1 00:00:00,012 --> 00:00:05,787 In this video we're going to talk about electronic circuits. This is the modern 2 00:00:05,799 --> 00:00:11,795 topic in electrical engineering, is how to build circuits that actually provide a 3 00:00:11,807 --> 00:00:17,222 power gain. We're going to talk about them, and we're going to have to talk 4 00:00:17,234 --> 00:00:21,842 about a new circuit element. This is called dependent sources, we'll 5 00:00:21,854 --> 00:00:26,466 introduce them. We're going to show how those are used in model, what are called 6 00:00:26,478 --> 00:00:30,990 operational amplifiers. These are the workhorses of what are called active 7 00:00:31,002 --> 00:00:35,796 filters. And the word active here has to do with providing a power gain. It's all 8 00:00:35,808 --> 00:00:42,915 about power. So let's see the difference between an electrical circuit and an 9 00:00:42,927 --> 00:00:49,295 electronic circuit. So an electrical circuit like we've been talking about 10 00:00:49,307 --> 00:00:56,345 can't provide any power gain. Now if you figure out the transfer function between 11 00:00:56,737 --> 00:01:04,055 some output variable and the source you may well have a case where the gain is 12 00:01:04,067 --> 00:01:12,415 bigger than one. However, when you attach a low resistor to it, for example, to try 13 00:01:12,427 --> 00:01:20,008 to get that power out, it turns out the power that's dissipated in the load has to 14 00:01:20,020 --> 00:01:25,977 be smaller than what's produced by the source. Easy way of seeing that, the only 15 00:01:25,989 --> 00:01:31,753 thing in the circuit that's producing power, is that source. And we know from 16 00:01:31,765 --> 00:01:37,781 the conservation of power, that all the other elements have to dissipate, or not. 17 00:01:37,972 --> 00:01:43,702 Dissipate at all but have to dissipate power, so you pick any one of 'em, that 18 00:01:43,714 --> 00:01:49,419 has to be less than what's the power that's being produced. So, no power gain. 19 00:01:49,540 --> 00:01:55,606 So it would seem like, when you look at the conservation of power, that there's no 20 00:01:55,618 --> 00:02:01,371 circuit that could produce a power gain, and that's true. Well we're going to 21 00:02:01,383 --> 00:02:07,823 introduce a new circuit element, that has that capability. And so our new circuit 22 00:02:07,835 --> 00:02:13,538 element, it's what's known as the dependent source. Okay, so here's the 23 00:02:13,550 --> 00:02:20,193 idea. We have a circuit, and I'm drawing an abstract line. Somewhere inside there's 24 00:02:20,205 --> 00:02:26,757 a circuit element. That has a voltage v defined for it. Somewhere else in the 25 00:02:26,769 --> 00:02:33,635 circuit is a voltage source whose voltage depends on what the voltage across that 26 00:02:33,647 --> 00:02:40,250 element was. How this happens is a mystery. Somehow this voltage source knows 27 00:02:40,262 --> 00:02:47,461 exactly what this voltage over here is. And produces something that's K times it. 28 00:02:47,602 --> 00:02:55,087 That's why this is a voltage dependent voltage source. Now this is clearly not 29 00:02:55,099 --> 00:03:03,100 something that you could build just out of raw elements. how in the world can you 30 00:03:03,392 --> 00:03:08,345 have some sort of thing and this is really points out that, these are ideal circuit 31 00:03:08,357 --> 00:03:14,135 elements and they're used to describe something a lot more complicated. Now this 32 00:03:14,147 --> 00:03:19,360 action gets built, I'll show you an example in a second, and it's not very 33 00:03:19,372 --> 00:03:25,250 simple but it can be well described by our ideal circuit elements, as we'll see in a 34 00:03:25,262 --> 00:03:31,489 second. I want to also mention that you can have all four possible configurations 35 00:03:31,501 --> 00:03:37,971 here. We have a voltage-dependent voltage source. You could have a current-dependent 36 00:03:37,983 --> 00:03:43,672 voltage source. You could have a voltage-dependent current source, and you 37 00:03:43,684 --> 00:03:50,102 could have a current Dependent current source. Turns out the last one is 38 00:03:50,114 --> 00:03:57,603 particularly important for modeling transistors. So they are fundamental to 39 00:03:57,615 --> 00:04:05,208 characterizing transistor circuits, in particular