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*Intersection of two networks* is the set of branches and nodes that are in both networks.
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The intersection of these two networks are the nodes *A*, *B*, *C*, and *E*,
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and all branches, except those connecting the nodes *A* and *D*, *D* and *C*, and *B* and *C*.
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Two Kirchhoff networks are equivalent
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if each branch of their intersection has the same current in both networks,
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and each node pair of the intersection has the same voltage in the two networks.
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These two networks are equivalent.
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Next we will show procedures
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to obtain Kirchhoff networks equivalent to a given network.
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The first is to replace branches in series by a single branch.
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And the current common to the original branches is assigned to the new branch,
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and the sum of the voltages of the branches in series is assigned as its voltage.
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The branch replacing the branches in series is called *resultant* of the series branches.
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Thus, the resultant of the series branches has the same current as the branches it replaces,
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and its voltage is the sum of the currents of those branches.
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The reverse process also provides equivalent Kirchhoff networks:
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the branch connected between nodes *A* and *C* can be replaced by two or more branches in series,
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whose currents are equal to the current of the original branch,
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and whose voltages must add to the voltage of the original branch.
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Two or more parallel branches connected between two nodes can be replaced by a branch
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whose assigned current is the sum of the currents of the substituted branches,
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and whose voltage is the voltage of the extreme nodes.
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The branch replacing the parallel branches is called *resultant*.
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Therefore, the resultant of parallel branches has the same voltage they have
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and its current is the sum off the currents of the original branches.
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The inverse operation gives equivalent networks too:
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a branch can be replaced by parallel branches connected to the same terminals as the first branch,
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with currents such that their sum is the current of the original branch.
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If a part of a Kirchhoff Network is replaced with a node with any potential,
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then we get another Kirchhoff network equivalent to the first one.
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Indeed, the new network is a Kirchhoff network,
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because Kirchhoff's first law is met at the *O* node,
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as the currents leaving the node
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are the currents of the cut-set that connects parts *A* and *B* in the original network,
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thus their sum is zero.
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Kirchhoff's second law continues to be fulfilled in the new network,
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because its voltages derive from node potentials.
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All branches of the new network have the same current as in the first network,
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and all nodes in the intersection have the same potentials in both networks,
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so that the two networks are equivalent.
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The inverse operation also creates a network equivalent to the first network.
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It consist in replacing a node by some part of a Kirchhoff network
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which connects to the initial part with the branches converging at the eliminated node,
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which are a cut-set.
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That is, a Kirchhoff network equivalent to other can be obtained
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by performing any modification that leaves unaltered
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the currents and voltages of the intersection of both networks.
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Subtitles and English translation: Roberto C. Redondo Melchor.