WEBVTT 00:00:05.600 --> 00:00:10.290 >> Now, I want to spend some time talking about operational amplifiers. 00:00:10.290 --> 00:00:15.645 Operational amplifiers are a device that gets used a lot in electrical circuits. 00:00:15.645 --> 00:00:18.460 You will see them over and over again this semester. 00:00:18.460 --> 00:00:20.670 They get used a lot in instrumentation systems. 00:00:20.670 --> 00:00:24.030 They get used a lot in control systems, etc, etc. 00:00:24.030 --> 00:00:27.450 They're quite often the basis for electrical circuits, 00:00:27.450 --> 00:00:30.210 which perform mathematical operations. 00:00:30.210 --> 00:00:33.450 That's why they're called operational amplifiers. 00:00:33.450 --> 00:00:37.595 Now, there's one point that has to be made very clear up front. 00:00:37.595 --> 00:00:40.055 These are not a passive device. 00:00:40.055 --> 00:00:43.370 So far, with the exception of our power sources, 00:00:43.370 --> 00:00:48.440 all of our circuit elements have been passive than resistors, essentially. 00:00:48.440 --> 00:00:55.115 That means that the energy delivered by the circuit to the element is non-negative. 00:00:55.115 --> 00:01:01.405 This element does not create power out of somewhere else and provide it to the circuit. 00:01:01.405 --> 00:01:06.180 Okay, it has to get any energy that it has from the circuit. 00:01:06.520 --> 00:01:11.255 Operational amplifiers are in active device. 00:01:11.255 --> 00:01:15.690 Okay. They will deliver power to your circuit. 00:01:16.210 --> 00:01:23.630 The way they deliver power to your circuit is because they have an external power supply. 00:01:23.630 --> 00:01:27.560 There's some other magical device somewhere that is feeding 00:01:27.560 --> 00:01:31.685 these guys power, which these guys can then provide to your circuit. 00:01:31.685 --> 00:01:37.415 I will tend to abbreviate operational amplifiers as op-amps. 00:01:37.415 --> 00:01:41.610 That's very common, primarily, just to save syllables. 00:01:41.860 --> 00:01:45.215 Quick overview of operational amplifiers. 00:01:45.215 --> 00:01:48.395 We're going to think of operational amplifiers as a device. 00:01:48.395 --> 00:01:51.835 It's something that performs some task. 00:01:51.835 --> 00:01:55.670 So, we're going to think of them as a black box. 00:01:55.670 --> 00:01:58.860 There is a bunch of internal circuitry in these guys. 00:01:58.860 --> 00:02:02.390 We won't be analyzing these on that level. 00:02:02.390 --> 00:02:04.340 Okay. They're going to be a black box that has 00:02:04.340 --> 00:02:06.665 essentially some input output characteristic. 00:02:06.665 --> 00:02:08.824 That's all we care about. 00:02:08.824 --> 00:02:13.670 One of the drawbacks of dealing with things this way is that it may 00:02:13.670 --> 00:02:18.230 appear as if KCL and KVL don't apply to these guys. 00:02:18.230 --> 00:02:19.745 That's not true. 00:02:19.745 --> 00:02:25.520 If you model the internal circuitry, these guys do satisfy KVL and KCL. 00:02:25.520 --> 00:02:29.575 It's just that there's something very complicated going on inside there. 00:02:29.575 --> 00:02:32.180 Number one, they've got an external power supply that's 00:02:32.180 --> 00:02:35.180 feeding them current or voltage or whatever, 00:02:35.180 --> 00:02:38.360 that we are generally not going to worry too much about when we're 00:02:38.360 --> 00:02:41.970 looking at the op-amp as part of an overall circuit. 00:02:41.970 --> 00:02:46.250 So, what we're going to end up with are several rules 00:02:46.250 --> 00:02:48.940 for how the op-amp is going to behave. 00:02:48.940 --> 00:02:53.705 Okay. Those are based on an analysis of the internal circuitry, 00:02:53.705 --> 00:02:56.270 but we aren't going to worry about that translation. 