those circuits that have a 40 00:04:05,220 --> 00:04:12,321 power gain. Let me show you how that's possible. So, here's a very simple 41 00:04:12,333 --> 00:04:20,880 circuit. It's split in 2 pieces as our voltage dependent and voltage source and 42 00:04:20,892 --> 00:04:27,931 because it's in two separate parts. they don't interact with each other, except by 43 00:04:27,943 --> 00:04:33,245 this mystical, dependent source that mystically somehow figures out what the 44 00:04:33,257 --> 00:04:38,231 voltage is across that resistor. Now, let's do a little power calculation. I 45 00:04:38,329 --> 00:04:43,875 think it's pretty clear that all the power dissipated in that resistor is produced by 46 00:04:43,887 --> 00:04:52,633 that source. And all the power dissipated in that resistor is produced by the 47 00:04:52,645 --> 00:05:02,273 dependent source, okay? Now let's draw a little box here, okay? So, what's the 48 00:05:02,285 --> 00:05:07,847 power It's produced, by the voltage source, but that's gotta be what's 49 00:05:07,859 --> 00:05:13,418 dissipating the resistor, so we know that's Vin squared over R. It has to be 50 00:05:13,430 --> 00:05:19,354 producing that much power. How much power is being consumed in our resistor over 51 00:05:19,366 --> 00:05:26,321 here? Well that's KVin squared. Over R. And I really do mean the same resistors 52 00:05:26,333 --> 00:05:33,068 here. So if K is bigger than 1 which is certainly is allowed to be. this circuit 53 00:05:33,080 --> 00:05:41,441 is producing more power than its, than is coming into it. There's a power game. So 54 00:05:41,453 --> 00:05:48,767 what's going on? I thought this wasn't possible and the answer is, it is possible 55 00:05:48,779 --> 00:05:56,096 if inside this circuit is somehow being provided with power that's not indicated. 56 00:05:56,236 --> 00:06:03,575 And this gets into the idea of what's called a power supply and these days for 57 00:06:03,587 --> 00:06:10,070 cellphones and things like that. You might think this has equivalent to a battery 58 00:06:10,082 --> 00:06:15,795 that provides power as part of it's functioning so that this circuit can 59 00:06:15,807 --> 00:06:22,085 provide the indicated power gain, because the power supply is an additional source 60 00:06:22,097 --> 00:06:28,963 of power For the output. how that happens, that's why the dependent source models are 61 00:06:28,975 --> 00:06:35,613 used, because they're very simple ways of describing somewhat complicated circuits. 62 00:06:35,734 --> 00:06:41,314 But in general, circuits containing dependent sources can produce a power 63 00:06:41,326 --> 00:06:45,870 gain. Okay. And here's an example of one. But here's a 64 00:06:45,882 --> 00:06:52,440 practical example, and this is the operational amplifier I mentioned earlier. 65 00:06:52,572 --> 00:06:59,565 So, the operational amplifier, it's, a jargon name is op-amp. It's, it's referred 66 00:06:59,577 --> 00:07:08,016 to as an op-amp. And the circuit symbol is indicated here to the left, has this 67 00:07:08,028 --> 00:07:17,324 triangle that has a plus and minus input which I've labelled. I'm thinking here of 68 00:07:17,336 --> 00:07:25,395 nodes so if that's node a and node b. And here's the reference. The output, comes 69 00:07:25,407 --> 00:07:31,540 out the end of the triangle. And it's node C. And that's the. We're going to plug 70 00:07:31,552 --> 00:07:37,865 that into a circuit in just a second. The model for it, in terms of ideal circuit 71 00:07:37,877 --> 00:07:46,456 elements, for the operational amplifier is as follows. There's a resistor Rn between 72 00:07:46,468 --> 00:07:53,018 the two input nodes. And here's our dependent source, voltage-dependent 73 00:07:53,030 --> 00:08:00,160 voltage source, that is sum gained times the difference of the two voltages here. 74 00:08:00,298 --> 00:08:06,564 And that's why Ea corresponds to plus And my e, b here correspondes to the negative 75 00:08:06,576 --> 00:08:11,835 side, because that's how the enter into this formulea for the voltage produced by 76 00:08:11,847 --> 00:08:17,153 this voltage dependant voltage source. It's all written in terms of node voltages 77 00:08:17,165 --> 00:08:22,186 that are referenc e to the taken with respect to that reference. There is a 78 00:08:22,198 --> 00:08:28,861 finally, there's