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cross-sectional_area

Cross-sectional area

Stan Zurek, Cross-sectional area, Encyclopedia Magnetica, E-Magnetica.pl

Cross-sectional area - a value of area of a given part or component, measured in the plane perpendicular (normal) to the axis of such component or direction of “flow” of a given quantity.1)

Cross-sectional area of an insulated copper wire

The amount of cross-sectional area is typically important for maintaining flux of a given quantity, such such electric current or magnetic flux. In order to maintain the same flux within a smaller area the flux density must increase proportionally.

The concept of cross-sectional area is important for wires and windings (electric current density) as well as magnetic cores (magnetic flux density).

In technical texts, cross-sectional area is often simply referred to as “area”, but the difference from a generic “surface area” is usually clear from the context.

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Insulated wire

For a conducting wire there is a maximum current that can be carried under given cooling conditions. Higher current density can be maintained for the same area if cooling is improved, for example by using forced or liquid cooling.

Limitation of maximum current density also applies to superconducting wires, but arises from the critical magnetic field limit (Silsbee's rule), because resistive losses are negligible in a superconductor, so no heat is dissipated as a result of the current flow.

Outer diameter of an insulated wire is always greater than the diameter of the conductor (e.g. Cu), so the useful cross-sectional area is proportionally smaller
Enamelled wire 0.01 mm grade 3 (G3) has much larger proportion of insulation in the cross-section area than 0.500 mm grade 1 (G1) wire2)3)
Enamelled wire, 0.43 mm outer diameter and 0.375 mm copper diameter (enamel thickness 0.0275 mm) enamelled_wire_em.jpg

Total cross-sectional area of a typical electric wire comprises the area of the conductor and the insulation. Hence, the effective cross-sectional area (the area available for conducting current) is appropriately smaller, related to the thickness of insulation. This less-than ideal amount of active area is related to window utilisation and fill factor.4)5)

Magnetic core

In a magnetic core, a sufficient cross-sectional area is required so that the magnetic flux density does not reach saturation. For this reason, for power applications the cores are typically designed so that the cross-sectional area is roughly constant throughout the magnetic path, such as in ETD cores.

Magnetic flux circulates along magnetic path, and flux density is inversely proportional to the area (ETD core)
Core design is typically optimised to have similar cross-sectional area throughout, even if the actual path is split into multiple sub-paths (ETD core)
Magnetic circuit with constant magnetic flux, in which magnetic flux density doubles where the cross-sectional area reduces by half
Core RM8 with the cross-sectional area balanced throughout its magnetic path ferrite_core_rm8_magnetica.jpg

If there is no risk of saturation, as for instance in sensing or shielding applications, the cross-sectional area of a given magnetic circuit can vary by a great amount.

Conversely, sensors such as fluxgate, rely on local saturation of the magnetic material, which can be achieved by using “magnetic probe”, whose cross-sectional area is smaller than the rest of the core, and therefore easier to saturate.6)

Other types of circuits

Electric flux density increases if the area of the high-permittivity dielectric is reduced in the same electric field

Similar relationship between cross-sectional area, flux and flux density apply to other physical quantities, such as electric field and heat.

For example, in the illustration on the right, a capacitor formed from two flat charged plates has a high-permittivity dielectric with changing cross-sectional area. The resulting electric flux density D increases in the part which has smaller cross-sectional area.

The flux of heat follows similar principles.

See also

References

cross-sectional_area.txt · Last modified: 2021/04/27 19:53 by stan_zurek

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