Current Optimization for Electrically Small Antennas Miloslav - - PowerPoint PPT Presentation

Current Optimization for Electrically Small Antennas

Miloslav ˇ Capek

Department of Electromagnetic Field CTU in Prague, Czech Republic miloslav.capek@fel.cvut.cz

Seminar at KTH Stockholm, Sweden January 18, 2017

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 1 / 34

Outline

1 Current Optimization 2 Minimum Quality Factor Q 3 Modal Approach 4 Optimal Composition of Modes 5 On the Natural Bases 6 Summary and Concluding Remarks In this talk: ◮ electric currents in vacuum, ◮ only surface regions are treated, ◮ time-harmonic quantities, i.e., A (r, t) = Re {A (r) exp (jωt)} are considered, ◮ be extremely careful when comparing different sources (papers):

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 2 / 34

different notation!

“Is it beneficial to be here today?”

Do you know the following publications well? ◮ Gustafsson, M., Tayli, D., Ehrenborg, C., Cismasu, M. and Norbedo, S.: “Antenna Current Optimization using MATLAB and CVX”, FERMAT, vol. 15, pp. 1-29, 2016. ◮ Capek, M. and Jelinek, L.: “Optimal Composition of Modal Currents For Minimal Quality Factor Q”, IEEE Trans. Antennas Propagation,

• vol. 64, no. 12, pp. 5230–5242, Dec. 2016.

◮ Jelinek, L. and Capek, M.: “Optimal Currents on Arbitrarily Shaped Surfaces”, IEEE Trans. Antennas Propagation, vol. 65, no. 1,

• pp. 329–341, Jan. 2017.

◮ Capek, M., Gustafsson, M., and Schab, K.: “Minimization of Antenna Quality Factor”, submitted to IEEE Trans. Antennas Propagation, arxiv: 1612.07676.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 3 / 34

Current Optimization

Antenna Analysis × Synthesis and Antenna Design

analysis synthesis

• 20
• 15
• 10
• 5

f0 s11 [dB]

Qmax = 7

Perfect Electric Conductor Feeding Point  Antenna characteristics electric current

Antenna analysis × antenna synthesis.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 4 / 34

Current Optimization

Antenna Analysis × Synthesis and Antenna Design

analysis synthesis

• 20
• 15
• 10
• 5

f0 s11 [dB]

Qmax = 7

Perfect Electric Conductor Feeding Point  Antenna characteristics electric current

Antenna analysis × antenna synthesis.

Antenna analysis studies. . . ◮ antenna parameters for a given antenna.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 4 / 34

Current Optimization

Antenna Analysis × Synthesis and Antenna Design

analysis synthesis

• 20
• 15
• 10
• 5

f0 s11 [dB]

Qmax = 7

Perfect Electric Conductor Feeding Point  Antenna characteristics electric current

Antenna analysis × antenna synthesis.

Antenna analysis studies. . . ◮ antenna parameters for a given antenna. Antenna synthesis seeks for. . . ◮ optimal currents, ◮ optimal feeding (placement), ◮ optimal material.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 4 / 34

Current Optimization

Antenna Analysis × Synthesis and Antenna Design

analysis synthesis

• 20
• 15
• 10
• 5

f0 s11 [dB]

Qmax = 7

Perfect Electric Conductor Feeding Point  Antenna characteristics electric current

Antenna analysis × antenna synthesis.

Antenna analysis studies. . . ◮ antenna parameters for a given antenna. Antenna synthesis seeks for. . . ◮ optimal currents, ◮ optimal feeding (placement), ◮ optimal material. Antenna design tries to find. . . ◮ the optimal combination of shape, material and feeding from infinitely many candidates.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 4 / 34

Current Optimization

Historical Overview

Former approaches to antenna design predominantly made use of ◮ circuit quantities1 (Vin, Zin, Γ, . . . ) → equivalent circuits,

• 1H. L. Thal, “Exact circuit analysis of spherical waves”,

IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287,

• 1978. doi: 10.1109/TAP.1978.1141822

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 5 / 34

Current Optimization

Historical Overview

Former approaches to antenna design predominantly made use of ◮ circuit quantities1 (Vin, Zin, Γ, . . . ) → equivalent circuits, ◮ field quantities2 (E, H).

• 1H. L. Thal, “Exact circuit analysis of spherical waves”,

IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287,

• 1978. doi: 10.1109/TAP.1978.1141822
• 2L. J. Chu, “Physical limitations of omni-directional antennas”,
• J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. doi:

10.1063/1.1715038

• R. E. Collin and S. Rothschild, “Evaluation of antenna Q”, , IEEE Trans. Antennas Propag., vol. 12, no.