a output resistor And this is the output. Okay, so, what does an 79 00:08:28,873 --> 00:08:37,137 op-amp look like physically? This is an op-amp. we'll do a little counting here. 80 00:08:37,282 --> 00:08:45,222 There are one, two, three and four inputs that have to be available for an op-amp. 81 00:08:45,542 --> 00:08:58,341 Well, this is a chip. That you will see in all kinds of circuits. And this is a short 82 00:08:58,353 --> 00:09:08,020 name for integrated circuit. Which, different names for the same thing, chip 83 00:09:08,032 --> 00:09:13,560 is what we commonly refer to it. And notice, it has four pins over here. And it 84 00:09:13,572 --> 00:09:19,555 turns out there are four pins on the other side, giving a total of eight. Well that's 85 00:09:19,567 --> 00:09:25,205 more than enough, it turns out it needs two other inputs to make the chip 86 00:09:25,217 --> 00:09:32,530 functioned correctly and two, the remaining two are not really used. So 87 00:09:33,077 --> 00:09:40,685 what's inside this chip that can provide the power gain and is modeled by our 88 00:09:40,697 --> 00:09:47,887 dependent source model? Well I think you'll be a little surprised by the 89 00:09:47,899 --> 00:09:55,632 complexity. So this is what's inside what's known as a 741 op-amp, operational 90 00:09:55,644 --> 00:10:03,711 amplifier. It contains 22 transistors, 11 resistors, 1 diode and 1 capacitor. So, 91 00:10:03,846 --> 00:10:10,629 this is the symbol for a transistor and here is another one, here's a really odd 92 00:10:10,641 --> 00:10:16,321 looking one over here. 22 total transistors and so it is a very 93 00:10:16,333 --> 00:10:23,629 complicated circuit inside to provide that game and let's go through what the inputs. 94 00:10:23,932 --> 00:10:32,279 So, so I N plus. That's node A in my diagram. I N minus is of course node B. 95 00:10:32,453 --> 00:10:41,908 Here is the output which is node C and these are offsets which are additional 96 00:10:41,920 --> 00:10:48,985 inputs. Which we're not going to talk about here that have to do with detail 97 00:10:48,997 --> 00:10:56,135 function of the output. But, what I really point out are these two, Vcc plus and Vcc 98 00:10:56,397 --> 00:11:02,357 minus, standard notation for the power supply. Turns out for this op-amp, the 99 00:11:02,369 --> 00:11:09,310 power supply has to be about 15 volts. so these are providing additional power 100 00:11:09,322 --> 00:11:16,263 through the circuit which allow us to use the deep ended source model. If these are 101 00:11:16,275 --> 00:11:21,865 turned off, you always take the battery out. It no longer looks like a dependent 102 00:11:21,877 --> 00:11:27,985 source. In fact it probably looks like very, somethin g very dull and boring. The 103 00:11:27,997 --> 00:11:34,045 output either is flat, which means it's 0, or it has some distorted waveform. It 104 00:11:34,057 --> 00:11:40,738 doesn't function like it was designed to. You have to turn on the power. In order to 105 00:11:40,750 --> 00:11:48,225 make the circuit come alive, and be described by our ideal circuit models. 106 00:11:48,379 --> 00:11:56,614 Now, I want to point out, that this diagram, the circuit diagram. What's 107 00:11:56,626 --> 00:12:03,465 inside came from a website at Texas Instruments, which is a well-known, 108 00:12:03,587 --> 00:12:09,865 worldwide producer of chips of all kinds, analog and digital. And this is a website 109 00:12:09,877 --> 00:12:15,850 they provide for you to download specification sheets, that's what these 110 00:12:15,862 --> 00:12:21,605 are called. I took this from a specification sheet for a 741 op-amp. And 111 00:12:21,617 --> 00:12:27,355 you can download and look at all the different kind of circuits and chips 112 00:12:27,367 --> 00:12:33,065 rather they provide and see how they operate. And we need to look at more 113 00:12:33,077 --> 00:12:38,660 detail to see what special about an op-amp, in other words I want to know 114 00:12:38,672 --> 00:12:45,036 something about Rin, Rout and G. I know the op-amp functions this way, but what 115 00:12:45,048 --> 00:12:51,809 are those parameters? Well, let's dive in to the specification, for our 741 op-amp. 116 00:12:51,938 --> 00:12:58,498 There it is and what you'll see listed in general are a bunch of different 117 00:12:58,510 --> 