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The values assigned to the node pairs in the figure are ‘*Kirchhoff voltages*’,
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because they meet Kirchhoff's second law.
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For example, the sum of the voltages of the closed path *ABCDA* is zero,
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as is in the other closed paths.
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The assignation of Kirchhoff voltages to a set of node pairs can be difficult,
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because we will have to find values that meet Kirchhoff's second law.
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But there is a safe and easy method to achieve this.
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It consists of assigning any value to each node.
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For example, node *A* is assigned ‘-5’. Node *B*, ‘j3’, and so on for each node.
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These arbitrary values assigned to each node will be called ‘*Kirchhoff potentials of the nodes*’.
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If we assign to each node pair a value equal to the potential difference of its nodes,
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we will say that these values ‘*derive from node potentials*’.
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Well then, the values derived from node potentials **always** meet Kirchhoff's second law.
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For example, in the closed path *ABEA* "(-j3-5)+(-2+j3)+7" equals zero.
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And so on for the rest of the closed paths.
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This property is demonstrated as follows:
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let the voltage of the node pair *AB* be ‘*v* sub *A*’ minus ‘*v* sub *B*’,
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the voltage of *BE* be ‘*v* sub *B*’ minus ‘*v* sub *E*’, and so on for the rest.
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The sum of the voltages in a closed path, such as *ABEA*, is ‘(*v* sub *A* minus *v* sub *B*), plus (*v* sub *B* minus *v* sub *E*), plus (*v* sub *E* minus *v* sub *A*)’, which is equal to zero.
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Which proves the property, which we will state as follows:
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“for the values assigned to the pairs of a node set to be Kirchhoff voltages, it is sufficient that they derive from a set of node potentials.”
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This theorem gives Kirchhoff's second law a great scope,
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because, according to it, any set of differences meets Kirchhoff's second law:
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the age differences of the students in a class,
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the differences of the account balances of the clients of a bank,
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temperature differences,
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pressure differences,
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and, of course, electric potential differences.
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So, Kirchhoff's second law is a particular case of a more general law that can be stated as follows:
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“the sum in each closed path of the differences of the values assigned to the elements of a set is zero.”
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Subtitles and English translation: Roberto C. Redondo Melchor.