Abstract
This review is devoted to the main results of the FRACTALCOMS international research program in the field of fractal electrodynamics and designing modern ultra-wideband or multiband fractal antennas and other basic electrodynamic structures for breakthrough information and communication technologies. Promising directions of further research are presented. The main results obtained by the author in research institutes of Russia and China in the fields of radio electronics and radio physics on the basis of the theory of fractals and scaling effects are outlined.
Similar content being viewed by others
REFERENCES
A. A. Pistol’kors, Antennas (Svyaz’izdat, Moscow, 1947) [in Russian].
Problems of Antenna Technology, Ed. by L. D. Bakhrakh and D. I. Voskresenskii (Radio i Svyaz’, Moscow, 1989) [in Russian].
L. D. Bakhrakh, L. S. Benenson, and E. G. Zelkin, Handbook on Antenna Technology, Ed. by Ya. N. Fel’d and E. G. Zelkin (IPRZhR, Moscow, 1997) [in Russian].
The Antenna Engineering Handbook, 4th ed., Ed. by J. L. Volakis (McGraw Hill, New York, 2007).
Ya. N. Fel’d and L. S. Benenson, Fundamentals of Antenna Theory (Drofa, Moscow, 2007) [in Russian].
E. I. Nefedov, Microwave Devices and Antennas (ITs Akademiya, Moscow, 2009) [in Russian].
Ultra-Wideband Antennas, Ed. by L. S. Benenson (Mir, Moscow, 1964).
V. H. Rumsey, Frequency Independent Antennas (Academic, New York, 1966; Mir, Moscow, 1968).
A. A. Potapov, Fractals in Radio Physics and Radar (Logos, Moscow, 2002) [in Russian].
S. A. Podosenov, A. A. Potapov, and A. A. Sokolov, Impulse Electrodynamics of Broadband Radio Systems and Fields of Coupled Structures, Ed. by A. A. Potapov (Radiotekhnika, Moscow, 2003) [in Russian].
A. A. Potapov, Fractals in Radio Physics and Radar: Topology of Sample, 2nd ed. (Universitetskaya Kniga, Moscow, 2005) [in Russian].
S. A. Podosenov, A. A. Potapov, Dzh. Foukzon, and E. R. Men’kova, Nonholonomic, Fractal, and Coupled Structures in Relativistic Continuous Environments, Electrodynamics, Quantum Mechanics, and Cosmology (LENAND, Moscow, 2015), Vols. 1−3 [in Russian].
Y. Kim and D. L. Jaggard, Proc. IEEE 74, 1278 (1986).
C. Puente, J. Romeu, R. Pous, and A. Cardama, Fractals in Engineering, Ed. by J. Levy Vehel, E. Lutton, and C. Tricot (Springer-Verlag, Berlin, 1997).
C. Puente, J. Romeu, and A. Cardama, in Frontiers in Electromagnetics, Ed. by D. H. Werner and R. Mittra (IEEE, New York, 2000), p. 48.
D. H. Werner, P. L. Werner, D. L. Jaggard, et al., in Frontiers in Electromagnetics, Ed. by D. H. Werner and R. Mittra (IEEE, New York, 2000), p. 94.
D. Werner, J. S. Petko, and T. G. Spence, in The Antenna Engineering Handbook, Ed. by J. L. Volakis (McGraw-Hill, New York, 2007).
X. Lianc, W. Zhensen, and W. Wenbing, J. Xidiun Univ. 21 (5), 80 (1994).
A. Boag, A. Boag, E. Michielssen, and R. Mittra, IEEE Trans. Antennas Propag. 44, 687 (1996).
N. Cohen, Commun. Quarterly 6 (3), 53 (1996).
C. Puente-Baliarda, J. Romeu, R. Pous, and A. Cardama, IEEE Trans. Antennas Propag. 46, 517 (1998).
R. Hohlfeld and N. Cohen, Fractals 7 (1), 79 (1999).
K. J. Vinoy and V. K. Varadan, Smart Mater. Struct. 10, 1211 (2001).
S. R. Best, IEEE Antennas Wireless Propag. Lett. 1 (1), 39 (2002).
D. H. Werner and S. Ganguly, IEEE Antennas Propag. Mag. 45 (1), 38 (2003).
B.-L. Ooi, IEEE Trans. Antennas Propag. 52, 1286 (2004).