00:13:04,940 parameters that describe how this circuit behaves. And we'll go through those but I 118 00:13:04,952 --> 00:13:10,053 want to point out something very important. Notice, that what's specified 119 00:13:10,065 --> 00:13:15,444 in this specification sheet is a minimum and maximum value, and a typical value, 120 00:13:15,557 --> 00:13:21,264 for every one of these parameters. So. The manufacturing process guarantees that 121 00:13:21,276 --> 00:13:26,902 the circuit will be within these minimum and maximum values, whichever parameter 122 00:13:26,914 --> 00:13:31,949 you're talking about. But what it is exactly will vary depending which 123 00:13:31,961 --> 00:13:37,785 operational amplifier you pick out of the batch. So this raises an issue in circuit 124 00:13:37,797 --> 00:13:43,010 design. Suppose I want to, and circuit with a very well-defined gain although we 125 00:13:43,022 --> 00:13:48,210 can't get it by just using an op amp as it is. We're going to have to put circuitry 126 00:13:48,222 --> 00:13:53,185 around it to make it function and behave exactly like we want, not like the 127 00:13:53,197 --> 00:13:58,558 manufacturer can sort of approximately tell us it's going to behave. We're going 128 00:13:58,558 --> 00:14:05,482 to see that in just a second. Let's g o through and see how the various parameters 129 00:14:05,494 --> 00:14:12,648 of our model for 741. So, I'm going to point out here ri, what the spec sheet 130 00:14:12,660 --> 00:14:20,238 calls ri, turns out, corresponds to Rn and the whole, and the specification sheet 131 00:14:20,250 --> 00:14:27,800 says that the typical value is two mega-ohms. And it could be as small as .3, 132 00:14:27,954 --> 00:14:35,997 and the maximum, it could be bigger. But the point is, is that this is big, that 133 00:14:36,009 --> 00:14:44,206 resistor is big and it's in the megaohm range, more or less, and that turns out to 134 00:14:44,218 --> 00:14:52,465 be a good thing, as we'll see. Okay? Let's go on to the output resistor, which is 135 00:14:52,866 --> 00:15:03,533 labeled Ro, output resistance. It is typically 75 ohms, so it's small, and 136 00:15:03,545 --> 00:15:16,207 that's good, okay? Now here comes the goody, and the large signal, differential 137 00:15:16,219 --> 00:15:24,402 voltage amplification constant, that's G. The typical value is 200, but look at the 138 00:15:24,414 --> 00:15:32,314 units, volts per millivolt. So what this is saying is that it's 200 times 10 to the 139 00:15:32,326 --> 00:15:40,953 3 typically, which means that the gain. And be about 2 times 10 to the 5th, 140 00:15:41,143 --> 00:15:50,430 200,000 but it could be as small as 50,000. well, this to me says that the 141 00:15:50,442 --> 00:15:59,050 gain is big, which has important consequences. So big gain, small incoming, 142 00:15:59,177 --> 00:16:05,430 big input resistors, small output resistor. There's something else about the 143 00:16:05,442 --> 00:16:11,500 operation of the amplifier, has to do with the power supply and this is worth 144 00:16:11,512 --> 00:16:18,517 pointing out. So, the vertical parameter here turns out its maximum output voltage 145 00:16:18,529 --> 00:16:26,514 swing. How big can this voltage at node C be? Okay. And it turns out it depends 146 00:16:26,526 --> 00:16:35,269 somewhat on the value of the load resistor which you're going to attach out here but 147 00:16:35,281 --> 00:16:43,203 notice it's about equal to the power supply voltage, plus and minus 14. Okay, 148 00:16:43,356 --> 00:16:52,014 let's take the big number. So suppose the output voltage, C so we get G times some 149 00:16:52,026 --> 00:17:00,002 voltage, equal to 14. And what's that voltage? Well if you divide this out and 150 00:17:00,014 --> 00:17:07,004 you use the V number, you get 70 microvolts. So, the largest that the input 151 00:17:07,016 --> 00:17:14,270 could be, if you stuck it in node a, is 70 microvolts. If you exceed that input, 152 00:17:14,414 --> 00:17:21,280 what's going to happen? If you put in a sign wave because we want the circuit to 153 00:17:21,292 --> 00:17:28,150 work in a linear way. Our output is going to be sign wav e, but if its amplitude is 154 00:17:28,162 --> 00:17:34,503 bigger than 14, what's going to happen is that the output's going to get what we 155 00:17:34,515 --> 00:17:41,605 