1 00:00:00,012 --> 00:00:04,982 In this video, we're going to extend our knowledge of Equivalent Circuits to the 2 00:00:04,982 --> 00:00:08,527 most general case. And last time we asked, what does the 3 00:00:08,527 --> 00:00:13,292 source see? And we figured out what we mean by sources seeing things, but in 4 00:00:13,292 --> 00:00:17,752 this video we're going to talk about what the circuit elements see. 5 00:00:17,752 --> 00:00:23,347 This is going to open a door to, the generalization of what we've already 6 00:00:23,347 --> 00:00:28,991 talked about namely Thevenin and the Mayer-Norton equivalent circuits. 7 00:00:28,991 --> 00:00:35,033 These are very important to understand and appreciate the knowledge they give 8 00:00:35,033 --> 00:00:38,860 us. So let's Talk about a little bit of an 9 00:00:38,860 --> 00:00:42,999 extension of what we've already talked about. 10 00:00:42,999 --> 00:00:50,266 You recall this was our original circuit. We had a source, and our system was a 11 00:00:50,266 --> 00:00:57,842 series resistance and we now, we know That, for that case V-out was R2/1+R2 * 12 00:00:57,842 --> 00:01:04,678 VN, we use voltage divider to find it. Well, if you're going to actually use 13 00:01:04,678 --> 00:01:10,282 that voltage for something it has to be attached to the sync. 14 00:01:10,282 --> 00:01:16,754 Remember our whole system thinking that we have to send our message to the sink. 15 00:01:16,754 --> 00:01:23,368 And the input to most circuits that we care about look like a simple resistor. 16 00:01:23,368 --> 00:01:27,392 And so the question is, if I attach my circuit to. 17 00:01:27,392 --> 00:01:34,294 The receiving circuit which is going to look like a resistor, does that change 18 00:01:34,294 --> 00:01:40,231 the output voltage V out. You may say well the voltage v out is 19 00:01:40,231 --> 00:01:44,845 across both the RL and across our original R2. 20 00:01:44,845 --> 00:01:50,412 But in reality it's really across the parallel combination. 21 00:01:50,412 --> 00:01:56,592 Well that changes the resistance, so you should expect there might be at least the 22 00:01:56,592 --> 00:02:01,982 possibility of some change in V out, so I want to explore that. 23 00:02:01,982 --> 00:02:07,617 By the way, RL, this is standard terminology, this L here stands for load, 24 00:02:07,617 --> 00:02:10,817 and that's a term that comes from the power in the circuit. 25 00:02:11,872 --> 00:02:20,126 Well, how do we think about this? And here's the way I want to do this. 26 00:02:20,126 --> 00:02:28,960 I want to try to figure out what does the load resistor see? What is the circuit 27 00:02:28,960 --> 00:02:35,683 that it sees looking back in that way? That's going to be a key for this, so 28 00:02:35,683 --> 00:02:39,493 that's what I meant by what are the circuit elements C. 29 00:02:39,493 --> 00:02:45,247 Well, it's not just a simple resistor of course, there's a voltage source in here, 30 00:02:45,247 --> 00:02:48,851 so the series and parallel rules don't quite apply. 31 00:02:48,851 --> 00:02:53,803 However, we can use the Principle of Superposition to figure this out. 32 00:02:53,803 --> 00:02:59,362 If you think about this for a little bit If I drew a box around here, this just 33 00:02:59,362 --> 00:03:04,982 looks like a regular old generic circuit with a current define going in the 34 00:03:04,982 --> 00:03:08,462 positive side, of the voltage defined for it. 35 00:03:08,462 --> 00:03:13,842 And I could figure out the, voltage if you just gave me the current. 36 00:03:13,842 --> 00:03:20,427 But because of the presence of the source There's going to be an additional term, 37 00:03:20,427 --> 00:03:24,337 which is the part of v that's due to the source. 38 00:03:24,337 --> 00:03:30,792 So the principal of superposition says that voltage V will be equal to the sum 39 00:03:30,792 --> 00:03:37,602 of the voltage you get when you set the voltage source that's inside, V in to 0. 40 00:03:37,602 --> 00:03:42,874 Plus the part you get when you set the input current to 0. 