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*Multi-terminal network* is a network that contains nodes at which only one branch arrives.
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Each node at which only one branch arrives is called *terminal* of the multi-terminal network.
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*Kirchhoff multi-terminal network* is a multi-terminal network with values assigned to its branches
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that meet Kirchhoff's first law at all non-terminal nodes,
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and with potentials assigned to its nodes.
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The currents entering the multi-terminal network through the terminals
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are called *currents of the terminals*.
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It results that the sum of the currents of the terminals of a Kirchhoff multi-terminal network is zero.
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Indeed, as the sum of the currents entering each non-terminal node is zero,
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the sum of these equalities is also zero.
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But in the sum the currents of the branches connecting those nodes are canceled,
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as each current has its opposite.
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And the only currents remaining are those entering from the terminals,
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whose sum is zero.
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Generally, for any Kirchhoff multi-terminal network
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the sum of the currents of its terminals is zero.
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The Kirchhoff power absorbed by a multi-terminal network
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is the sum of the Kirchhoff powers absorbed by its branches.
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The Kirchhoff power delivered by a multi-terminal network is the opposite of the power it absorbs.
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Well, the multi-terminal network power theorem states
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that the power absorbed by a multi-terminal network is the sum of the products
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of the potentials of its terminals by the respective currents.
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To prove this theorem let us get a Kirchhoff network that is equivalent to the multi-terminal network,
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by adding *t* branches that start at its terminals,
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and gather at the node *O*, which is assigned a potential zero.
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The voltages of these *t* new branches are the potentials of the terminals.
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Each of these branches is assigned a current
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opposite to that entering the multi-terminal network through the corresponding terminal.
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The resulting network is a Kirchhoff network,
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because the sum of the currents of the branches that reach all the nodes is zero,
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*O* included, for the currents that arrive at it are opposed to the currents of the terminals,
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which we saw that sum to zero.
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Tellegen's theorem, when applied to this Kirchhoff network, states that
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the sum of the Kirchhoff powers absorbed by all of its branches is zero.
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If we isolate the power absorbed by the multi-terminal network, the theorem is proved.
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In other words, the power absorbed by a multi-terminal network
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is the sum of the products of the potentials of its terminals
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by their respective currents.
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We call this property the **multi-terminal network power theorem**.
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We can consider any Kirchhoff network as consisting of two multi-terminal networks.
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The Kirchhoff power absorbed by the multi-terminal network *B*
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is opposite to that absorbed by the multi-terminal network *A*.
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Therefore, the Kirchhoff power absorbed by the multi-terminal network *B*
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is the Kirchhoff power delivered by the multi-terminal network *A*.
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Subtitles and English translation: Roberto C. Redondo Melchor.