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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

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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

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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 (jωt)}.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 2 / 23

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Electrically Small Antennas

Electrically Small Antennas

Directivity

D

Efficiency

η

Bandwidth

FBW

Electrical size

ka < 1/2

◮ ESAs (ka < 1/2) mainly suffer from restrictions on directivity D, efficiency η, and bandwidth ∝ 1/Q. ◮ Each antenna parameter can independently be boosted (but price is paid)

  • superdirectivity,
  • 3D/volumetric antennas,
  • active antennas – any bounds?

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 3 / 23

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Electrically Small Antennas

Electrically Small Antennas

Directivity

D

Efficiency

η

Bandwidth

FBW

Electrical size

ka < 1/2

◮ ESAs (ka < 1/2) mainly suffer from restrictions on directivity D, efficiency η, and bandwidth ∝ 1/Q. ◮ Each antenna parameter can independently be boosted (but price is paid)

  • superdirectivity,
  • 3D/volumetric antennas,
  • active antennas – any bounds?

◮ 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

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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

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Fundamental Bounds

Optimal Currents and Optimal Antennas

What is the best current distribution in principle? “The best” in what sense? ◮ Boundary region Ω → ΩN, ◮ angular frequency ω (el. size ka), ◮ selected antenna metric(s). Ω ΩN

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23

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Fundamental Bounds

Optimal Currents and Optimal Antennas

What is the best current distribution in principle? What is the best reachable current distribution? “The best” in what sense? ◮ Boundary region Ω → ΩN, ◮ angular frequency ω (el. size ka), ◮ selected antenna metric(s). Ω ΩN

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23

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Fundamental Bounds

Optimal Currents and Optimal Antennas

What is the best current distribution in principle? What is the best reachable current distribution? “The best” in what sense? ◮ Boundary region Ω → ΩN, ◮ angular frequency ω (el. size ka), ◮ selected antenna metric(s). “The best” in what sense? ◮ ← Same as on the left and ◮ feeding. Ω ΩN

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 4 / 23

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Fundamental Bounds

Determination of Fundamental Bounds

Minimization of bilinear form κ = IHAI IHBI, A ≻ 0, B ≻ 0 (3) solved via generalized eigenvalue problem AI = κBI. (4)

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 5 / 23

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Fundamental Bounds

Determination of Fundamental Bounds

Minimization of bilinear form κ = IHAI IHBI, A ≻ 0, B ≻ 0 (3) solved via generalized eigenvalue problem AI = κBI. (4) Multi-objective form A =

  • i

αiAi,

  • i

αi = 1. (5) ◮ No feeding, ◮ no geometry modifications

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 5 / 23

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Fundamental Bounds

Determination of Fundamental Bounds

Example: Minimum Q-factor1 A = (1 − ν) Xm + νXe = W, (1) B = R, (2) where Z = R + jX is impedance matrix. Minimization of bilinear form κ = IHAI IHBI, A ≻ 0, B ≻ 0 (3) solved via generalized eigenvalue problem AI = κBI. (4) Multi-objective form A =

  • i

αiAi,

  • i

αi = 1. (5) ◮ No feeding, ◮ no geometry modifications

  • 1M. 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

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Fundamental Bounds

Antenna Synthesis: Combinatorial Explosion

How much DOF we have?

W

N (unknowns) 28 52 120 ∞ possibilities 5.24 · 1029 1.39 · 1068 1.15 · 10199 ∞ unique solutions 2.68 · 108 4.50 · 1015 1.33 · 1036 ∞

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

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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

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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 Q = IHWI IHRI =

  • Z−1V

H W

  • Z−1V
  • (Z−1V)H R (Z−1V)

= VHWZV VHRZV , (6) with ZI = V, AZ ≡ Z−HAZ−1, A ∈ RN×N, and since vector of excitation coefficients is full of zero except one position with Vn = 1, we get optimal position as n : min {diag (WZ) ⊘ diag (RZ)} (7)

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 7 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Initial Geometry

PEC rectangular plate of L × L/2 size.

◮ Rectangular plate L × L/2, ◮ electrical size ka = 0.5, ◮ k is wavenumber, ◮ a = √ 5L/4. Number of possibilities: ∞. Our ambition: Reduce the complexity.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 8 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Initial Geometry

PEC rectangular plate of L × L/2 size.

◮ Rectangular plate L × L/2, ◮ electrical size ka = 0.5, ◮ k is wavenumber, ◮ a = √ 5L/4. Number of possibilities: 4.83 · 10170. Our ambition: Reduce the complexity.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 8 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Reducing Geometrical Complexity

Reducing complexity of the shape to be optimized.

◮ Number of RWG is drastically decreased, ◮ shorts are eliminated, ◮ symmetry is preserved for MoM acceleration.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 9 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Discretization Grid

Discretized model.

◮ Uniform grid to preserve symmetries and improve convergence. Number of possibilities: 2.16 · 10127.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 10 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Feedable Edges

Edges to be potentially fed.

