Non-Linear Synthesis Problems for Plane Radiating Systems According to the Prescribed Power Directivity Pattern

Affiliation(s)

Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, NASU, Lviv, Ukraine.

Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, NASU, Lviv, Ukraine.

ABSTRACT

A variational formulation of the synthesis problem for plane radiating systems according to the prescribed power directivity pattern (DP) is considered. The function representing the mean-square deviation of the prescribed and synthesized power DPs and containing the additional term with squared norm of the current or field in the antenna aperture is considered as the criterion of optimization. Freedom to choose the phase DP is used to improve the proximity of the prescribed and synthesized DPs. In such formulation, the classes of non-linear problems, for which the non-uniqueness of solutions, their branching and bifurcation are characteristic, arise. The properties of solutions depend on the electric size of radiating system and prescribed power DP. From a practical point of view, the existence of different solutions creating the same or similar DPs, gives the opportunity to choose the solution that has a simpler implementation. The synthesis problems for plane radiating systems and plane arrays are considered.

A variational formulation of the synthesis problem for plane radiating systems according to the prescribed power directivity pattern (DP) is considered. The function representing the mean-square deviation of the prescribed and synthesized power DPs and containing the additional term with squared norm of the current or field in the antenna aperture is considered as the criterion of optimization. Freedom to choose the phase DP is used to improve the proximity of the prescribed and synthesized DPs. In such formulation, the classes of non-linear problems, for which the non-uniqueness of solutions, their branching and bifurcation are characteristic, arise. The properties of solutions depend on the electric size of radiating system and prescribed power DP. From a practical point of view, the existence of different solutions creating the same or similar DPs, gives the opportunity to choose the solution that has a simpler implementation. The synthesis problems for plane radiating systems and plane arrays are considered.

Cite this paper

Andriychuk, M. , Savenko, P. and Tkach, M. (2013) Non-Linear Synthesis Problems for Plane Radiating Systems According to the Prescribed Power Directivity Pattern.*Open Journal of Antennas and Propagation*, **1**, 23-34. doi: 10.4236/ojapr.2013.12006.

Andriychuk, M. , Savenko, P. and Tkach, M. (2013) Non-Linear Synthesis Problems for Plane Radiating Systems According to the Prescribed Power Directivity Pattern.

References

[1] C. A. Balanis, “Antenna Theory: Analysis and Design,” 3rd Edition, Wiley-Interscience, New York, 2005.

[2] L. D. Bakhrakh and S. D. Kremenetskyi, “Synthesis of Radiating Systems (Theory and Methods of Design),” Sov. Radio, Moscow, 1974.

[3] B. M. Minkovich and V. P. Yakovlev, “Theory of Antenna Synthesis,” Sov. Radio, Moscow, 1969.

[4] Ya. S. Shifrin, “Problems of Statistical Theory of Antennas,” Sov. Radio, Moscow, 1970.

[5] W. L Stutzman and G. A. Thiele, “Antenna Theory and Design,” 3rd Edition, Wiley, London, 2013.

[6] E. G. Zelkin and V. G. Sokolov, “Methods of Antenna Synthesis. Phased Arrays and Antennas with Continuous Aperture,” Sov. Radio, Moscow, 1980.

[7] J. N. Sahalos, “Orthogonal Methods for Array Synthesis,” Wiley, New York, 2006, doi:10.1002/0470028548

[8] E. G. Zelkin, V. F. Kravchenko and V. I. Gusevskyi, “Constructive Methods of Approximation in Antenna Theory,” Science Press, Moscow, 2005.

[9] V. F. Kravchenko, “Atomic and R-Functions in Radiophysical Applications (Review),” Proceedings of 7th International Conference on Antenna Theory and Techniques (ICATT-2009), Lviv, 6-9 October 2009, pp. 3-12.

[10] M. I. Andriychuk, N. N. Voitovich, P. A. Savenko and V. P. Tkachuk, “Antenna Synthesis according to Amplitude Radiation Pattern: Numerical Methods and Algorithms,” Nauk. Dumka, Kiev, 1993.

