WEBVTT 00:00:00.000 --> 00:00:04.562 In this video we're gonna take a look at using node analysis techniques to analyze 00:00:04.562 --> 00:00:06.952 a circuit that's got a dependent source. 00:00:06.952 --> 00:00:11.895 You'll notice right off the bat that we got a dependent current source 00:00:11.895 --> 00:00:13.413 here in this branch. 00:00:13.413 --> 00:00:18.187 This dependent current source, the amount of current that this current is 00:00:18.187 --> 00:00:23.037 producing depends upon the current i flowing through the 6 ohm resistor and 00:00:23.037 --> 00:00:26.238 this current source, then, is two times that I. 00:00:26.238 --> 00:00:30.226 We're gonna take that into account as we go through and define our node voltages 00:00:30.226 --> 00:00:33.870 and we'll come up with an expression for I in terms of our node voltages. 00:00:33.870 --> 00:00:36.028 So let's go ahead and get started. 00:00:36.028 --> 00:00:37.535 We've got three critical nodes again. 00:00:37.535 --> 00:00:40.108 One here, one here, and one along the bottom. 00:00:40.108 --> 00:00:43.699 Once again, we'll go ahead and call this our reference node, 00:00:43.699 --> 00:00:45.409 where we've got v equals 0. 00:00:45.409 --> 00:00:50.710 And once again, we've got a voltage source tied to the reference, 00:00:50.710 --> 00:00:56.941 so we know the voltage here at this point of our circuit is equal to 5.3 volts. 00:00:56.941 --> 00:01:03.747 We'll define this node here to be V1, this node here to be V2. 00:01:03.747 --> 00:01:08.575 And with those two defined voltages now we can express i, 00:01:08.575 --> 00:01:11.702 the current i In terms of V1 and V2. 00:01:11.702 --> 00:01:16.685 In fact I is equal to V1 minus V2 divided by 6 ohms, 00:01:16.685 --> 00:01:20.878 it's just the voltage here on the left minus 00:01:20.878 --> 00:01:25.091 the voltage on the right divided by 6 ohms. 00:01:28.238 --> 00:01:31.040 When we're all done with this analysis, 00:01:31.040 --> 00:01:35.126 we're going to use that expression to calculate I let's say 00:01:35.126 --> 00:01:39.547 to calculate the power being produced by this dependent source. 00:01:39.547 --> 00:01:41.818 With those definitions, let's go ahead and 00:01:41.818 --> 00:01:44.096 write the node equations at these two nodes. 00:01:44.096 --> 00:01:46.893 First one here on the left the current leaving. 00:01:46.893 --> 00:01:51.070 The first node here going to the left is going to be, 00:01:51.070 --> 00:01:55.734 V1-5.3 divided by 4 ohms + the current coming down 00:01:55.734 --> 00:02:00.590 here through the 3 ohm resistor is going to be V1 divided by 00:02:00.590 --> 00:02:05.084 3 + the current leaving this node going to the right. 00:02:05.084 --> 00:02:09.395 Well notice that that's that current I has been defined for us. 00:02:09.395 --> 00:02:13.982 We don't want to use I we want to use or we want an expression for 00:02:13.982 --> 00:02:18.306 that current in terms of the node voltages V1 and V2, and 00:02:18.306 --> 00:02:23.706 we've already expressed that I over here as V1 minus V2 divided by 6. 00:02:23.706 --> 00:02:28.896 So plus V1-V2 divided by 6. 00:02:28.896 --> 00:02:32.174 The sum of those three currents has to equal 0. 00:02:32.174 --> 00:02:35.690 Now let's write the node equation of the V2 node. 00:02:35.690 --> 00:02:40.736 Current going to the left is going to 00:02:40.736 --> 00:02:48.319 be V2-V1 divided by 6 + V2-0 divided by 12. 00:02:49.979 --> 00:02:53.415 Minus, okay, now let's be careful. 00:02:53.415 --> 00:02:59.583 Minus two times I, it's minus because it's referenced into the node. 00:02:59.583 --> 00:03:05.079 Two times I is this dependent source, but I don't want to put I in here. 00:03:05.079 --> 00:03:08.559 We want to write everything in terms of our node variables or 00:03:08.559 --> 00:03:10.727 node voltage variables V1 and V2. 00:03:10.727 --> 00:03:16.754 So I've got i is equal to two, or v1 minus v2 divided by 6. 