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To analyze an electrical network, or a Kirchhoff network,
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is to derive a set of variables of that network
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from another set of those variables.
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The equation system that provides
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the unknown variables of the network
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comes from the two Kirchhoff's Laws,
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and the voltage-current relationships of the network branches.
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As we did on the course Theory of Kirchhoff Networks,
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we will call *n*-sub-*t* to the number of network nodes,
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*r* to the number of non-oriented network branches,
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*n* to the number of non-oriented branches of this network's trees
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and *l* to the number of non-oriented links of the network.
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We also have to remember that the number of independent equations
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obtained by applying Kirchhoff's First Law
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is the number of nodes minus one.
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By applying Kirchhoff's Second Law,
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we obtain as many independent equations
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as the number of non-oriented links in the network.
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And the voltage-current relationships of the *r* non-oriented branches of the network
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give us *r* independent equations.
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Therefore, in total, the two Kirchhoff's Laws,
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and the voltage-current relationships of the branches
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create a system with 2*r* independent equations.
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This system is called
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the **equilibrium equation system** of the network.
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Let us use what we have seen to analyze a network.
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This network has 5 non-oriented branches.
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Therefore it gives 10 independent equations.
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We obtain 3 equations by applying Kirchhoff's First Law to
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all four nodes except one.
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We obtain 2 equations by applying Kirchhoff's Second Law to
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two independent loops.
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And the voltage-current relationships of the branches
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give us the last 5 equations.
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Like any system of differential equations,
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this one has infinite solutions.
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The choice of a specific solution
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is made through integration constants.
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If the network was working for a long time
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it is said to be in *steady state*, and then
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the specific solution of the system is the one appearing on the screen.
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Once the values of the currents and voltages are found
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we consider the network as analyzed,
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since with them it is possible to obtain the powers
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and the energies absorbed by the branches.
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This is actually the only method
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of analysis of Kirchhoff networks, and,
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therefore, of electrical networks.
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But it also has two variations that let us simplify it.
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These variations will be discussed in the next lesson.
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Subtitles: Roberto C. Redondo Melchor.