00:02:56.270 --> 00:02:58.100 We're just going to have a few rules that we're going to say 00:02:58.100 --> 00:03:00.290 this is the way this device behaves, 00:03:00.290 --> 00:03:04.740 we're going to forget about it until a 400-level class later on. 00:03:04.740 --> 00:03:09.200 So, we're going to use op-amps to perform operations, 00:03:09.200 --> 00:03:13.370 but we don't need to actually design and build the operational amplifiers 00:03:13.370 --> 00:03:19.050 themselves or analyze them on a detailed level at this stage in our career. 00:03:19.370 --> 00:03:26.465 Here's a schematic of a very common 741 operational amplifier. 00:03:26.465 --> 00:03:28.370 You can see that it's pretty complex. 00:03:28.370 --> 00:03:31.895 It has a whole bunch of bipolar junction transistors in it. 00:03:31.895 --> 00:03:36.410 It has a bunch of resistors. It has a couple of inputs. 00:03:36.410 --> 00:03:40.850 It has an output. It has a couple of external power supplies and 00:03:40.850 --> 00:03:44.405 it's got some stuff here that we don't even need to worry about yet. 00:03:44.405 --> 00:03:47.810 But we aren't going to deal with this internal view of 00:03:47.810 --> 00:03:52.320 the operational amplifier, we're going to treat it as a black box. 00:03:52.360 --> 00:03:57.200 Okay, our high level view of an operational amplifier is going to be to 00:03:57.200 --> 00:04:01.570 represent it just as this rightward pointing triangle. 00:04:01.570 --> 00:04:07.280 This device has three terminals. There are two input terminals. 00:04:07.280 --> 00:04:10.190 They have a positive and a negative sign associated with 00:04:10.190 --> 00:04:13.775 them and one output terminal here. 00:04:13.775 --> 00:04:20.149 V_n is the voltage applied at the inverting or negative input terminal. 00:04:20.149 --> 00:04:26.525 V_p is the voltage applied at the non-inverting or positive input terminal. 00:04:26.525 --> 00:04:32.020 The out comes at the output of the operational amplifier. 00:04:32.020 --> 00:04:34.730 Now, there are a number of parameters that 00:04:34.730 --> 00:04:39.900 this operational amplifiers operation is going to be characterized relative to. 00:04:39.900 --> 00:04:44.330 They're not necessarily these individual values, they're something else. 00:04:44.330 --> 00:04:47.210 The first of these is the difference in 00:04:47.210 --> 00:04:50.605 voltage between the inverting and non-inverting terminals. 00:04:50.605 --> 00:04:54.060 The change in voltage between V_p and V_n. 00:04:54.060 --> 00:05:02.555 So, Delta V_in is V_p minus V sub n. That's the voltage difference. 00:05:02.555 --> 00:05:06.950 Keep in mind that generally, according to our operational amplifier behavior, 00:05:06.950 --> 00:05:09.650 we don't care what these individual voltages are, 00:05:09.650 --> 00:05:13.105 we just care what the difference is between them. 00:05:13.105 --> 00:05:15.800 The other thing that you use to characterize 00:05:15.800 --> 00:05:20.650 operational amplifier behavior are the currents into the input terminals. 00:05:20.650 --> 00:05:25.865 We'll have some current into the positive or non-inverting terminal and 00:05:25.865 --> 00:05:29.680 some other current into the negative or inverting terminal. 00:05:29.680 --> 00:05:32.390 Okay, these parameters are what we're going to base 00:05:32.390 --> 00:05:36.230 our rules of operational amplifier behavior on. 00:05:36.230 --> 00:05:39.920 Now, I want to provide the rules by which we will 00:05:39.920 --> 00:05:43.100 characterize the operational amplifiers behavior. 00:05:43.100 --> 00:05:45.740 In order to do that, I want to give a slightly more 00:05:45.740 --> 00:05:49.295 complete symbol for the operational amplifier. 