[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.52,Default,,0000,0000,0000,,>> This amplifier circuit is referred to as a difference amplifier.
Dialogue: 0,0:00:04.52,0:00:07.98,Default,,0000,0000,0000,,As the name suggests, we're going to see that at the output,
Dialogue: 0,0:00:07.98,0:00:15.57,Default,,0000,0000,0000,,we're going to find the difference or the subtraction of V2 and V1 or V2 minus V1.
Dialogue: 0,0:00:15.57,0:00:17.40,Default,,0000,0000,0000,,Before we get started on this circuit,
Dialogue: 0,0:00:17.40,0:00:18.81,Default,,0000,0000,0000,,and before you analyze this circuit,
Dialogue: 0,0:00:18.81,0:00:21.39,Default,,0000,0000,0000,,let's just make a few observations.
Dialogue: 0,0:00:21.39,0:00:25.100,Default,,0000,0000,0000,,First of all, in the circuit that we've analyzed involving Op-amps until now,
Dialogue: 0,0:00:25.100,0:00:29.55,Default,,0000,0000,0000,,the sources have been on either the inverting terminal,
Dialogue: 0,0:00:29.55,0:00:31.62,Default,,0000,0000,0000,,or the non-inverting terminal.
Dialogue: 0,0:00:31.62,0:00:37.39,Default,,0000,0000,0000,,This configuration has sources V1 connected to the inverting terminal and V2
Dialogue: 0,0:00:37.39,0:00:40.52,Default,,0000,0000,0000,,connected to the non-inverting terminal albeit through
Dialogue: 0,0:00:40.52,0:00:44.93,Default,,0000,0000,0000,,this resistive network before it gets to the non-inverting terminal.
Dialogue: 0,0:00:44.93,0:00:49.47,Default,,0000,0000,0000,,That points out the fact that VP,
Dialogue: 0,0:00:49.47,0:00:53.30,Default,,0000,0000,0000,,the voltage at the inverting terminal is not exactly V2.
Dialogue: 0,0:00:53.30,0:00:56.32,Default,,0000,0000,0000,,So, what effect is this resistive network have?
Dialogue: 0,0:00:56.32,0:01:01.34,Default,,0000,0000,0000,,Well, we know that the current going into the input of the Op-amp is zero.
Dialogue: 0,0:01:01.34,0:01:07.66,Default,,0000,0000,0000,,Therefore, any current coming from V2 going through R3 will also go through R4.
Dialogue: 0,0:01:07.66,0:01:09.78,Default,,0000,0000,0000,,R3 and R4 are in series with each other,
Dialogue: 0,0:01:09.78,0:01:12.46,Default,,0000,0000,0000,,we have a voltage divider taking place here.
Dialogue: 0,0:01:12.46,0:01:19.85,Default,,0000,0000,0000,,Thus, the voltage at the non-inverting terminal is the voltage across to R4.
Dialogue: 0,0:01:19.85,0:01:24.52,Default,,0000,0000,0000,,It's not exactly V2 but a subdivided portion of V2.
Dialogue: 0,0:01:24.52,0:01:26.24,Default,,0000,0000,0000,,Lets just go ahead and make that observation,
Dialogue: 0,0:01:26.24,0:01:27.45,Default,,0000,0000,0000,,we'll do it up here because we need to keep room,
Dialogue: 0,0:01:27.45,0:01:29.48,Default,,0000,0000,0000,,this is a fairly long derivation.
Dialogue: 0,0:01:29.48,0:01:31.80,Default,,0000,0000,0000,,Let's just say that V sub p,
Dialogue: 0,0:01:31.80,0:01:34.51,Default,,0000,0000,0000,,the voltage of the non-inverting terminal is going to equal
Dialogue: 0,0:01:34.51,0:01:43.28,Default,,0000,0000,0000,,V2 times R4 over R3 plus R4.
Dialogue: 0,0:01:43.28,0:01:48.58,Default,,0000,0000,0000,,So, V sub p is just a little less than V2.
Dialogue: 0,0:01:48.58,0:01:52.82,Default,,0000,0000,0000,,Now keeping in mind, we have the virtual short between the terminals,
Dialogue: 0,0:01:52.82,0:01:59.42,Default,,0000,0000,0000,,that then means that V sub n is equal to V sub p which is equal to that.
