1 00:00:00,000 --> 00:00:04,562 In this video we're gonna take a look at using node analysis techniques to analyze 2 00:00:04,562 --> 00:00:06,952 a circuit that's got a dependent source. 3 00:00:06,952 --> 00:00:11,895 You'll notice right off the bat that we got a dependent current source 4 00:00:11,895 --> 00:00:13,413 here in this branch. 5 00:00:13,413 --> 00:00:18,187 This dependent current source, the amount of current that this current is 6 00:00:18,187 --> 00:00:23,037 producing depends upon the current i flowing through the 6 ohm resistor and 7 00:00:23,037 --> 00:00:26,238 this current source, then, is two times that I. 8 00:00:26,238 --> 00:00:30,226 We're gonna take that into account as we go through and define our node voltages 9 00:00:30,226 --> 00:00:33,870 and we'll come up with an expression for I in terms of our node voltages. 10 00:00:33,870 --> 00:00:36,028 So let's go ahead and get started. 11 00:00:36,028 --> 00:00:37,535 We've got three critical nodes again. 12 00:00:37,535 --> 00:00:40,108 One here, one here, and one along the bottom. 13 00:00:40,108 --> 00:00:43,699 Once again, we'll go ahead and call this our reference node, 14 00:00:43,699 --> 00:00:45,409 where we've got v equals 0. 15 00:00:45,409 --> 00:00:50,710 And once again, we've got a voltage source tied to the reference, 16 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. 17 00:00:56,941 --> 00:01:03,747 We'll define this node here to be V1, this node here to be V2. 18 00:01:03,747 --> 00:01:08,575 And with those two defined voltages now we can express i, 19 00:01:08,575 --> 00:01:11,702 the current i In terms of V1 and V2. 20 00:01:11,702 --> 00:01:16,685 In fact I is equal to V1 minus V2 divided by 6 ohms, 21 00:01:16,685 --> 00:01:20,878 it's just the voltage here on the left minus 22 00:01:20,878 --> 00:01:25,091 the voltage on the right divided by 6 ohms. 23 00:01:28,238 --> 00:01:31,040 When we're all done with this analysis, 24 00:01:31,040 --> 00:01:35,126 we're going to use that expression to calculate I let's say 25 00:01:35,126 --> 00:01:39,547 to calculate the power being produced by this dependent source. 26 00:01:39,547 --> 00:01:41,818 With those definitions, let's go ahead and 27 00:01:41,818 --> 00:01:44,096 write the node equations at these two nodes. 28 00:01:44,096 --> 00:01:46,893 First one here on the left the current leaving. 29 00:01:46,893 --> 00:01:51,070 The first node here going to the left is going to be, 30 00:01:51,070 --> 00:01:55,734 V1-5.3 divided by 4 ohms + the current coming down 31 00:01:55,734 --> 00:02:00,590 here through the 3 ohm resistor is going to be V1 divided by 32 00:02:00,590 --> 00:02:05,084 3 + the current leaving this node going to the right. 33 00:02:05,084 --> 00:02:09,395 Well notice that that's that current I has been defined for us. 34 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 35 00:02:13,982 --> 00:02:18,306 that current in terms of the node voltages V1 and V2, and 36 00:02:18,306 --> 00:02:23,706 we've already expressed that I over here as V1 minus V2 divided by 6. 37 00:02:23,706 --> 00:02:28,896 So plus V1-V2 divided by 6. 38 00:02:28,896 --> 00:02:32,174 The sum of those three currents has to equal 0. 39 00:02:32,174 --> 00:02:35,690 Now let's write the node equation of the V2 node. 40 00:02:35,690 --> 00:02:40,736 Current going to the left is going to 41 00:02:40,736 --> 00:02:48,319 be V2-V1 divided by 6 + V2-0 divided by 12. 42 00:02:49,979 --> 00:02:53,415 Minus, okay, now let's be careful. 43 00:02:53,415 --> 00:02:59,583 Minus two times I, it's minus because it's referenced into the node. 44 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. 45 00:03:05,079 --> 00:03:08,559 We want to write everything in terms of our node variables or 46 00:03:08,559 --> 00:03:10,727 node voltage variables V1 and V2. 47 00:03:10,727 --> 00:03:16,754 So I've got i is equal to two, or v1 minus v2 divided by 6. 