41 00:03:42,874 --> 00:03:46,441 And that will give you the, the part of V, this is to the V in. 42 00:03:48,196 --> 00:03:54,663 Okay? So, let's do the first one. When we set the voltage source to 0, 43 00:03:54,663 --> 00:04:01,212 that's the same as a short circuit. So remember our voltage source supplies 44 00:04:01,212 --> 00:04:06,742 some voltage for any current. Well, if you set that source to zero now 45 00:04:06,742 --> 00:04:12,812 the voltage is zero for any current. That's the definition of an ideal wire, 46 00:04:12,812 --> 00:04:17,517 or a short circuit. So now, what do we see looking into The 47 00:04:17,517 --> 00:04:23,767 circuit, we see R1 in parallel with R2. And so we know that at least that 48 00:04:23,767 --> 00:04:30,217 component of the output voltage is given by a simple resistive like term. 49 00:04:30,217 --> 00:04:36,782 And now, if you go on to setting I to 0. So I'm going to turn the source back on. 50 00:04:36,782 --> 00:04:42,138 Our source is alive and well. In a setting i=0, means there's nothing 51 00:04:42,138 --> 00:04:47,855 attached to the terminals, and what we have is the same circuit that we had 52 00:04:47,855 --> 00:04:53,388 before that we already analyzed. You simply use voltage divider to figure 53 00:04:53,388 --> 00:04:57,310 out what vout is and this is what we got. Last time. 54 00:04:57,310 --> 00:05:03,267 So, v, is going to be the sum of 2 terms. If you look, think about, that, as a 55 00:05:03,267 --> 00:05:09,049 system, and you want to know what the v i relation is, you can use this super 56 00:05:09,049 --> 00:05:13,749 position principle. And we see that, at least for this case, 57 00:05:13,749 --> 00:05:19,214 that's what we get. We get that v is equal to some resistance 58 00:05:19,214 --> 00:05:24,612 linked term times the current plus a source linked term. 59 00:05:24,612 --> 00:05:31,200 Well, that turns out, opens the door to a really interesting result. 60 00:05:31,200 --> 00:05:38,127 So, let's define something. I'm going to call that r e q, equivalent 61 00:05:38,127 --> 00:05:43,627 resistance. I'm going to call the second term v e q, 62 00:05:43,627 --> 00:05:50,427 equivalent voltage. And I want to point out, that something, 63 00:05:50,427 --> 00:05:56,002 that of this form. Which can be written like that, that 64 00:05:56,002 --> 00:06:02,105 looks, that is the same via our relationship that applies to this 65 00:06:02,105 --> 00:06:06,849 circuit. And of course the subscripts EQ here mean 66 00:06:06,849 --> 00:06:11,914 equivalent. So these two circuits cannot, you cannot 67 00:06:11,914 --> 00:06:18,439 tell them apart Just from the terminals. They look identical because their VI 68 00:06:18,439 --> 00:06:24,547 relationships are exactly the same. So I can express in particular as a very 69 00:06:24,547 --> 00:06:30,622 simple circuit, a very complicated circuit, containing voltage sources. 70 00:06:30,622 --> 00:06:34,567 And resistors. We're going to show you in a second that 71 00:06:34,567 --> 00:06:39,721 you can in addition to alter sources, you can have current sources too. 72 00:06:39,721 --> 00:06:45,548 So this is called the Thevenin equivalent circuit, and it's named for a late 19th 73 00:06:45,548 --> 00:06:51,365 century enigneer Who developed the equivalent circuit idea, while he was 74 00:06:51,365 --> 00:06:55,937 learning to teach a course at the eco polytectic in Paris. 75 00:06:55,937 --> 00:07:00,534 So, that's quite interesting. But there's even more. 76 00:07:00,534 --> 00:07:05,932 So, here's our simple little circuit that we've been talking about. 77 00:07:05,932 --> 00:07:11,900 And here's the, Thevenin equivalent. relationship for it. 78 00:07:11,900 --> 00:07:18,406 I'm going to turn that equation around. I'm going to write this as i equals 79 00:07:18,406 --> 00:07:23,483 something, and I'm going to call that last term, IEQ. 80 00:07:23,483 --> 00:07:29,978 And I want to point out that that equisite of that equation Describes 81 00:07:29,978 --> 00:07:35,352 behavior of this circuit. Where there is a current source, in 82 00:07:35,352 --> 00:07:42,534 parallel with REQ, it's the same REQ that we had, when we talked about the Thevenin 83 00:07:42,534 --> 00:07:48,712 equivalent, there's no difference. This is called the Mayer-Norton. 84 00:07:48,712 --> 00:07:54,156 Equivalent circuit. Mayer-Norton is named for two engineers, 85 00:07:54,156 --> 00:08:01,245 Edward Norton who is from the US and Hans Ferdinand Mayer who is from Germany, they 86 00:08:01,245 --> 00:08:08,142 developed this idea, at the, almost exactly same time, but independently. 