Rating of Reactors as Functions of Volume

For given values of frequency, flux density, and current density, the volts per turn are proportional to the cross-sectional area of the core and the ampere turns are proportional to the area occupied by the winding, so that

[5-85]

where A_core is the cross-sectional area of the core or of the air gap corrected for fringing, and

[5-86]

where J is the rated current density in the winding. The volt-ampere rating of the reactor is the product of Eqs. 5-85 and 5-86

[5-87]

For a given configuration, an increase in all linear dimensions by a factor k increases the product A_wA_core by k⁴ and the volt-ampere rating of the reactor increases as the f power of the volume or of the weight. Therefore

from which it follows that the volt-amperes per unit weight or volt-amperes per pound is related to weight as

[5-88]

The rated volt-amperes per pound then varies directly as the linear dimension. With fixed current density the I²R losses in the winding vary at a given temperature directly as the volume of the winding, and when the flux density and frequency are fixed, the core loss varies directly as the volume of the core. It follows, therefore, that under rated conditions, the heat generated in the reactor is proportional to the volume of the reactor, or k³. However, the radiating surface varies as k² so that the temperature rise for a given rating varies directly as the linear dimension, or as k. This indicates that if frequency, flux density, current density, and configuration remain fixed, greater provision in proportion to k must be made to maintain the same temperature rise in the core and windings as the size and rating of the reactor are increased. For small ratings, normal radiation and convection dissipate the heat generated in the core and windings without excessive temperature rise, but beyond a certain rating it becomes necessary to facilitate cooling by means of forced air, circulation of oil, or some other method, when operating at the same current density and flux density.