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# Calculation Inductance of Toroidal Inductor Wound by Rectangular Cross-Sectional Wire

### Descripción

Calculation Inductance of Toroidal Inductor Wound by Rectangular Cross-Sectional Wire

In this article, we present two methods for calculation of the inductance of toroidal core power inductors wound by rectangular cross-sectional wire, considering that the current density is inversely proportional to the circular coil radius. The first method is to simplify the helical toroidal coil into a thick-walled toroidal, and based on Grover’s toroidal inductor formula, the inductance is obtained by calculation the magnetic flux and the calculation method is simple, but the applicability is poor. The second method is to simplify the helical toroidal coil into a collection of self-closing circular coils, the calculation method is complex but has high accuracy, and the mutual inductance between the circular coils is calculated by the filament method based on the adjusted Grover’s mutual inductance of circular coils with inclined axes. We verify the adjusted Grover’s mutual inductance of filamentary circular coils with inclined axes and the mutual inductance between inclined circular coils with a rectangular cross section. Finally, we compared and analyzed the results calculated by the two methods proposed in this article and the results calculated by the finite element method.

The various advantages of toroidal inductors, which are cooler, smaller and more EMI-resistant are discussed. With toroidal inductors, there is an advantage to maintaining a single layer of windings due to which the inductor behaves closer to an ideal component of lower levels of parasitic capacitance. Multi-layer toroidal inductors involve both turn-to-turn capacitance and layer-to-layer capacitance and a very significat start/finish gap capacitance since there is no start/finish gap. This increases the total amount of parasitic capacitance by orders of magnitude.

This paper presents a micromachined implementation of embedded toroidal solenoids for high-performance on-chip inductors and transformers, which is highly demanded in radio-frequency integrated circuits (RFICs). Microfabricated on CMOS compatible silicon wafers with post-CMOS micromachining techniques, the RF toroidal components can constrain the magnetic flux into a well-defined path and away from other on-chip RF devices, thereby, being in favor of decrease in RF loss, increase in Q-factor and elimination of electromagnetic interference. By using a technical combination of an anisotropic wet etch and an isotropic dry etc., the micromachined toroidal structure can be used for the formation of metal solenoid by copper electroplating. Processed under low temperature (Max 120 °C for photoresist hard-baking), the three mask microfabrication can be compatible with CMOS IC fabrication in a post-process way. The formed toroidal inductors with 4.92 nH and 8.48 nH inductance are tested, and we obtain maximum Q-factors of 25.7 and 17.8 at 3.6 GHz and 3.1 GHz, while the self-resonant frequencies are 17.3 GHz and 7.4 GHz, respectively. On the other hand, two types of toroidal transformers are also formed and tested, resulting in satisfactory RF-performance. Therefore, the novel techniques for close-loop solenoid inductors are promising for high-performance RF ICs.

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