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This lesson marks the beginning of the subject we call *Theory of Kirchhoff's Networks*.
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In 1845 Gustav Robert Kirchhoff enunciated his two laws as properties of electrical currents and voltages.
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The first law could be enunciated as follows:
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the sum of currents flowing **into** a node through each branch of an electrical network is zero.
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For example, if this network is an electrical network, it fulfills Kirchhoff's first law:
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at the *A* node we have that ‘*i* sub 1’, plus ‘*i* sub 2’, plus ‘*i* sub 5’, equals zero.
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At the *B* node, ‘minus *i* sub 2’, ‘minus *i* sub 3’, plus ‘*i* sub 6’, equals zero.
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And so on for the rest of the nodes.
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If the equalities are multiplied by ‘negative one’, the addends change their sign.
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So, the equality obtained at the *A* node becomes ‘minus *i* sub 1’, ‘minus *i* sub 2’, ‘minus *i* sub 5’, equals zero.
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The currents ‘minus *i* sub 1’, ‘minus *i* sub 2’, and ‘minus *i* sub 5’, flow **out** of the *A* node.
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That is, multiplying by ‘negative one’ changes the currents from those that flow **into** the *A* node to those that flow **out** of it.
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Therefore *Kirchhoff's first law* can also be expressed by saying that
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the sum of currents flowing **out** of a node through each branch of an electrical network is zero.
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To enunciate *Kirchhoff's second law* we should first specify the meaning of *path*.
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**Path** is each sequence of ordered pairs in which the first component of each next-pair is the second component of the preceding-pair.
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The sequence of ordered pairs {(*A*,*B*),(*B*,*C*),(*C*,*E*)} is a path.
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If the beginning of the path *A* coincides with the end, it is a **closed** path.
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So, the path {(*A*,*B*),(*B*,*C*),(*C*,*E*),(*E*,*A*)} is a **closed path of node pairs**.
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It is usually designated as the path *ABCEA*.
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Other closed path is *ECDE*, another *ABCDA*.
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And so on.
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A *closed path of branches* is called **loop**.
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Well, *Kirchhoff's second law* can be stated as follows:
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the sum of the voltages of each closed path of node pairs in an electrical network is zero.
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‘*v* sub 1’ is the voltage of the node pair *DA*, which means that the voltage of *AD* is ‘minus *v* sub 1’.
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And so on for the rest of the nodes.
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Therefore, *Kirchhoff's second law* for the closed path *DAED* states that ‘*v* sub 1’, ‘minus *v* sub 5’, plus ‘*v* sub 8’, equals zero.
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For the closed path *DABCD*, ‘*v* sub 1’, ‘minus *v* sub 2’, plus ‘*v* sub 3’, ‘minus *v* sub 4’, equals zero.
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And so on for the entire network.
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But in this course we will not show *Kirchhoff's laws* as properties of electrical variables, but as **topological relationships**.
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For example, each branch of this network has been assigned a real number.
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The *BA* branch is assigned ‘2’. That is, the *AB* branch is assigned ‘minus 2’.
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And so on for each branch.
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These numbers were chosen to meet *Kirchhoff's first law*.
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For example, at the *A* node, ‘2+(-3)+1=0’.
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And so on for the rest of the nodes.
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These values are not measures of *physical variables*, but rather they were assigned taking into account just the **relative position** of branches and nodes.
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This is expressed by saying that “*Kirchhoff's first law is a ***topological property**”, a position property.
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Similarly, real numbers are assigned to node pairs.
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For example, the *BA* pair is assigned ‘minus 6’, which also means that the *AB* pair is assigned ‘6’.
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These values meet *Kirchhoff's second law*.
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For example, for the closed path *ABEA*, ‘-(-6)-10+4=0’.
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And so on for all other *closed paths* of node pairs.
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Again, these values are not measures of any *physical variable*, but rather are chosen to meet *Kirchhoff's second law* taking into account the **relative position** of the node pairs,
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which also makes *Kirchhoff's second law* a position property, a **topological property**.
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We will see that presenting the *two Kirchhoff laws* as *topological properties* facilitates the extraction of other properties derived just from them, without resorting to any physical law.
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It also facilitates the identification of other networks, besides electrical networks, that meet these properties.
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Subtitles and English translation: Roberto C. Redondo Melchor.