F. Frezza, L. Pajewski, and G. Schettini, IEEE Trans. Microwave Theory Tech. 52, 220 (2004).
G. Konstantatos, C. Soras, G. Tsachtsiris, et al., Electromagnetics 24 (1–2), 81 (2004).
A. A. Potapov, in Use of Methods of Artificial Intelligence and High-Performance Computations in Space Researches (Proc. 3rd Extend. Seminar, Pereslavl-Zalessky, Nov. 26–27 2003) (Fizmatlit, Moscow, 2003), p. 76.
H. A. Ghali, A. Tarek, and T. A. Moselhy, IEEE Trans. Microwave Theory Tech. 53, 1946 (2005).
Yu. V. Gulyaev, S. A. Nikitov, A. A. Potapov, and A. G. Davydov, J. Commun. Technology and Electronics 50, 988 (2005).
A. A. Potapov, in Proc. Int. Seminar “Days on Diffraction-2006,” St. Petersburg, May 30–June 2, 2006 (SPb Univ., St. Petersburg, 2006).
A. A. Potapov, Nelin. Mir 12 (4), 3 (2014).
Hongming Zang, Xiaoying Zhao, Xiong Xu, et al., in Proc. Int. Conf. on Future Computer and Communication Engineering - (ICFCCE-2014), Vietnam, Da Nang, July 30−August 1, 2014 (ICFCCE, 2014), p. 15.
P. Jarry and J. Beneat, Design and Realizations of Miniaturized Fractal Microwave and RF Filters (Wiley, New York, 2009).
A. A. Potapov, in Applied Synergetics-II (Proc. Int. Sci.-Tech. Conf., Devoted to Memory I. Prigozhina, Ufa, Oct. 20−22, 2004) (UGNTU, Ufa, 2004), Vol. 2, p. 100.
J. S. Petko and D. H. Werner, IEEE Trans. Antennas Propag. 56, 97 (2008).
A. A. Potapov, Tekhnol. Konstruir. Elektron. Apparat., No. 5, 3 (2008).
J. J. Wang, Y. Z. Yin, and X. W. Dai, J. Electromagn. Waves Appl. 23, 1313 (2009).
Rui Guo, Xing Chen, Kama Huang, Electromagnetics 29, 283 (2009).
A. A. Potapov, in Innovations in Radio Engineering Information and Telecommunication Technologies (Proc. Sci.-Tech. Conf. Devoted to 60th Anniversary of the A. L. Mintz Radio Engineering Institute and Faculty of Radio Electronics of Aircraft of MAI, Moscow, Oct. 24−26, 2006 (Ekstra Print, Moscow, 2006), Part 1, p. 66.
E. N. Matveev and A. A. Potapov, in Proc. Progress in Electromagnetics Research Symp. (PIERS 2009), Aug., Moscow, 18−21, 2009 (Electromagn. Academy, Cambridge, MA, 2009), p. 1933.
E. N. Matveev and A. A. Potapov, in Proc. Int. Radar Symp. (IRS-2009), Hamburg, Germany, Sept. 09–11, 2009 (IRS, Hamburg, 2009), p. 465.
A. A. Potapov, in Nonlinear Radar: Collected Articles (Library of the Nelineinyi Mir Magazine: Scientific Series “Fractals. Chaos. Probability”) (Radiotekhnika, Moscow, 2007), Part 3, p. 5.
A. A. Potapov, Problems of Non-Linear Analysis in Engineering Systems 14 (1), 165 (2008);
http://www.kcn.ru/tat_en/science/ans/journals/ansj_cnt/08_1_13.html).
P. Xu, H. Tian, and Y. Ji, J. Opt. Soc. Am. B 27, 640 (2010).
E. E. C. de Oliveira, P. H. F. da Silva, A. L. P. S. Campos, and S. G. da Silva, Microwave Opt. Technol. Lett. 51, 671 (2009).
