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SelfInductance
The voltage impressed on a series inductive circuit in which the capacitance effects are negligible and in which there are no thermal or electrolytic sources of emf, is expressed by
where v = the applied voltage in volts The flux linkage λ may be due only to the current i in the circuit itself; in addition it may be due to currents in other circuits that are coupled magnetically with the circuit under discussion. For that matter a permanent magnet in the vicinity of the circuit may contribute flux linkage also. If, however, the flux linkage λ is produced solely by the current i in the circuit itself and if the relationship between λ. and i is a linear function, then the expression for the applied voltage becomes
where L = selfinductance in henries. Figure 41 shows a schematic diagram of an inductive circuit. The dotted lines indicate the magnetic flux.
The voltage that is induced by the time variations in the current of a circuit is called the electromotive force of selfinduction, or simply the emf of selfinduction, and is expressed in terms of the selfinductance by
The flux linkage λ can be a linear function of the current i only if the selfinductance L is constant. In circuits where the selfinductance is a variable, as for example in the case of electromagnets with variable length of air gaps as well as in some generators, the time variation of the selfinductance will affect the voltage and must be taken into account. Circuits with variable inductance will be discussed later in this chapter. The emf induced in a circuit can always be expressed by
This is true whether L is a constant or a variable, or whether the flux linkage λ is produced solely by the current in that circuit or whether currents in other circuits contribute to the flux linkage as well. However, if the selfinductance is constant and the only flux linkage is the one resulting from the current i in the circuit itself, then Eq. 42 can be equated to Eq. 43 with the result that
from which
and
Equation 46 defines inductance as the flux linkage per ampere. Consider the toroid shown in Fig. 39 and assume the thickness R_{2}  R_{1} to be small compared with R_{1}. Then the expression for the flux produced by the current i in the winding is
which can be reduced to the more general form
where A is the crosssectional area of the flux path and l is the length of the flux path. When all of the flux Φ links all the turns N the flux linkage is
The flux Φ in the toroid links all N turns, so that on the basis of Eqs. 46 and 48 the selfinductance is
The permeability for an air core is μ = μ_{0} = 4π x 10^{7} h per m, and the expression for selfinductance if the core has a uniform crosssectional area becomes
If the core material has constant relative permeability of μ_{r} then the selfinductance is given by
When both sides of Eq. 49 are multiplied by i^{2}/2 the result expresses the energy stored in the field thus
where
Substitution of Eq. 413 in Eq. 412 yields
The term Vol HB/2 is the value of the energy stored magnetically in volume Vol. When Eq. 46 is substituted in Eq. 414 the energy that is stored magnetically can be expressed in terms of flux linkage and current as follows
Equations 414 and 415 apply only to the case of core materials that have constant permeability. There is the further restriction on Eq. 414 that the flux density B is practically uniform throughout the core.


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