Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic...

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Characteristic Modes Part I: Introduction Miloslav ˇ Capek Department of Electromagnetic Field Czech Technical University in Prague, Czech Republic [email protected] Seminar May 2, 2018 ˇ Capek, M. Characteristic Modes – Part I: Introduction 1 / 39

Transcript of Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic...

Page 1: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Characteristic ModesPart I: Introduction

Miloslav Capek

Department of Electromagnetic FieldCzech Technical University in Prague, Czech Republic

[email protected]

SeminarMay 2, 2018

Capek, M. Characteristic Modes – Part I: Introduction 1 / 39

Page 2: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Characteristic Modes

Conventionally, characteristic modes In aredefined as

XIn = λnRIn,

in which Z = R + jX is the impedance matrix.

However, who knows:

I What is impedance matrix and how to get it?

I What the hell are the characteristic modes?

I Why are they of our interest?

Dominant characteristic mode of helicoptermodel discretized into 18989 basis functions,ka = 1/2, decomposed into characteristic

modes in AToM in 47 s.

Therefore, . . .

. . . the characteristic mode theory is to be systematically derived.

Disclaimer: There will be equations! Brace yourself and be prepared. . .

Capek, M. Characteristic Modes – Part I: Introduction 2 / 39

Page 3: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Characteristic Modes

Conventionally, characteristic modes In aredefined as

XIn = λnRIn,

in which Z = R + jX is the impedance matrix.

However, who knows:

I What is impedance matrix and how to get it?

I What the hell are the characteristic modes?

I Why are they of our interest?

Dominant characteristic mode of helicoptermodel discretized into 18989 basis functions,ka = 1/2, decomposed into characteristic

modes in AToM in 47 s.

Therefore, . . .

. . . the characteristic mode theory is to be systematically derived.

Disclaimer: There will be equations! Brace yourself and be prepared. . .

Capek, M. Characteristic Modes – Part I: Introduction 2 / 39

Page 4: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Characteristic Modes

Conventionally, characteristic modes In aredefined as

XIn = λnRIn,

in which Z = R + jX is the impedance matrix.

However, who knows:

I What is impedance matrix and how to get it?

I What the hell are the characteristic modes?

I Why are they of our interest?

Dominant characteristic mode of helicoptermodel discretized into 18989 basis functions,ka = 1/2, decomposed into characteristic

modes in AToM in 47 s.

Therefore, . . .

. . . the characteristic mode theory is to be systematically derived.

Disclaimer: There will be equations! Brace yourself and be prepared. . .

Capek, M. Characteristic Modes – Part I: Introduction 2 / 39

Page 5: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Characteristic Modes

Conventionally, characteristic modes In aredefined as

XIn = λnRIn,

in which Z = R + jX is the impedance matrix.

However, who knows:

I What is impedance matrix and how to get it?

I What the hell are the characteristic modes?

I Why are they of our interest?

Dominant characteristic mode of helicoptermodel discretized into 18989 basis functions,ka = 1/2, decomposed into characteristic

modes in AToM in 47 s.

Therefore, . . .

. . . the characteristic mode theory is to be systematically derived.

Disclaimer: There will be equations! Brace yourself and be prepared. . .

Capek, M. Characteristic Modes – Part I: Introduction 2 / 39

Page 6: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Outline

1 Necessary Background2 Discretization and Method of Moments3 Definition of Characteristic Modes4 Properties of Characteristic Modes5 Activities at the Department6 Concluding Remarks

This talk concerns:

I electric currents in vacuum (generalization is, however,straightforward),

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

Capek, M. Characteristic Modes – Part I: Introduction 3 / 39

Page 7: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation1

Ω

σ →∞(PEC)

k Ei (r)

k Es (r)

Original problem.

Ω

ε0, µ0

J (r′)

k Es (r)

Equivalent problem.

n×(Es

(r′)

+Ei

(r′) )

= 0, r′ ∈ Ω −n× n×Ei

(r′)

= Z (J) , J = J(r′)

1R. F. Harrington, Time-Harmonic Electromagnetic Fields, 2nd ed. Wiley – IEEE Press, 2001

Capek, M. Characteristic Modes – Part I: Introduction 4 / 39

Page 8: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation1

Ω

σ →∞(PEC)

k Ei (r)

k Es (r)

Original problem.