F. Frezza, L. Pajewski, and G. Schettini, IEEE Trans. Microwave Theory Tech. 52, 220 (2004).
K. Sarabandi and Behdad N. Nader, IEEE Trans. Antennas Propag. 55, 1239 (2007).
G. Stojanović, M. Radovanović, and V. Radonić, in Active and Passive Electron. Compon. (USA, 2008), Article ID 134805.
Y. Xiang, X. Dai, S. Wen, and D. Fan, Opt. Lett. 33, 1255 (2008).
B. Ghosh, S. N. Sinha, and M. V. Kartikeyan, Int. J. RF and Microwave Comput.-Aided Eng. 20, 209 (2010).
B. Hou, X. Q. Liao, and J. K. S. Poon, Opt. Express 18, 3946 (2010).
T. L. Thomas, Smart Skins Program Boeing Military Airplanes (Boeing Comp., Seattle, 1989).
G. Hanson, IEEE Antennas Propagation Mag. 50 (3), 66 (2008).
Y. Huang, W.-Y. Yin, and Q. H. Liu, IEEE Trans. Nanobiosci. 7, 331 (2008).
A. M. Attiya, Progress in Electromagnetics Research, PIERS 94, 419 (2009).
A. A. Potapov and E. N. Matveev, J. Commun. Technology and Electronics 55, 1083 (2010).
A. M. Bobreshov, A. E. Kalashnikov, and A. A. Potapov, Usp. Sovr. Radioelektron., No. 7, 47 (2010).
A. M. Bobreshov, A. E. Kalashnikov, and A. A. Potapov, Usp. Sovr. Radioelektron., No. 1, 3 (2014).
A. A. Potapov, in Proc. IX Int. Conf. on Antenna Theory and Techniques (ICATT-13), Odessa, Ukraine, Sept. 16–20, 2013 (ICATT, Odessa, 2013), p. 62.
http://www.tsc.upc.es/fractalcoms/
J. M. Rius, Exploring the Limits of Fractal Electrodynamics for the Future Telecommunication Technologies (Ann. Report T0+12.IST-2001-33055; FRACTALCOMS, 2003).
B. Mandelbrot, Les Objects Fractals: Forme, Hasardet Dimension (Flammarion, Paris, 1975).
B. Mandelbrot, Fractals: Forme, Chance and Dimension (Freeman, San-Francisco, 1977).
B. Mandelbrot, The Fractals Geometry of Nature (Freeman, New York, 1982).
B. Mandelbrot, The Fractalist. Memoir of a Scientific Maverick (Vintage Books, New York, 2013).
A. A. Potapov, Yu. V. Gulyaev, S. A. Nikitov, et al., Newest Methods of Image Processing, Ed. by A. A. Potapov (Fizmatlit, Moscow, 2008) [in Russian].
D. L. Jaggard, Recent Advances in Electromagnetic Theory, Ed. by H. N. Kritikos and D. L. Jaggard (Springer-Verlag, New York, 1990), p. 183.
D. L. Jaggard in Directions in Electromagnetic Wave Modeling, Ed. by H. L. Bertoni and L. B. Felsen (Plenum, New York, 1991), p. 435.
D. L. Jaggard, Electromagnetic Symmetry, Ed. by C. Baum and H. N. Kritikos (Taylor&Francis, London, 1995), p. 231.
D. L. Jaggard, Fractals in Engineering, Ed. by J. L. Vehel, E. Lutton, and C. Tricot (Springer-Verlag, Berlin, 1997), p. 204.
D. L. Jaggard, Frontiers in Electromagnetics, Ed. by D. H. Werner and R. Mittra (IEEE, New York, 2000), p. 2.
A. A. Potapov in Introduction to Fractals and Chaos, Ed. by M. Crownover (Jones and Bartlett, Sudbury, MA, 1995; Tekhnosfera, Moscow, 2006) p. 374.
A. A. Potapov in Proc. XIV Int. Conf. Haritonovsky Thematic Scientific Readings “High-Power Pulsed Electrophysics,” Devoted to the 110th Anniversary of the Birth of Academic Yu. B. Khariton, Sarov, Apr. 21−25, 2014 (RFYaTs-VNIIEF, Sarov, 2014), p. 80.
A. A. Potapov, in Radiolocation, Navigation, and Communications: RLNC-2011 (Proc. 17 Int. Sci. Eng. Conf., Voronezh, Russia, April 12−14, 2011) (NPF “Sakvoee”, Voronezh, 2011), Vol. 3, p. 2536.
A. A. Potapov, Fiz. Voln. Prots. &Radiotekh. Sist. 14 (3), 54 (2011).
A. A. Potapov, in Irreversible Processes in Nature and Technology: Collection of Proceedings, Ed. by V. S. Gorelik and A. N. Morozov (Bauman MGTU, Moscow, 2012), Vol. 4, p. 5.
