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Figure 165 Concept of ux linkage
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(168)
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Note that equation 168, relating the derivative of the ux linkage to the induced emf, is analogous to the equation describing current as the derivative of charge: dq i= (169) dt
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In other words, ux linkage can be viewed as the dual of charge in a circuit analysis sense, provided that we are aware of the simplifying assumptions just stated in the preceding paragraphs, namely, a uniform magnetic eld perpendicular to the area delimited by a tightly wound coil These assumptions are not at all unreasonable when applied to the inductor coils commonly employed in electric circuits What, then, are the physical mechanisms that can cause magnetic ux to change, and therefore to induce an electromotive force Two such mechanisms are possible The rst consists of physically moving a permanent magnet in the vicinity of a coil for example, so as to create a time-varying ux The second requires that we rst produce a magnetic eld by means of an electric current (how this can be accomplished is discussed later in this section) and then vary the current, thus varying the associated magnetic eld The latter method is more practical in many circumstances, since it does not require the use of permanent magnets and allows variation of eld strength by varying the applied current; however, the former method is conceptually simpler to visualize The voltages induced by a moving magnetic eld are called motional voltages; those generated by a timevarying magnetic eld are termed transformer voltages We shall be interested in both in this chapter, for different applications In the analysis of linear circuits in 4, we implicitly assumed that the relationship between ux linkage and current was a linear one: = Li (1610)
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so that the effect of a time-varying current was to induce a transformer voltage across an inductor coil, according to the expression v=L di dt (1611)
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This is, in fact, the de ning equation for the ideal self-inductance, L In addition to self-inductance, however, it is also important to consider the magnetic coupling that can occur between neighboring circuits Self-inductance measures the voltage induced in a circuit by the magnetic eld generated by a current owing in the same circuit It is also possible that a second circuit in the vicinity of the rst may experience an induced voltage as a consequence of the magnetic eld generated in the rst circuit As we shall see in Section 164, this principle underlies the operation of all transformers Self- and Mutual Inductance
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Figure 166 depicts a pair of coils, one of which, L1 , is excited by a current, i1 , and therefore develops a magnetic eld and a resulting induced voltage, v1 The second coil, L2 , is not energized by a current, but links some of the ux generated by the current i1 around L1 because of its close proximity to the rst coil The magnetic coupling between the coils established by virtue of their proximity is described by a quantity called mutual inductance and de ned by the symbol M The mutual inductance is de ned by the equation di1 v2 = M dt (1612)
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+ i1 ~ v1 L1 M + i1 ~ v1 L1 L2 L2
+ v2
v2 +
The dots shown in the two gures indicate the polarity of the coupling between the coils If the dots are at the same end of the coils, the voltage induced in coil 2 by a current in coil 1 has the same polarity as the voltage induced by the same current
Figure 166 Mutual inductance
16
Principles of Electromechanics
in coil 1; otherwise, the voltages are in opposition, as shown in the lower part of Figure 166 Thus, the presence of such dots indicates that magnetic coupling is present between two coils It should also be pointed out that if a current (and therefore a magnetic eld) were present in the second coil, an additional voltage would be induced across coil 1 The voltage induced across a coil is, in general, equal to the sum of the voltages induced by self-inductance and mutual inductance
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