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In the previous lesson we saw that any set of differences meets Kirchhoff's second law.
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Or, more precisely,
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that the values assigned to the node pairs, and which derive from node potentials, are Kirchhoff voltages.
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Here we will see that the reciprocal statement is also true.
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That is, every set of Kirchhoff voltages derives from node potentials.
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Indeed, let us suppose that the pairs from a node set have Kirchhoff voltages assigned.
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This means that the values of any closed path sum up to zero.
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For example, ‘*v* sub *AB*’, plus ‘*v* sub *BD*’, plus ‘*v* sub *DA*’, equals zero.
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If we isolate ‘*v* sub *BD*’, it follows that
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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*.
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With this in mind, in a network with voltages, we assign potential to the nodes as follows:
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the node *A* is assigned any potential ‘*v* sub *A*’,
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and each of the other nodes, such as *D*,
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is assigned the potential ‘*v* sub *D*’ equal to ‘*v* sub *A*’ plus ‘*v* sub *DA*’.
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And so on for the rest of the nodes.
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Now, any voltage such as ‘*v* sub *BD*’ is ‘*v* sub *BA*’ plus ‘*v* sub *AD*’,
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that is, ‘*v* sub *BA*’, ‘minus *v* sub *DA*’.
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If we add and subtract ‘*v* sub *A*’ it results:
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‘*v* sub *A*’ plus ‘*v* sub *BA*’, minus, ‘*v* sub *A*’ plus ‘*v* sub *DA*’.
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The first parenthesis is the potential of the *B* node, and the second is the potential of the *D* node,
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so the voltage ‘*v* sub *BD*’ is ‘*v* sub *B*’, minus ‘*v* sub *D*’.
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That is, the voltage of any node pair is the difference of the assigned potentials.
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So proving the theorem.
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Note that, in order to assign potentials to the nodes, we have started by assigning a random potential to the node *A*,
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from which the potential of all other nodes is obtained.
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That means that,
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“for any given Kirchhoff voltages, there are infinite sets of node potentials from which those voltages derive.”
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In the previous lesson we showed that
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for the values assigned to node pairs to be Kirchhoff voltages it suffices that they derive from node potentials.
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In this lesson we have shown the necessary condition, that is,
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that each set of Kirchhoff voltages derives from a set of node potentials.
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The two properties can be jointly stated as follows:
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“in order for the values assigned to the pairs of a node set to be Kirchhoff voltages
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it is a necessary and sufficient condition that they derive from a set of potentials of those nodes.”
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We called this property *characterization theorem of Kirchhoff voltages*.
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Subtitles and English translation: Roberto C. Redondo Melchor.