Ω

ε0, µ0

J (r′)

k Es (r)

Equivalent problem.

n×(Es

(r′)

+Ei

(r′) )

= 0, r′ ∈ Ω

−n× n×Ei

(r′)

= Z (J) , J = J(r′)

1R. F. Harrington, Time-Harmonic Electromagnetic Fields, 2nd ed. Wiley – IEEE Press, 2001

Capek, M. Characteristic Modes – Part I: Introduction 4 / 39

Page 9: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation1

Ω

σ →∞(PEC)

k Ei (r)

k Es (r)

Original problem.

Ω

ε0, µ0

J (r′)

k Es (r)

Equivalent problem.

n×(Es

(r′)

+Ei

(r′) )

= 0, r′ ∈ Ω

−n× n×Ei

(r′)

= Z (J) , J = J(r′)

1R. F. Harrington, Time-Harmonic Electromagnetic Fields, 2nd ed. Wiley – IEEE Press, 2001

Capek, M. Characteristic Modes – Part I: Introduction 4 / 39

Page 10: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation1

Ω

σ →∞(PEC)

k Ei (r)

k Es (r)

Original problem.

Ω

ε0, µ0

J (r′)

k Es (r)

Equivalent problem.

n×(Es

(r′)

+Ei

(r′) )

= 0, r′ ∈ Ω −n× n×Ei

(r′)

= Z (J) , J = J(r′)

1R. F. Harrington, Time-Harmonic Electromagnetic Fields, 2nd ed. Wiley – IEEE Press, 2001

Capek, M. Characteristic Modes – Part I: Introduction 4 / 39

Page 11: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation – Problem Formalization

Key role of the impedance operator Z (J)

n× n×Es

(r′)

= Z (J) = −n× n× (jωA+∇ϕ) .

Substituting for Lorenz gauge-calibrated potentials2 A and ϕ gives

Z (J) = jkZ0

Ω

G(r, r′

)·J(r′)

dS

= jkZ0

Ω

(1 +

1

k2∇∇

)·J(r′) e−jk|r

′−r|

4π |r′ − r| dS,

I Impedance operator Z is linear, symmetric (reciprocal, thus non-Hermitian).

I Alternative formulation MFIE3, common extension towards CFIE3.

2J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 1998

Capek, M. Characteristic Modes – Part I: Introduction 5 / 39

Page 12: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Necessary Background

Electric Field Integral Equation – Problem Formalization

Key role of the impedance operator Z (J)

n× n×Es

(r′)

= Z (J) = −n× n× (jωA+∇ϕ) .

Substituting for Lorenz gauge-calibrated potentials2 A and ϕ gives

Z (J) = jkZ0

Ω

G(r, r′

)·J(r′)

dS = jkZ0

Ω

(1 +

1

k2∇∇

)·J(r′) e−jk|r

′−r|

4π |r′ − r| dS,

I Impedance operator Z is linear, symmetric (reciprocal, thus non-Hermitian).

I Alternative formulation MFIE3, common extension towards CFIE3.

2J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 19983W. C. Gibson, The Method of Moments in Electromagnetics, 2nd ed. Chapman and Hall/CRC, 2014

Capek, M. Characteristic Modes – Part I: Introduction 5 / 39

Page 13: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Dicretization of the Problem

Only canonical bodies can typically be evaluated analytically.

Problem −n× n×Ei (r′) = Z (J) has to be solved numerically!

I Discretization4 Ω → ΩT is needed (nontrivial task!)

Ω

ε0, µ0

Equivalent problem.

ΩT

Triangularized domain ΩT .

4J. A. De Loera, J. Rambau, and F. Santos, Triangulations – Structures for Algorithms and Applications. Berlin,Germany: Springer, 2010

Capek, M. Characteristic Modes – Part I: Introduction 6 / 39

Page 14: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Dicretization of the Problem

Only canonical bodies can typically be evaluated analytically.

Problem −n× n×Ei (r′) = Z (J) has to be solved numerically!

I Discretization4 Ω → ΩT is needed (nontrivial task!)

Ω

ε0, µ0

Equivalent problem.

ΩT

Triangularized domain ΩT .

