excitation of optimal and suboptimal currents - quality factor q minimization of quality factor q...

Excitation of Optimal and Suboptimal Currents

Miloslav Capek1 Lukas Jelınek1 Petr Kadlec2 Martin Strambach3

1Department of Electromagnetic FieldCzech Technical University in Prague, Czech Republic

[email protected]

2Department of Radio ElectronicsBrno University of Technology, Czech Republic

3Faculty of Information TechnologyCzech Technical University in Prague, Czech Republic

The 11th European Conference on Antennas and PropagationParis, France

March 23, 2017

Capek, M., et al. Excitation of Optimal and Suboptimal Currents 1 / 18

Outline

1 Optimal Currents2 Minimum Quality Factor Q3 Solution Expressed in Characteristic Modes4 Alternative Bases5 Excitation – Sub-optimal Currents6 Structure of the Solution Space

This talk concerns:

I electric currents in vacuum,I time-harmonic quantities, i.e., A (r, t) = Re A (r) exp (jωt).

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)〉 = minJ〈J , L (J)〉 = pmin, (1)

〈Jopt,Mn (Jopt)〉 = qn, (2)

〈Jopt,Nn (Jopt)〉 ≤ rn. (3)

What are the optimal currents good for?

I 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.

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)〉 = minJ〈J , L (J)〉 = pmin, (1)

〈Jopt,Mn (Jopt)〉 = qn, (2)

〈Jopt,Nn (Jopt)〉 ≤ rn. (3)

What are the optimal currents good for?

I 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.

Minimum Quality Factor Q

Minimization of Quality Factor Q

Current Jopt minimizing quality factor Q of a given shape Ω:

Q (Jopt) = minJQ (J) (4)

ρ+nρ−n

xψn (r) =

2A±nρ±n

RWG basis functions.

Rao-Wilton-Glisson basis functions

J (r) ≈∑n

Inψn (r) (5)

Q (I) =2ωmax Wm,We

maxIHXmI, I

IHRI(6)

We know several efficient minimization procedures2.

ρ+nρ−n

xψn (r) =

2A±nρ±n

J (r) ≈∑n

Inψn (r) (5)

Q (I) =2ωmax Wm,We

maxIHXmI, I

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

ρ+nρ−n

xψn (r) =

2A±nρ±n

J (r) ≈∑n

Inψn (r) (5)

Q (I) =2ωmax Wm,We

maxIHXmI, I

IHRI(6)

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

Basis of Characteristic Modes

Diagonalization of impedance matrix Z = R + jX as3

XIm = λmRIm (7)

I useful set of entire-domain basis functions,

I =∑m

αmIm (8)

I only few modes needed to represent ESAs

(1 + jλm) δmn =1

2IHmZIn. (9)

I meant originally for scattering problems4.

3R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies”, IEEE Trans. AntennasPropag., 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. AntennasPropag., vol. 19, no. 3, pp. 348–358, 1971. doi: 10.1109/TAP.1971.1139935

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 =

√−λ1λ2

e−jϕ =

√−IT1 XI1

IT2 XI2e−jϕ, ϕ ∈ [−π, π] (11)

I The optimization problem can be advantageouslysolved in other bases as well!

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.

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.

Alternative Bases

I Stored energy modes6

ω∂X

∂ωIm = qmRIm, (12)

I minimum quality factor Q modes7

((1− ν)Xm + νXe) Im = QνmRIm, (13)

I optimal gain G including losses in metalization8

U (e, r) Im = ζm1

8π(R + Rρ) Im, (14)

I optimal radiation efficiency8

RIm = ζm (R + Rρ) . (15)

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

Excitation – Sub-optimal Currents

Excitation of Optimal Currents

Optimal current Iopt for minimal quality factor Q.

Feeding map (abs values) for optimal current Iopt.

I How to feed optimal currents?

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

I =∑n

1 + jλn

InIHnRIn

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

I Vopt = ZIopt

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

I =∑n

1 + jλn

InIHnRIn

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

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

I =∑n

1 + jλn

InIHnRIn

How to Excite the Optimal Currents

00 2 4 6 8 10 12 14 16

number of feeding edges

fed current optimal current

optimal positionsof four feeders

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

I Let us try to modifystructure manually.

• A loop.• 2 modes = at least

two feeders?