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.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 11 / 23

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Synthesis: Reduction of Degrees of Freedom

Geometry

Pixelized Structure ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ✺✶ ✺✷ ✺✸ ✺✹ ✺✺ ✺✻ ✺✼ ✺✽ ✺✾ ✻✵ ✻✶ ✻✷ ✻✸ ✻✹ ✻✺ ✻✻ ✻✼ ✻✽ ✻✾ ✼✵ ✼✶ ✼✷ ✼✸ ✼✹ ✼✺ ✼✻ ✼✼ ✼✽ ✼✾ ✽✵

Pixelization of rectangle into 80 unknowns.

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 · 1024.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 12 / 23

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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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Synthesis: Results

What Can Be Optimized?

Following criteria are always calculated and optimized2 only when chosen so: ◮ minimum Q-factor, ◮ external tuning |Xin| Rin = 2Qext, ◮ input resistance |R0 − Rin| R0 , ◮ radiation (in)efficiency 1 − ηrad, ◮ total area spanned by the structure Aused Atot . Notes: ◮ Arbitrary number of criteria can be optimized (recommended: 2–4). ◮ All quantities normalized (no units) and to be minimized.

2(2017). Fast Optimization Procedures (FOPS), , Czech Technical University in Prague, [Online]. Available:

www.antennatoolbox.com/fops

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 14 / 23

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Synthesis: Results

Multiobjective Optimization

Quality factor Q vs tuning element.

50 55 60 65 70 75 5 10 15 20 25 30 min

Ω,V {Q}

min

Ω,V {|Xin| /Rin}

H = 6, 500 agents, 1500 iterations cut from Q/Qext/Rin/A optimization

◮ Expected result. ◮ External tuning helps a lot.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 15 / 23

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Synthesis: Results

Multiobjective Optimization

Quality factor Q vs required area A.

102 103 104 0.1 0.2 0.3 0.4 min

Ω,V {Q}

min

Ω,V {Aused/Atot}

250 agents, 1000 iterations 6 × 3, 1040 s 8 × 4, 2397 s 10 × 5, 3860 s 8 × 4 (50 iters., 250 ags.)

◮ The granularity of the grid causes big difference in Q-factor.

  • What about convergence?
  • ↑ N →↓ Q → combinatorial

explosion

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 16 / 23

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Synthesis: Results

Multiobjective Optimization

Sample of 3(4)-criteria optimization.

260 270 220 225 230 235 240 245 250 0.96 0.96 0.96 0.96 0.96 0.97 min

Ω {Q}

min

Ω,V {|Xin| /Rin}

min

Ω,V {|R0 − Rin| /R0}

1 Area

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 17 / 23

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Synthesis: Results

Optional Features – Flood-Filling (FF) Algorithm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

Green: kept, red: removed by GA, yellow: removed by FF.

Flood-filling implemented: ◮ All isolated pixels are removed by FF algorithm before physics is evaluated. ◮ Kind of penalization.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 18 / 23

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Synthesis: Results

Optional Features – Probability Map

1

Statistics how often were the pixels used.

Probability map of how often were various pixels used. ◮ Reveals trends.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 19 / 23

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Synthesis: Results

Multiobjective Optimization (6 × 3 test case)

Q-factor vs tuning element (currents).

Optimal current with respect to min

Ω,V {Q}.

Optimal current with respect to min

Ω,V {|Xin| /Rin}. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 20 / 23

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Synthesis: Results

Multiobjective Optimization (10 × 5 fine case)

Q-factor vs radiation efficiency ηrad (currents).

Optimal current with respect to min

Ω,V {Q}.

Optimal current with respect to max

Ω,V {ηrad}. Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 21 / 23

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Concluding Remarks

Conclusions

What is the best current distribution in principle? What is the best reachable current distribution?

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 22 / 23

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Concluding Remarks

Conclusions

What is the best current distribution in principle? What is the best reachable current distribution? ◮ Optimum is practically unreachable. ◮ Problem is solvable straightforwardly via convex optimization. ◮ Problem cannot be exactly solved (at least now). ◮ Suboptimality as problem of geometry (integer) optimization.

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 22 / 23

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Concluding Remarks

Conclusions

What is the best current distribution in principle? What is the best reachable current distribution? ◮ Optimum is practically unreachable. ◮ Problem is solvable straightforwardly via convex optimization. ◮ Problem cannot be exactly solved (at least now). ◮ Suboptimality as problem of geometry (integer) optimization. What next? ◮ further reduction of the solution space (proper parametrization), ◮ investigation of the solution space properties (sub-optimality), ◮ mixture of heuristic and gradient techniques (no-free-lunch, topology deriv.).

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 22 / 23

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Questions?

For a complete PDF presentation see

capek.elmag.org

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz

  • 10. 04. 2018, v1.0

Capek, M. and Jelinek, L. Optimal Currents and Optimal Antennas 23 / 23