Excitation of Optimal and Suboptimal Currents Miloslav ˇ Martin ˇ Capek 1 ınek 1 Petr Kadlec 2 Strambach 3 Luk´ aˇ s Jel´ 1 Department of Electromagnetic Field Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz 2 Department of Radio Electronics Brno University of Technology, Czech Republic 3 Faculty of Information Technology Czech Technical University in Prague, Czech Republic The 11th European Conference on Antennas and Propagation Paris, France March 23, 2017 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 1 / 18
Outline Optimal Currents 1 Minimum Quality Factor Q 2 Solution Expressed in Characteristic Modes 3 Alternative Bases 4 Excitation – Sub-optimal Currents 5 Structure of the Solution Space 6 This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e. , A ( r , t ) = Re { A ( r ) exp (j ωt ) } . ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 2 / 18
Optimal Currents Optimal Currents – What Are They? A current J = J ( r , ω ), r ∈ Ω , is denoted J opt and called as optimal current 1 if � J opt , L ( J opt ) � = min J � J , {L ( J ) �} = p min , (1) � J opt , M n ( J opt ) � = q n , (2) � J opt , N n ( J opt ) � ≤ r n . (3) 1 L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces”, IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. doi : 10.1109/TAP.2016.2624735 . ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 3 / 18
Optimal Currents Optimal Currents – What Are They? A current J = J ( r , ω ), r ∈ Ω , is denoted J opt and called as optimal current 1 if � J opt , L ( J opt ) � = min J � J , {L ( J ) �} = p min , (1) � J opt , M n ( J opt ) � = q n , (2) � J opt , N n ( J opt ) � ≤ r n . (3) What are the optimal currents good for? ◮ They establish fundamental bounds of p = � J , L ( J ) � for a given Ω and ω . Use case: Minimum quality factor Q for electrically small antennas. 1 L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces”, IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. doi : 10.1109/TAP.2016.2624735 . ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 3 / 18
Minimum Quality Factor Q Minimization of Quality Factor Q Current J opt minimizing quality factor Q of a given shape Ω : Q ( J opt ) = min J { Q ( J ) } (4) ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 4 / 18
Minimum Quality Factor Q Minimization of Quality Factor Q Current J opt minimizing quality factor Q of a given shape Ω : Q ( J opt ) = min J { Q ( J ) } (4) Rao-Wilton-Glisson basis functions l n P − T + ρ − n n n � ρ + J ( r ) ≈ I n ψ n ( r ) (5) n P + A − n A + n T − n n n r z l n ρ ± � I H X m I , I H X e I � ψ n ( r ) = = max Q ( I ) = 2 ω max { W m , W e } y 2 A ± n x O n (6) I H RI P r RWG basis functions. 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , IEEE Trans. Antennas Propag. , vol. 64, no. 12, pp. 5230–5242, 2016. doi : 10.1109/TAP.2016.2617779 L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces”, IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. doi : 10.1109/TAP.2016.2624735 M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017, eprint arXiv: 1612.07676. [Online]. Available: https://arxiv.org/abs/1612.07676 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 4 / 18
Minimum Quality Factor Q Minimization of Quality Factor Q Current J opt minimizing quality factor Q of a given shape Ω : Q ( J opt ) = min J { Q ( J ) } (4) Rao-Wilton-Glisson basis functions l n P − T + ρ − n n n � ρ + J ( r ) ≈ I n ψ n ( r ) (5) n P + A − n A + n T − n n n r z l n ρ ± � I H X m I , I H X e I � ψ n ( r ) = = max Q ( I ) = 2 ω max { W m , W e } y 2 A ± n x O n (6) I H RI P r RWG basis functions. We know several efficient minimization procedures 2 . 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , IEEE Trans. Antennas Propag. , vol. 64, no. 12, pp. 5230–5242, 2016. doi : 10.1109/TAP.2016.2617779 L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces”, IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. doi : 10.1109/TAP.2016.2624735 M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017, eprint arXiv: 1612.07676. [Online]. Available: https://arxiv.org/abs/1612.07676 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 4 / 18