call clipped. Is the amplitude will be chopped off. So, that's not good. That 156 00:17:41,617 --> 00:17:49,105 means sign wave in something looking more like a square laid out and it's a clear 157 00:17:49,117 --> 00:17:56,179 sign of distortion. So, how do we handle this? The point is the threshold voltage 158 00:17:56,191 --> 00:18:01,191 for the clipping phenomenon is pretty small and that's where the external 159 00:18:01,203 --> 00:18:06,125 circuitry comes in. We have to learn how to use the op amp in the context of 160 00:18:06,137 --> 00:18:11,263 surrounding circuitry to control it. But the big thing that's going to be very 161 00:18:11,275 --> 00:18:14,781 important to understanding that is that the gain is big. 162 00:18:14,784 --> 00:18:20,579 We'll see that, how that works in just a second. Okay, so here is the classic 163 00:18:20,591 --> 00:18:26,777 op-amp circuit, the reason op-amps were developed. Here's our circuit model and 164 00:18:26,789 --> 00:18:32,609 I've now stuck that in the circuit. And it's, I have a source and I have my own 165 00:18:32,621 --> 00:18:40,775 resistor that I'm using, here's the load. And here is our F. What does the F mean? 166 00:18:40,961 --> 00:18:50,215 What does the F signify? What do you think? Well F, is for feedback. It's 167 00:18:50,227 --> 00:18:59,876 where, it means it's regarding the input to the output. Now one thing to notice 168 00:18:59,888 --> 00:19:08,255 here, is that, the op-amp is put in upside down. So, you'll notice this is plus and 169 00:19:08,267 --> 00:19:17,010 minus up top, notice here is minus plus and that's intentional, you'll see that in 170 00:19:17,022 --> 00:19:26,596 a second. , also note that the plus side, node a has been tied down to the 171 00:19:26,608 --> 00:19:33,313 reference, okay? And node b is where the input voltage is 172 00:19:33,325 --> 00:19:38,222 coming in. Okay. So, maybe it becomes a little bit clearer 173 00:19:38,234 --> 00:19:43,685 when we use the circuit model for the op-amp, and stick it into our circuit, 174 00:19:43,685 --> 00:19:47,023 okay. So, the way that works here is Rn, and 175 00:19:47,035 --> 00:19:52,752 we'll define a voltage V across it, and because the op-amp, if you will, is 176 00:19:52,764 --> 00:19:58,800 plugged in upside down. The output is minus Gb where G is this big, positive 177 00:19:58,812 --> 00:20:03,440 gain, that we, I had showed you in the previous slide. Okay. 178 00:20:03,442 --> 00:20:09,665 Now that I have the model for my op-amp that's been turned on, I need to find how 179 00:20:09,677 --> 00:20:15,387 the input and output voltages are related to each other. We need to solve the 180 00:20:15,399 --> 00:20:19,715 circuit. What we're going to use is the node metho d because it can solve 181 00:20:19,727 --> 00:20:25,140 anything. The dependent source means, the presence of the dependent source means we 182 00:20:25,152 --> 00:20:29,209 can't use the serial and parallel rule This voltage is either not current while 183 00:20:28,593 --> 00:20:33,093 they're gone. So we have to use node method, that's why we introduced it so we 184 00:20:33,105 --> 00:20:41,610 could talk about this circuit. Okay. How many nodes are there? Okay. What I 185 00:20:41,622 --> 00:20:52,715 get. there's 1, 2, 3, and 4 and of course, here's our reference node, pretty obvious. 186 00:20:52,715 --> 00:20:58,544 Well, this node has a voltage source attached to it so we don't' need to define 187 00:20:58,556 --> 00:21:04,036 a node voltage there. It turns out the same thing applies to the dependent source 188 00:21:04,048 --> 00:21:10,009 because there's a voltage source between that node and the reference. we don't need 189 00:21:10,021 --> 00:21:16,499 it to find the node voltages, so the only voltages we need to define are. v, at node 190 00:21:16,511 --> 00:21:24,193 voltage, and that node voltage which turns out to be v out. So we need to write the 191 00:21:24,205 --> 00:21:32,180 node equations. So, here they are. And let's go over one of them, I've writ, the 192 00:21:32,192 --> 00:21:39,247 first one I've written applies to this node right there, so let's see. The 193 00:21:39,259 --> 00:21:46,378 current going south, is V out over RL. And the node voltage headed west is V out 194 00:21:46,390 --> 00:21:53,638 minus, the minus GV, which is