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In the previous lesson we saw that the values that are assigned to the branches of a network, and that derive from loop currents, are *Kirchhoff currents*.
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Here we will see that the reciprocal statement is also true.
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That is, that “the currents of a Kirchhoff current network always derive from a set of loop currents”.
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The demonstration of this theorem requires the use of the concept of ‘*tree of a network*’.
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**Tree of a network** is the smallest set of branches that connects all nodes.
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To draw the branches of a tree we start from a node to another.
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We draw the second branch from either of the two nodes to another node,
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and so on until we reach the last node.
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The first branch connected two nodes, and each next branch connected another,
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so that the number of *non-oriented branches* in a tree is the number of nodes minus one.
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For a given tree, the branches that do not belong to it are called ‘**links of that tree**’.
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The branches *AB*, *BC*, *AE* and *AD*, and their opposite, are branches of a tree.
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The branches *BE*, *CE*, *CD* and *DE*, and their opposite, are the links of that tree.
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Therefore, if the ‘*n*’ branches of a tree are subtracted from all the ‘*r*’ branches of the network, we obtain the number of non-oriented links.
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Note that there are no loops in a tree.
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But if one single link is added to a tree, a loop appears, which is called ‘*loop of this link*’.
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The loop of the link *DC* is *DCBAD*.
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And so on for the rest of the links.
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In a network of Kirchhoff currents, “the currents of the branches of a tree can be written as sums of the currents of the links”.
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To do this we simply have to apply Kirchhoff's first law to the nodes, starting from the ends:
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For node *C*, ‘j sub 2’, plus ‘j sub 7’, minus ‘j sub 3’, equals zero.
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That is, ‘j sub 2’ equals ‘j sub 3’, minus ‘j sub 7’.
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At *B*, ‘j sub 1’, minus ‘j sub 2’, plus ‘j sub 6’, equals zero.
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Hence ‘j sub 1’ equals ‘j sub 3’, minus ‘j sub 7’, minus ‘j sub 6’.
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And so on.
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There is no need to use the last node, as there is only ‘*n*’ equals ‘*n* sub *t*’ minus ‘1’, currents of the branches of the tree.
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But even if the equation of the last node is written, it would be a linear combination of the previous equations.
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Therefore, for any given tree, “all the currents in the network can be written as sums of the currents of the links of that tree”.
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The loop of the link *EB* is *EBAE*.
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We will assign the current of its link as its loop current, that is, ‘j sub 6’.
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The loop of the link *EC*, which is *ECBAE*, will be assigned the current of its link, ‘j sub 7’.
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The loops of the other links will be assigned the currents of their links as loop currents.
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The rest of the loops, which are not link loops, will be assigned a current zero.
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Well, as we have seen, all the currents of this network of Kirchhoff currents
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are the sum of these loop currents, because they are the currents of the links.
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And thus it is demonstrated that all “set of Kirchhoff currents derives from a set of loop currents”.
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In the previous lesson we demonstrated the reciprocal statement,
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that is “the values derived from a set of loop currents are Kirchhoff currents”.
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Both theorems can be stated as one as follows:
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“in order for the values assigned to the branches of a network to be Kirchhoff currents, it is a necessary and sufficient condition that they derive from a set of loop currents of that network.”
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We will call this theorem ‘**characterization theorem for Kirchhoff currents**’.
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Subtitles and English translation: Roberto C. Redondo Melchor.