Optimal Currents and Optimal Antennas Preliminary Results Miloslav Capek Lukas Jelinek Department of Electromagnetic Field Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz The 12th European Conference on Antennas and Propagation London, United Kingdom April 10, 2018 Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 1 / 23
Outline 1 Electrically Small Antennas 2 Fundamental Bounds 3 Synthesis: Determination of Feeder’s Position 4 Synthesis: Reduction of Degrees of Freedom 5 Synthesis: Results 6 Concluding Remarks This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e. , A ( r , t ) = Re { A ( r ) exp (j ωt ) } . Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 2 / 23
Electrically Small Antennas Electrically Small Antennas ◮ ESAs ( ka < 1 / 2) mainly suffer from restrictions on directivity D , efficiency η , Directivity and bandwidth ∝ 1 /Q . D ◮ Each antenna parameter can independently be boosted ( but price is paid) Electrical • superdirectivity, size • 3D/volumetric antennas, ka < 1 / 2 • active antennas – any bounds? Efficiency Bandwidth η FBW Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 3 / 23
Electrically Small Antennas Electrically Small Antennas ◮ ESAs ( ka < 1 / 2) mainly suffer from restrictions on directivity D , efficiency η , Directivity and bandwidth ∝ 1 /Q . D ◮ Each antenna parameter can independently be boosted ( but price is paid) Electrical • superdirectivity, size • 3D/volumetric antennas, ka < 1 / 2 • active antennas – any bounds? Efficiency Bandwidth η FBW ◮ On the fundamental level, the optimal distribution of the sources is sought for. . . • fundamental bounds. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 3 / 23
Fundamental Bounds Optimal Currents and Optimal Antennas What is the best current distribution in principle? Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23
Fundamental Bounds Optimal Currents and Optimal Antennas What is the best current distribution in principle? “The best” in what sense? ◮ Boundary region Ω → Ω N , ◮ angular frequency ω (el. size ka ), ◮ selected antenna metric(s). Ω Ω N Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23
Fundamental Bounds Optimal Currents and Optimal Antennas What is the best current distribution in What is the best reachable current principle? distribution? “The best” in what sense? ◮ Boundary region Ω → Ω N , ◮ angular frequency ω (el. size ka ), ◮ selected antenna metric(s). Ω Ω N Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23
Fundamental Bounds Optimal Currents and Optimal Antennas What is the best current distribution in What is the best reachable current principle? distribution? “The best” in what sense? “The best” in what sense? ◮ Boundary region Ω → Ω N , ◮ ← Same as on the left and ◮ angular frequency ω (el. size ka ), ◮ feeding. ◮ selected antenna metric(s). Ω Ω N Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23
Fundamental Bounds Determination of Fundamental Bounds Minimization of bilinear form κ = I H AI I H BI , A ≻ 0 , B ≻ 0 (3) solved via generalized eigenvalue problem AI = κ BI . (4) Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 5 / 23
Fundamental Bounds Determination of Fundamental Bounds Minimization of bilinear form κ = I H AI I H BI , A ≻ 0 , B ≻ 0 (3) solved via generalized eigenvalue problem AI = κ BI . (4) Multi-objective form � � A = α i A i , α i = 1 . (5) i i ◮ No feeding, ◮ no geometry modifications Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 5 / 23
Fundamental Bounds Determination of Fundamental Bounds Minimization of bilinear form κ = I H AI I H BI , A ≻ 0 , B ≻ 0 (3) solved via generalized eigenvalue problem AI = κ BI . (4) Multi-objective form Example: Minimum Q-factor 1 � � A = α i A i , α i = 1 . (5) i i A = (1 − ν ) X m + ν X e = W , (1) B = R , (2) ◮ No feeding, where Z = R + j X is impedance matrix. ◮ no geometry modifications 1 M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, IEEE Trans. Antennas Propag. , vol. 65, no. 8, pp. 4115–4123, 2017. doi : 10.1109/TAP.2017.2717478 Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 5 / 23
