The Spiral Solenoids and the Leaf Antenna in Phyllotaxis Differential Geometry

  • I.M. Fabbri Department of Physics, University of Milan, via Celoria 16, 20133 Milan, Italy
Keywords: Spiral solenoids, leaf antenna, phyllotaxis differential geometry, magnetic coils.


In this paper new classes of spiral thin filamentary wire magnetic coils and antennas are introduced theoretically. This study shows that the thin wire circular loop coil as well as the cylindrical solenoid are particular cases of these classes. If the small spiral filamentary wire leaf loop coil is driven by an alternating electrical current, it will radiate as an antenna, whose theoretical analysis is provided throughout the paper. The fundamental spiral coil named leaf coil is obtained by joining together the so called forward and backward spiral coils which define the new concept of the Phyllotaxis differential geometry. The other spiral coils are composed of symmetrically rotated leaf coils, combined together in such a way to eliminate the transverse magnetic components along the longitudinal z axis. The magnetic moments and the multi-pole expansions of the various types of spiral loop coils are obtained by using the Taylor series of the spiral vector magnetic potentials.
According to the formulas and simulations, the magnetic flux density created by the long spiral solenoids is uniform and dovetails with the one created by the long cylindrical solenoid.
The bifolium coil may be flattened along one axis to meet the requirements of specific geometrical needs in space technology. Furthermore the crown solenoids may be cooled down to low temperatures by exploiting
the holes in between their coils.
Applications of the spiral induction coils are countless and can be found in space technology measurements, spacecraft magnetic shielding, telemetry, telecommunication, electron optics, physics and engineer.


Download data is not yet available.


E. Waffenschmidt,”Wireless power for mobile devices”, Telecommunications Energy Conference (INTELEC), 33rd International IEEE, 2011.

H. Lenz, ”Ueber die Bestimmung der Richtung der durch elekrodynamische Vertheilung erregten galvanischen Ströme”, Annalen der Physik und Chemie, vol. 31, pp. 483, 1834.

N. Tesla, ”Coil for electro magnets”, Patent no. 512, 340, Jan 9, 1894.

M. S. Karoul et al., ”Study and design of a loop antenna for medical telemetry application”, Sensors, Circuits & Instrumentation Systems Third International Conference on System, Signals & Devices March, Sousse, Tunisia, Vol. IV, pp. 21-24, 2005.

T.K. Sarkar et al., ”History of Wireless”, John Wiley & Sons, 2006.

L.M.B. Campos and P.J.S.Gil, ”On spiral coordinates with application to wave propagation,” J. Fluid Mech., Cambridge University Press, vol. 301, pp. 153-173, 1995.

I.M.Fabbri,”The Spiral Coaxial Cable”, International Journal of Microwave Science and Technology Hindawi Pub. Corp., 2015.

E.M Wassermann, C. M. Epstein, U. Ziemann, V. Walsh, T. Paus, ”The Oxford Handbook of Transcranial Stimulation”, Oxford University Press, 2008.

A.Maravita, N. Bolognini, ”Updates on multisensory perception: from neurons to cognition”, Frontiers Media SA e-book, 2012.

L.G. Cohen et al., ”Effects of coil design on delivery of focal magnetic stimulation. Technical considerations”, Electroencephalography and clinical Neurophysiology, Elsevier pub., vol. 75, pp. 350-357, 1990.

G. Xu et al., ”The Optimal Design of Magnetic Coil in Transcranial Magnetic Stimulation”, Proc. of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September, pp.1-4, 2005.

V. Kuperman, ”MRI, Physical principles and applications”, Academic Press, 2000.

Raja Parasuraman, M. Rizzo, ”Neuroergonomics: The Brain at Work: The Brain at Work”, Oxford University Press, USA, 2006.

P. R. Troyk, G.A. DeMichele, D.A. Kerns, R.F. Weir, ”IMES: An Implantable Myoelectric Sensor”, Proceedings of the 29th Annual International Conference of the IEEE EMBS, Lyon, France, 2007.

R. Melik, N. K. Perkgoz, E. Unal, C. Puttlitz and H. V. Demir, ”Bio-implantable passive on-chip RF-MEMS strain sensing resonators for orthopaedic applications”, J. Micromech. Microeng. vol. 18 pp. 115017, 2008.

R.N. Simons, D.G.Hall, and F. A. Miranda, ”Spiral Chip Implantable Radiator and Printed Loop External Receptor for RF Telemetry in Bio-Sensor Systems”, NASA/TM-2004-213217.

R. Bancroft, ”Microstrip and Printed Antenna Design”, Noble Publishing, 2004.

J.M. Schelmann, ”Multi-user MIMO-OFDM in Practice Enabling Spectrally Efficient Transmission Over Timevarying Channels”, Südwestdeutscher Verlag, 2009.

C.A. Kleitzing, ”The Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) on RBSP”, Space Sci. Rev., Springer, vol. 179, pp. 127-181, 2013.

S. Tumanski, ”Induction Coil Sensors - a Review”, Meas. Sci. Technol., IOP publishing group, vol. 18, pp. R31-R46, 2007.

A. Roux, ”The Search Coil magnetometer for THEMIS”, Space Science Reviews vol.141, iss. 1, pp. 265-275, dec. 2008.

B.Romanowicz, A. Dziewonski, ”Treatise on Geophysics”, Elsevier, 2015.

D.Cohen E.Halgren, ” Magnetoencephalography”, in G. Adelman, B. Smith Encyclopedia of Neuroscience, Elseviers, 2004.

J.C.Lin, ”Electromagnetic Fields in Biological Systems”, CRC, 2011.

J.H.Schultz, T. Antaya, J. Feng, C.Y.Gung, N. Martovetsky,

J.V.Minervini, P. Michael, A. Radovinsky, P. Titus, ”The ITER Central Solenoid”, Fusion Engineering, Twenty-First IEEE/NPS Symposium on, 2005.

P. Vedrine, ”Large Superconducting Magnet systems”, CERN Yellow Report CERN-2014-005, pp. 559-583, 2014.

R. Battiston et al., ”Active Radiation Shield for Space Exploration Missions, ARSSEM”, Final Report ESTEC Contract Nr. 4200023097/10/NL/AF, 2012.

S. Tumanski, ”Handbook of Magnetic Measurements”, CRC Press, 2011.

N. K. Das, P. Barat, S. Dey and T. Jayakumar ”, Design of Miniature Coil to Generate Uniform Magnetic Field,” Progress In Electromagnetics Research M, Vol. 34, 99-105, 2014.

R. Beck, ”Magnetic fields in the nearby spiral galaxy IC342: A multi-frequency radio polarization study”, A&A, 578, A93, 2015.

J. D. Jackson, ”Classical electrodynamics”, John Wiley & Sons, Inc. New York, Sydney, London, 1962.

G Arfken, ”University Physics”, Academic Press, 2012.

D.J. Griffiths, ”Introduction to Electrodynamics”, Prentice Hall New Jersey 3rd. ed., 1999.

N. Derby, S. Olbert, ”Cylindrical Magnets and Ideal Solenoids”, American Journal of Physics, vol. 78, pp. 229-235, 2010.

How to Cite
Fabbri, I. (2018). The Spiral Solenoids and the Leaf Antenna in Phyllotaxis Differential Geometry. Boson Journal of Modern Physics, 4(2), 338-360. Retrieved from