WEBVTT FILE 00:09.447 --> 00:14.027 In the previous lesson we saw that any set of differences meets Kirchhoff's second law. 00:14.027 --> 00:15.802 Or, more precisely, 00:15.802 --> 00:22.076 that the values assigned to the node pairs, and which derive from node potentials, are Kirchhoff voltages. 00:22.076 --> 00:25.147 Here we will see that the reciprocal statement is also true. 00:25.147 --> 00:30.939 That is, every set of Kirchhoff voltages derives from node potentials. 00:30.939 --> 00:36.319 Indeed, let us suppose that the pairs from a node set have Kirchhoff voltages assigned. 00:36.319 --> 00:40.285 This means that the values of any closed path sum up to zero. 00:40.285 --> 00:46.564 For example, ‘v sub AB’, plus ‘v sub BD’, plus ‘v sub DA’, equals zero. 00:46.564 --> 00:49.697 If we isolate ‘v sub BD’, it follows that 00:49.697 --> 00:58.048 the voltage between two nodes such as B and D is found by adding the voltages of the node pairs that link B and D. 00:58.048 --> 01:03.792 With this in mind, in a network with voltages, we assign potential to the nodes as follows: 01:03.792 --> 01:07.524 the node A is assigned any potential ‘v sub A’, 01:07.524 --> 01:09.912 and each of the other nodes, such as D, 01:09.912 --> 01:16.283 is assigned the potential ‘v sub D’ equal to ‘v sub A’ plus ‘v sub DA’. 01:16.283 --> 01:20.121 And so on for the rest of the nodes. 01:20.121 --> 01:26.900 Now, any voltage such as ‘v sub BD’ is ‘v sub BA’ plus ‘v sub AD’, 01:26.900 --> 01:32.101 that is, ‘v sub BA’, ‘minus v sub DA’. 01:32.101 --> 01:35.366 If we add and subtract ‘v sub A’ it results: 01:35.366 --> 01:42.438 ‘v sub A’ plus ‘v sub BA’, minus, ‘v sub A’ plus ‘v sub DA’. 01:42.438 --> 01:49.220 The first parenthesis is the potential of the B node, and the second is the potential of the D node, 01:49.220 --> 01:55.939 so the voltage ‘v sub BD’ is ‘v sub B’, minus ‘v sub D’. 01:55.939 --> 02:02.248 That is, the voltage of any node pair is the difference of the assigned potentials. 02:02.248 --> 02:06.846 So proving the theorem. 02:06.846 --> 02:12.340 Note that, in order to assign potentials to the nodes, we have started by assigning a random potential to the node A, 02:12.340 --> 02:16.262 from which the potential of all other nodes is obtained. 02:16.262 --> 02:17.406 That means that, 02:17.406 --> 02:25.700 “for any given Kirchhoff voltages, there are infinite sets of node potentials from which those voltages derive.” 02:25.700 --> 02:27.328 In the previous lesson we showed that 02:27.328 --> 02:33.982 for the values assigned to node pairs to be Kirchhoff voltages it suffices that they derive from node potentials. 02:33.982 --> 02:37.214 In this lesson we have shown the necessary condition, that is, 02:37.214 --> 02:42.857 that each set of Kirchhoff voltages derives from a set of node potentials. 02:42.857 --> 02:46.052 The two properties can be jointly stated as follows: 02:46.052 --> 02:50.188 “in order for the values assigned to the pairs of a node set to be Kirchhoff voltages 02:50.188 --> 02:56.802 it is a necessary and sufficient condition that they derive from a set of potentials of those nodes.” 02:56.802 --> 03:03.239 We called this property characterization theorem of Kirchhoff voltages. 03:08.300 --> 03:10.300 Subtitles and English translation: Roberto C. Redondo Melchor.