00:03:16.754 --> 00:03:21.972 This current here is two times I so it's going 00:03:21.972 --> 00:03:27.051 to be -2 times v1 minus v2 divided by 6, 00:03:27.051 --> 00:03:32.011 the sum of those three currents equals 0. 00:03:32.011 --> 00:03:35.813 I suggest that you stop the video right now and just take a look at that, 00:03:35.813 --> 00:03:38.147 there's a couple of things going on here. 00:03:38.147 --> 00:03:41.612 The fact that it's going into the node gives us the minus sign. 00:03:41.612 --> 00:03:43.496 It's gonna be minus the current flowing in. 00:03:43.496 --> 00:03:47.698 Well, the current flowing in is 2 times or 00:03:47.698 --> 00:03:51.333 has a value of 2 times whatever I is. 00:03:51.333 --> 00:03:55.361 We defined I as V1- V2 divided by 6. 00:03:55.361 --> 00:04:02.760 So we have, then, -2 times V1- V2 divided by 6. 00:04:02.760 --> 00:04:06.943 Alrighty, now that you've taken a moment to make sure you understand what we did 00:04:06.943 --> 00:04:09.096 there with the signs, let's go ahead and 00:04:09.096 --> 00:04:13.176 solve this system of two equations with two unknowns by combining like terms. 00:04:13.176 --> 00:04:19.741 For the first equation we've got 00:04:19.741 --> 00:04:27.012 V1 times one-fourth + one-third 00:04:27.012 --> 00:04:32.648 + one- sixth + V2 times. 00:04:32.648 --> 00:04:38.323 This is just one V2 term that's got a negative one- sixth with it is 00:04:38.323 --> 00:04:45.209 equal to negative 5 divided by 4 take it to the other side as I'm sorry negative. 00:04:45.209 --> 00:04:49.303 Let's see, it's -5.3 divided by 4, 00:04:49.303 --> 00:04:53.950 take it to the other side is a +5.3 divided by 4 00:04:53.950 --> 00:04:58.836 will write it like that for now 5.3 divided by 4. 00:04:58.836 --> 00:05:00.549 Alrighty, the second equation and 00:05:00.549 --> 00:05:04.204 here we're going to need to be careful because we've got this stuff going on 00:05:04.204 --> 00:05:07.479 there with the last term but once again factoring out the V1 terms. 00:05:07.479 --> 00:05:12.559 Here I've got a V1 term and I've got a V1 term here, let's be careful there. 00:05:12.559 --> 00:05:16.105 The first one is negative one- sixth. 00:05:16.105 --> 00:05:18.785 But what do I actually have over here? 00:05:18.785 --> 00:05:24.457 I've got -2 times V1 divided by 6. 00:05:24.457 --> 00:05:28.759 So that's negative -2 over 6, that's negative one third V1. 00:05:28.759 --> 00:05:33.495 Factor out the V1, and that leaves me 00:05:33.495 --> 00:05:37.935 with a negative one-third there. 00:05:37.935 --> 00:05:42.855 Alrighty, now for the V2 terms, + V2 times, 00:05:42.855 --> 00:05:49.939 I've got a V2 over 6, factoring out the V2 leaves me a one-sixth. 00:05:49.939 --> 00:05:54.332 I've got a V2 over 12, so + one-twelfth. 00:05:54.332 --> 00:05:55.969 Now once again, slow down here. 00:05:55.969 --> 00:05:58.337 You've got signs and signs like red flags waving. 00:05:58.337 --> 00:06:02.016 That means slow down, you've got an opportunity to make a sign error and 00:06:02.016 --> 00:06:03.476 we don't like sign errors. 00:06:03.476 --> 00:06:07.469 We've got -2 times a -V2 over 6. 00:06:07.469 --> 00:06:12.287 Negative times a negative is a positive to V2 over 6, 00:06:12.287 --> 00:06:16.582 so that's gonna be positive factor out the V2, 00:06:16.582 --> 00:06:23.729 I've got 6 that's one-third, and the sum of those three currents equals 0. 00:06:23.729 --> 00:06:26.534 Once again, I'd encourage you to stop the video at this point and 00:06:26.534 --> 00:06:29.191 just make sure that you understand what we did with the signs. 00:06:29.191 --> 00:06:36.770 Alrighty, let's just Be careful here that we keep our equations separate, 00:06:36.770 --> 00:06:41.797 and now let's go ahead and combine the fractions. 00:06:41.797 --> 00:06:46.727 We've got here then for the first equation we have v1 times 00:06:46.727 --> 00:06:52.159 one-fourth plus one-third plus one-sixth is three-forth. 00:06:55.355 --> 00:07:01.517 Plus V2 time negative one-sixth 00:07:01.517 --> 00:07:08.534 = 5.3 divided by 4 is 1.325. 