the voltage over there, divided by R out. And then 195 00:21:53,650 --> 00:22:00,555 finally, V out minus V divided by RF, so that's the node equation. At this node, 196 00:22:00,661 --> 00:22:05,421 and, you can check to make sure I've got the right node equation for that node. 197 00:22:05,421 --> 00:22:08,136 Okay. So, it's pretty easy to write. It's 198 00:22:08,148 --> 00:22:13,115 always, the node voltage has to come in with a plus sign for that particular node 199 00:22:13,127 --> 00:22:18,430 equation, and the algorithms have to come in with a minus sign. So that all checks, 200 00:22:18,545 --> 00:22:24,181 so we're ready to go. We want to find the relationship to VN and VL. Which means we 201 00:22:24,193 --> 00:22:29,551 have to eliminate the voltage V by combining these equations and simplifying. 202 00:22:29,666 --> 00:22:36,483 And the results you get unfortunately, is not something simple but that's okay. So, 203 00:22:36,638 --> 00:22:45,243 after some manipulation what you get is the following result. So, as you might 204 00:22:45,255 --> 00:22:53,717 expect, the result is going to depend on every circuit element, and the gain of the 205 00:22:53,717 --> 00:22:58,197 op-amp. This is where the fun comes. What do we 206 00:22:58,209 --> 00:23:05,049 know about the parameters of the op-amp? The circuit model. We know that R out is 207 00:23:05,061 --> 00:23:12,001 small, and we know that R in is big. Now, the critical thing is right over here, so 208 00:23:12,013 --> 00:23:19,346 R out is small, and G is big. So GRf is certainly bigger than RL so we can forget 209 00:23:19,358 --> 00:23:27,455 about the RL. Well that simplifies things. That means the RL cancels so this term in 210 00:23:27,467 --> 00:23:33,991 turn is Rout over G times these, this complicated expression. Raw. 211 00:23:33,992 --> 00:23:40,890 Because G is big, and all resistors have values basically bigger than one, all 212 00:23:40,902 --> 00:23:47,370 these things are small, and further more dividing them by G is going to make them 213 00:23:47,382 --> 00:23:54,255 smaller cause G is huge. Which means we can forget about everything here. All that 214 00:23:54,267 --> 00:24:01,631 is Negligible compared to this, which leaves us with a very simple answer. So, 215 00:24:01,776 --> 00:24:09,508 we have, we have found the input output relationship, under the assumption that G 216 00:24:09,520 --> 00:24:17,171 is big. So, the output is equal to the input voltage times a negative number. It 217 00:24:17,183 --> 00:24:26,096 is given by RF over R the ratio of these two resistor values. This is a important 218 00:24:26,108 --> 00:24:34,381 result because now we control the gain of our amplifier by specifying the feedback 219 00:24:34,393 --> 00:24:42,308 and input resistor values, we control the gain. So we're not, sensitive anymore to 220 00:24:42,320 --> 00:24:49,496 the variations in G of Rn or RF. They don't matter, as long as we have a big 221 00:24:49,508 --> 00:24:57,148 input a big gain. This result applies so we control the gain. Do note that it 222 00:24:57,160 --> 00:25:06,000 inverts. Remember, a negative sign corresponds to inversion in electrode 223 00:25:06,012 --> 00:25:13,587 engineering. So, this is a inverting amplifier. That's the name for it. I do 224 00:25:13,599 --> 00:25:19,897 want to point out that if you had put the op-amp in the other way, if you had put it 225 00:25:19,909 --> 00:25:26,043 in plus to minus like this, this answer would not have changed. If you note how 226 00:25:26,055 --> 00:25:31,744 you can see that mathematically, if I put in minus G in place of G which corresponds 227 00:25:31,756 --> 00:25:38,596 of putting it in right side up, if you will. This term is still small. And you 228 00:25:38,608 --> 00:25:43,643 still would have forgotten about it. However, the result would not have been 229 00:25:43,655 --> 00:25:48,790 consistent. The input is going in this side. And, if you had plus to minus like 230 00:25:48,802 --> 00:25:54,242 this, and built it, what would have happened, is that you would, it would not 231 00:25:54,254 --> 00:25:59,790 have inverted, instead, would it happen , the output would hit the power supply 232 00:25:59,802 --> 00:26:05,569 values, positive or negative and it would not have looked like a sign wave. So it is 233 