E-StoreInductance and Inductor
Inductance and Inductor:
Resistance, capacitance and inductance are the three familiar parameters from circuit theory. We have already discussed about the parameters resistance and capacitance in the earlier chapters. In this section, we discuss about the parameter inductance. Before we start our discussion, let us first introduce the concept of flux linkage. If in a coil with N closely wound turns around where a current I produces a flux 
and this flux links or encircles each of the N turns, the flux linkage 
is defined as 
. In a linear medium, where the flux is proportional to the current, we define the self inductance L as the ratio of the total flux linkage to the current which they link.
i.e., 
...................................(4.47)
To further illustrate the concept of inductance, let us consider two closed loops C1 and C2 as shown in the figure 4.10, S1 and S2 are respectively the areas of C1 and C2 . If a current I1 flows in C1 , the magnetic flux B1 will be created part of which will be linked to C2 as shown in Figure 4.10. In a linear medium, where L12 is the mutual inductance. For a more general case, if C2 has N2 turns then and or i.e., the mutual inductance can be defined as the ratio of the total flux linkage of the second circuit to the current flowing in the first circuit. As we have already stated, the magnetic flux produced in C1 gets linked to itself and if C1 has N1 turns then Therefore, self inductance As some of the flux produced by I1 links only to C1 & not C2. Further in general, in a linear medium, Example 1: Inductance per unit length of a very long solenoid:
Fig 4.10
...................................(4.48)
is proportional to I 1. Therefore, we can write
...................................(4.49)
...................................(4.50)
...................................(4.51)
, where 
is the flux linkage per turn.
= 
...................................(4.52) 
...................................(4.53)
and 
Let us consider a solenoid having n turns/unit length and carrying a current I. The solenoid is air cored.
Fig 4.11: A long current carrying solenoid
The magnetic flux density inside such a long solenoid can be calculated as
..................................(4.54)
where the magnetic field is along the axis of the solenoid.
If S is the area of cross section of the solenoid then
..................................(4.55)
The flux linkage per unit length of the solenoid
..................................(4.56) 
The inductance per unit length of the solenoid 
..................................(4.57)
Example 2: Self inductance per unit length of a coaxial cable of inner radius 'a' and outer radius 'b'. Assume a current I flows through the inner conductor.
Solution:
Let us assume that the current is uniformly distributed in the inner conductor so that inside the inner conductor.
i.e., 
..................................(4.58)
and in the region , 
..................................(4.59)
Let us consider the flux linkage per unit length in the inner conductor. Flux enclosed between the region 
and 
( and unit length in the axial direction).
..................................(4.60)
Fraction of the total current it links is 
..................................(4.61)
Similarly for the region 
..................................(4.62)
& 
.................................(4.63)
Total linkage 
..................................(4.64)
The self inductance, 
..................................(4.65)
Here, the first term 
arises from the flux linkage internal to the solid inner conductor and is the internal inductance per unit length.
In high frequency application and assuming the conductivity to be very high, the current in the internal conductor instead of being distributed throughout remain essentially concentrated on the surface of the inner conductor ( as we shall see later) and the internal inductance becomes negligibly small.
Example 3: Inductance of an N turn toroid carrying a filamentary current I.
Fig 4.12: N turn toroid carrying filamentary current I.
Solution: Magnetic flux density inside the toroid is given by 
..................................(4.66)
Let the inner radius is 'a' and outer radius is 'b'. Let the cross section area 'S' is small compared to the mean radius of the toroid 
Then total flux
..................................(4.67)
and flux linkage
..................................(4.68)
The inductance
..................................(4.69)
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