00:05:49.295 --> 00:05:55.190 I said that the operational amplifier requires an external power supply to do its job. 00:05:55.190 --> 00:05:57.890 The power supplies are provided here and here. 00:05:57.890 --> 00:06:00.860 There are actually two additional terminals 00:06:00.860 --> 00:06:04.295 that we need to worry about for five terminals in all. 00:06:04.295 --> 00:06:08.480 Generally, when I'm analyzing a circuit, I'll leave these off 00:06:08.480 --> 00:06:12.080 and just in the back of my mind, recognize that they're there, 00:06:12.080 --> 00:06:14.875 but for right now, I need to put them back in. 00:06:14.875 --> 00:06:19.610 Now, I said earlier, that the operation is going to be characterized by this difference in 00:06:19.610 --> 00:06:27.230 voltage delta V_in and the currents into the non-inverting and inverting input terminals. 00:06:27.230 --> 00:06:30.380 So, when we see one of these devices, 00:06:30.380 --> 00:06:32.615 we're going to make the following assumptions. 00:06:32.615 --> 00:06:37.820 These assumptions are relative to ideal operational amplifiers. 00:06:37.820 --> 00:06:42.110 We will assume that the current into 00:06:42.110 --> 00:06:47.270 the non-inverting and the inverting input terminals are both zero. 00:06:47.270 --> 00:06:52.075 This device is not accepting any power into these terminals. 00:06:52.075 --> 00:06:57.080 So, any power that comes into the output comes from the power 00:06:57.080 --> 00:07:00.560 supplies that you've connected up here and here. 00:07:00.560 --> 00:07:05.000 We're also going to assume that the difference in voltage 00:07:05.000 --> 00:07:10.500 between the positive and negative terminals has to be zero. 00:07:10.600 --> 00:07:23.880 Therefore, V_p is going to be equal to V_n. 00:07:23.880 --> 00:07:30.995 One other thing has to be true for operational amplifiers, this output voltage. 00:07:30.995 --> 00:07:39.725 The amount of output voltage you can get here is constrained by these two power supplies. 00:07:39.725 --> 00:07:44.810 This voltage has to be greater than the negative power supply 00:07:44.810 --> 00:07:48.520 and less than the positive power supply, 00:07:48.520 --> 00:07:50.760 and those are strict inequalities. 00:07:50.760 --> 00:07:54.275 Generally, most operational amplifiers you can only get to within 00:07:54.275 --> 00:07:59.370 a volt or two of your power supply voltages. 00:07:59.660 --> 00:08:03.965 Okay, a few points about operational amplifier behavior. 00:08:03.965 --> 00:08:07.460 The output current is generally not known, 00:08:07.460 --> 00:08:10.945 you cannot make any assumptions relative to the output current. 00:08:10.945 --> 00:08:14.615 Right? It's provided by the external power supplies. 00:08:14.615 --> 00:08:17.600 That's where people sometimes early on get themselves into trouble. 00:08:17.600 --> 00:08:20.400 They say, "Okay, there's no current into the input terminals, 00:08:20.400 --> 00:08:22.730 but I've got some current out of the output terminals. 00:08:22.730 --> 00:08:25.385 KCL doesn't apply." Well, it does. 00:08:25.385 --> 00:08:27.830 The current coming out of the operational amplifier is 00:08:27.830 --> 00:08:29.930 coming from the external power supplies, 00:08:29.930 --> 00:08:32.000 you know nothing about those. 00:08:32.000 --> 00:08:35.090 You don't know anything about the output current unless you 00:08:35.090 --> 00:08:38.809 analyze the circuit to determine what that is. 00:08:38.809 --> 00:08:42.770 In general, when I'm analyzing an operational amplifier, 00:08:42.770 --> 00:08:47.795 I will start out by applying KCL at the input nodes. 00:08:47.795 --> 00:08:52.255 Okay. It doesn't always cure all of your problems, 00:08:52.255 --> 00:08:55.730 but it's generally a good place to start. 