87 00:08:08,142 --> 00:08:12,043 So. We have two, different ways, of thinking 88 00:08:12,043 --> 00:08:13,982 about, a circuit. So. 89 00:08:13,982 --> 00:08:20,581 We think about these, terminals, this pair of terminals out here, as a place 90 00:08:20,581 --> 00:08:25,854 where you're about to attach something, some other circuit. 91 00:08:25,854 --> 00:08:30,682 And you want to know. What does that circuit look like? In 92 00:08:30,682 --> 00:08:36,664 order to simplify your calculations, you can describe it either as a voltage 93 00:08:36,664 --> 00:08:42,721 source in series with a resistor, or you can describe it as a current source in 94 00:08:42,721 --> 00:08:47,822 parallel with the same resistor. Whichever seems to be the easiest 95 00:08:47,822 --> 00:08:51,327 approach for your application. They both apply. 96 00:08:51,327 --> 00:08:55,942 They, you can do either one. So, this really simplifies things. 97 00:08:55,942 --> 00:09:01,512 In fact, this is a profound result. This says that no matter how complicated 98 00:09:01,512 --> 00:09:07,647 A circuit is in side consisting of sources, current sources, voltage sources 99 00:09:07,647 --> 00:09:11,660 and resistors. It all can be summarized as a simple 100 00:09:11,660 --> 00:09:16,043 voltage source and series with equivalent resistance. 101 00:09:16,043 --> 00:09:22,282 This equivalent resistance is sometimes cal, called the output resistance. 102 00:09:22,282 --> 00:09:30,624 And we'll see that this is an important quantity in just a second. 103 00:09:30,624 --> 00:09:41,236 Now, and these, this equation ties these two equivalent circuits together, so that 104 00:09:41,236 --> 00:09:48,922 VEQ is REQ times Ieq so if you find one. You, you've got all you need to find the 105 00:09:48,922 --> 00:09:54,022 other, no problem. So, one issue with finding equivalent 106 00:09:54,022 --> 00:10:01,647 circuits is, can you really, do it. Find the, equivalent voltage, equivalent 107 00:10:01,647 --> 00:10:05,525 resistance. Without looking inside the box can you 108 00:10:05,525 --> 00:10:11,247 make measurements just at the terminals to figure out what those quantities are? 109 00:10:11,247 --> 00:10:15,518 And the answer is yes. The first thing we do is to find what we 110 00:10:15,518 --> 00:10:20,762 call the open-circuit voltage and that means we set this current to zero. 111 00:10:20,762 --> 00:10:26,545 And we look at the voltage we get. Well, if you look at this equivalent 112 00:10:26,545 --> 00:10:30,605 circuit. If you set that current to 0, that means 113 00:10:30,605 --> 00:10:36,739 there's no voltage drop across that resistor, so v is just equal to v(eq). 114 00:10:36,739 --> 00:10:42,881 So if you do that for a real circuit You just open-circuit the terminals, don't 115 00:10:42,881 --> 00:10:46,657 attach anything, and you just measure the voltage. 116 00:10:46,657 --> 00:10:52,041 That voltage that you measure is the equivalent, the Thevenin equivalent 117 00:10:52,041 --> 00:10:55,521 voltage. Similarly, you can short-circuit the 118 00:10:55,521 --> 00:11:01,592 current, so what that means Is that you short circuit these terminals, you put a 119 00:11:01,592 --> 00:11:06,472 short across there. That sets the voltage to, zero of course 120 00:11:06,472 --> 00:11:11,937 and then you measure the current. And, when you do that, you look at this 121 00:11:11,937 --> 00:11:17,692 equivalent, circuit, that short circuit current, is just going to be minus VEQ, 122 00:11:17,692 --> 00:11:23,741 over REQ, and that's the negative. Of the Myer-Norton equivalent current 123 00:11:23,741 --> 00:11:30,780 source so, I'm making measurements at the terminal, the open circuit voltage and 124 00:11:30,780 --> 00:11:36,447 the short circuit current. You do not have to look inside the box, 125 00:11:36,447 --> 00:11:41,712 to figure out what the equivalent circuits are, so let's 126 00:11:41,712 --> 00:11:49,802 go on, and to a simple example. And so, to show these ideas, I want to 127 00:11:49,802 --> 00:11:57,017 find the equivalent circuits for this example circuit. 128 00:11:57,017 --> 00:12:03,419 And I could find, explicitly, the. Thevenin equivalent. 