4J. A. De Loera, J. Rambau, and F. Santos, Triangulations – Structures for Algorithms and Applications. Berlin,Germany: Springer, 2010

Capek, M. Characteristic Modes – Part I: Introduction 6 / 39

Page 15: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Representation of the Operator (Recap.)

Engineers like linear systems

L (f) = h.

I Typically unsolvable for f in the present state(how to invert L?).

f L h

Linear system with input f andoutput h.

Representation in a basis ψn and linearity of operator L readily gives5

N∑

n=1

InL (ψn) = h.

I One equation for N unknowns → still unsolvable.

Capek, M. Characteristic Modes – Part I: Introduction 7 / 39

Page 16: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Representation of the Operator (Recap.)

Engineers like linear systems

L (f) = h.

I Typically unsolvable for f in the present state(how to invert L?).

f L h

Linear system with input f andoutput h.

Representation in a basis ψn and linearity of operator L readily gives5

N∑

n=1

InL (ψn) = h.

I One equation for N unknowns → still unsolvable.

5R. F. Harrington, Field Computation by Moment Methods. Wiley – IEEE Press, 1993

Capek, M. Characteristic Modes – Part I: Introduction 7 / 39

Page 17: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Representation of the Operator (Recap.)

Using proper inner product 〈·, ·〉 and N tests from left, we get

N∑

n=1

In〈χn,L (ψn)〉 = 〈χn,h〉,

i.e., in matrix form the method of moments5 relation reads

LI = H.

5R. F. Harrington, Field Computation by Moment Methods. Wiley – IEEE Press, 1993

Capek, M. Characteristic Modes – Part I: Introduction 8 / 39

Page 18: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6

ψn (r) =ln

2A±nρ± (r)

are applied to approximate J (r) as

J (r) ≈N∑

n=1

Inψn (r) .

P+n

P−n

ρ+nρ−n

A+n

A−n

lnT+n

T−n

O

r

y

z

x

RWG basis function ψn.

Galerkin testing7, i.e., χn = ψn, is performed as∫

Ω

ψ · Z (ψ) dS = 〈ψ,Z (ψ)〉 ≡ Z = [Zpq] ∈ CN×N .

6S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Trans.Antennas Propag., vol. 30, no. 3, pp. 409–418, 1982. doi: 10.1109/TAP.1982.1142818

Capek, M. Characteristic Modes – Part I: Introduction 9 / 39

Page 19: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6

ψn (r) =ln

2A±nρ± (r)

are applied to approximate J (r) as

J (r) ≈N∑

n=1

Inψn (r) .

P+n

P−n

ρ+nρ−n

A+n

A−n

lnT+n

T−n

O

r

y

z

x

RWG basis function ψn.

Galerkin testing7, i.e., χn = ψn, is performed as∫

Ω

ψ · Z (ψ) dS = 〈ψ,Z (ψ)〉 ≡ Z = [Zpq] ∈ CN×N .

6S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Trans.Antennas Propag., vol. 30, no. 3, pp. 409–418, 1982. doi: 10.1109/TAP.1982.1142818

Capek, M. Characteristic Modes – Part I: Introduction 9 / 39

Page 20: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6

ψn (r) =ln

2A±nρ± (r)

are applied to approximate J (r) as

J (r) ≈N∑

n=1

Inψn (r) .

P+n

P−n

ρ+nρ−n

A+n

A−n

lnT+n

T−n

O

r

y

z

x

RWG basis function ψn.

Galerkin testing7, i.e., χn = ψn, is performed as∫

Ω

ψ · Z (ψ) dS = 〈ψ,Z (ψ)〉 ≡ Z = [Zpq] ∈ CN×N .

6S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Trans.Antennas Propag., vol. 30, no. 3, pp. 409–418, 1982. doi: 10.1109/TAP.1982.1142818

7P. M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw-Hill, 1953

Capek, M. Characteristic Modes – Part I: Introduction 9 / 39

Page 21: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

From Impedance Operator Z to Impedance Matrix Z

The impedance matrix Z reads

Zpq =

Ω

ψp · Z(ψq)

dS = jkZ0

Ω

Ω

ψp (r1) ·G (r1, r2) ·ψq (r2) dS1 dS2.

