Excitation of Optimal and Suboptimal Currents Miloslav Martin - - PowerPoint PPT Presentation

Excitation of Optimal and Suboptimal Currents

Miloslav ˇ Capek1 Luk´ aˇ s Jel´ ınek1 Petr Kadlec2 Martin ˇ Strambach3

1Department of Electromagnetic Field

Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz

2Department of Radio Electronics

Brno University of Technology, Czech Republic

3Faculty 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

1

Optimal Currents

2

Minimum Quality Factor Q

3

Solution Expressed in Characteristic Modes

4

Alternative Bases

5

Excitation – Sub-optimal Currents

6

Structure of the Solution Space

This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e., A (r, t) = Re {A (r) exp (jωt)}.

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 Jopt and called as optimal current1 if Jopt, L (Jopt) = min

J J, {L (J)} = pmin,

(1) Jopt, Mn (Jopt) = qn, (2) Jopt, Nn (Jopt) ≤ rn. (3)

• 1L. 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 Jopt and called as optimal current1 if Jopt, L (Jopt) = min

J J, {L (J)} = pmin,

(1) Jopt, Mn (Jopt) = qn, (2) Jopt, Nn (Jopt) ≤ rn. (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.

• 1L. 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 Jopt minimizing quality factor Q of a given shape Ω: Q (Jopt) = 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 Jopt minimizing quality factor Q of a given shape Ω: Q (Jopt) = min

J {Q (J)}

(4)

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x ψn (r) = ln 2A±

n

ρ±

n

RWG basis functions.

Rao-Wilton-Glisson basis functions J (r) ≈

• n

In ψn (r) (5) Q (I) = 2ω max {Wm, We} Pr = max

• IHXmI, IHXeI
• IHRI

(6)

• 2M. 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 Jopt minimizing quality factor Q of a given shape Ω: Q (Jopt) = min

J {Q (J)}

(4)

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x ψn (r) = ln 2A±

n

ρ±

n

RWG basis functions.

Rao-Wilton-Glisson basis functions J (r) ≈

• n

In ψn (r) (5) Q (I) = 2ω max {Wm, We} Pr = max

• IHXmI, IHXeI
• IHRI

(6) We know several efficient minimization procedures2.

• 2M. 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 + jX as3 XIm = λmRIm (7) ◮ useful set of entire-domain basis functions, I =

• m

αmIm (8) ◮ only few modes needed to represent ESAs (1 + jλm) δmn = 1 2IH

mZIn.

(9) ◮ meant originally for scattering problems4.

• 3R. 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.

• 4R. 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

Two different optimal currents for Qmin.

Optimal current can be approximated5 by Q (Iopt) ≈ Q (I1 + αoptI2) (10) αopt =

• −λ1

λ2 e−jϕ =

• −IT

1 XI1

IT

2 XI2

e−jϕ, ϕ ∈ [−π, π] (11) ◮ The optimization problem can be advantageously solved in other bases as well!

• 5M. 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 Jopt

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 Jopt

Optimal current with respect to minimum quality factor Q. Dominant (dipole-like) characteristic mode J1.

+

First inductive (loop-like) mode J2, α2 = 0.4553.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 7 / 18

Alternative Bases

Alternative Bases

◮ Stored energy modes6 ω∂X ∂ω Im = qmRIm, (12) ◮ minimum quality factor Q modes7 ((1 − ν) Xm + νXe) Im = QνmRIm, (13) ◮ optimal gain G including losses in metalization8 U (ˆ e, ˆ r) Im = ζm 1 8π (R + Rρ) Im, (14) ◮ optimal radiation efficiency8 RIm = ζm (R + Rρ) . (15)

• 6M. 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

• 7M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017, eprint arXiv: 1612.07676.

[Online]. Available: https://arxiv.org/abs/1612.07676

• 8L. 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 Iopt 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 Iopt for minimal quality factor Q. Feeding map (abs values) for optimal current Iopt.

◮ How to feed optimal currents? ◮ Vopt = ZIopt I =

• n

IH

n V

1 + jλn In IH

n RIn

(16)

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 9 / 18

Excitation – Sub-optimal Currents

Excitation of Optimal Currents

Optimal current Iopt for minimal quality factor Q. Feeding map (abs values) for optimal current Iopt.