00:26:05,581 --> 00:26:10,986 important that we, you put the op-amp in right away. In mathematics, when you 234 00:26:10,998 --> 00:26:16,505 figure it out tells you which way it had to be put in. It has to be put in. If you 235 00:26:16,517 --> 00:26:22,150 will, upside down because of that minus sign. So, that minus sign is very 236 00:26:22,162 --> 00:26:27,624 important. It tells us that the input had to go in the minus side. Okay. 237 00:26:27,624 --> 00:26:34,295 So, there's another thing to point out. That this result does not depend on RL. So 238 00:26:34,307 --> 00:26:41,250 again, it's only these two resistors that determine the gain. And those two values 239 00:26:41,262 --> 00:26:48,540 control the negative gain of the inverted amplifier. Now you may not like the fact 240 00:26:48,552 --> 00:26:55,620 that it inverts, that it negates. And I'll show you a little bit later how you can 241 00:26:55,632 --> 00:27:04,911 fix that. Turns out to be pretty easy. Alright so, we now because of impedances 242 00:27:04,923 --> 00:27:14,369 that everything follows. If I replace my resistors in my circuit, IRLC circuits. 243 00:27:14,369 --> 00:27:17,839 Okay. I haven't shown an RL here or a ZL, if you 244 00:27:17,851 --> 00:27:23,527 will, because the answer as we just found for the transfer function doesn't depend 245 00:27:23,539 --> 00:27:28,930 on that. And we know right away, I don't have to do any more analysis because in 246 00:27:28,942 --> 00:27:34,593 the analogy with resistor circuits that the transfer function for this is minus ZF 247 00:27:34,605 --> 00:27:40,807 over Z. The word active filter comes in because we now can provide a gain that is 248 00:27:40,819 --> 00:27:47,040 active. It has to have the power supply. That's where the word active comes from. 249 00:27:47,165 --> 00:27:53,725 So, let's suppose I wanted to design a lowpass filter. And for the simple kind of 250 00:27:53,737 --> 00:27:59,969 filters we've been talking about, this is the trans, let's say that this is the 251 00:27:59,981 --> 00:28:05,004 transfer function we want. Okay. So let's go over it a little bit. So, the 252 00:28:05,016 --> 00:28:09,964 negative sign comes in because we know this is going to invert, so we have to 253 00:28:09,976 --> 00:28:15,505 live with that. K. What's K? Well if you look at the denominator when f is equal to 254 00:28:15,517 --> 00:28:24,598 0 then the, the gain of our of our lowpass filter at the origin, is now K in 255 00:28:24,610 --> 00:28:33,753 magnitude, so our transfer function starts out at K. You look at the rest of the 256 00:28:33,765 --> 00:28:41,428 formula, when F is equal to FC that makes this imaginaary number here component one, 257 00:28:41,563 --> 00:28:48,640 which mean that corresponds to the cut-off frequency, so. We have some FC out here 258 00:28:48,652 --> 00:28:55,214 and so in between we get our nice low pass filter where as 1 over the square root of 259 00:28:55,226 --> 00:29:01,311 two times K, So this formula is a little bit different for the RC filter because 260 00:29:01,323 --> 00:29:07,189 now I'm going to have to pick element values to fit this transfer function that 261 00:29:07,201 --> 00:29:13,408 hasn't gained but has a cut off frequency of f c. So how would we do that? Well one 262 00:29:13,420 --> 00:29:21,222 way is just to mimic the formula that we have. We could set ZF equal to our gain, 263 00:29:21,373 --> 00:29:28,918 because it's upstairs. And we'll set Z equal to that, because it's downstairs. 264 00:29:29,461 --> 00:29:36,859 what's the impedance 1 plus j times f, constant times f. And that looks like a 265 00:29:36,871 --> 00:29:44,093 resistor in series with an inductor. Now for all kinds of reasons, inductors are 266 00:29:44,105 --> 00:29:50,819 not used very frequently and filter circuits can be larger than capacitors. 