Dialogue: 0,0:01:59.42,0:02:02.21,Default,,0000,0000,0000,,This should give us a little bit of flavor of what's going on here.
Dialogue: 0,0:02:02.21,0:02:04.43,Default,,0000,0000,0000,,We know that sources connected to
Dialogue: 0,0:02:04.43,0:02:09.17,Default,,0000,0000,0000,,the non-inverting terminal in our non-inverting amplifier configuration had a gain
Dialogue: 0,0:02:09.17,0:02:11.01,Default,,0000,0000,0000,,that is slightly larger than
Dialogue: 0,0:02:11.01,0:02:16.55,Default,,0000,0000,0000,,the gain experienced by sources connected to the inverting terminal.
Dialogue: 0,0:02:16.55,0:02:22.22,Default,,0000,0000,0000,,The non-inverting gain was one plus R2 over R1,
Dialogue: 0,0:02:22.22,0:02:25.37,Default,,0000,0000,0000,,whereas the inverting gain was negative of course,
Dialogue: 0,0:02:25.37,0:02:27.44,Default,,0000,0000,0000,,just R2 over R1.
Dialogue: 0,0:02:27.44,0:02:32.15,Default,,0000,0000,0000,,So, this voltage divider circuit is intended to
Dialogue: 0,0:02:32.15,0:02:37.93,Default,,0000,0000,0000,,reduce V2 down just a little bit so that at the output,
Dialogue: 0,0:02:37.93,0:02:42.52,Default,,0000,0000,0000,,V2 and V1 will experience the same gain,
Dialogue: 0,0:02:42.52,0:02:47.80,Default,,0000,0000,0000,,and this circuit then we'll implement a true subtraction.
Dialogue: 0,0:02:47.80,0:02:53.76,Default,,0000,0000,0000,,All right. So, with that,
Dialogue: 0,0:02:53.76,0:02:56.77,Default,,0000,0000,0000,,let's go ahead and start our analysis of this by doing
Dialogue: 0,0:02:56.77,0:03:00.76,Default,,0000,0000,0000,,exactly what we've done in the other circuits involving amplifiers,
Dialogue: 0,0:03:00.76,0:03:05.92,Default,,0000,0000,0000,,and that is write a node equation summing the currents leaving the inverting terminal.
Dialogue: 0,0:03:05.92,0:03:09.94,Default,,0000,0000,0000,,Again, we'll do it in terms of V sub n until we get things simplify then will substitute
Dialogue: 0,0:03:09.94,0:03:14.72,Default,,0000,0000,0000,,V sub n for V sub p which is going to be V2 times R4 over R3 plus R4.
Dialogue: 0,0:03:14.72,0:03:17.66,Default,,0000,0000,0000,,So, KCL at the node.
Dialogue: 0,0:03:17.66,0:03:19.98,Default,,0000,0000,0000,,We got the current leaving the node going this way.
Dialogue: 0,0:03:19.98,0:03:25.52,Default,,0000,0000,0000,,That's going to be V sub n minus V1 divided by R1,
Dialogue: 0,0:03:25.52,0:03:31.18,Default,,0000,0000,0000,,plus the current going this way through the feedback loop is going to be,
Dialogue: 0,0:03:31.18,0:03:33.06,Default,,0000,0000,0000,,and we're adding that in,
Dialogue: 0,0:03:33.06,0:03:38.04,Default,,0000,0000,0000,,V sub n minus Vout divided by R2.
Dialogue: 0,0:03:38.04,0:03:39.98,Default,,0000,0000,0000,,There's no current going into the inverting terminal,
Dialogue: 0,0:03:39.98,0:03:44.42,Default,,0000,0000,0000,,so we only have those two terms that must add to give us zero.
Dialogue: 0,0:03:44.42,0:03:47.27,Default,,0000,0000,0000,,Now, combining terms and factoring out the V sub n,
Dialogue: 0,0:03:47.27,0:03:56.30,Default,,0000,0000,0000,,then we've got V sub n times one over R1 plus one over R2,
Dialogue: 0,0:03:56.30,0:04:06.88,Default,,0000,0000,0000,,and then we have minus V1 over R1.