00:08:55.730 --> 00:09:03.065 The operation of the operational amplifiers is generally based on Delta V_in. 00:09:03.065 --> 00:09:09.035 If I look at this amplifier as an input-output relationship, 00:09:09.035 --> 00:09:14.150 what it looks like is that I have some Delta V_in, 00:09:14.150 --> 00:09:19.505 I multiply that by some large number K. 00:09:19.505 --> 00:09:28.230 That gives me V_out. 00:09:28.230 --> 00:09:32.160 Now, for an ideal operational amplifier, 00:09:35.810 --> 00:09:40.095 we assume that K goes to infinity. 00:09:40.095 --> 00:09:43.730 Okay. For non-ideal or realistic operational amplifiers, 00:09:43.730 --> 00:09:45.635 K is on the order of millions. 00:09:45.635 --> 00:09:48.050 We'll assume it's infinite. 00:09:48.260 --> 00:09:55.190 Also, I mentioned earlier the output voltage is limited by the external power supplies. 00:09:55.190 --> 00:10:00.740 Okay, your output voltage must be lower than the positive power supply. 00:10:00.740 --> 00:10:04.625 It must be higher than the negative power supply. 00:10:04.625 --> 00:10:08.900 These two things in conjunction with one another lead 00:10:08.900 --> 00:10:13.225 us to the conclusion that Delta V_in has to be equal to zero, 00:10:13.225 --> 00:10:20.530 because if V_out is infinity times Delta V_in and V_out must be finite. 00:10:24.690 --> 00:10:30.310 Right, we can only apply so much voltage at the power supply terminals. 00:10:30.310 --> 00:10:32.785 In order to make this a finite number, 00:10:32.785 --> 00:10:37.010 if this is infinite, this guy has to be zero. 00:10:39.900 --> 00:10:47.320 Okay, let's do an example of analyzing an operational amplifier based circuit. 00:10:47.320 --> 00:10:52.960 Here is my op-amp. It has two inputs and one output, okay? 00:10:52.960 --> 00:10:57.760 It also has some other voltage supplies and some resistors hanging around here, 00:10:57.760 --> 00:11:01.255 and what I haven't shown are the external power supplies. 00:11:01.255 --> 00:11:07.090 I generally won't show those. Generally, when I start out analyzing one of these, 00:11:07.090 --> 00:11:12.775 the first thing I'm going to do is employ my op-amp rules, okay? 00:11:12.775 --> 00:11:17.590 There is no voltage difference between the inverting and non-inverting terminals. 00:11:17.590 --> 00:11:23.470 So this delta V is zero. I've tied the non-inverting terminal to ground. 00:11:23.470 --> 00:11:29.500 So, this voltage is zero volts. That means that since I 00:11:29.500 --> 00:11:32.515 can't have a voltage difference between here and here, 00:11:32.515 --> 00:11:35.810 this voltage is zero volts. 00:11:37.110 --> 00:11:43.690 Likewise, the current here into the non-inverting terminal is zero, 00:11:43.690 --> 00:11:48.760 the current here into the inverting terminal is also zero. 00:11:48.760 --> 00:11:53.440 Now, I've labeled everything that I know about this operational amplifier. 00:11:53.440 --> 00:11:59.590 I can go ahead and analyze it to determine V_out, okay? 00:11:59.590 --> 00:12:04.420 As usual, I'll start out applying KCL at an input node. 00:12:04.420 --> 00:12:07.000 KCL at an input node is kind of a good idea, 00:12:07.000 --> 00:12:08.560 because you already know something's there. 00:12:08.560 --> 00:12:11.635 You know there's no current into the op-amp itself. 00:12:11.635 --> 00:12:18.470 So if I call this node A and do KCL at A, 00:12:19.380 --> 00:12:24.070 the current through R_in is this voltage, 00:12:24.070 --> 00:12:31.735 V_in minus this voltage which the op amp is constraining to be zero volts over RN. 00:12:31.735 --> 00:12:36.325 So V_in minus zero over R_in, 00:12:36.325 --> 00:12:41.110 this current into this node is equal to this current out of the node, 00:12:41.110 --> 00:12:44.200 because there's no current flow through this leg, here. 00:12:44.200 --> 00:12:49.930 This current is this voltage minus this voltage over R_f. 00:12:49.930 --> 00:12:58.645 So that's