The electrically small limit of fractal element antennas

Small antennas pose severe practical restrictions in tradeoffs of directivity, bandwidth, and radiative efficiency. Chu (1948) presented an analytical basis for these restrictions. We demonstrate that the applicability of Chu's analysis appears to be overgeneralized when complex (fractal) antennas are considered. We derive a spherical harmonic expansion of the current distribution of a simple fractal loop, and show that this falls outside of the expansion's truncation assumption of Chu. This affirms that Chu's analysis is valid only for a very specific example of a conventional monopole or dipole which is electrically small. Presently-used electrically small guidelines are thus poor at predicting the performance of complex antenna shapes, such as fractal elements in the 'just electrically small' limit.