Inductance
Inductance1 is the main property of each coil that indicates the coil’s capacity. It is the main indicator of the coil’s capability in energy storage and refers to its ability to resist or oppose the sudden change of current flowing through it.
The inductance of a coil is the ratio of the total linked magnetic flux flowing through the coil to the current flowing through the coil’s conductors. The standard unit of a coil’s inductance is henry. However, due to the larger values of a henry, smaller units (e.g., millihenry and microhenry) are used in practice.
The inductance value of each coil mostly depends on its structural features. Ampere’s circuital law relates inductance to the structural properties of the coil. In calculations, the coil’s length is considered much higher than its cross-section to avoid complexity while providing a general equation. This way, the magnetic field passing through its cross-section will almost be equal in all areas, delivering a vector that is parallel to the cross-section plane vector. Thus, the resulting equation is valid solely for a long coil, not for a coil whose cross-section is not smaller than its length.
Additionally, in calculations, the core of a coil is of air (air core) to allow formulating the inductance of a long coil. For non-air core coils, the value obtained from the air-core long coil formula is multiplied by the relative permeability value of the coil core.
As mentioned, the equation obtained for a coil inductance is valid solely for long coils, not for short ones – provided that the coil not to be too lengthy in relation to its diameter. For example, in an induction furnace, the coil is of a short type. The reason why the obtained equation is not valid for short coils is that the magnetic field produced in long coils is distributed almost uniformly throughout the coil’s inner cross-section. Thus, during integration, the equation is supposed to be independent of the integration course. For short coils, however, due to the low length of the coil than its cross-section, the magnetic field produced for loop formation is bent in the edges of the coil and includes both horizontal and vertical dimensions, while it only contains horizontal components in the centers. Thus, the density of the magnetic field produced by the coil is not constant throughout the coil’s length.
The image below compares two long and short coils.
Long coil
Short coil
Measurement of inductance for a short coil requires special and convoluted calculations. Its value is obtained using Nagaoka’s coefficient. Simply put, to calculate the inductance of a short coil, the coil is first assumed to be long and its inductance is measured using an equation obtained for catching the inductance of a long coil. The obtained value is then multiplied by Nagaoka’s coefficient2 . Nagaoka’s coefficient is a reduction factor with a value of less than one and is obtained from other physical and structural parameters of the coil through complex integral calculations. This value tends toward one if the coil’s length is much higher than its cross-section through which the magnetic flux flows. The equations below describe Nagaoka’s coefficient calculation to measure the inductance of a short coil.
Where “R” denotes the radius, “D” is the diameter, “A” is the area of the cross-section through which the magnetic flux flows in the coil, l is the length of the coil, and N is the number of turns. Additionally, K(k) and E(k) are, respectively, elliptic integral functions types I3 and II4 obtained from the equation below:
The graph below intended based on the above calculations for Nagaoka’s coefficient illustrates the Nagaoka’s coefficient value for a coil in terms of the ratio of the coil’s length to the diameter of the cross-section through which the magnetic flux flows. As shown, Nagaoka’s coefficient value tends toward one when the coil’s length is bigger than its cross-section. Thus, the coil inductance measured through the simple equation obtained in the first equation will be more accurate. Notably, Nagaoka’s coefficient is too close to one from the point where the coil’s length is almost three times that of its cross-section.
Due to the voluminous, the above equations to obtain the inductance of a long coil requires special software. For simplicity, however, a reality approximation is utilized, though the software result will not specify the accurate value of the inductance of a short coil. However, the result obtained using the equation developed for a long coil will be more accurate.
The approximation equation to measure the inductance of a short coil is as follows:
