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2.6 Electrical Circuits                                                             73
                   2.6 ELECTRICAL CIRCUITS



                                    The theory of electrical circuits is an important application that naturally gives
                                    rise to rectangular systems of linear equations. Because the underlying mathe-
                                    matics depends on several of the concepts discussed in the preceding sections,
                                    you may find it interesting and worthwhile to make a small excursion into the
                                    elementary mathematical analysis of electrical circuits. However, the continuity
                                    of the text is not compromised by omitting this section.
                                        In a direct current circuit containing resistances and sources of electromo-
                                    tive force (abbreviated EMF) such as batteries, a point at which three or more
                                    conductors are joined is called a node or branch point of the circuit, and a
                                    closed conduction path is called a loop. Any part of a circuit between two ad-
                                    joining nodes is called a branch of the circuit. The circuit shown in Figure 2.6.1
                                    is a typical example that contains four nodes, seven loops, and six branches.
                                                           E 1             E 2
                                                                     1
                                                                 R 1     R 2
                                                               I 1     I 2
                                                                A         B
                                                                      R 5
                                                            E 3
                                                                    I 5
                                                                 R 3     R 6
                                                          2                     4
                                                                     3
                                                               I 3         I 6
                                                                     C
                                                                            I 4
                                                                        R 4
                                                                E 4
                                                                  Figure 2.6.1
                                        The problem is to relate the currents I k in each branch to the resistances R k
                                                    15
                                    and the EMFs E k .  This is accomplished by using Ohm’s law in conjunction
                                    with Kirchhoff’s rules to produce a system of linear equations.


                                                               Ohm’s Law

                                       Ohm’s law states that for a current of I amps, the voltage drop (in
                                       volts) across a resistance of R ohms is given by V = IR.



                                        Kirchhoff’s rules—formally stated below—are the two fundamental laws
                                    that govern the study of electrical circuits.

                                 15
                                    For an EMF source of magnitude E and a current I, there is always a small internal resistance
                                    in the source, and the voltage drop across it is V = E−I ×(internal resistance). But internal
                                    source resistance is usually negligible, so the voltage drop across the source can be taken as
                                    V = E. When internal resistance cannot be ignored, its effects may be incorporated into
                                    existing external resistances, or it can be treated as a separate external resistance.
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