I We say8: “Matrix Z is the impedance operator Z represented in ψn basis.”

I Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

I Generally11, impedance matrix Z inherits properties of impedance operator Z.

• Symmetric, complex-valued.

I Matrix Z completely describe the scattering properties of radiator ΩT .

Capek, M. Characteristic Modes – Part I: Introduction 10 / 39

Page 22: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

From Impedance Operator Z to Impedance Matrix Z

The impedance matrix Z reads

Zpq =

Ω

ψp · Z(ψq)

dS = jkZ0

Ω

Ω

ψp (r1) ·G (r1, r2) ·ψq (r2) dS1 dS2.

I We say8: “Matrix Z is the impedance operator Z represented in ψn basis.”

I Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

I Generally11, impedance matrix Z inherits properties of impedance operator Z.

• Symmetric, complex-valued.

I Matrix Z completely describe the scattering properties of radiator ΩT .

8C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 vol. set), 1st ed. Wiley, 1992

Capek, M. Characteristic Modes – Part I: Introduction 10 / 39

Page 23: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

From Impedance Operator Z to Impedance Matrix Z

The impedance matrix Z reads

Zpq =

Ω

ψp · Z(ψq)

dS = jkZ0

Ω

Ω

ψp (r1) ·G (r1, r2) ·ψq (r2) dS1 dS2.

I We say8: “Matrix Z is the impedance operator Z represented in ψn basis.”

I Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

I Generally11, impedance matrix Z inherits properties of impedance operator Z.

• Symmetric, complex-valued.

I Matrix Z completely describe the scattering properties of radiator ΩT .

8C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 vol. set), 1st ed. Wiley, 19929(2017). Antenna Toolbox for MATLAB (AToM), Czech Technical University in Prague, [Online]. Available:

www.antennatoolbox.com

10A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Wiley – IEEE Press, 1998

Capek, M. Characteristic Modes – Part I: Introduction 10 / 39

Page 24: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

From Impedance Operator Z to Impedance Matrix Z

The impedance matrix Z reads

Zpq =

Ω

ψp · Z(ψq)

dS = jkZ0

Ω

Ω

ψp (r1) ·G (r1, r2) ·ψq (r2) dS1 dS2.

I We say8: “Matrix Z is the impedance operator Z represented in ψn basis.”

I Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

I Generally11, impedance matrix Z inherits properties of impedance operator Z.• Symmetric, complex-valued.

I Matrix Z completely describe the scattering properties of radiator ΩT .

8C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 vol. set), 1st ed. Wiley, 19929(2017). Antenna Toolbox for MATLAB (AToM), Czech Technical University in Prague, [Online]. Available:

www.antennatoolbox.com

10A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Wiley – IEEE Press, 199811N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 2nd ed. Dover, 1993

Capek, M. Characteristic Modes – Part I: Introduction 10 / 39

Page 25: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

From Impedance Operator Z to Impedance Matrix Z

The impedance matrix Z reads

Zpq =

Ω

ψp · Z(ψq)

dS = jkZ0

Ω

Ω

ψp (r1) ·G (r1, r2) ·ψq (r2) dS1 dS2.

I We say8: “Matrix Z is the impedance operator Z represented in ψn basis.”

I Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

I Generally11, impedance matrix Z inherits properties of impedance operator Z.• Symmetric, complex-valued.

I Matrix Z completely describe the scattering properties of radiator ΩT .

8C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 vol. set), 1st ed. Wiley, 19929(2017). Antenna Toolbox for MATLAB (AToM), Czech Technical University in Prague, [Online]. Available:

www.antennatoolbox.com

10A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Wiley – IEEE Press, 199811N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 2nd ed. Dover, 1993

Capek, M. Characteristic Modes – Part I: Introduction 10 / 39

Page 26: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Two Hilbert Space Representations12

analytics numerics

govering expression −n× n×Ei = Z (J) V = ZI

type of solution exact approximate

quantities operators, functions matrices, vectors

usage limited general

solution of the system no inverse∗ I = Z−1V

representation Z = 〈r,Z (r)〉 Z = 〈ψ,Z (ψ)〉bilinear form (for Z) p = 〈J ,Z (J)〉 p ≈ IHZI

〈f , g〉 =

Ω

f∗ (x) · g (x) dx, AH =(AT)∗

12N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 2nd ed. Dover, 1993

Capek, M. Characteristic Modes – Part I: Introduction 11 / 39

Page 27: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following.