00:07:08.534 --> 00:07:13.496 And on the second equation we have V1 times negative 00:07:13.496 --> 00:07:18.813 one-sixth minus one-third is a negative one-half. 00:07:21.299 --> 00:07:26.348 Plus V2 times one-sixth + one-twelfths 00:07:26.348 --> 00:07:33.859 + one-third is seven-twelfths and it's positive equals 0. 00:07:35.075 --> 00:07:39.686 So we've got our two equations there. 00:07:39.686 --> 00:07:41.703 Once again, plug that into your matrix solver, 00:07:41.703 --> 00:07:44.915 use the solve button on your calculator or you can use MATLAB for that matter. 00:07:44.915 --> 00:07:49.453 And when you do, you'll get 00:07:49.453 --> 00:07:54.561 that V1 equals 2.1824, 00:07:54.561 --> 00:08:01.569 2.1824 and V2 is equal to 1.87. 00:08:01.569 --> 00:08:06.551 Now what can we do with V1 and V2 while we can do anything we need to do. 00:08:06.551 --> 00:08:10.933 Just for example in this one, why do we calculate the power of this being 00:08:10.933 --> 00:08:13.418 generated by this dependency source and 00:08:13.418 --> 00:08:18.185 let's be careful here because we've got issues with reference direction. 00:08:18.185 --> 00:08:23.755 There is a dependence source it has a value 00:08:23.755 --> 00:08:28.537 of 2 times I, reference going up. 00:08:28.537 --> 00:08:34.784 And what is the voltage across that power supply, well is the reference plus to 00:08:34.784 --> 00:08:41.033 minus V2 that power supply that dependent current source has V2 on the top and 00:08:41.033 --> 00:08:47.394 it's tied on the bottom to our reference so the voltage across that is just V2. 00:08:47.394 --> 00:08:53.039 Now we know that power is equal to I times V. 00:08:55.068 --> 00:09:00.371 And we also know that we got to be careful about the sign in front of this. 00:09:00.371 --> 00:09:04.835 Our passive sign convention says that if current is referenced going from 00:09:04.835 --> 00:09:09.016 the positive to the negative terminal, it will be a positive power. 00:09:09.016 --> 00:09:12.765 That's not what we have here, in this case, we have the current going up, 00:09:12.765 --> 00:09:16.350 referenced into the negative terminal and out the positive terminal. 00:09:16.350 --> 00:09:19.632 So that means that we need a minus sign there. 00:09:19.632 --> 00:09:21.490 Now, what is I? 00:09:21.490 --> 00:09:25.682 Well, we've got I up here, I is V1- V2 divided by 6. 00:09:25.682 --> 00:09:27.963 So let's go ahead and calculate that. 00:09:27.963 --> 00:09:35.380 V1 was 2.1824 minus V2. 00:09:35.380 --> 00:09:40.422 Which was 1.87-1.87, 00:09:40.422 --> 00:09:45.648 and the whole thing divided by 6. 00:09:45.648 --> 00:09:50.111 That gives us a value for 00:09:50.111 --> 00:09:54.798 I of 0.052 amps, or 00:09:54.798 --> 00:10:02.174 52 milliamps, so 0.052 amps. 00:10:02.174 --> 00:10:08.673 Now this current source right here is equal to if I sub x, 00:10:08.673 --> 00:10:14.758 the current that it generates is equal to 2 times I, 00:10:14.758 --> 00:10:18.100 we now know I is 0.052. 00:10:18.100 --> 00:10:22.513 So that current source is producing 00:10:22.513 --> 00:10:27.078 2 times .052 or .104 amps. 00:10:27.078 --> 00:10:31.664 That's the current that we need down here to calculate the power. 00:10:31.664 --> 00:10:38.569 This current source is producing 0.104 amps going up. 00:10:38.569 --> 00:10:45.324 So the power down here is going to be equal to negative I is 0.104. 00:10:48.760 --> 00:10:50.089 Times V, which V did we say it was? 00:10:50.089 --> 00:10:53.160 It was V2. 00:10:53.160 --> 00:10:59.284 So times V2, which is 1.87 volts, 00:10:59.284 --> 00:11:05.767 and when you multiple those two together, 00:11:05.767 --> 00:11:11.541 you get a negative 0.194 watts. 00:11:11.541 --> 00:11:14.473 Negative, what does the negative sign mean on power? 00:11:14.473 --> 00:11:17.197 It means that it's producing energy, 00:11:17.197 --> 00:11:20.919 it's the source is putting energy into the circuit.