129 00:12:03,419 --> 00:12:10,005 I could find the minor Norton equivalent, but I'm just going to find, the 130 00:12:10,005 --> 00:12:14,698 equivalent, quantities, the Veq, Req, and Ieq. 131 00:12:14,698 --> 00:12:20,352 so, the first thing we can do is set all the sources to zero. 132 00:12:20,352 --> 00:12:25,749 If you set the source to zero in the Thevenin equivalent, that's your short 133 00:12:25,749 --> 00:12:30,635 circuit, and what you see from the terminals is just that resistor. 134 00:12:30,635 --> 00:12:36,026 If you set the current source to zero, that's the same as opening it up, and 135 00:12:36,026 --> 00:12:41,072 then again, you also see Req. So, here if you look inside the box we 136 00:12:41,072 --> 00:12:45,611 see we have a current source, a zero valued current source. 137 00:12:45,611 --> 00:12:51,060 It's the same as an open circuit, no current flows through any voltage. 138 00:12:51,060 --> 00:12:56,942 That's the definition of an open circuit. And what I see when I look this way. 139 00:12:56,942 --> 00:13:03,880 Into there, I see R3 is in parallel with this series combination. 140 00:13:03,880 --> 00:13:11,650 So we can use our series and parallel formulas and we get that for Req. 141 00:13:11,650 --> 00:13:18,002 So, that's the source resistance of the Rs, of our circuit. 142 00:13:18,002 --> 00:13:24,903 Now, suppose we want to find VEQ. I'm just trying to use our terminal only 143 00:13:24,903 --> 00:13:31,679 results, I'm going to open circuit voltage so there's nothing across here, 144 00:13:31,679 --> 00:13:35,422 this is 0. And now we measure the voltage. 145 00:13:35,422 --> 00:13:41,201 Well this voltage. Is, going to be given by some dependence 146 00:13:41,201 --> 00:13:48,490 on this current source, and since it's a current source, it's a whole lot easier 147 00:13:48,490 --> 00:13:55,467 to find the current through here, and then I'll multiply that by R3 to get the 148 00:13:55,467 --> 00:13:57,042 voltage. Well. 149 00:13:57,042 --> 00:14:02,675 Because we have this in parallel with this now, from the point of view of the 150 00:14:02,675 --> 00:14:07,620 volt of the current source. I'm going to use current divider which is 151 00:14:07,620 --> 00:14:13,119 going to be the other resistor divided by the sum of that one plus this series 152 00:14:13,119 --> 00:14:16,754 combination. And when you all get said and done, 153 00:14:16,754 --> 00:14:21,322 that's what you get. So, here's the, current divider part, and 154 00:14:21,322 --> 00:14:24,409 then I just multiply by R3 to get our voltage. 155 00:14:24,409 --> 00:14:28,762 So that is the Thevinin equivalent, voltage, for this circuit. 156 00:14:28,762 --> 00:14:33,925 It's a little more complicated than you might think, but it's pretty easy to 157 00:14:33,925 --> 00:14:36,835 find. And then finally I'm going to find IEQ, 158 00:14:36,835 --> 00:14:40,772 which I'm going to do by finding the short circuit current. 159 00:14:40,772 --> 00:14:50,141 And I just put a wire across here, and, so what that does by setting the voltage 160 00:14:50,141 --> 00:14:59,146 to zero, that shorts out this, so that resistor isn't there, doesn't matter if 161 00:14:59,146 --> 00:15:04,227 it's there or not. And so, again, I'm going to use current 162 00:15:04,227 --> 00:15:07,909 divider, because I have a current source here. 163 00:15:07,909 --> 00:15:13,752 Well, the current i is going to be, i is going to be the negative of that current, 164 00:15:13,752 --> 00:15:19,847 and I used current divider, so current divider says that current is R1 divided 165 00:15:19,847 --> 00:15:25,910 by the sum of the resistances times I in. And I stick a negative sign in front 166 00:15:25,910 --> 00:15:33,926 because of the directionality of the current and I get this for isc, the short 167 00:15:33,926 --> 00:15:39,606 circuit current which, I know, is the negative of Ieq. 168 00:15:39,606 --> 00:15:47,784 So, we could have divided The equivalent voltage, by the equivalent, current, 169 00:15:47,784 --> 00:15:51,742 defined req. We u-, in finding req up here, what we 170 00:15:51,742 --> 00:15:56,718 did was look inside the box. And we don't really have to do that. 171 00:15:56,718 --> 00:16:01,380 But it's nice to know we have a way of checking your answers. 