[11] P. O. Savenko, “Nonlinear Problems of Radiating Systems,” Publ. IAPMM of NASU, Lviv, 2002.

[12] W. F. Crosswell, T. Durham, M. Jones, D. Shaubert, P. Friederich and J. Maloney, “Wideband Arrays,” In: Modern Antenna Handbook, Wiley, Interscience, New York, 2008, pp. 581-607.

[13] S. P. Skobelev, “Phased Array Antennas with Optimized Element Pattern,” Artech House, London, 2011.

[14] R. G. Ayestarana, J. Laviadab and F. Las-Herasc, “Realistic Antenna Array Synthesis in Complex Environments Using a MOM-SVR Approach,” Journal of Electromagnetic Waves and Applications, Vol. 23, No. 1, 2009, pp. 97-108. doi:10.1002/0470028548

[15] P. A. Savenko and L. M. Pasnak, “Numerical-Analytical Method of Synthesis for Linear Array of Vibrators with due to Mutual Coupling according to Prescribed Amplitude Radiation Pattern,” Izvestiya Vuzov. Radioelectronics, Vol. 40, No. 12, 1997, pp. 11-25.

[16] D. A. Morgan, “Handbook for EMC Testing and Measurement,” IET Press, London, 2007.

[17] V. I. Dmitriev and N. I. Berezina, “Numerical Methods of Solution of Synthesis Problems for Radiating Systems,” Publ. MSU, Moscow, 1986.

[18] P. A. Savenko and M. D. Tkach, “Structure of Solutions of the Synthesis Problem for Linear Microstrip Array Using the Power Criterion,” Izvestiya Vuzov. Radioelectronics, Vol. 46, No. 1, 2004, pp. 38-49.

[19] M. A. Krasnoselskyi, G. M. Vainikko and P. P. Zabreyko, “Approximate Solution of Operator Equations,” Nauka, Moscow, 1969.

[20] P. A. Savenko, “Synthesis of Linear Arrays according to Prescribed Amplitude Radiation Pattern,” Izvestiya Vuzov. Radioelectronics, Vol. 22, No. 12, 1979, pp. 1498-1504.

[21] V. F. Kravchenko, L. P. Protsakh, P. A. Savenko and M. D. Tkach, “Mathematical Features of Synthesis of Plane Equidistant Arrays according to Prescribed Amplitude Radiation Pattern,” Antennas, Vol. 3, No. 154, 2010, pp. 34-48.

[22] P. A. Savenko and L. P. Protsakh, “Method of Implicit Function Applying to Solution of Two-Dimensional Nonlinear Spectral Problem,” Izvestiya Vuzov. Radioelectronics, Vol. 11, No. 546, 2007, pp. 41-44.

[23] M. M. Veinberg and V. A. Trenogin, “Theory of Branching of Solutions to Nonlinear Equations,” Nauka, Moscow, 1969.

[24] P. O. Savenko and M. D. Tkach, “Antenna Synthesis with Flat Radiating Aperture according to the Prescribed Directivity Pattern by Power,” Proceedings of 17th International Seminar/Workshop DIPED-2012, Tbilisi, 24-27 September 2012, pp. 75-80.

[1] C. A. Balanis, “Antenna Theory: Analysis and Design,” 3rd Edition, Wiley-Interscience, New York, 2005.

[2] L. D. Bakhrakh and S. D. Kremenetskyi, “Synthesis of Radiating Systems (Theory and Methods of Design),” Sov. Radio, Moscow, 1974.

[3] B. M. Minkovich and V. P. Yakovlev, “Theory of Antenna Synthesis,” Sov. Radio, Moscow, 1969.

[4] Ya. S. Shifrin, “Problems of Statistical Theory of Antennas,” Sov. Radio, Moscow, 1970.

[5] W. L Stutzman and G. A. Thiele, “Antenna Theory and Design,” 3rd Edition, Wiley, London, 2013.

[6] E. G. Zelkin and V. G. Sokolov, “Methods of Antenna Synthesis. Phased Arrays and Antennas with Continuous Aperture,” Sov. Radio, Moscow, 1980.

