:31.039
because it's a plus sign,

00:09:31.039 --> 00:09:34.039
that means that v out goes this way.

00:09:35.348 --> 00:09:38.818
Now, if I change it over to v minus,

00:09:38.818 --> 00:09:41.588
v minus is on the negative input.

00:09:41.588 --> 00:09:43.171
If v minus goes up,

00:09:44.658 --> 00:09:46.408
then v out goes down,

00:09:47.332 --> 00:09:50.268
so that's the inverting, it's
called the inverting input,

00:09:50.268 --> 00:09:53.321
and this is called the
non-inverting input.

00:09:53.321 --> 00:09:56.654
So on a non-inverting input, up goes up,

00:09:57.555 --> 00:10:00.048
and on the inverting input,

00:10:00.048 --> 00:10:02.243
if you go up on the inverting input,

00:10:02.243 --> 00:10:06.160
you go in the opposite
direction on the output.

00:10:07.750 --> 00:10:11.678
Let's say that the positive
input, the non-inverting input,

00:10:11.678 --> 00:10:15.278
went down this time, and that means what?

00:10:15.278 --> 00:10:18.923
That means that v out will go down,

00:10:18.923 --> 00:10:21.340
and just do the same thing over here,

00:10:21.340 --> 00:10:22.902
gonna run out of colors.

00:10:22.902 --> 00:10:25.755
Let's do the same thing
for the inverting input.

00:10:25.755 --> 00:10:30.564
If the inverting input goes
down, what does the output do?

00:10:30.564 --> 00:10:34.397
It goes up, it goes in
the opposite direction.

00:10:36.085 --> 00:10:37.394
That's the way to think about this symbol

00:10:37.394 --> 00:10:39.479
when you see it on a schematic page,

00:10:39.479 --> 00:10:43.660
is how do these signals
translate through the device?

00:10:43.660 --> 00:10:47.517
Positive, non-inverting signals
go in the same direction,

00:10:47.517 --> 00:10:51.991
inverting signals go in
the opposite directions.

00:10:51.991 --> 00:10:54.389
Okay, and here's one final
trick I wanna share with you,

00:10:54.389 --> 00:10:56.993
something to be aware of.

00:10:56.993 --> 00:11:00.090
You're gonna see a symbol
like this on a page,

00:11:00.090 --> 00:11:01.877
the same op-amp.

00:11:01.877 --> 00:11:05.867
It's the same op-amp, but
it's written on the page

00:11:05.867 --> 00:11:09.051
with the negative, with the
inverting symbol on the top

00:11:09.051 --> 00:11:11.588
and the non-inverting
symbol on the bottom.

00:11:11.588 --> 00:11:15.930
So, as you look at a schematic
that has an op-amp in it,

00:11:15.930 --> 00:11:18.446
one of the first things you
wanna do is just glance and see

00:11:18.446 --> 00:11:21.716
what order these symbols are in.

00:11:21.716 --> 00:11:22.931
Does it look like that

00:11:22.931 --> 00:11:24.293
or does it look like that?

00:11:24.293 --> 00:11:26.915
And keep that in mind as
you're reading the circuit

00:11:26.915 --> 00:11:29.788
and trying to understand what it does.

00:11:29.788 --> 00:11:33.042
Okay, let's move on and build
something with our op-amp.