Minimum Quality Factor Q Basis of Characteristic Modes Diagonalization of impedance matrix Z = R + j X as 3 XI m = λ m RI m (7) ◮ useful set of entire-domain basis functions, � I = α m I m (8) m ◮ only few modes needed to represent ESAs (1 + j λ m ) δ mn = 1 2 I H m ZI n . (9) ◮ meant originally for scattering problems 4 . 3 R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies”, IEEE Trans. Antennas Propag. , vol. 19, no. 5, pp. 622–628, 1971. doi : 10.1109/TAP.1971.1139999 . 4 R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated and scattered fields”, IEEE Trans. Antennas Propag. , vol. 19, no. 3, pp. 348–358, 1971. doi : 10.1109/TAP.1971.1139935 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 5 / 18
Solution Expressed in Characteristic Modes Approximative Solution in CM Basis Optimal current can be approximated 5 by Q ( I opt ) ≈ Q ( I 1 + α opt I 2 ) (10) � � − I T − λ 1 1 XI 1 e − j ϕ = e − j ϕ , α opt = ϕ ∈ [ − π, π ] (11) I T λ 2 2 XI 2 ◮ The optimization problem can be advantageously solved in other bases as well! Two different optimal currents for Q min . 5 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , IEEE Trans. Antennas Propag. , vol. 64, no. 12, pp. 5230–5242, 2016. doi : 10.1109/TAP.2016.2617779 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 6 / 18
Solution Expressed in Characteristic Modes Modal Composition of the Optimal Current J opt Optimal current with respect to minimum quality factor Q . ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 7 / 18
Solution Expressed in Characteristic Modes Modal Composition of the Optimal Current J opt Optimal current with respect to minimum quality factor Q . + Dominant (dipole-like) characteristic mode J 1 . First inductive (loop-like) mode J 2 , α 2 = 0 . 4553. ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 7 / 18
Alternative Bases Alternative Bases ◮ Stored energy modes 6 ω∂ X ∂ω I m = q m RI m , (12) ◮ minimum quality factor Q modes 7 ((1 − ν ) X m + ν X e ) I m = Q νm RI m , (13) ◮ optimal gain G including losses in metalization 8 1 U (ˆ e , ˆ r ) I m = ζ m 8 π ( R + R ρ ) I m , (14) ◮ optimal radiation efficiency 8 RI m = ζ m ( R + R ρ ) . (15) 6 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , IEEE Trans. Antennas Propag. , vol. 64, no. 12, pp. 5230–5242, 2016. doi : 10.1109/TAP.2016.2617779 7 M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017, eprint arXiv: 1612.07676. [Online]. Available: https://arxiv.org/abs/1612.07676 8 L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces”, IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. doi : 10.1109/TAP.2016.2624735 ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 8 / 18
Excitation – Sub-optimal Currents Excitation of Optimal Currents Optimal current I opt for minimal quality factor Q . ◮ How to feed optimal currents? ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 9 / 18
Excitation – Sub-optimal Currents Excitation of Optimal Currents Optimal current I opt for minimal quality factor Q . Feeding map (abs values) for optimal current I opt . ◮ How to feed optimal currents? I H ◮ V opt = ZI opt n V I n � I = (16) I H 1 + j λ n n RI n n ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 9 / 18
Excitation – Sub-optimal Currents Excitation of Optimal Currents Optimal current I opt for minimal quality factor Q . Feeding map (abs values) for optimal current I opt . ◮ How to feed optimal currents? I H ◮ V opt = ZI opt n V I n � I = (16) I H 1 + j λ n n RI n • Impressed currents in vacuum. n • Shape has to be modified. • Can modal techniques help? ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 9 / 18
Excitation – Sub-optimal Currents How to Excite the Optimal Currents 240 optimal positions 180 quality factor Q of four feeders ◮ Let us try to modify structure manually. 120 • A loop. • 2 modes = at least two feeders? 60 fed current optimal current 0 0 2 4 6 8 10 12 14 16 number of feeding edges Dependence of Q min on number of (optimally placed) feeders. ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 10 / 18
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