48 00:03:16,754 --> 00:03:21,972 This current here is two times I so it's going 49 00:03:21,972 --> 00:03:27,051 to be -2 times v1 minus v2 divided by 6, 50 00:03:27,051 --> 00:03:32,011 the sum of those three currents equals 0. 51 00:03:32,011 --> 00:03:35,813 I suggest that you stop the video right now and just take a look at that, 52 00:03:35,813 --> 00:03:38,147 there's a couple of things going on here. 53 00:03:38,147 --> 00:03:41,612 The fact that it's going into the node gives us the minus sign. 54 00:03:41,612 --> 00:03:43,496 It's gonna be minus the current flowing in. 55 00:03:43,496 --> 00:03:47,698 Well, the current flowing in is 2 times or 56 00:03:47,698 --> 00:03:51,333 has a value of 2 times whatever I is. 57 00:03:51,333 --> 00:03:55,361 We defined I as V1- V2 divided by 6. 58 00:03:55,361 --> 00:04:02,760 So we have, then, -2 times V1- V2 divided by 6. 59 00:04:02,760 --> 00:04:06,943 Alrighty, now that you've taken a moment to make sure you understand what we did 60 00:04:06,943 --> 00:04:09,096 there with the signs, let's go ahead and 61 00:04:09,096 --> 00:04:13,176 solve this system of two equations with two unknowns by combining like terms. 62 00:04:13,176 --> 00:04:19,741 For the first equation we've got 63 00:04:19,741 --> 00:04:27,012 V1 times one-fourth + one-third 64 00:04:27,012 --> 00:04:32,648 + one- sixth + V2 times. 65 00:04:32,648 --> 00:04:38,323 This is just one V2 term that's got a negative one- sixth with it is 66 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. 67 00:04:45,209 --> 00:04:49,303 Let's see, it's -5.3 divided by 4, 68 00:04:49,303 --> 00:04:53,950 take it to the other side is a +5.3 divided by 4 69 00:04:53,950 --> 00:04:58,836 will write it like that for now 5.3 divided by 4. 70 00:04:58,836 --> 00:05:00,549 Alrighty, the second equation and 71 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 72 00:05:04,204 --> 00:05:07,479 there with the last term but once again factoring out the V1 terms. 73 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. 74 00:05:12,559 --> 00:05:16,105 The first one is negative one- sixth. 75 00:05:16,105 --> 00:05:18,785 But what do I actually have over here? 76 00:05:18,785 --> 00:05:24,457 I've got -2 times V1 divided by 6. 77 00:05:24,457 --> 00:05:28,759 So that's negative -2 over 6, that's negative one third V1. 78 00:05:28,759 --> 00:05:33,495 Factor out the V1, and that leaves me 79 00:05:33,495 --> 00:05:37,935 with a negative one-third there. 80 00:05:37,935 --> 00:05:42,855 Alrighty, now for the V2 terms, + V2 times, 81 00:05:42,855 --> 00:05:49,939 I've got a V2 over 6, factoring out the V2 leaves me a one-sixth. 82 00:05:49,939 --> 00:05:54,332 I've got a V2 over 12, so + one-twelfth. 83 00:05:54,332 --> 00:05:55,969 Now once again, slow down here. 84 00:05:55,969 --> 00:05:58,337 You've got signs and signs like red flags waving. 85 00:05:58,337 --> 00:06:02,016 That means slow down, you've got an opportunity to make a sign error and 86 00:06:02,016 --> 00:06:03,476 we don't like sign errors. 87 00:06:03,476 --> 00:06:07,469 We've got -2 times a -V2 over 6. 88 00:06:07,469 --> 00:06:12,287 Negative times a negative is a positive to V2 over 6, 89 00:06:12,287 --> 00:06:16,582 so that's gonna be positive factor out the V2, 90 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. 91 00:06:23,729 --> 00:06:26,534 Once again, I'd encourage you to stop the video at this point and 92 00:06:26,534 --> 00:06:29,191 just make sure that you understand what we did with the signs. 93 00:06:29,191 --> 00:06:36,770 Alrighty, let's just Be careful here that we keep our equations separate, 94 00:06:36,770 --> 00:06:41,797 and now let's go ahead and combine the fractions. 95 00:06:41,797 --> 00:06:46,727 We've got here then for the first equation we have v1 times 96 00:06:46,727 --> 00:06:52,159 one-fourth plus one-third plus one-sixth is three-forth. 97 00:06:55,355 --> 00:07:01,517 Plus V2 time negative one-sixth 98 00:07:01,517 --> 00:07:08,534 = 5.3 divided by 4 is 1.325. 