[7] J. N. Sahalos, “Orthogonal Methods for Array Synthesis,” Wiley, New York, 2006, doi:10.1002/0470028548

[8] E. G. Zelkin, V. F. Kravchenko and V. I. Gusevskyi, “Constructive Methods of Approximation in Antenna Theory,” Science Press, Moscow, 2005.

[9] V. F. Kravchenko, “Atomic and R-Functions in Radiophysical Applications (Review),” Proceedings of 7th International Conference on Antenna Theory and Techniques (ICATT-2009), Lviv, 6-9 October 2009, pp. 3-12.

[10] M. I. Andriychuk, N. N. Voitovich, P. A. Savenko and V. P. Tkachuk, “Antenna Synthesis according to Amplitude Radiation Pattern: Numerical Methods and Algorithms,” Nauk. Dumka, Kiev, 1993.

[11] P. O. Savenko, “Nonlinear Problems of Radiating Systems,” Publ. IAPMM of NASU, Lviv, 2002.

[12] W. F. Crosswell, T. Durham, M. Jones, D. Shaubert, P. Friederich and J. Maloney, “Wideband Arrays,” In: Modern Antenna Handbook, Wiley, Interscience, New York, 2008, pp. 581-607.

[13] S. P. Skobelev, “Phased Array Antennas with Optimized Element Pattern,” Artech House, London, 2011.

[14] R. G. Ayestarana, J. Laviadab and F. Las-Herasc, “Realistic Antenna Array Synthesis in Complex Environments Using a MOM-SVR Approach,” Journal of Electromagnetic Waves and Applications, Vol. 23, No. 1, 2009, pp. 97-108. doi:10.1002/0470028548

[15] P. A. Savenko and L. M. Pasnak, “Numerical-Analytical Method of Synthesis for Linear Array of Vibrators with due to Mutual Coupling according to Prescribed Amplitude Radiation Pattern,” Izvestiya Vuzov. Radioelectronics, Vol. 40, No. 12, 1997, pp. 11-25.

[16] D. A. Morgan, “Handbook for EMC Testing and Measurement,” IET Press, London, 2007.

[17] V. I. Dmitriev and N. I. Berezina, “Numerical Methods of Solution of Synthesis Problems for Radiating Systems,” Publ. MSU, Moscow, 1986.

[18] P. A. Savenko and M. D. Tkach, “Structure of Solutions of the Synthesis Problem for Linear Microstrip Array Using the Power Criterion,” Izvestiya Vuzov. Radioelectronics, Vol. 46, No. 1, 2004, pp. 38-49.

[19] M. A. Krasnoselskyi, G. M. Vainikko and P. P. Zabreyko, “Approximate Solution of Operator Equations,” Nauka, Moscow, 1969.

[20] P. A. Savenko, “Synthesis of Linear Arrays according to Prescribed Amplitude Radiation Pattern,” Izvestiya Vuzov. Radioelectronics, Vol. 22, No. 12, 1979, pp. 1498-1504.

[21] V. F. Kravchenko, L. P. Protsakh, P. A. Savenko and M. D. Tkach, “Mathematical Features of Synthesis of Plane Equidistant Arrays according to Prescribed Amplitude Radiation Pattern,” Antennas, Vol. 3, No. 154, 2010, pp. 34-48.

[22] P. A. Savenko and L. P. Protsakh, “Method of Implicit Function Applying to Solution of Two-Dimensional Nonlinear Spectral Problem,” Izvestiya Vuzov. Radioelectronics, Vol. 11, No. 546, 2007, pp. 41-44.

[23] M. M. Veinberg and V. A. Trenogin, “Theory of Branching of Solutions to Nonlinear Equations,” Nauka, Moscow, 1969.

[24] P. O. Savenko and M. D. Tkach, “Antenna Synthesis with Flat Radiating Aperture according to the Prescribed Directivity Pattern by Power,” Proceedings of 17th International Seminar/Workshop DIPED-2012, Tbilisi, 24-27 September 2012, pp. 75-80.