172 00:16:01,380 --> 00:16:07,445 So if the source is to zero to find Req. Find the open circuit voltage to find an 173 00:16:07,445 --> 00:16:12,200 evident equivalent voltage. find the short circuit current. 174 00:16:12,200 --> 00:16:18,139 Find the Meyer-Norton equivalent current source, and then your ratios should work 175 00:16:18,139 --> 00:16:21,467 out to be the other. Everything should check. 176 00:16:21,467 --> 00:16:24,657 Okay. So, what did we discover? Well, we 177 00:16:24,657 --> 00:16:31,052 discovered that all circuits consisting of sources and resistors are equivalent 178 00:16:31,052 --> 00:16:37,697 to one source, be it a voltage source or a current source, that's in series or in 179 00:16:37,697 --> 00:16:43,120 parallel with a single resistor. No matter how complicated that circuit 180 00:16:43,120 --> 00:16:46,106 is. From the point of view of the connection 181 00:16:46,106 --> 00:16:50,104 points, the terminals. It is a very, very simple circuit. 182 00:16:50,104 --> 00:16:55,468 So, that means that, we can use either of these equivalent, circuits. 183 00:16:55,468 --> 00:17:00,657 The Thévenin or Mayer-Norton form. The voltage source or the current source 184 00:17:00,657 --> 00:17:03,517 form. Whichever seems to be appropriate. 185 00:17:03,517 --> 00:17:07,262 It makes solving the problem easy. What I mean by that. 186 00:17:07,262 --> 00:17:12,787 Is you're now going to attach something to the terminals, and, and des, determine 187 00:17:12,787 --> 00:17:16,162 things. And, depending on what you're interested 188 00:17:16,162 --> 00:17:21,462 in finding, how complicated the circuit is you attach, you would use one or the 189 00:17:21,462 --> 00:17:22,312 other. Now. 190 00:17:22,312 --> 00:17:27,946 Remember how we got started on this. Remember we were attaching our, Sync to 191 00:17:27,946 --> 00:17:32,506 our original circuit That we've been talking about. 192 00:17:32,506 --> 00:17:40,150 And the issue was what's called Loading. And, Loading is a term that means, that 193 00:17:40,150 --> 00:17:47,065 if I attach to the wrong kind of load resistor for the given application. 194 00:17:47,065 --> 00:17:53,404 This V out will change. That's called loading down the original 195 00:17:53,404 --> 00:17:58,337 circuit. Well, I want to figure out what values of 196 00:17:58,337 --> 00:18:06,220 RL will not contribute to loading. What kind of resistor values can the sink 197 00:18:06,220 --> 00:18:15,726 have so the The alk doesn't change from what it is if nothing is attached. 198 00:18:15,726 --> 00:18:23,952 Well, the, we now know that our system has, looks like this. 199 00:18:23,952 --> 00:18:28,576 This, which is, I'm going to use the Thevenin equivalent. 200 00:18:28,576 --> 00:18:35,017 Because I have voltage that's the output. And the voltage I want is across RL. 201 00:18:35,017 --> 00:18:41,176 Well, what is v out for this really simple circuit? Well, it's the voltage 202 00:18:41,176 --> 00:18:43,914 divider. It's very easy to find. 203 00:18:43,914 --> 00:18:49,181 And, what values of RL. We'll allow V out to be = VEQ. 204 00:18:49,181 --> 00:18:55,579 Because when RL is infinity, that's the same as attaching nothing. 205 00:18:55,579 --> 00:19:01,811 and that's what we started. So I know that the output that I want is 206 00:19:01,811 --> 00:19:08,112 VEQ, so I think it's pretty clear that what we want is RL to be big. 207 00:19:08,112 --> 00:19:14,973 But big relative to what? And the answer is it has to be big 208 00:19:14,973 --> 00:19:21,056 relative to Req, which is R1 in parallel with r 2. 209 00:19:21,056 --> 00:19:29,569 So as long as the load is big compared to the Whatever this parallel, resistance 210 00:19:29,569 --> 00:19:34,880 turns out to be, then the, you can attach the sync to the circuit and it won't 211 00:19:34,880 --> 00:19:38,456 change V out. However, if RL is smaller than that 212 00:19:38,456 --> 00:19:43,486 quantity or comparable to it, that outputs going to change and that may not 213 00:19:43,486 --> 00:19:46,732 be what you want It depends on the application. 214 00:19:46,732 --> 00:19:51,595 This little, simple example points out the power of using equivalent circuits.