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Premise
We have already seen that any circuit, composed of a combination of resistors in series and/or in parallel,
however complex it may be, can be simplified till obtaining a single equivalent resistance. Done that,
with Ohm's law is possible to calculate the value of the total current and the current for each circuit branch.
This applies to circuits with a single source of power (EMF, electromotive force), but when there is two or
more electricity source, Ohm's law is no longer sufficient to calculate electric parameters. For these kind of
circuits it is necessary to use Kirchhoff's laws or principles. The principles of Kirchhoff for electrical
circuits, are two: one for nodes, and one for meshes.
Knots and Meshes:
Node is considered the meeting of three or more branches of a circuit, while a mesh is defined
as a closed circuit, part of a circuit more complex. A complex circuit has at least three meshes.
To solve any problem with the principles of Kirchhoff, it is necessary to set up a system
of equations consisting of n-1 nodes and n-1 mesh. In the circuit below, the same is composed
of two nodes and three meshes, you must set up a system of three equations (one node and two meshes).
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a |
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i1  |
b |
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c |
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| R1 |
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I |
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R3 |
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d |
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I |
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E2 |
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g |
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i1 |
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i2 |
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e |
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With the definitions before stated, in the example circuit on the left,
(object of a subsequent exercise):
Nodes are:
- the joint b
- the joint f
on the contrary, nodes aren't the joints d and h
Meshes are the circuits:
- a b f g
- b c e f
- a c e g
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Kirchhoff's Law n.1:
In a node, the sum of the currents entering is equal to the sum of exit currents.
For an example, in mathematical terms, for the node b of the above circuit:
I = i1 + i2
or also
i1 + i2 - I = 0
Kirchhoff's Law n.2:
In a mesh, the sum of e.m.f. (algebraic) is equal to the sum of falling voltage.
Referring to the mesh a b f g of the same circuit, in mathematical terms we have:
E1 = R1 * i1 + R2 * I
or also
R1 * i1 + R2 * I - E1 = 0
EXERCISE
In the following figure, in an interactive manner, the circuit previously given as an example is proposed.
By entering, in the appropriate fields, the values: the emf in volts for the DC generators
E1 and E2, the electrical resistance in ohms for R1 and R2,
R3, and remembering that you must type (.) to express decimal point, instead of a comma (, used in Europe)..
By pressing the "submit" button will appear the circuit with the values and direction of the current intensity that would flow
in the various branches (the resistance of the conductor are neglected). Your task, to exercise, is to calculate values and
direction to the current intensity according to Kirchhoff's laws, and then compare them with those appearing in interactive circuit.
You can also see how the currents vary by changing the circuit's parameters entered.
Circuit number 1
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Circuit number 2
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Circuit number 3
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