A. A. Potapov, in Proc. XI Int. Conf. Zababakhin Scientific Readings Devoted to the 95 Anniversary since the Birth of E. I. Zababakhin, Snezhinsk, Chelyabinsk. Region, Apr. 16–20, 2012 (RFYaTs-VNIITF, Snezhinsk, 2012), p. 311.
A. A. Potapov, in Proc. Int. Conf. on Dynamics, Bifurcations and Strange Attractors, Dedicated to the Memory of L.P. Shil’nikov, (1934-2011), Nizhny Novgorod, Russia, July 1–5, 2013 (Lobachevsky Gos. Univ., Nizhny Novgorod, 2013), p. 98.
A. A. Potapov, Vestn. Nizhegorod. Univ. im. N.I. Lobachevskogo. Ser. Mat. Modelir. Opt. Uprav., No. 5, 172 (2012).
A. A. Potapov, in Vestnik Tambov. Univ., Ser. Estestv. i Tekh. Nauki (Proc. Int. Conf. Kolmogorov Readings-VI: General Problems of Control and Their Applications, Devoted to the 95 Anniversary since the Birth of A.N. Kolmogorov, Tambov, Oct. 07–11, 2013) 18, 2645 (2013).
A. A. Potapov, in Element Base of the Domestic Radio Electronics (Proc. Russia-Belarus Sci.-Tekh. Conf., Devoted to the 110th Anniversary since the Birth of O.V. Losev, Nizhny Novgorod, Sept. 11–14, 2013) (NRO RNTORES im. A.S. Popova, Moscow, 2013), Vol. 1, p. 139.
A. A. Potapov, in Astronautics. Radio Electronics. Geoinformatic, Devoted to the 90th Anniversary since the Birth of the Academician V. F. Utkin (Ryazan’, Oct. 1−3, 2013) (Ryazan. Gos. Radio-Tekh. Univ., Ryazan’, 2013), p. 203.
A. A. Potapov, Turbulence and Wave Processes (Conf., Dedicated to the Centenary of Mikhail D. Millionshchikov (1913–1973), Moscow, MGU, Nov. 26−28, 2013) (INTUIT.RU Ltd, Moscow, 2013).
A. A. Potapov, Elektromagn. Volny Elektron. Sist. 19 (2), 24 (2014).
A. A. Potapov, Ya. S. Shifrin, and R. R. Kuzeev, Antenny, No. 3, 25 (2014).
A. A. Potapov, in Zbornik Radova Konferencije MIT – Matematicke i Informacione Tehnologije (Vrnjackoj Banji 5−9 septembrai u Becicima 10−14 septembra 2013 godine) (Prirodno-Mat. Fakultet Ulverziteta u Pristini (Serbia), Kosovska Mitrovica, 2014), p. 530.
A. A. Potapov, Vestn. Chuvash. Univ., No. 1, 189 (2015).
A. A. Potapov, Vestn. SibGUTI, No. 2, 108 (2015).
A. A. Potapov, Vestn. RGRTU, No. 2, 28 (2015).
A. A. Potapov, Radiotekhnika, No. 8, 95 (2015).
A. A. Potapov, Fiz. Voln. Prots. i Radiotekh. Sist. 18 (3), 59 (2015).
A. A. Potapov, Vestn. Karagandin. Univ., Ser. Fiz. No. 3, 4 (2015).
A. A. Potapov, in Prospects of Development of Long-Range Radar, Integrated Information Support Systems and Complexes for Space Defense, and Complexes for Control and Information Processing, RTI System VKO‑2015 (Proc. III All-Russian Sci.-Tech. Conf., Moscow, May 28, 2015) (Bauman MGTU, Moscow, 2015), p. 573.
A. A. Potapov, in Radar, Navigation, and Communications: RLNC-2010 (Proc. 21st Int. Sci. Eng. Conf., Voronezh, Russia, Apr. 14−16, 2015) (NPF Sakvoee, Voronezh, 2015), Vol. 2, p. 560.
A. A. Potapov in Proc. 8th Int. Conf. on Chaotic Modeling, Simulation and Applications (CHAOS’2015), Paris, May 26–29, 2015 (Henri Poincaré Institute, Paris, 2015), p. 102.