1, pp. 23–27, 1964. doi: 10.1109/TAP.1964.1138151

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 5 / 34

Current Optimization

Historical Overview

Former approaches to antenna design predominantly made use of ◮ circuit quantities1 (Vin, Zin, Γ, . . . ) → equivalent circuits, ◮ field quantities2 (E, H). However, ◮ all antenna parameters can be inferred from source current3 (J, M) p = J, L (J) .

f, L (g) =

• Ω

f ∗ (r) · L (g (r)) dV

• 1H. L. Thal, “Exact circuit analysis of spherical waves”,

IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287,

• 1978. doi: 10.1109/TAP.1978.1141822
• 2L. J. Chu, “Physical limitations of omni-directional antennas”,
• J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. doi:

10.1063/1.1715038

• R. E. Collin and S. Rothschild, “Evaluation of antenna Q”, , IEEE Trans. Antennas Propag., vol. 12, no.

1, pp. 23–27, 1964. doi: 10.1109/TAP.1964.1138151

• 3J. Schwinger, “Sources and electrodynamics”,
• Phys. Rev., vol. 158, no. 5, pp. 1391–1407, 1967. doi:

10.1103/PhysRev.158.1391

• R. F. Harrington, Field computation by moment methods. Wiley – IEEE Press, 1993

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 5 / 34

Current Optimization

Operators to Rule Them All.. .

All antenna parameters can be inferred directly from source current p = J, L (J) . (1)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 6 / 34

Current Optimization

Operators to Rule Them All.. .

All antenna parameters can be inferred directly from source current p = J, L (J) . (1)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 6 / 34

Current Optimization

Operators to Rule Them All.. .

All antenna parameters can be inferred directly from source current p = J, L (J) . (1) Observations: ◮ only properties of the operators are important, ◮ physics is imprinted in their structure, ◮ can be represented in many different ways, e.g., [ ψm, L ψn], [Jp, LJq], ◮ as compared to fields, the current support is limited.

f, g =

• Ω

f ∗ (r) · g (r) dV

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 6 / 34

Current Optimization

Example: Radiated and Reactive Power

Consider Electric Field Integral Equation4 written as Z (J) = −ˆ n × ˆ n × Ei (J) (2)

• 4W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves. Morgan &

Claypool, 2009.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 7 / 34

Current Optimization

Example: Radiated and Reactive Power

Consider Electric Field Integral Equation4 written as Z (J) = −ˆ n × ˆ n × Ei (J) (2) and let us represent it in a basis { ψn}, n ∈ {1, . . . , N} as Z = [Zmn], in which Zmn = ψm, Z ( ψn) = −jZ0 4π

• Ω
• Ω′
• k ψ∗

m · ψn − 1

k∇ · ψ∗

m∇′ · ψn

e−jk|r−r′| |r − r′| dΩ′ dΩ (3)

• 4W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves. Morgan &

Claypool, 2009.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 7 / 34

Current Optimization

Example: Radiated and Reactive Power

Consider Electric Field Integral Equation4 written as Z (J) = −ˆ n × ˆ n × Ei (J) (2) and let us represent it in a basis { ψn}, n ∈ {1, . . . , N} as Z = [Zmn], in which Zmn = ψm, Z ( ψn) = −jZ0 4π

• Ω
• Ω′
• k ψ∗

m · ψn − 1

k∇ · ψ∗

m∇′ · ψn

e−jk|r−r′| |r − r′| dΩ′ dΩ (3)

• r even more as

(1 + jλm) δmn = 1 2 Im, ZIn = 1 2IH

mZIn.

(4)

• 4W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves. Morgan &

Claypool, 2009.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 7 / 34

Current Optimization

Example: Radiated and Reactive Power

Consider Electric Field Integral Equation4 written as Z (J) = −ˆ n × ˆ n × Ei (J) (2) and let us represent it in a basis { ψn}, n ∈ {1, . . . , N} as Z = [Zmn], in which Zmn = ψm, Z ( ψn) = −jZ0 4π

• Ω
• Ω′
• k ψ∗

m · ψn − 1

k∇ · ψ∗

m∇′ · ψn

e−jk|r−r′| |r − r′| dΩ′ dΩ (3)

• r even more as

(1 + jλm) δmn = 1 2 Im, ZIn = 1 2IH

mZIn.

(4) ◮ All common algebraic operations are available for (3). ◮ Representation (4) profitably diagonalizes impedance operator (matrix).

• 4W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves. Morgan &

Claypool, 2009.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 7 / 34

Current Optimization

Source Concept (J , M)

Integral and variational methods Modal de- compositions Perspective topol-

• gy and

geometry HPC, algorithm efficiency Heuristic

• r convex
• ptimization

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 8 / 34

Current Optimization

Source Concept

Source Concept (J , M)

Integral and variational methods Modal de- compositions Perspective topol-

• gy and

geometry HPC, algorithm efficiency Heuristic

• r convex
• ptimization

Sketch of main fields of the source concept.

Source concept ◮ represents a radiator(s) completely by a source currents J (and M), ◮ J ≡ M ≡ 0 ⇔ r / ∈ Ω.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 9 / 34

Current Optimization

Optimal Currents – What They Are?