Fundamental Bounds Antenna Synthesis: Combinatorial Explosion How much DOF we have? W N (unknowns) 28 52 120 ∞ 5 . 24 · 10 29 1 . 39 · 10 68 1 . 15 · 10 199 possibilities ∞ 2 . 68 · 10 8 4 . 50 · 10 15 1 . 33 · 10 36 unique solutions ∞ Complexity of geometrical optimization for given voltage gap (red line) and N unknowns. ◮ Combinatorial explosion → curse of complexity → NP-hard problem, ◮ good parametrization is needed, reduction of DOFs, application of GA. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 6 / 23
Synthesis: Determination of Feeder’s Position Determination of Feeder’s Position ◮ For one feeder, the optimal placement with respect to a given quantity can be found directly (no heuristics)! Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 7 / 23
Synthesis: Determination of Feeder’s Position Determination of Feeder’s Position ◮ For one feeder, the optimal placement with respect to a given quantity can be found directly (no heuristics)! Example: minimum Q-factor � H W Z − 1 V Z − 1 V Q = I H WI � � � = V H W Z V I H RI = V H R Z V , (6) ( Z − 1 V ) H R ( Z − 1 V ) with ZI = V , A Z ≡ Z − H AZ − 1 , A ∈ R N × N , and since vector of excitation coefficients is full of zero except one position with V n = 1, we get optimal position as n : min { diag ( W Z ) ⊘ diag ( R Z ) } (7) Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 7 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Initial Geometry ◮ Rectangular plate L × L/ 2, ◮ electrical size ka = 0 . 5, ◮ k is wavenumber, √ ◮ a = 5 L/ 4. Number of possibilities: ∞ . Our ambition: PEC rectangular plate of L × L/ 2 size. Reduce the complexity. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 8 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Initial Geometry ◮ Rectangular plate L × L/ 2, ◮ electrical size ka = 0 . 5, ◮ k is wavenumber, √ ◮ a = 5 L/ 4. Number of possibilities: 4 . 83 · 10 170 . Our ambition: PEC rectangular plate of L × L/ 2 size. Reduce the complexity. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 8 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Reducing Geometrical Complexity ◮ Number of RWG is drastically decreased, ◮ shorts are eliminated, ◮ symmetry is preserved for MoM acceleration. Reducing complexity of the shape to be optimized. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 9 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Discretization Grid ◮ Uniform grid to preserve symmetries and improve convergence. Number of possibilities: 2 . 16 · 10 127 . Discretized model. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 10 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Feedable Edges To calculate optimal feeding, not all edges have to be taken into account: ◮ Some can cause shorts. Number of potent. feeders: 103. One feeder preserves symmetry: 4 possibilities. Edges to be potentially fed. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 11 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Pixelized Structure To compress the optimization ✼ ✶✶ ✶✽ ✷✷ ✷✾ ✸✸ ✹✵ ✹✼ ✺✶ ✺✽ ✻✷ ✻✾ ✼✸ ✽✵ problem, map from RWG to GA ✻ ✶✼ ✷✽ ✸✾ ✹✻ ✺✼ ✻✽ ✼✾ “pixels” is done: ✺ ✶✵ ✶✻ ✷✶ ✷✼ ✸✷ ✸✽ ✹✺ ✺✵ ✺✻ ✻✶ ✻✼ ✼✷ ✼✽ ◮ pixel enabled (1) = all RWG ✹ ✶✺ ✷✻ ✸✼ ✹✹ ✺✺ ✻✻ ✼✼ edges present, ✸ ✾ ✶✹ ✷✵ ✷✺ ✸✶ ✸✻ ✹✸ ✹✾ ✺✹ ✻✵ ✻✺ ✼✶ ✼✻ ◮ pixel disabled (0) = all RWG ✷ ✶✸ ✷✹ ✸✺ ✹✷ ✺✸ ✻✹ ✼✺ edges removed. ✶ ✽ ✶✷ ✶✾ ✷✸ ✸✵ ✸✹ ✹✶ ✹✽ ✺✷ ✺✾ ✻✸ ✼✵ ✼✹ Number of possibilities: 1 . 21 · 10 24 . Pixelization of rectangle into 80 unknowns. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 12 / 23
Synthesis: Reduction of Degrees of Freedom Geometry Optimized Structures and Their Reduction H Holes Grid RWGs RWGs (reduced) Feedable edges GA pixels 6 6 × 3 14 × 7 564 423 103 80 8 8 × 4 18 × 9 945 689 169 130 10 10 × 5 22 × 11 1419 1019 192 192 Comparison of optimized structures. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 13 / 23
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