A. A. Potapov, in Chaos and Structures in Nonlinear Systems. Theory and Experiment (Proc. 9th Int. Sci. Conf., Devoted to the 90th Anniversary since the Birth of Academician E. A. Buketov, Kazakhstan, Karaganda, June 18−20, 2015 (Buketov Kar. Gos. Univ., Karaganda, 2015), p. 3.
A. A. Potapov, in The Modeling of Nonlinear Processes and Systems (Proc. Third Int. Scientific Symp. MNPS−2015, Russia, June 22−26, 2015) (Yanus-K, Moscow, 2015), p. 34.
A. A. Potapov, in Communication System and Radio Navigation (Proc. 2 All-Russian Sci.-Tech. Conf., Krasnoyarsk, Aug. 27–28, 2015 (Radiosvyaz’, Krasnoyarsk, 2015), p. 175.
A. A. Potapov, in Microwave Devices and Telecommunication Technologies, KryMiKo’2015 (Proc. 25th Int. Crimean Conf., KryMiKo’2015, Sevastopol’, Sept. 6−12, 2015) (Veber, Sevastopol’, 2015), Vol. 1, p. 65.
Yu.V. Gulyaev and A. A. Potapov, in Problems of Transfer and Information Processing in Telecommunication Networks and Systems (Proc. 18th Int. Sci.-Tech. Conf., Ryazan’, Oct. 26−28, 2015) (Goryachaya Liniya-Telekom, Ryazan’, 2015), p. 3.
A. A. Potapov, in Element Base of Domestic Radio Electronics: Import Substitution and Application (Proc. Russia–Belarus Sci.-Tech. Conf. im. Loseva, Nizhny Novgorod, Nov. 17−19, 2015) (NNGU im. N.I. Lobachevskogo, N. Novgorod, 2015), p. 346.
Yu. V. Opalenov, A. A. Potapov, S. Yu. Fedyunin, in Formation of Complex Signals (Proc. Sci.-Tech. Conf., Suzdal’, Nov. 28−Dec. 1, 1988) (Soyuz NIO SSSR, Moscow, 1988), p. 49.
Yu. V. Opalenov and A. A. Potapov, J. Commun. Technol. Electron. 45, 1311 (2000).
Yu. V. Opalenov and A. A. Potapov, Zarubezh. Radioelektron. Usp. Sovr. Radioelektron., No. 4, 3 (2001).
Yu. V. Opalenov and A. A. Potapov, Questions of Prospective Radar (Collective Monograph) (Radiotekhnika, Moscow, 2003), pp. 132, 148 [in Russian].
Yu. V. Opalenov and A. A. Potapov, in Radar, Navigation, and Communications: RLNC-2005 (Proc. 11th Int. Sci. Eng. Conf., Voronezh, Russia, April 12–14, 2005) (Sakvoee, Voronezh, 2005), Vol. 3, p. 1709.
Yu. V. Opalenov and A. A. Potapov, Electromagn. Phenom. 5, 143 (2005).
A. A. Potapov, J. Commun. Technol. Electron. 52, 245 (2007).
Salim Ali Jabbar, Ahmad Adil Hameed, and Fyath Raad Sami, MIMO Fractal Antennas (Lambert Acad., Saarbrucken, 2013).
M. N. Andrianov and A. A. Potapov, Nelin. Mir 11, 759 (2013).
A. A. Potapov, in The Modeling of Nonlinear Processes and Systems (MNPS-2015) (Proc. Third Int. Scientific Symp. MNPS-2015, Russia, June 22–26, 2015) (Yanus-K, Moscow, 2015), p. 33.
http://www.upc.es/.
http://www.uniroma1.it/.
http://www.epfl.ch/.
http://www.ugr.es/.
http://www.cimne.upc.es/.
https://fractalcoms.upc.edu/.
W. A. Schwalm, B. Moritz, M. Giona, and M. K. Schwalm, Phys. Rev. E 59, 1217 (1999).
M. Giona, W. A. Schwalm, M. K. Schwalm, and A. Adrover, Chem. Eng. Sci. 51, 4717 (1996).
A. Adrover, W. A. Schwalm, M. Giona, and D. S. Bachand, Phys. Rev. E 55, 7304 (1997).
M. Giona, Chaos, Solitons Fractals 7, 1371 (1996).
M. J. Giona, Phys. A 30, 4293 (1997).
J. Zhou, M. J. Lancaster, and F. Huang, Electron. Lett. 39, 665 (2003).
Z. M. Hejazi and P. S. Excell, IEEE Trans. Antennas Propag. 12, 1813 (2002).
C. Y. Tan, Linfeng Chen, Jian Lu et al., IEEE Microwave Wireless Compon. Lett. 13, 247 (2003).
P. Jarry and J. Beneat, Design and Realizations of Miniaturized Fractal Microwave and RF Filters (Wiley, New York, 2009).