A current J = J (r, ω), r ∈ Ω, is denoted Jopt and called as optimal current iff Jopt, L (Jopt) = min

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

(5) Jopt, Mn (Jopt) = qn, (6) Jopt, Nn (Jopt) ≤ rn. (7)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 10 / 34

Current Optimization

Optimal Currents – What They Are?

A current J = J (r, ω), r ∈ Ω, is denoted Jopt and called as optimal current iff Jopt, L (Jopt) = min

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

(5) Jopt, Mn (Jopt) = qn, (6) Jopt, Nn (Jopt) ≤ rn. (7) ◮ What are the optimal currents good for?

• They establish fundamental bounds of p = L (J) for a given Ω, ω.

◮ How to find them?

• You will see. . .

◮ Can they be realized?

• Only as impressed currents (they are unrealistic).

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 10 / 34

Current Optimization

Optimal Currents – What They Are?

A current J = J (r, ω), r ∈ Ω, is denoted Jopt and called as optimal current iff Jopt, L (Jopt) = min

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

(5) Jopt, Mn (Jopt) = qn, (6) Jopt, Nn (Jopt) ≤ rn. (7) ◮ What are the optimal currents good for?

• They establish fundamental bounds of p = L (J) for a given Ω, ω.

◮ How to find them?

• You will see. . .

◮ Can they be realized?

• Only as impressed currents (they are unrealistic).

The rest of the presentation is about L and techniques how to find Jopt.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 10 / 34

Minimum Quality Factor Q

Case Study: Minimization of Quality Factor Q

Quality factor Q ◮ is (generally) inversely proportional to fractional bandwidth (FBW), ◮ therefore, of interest for electrically small antennas (ESA, ka < 1/2).

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 11 / 34

Minimum Quality Factor Q

Case Study: Minimization of Quality Factor Q

Quality factor Q ◮ is (generally) inversely proportional to fractional bandwidth (FBW), ◮ therefore, of interest for electrically small antennas (ESA, ka < 1/2). Current Jopt minimizing quality factor Q of a given shape Ω: Q (Jopt) = min

J {Q (J)}

(8) How to find Jopt for a given Ω?

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 11 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices,

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices, STEP 2 definition of quality factor Q and stored energy Wsto,

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices, STEP 2 definition of quality factor Q and stored energy Wsto, STEP 3 formulation of optimization task related to (8),

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices, STEP 2 definition of quality factor Q and stored energy Wsto, STEP 3 formulation of optimization task related to (8), STEP 4 representation of a solution in an appropriate basis,

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices, STEP 2 definition of quality factor Q and stored energy Wsto, STEP 3 formulation of optimization task related to (8), STEP 4 representation of a solution in an appropriate basis, STEP 5 optimal composition of modal currents,

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Procedure Undertaken in This Presentation

Procedure followed in this talk: STEP 1 representation of operators as matrices, STEP 2 definition of quality factor Q and stored energy Wsto, STEP 3 formulation of optimization task related to (8), STEP 4 representation of a solution in an appropriate basis, STEP 5 optimal composition of modal currents, 1st method: combination of 2 modes5, 2nd method: composition of n modes6, 3rd method: solution with 1 mode7.

• 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

• 6L. 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
• 7M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017. [Online]. Available:

https://arxiv.org/abs/1612.07676

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 12 / 34

Minimum Quality Factor Q

Source Concept Based Definition – Operators

STEP 1 ◮ assume operators in their matrix forms, i.e., Z → Z, ◮ assume functions in their vector forms, i.e., J → I, Z = R + jX = R + j (Xm − Xe) , J ≈

• n

In ψn, (9) therefore, we get Pr = 1 2IHRI, 2ω (Wm − We) = 1 2IH (Xm − Xe) I (10)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 13 / 34

Minimum Quality Factor Q

Source Concept Based Definition – Operators

STEP 1 ◮ assume operators in their matrix forms, i.e., Z → Z, ◮ assume functions in their vector forms, i.e., J → I, Z = R + jX = R + j (Xm − Xe) , J ≈

• n

In ψn, (9) therefore, we get Pr = 1 2IHRI, 2ω (Wm − We) = 1 2IH (Xm − Xe) I (10) STEP 2A ◮ definition of quality factor Q Q (I) = 2ω max {Wm, We} Pr = ω (Wm + We) Pr + ω |Wm − We| Pr (11)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 13 / 34

Minimum Quality Factor Q

Stored Energy Operator and Its Minimization

STEP 2B ◮ frankly speaking we still do not have complete idea what the stored energy is

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 14 / 34

Minimum Quality Factor Q

Stored Energy Operator and Its Minimization

STEP 2B ◮ frankly speaking we still do not have complete idea what the stored energy is

• but. . . we have many papers attempting to define it8
• one of the best possibilities for small radiators, ka < 1, is9:

Wm + We = Wsto ≈ 1 4

• J, ∂Im {Z}

∂ω J

• ≈ 1

4IH ∂X ∂ω I = 1 4IHX′I = 1 4ωIH (Xm + Xe) I (12)

• 8G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”,

IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1112–1127, 2010. doi: 10.1109/TAP.2010.2041166

• 9M. Gustafsson and B. L. G. Jonsson, “Antenna Q and stored energy expressed in the fields, currents, and input

impedance”, IEEE Trans. Antennas Propag., vol. 63, no. 1, pp. 240–249, 2015. doi: 10.1109/TAP.2014.2368111

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 14 / 34

Minimum Quality Factor Q

Stored Energy Operator and Its Minimization

STEP 2B ◮ frankly speaking we still do not have complete idea what the stored energy is

• but. . . we have many papers attempting to define it8
• one of the best possibilities for small radiators, ka < 1, is9:

Wm + We = Wsto ≈ 1 4

• J, ∂Im {Z}

∂ω J

• ≈ 1

4IH ∂X ∂ω I = 1 4IHX′I = 1 4ωIH (Xm + Xe) I (12) STEP 3 ◮ find Iopt so that minimize quality factor Q (I) , (13) subject to Wm (I) − We (I) = 0. (14)

• 8G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”,

IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1112–1127, 2010. doi: 10.1109/TAP.2010.2041166

• 9M. Gustafsson and B. L. G. Jonsson, “Antenna Q and stored energy expressed in the fields, currents, and input

impedance”, IEEE Trans. Antennas Propag., vol. 63, no. 1, pp. 240–249, 2015. doi: 10.1109/TAP.2014.2368111

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 14 / 34

Modal Approach

Modal Approach: Combining Two Proper Modes

STEP 4 ◮ let us decompose the current into (yet unknown) modes such that I =

N

• n=1

αnIn (15)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 15 / 34

Modal Approach

Modal Approach: Combining Two Proper Modes

STEP 4 ◮ let us decompose the current into (yet unknown) modes such that I =

N

• n=1

αnIn (15) ◮ then, substituting (10), (12), and (15) into (11), the quality factor Q reads Q (I) = ω

V

• v=1

U

• u=1

α∗

uαvIH u X′Iv +

• V
• v=1

U

• u=1

α∗

uαvIH u XIv

• 2

V

• v=1

U

• u=1

α∗

uαvIH u RIv

. (16)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 15 / 34

Modal Approach

Modal Approach: Combining Two Proper Modes

STEP 4 ◮ let us decompose the current into (yet unknown) modes such that I =

N

• n=1

αnIn (15) ◮ then, substituting (10), (12), and (15) into (11), the quality factor Q reads Q (I) = ω

V

• v=1

U

• u=1

α∗

uαvIH u X′Iv +

• V
• v=1

U

• u=1

α∗

uαvIH u XIv

• 2

V

• v=1

U

• u=1

α∗

uαvIH u RIv

. (16) ◮ analytical solution can easily be found as a combination of two modes iff IH

u RIv = δuv,

IH

u XIv = Λuvδuv,

ωIH

u X′Iv = χuvδuv.

(17)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 15 / 34

Modal Approach

Why to Combine Two Modes?

QU(I) wWm  wWe  Pr QU(I)+Qext wWm  wWe  Pr

max{wWe, wWm}  

30 15 20 30 20 30

Tuning by external lumped element (localized current).

w  w  w  w  w  w 

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 16 / 34

Modal Approach

Why to Combine Two Modes?

QU(I) wWm  wWe  Pr QU(I)+Qext wWm  wWe  Pr

max{wWe, wWm}  

30 15 20 30 20 30

Tuning by external lumped element (localized current).

Q (Iopt) wWm  wWe  Pr wWm  wWe  Pr wWm  wWe  Pr

current #1 current #2

+ =

30 15 20 5 20 35 32.5 32.5 30

Tuning by distributive current.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 16 / 34

Modal Approach

How to Combine Two Modes?

Normalizing α1 = 1 and selecting proper mode(s), we get10 Q (Iopt) = ω

• IH

1 X′I1 + |αopt|2 IH 2 X′I2

• 2
• 1 + |αopt|2

. (18)

• 10M. 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., CTU in Prague Current Optimization for Electrically Small Antennas 17 / 34

Modal Approach

How to Combine Two Modes?

Normalizing α1 = 1 and selecting proper mode(s), we get10 Q (Iopt) = ω

• IH

1 X′I1 + |αopt|2 IH 2 X′I2

• 2
• 1 + |αopt|2

. (18) STEP 5 To diagonalize at least two of R, X, and X′ matrices we can choose: XIu = λuRIu, (19) ω 2 X′Iu = quRIu, (20) XIu = ξu ω 2 X′Iu. (21)

• 10M. 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., CTU in Prague Current Optimization for Electrically Small Antennas 17 / 34

Modal Approach

How to Combine Two Modes?

Normalizing α1 = 1 and selecting proper mode(s), we get10 Q (Iopt) = ω

• IH

1 X′I1 + |αopt|2 IH 2 X′I2

• 2
• 1 + |αopt|2

. (18) STEP 5 To diagonalize at least two of R, X, and X′ matrices we can choose: XIu = λuRIu, (19) ω 2 X′Iu = quRIu, (20) XIu = ξu ω 2 X′Iu. (21)

• 10M. 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., CTU in Prague Current Optimization for Electrically Small Antennas 17 / 34

Modal Approach

Example: Optimal Current For Minimal Quality Factor Q

Take the pen and try to draw a current possessing minimum quality factor Q. . .

PEC plate L × L/2, ka = 0.5.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 18 / 34

Modal Approach

Example: Optimal Current For Minimal Quality Factor Q

. . . here is the correct answer.

Optimal current with respect to minimum quality factor Q.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 18 / 34

Modal Approach

Modal Composition of the Optimal Current

Dominant (dipole-like) characteristic mode J1, α1 = 1.

+

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

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 19 / 34

Modal Approach

Modal Composition of the Optimal Current

Dominant (dipole-like) characteristic mode J1, α1 = 1.

+

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

◮ characteristic modes (19) are quite convenient to define and interpret optimal currents11, e.g., for ka < 1 we have Q (Jopt) ≈ Q (J1 + α2J2) ◮ however, for higher ka or for highly irregular shapes, the energy cross-terms

• ccur, i.e., ωIH

u X′Iv = 0

• 11M. 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., CTU in Prague Current Optimization for Electrically Small Antennas 19 / 34

Modal Approach

Convexity of Two-Mode Combination

40 80 ka = 0.5 a2

• 1
• 0.5

0.5 1

QU(I) Q (I)

IC1

IC1+a2II1 IC2+a2II1 IC1+a2IC2 IC1+a2II2

IC2 II1

QU(I), Q (I)

4 3 . 8 9 3 5 . 6 2 4 . 4 2 5 4 . 7 3

II2

|aopt|

Quality factor Q as a combination of two modes for L × L/2 PEC plate, QU (I) = ωIHX′I/2IHRI, I = I1 + α2I2.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 20 / 34

Modal Approach

Convexity of Two-Mode Combination

40 80 ka = 0.5 a2

• 1
• 0.5

0.5 1

QU(I) Q (I)

IC1

IC1+a2II1 IC2+a2II1 IC1+a2IC2 IC1+a2II2

IC2 II1

QU(I), Q (I)

4 3 . 8 9 3 5 . 6 2 4 . 4 2 5 4 . 7 3

II2

|aopt|

Quality factor Q as a combination of two modes for L × L/2 PEC plate, QU (I) = ωIHX′I/2IHRI, I = I1 + α2I2.

◮ extremely straightforward analytical solution for spherical shell12 ◮ sub-optimality for G/Q ratio ◮ alternative bases can be used to reduce the effect of cross-terms ◮ intuitive, however, non-convex and only approximative procedure (non-zero cross-terms)

• 12M. 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., CTU in Prague Current Optimization for Electrically Small Antennas 20 / 34

Optimal Composition of Modes

Combining More Modes

Lessons Learned ◮ More modes than two are needed for a given set of operators. ◮ Particular approach to quality factor Q needs to be generalized.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 21 / 34

Optimal Composition of Modes

Combining More Modes

Lessons Learned ◮ More modes than two are needed for a given set of operators. ◮ Particular approach to quality factor Q needs to be generalized. For now, let us start with the following general optimization problem13: min

I

• IHAI
• ,

(22) IHBI = 1, (23) IHCI = γ. (24) ◮ any problem expressible in bilinear form can be solved, ◮ matrices A, B and C cannot be generally diagonalized simultaneously.

• 13L. 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., CTU in Prague Current Optimization for Electrically Small Antennas 21 / 34

Optimal Composition of Modes

Optimization Problem to Be Solved

◮ represent our solution in a basis (same as before) I =

N

• n=1

αnIn (25)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 22 / 34

Optimal Composition of Modes

Optimization Problem to Be Solved

◮ represent our solution in a basis (same as before) I =

N

• n=1

αnIn (25) ◮ so that the modal currents fulfill (before: S ≡ X/2, T ≡ R/2) SIn = ζnTI, IH

m (T + jS) In = (1 + jζn) δmn

(26) with T = TH, S = SH and ζn ∈ R

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 22 / 34

Optimal Composition of Modes

Optimization Problem to Be Solved

◮ represent our solution in a basis (same as before) I =

N

• n=1

αnIn (25) ◮ so that the modal currents fulfill (before: S ≡ X/2, T ≡ R/2) SIn = ζnTI, IH

m (T + jS) In = (1 + jζn) δmn

(26) with T = TH, S = SH and ζn ∈ R ◮ finally, represent operators A, B and C in basis (25) as AGEP

mn

= Im, AIn = IH

mAIn,

AGEP =

• AGEP

mn

• (27)

and same for B → BGEP and C → CGEP

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 22 / 34

Optimal Composition of Modes

Optimization Procedure

Problem (22)–(24) is transformed to14 min

α

• αHAGEPα
• ,

(28) αHBGEPα = 1, (29) αHCGEPα = γ, (30)

14Be careful here with λ1, λ2: here, they present the Lagrange multipliers not the characteristic numbers as in the

previous method.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 23 / 34

Optimal Composition of Modes

Optimization Procedure

Problem (22)–(24) is transformed to14 min

α

• αHAGEPα
• ,

(28) αHBGEPα = 1, (29) αHCGEPα = γ, (30) and solved using Lagrange multipliers AGEPα = λ1CGEPα + λ2BGEPα. (31)

14Be careful here with λ1, λ2: here, they present the Lagrange multipliers not the characteristic numbers as in the

previous method.

• 15L. 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., CTU in Prague Current Optimization for Electrically Small Antennas 23 / 34

Optimal Composition of Modes

Optimization Procedure

Problem (22)–(24) is transformed to14 min

α

• αHAGEPα
• ,

(28) αHBGEPα = 1, (29) αHCGEPα = γ, (30) and solved using Lagrange multipliers AGEPα = λ1CGEPα + λ2BGEPα. (31) Procedure proceed in following steps15:

• 1. Choose λ2 and solve (31).
• 2. Normalize all solutions to satisfy (29).
• 3. Check the constraint (30).
• 4. Vary λ2 and find solution to (30).
• 5. From candidates satisfying (29)–(31) select the one fulfilling (28).

14Be careful here with λ1, λ2: here, they present the Lagrange multipliers not the characteristic numbers as in the

previous method.

• 15L. 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., CTU in Prague Current Optimization for Electrically Small Antennas 23 / 34

Optimal Composition of Modes

Minimum Quality Factor Q and Lagrange Multipliers

Problem to be solved A = ωX′/4, (32) B = R/2, (33) C = X/2, (34) Representation S = X/2, (35) T = R/2, (36) Consequences γ = 0, (37) BGEP = I, (38) λ2 = αHAGEPα = Q, (39)

AGEPα = λ1CGEPα + λ2BGEPα αHAGEPα = λ1αHCGEPα + λ2αHBGEPα

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 24 / 34

Optimal Composition of Modes

Minimum Quality Factor Q and Lagrange Multipliers

Problem to be solved A = ωX′/4, (32) B = R/2, (33) C = X/2, (34) Representation S = X/2, (35) T = R/2, (36) Consequences γ = 0, (37) BGEP = I, (38) λ2 = αHAGEPα = Q, (39) Observations: ◮ γ = 0 is the resonant condition, ◮ with γ = 0 the multiplier λ2 is directly equal to the optimized quantity.

AGEPα = λ1CGEPα + λ2BGEPα αHAGEPα = λ1αHCGEPα + λ2αHBGEPα

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 24 / 34

Optimal Composition of Modes

Procedure and Its Results

−0.3 −0.2 −0.1 0.1 0.2 0.3 50 100 150 200 λ1 λ2 = Q ka = 0.5 λ2 min {λ2} L L/2

Optimization done as λ2 (λ1) = Q.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 25 / 34

Optimal Composition of Modes

Procedure and Its Results

−0.3 −0.2 −0.1 0.1 0.2 0.3 50 100 150 200 λ1 λ2 = Q ka = 0.5 λ2 min {λ2} L L/2

Optimization done as λ2 (λ1) = Q.

◮ procedure converges well to the correct result ◮ two options how to solve the problem

• 1. vary λ1, find λ2, i.e.,

tuning for minimum quality factor Q

• 2. vary λ2, find λ1, i.e.,

sweeping λ2 = Q for proper tuning

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 25 / 34

On the Natural Bases

What Is the Best Basis Possible?

Lessons Learned ◮ Lagrange multipliers constitute powerful method. ◮ It is, however, both theoretically and practically quite complicated. ◮ Is there any basis in which Iopt = I1?

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 26 / 34

On the Natural Bases

What Is the Best Basis Possible?

Lessons Learned ◮ Lagrange multipliers constitute powerful method. ◮ It is, however, both theoretically and practically quite complicated. ◮ Is there any basis in which Iopt = I1? Procedure not relying on optimal composition of modes exists16 ((1 − ν) Xm + νXe) In = QνRIn, (40) constituting a dual problem which can be easily solved (since its convex nature): min

I

{Q} = max

ν

• Qν
• .

(41)

• 16M. Gustafsson, D. Tayli, C. Ehrenborg, et al., “Antenna current optimization using MATLAB and CVX”, , FERMAT,
• vol. 15, no. 5, pp. 1–29, 2016

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 26 / 34

On the Natural Bases

What Is the Best Basis Possible?

Lessons Learned ◮ Lagrange multipliers constitute powerful method. ◮ It is, however, both theoretically and practically quite complicated. ◮ Is there any basis in which Iopt = I1? Procedure not relying on optimal composition of modes exists16 ((1 − ν) Xm + νXe) In = QνRIn, (40) constituting a dual problem which can be easily solved (since its convex nature): min

I

{Q} = max

ν

• Qν
• .

(41) ◮ based only on convex combination of Xm and Xe operators ◮ needs positive-definite operator in RHS of (40)

• 16M. Gustafsson, D. Tayli, C. Ehrenborg, et al., “Antenna current optimization using MATLAB and CVX”, , FERMAT,
• vol. 15, no. 5, pp. 1–29, 2016

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 26 / 34

On the Natural Bases

Does It Work?

0.2 0.4 0.6 0.8 1 50 100 150 200 ν quality factors Q

• Qν

max {Qmν, Qeν}

min {Q}

kL ≈ 0.628

Qm/eν =

IH

ν Xm/eIν

IH

ν RIν

L L/2 Originally concluded17 that a non-zero dual gap exists. . .

• 17M. Gustafsson, D. Tayli, C. Ehrenborg, et al., “Antenna current optimization using MATLAB and CVX”, , FERMAT,
• vol. 15, no. 5, pp. 1–29, 2016

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 27 / 34

On the Natural Bases

Does It Work?

0.2 0.4 0.6 0.8 1 50 100 150 200 ν quality factors Q

• Qν

max {Qmν, Qeν}

min {Q}

kL ≈ 0.628

Qm/eν =

IH

ν Xm/eIν

IH

ν RIν

L L/2 Originally concluded17 that a non-zero dual gap exists. . . 0.2 0.4 0.6 0.8 1 50 100 150 200 ν quality factors Q

• Qν

max {Qmν, Qeν}

min {Q}

ν = 0.75 ν = 0.95

kL ≈ 0.628

Qm/eν =

IH

ν Xm/eIν

IH

ν RIν

L L/2 L/2 L/4 . . . or not18?

• 17M. Gustafsson, D. Tayli, C. Ehrenborg, et al., “Antenna current optimization using MATLAB and CVX”, , FERMAT,
• vol. 15, no. 5, pp. 1–29, 2016
• 18M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017. [Online]. Available:

https://arxiv.org/abs/1612.07676

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 27 / 34

On the Natural Bases

Does It Work?

0.2 0.4 0.6 0.8 1 50 100 150 200 ν quality factors Q

• Qν

max {Qmν, Qeν}

min {Q}

kL ≈ 0.628

Qm/eν =

IH

ν Xm/eIν

IH

ν RIν

L L/2 Originally concluded17 that a non-zero dual gap exists. . . 0.2 0.4 0.6 0.8 1 50 100 150 200 ν quality factors Q

• Qν

max {Qmν, Qeν}

min {Q}

ν = 0.75 ν = 0.95

kL ≈ 0.628

Qm/eν =

IH

ν Xm/eIν

IH

ν RIν

L L/2 L/2 L/4 . . . or not18?

◮ Do you have any idea where is the problem?

• 17M. Gustafsson, D. Tayli, C. Ehrenborg, et al., “Antenna current optimization using MATLAB and CVX”, , FERMAT,
• vol. 15, no. 5, pp. 1–29, 2016
• 18M. Capek, M. Gustafsson, and K. Schab, “Minimization of antenna quality factor”, , 2017. [Online]. Available:

https://arxiv.org/abs/1612.07676

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 27 / 34

On the Natural Bases

Degenerated Eigenspace: Let’s Make It Work. . .

◮ if dual gap exists then only because of internal symmetries and existence of degenerated eigenspace19

• the internal symmetries must be preserved by the used discretization scheme

(e.g., rectangular plate is discretized with rooftop basis functions)

• 19J. von Neumann and E. Wigner, “On the behaviour of eivenvalues in adiabatic processes”,

in Quantum Chemistry: Classic Scientific Papers. in Quantum Chemistry: Classic Scientific Papers. Singapore: World Scientific, 2000

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 28 / 34

On the Natural Bases

Degenerated Eigenspace: Let’s Make It Work. . .

◮ if dual gap exists then only because of internal symmetries and existence of degenerated eigenspace19

• the internal symmetries must be preserved by the used discretization scheme

(e.g., rectangular plate is discretized with rooftop basis functions)

◮ it can be shown the that dual gap can always be done zero

• for two degenerated solutions I1 and I2 find c2 ∈ C such that20

IH

ν (Xm − Xe) Iν = 0,

Iν = I1 + c2I2, (42) i.e., the optimal current Iν is self-resonant

• 19J. von Neumann and E. Wigner, “On the behaviour of eivenvalues in adiabatic processes”,

in Quantum Chemistry: Classic Scientific Papers. in Quantum Chemistry: Classic Scientific Papers. Singapore: World Scientific, 2000

19Notice that I1 and I2 in (42) are different from those in (18). ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 28 / 34

On the Natural Bases

What Can Be Solved Now?

RHS of (40) Operator Radiated power R Mode expansion 1 Z0

• τ

FH

τ Fτ

Radiation intensity U (ˆ r, ˆ e) = 1 Z0 FH (ˆ r, ˆ e) F (ˆ r, ˆ e) Antenna gain 4π

• U
• ˆ

r, ˆ θ

• + U
• ˆ

r, ˆ φ

• Ohmic losses

Rσ Field intensity NH

e/mNe/m

. . . . . .

As far as the operator on the RHS of (40) is positive definite the procedure works well.

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 29 / 34

Summary and Concluding Remarks

Grand Unification

Recall the method of Lagrange multipliers from (22)–(24) as AI = λ1CI + λ2BI. (43)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 30 / 34

Summary and Concluding Remarks

Grand Unification

Recall the method of Lagrange multipliers from (22)–(24) as AI = λ1CI + λ2BI. (43) Minimization of quality factor Q (proportional to λ2) reads ω 4 X′I = λ1 2 XI + λ2 2 RI. (44)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 30 / 34

Summary and Concluding Remarks

Grand Unification

Recall the method of Lagrange multipliers from (22)–(24) as AI = λ1CI + λ2BI. (43) Minimization of quality factor Q (proportional to λ2) reads ω 4 X′I = λ1 2 XI + λ2 2 RI. (44) Substituting X′ = (Xm + Xe) /ω from (12) and X = Xm − Xe from (9) yields 1 4 (Xm + Xe) I = λ1 2 (Xm − Xe) I + λ2 2 RI, (45) 1 2 − λ1

• Xm +

1 2 + λ1

• Xe
• I = λ2RI,

(46)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 30 / 34

Summary and Concluding Remarks

Grand Unification

Recall the method of Lagrange multipliers from (22)–(24) as AI = λ1CI + λ2BI. (43) Minimization of quality factor Q (proportional to λ2) reads ω 4 X′I = λ1 2 XI + λ2 2 RI. (44) Substituting X′ = (Xm + Xe) /ω from (12) and X = Xm − Xe from (9) yields 1 4 (Xm + Xe) I = λ1 2 (Xm − Xe) I + λ2 2 RI, (45) 1 2 − λ1

• Xm +

1 2 + λ1

• Xe
• I = λ2RI,

(46) Substituting ν = 1/2 + λ1 finally creates the link between two presented methods ((1 − ν) Xm + νXe) I = λ2RI. (47)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 30 / 34

Summary and Concluding Remarks

Summary

Part 1 – Minimization of quality factor Q

Representation of continuous int.-dif. operators:

• 1. make the problem algebraic
• our choice: piecewise basis functions { ψn} with ψm, L ( ψn)
• 2. make the problem feasible
• our choice: entire domain basis functions {In} with Im, L (In)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 31 / 34

Summary and Concluding Remarks

Summary

Part 1 – Minimization of quality factor Q

Representation of continuous int.-dif. operators:

• 1. make the problem algebraic
• our choice: piecewise basis functions { ψn} with ψm, L ( ψn)
• 2. make the problem feasible
• our choice: entire domain basis functions {In} with Im, L (In)

Quality factor Q without external tuning 1st method we saw that the self-resonant current is optimal 2nd method the resonant (or other) constraint can easily be set 3rd method yields the self-resonant current automatically

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 31 / 34

Summary and Concluding Remarks

Summary

Part 2 – Optimal currents

◮ determination of the optimal currents is well-established ◮ more challenging problems can be now studied ◮ all concepts are still half-way to their applicability (feeding)

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 32 / 34

Summary and Concluding Remarks

Summary

Part 2 – Optimal currents

◮ determination of the optimal currents is well-established ◮ more challenging problems can be now studied ◮ all concepts are still half-way to their applicability (feeding) Future work ◮ arrays/scatterers ◮ excitation placement, number of feeders ◮ shape modifications

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 32 / 34

Summary and Concluding Remarks

About Tomorrow’s Seminar

Today’s presentation will be followed by the second part: ◮ Capek, M.: Implementation of Source Concept in Matlab, (Jan. 19 Thu 11 AM). You will learn about ◮ implementation of the Source Concept, ◮ (problematic) feeding of optimal currents, ◮ Matlab (proc and cons), ◮ new features in Matlab, ◮ developing big project in Matlab (how to stay sane).

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 33 / 34

Questions?

For complete PDF presentation see

capek.elmag.org

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

• 17. 1. 2017, v1.01

ˇ Capek, M., CTU in Prague Current Optimization for Electrically Small Antennas 34 / 34