Dialogue: 0,0:04:06.88,0:04:09.66,Default,,0000,0000,0000,,We still have this minus Vout over R2,
Dialogue: 0,0:04:09.66,0:04:16.32,Default,,0000,0000,0000,,lets take it to the other side as a positive, Vout over R2.
Dialogue: 0,0:04:16.32,0:04:19.07,Default,,0000,0000,0000,,Now, let's get a common denominator here.
Dialogue: 0,0:04:19.07,0:04:28.77,Default,,0000,0000,0000,,So, we have V sub n times R1 plus R2 over R1 times R2,
Dialogue: 0,0:04:28.77,0:04:35.86,Default,,0000,0000,0000,,minus V1 over R1 equals Vout over R2.
Dialogue: 0,0:04:36.59,0:04:42.09,Default,,0000,0000,0000,,Now, we're going to multiply both sides of the equation by R2,
Dialogue: 0,0:04:42.09,0:04:46.13,Default,,0000,0000,0000,,to solve for Vout and we'll reverse the order of the equation.
Dialogue: 0,0:04:46.13,0:04:50.66,Default,,0000,0000,0000,,We're left with then Vout is equal to,
Dialogue: 0,0:04:50.66,0:04:53.65,Default,,0000,0000,0000,,multiplying both sides by R2,
Dialogue: 0,0:04:53.65,0:04:57.03,Default,,0000,0000,0000,,we have an R2 here that's kind of cancel that R2.
Dialogue: 0,0:04:57.03,0:05:07.13,Default,,0000,0000,0000,,So, we're left with then V sub n times R1 plus R2 over R1,
Dialogue: 0,0:05:07.13,0:05:17.64,Default,,0000,0000,0000,,minus V1 times R2 over R1.
Dialogue: 0,0:05:18.29,0:05:20.79,Default,,0000,0000,0000,,Okay, so we've got the two terms.
Dialogue: 0,0:05:20.79,0:05:23.86,Default,,0000,0000,0000,,We have the inverting term here,
Dialogue: 0,0:05:23.86,0:05:26.22,Default,,0000,0000,0000,,V1 times R2 over R1.
Dialogue: 0,0:05:26.22,0:05:28.10,Default,,0000,0000,0000,,We have V sub n,
Dialogue: 0,0:05:28.10,0:05:34.38,Default,,0000,0000,0000,,now we need to replace V sub n with V2 over
Dialogue: 0,0:05:34.38,0:05:41.48,Default,,0000,0000,0000,,R4 over R3 plus R4 so that we can see the dependency of the output on V2.
Dialogue: 0,0:05:41.48,0:05:43.32,Default,,0000,0000,0000,,Let's do that on up here.
Dialogue: 0,0:05:43.32,0:05:47.70,Default,,0000,0000,0000,,We've got the Vout is equal to V sub n,
Dialogue: 0,0:05:47.70,0:05:56.16,Default,,0000,0000,0000,,then V sub n is equal to V2 times R4 over R3 plus R4.
Dialogue: 0,0:05:56.16,0:06:05.36,Default,,0000,0000,0000,,So that's V sub n times R1 plus R2 over R1.
Dialogue: 0,0:06:05.36,0:06:14.88,Default,,0000,0000,0000,,That's our first term, minus V1 times R2 over R1.
Dialogue: 0,0:06:14.88,0:06:18.06,Default,,0000,0000,0000,,Thus, we see that we have different gain terms.
Dialogue: 0,0:06:18.06,0:06:21.87,Default,,0000,0000,0000,,We've got this multiplying effect here,
Dialogue: 0,0:06:21.97,0:06:26.75,Default,,0000,0000,0000,,minus V1 times R2 over R1.
Dialogue: 0,0:06:26.75,0:06:29.29,Default,,0000,0000,0000,,We've got a couple more steps to take until we get there,
Dialogue: 0,0:06:29.29,0:06:31.62,Default,,0000,0000,0000,,but let's be explicit and show all of our work here.
Dialogue: 0,0:06:31.62,0:06:37.50,Default,,0000,0000,0000,,First of all, let's notice that this term here R1 plus R2 over R1,
Dialogue: 0,0:06:37.50,0:06:40.03,Default,,0000,0000,0000,,R1 is a common denominator to both of them.