267 00:29:51,072 --> 00:29:56,394 And we tend to want to shy away from using the inductor unless we really have to. So 268 00:29:56,678 --> 00:30:02,247 this may satisfy the if you built this with a, inductor and resistor in series in 269 00:30:02,259 --> 00:30:07,927 the feedback part I'm sorry, in the input part, you certainly would come up with a 270 00:30:07,939 --> 00:30:13,930 low-pass, but it's not the best way to do it. Let's think about this a little. Okay. 271 00:30:13,932 --> 00:30:21,420 How about this suggestion? Suppose that feedback had this for impedance. And 272 00:30:21,432 --> 00:30:29,485 suppose the input of impedance was 1 over K, well that means that it's a resistor 273 00:30:29,497 --> 00:30:36,492 with a value of 1 over K. And the question is, how do you interpret this thing? And 274 00:30:36,504 --> 00:30:43,014 what that's telling me is that I have two elements in parallel because 1 over, you 275 00:30:43,026 --> 00:30:49,213 have a parallel combination of two elements, two impedances. That's 1 over 1 276 00:30:49,225 --> 00:30:57,520 over Z1 plus 1 over Z2. Which will give us the old parallel form of the product where 277 00:30:57,532 --> 00:31:05,864 the sum. Okay? So, what this looks like is that Z1 is a 1, so it's a resistor. And 278 00:31:05,876 --> 00:31:13,904 Z2, at 1 over Z2 looks like that which means it's a capacitor. So, this is what I 279 00:31:13,916 --> 00:31:19,855 would like to put in the feedback circuit, is a resistor in parallel with a 280 00:31:20,190 --> 00:31:26,568 capacitor. Right now we don't have very good values here, if we want to gain 281 00:31:26,580 --> 00:31:32,582 bigger than 1, that means the input here is a really s mall resistor. And I 282 00:31:32,594 --> 00:31:38,543 certainly don't like this 1 ohm resistor sitting here. But I know the structure I 283 00:31:38,555 --> 00:31:44,579 want to head for, I'm going to put that in the feedback of the resistor in input and 284 00:31:44,591 --> 00:31:50,233 let's just reanalyze it with those, so we can make our choices. So here's the 285 00:31:50,245 --> 00:31:57,008 circuit I want to consider. There's the parallel combination of the resistor and 286 00:31:57,020 --> 00:32:03,712 capacitor on the feedback part. And here's the input resistor. And we know that the 287 00:32:03,724 --> 00:32:09,904 transfer function using s is got RF in parallel with a capacitor Okay, you see 288 00:32:09,916 --> 00:32:15,139 it. That's the shorthand notation. And it's going to be divided by R, and there's 289 00:32:15,151 --> 00:32:20,522 our minus sign. And we start plugging things in, like, particularly, what's the 290 00:32:20,534 --> 00:32:26,360 impedance of the capacitor, and simplify, you'll finally get this. Okay, so now 291 00:32:26,372 --> 00:32:35,353 let's see in terms of these three circuit element values the transfer function 292 00:32:35,365 --> 00:32:43,613 changes. So clearly at 0 frequency, the gain k is r f over r. So those two element 293 00:32:43,625 --> 00:32:51,520 values control the gain at 0 frequency. The cutoff frequency is 1 over 2 pi RF CF 294 00:32:51,532 --> 00:32:58,997 so these two control the cutoff frequency and you should not be surprised by that 295 00:32:59,009 --> 00:33:06,649 this happened for the RC low pass 4 over 2 there was a product of R and C that gave 296 00:33:06,661 --> 00:33:12,923 us what we want. So, we can pick element values. And we have three elements, but 297 00:33:12,935 --> 00:33:18,951 we're only constrained by two values, so we have some flexibility to pick and get 298 00:33:18,963 --> 00:33:24,802 the gain we want, and to pick the cutoff frequency we want for our low pass. And 299 00:33:24,814 --> 00:33:29,691 this is a very commonly used circuit. For low-pass filters. Okay. 300 00:33:29,691 --> 00:33:31,041 So. Electronics. 301 00:33:31,045 --> 00:33:36,798 What are electronic circuits? They're the ones that provide power gains. They have 302 00:33:36,810 --> 00:33:41,920 to have a power supply, or a battery, that's in the circuit to provide that 303 00:33:41,932 --> 00:33:47,506 power gain. Otherwise you cannot have a power gain with what's called electrical 304 00:33:47,518 --> 00:33:52,771 circuits. And that's only possible to power supply. And most circuits today are 305 00:33:52,783 --> 00:33:57,945 designed based on transistors. They could be integrated circuits or chips, but 306 00:33:57,957 --> 00:34:03,263 inside they're based on transistor theory. You have to learn about transistors in 307 00:34:03,275 --> 00:34:08,199 another course. We're not going to talk about them in any great detail at all. 308 00:34:08,461 --> 00:34:14,357 With other things we want to do, and that is to understand signals and systems based 309 00:34:14,369 --> 00:34:17,099 on designing and using active filters.