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This lesson is devoted to show a process for obtaining
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Kirchhoff networks from another given one.
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Before we start, let us recall that a linear transformation is each application T between two vector spaces
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over the same commutative field that meets the equality that appears on the screen.
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An example of linear transformation is the function derivation,
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because it meets the previous equality.
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A property of linear transformations is that the transform of zero is zero.
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We will use this property to prove the following theorem:
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“the images obtained with any linear transformation of Kirchhoff currents are Kirchhoff currents.”
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Indeed, as the currents arriving at each node of a network of Kirchhoff currents add to zero,
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by the previous theorem the image of the sum obtained with a linear transformation T is zero
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and it results that the transforms of the currents that arrive at each node add to zero,
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therefore they are Kirchhoff currents too,
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thus the theorem is proved.
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Therefore, for any given network of Kirchhoff currents,
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the derivatives of these currents are Kirchhoff currents.
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Also their products by the same real or complex number are Kirchhoff currents.
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Or their products by the same real or complex function.
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Also their Laplace transform is a linear application,
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so that the Laplace transforms of the Kirchhoff currents of a network
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are Kirchhoff currents of that network.
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If the Kirchhoff currents of a network are sinusoidal functions of the same frequency,
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then finding their phasors or phasor conjugates are linear transformations,
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thus phasors and their conjugates are Kirchhoff currents.
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Finding the average value of functions at the same interval is a linear transformation.
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Therefore the average values of Kirchhoff currents are Kirchhoff currents.
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However, the effective values of Kirchhoff currents are **not** Kirchhoff currents.
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The reason is that the effective values are positive real numbers,
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therefore their sum can not be zero, unless all of them are zero.
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Obviously, calculating the root-mean-square value is not a linear transformation.
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The previous theorem is also true for Kirchhoff voltages:
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“the images obtained through any linear transformation of Kirchhoff voltages are Kirchhoff voltages.”
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The demonstration is the same:
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as the voltages of every closed path add to zero, their linear transformations add to zero too.
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And the same examples we gave for Kirchhoff currents are valid for Kirchhoff voltages:
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their multiplication by numbers of functions give Kirchhoff voltages,
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their derivatives are Kirchhoff voltages,
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so are their average values,
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the Laplace transforms of Kirchhoff voltages are Kirchhoff voltages, etcetera.
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The effective values of Kirchhoff voltages are **not** Kirchhoff voltages.
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Although obtaining linear transformations is a good method to get other Kirchhoff networks,
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two sets of Kirchhoff currents of the same network
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are not necessarily linear transformations of each other.
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For example, the currents of the right figure
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are not obtained from any linear transformation of the currents of the left figure.
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Also, two sets of Kirchhoff voltages for the same node set
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are not necessarily linear transformations of each other.
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A Kirchhoff network that is obtained from another by means of a linear transformation
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is called *transformed Kirchhoff network* of the first one.
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Subtitles and English translation: Roberto C. Redondo Melchor.