Dialogue: 0,0:06:40.03,0:06:42.39,Default,,0000,0000,0000,,That then gives us,
Dialogue: 0,0:06:42.39,0:06:44.27,Default,,0000,0000,0000,,if we identify that as such,
Dialogue: 0,0:06:44.27,0:06:50.84,Default,,0000,0000,0000,,let's just write this then as R4 over R3 plus R4,
Dialogue: 0,0:06:50.84,0:06:53.68,Default,,0000,0000,0000,,times R1 over R1.
Dialogue: 0,0:06:53.68,0:07:00.08,Default,,0000,0000,0000,,That's just one plus R2 over R1,
Dialogue: 0,0:07:00.08,0:07:06.07,Default,,0000,0000,0000,,minus V1 times R2 over R1.
Dialogue: 0,0:07:09.92,0:07:14.30,Default,,0000,0000,0000,,We see that this term right here really is just
Dialogue: 0,0:07:14.30,0:07:18.30,Default,,0000,0000,0000,,that the gain term for the non-inverting amplifier,
Dialogue: 0,0:07:18.30,0:07:20.75,Default,,0000,0000,0000,,there's the gain term for the inverting amplifier.
Dialogue: 0,0:07:20.75,0:07:25.94,Default,,0000,0000,0000,,Here's this voltage divider term which is taking V2 and subdividing it just a little bit.
Dialogue: 0,0:07:25.94,0:07:31.04,Default,,0000,0000,0000,,Now, what we want to do is determine the relationships between R1, R2, R3,
Dialogue: 0,0:07:31.04,0:07:36.49,Default,,0000,0000,0000,,and R4 that will make it so that this gain term right here,
Dialogue: 0,0:07:36.49,0:07:41.21,Default,,0000,0000,0000,,multiplying V2, is the same as the gain term
Dialogue: 0,0:07:41.21,0:07:46.50,Default,,0000,0000,0000,,multiplying V1 with the obvious exception that V1 it's got the minus sign on it.
Dialogue: 0,0:07:46.50,0:07:49.86,Default,,0000,0000,0000,,All right. In order to do that,
Dialogue: 0,0:07:49.86,0:07:57.42,Default,,0000,0000,0000,,we're going to factor in R3 out of the denominator of this term right here.
Dialogue: 0,0:07:58.18,0:08:01.80,Default,,0000,0000,0000,,So, we have then equals
Dialogue: 0,0:08:01.80,0:08:07.04,Default,,0000,0000,0000,,V2 factoring in R3 out and I'm also going to bring the R4 out in front.
Dialogue: 0,0:08:07.04,0:08:09.48,Default,,0000,0000,0000,,So, we'll have, let's see,
Dialogue: 0,0:08:09.48,0:08:15.74,Default,,0000,0000,0000,,R4 over R3 times one
Dialogue: 0,0:08:15.74,0:08:19.86,Default,,0000,0000,0000,,over factor in the R3 that leaves
Dialogue: 0,0:08:19.86,0:08:24.07,Default,,0000,0000,0000,,as a one plus factoring in R4 out there it leads me to R3,
Dialogue: 0,0:08:24.07,0:08:27.74,Default,,0000,0000,0000,,refactoring R3 out of this term here leaves me
Dialogue: 0,0:08:27.74,0:08:32.47,Default,,0000,0000,0000,,an R3 in the denominator or I have a R4 over
Dialogue: 0,0:08:32.47,0:08:38.86,Default,,0000,0000,0000,,R3 times one plus R2 over
Dialogue: 0,0:08:38.86,0:08:47.31,Default,,0000,0000,0000,,R1 minus V1 times R2 over R1.
Dialogue: 0,0:08:47.31,0:08:52.64,Default,,0000,0000,0000,,Kind of a mess but we've got this V2 times R4 over R3 then
Dialogue: 0,0:08:52.64,0:08:58.36,Default,,0000,0000,0000,,multiplying this one divided by one plus R4 over R3.
Dialogue: 0,0:08:58.36,0:09:02.84,Default,,0000,0000,0000,,Then multiplying this one plus R2 over R1.
Dialogue: 0,0:09:02.84,0:09:05.76,Default,,0000,0000,0000,,Just a second, I'm going to ask you to stop the video.