◮ How to feed optimal currents? ◮ Vopt = ZIopt

• Impressed currents in vacuum.
• Shape has to be modified.
• Can modal techniques help?

I =

• n

IH

n V

1 + jλn In IH

n RIn

(16)

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 9 / 18

Excitation – Sub-optimal Currents

How to Excite the Optimal Currents

240 180 120 60 2 4 6 8 10 12 14 16 number of feeding edges quality factor Q

fed current

• ptimal current
• ptimal positions
• f four feeders

Dependence of Qmin on number of (optimally placed) feeders.

◮ Let us try to modify structure manually.

• A loop.
• 2 modes = at least

two feeders?

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 10 / 18

Excitation – Sub-optimal Currents

How to Excite the Optimal Currents

240 180 120 60 2 4 6 8 10 12 14 16 number of feeding edges quality factor Q

fed current

• ptimal current
• ptimal positions
• f four feeders

Dependence of Qmin on number of (optimally placed) feeders.

◮ Let us try to modify structure manually.

• A loop.
• 2 modes = at least

two feeders?

◮ Rectangle: Qmin = 69.5 ◮ Loop: Qmin = 78.9

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 10 / 18

Excitation – Sub-optimal Currents

Excited Characteristic Modes

1.0 0.8 0.6 0.4 0.2 0.00 2 4 6 8 10 12 14 16 number of feeding edges

mode indices

1 6

½an½

Dependence of ME coef. |αn| on number of (optimally placed) feeders.

◮ As expected, solution represented by two CMs. ◮ Even to excite two CMs properly, many feeders needed.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 11 / 18

Excitation – Sub-optimal Currents

Pixeling with Heuristic Optimization

Antenna synthesis – how far can we go? ◮ On the present, only the heuristic optimization9,

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 12 / 18

Excitation – Sub-optimal Currents

Pixeling with Heuristic Optimization

Antenna synthesis – how far can we go? ◮ On the present, only the heuristic optimization9, ◮ triangles and edges can be subjects of pixelization. Computational time: 12116 s

Result of heuristic structural optimization using MOGA NSGAII from AToM-FOPS.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 12 / 18

Excitation – Sub-optimal Currents

Pixeling with Heuristic Optimization

Antenna synthesis – how far can we go? ◮ On the present, only the heuristic optimization9, ◮ triangles and edges can be subjects of pixelization. Computational time: 12116 s

Result of heuristic structural optimization using MOGA NSGAII from AToM-FOPS.

Q (I) /Q (Iopt) = 1.811

Resulting sub-optimal current approaching minimal value of quality factor Q.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 12 / 18

Structure of the Solution Space

Complexity of the Problem

◮ shape modification resembles NP-hard problem ◮ any extra feeder levels up the complexity enormously

W

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18

Structure of the Solution Space

Complexity of the Problem

◮ shape modification resembles NP-hard problem ◮ any extra feeder levels up the complexity enormously

How much DOF we have?

W

N (unknowns) 28 52 120 ∞ possibilities unique solutions

Complexity of geometrical optimization for given voltage gap (red line) and N unknowns.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18

Structure of the Solution Space

Complexity of the Problem

◮ shape modification resembles NP-hard problem ◮ any extra feeder levels up the complexity enormously

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.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18

Structure of the Solution Space

Structure of Solution Space

◮ all combinations for N = 28 edges (5.24 · 1029) calculated in Matlab

10

• 3 days on supercomputer, 2 resulting vectors + permutation table ≈ 55 GB

× × × × × × ×

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 14 / 18

Structure of the Solution Space

Structure of Solution Space

◮ all combinations for N = 28 edges (5.24 · 1029) calculated in Matlab

10

• 3 days on supercomputer, 2 resulting vectors + permutation table ≈ 55 GB

280 300 320 340 360 380 400 7×104 6×104 5×104 4×104 3×104 2×104 1×104 quality factor Q number of solutions

best solution 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

Structure of all suboptimal solution within 2 % tolerance to the best found candidate. Edge no. 18 is fed.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 14 / 18

