The passive components such as inductors and capacitors are bulky parts on circuit boards. Researchers in academia, government, and industry have been searching for ways to improve the magnetic energy density and reduce the package size of magnetic parts. The "constant-flux" concept discussed herein is leveraged to achieve high magnetic-energy density by distributing the magnetic flux uniformly, leading to inductor geometries with a volume significantly lower than that of conventional products. A relatively constant flux distribution is advantageous not only from the density standpoint, but also from the thermal standpoint via the reduction of hot spots, and from the reliability standpoint via the suppression of flux crowding. For toroidal inductors, adding concentric toroidal cells of magnetic material and distributing the windings properly can successfully make the flux density distribution uniform and thus significantly improve the power density. Compared with a conventional toroidal inductor, the constant-flux inductor introduced herein has an enclosed-winding geometry. The winding layout inside the core is configured to distribute the magnetic flux relatively uniformly throughout the magnetic volume to obtain a higher energy density and smaller package volume than those of a conventional toroidal inductor. Techniques to shape the core and to distribute the winding turns to form a desirable field profile is described for one class of magnetic geometries with the winding enclosed by the core. For a given set of input parameters such as the inductor's footprint and thickness, permeability of the magnetic material, maximum permissible magnetic flux density for the allowed core loss, and current rating, the winding geometry can be designed and optimized to achieve the highest time constant, which is the inductance divided by resistance (L/Rdc). The design procedure is delineated for the constant-flux inductor design together with an example with three winding windows, an inductance of 1.6 µH, and a resistance of 7 mΩ. The constant-flux inductor designed has the same inductance, dc resistance, and footprint area as a commercial counterpart, but half the height. The uniformity factor α is defined to reflect the uniformity level inside the core volume. For each given magnetic material and given volume, an optimal uniformity factor exists, which has the highest time constant. The time constant varies with the footprint area, inductor thickness, relative permeability of the magnetic material, and uniformity factor. Therefore, the objective for the constant-flux inductor design is to seek the highest possible time constant, so that the constant-flux inductor gives a higher inductance or lower resistance than commercial products of the same volume. The calculated time-constant-density of the constant-flux inductor designed is 4008 s/m3, which is more than two times larger than the 1463 s/m3 of a commercial product. To validate the concept of constant-flux inductor, various ways of fabrication for the core and the winding were explored in the lab, including the routing process, lasing process on the core, etching technique on copper, and screen printing with silver paste. The most successful results were obtained from the routing process on both the core and the winding. The core from Micrometals has a relative permeability of around 22, and the winding is made of copper sheets 0.5 mm thick. The fabricated inductor prototype shows a significant improvement in energy density: at the same inductance and resistance, the volume of the constant-flux inductor is two times smaller than that of the commercial counterpart. The constant-flux inductor shows great improvement in energy density and the shrinking of the total size of the inductor below that of the commercial products. Reducing the volume of the magnetic component is beneficial to most power. The study of the constant-flux inductor is currently focused on the dc analysis, and the ac analysis is the next step in the research. / Master of Science