J. McVay, N. Engheta, and A. Hoorfar, IEEE Microwave Wireless Compon. Lett. 14, 130 (2004).
V. Crnojevic-Bengin and D. Budimir, Asia-Pacific Microwave Conf. (APMC’, 2004), New Delhi, India, Dec. 2004.
V. Crnojevic-Bengin and D. Budimir, Microwave Opt. Technol. Lett. 46, 195 (2005).
G. A. Lee, M. Megahed, and F. De Flaviis, in Proc. IEEE MTT-S Int. Microwave Symp. Philadelphia, Pennsylvania, June, 8−13, 2003 (IEEE, New York, 2003), p. 527.
A. A. Potapov, in Nanostructural Materials-2008: Belarus–Russia–Ukraine (NANO-2008) (Proc. 1st Int. Conf., Minsk, Apr. 22−25, 2008) (Belorus. Nauka, Minsk, 2008), p. 532.
L. L. Cristea and B. Steinsky, Geom. Dedicata 141, 1 (2009).
A. A. Potapov, in Mathematical Physics and Its Applications (Proc. 3rd Int. Conf., Samara, Aug. 27–Sept. 1, 2012) (SamGTU, Samara, 2012), p. 239.
A. A. Potapov, in Proc. 4th Ann. Conf. Nanotechnol. Soc. Russia, Moscow, Dec. 20, 2012 (Kurchatov Inst., Moscow, 2012), p. 121.
A. A. Potapov and D. V. Slezkin, in Mathematical and Informational Technologies, MIT-2013 (Proc. Int. Conf., Vrnjacka Banja, Serbia Sept. 5−8, and Budva, Montenegro, Sept. 9–14, 2013) (Univ. Pristina, Kosovska Mitrovica, 2013), p. 124.
A. A. Potapov, V. A. Potapov, and R. R. Kuzeev, in Nigmatullin Readings-2013 (Proc. Int. Conf., Kazan’, Nov. 19–21, 2013) (KNITU-KAI, Kazan’, 2013), p. 82.
A. A. Potapov, D. V. Slezkin, and V. A. Potapov, Radiotekhnika, No. 8, 31 (2013).
A. A. Potapov, in High-Power Pulse Electrophysics (Proc. 14th Int. Ann. Khariton. Sci. Reading, Devoted to the 110th Anniversary since the Birth of Academician Yu.B. Khariton, Sarov, Apr. 21–25, 2014) (RFYaTs-VNIIEF, Sarov, 2014), p. 80.
A. A. Potapov, in Proc. XII Int. Conf. Zababakhin Sci. Readings, Snezhinsk, Chelyabinsk. Region, June 2–6, 2014 (RFYaTs-VNIITF, Snezhinsk, 2014), p. 317.
K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, New York, 1974).
Applications of Fractional Calculus in Physics, Ed. by R. Hilfer (World Scientific, Singapore, 1999).
R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
V. V. Uchaikin, Method of Fractional Derivatives (Artishok, Ul’yanovsk, 2008) [in Russian].
L. H. Randy and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, New York, 2007).
The Modeling of Nonlinear Processes and Systems (MNPS-2015) (Proc. 3rd Int. Sci. Symp., Moscow, June 22–26, 2015) (Yanus, Moscow, 2015), p. 151.
A. A. Potapov, Proc. SPIE 7374, 73740 (2009).
A. A. Potapov, Fractal Method and Fractal Paradigm in Modern Natural Sciences (Nauchnaya Kniga, Voronezh, 2012) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by E. Chernokozhin
Rights and permissions
About this article
Cite this article
Potapov, A.A. Fractal Electrodynamics: Numerical Modeling of Small Fractal Antenna Devices and Fractal 3D Microwave Resonators for Modern Ultra-Wideband or Multiband Radio Systems. J. Commun. Technol. Electron. 64, 629–663 (2019). https://doi.org/10.1134/S1064226919060068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064226919060068