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# Fractal inductor

Stan Zurek, Fractal inductor, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/fractal_inductor |

**Fractal inductor**^{1)} (also **fractal loop inductor**^{2)}) - a type of inductor made by shaping an electrical conductor in a pattern resembling mathematical fractal.

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In high-frequency electronic circuits (up to GHz range) inductors can be made as a single-turn loops, for instance deposited as a thin layer of conductor (even below 1 µm) on a substrate.

In a planar arrangement, spiral inductors require at least two layers, or an out-of-plane return connection.

A single-turn loop inductors can be made within a single layer of conductor, but they might require significant surface area (footprint) to represent sufficient value of inductance. Fractal inductors are capable of representing increased value of inductance without increasing the footprint.

A higher-level fractal will have more features fit into the same space. For instance, the 3O fractal is equivalent to three base features positioned in a shape resembling the base configuration.^{4)}^{5)}

In the study presented by Shoute and Barlage^{6)} the inductance of the base structure was 4.6 nH with resistance of 5 Ω. However, the values for the 3^{rd}-3O configuration was 44.7 nH (increase by 9.7x) and 63.5 Ω (increase by 12.7x), respectively. According to the authors the self-inductance (due to increased length of conductor) does not sufficiently explain the increase, which must be due to some amount of mutual inductance between the neighbouring lobes of the structure. Even some professional modelling electromagnetic software is unable to model this effect seen in experiments.^{7)}

Fractal inductors can exhibit Q factor greater than 40 at the operating frequency. The actual design depends on the type of fractal pattern used, as well as compromises in the design process (e.g. joining the neighbouring lobes). From practical viewpoint it might be more beneficial to implement fewer lobes (thus easier to manufacture) which would give equivalent inductance as more intricate pattern. This less-than-ideal behaviour can be referred to as ** fractal dilution**.

Fractal inductors are especially useful for single-layer applications, and can be used on flexible or stretchable substrates.^{8)} A patent application for this technology was submitted.^{9)}

The fractal approach can be also applied to other devices: fractal transformer and fractal capacitor.

## See also

## References

^{2), 3), 4), 6), 7), 8)}Shoute G., Barlage D.W., Fractal Loop Inductors, IEEE Transactions on Magnetics, Vol. 51 (6), 2015, art. 2300108

^{5)}K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2003, ISBN 9780470848616, p. 36–43