equal to zero minus V_out over R_f. 00:12:58.645 --> 00:13:01.120 V_in, as my input voltage, 00:13:01.120 --> 00:13:02.140 I don't know what it is, 00:13:02.140 --> 00:13:05.830 but I have to be told it before I can determine a number for V_out. 00:13:05.830 --> 00:13:12.865 So V_out, let's multiply this by R_f, 00:13:12.865 --> 00:13:21.460 is R_f over Rn times V_in taking this negative sign over here. 00:13:21.460 --> 00:13:28.270 V_out is equal to minus R_f over R_in times V_in. 00:13:28.270 --> 00:13:31.660 So whatever you give me for V_in, 00:13:31.660 --> 00:13:35.155 I'm going to multiply that by a number, 00:13:35.155 --> 00:13:37.330 take the negative of that, 00:13:37.330 --> 00:13:41.690 and this op-amp will give you that as V_out. 00:13:42.570 --> 00:13:46.450 This has a particular name. 00:13:46.450 --> 00:14:02.260 This is an inverting voltage amplifier, okay? 00:14:02.260 --> 00:14:04.585 It's amplifying voltage. 00:14:04.585 --> 00:14:05.905 You give it a voltage in, 00:14:05.905 --> 00:14:07.345 it gives you a voltage out. 00:14:07.345 --> 00:14:08.605 It inverts that. 00:14:08.605 --> 00:14:12.235 The voltage out you get as the negative of the voltage in. 00:14:12.235 --> 00:14:14.440 It is also amplifying that, 00:14:14.440 --> 00:14:18.415 according to whatever you choose for R_f and R_in. 00:14:18.415 --> 00:14:21.790 If R_f is 10 ohms and R_in is one ohms, 00:14:21.790 --> 00:14:24.910 then this is going to be 10 and the output voltage is 00:14:24.910 --> 00:14:29.030 going to be negative 10 times whatever the input voltage is. 00:14:29.220 --> 00:14:33.130 Let's analyze another operational amplifier circuit. 00:14:33.130 --> 00:14:38.380 I want to find V_out with this operational amplifier based circuit. 00:14:38.380 --> 00:14:41.890 Notice, again, that I have my three terminal device. 00:14:41.890 --> 00:14:47.470 It has some non-inverting and inverting terminal in and output terminal. 00:14:47.470 --> 00:14:49.690 I'm not showing my power supplies, 00:14:49.690 --> 00:14:51.805 but if I wired this circuit up in the lab, 00:14:51.805 --> 00:14:55.150 I would need to provide power to it, okay? 00:14:55.150 --> 00:14:57.565 So let's find V_out as a function of V_in. 00:14:57.565 --> 00:15:02.380 The first thing I want to do is apply my op-amp rules. 00:15:02.380 --> 00:15:07.645 This voltage source is insisting that 00:15:07.645 --> 00:15:13.495 the voltage at the non-inverting terminal is going to be set to be V_in. 00:15:13.495 --> 00:15:17.110 The op-amp itself is insisting that 00:15:17.110 --> 00:15:20.305 there is no voltage difference between these two terminals, 00:15:20.305 --> 00:15:24.550 therefore I must have voltage V_in at this terminal. 00:15:24.550 --> 00:15:29.990 Therefore, this voltage here is V_in. 00:15:30.180 --> 00:15:36.220 Now, I have no current into these terminals. 00:15:36.220 --> 00:15:44.055 Notice, very importantly, this voltage source is not providing any power, okay? 00:15:44.055 --> 00:15:46.230 The current out of the voltage source is zero. 00:15:46.230 --> 00:15:50.165 It's not providing any power in order to create this output voltage. 00:15:50.165 --> 00:15:55.360 Any power in this output signal is coming from the external power supplies. 00:15:55.360 --> 00:15:57.385 So these currents are zero. 00:15:57.385 --> 00:15:59.425 I know something about that current. 00:15:59.425 --> 00:16:04.060 Now I'm going to my old fallback standard position. 00:16:04.060 --> 00:16:06.865 I'm going to do KCL at the input nodes. 00:16:06.865 --> 00:16:11.515 Let me call this node A. I'll apply a KCL there. 00:16:11.515 --> 00:16:18.085 Let me say that this current through the resistor Rf is I_f, 00:16:18.085 --> 00:16:21.520 and this current through R_1 is I_1, 00:16:21.520 --> 00:16:24.475 and those are going to be my positive directions. 