99 00:07:08,534 --> 00:07:13,496 And on the second equation we have V1 times negative 100 00:07:13,496 --> 00:07:18,813 one-sixth minus one-third is a negative one-half. 101 00:07:21,299 --> 00:07:26,348 Plus V2 times one-sixth + one-twelfths 102 00:07:26,348 --> 00:07:33,859 + one-third is seven-twelfths and it's positive equals 0. 103 00:07:35,075 --> 00:07:39,686 So we've got our two equations there. 104 00:07:39,686 --> 00:07:41,703 Once again, plug that into your matrix solver, 105 00:07:41,703 --> 00:07:44,915 use the solve button on your calculator or you can use MATLAB for that matter. 106 00:07:44,915 --> 00:07:49,453 And when you do, you'll get 107 00:07:49,453 --> 00:07:54,561 that V1 equals 2.1824, 108 00:07:54,561 --> 00:08:01,569 2.1824 and V2 is equal to 1.87. 109 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. 110 00:08:06,551 --> 00:08:10,933 Just for example in this one, why do we calculate the power of this being 111 00:08:10,933 --> 00:08:13,418 generated by this dependency source and 112 00:08:13,418 --> 00:08:18,185 let's be careful here because we've got issues with reference direction. 113 00:08:18,185 --> 00:08:23,755 There is a dependence source it has a value 114 00:08:23,755 --> 00:08:28,537 of 2 times I, reference going up. 115 00:08:28,537 --> 00:08:34,784 And what is the voltage across that power supply, well is the reference plus to 116 00:08:34,784 --> 00:08:41,033 minus V2 that power supply that dependent current source has V2 on the top and 117 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. 118 00:08:47,394 --> 00:08:53,039 Now we know that power is equal to I times V. 119 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. 120 00:09:00,371 --> 00:09:04,835 Our passive sign convention says that if current is referenced going from 121 00:09:04,835 --> 00:09:09,016 the positive to the negative terminal, it will be a positive power. 122 00:09:09,016 --> 00:09:12,765 That's not what we have here, in this case, we have the current going up, 123 00:09:12,765 --> 00:09:16,350 referenced into the negative terminal and out the positive terminal. 124 00:09:16,350 --> 00:09:19,632 So that means that we need a minus sign there. 125 00:09:19,632 --> 00:09:21,490 Now, what is I? 126 00:09:21,490 --> 00:09:25,682 Well, we've got I up here, I is V1- V2 divided by 6. 127 00:09:25,682 --> 00:09:27,963 So let's go ahead and calculate that. 128 00:09:27,963 --> 00:09:35,380 V1 was 2.1824 minus V2. 129 00:09:35,380 --> 00:09:40,422 Which was 1.87-1.87, 130 00:09:40,422 --> 00:09:45,648 and the whole thing divided by 6. 131 00:09:45,648 --> 00:09:50,111 That gives us a value for 132 00:09:50,111 --> 00:09:54,798 I of 0.052 amps, or 133 00:09:54,798 --> 00:10:02,174 52 milliamps, so 0.052 amps. 134 00:10:02,174 --> 00:10:08,673 Now this current source right here is equal to if I sub x, 135 00:10:08,673 --> 00:10:14,758 the current that it generates is equal to 2 times I, 136 00:10:14,758 --> 00:10:18,100 we now know I is 0.052. 137 00:10:18,100 --> 00:10:22,513 So that current source is producing 138 00:10:22,513 --> 00:10:27,078 2 times .052 or .104 amps. 139 00:10:27,078 --> 00:10:31,664 That's the current that we need down here to calculate the power. 140 00:10:31,664 --> 00:10:38,569 This current source is producing 0.104 amps going up. 141 00:10:38,569 --> 00:10:45,324 So the power down here is going to be equal to negative I is 0.104. 142 00:10:48,760 --> 00:10:50,089 Times V, which V did we say it was? 143 00:10:50,089 --> 00:10:53,160 It was V2. 144 00:10:53,160 --> 00:10:59,284 So times V2, which is 1.87 volts, 145 00:10:59,284 --> 00:11:05,767 and when you multiple those two together, 146 00:11:05,767 --> 00:11:11,541 you get a negative 0.194 watts. 147 00:11:11,541 --> 00:11:14,473 Negative, what does the negative sign mean on power? 148 00:11:14,473 --> 00:11:17,197 It means that it's producing energy, 149 00:11:17,197 --> 00:11:20,919 it's the source is putting energy into the circuit.