I Continuous form13 using operator Z

−1

2

Ω

J∗ ·Es dS =1

2

Ω

J∗ · Z (J) dS = Prad + 2jω (Wm −We) .

I Algebraic form14 using matrix Z

1

2

Ω

J∗ · Z (J) dS ≈ 1

2IHZI = Prad + 2jω (Wm −We) .

Capek, M. Characteristic Modes – Part I: Introduction 12 / 39

Page 28: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following.

I Continuous form13 using operator Z

−1

2

Ω

J∗ ·Es dS =1

2

Ω

J∗ · Z (J) dS = Prad + 2jω (Wm −We) .

I Algebraic form14 using matrix Z

1

2

Ω

J∗ · Z (J) dS ≈ 1

2IHZI = Prad + 2jω (Wm −We) .

13J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 1998

Capek, M. Characteristic Modes – Part I: Introduction 12 / 39

Page 29: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following.

I Continuous form13 using operator Z

−1

2

Ω

J∗ ·Es dS =1

2

Ω

J∗ · Z (J) dS = Prad + 2jω (Wm −We) .

I Algebraic form14 using matrix Z

1

2

Ω

J∗ · Z (J) dS ≈ 1

2IHZI = Prad + 2jω (Wm −We) .

13J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 199814R. F. Harrington, Field Computation by Moment Methods. Wiley – IEEE Press, 1993

Capek, M. Characteristic Modes – Part I: Introduction 12 / 39

Page 30: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Diagonalization of the Impedance Operator/Matrix

Motivation

Describe behavior of a scatterer Ω without feeding considered.

Diagonalization of the impedance matrix Z:

Z11 Z12 Z13 · · · Z1N

Z21 Z22 Z23 · · · Z2N

Z31 Z32 Z33 · · · Z3N

......

.... . .

...

ZN1 ZN2 ZN3 · · · ZNN

Impedance matrix Z.

ν1 0 0 · · · 0

0 ν2 0 · · · 0

0 0 ν3 · · · 0

......

.... . .

...

0 0 0 · · · νN

Yet-unknown diagonalization of impedance matrix Z.

Capek, M. Characteristic Modes – Part I: Introduction 13 / 39

Page 31: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Diagonalization of the Impedance Operator/Matrix

Motivation

Describe behavior of a scatterer Ω without feeding considered.

Diagonalization of the impedance matrix Z:

Z11 Z12 Z13 · · · Z1N

Z21 Z22 Z23 · · · Z2N

Z31 Z32 Z33 · · · Z3N

......

.... . .

...

ZN1 ZN2 ZN3 · · · ZNN

Impedance matrix Z.

ν1 0 0 · · · 0

0 ν2 0 · · · 0

0 0 ν3 · · · 0

......

.... . .

...

0 0 0 · · · νN

Yet-unknown diagonalization of impedance matrix Z.

Capek, M. Characteristic Modes – Part I: Introduction 13 / 39

Page 32: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Impedance Operator Represented in Spherical Harmonics

Spherical shell of radius a.

Example: Let us represent the impedance operator Zin a basis15 of spherical harmonics

J shn (ϑ, ϕ, a)

∈ R

⟨J shm ,Z

(J shn

)⟩=

Ω

J shm · Z

(J shn

)dS,

This representation gives diagonal matrix16, i.e.,

⟨J shm ,Z

(J shn

)⟩= 2

(P shrad,n + 2jω

(W sh

m,n −W she,n

))δmn,

δmn = 1⇔ m = n ∧ δmn = 0⇔ m 6= n.

15This can be understood as solving method of moments analytically with spherical harmonics as basis functions.16J. A. Stratton, Electromagnetic Theory. Wiley – IEEE Press, 2007

Capek, M. Characteristic Modes – Part I: Introduction 14 / 39

Page 33: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Impedance Operator Represented in Spherical Harmonics

Spherical shell of radius a.