00:16:24.475 --> 00:16:31.330 So KCL, at A, tells me that the current going into node A 00:16:31.330 --> 00:16:33.760 is equal to the current coming out of node A, 00:16:33.760 --> 00:16:37.915 so I_f is equal to I_1. 00:16:37.915 --> 00:16:42.760 This current is zero, right? 00:16:42.760 --> 00:16:45.970 I don't need to list it in my KCL. 00:16:45.970 --> 00:16:51.190 Now, I_f is this voltage minus this voltage, over R_f. 00:16:51.190 --> 00:17:01.450 So, V_out minus V_in over R_f is equal to the current going through here, 00:17:01.450 --> 00:17:03.775 which is just V_in minus, 00:17:03.775 --> 00:17:05.290 I'm going to take this as my reference, 00:17:05.290 --> 00:17:07.819 it's going to be zero volts. 00:17:08.579 --> 00:17:15.040 V_in minus zero over R_1. 00:17:15.040 --> 00:17:21.040 So V_out over R_f is equal 00:17:21.040 --> 00:17:27.880 to V_in over R_f taking this term over to the other side, 00:17:27.880 --> 00:17:32.635 plus V_in over R_1. 00:17:32.635 --> 00:17:39.805 Grouping terms, V_out is equal to R_f 00:17:39.805 --> 00:17:48.100 times one over R_f plus one over R_1 times V_in. 00:17:48.100 --> 00:17:52.105 This becomes a one plus R_f over R_1. 00:17:52.105 --> 00:18:00.100 So therefore, V_out is equal 00:18:00.100 --> 00:18:06.740 to one plus R_f over R_1 times V_in. 00:18:10.860 --> 00:18:16.015 This device takes a voltage V_in, 00:18:16.015 --> 00:18:19.375 multiplies it by a number, 00:18:19.375 --> 00:18:22.225 and actually a positive number. 00:18:22.225 --> 00:18:24.250 In order to get V_out, 00:18:24.250 --> 00:18:33.440 this is a non-inverting voltage amplifier. 00:18:36.990 --> 00:18:41.830 Okay, you're still amplifying voltage by taking your input voltage, 00:18:41.830 --> 00:18:44.635 multiplying it by a number to get the output voltage, 00:18:44.635 --> 00:18:48.010 but you're not changing the sign, it not-inverting. 00:18:48.010 --> 00:18:54.415 Now one quick thing I want to point out about both this example and the previous one. 00:18:54.415 --> 00:18:57.820 Both of these had a resistor which was 00:18:57.820 --> 00:19:01.510 feeding back from the output to one of the input terminals. 00:19:01.510 --> 00:19:04.720 That is very typical of op-amp based circuits. 00:19:04.720 --> 00:19:07.315 It's generally called the feedback loop. 00:19:07.315 --> 00:19:14.425 Almost invariably you will feedback from the output to the inverting input terminal. 00:19:14.425 --> 00:19:19.145 That's necessary in order to keep this entire device stable. 00:19:19.145 --> 00:19:22.020 If I feedback to the positive terminal, 00:19:22.020 --> 00:19:25.485 generally this device will do what is called going unstable, 00:19:25.485 --> 00:19:28.470 the output will try to go to infinity. 00:19:28.470 --> 00:19:32.190 So what you'll have happen is that your output voltage will either 00:19:32.190 --> 00:19:36.130 go to the positive or the negative voltage rail, and stay there. 00:19:36.130 --> 00:19:37.540 It can't get to infinity, 00:19:37.540 --> 00:19:40.420 but it's going to go as high as it can go, 00:19:40.420 --> 00:19:42.800 and it's not going to come back. 00:19:42.990 --> 00:19:46.015 Okay, this concludes Lecture 12. 00:19:46.015 --> 00:19:50.185 Next lecture, we'll do some more work with operational amplifiers. 00:19:50.185 --> 00:19:52.810 We'll look at their operation in a little bit more depth, 00:19:52.810 --> 00:19:58.240 we'll talk more about the voltage rails applied by the power supplies, 00:19:58.240 --> 00:20:02.515 and we'll start looking at them as dependent sources. 00:20:02.515 --> 00:20:06.340 Okay, later on in your schooling career you'll see a lot of dependent sources. 00:20:06.340 --> 00:20:11.090 We'll just start kind of looking at things in that way in this class.