Structure of the Solution Space

Structure of Solution Space

◮ all combinations for N = 28 edges (5.24 · 1029) calculated in Matlab

10

• 3 days on supercomputer, 2 resulting vectors + permutation table ≈ 55 GB

280 300 320 340 360 380 400 7×104 6×104 5×104 4×104 3×104 2×104 1×104 quality factor Q number of solutions

best solution 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

Structure of all suboptimal solution within 2 % tolerance to the best found candidate. Edge no. 18 is fed. 1 29 5 10 20 25 edge solution

fed edge removed edge retained edge best solution

1 25 75 Number of solutions in dependence on their quality factor Q. The best solution reaches Q (Ωopt) ≈ 292.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 14 / 18

Structure of the Solution Space

Naive Alternative to Heuristic Algorithms

Deterministic algorithm dealing with shape optimization ◮ The worst edge (causing high quality factor Q) is iteratively removed.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 15 / 18

Structure of the Solution Space

Naive Alternative to Heuristic Algorithms

Deterministic algorithm dealing with shape optimization ◮ The worst edge (causing high quality factor Q) is iteratively removed. Computational time: 1155 s

Result of deterministic in-house algorithm removing in each iteration “the worst” edge.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 15 / 18

Structure of the Solution Space

Naive Alternative to Heuristic Algorithms

Deterministic algorithm dealing with shape optimization ◮ The worst edge (causing high quality factor Q) is iteratively removed. Computational time: 1155 s

Result of deterministic in-house algorithm removing in each iteration “the worst” edge.

Q (I) /Q (Iopt) = 1.813

Resulting current given by in-house deterministic algorithm.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 15 / 18

Structure of the Solution Space

Formal Simplification of the Problem

W WN, N=5 WN, N=13

y y

Mesh grid converted to graph.

W

◮ Longest cycle (loop) or path (dipole) in a mesh are NP hard. ◮ Can adaptive meshing help? ◮ Convergence of mesh grid has to be controlled.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 16 / 18

Structure of the Solution Space

Formal Simplification of the Problem

W WN, N=5 WN, N=13

y y

Mesh grid converted to graph.

WN, N=5

1111 0110 1010 0101 0011 1001 1100 1110 1101 1011 0111 0000 1000 0100 0010 0001 fed edge fixed edge free edge Can we somehow combine heuristic and our knowledge?

◮ Longest cycle (loop) or path (dipole) in a mesh are NP hard. ◮ Can adaptive meshing help? ◮ Convergence of mesh grid has to be controlled.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 16 / 18

Structure of the Solution Space

Current and Antenna Optimization

Current optimization ◮ lower bounds, Antenna optimization ◮ real performance,

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18

Structure of the Solution Space

Current and Antenna Optimization

Current optimization ◮ lower bounds, ◮ can be calculated “for free”, Antenna optimization ◮ real performance, ◮ NP-hard (NP-complete),

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18

Structure of the Solution Space

Current and Antenna Optimization

Current optimization ◮ lower bounds, ◮ can be calculated “for free”, ◮ convex optimization, Antenna optimization ◮ real performance, ◮ NP-hard (NP-complete), ◮ heuristic optimization,

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18

Structure of the Solution Space

Current and Antenna Optimization

Current optimization ◮ lower bounds, ◮ can be calculated “for free”, ◮ convex optimization, ◮ no support, only current, Antenna optimization ◮ real performance, ◮ NP-hard (NP-complete), ◮ heuristic optimization, ◮ (modified) shape,

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18

Structure of the Solution Space

Current and Antenna Optimization

Current optimization ◮ lower bounds, ◮ can be calculated “for free”, ◮ convex optimization, ◮ no support, only current, ◮ N feeders.

GAP GAP

Antenna optimization ◮ real performance, ◮ NP-hard (NP-complete), ◮ heuristic optimization, ◮ (modified) shape, ◮ n ≪ N feeders. Can modal techniques help? ◮ Understanding and interpretation of the solution. ◮ For matrix compression, i.e., AI

red =

• IH

mAfullIn

• .

◮ New operators → new decompositions.

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18

Questions?

For a complete PDF presentation see

capek.elmag.org

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

• 23. 03. 2017, v1.0

ˇ Capek, M., et al. Excitation of Optimal and Suboptimal Currents 18 / 18