I Rectangle: Qmin = 69.5

I Loop: Qmin = 78.9

How to Excite the Optimal Currents

00 2 4 6 8 10 12 14 16

fed current optimal current

optimal positionsof four feeders

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

I Let us try to modifystructure manually.

• A loop.• 2 modes = at least

two feeders?

I Rectangle: Qmin = 69.5

I Loop: Qmin = 78.9

Excited Characteristic Modes

0.00 2 4 6 8 10 12 14 16

modeindices

½an½

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

I As expected, solutionrepresented by twoCMs.

I Even to excite two CMsproperly, many feedersneeded.

Pixeling with Heuristic Optimization

Antenna synthesis – how far can we go?

I On the present, only the heuristic optimization9,

I triangles and edges can be subjects of pixelization.

Computational time: 12116 s

Result of heuristic structural optimization using MOGANSGAII from AToM-FOPS.

Q (I) /Q (Iopt) = 1.811

Resulting sub-optimal current approaching minimalvalue of quality factor Q.

Q (I) /Q (Iopt) = 1.811

Structure of the Solution Space

Complexity of the Problem

I shape modification resembles NP-hard problem

I any extra feeder levels up the complexity enormously

How much DOF we have?

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.

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 ∞

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 ∞

Structure of Solution Space

I all combinations for N = 28 edges (5.24 · 1029) calculated in Matlab10

• 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

bestsolution

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

1 295 10 20 25edge

fededge

removededge

retainededge

bestsolution

Number of solutions in dependence on their qualityfactor Q. The best solution reaches Q (Ωopt) ≈ 292.

280 300 320 340 360 380 400

7×104

6×104

5×104

4×104

3×104

2×104

1×104

quality factor Q

bestsolution

1 295 10 20 25edge

fededge

removededge

retainededge

bestsolution

280 300 320 340 360 380 400

7×104

6×104

5×104

4×104

3×104

2×104

1×104

quality factor Q

bestsolution

1 295 10 20 25edge

fededge

removededge

retainededge

bestsolution

Naive Alternative to Heuristic Algorithms

Deterministic algorithm dealing with shape optimization

I The worst edge (causing high quality factor Q) is iteratively removed.

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

Q (I) /Q (Iopt) = 1.813

Resulting current given by in-house deterministicalgorithm.

Q (I) /Q (Iopt) = 1.813

Formal Simplification of the Problem

W WN, N=5 WN, N=13

Mesh grid converted to graph.

WN, N=5

01101010 0101 001110011100

1110 1101 1011 0111

1000 0100 0010 0001

fed edgefixed edgefree edge

Can we somehow combine heuristic and our knowledge?

I Longest cycle (loop) or path (dipole) in a mesh are NP hard.

I Can adaptive meshing help?

I Convergence of mesh grid has to be controlled.

Formal Simplification of the Problem

W WN, N=5 WN, N=13

Mesh grid converted to graph.

WN, N=5

01101010 0101 001110011100

1110 1101 1011 0111

1000 0100 0010 0001

fed edgefixed edgefree edge

Can we somehow combine heuristic and our knowledge?

I Longest cycle (loop) or path (dipole) in a mesh are NP hard.

I Can adaptive meshing help?

I Convergence of mesh grid has to be controlled.

Current and Antenna Optimization

Current optimization

I lower bounds,

I can be calculated “for free”,

I convex optimization,

I no support, only current,

I N feeders.

Antenna optimization

I real performance,

I NP-hard (NP-complete),

I heuristic optimization,

I (modified) shape,

I n N feeders.

Can modal techniques help?

I Understanding and interpretation of the solution.

I For matrix compression, i.e., AIred =

[IHmAfullIn

I New operators → new decompositions.

I lower bounds,

I N feeders.

I real performance,

I (modified) shape,

I n N feeders.

[IHmAfullIn

I lower bounds,

I N feeders.

I real performance,

I (modified) shape,

I n N feeders.

[IHmAfullIn

I lower bounds,

I N feeders.

I real performance,

I (modified) shape,

I n N feeders.

[IHmAfullIn

I lower bounds,

I N feeders.

I real performance,

I (modified) shape,

I n N feeders.

[IHmAfullIn

Questions?

For a complete PDF presentation see capek.elmag.org

Miloslav [email protected]

23. 03. 2017, v1.0

excitation of optimal and suboptimal currents - quality factor q minimization of quality factor q...

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