Excitation of Optimal and Suboptimal Currents
Miloslav Capek1 Lukas Jelınek1 Petr Kadlec2 Martin Strambach3
1Department of Electromagnetic FieldCzech Technical University in Prague, Czech Republic
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).
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)〉 = 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.
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)〉 = 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.
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) = minJQ (J) (4)
P+n
P−n
ρ+nρ−n
A+n
A−n
lnT+n
T−n
O
r
y
z
xψn (r) =
ln
2A±nρ±n
RWG basis functions.
Rao-Wilton-Glisson basis functions
J (r) ≈∑n
Inψn (r) (5)
Q (I) =2ωmax Wm,We
Pr=
maxIHXmI, I
HXeI
IHRI(6)
We know several efficient minimization procedures2.
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) = minJQ (J) (4)
P+n
P−n
ρ+nρ−n
A+n
A−n
lnT+n
T−n
O
r
y
z
xψn (r) =
ln
2A±nρ±n
RWG basis functions.
Rao-Wilton-Glisson basis functions
J (r) ≈∑n
Inψn (r) (5)
Q (I) =2ωmax Wm,We
Pr=
maxIHXmI, I
HXeI
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
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) = minJQ (J) (4)
P+n
P−n
ρ+nρ−n
A+n
A−n
lnT+n
T−n
O
r
y
z
xψn (r) =
ln
2A±nρ±n
RWG basis functions.
Rao-Wilton-Glisson basis functions
J (r) ≈∑n
Inψn (r) (5)
Q (I) =2ωmax Wm,We
Pr=
maxIHXmI, I
HXeI
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
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)
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
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 =
√−λ1λ2
e−jϕ =
√−IT1 XI1
IT2 XI2e−jϕ, ϕ ∈ [−π, π] (11)
I The optimization problem can be advantageouslysolved 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
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.
Dominant (dipole-like) characteristic mode J1.
+
First inductive (loop-like) mode J2, α2 = 0.4553.
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 7 / 18
Alternative Bases
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)
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
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.
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
IHnV
1 + jλn
InIHnRIn
(16)
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.
I How to feed optimal currents?
I Vopt = ZIopt
• Impressed currents in vacuum.• Shape has to be modified.• Can modal techniques help?
I =∑n
IHnV
1 + jλn
InIHnRIn
(16)
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.
I How to feed optimal currents?
I Vopt = ZIopt• Impressed currents in vacuum.• Shape has to be modified.• Can modal techniques help?
I =∑n
IHnV
1 + jλn
InIHnRIn
(16)
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
00 2 4 6 8 10 12 14 16
number of feeding edges
qual
ity
fact
or Q
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
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
00 2 4 6 8 10 12 14 16
number of feeding edges
qual
ity
fact
or Q
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
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
modeindices
1
6
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.
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?
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.
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?
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.
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?
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 12 / 18
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?
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18
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?
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18
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?
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 13 / 18
Structure of the Solution Space
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
0
quality factor Q
num
ber
of s
olut
ions
bestsolution
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 % toleranceto the best found candidate. Edge no. 18 is fed.
1 295 10 20 25edge
solu
tion
fededge
removededge
retainededge
bestsolution
1
25
75
Number of solutions in dependence on their qualityfactor Q. The best solution reaches Q (Ωopt) ≈ 292.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 14 / 18
Structure of the Solution Space
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
0
quality factor Q
num
ber
of s
olut
ions
bestsolution
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 % toleranceto the best found candidate. Edge no. 18 is fed.
1 295 10 20 25edge
solu
tion
fededge
removededge
retainededge
bestsolution
1
25
75
Number of solutions in dependence on their qualityfactor Q. The best solution reaches Q (Ωopt) ≈ 292.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 14 / 18
Structure of the Solution Space
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
0
quality factor Q
num
ber
of s
olut
ions
bestsolution
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 % toleranceto the best found candidate. Edge no. 18 is fed.
1 295 10 20 25edge
solu
tion
fededge
removededge
retainededge
bestsolution
1
25
75
Number of solutions in dependence on their qualityfactor Q. The best solution reaches Q (Ωopt) ≈ 292.
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
I The worst edge (causing high quality factor Q) is iteratively removed.
Computational time: 1155 s
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.
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
I The worst edge (causing high quality factor Q) is iteratively removed.
Computational time: 1155 s
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.
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
I The worst edge (causing high quality factor Q) is iteratively removed.
Computational time: 1155 s
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.
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
yy
Mesh grid converted to graph.
WN, N=5
1111
01101010 0101 001110011100
1110 1101 1011 0111
0000
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.
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
yy
Mesh grid converted to graph.
WN, N=5
1111
01101010 0101 001110011100
1110 1101 1011 0111
0000
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 16 / 18
Structure of the Solution Space
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.
GAP
GAP
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18
Structure of the Solution Space
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.
GAP
GAP
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18
Structure of the Solution Space
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.
GAP
GAP
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18
Structure of the Solution Space
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.
GAP
GAP
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18
Structure of the Solution Space
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.
GAP
GAP
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.
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 17 / 18
Questions?
For a complete PDF presentation see capek.elmag.org
Miloslav [email protected]
23. 03. 2017, v1.0
Capek, M., et al. Excitation of Optimal and Suboptimal Currents 18 / 18