Optimal Composition of Characteristic Modes For Minimal Quality Factor Q Miloslav ˇ Capek Luk´ aˇ s Jel´ ınek Department of Electromagnetic Field CTU in Prague, Czech Republic miloslav.capek@fel.cvut.cz 2016 IEEE International Symposium on Antennas and Propagation/USNC-URSI National Radio Science meeting Fajardo, Puerto Rico June 27, 2016 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 1 / 20
Outline 1 Quality factor Q Minimization of quality factor Q 2 3 Results: Quality factor Q Results: Sub-optimality of G/Q 4 5 Excitation of optimal currents Conclusion 6 In this talk: ◮ electric currents in vacuum, ◮ only surface regions are treated, ◮ all quantities in their matrix form, i.e. operators → matrices, functions → vectors, ◮ small electrical size is considered, i.e. ka < 1, √ ◮ time-harmonic quantities, i.e., A ( r , t ) = 2 Re { A ( r ) exp (j ωt ) } are considered. ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 2 / 20
Quality factor Q Minimization of quality factor Q Quality factor Q . . . ◮ is (generally) proportional to FBW, ◮ therefore, of interest for ESA ( ka < 1). Fundamental bounds of quality factor Q ◮ are known for several canonical bodies, ◮ many interesting works recently appeared 1 , • still, they are unknown for arbitrarily shaped bodies. 1 M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape”, Proc. of Royal Soc. A , vol. 463, pp. 2589–2607, 2007. doi : 10.1098/rspa.2007.1893 M. Gustafsson, D. Tayli, C. Ehrenborg, et al. , “Tutorial on antenna current optimization using MATLAB and CVX”, , FERMAT , 2015 O. S. Kim, “Lower bounds on Q for finite size antennas of arbitrary shape”, IEEE Trans. Antennas Propag. , vol. 64, no. 1, pp. 146–154, 2016. doi : 10.1109/TAP.2015.2499764 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 3 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , STEP 2 definition of stored energy � W sto , 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , STEP 2 definition of stored energy � W sto , STEP 3 formulation of optimization task related to (1), 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , STEP 2 definition of stored energy � W sto , STEP 3 formulation of optimization task related to (1), STEP 4 representation of I opt in an appropriate basis, 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , STEP 2 definition of stored energy � W sto , STEP 3 formulation of optimization task related to (1), STEP 4 representation of I opt in an appropriate basis, STEP 5 optimal composition of modal currents forming I opt . 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Minimization of quality factor Q Current I opt minimizing quality factor Q of a given shape Ω: Q ( I opt ) = min I { Q ( I ) } (1) How to find I opt for a given Ω? Procedure followed in this talk 2 : STEP 1 definition of quality factor Q , STEP 2 definition of stored energy � W sto , STEP 3 formulation of optimization task related to (1), STEP 4 representation of I opt in an appropriate basis, STEP 5 optimal composition of modal currents forming I opt . 2 M. Capek and L. Jelinek, “Optimal composition of modal currents for minimal quality factor Q”, , 2016, arXiv:1602.04808 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 4 / 20
Quality factor Q Step 1+2: Definition of Q and � W sto Quality factor Q defined by parts as Q ( I ) = Q U ( I ) + Q ext ( I ) (2) using stored energy 3 I H ω∂ X ∂ω I Q U ( I ) = ω � = I H X ′ I W sto 2 I H RI = 2 I H RI , (3) P r and tuning � � � I H XI � Q ext ( I ) = 2 I H RI . (4) � J ≈ I n f n , Z = R + j X n 3 M. Cismasu and M. Gustafsson, “Antenna bandwidth optimization with single freuquency simulation”, IEEE Trans. Antennas Propag. , vol. 62, no. 3, pp. 1304–1311, 2014, R. F. Harrington and J. R. Mautz, “Control of radar scattering by reactive loading”, IEEE Trans. Antennas Propag. , vol. 20, no. 4, pp. 446–454, 1972. doi : 10.1109/TAP.1972.1140234 , G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”, IEEE Trans. Antennas Propag. , vol. 58, no. 4, pp. 1112–1127, 2010. doi : 10.1109/TAP.2010.2041166 . ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 5 / 20
Minimization of quality factor Q Step 3: Formulation of the problem Find I opt so that minimize quality factor Q, (5) � W m − � subject to W e = 0 . (6) ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 6 / 20
Minimization of quality factor Q Step 3: Formulation of the problem Find I opt so that minimize quality factor Q, (5) � W m − � subject to W e = 0 . (6) Searching for self-resonant current I opt fulfilling (5)–(6) is not a convex problem. ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 6 / 20
Minimization of quality factor Q Step 4: Representation of I opt Current decomposition Let us decompose the current into (yet unknown) modes such that N � I = α n I n . (7) n =1 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 7 / 20
Minimization of quality factor Q Step 4: Representation of I opt Current decomposition Let us decompose the current into (yet unknown) modes such that N � I = α n I n . (7) n =1 Then, the quality factor Q reads � � � � � V � U � V � U � � α ∗ u α v I H u X ′ I v + α ∗ u α v I H u XI v � � v =1 u =1 v =1 u =1 Q ( I ) = . (8) � � V U α ∗ u α v I H 2 u RI v v =1 u =1 ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 7 / 20
Minimization of quality factor Q Step 4: Representation of I opt Current decomposition Let us decompose the current into (yet unknown) modes such that N � I = α n I n . (7) n =1 Then, the quality factor Q reads � � � � � V � U � V � U � � α ∗ u α v I H u X ′ I v + α ∗ u α v I H u XI v � � v =1 u =1 v =1 u =1 Q ( I ) = . (8) � � V U α ∗ u α v I H 2 u RI v v =1 u =1 Analytical solution can easily be found if I H u RI v = δ uv , (9) I H u XI v = A uv δ uv , (10) I H u X ′ I v = B uv δ uv . (11) ˘ Capek, Jel´ ınek – AP-S/URSI 2016 Optimal Composition of CMs For Minimal Q 7 / 20
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