Dialogue: 0,0:09:05.76,0:09:07.44,Default,,0000,0000,0000,,Let me observe first of all,
Dialogue: 0,0:09:07.44,0:09:13.04,Default,,0000,0000,0000,,that this gain term here will equal this gain term here,
Dialogue: 0,0:09:13.04,0:09:21.93,Default,,0000,0000,0000,,if the ratio of R4 over R3 equals the ratio of R2 over R1.
Dialogue: 0,0:09:22.16,0:09:29.70,Default,,0000,0000,0000,,If the ratio R4 over R3 equals R2 over R1,
Dialogue: 0,0:09:29.70,0:09:33.56,Default,,0000,0000,0000,,then this term here in the denominator is the same as
Dialogue: 0,0:09:33.56,0:09:38.53,Default,,0000,0000,0000,,this term here in the numerator and they will cancel.
Dialogue: 0,0:09:39.50,0:09:45.71,Default,,0000,0000,0000,,We're left with simply then Vout is equal
Dialogue: 0,0:09:45.71,0:09:52.40,Default,,0000,0000,0000,,to V2 times again R4 over R3 equals R2 over R1.
Dialogue: 0,0:09:52.40,0:09:57.20,Default,,0000,0000,0000,,So, let's replace it here also, R2 over R1,
Dialogue: 0,0:09:57.20,0:10:04.48,Default,,0000,0000,0000,,minus V1 times R2 over R1.
Dialogue: 0,0:10:06.29,0:10:09.75,Default,,0000,0000,0000,,Now, if we factor out this R2 over R1,
Dialogue: 0,0:10:09.75,0:10:12.94,Default,,0000,0000,0000,,we get then and this is where we've been trying to get this whole way.
Dialogue: 0,0:10:12.94,0:10:23.04,Default,,0000,0000,0000,,Vout then is equal to R2 over R1 times V2 minus V1.
Dialogue: 0,0:10:23.04,0:10:27.70,Default,,0000,0000,0000,,From here, you can see then why this is called a difference amplifier.
Dialogue: 0,0:10:27.70,0:10:31.32,Default,,0000,0000,0000,,The output voltage is equal to a scaled version of
Dialogue: 0,0:10:31.32,0:10:37.36,Default,,0000,0000,0000,,this source voltage minus this source voltage.
Dialogue: 0,0:10:37.36,0:10:41.66,Default,,0000,0000,0000,,The usefulness of this amplifier configuration can't be overstated.
Dialogue: 0,0:10:41.66,0:10:44.27,Default,,0000,0000,0000,,What this allows us to do and it's sometimes
Dialogue: 0,0:10:44.27,0:10:47.48,Default,,0000,0000,0000,,also referred to as a differential amplifier.
Dialogue: 0,0:10:47.48,0:10:52.78,Default,,0000,0000,0000,,It allows or this circuit will sample voltages at
Dialogue: 0,0:10:52.78,0:10:58.36,Default,,0000,0000,0000,,two different points in a circuit and subtract one from the other.
Dialogue: 0,0:10:58.36,0:11:04.83,Default,,0000,0000,0000,,In control circuits, this comparison is used to or they'll
Dialogue: 0,0:11:04.83,0:11:11.46,Default,,0000,0000,0000,,wraps circuitry around it to drive that difference to zero.
Dialogue: 0,0:11:11.46,0:11:15.66,Default,,0000,0000,0000,,This type of an amplifier becomes the core of
Dialogue: 0,0:11:15.66,0:11:18.25,Default,,0000,0000,0000,,feedback systems or control systems where
Dialogue: 0,0:11:18.25,0:11:21.64,Default,,0000,0000,0000,,one voltage is meant to control another voltage.
Dialogue: 0,0:11:21.64,0:11:26.98,Default,,0000,0000,0000,,It also performs the obvious operation of taking one source and
Dialogue: 0,0:11:26.98,0:11:33.15,Default,,0000,0000,0000,,subtracting it from another or adding in depending upon the size of the V1 and V2.
Dialogue: 0,0:11:33.15,0:11:37.30,Default,,0000,0000,0000,,So, if I would say for now this is a very useful amplifier configuration
Dialogue: 0,0:11:37.30,0:11:40.55,Default,,0000,0000,0000,,and you'll see more of it as you go on with
Dialogue: 0,0:11:40.55,0:11:44.87,Default,,0000,0000,0000,,this course and then also into your linear electronics classes.