Example: Let us represent the impedance operator Zin a basis15 of spherical harmonics

J shn (ϑ, ϕ, a)

∈ R

⟨J shm ,Z

(J shn

)⟩=

Ω

J shm · Z

(J shn

)dS,

This representation gives diagonal matrix16, i.e.,

⟨J shm ,Z

(J shn

)⟩= 2

(P shrad,n + 2jω

(W sh

m,n −W she,n

))δmn,

δmn = 1⇔ m = n ∧ δmn = 0⇔ m 6= n.

15This can be understood as solving method of moments analytically with spherical harmonics as basis functions.16J. A. Stratton, Electromagnetic Theory. Wiley – IEEE Press, 2007

Capek, M. Characteristic Modes – Part I: Introduction 14 / 39

Page 34: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

Formula for spherical shell normalized to unitary radiated power (no units)

⟨J shm ,Z

(J shn

)⟩⟨J shm ,R

(J shn

)⟩ =

(1 + j

2ω(W sh

m,n −W she,n

)

P shrad,n

)δmn,

where Z = R+ jX

or, alternatively (without problems due to division by zero),⟨J shm ,R

(J shn

)+ jX

(J shn

)⟩=(

1 + jλshn

)⟨J shm ,R

(J shn

)⟩δmn.

Linearity of the impedance operator allows to write⟨J shm ,X

(J shn

)⟩= λshn

⟨J shm ,R

(J shn

)⟩δmn,

which is solved for all n ∈ 1, . . . ,∞ via generalized eigenvalue problem (GEP)

X(J shn

)= λshn R

(J shn

).

Capek, M. Characteristic Modes – Part I: Introduction 15 / 39

Page 35: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

Formula for spherical shell normalized to unitary radiated power (no units)

⟨J shm ,Z

(J shn

)⟩⟨J shm ,R

(J shn

)⟩ =

(1 + j

2ω(W sh

m,n −W she,n

)

P shrad,n

)δmn,

where Z = R+ jX or, alternatively (without problems due to division by zero),⟨J shm ,R

(J shn

)+ jX

(J shn

)⟩=(

1 + jλshn

)⟨J shm ,R

(J shn

)⟩δmn.

Linearity of the impedance operator allows to write⟨J shm ,X

(J shn

)⟩= λshn

⟨J shm ,R

(J shn

)⟩δmn,

which is solved for all n ∈ 1, . . . ,∞ via generalized eigenvalue problem (GEP)

X(J shn

)= λshn R

(J shn

).

Capek, M. Characteristic Modes – Part I: Introduction 15 / 39

Page 36: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

Formula for spherical shell normalized to unitary radiated power (no units)

⟨J shm ,Z

(J shn

)⟩⟨J shm ,R

(J shn

)⟩ =

(1 + j

2ω(W sh

m,n −W she,n

)

P shrad,n

)δmn,

where Z = R+ jX or, alternatively (without problems due to division by zero),⟨J shm ,R

(J shn

)+ jX

(J shn

)⟩=(

1 + jλshn

)⟨J shm ,R

(J shn

)⟩δmn.

Linearity of the impedance operator allows to write⟨J shm ,X

(J shn

)⟩= λshn

⟨J shm ,R

(J shn

)⟩δmn,

which is solved for all n ∈ 1, . . . ,∞ via generalized eigenvalue problem (GEP)

X(J shn

)= λshn R

(J shn

).

Capek, M. Characteristic Modes – Part I: Introduction 15 / 39

Page 37: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

Formula for spherical shell normalized to unitary radiated power (no units)

⟨J shm ,Z

(J shn

)⟩⟨J shm ,R

(J shn

)⟩ =

(1 + j

2ω(W sh

m,n −W she,n

)

P shrad,n

)δmn,

where Z = R+ jX or, alternatively (without problems due to division by zero),⟨J shm ,R

(J shn

)+ jX

(J shn

)⟩=(

1 + jλshn

)⟨J shm ,R

(J shn

)⟩δmn.

Linearity of the impedance operator allows to write⟨J shm ,X

(J shn

)⟩= λshn

⟨J shm ,R

(J shn

)⟩δmn,

which is solved for all n ∈ 1, . . . ,∞ via generalized eigenvalue problem (GEP)

X(J shn

)= λshn R

(J shn

).

Capek, M. Characteristic Modes – Part I: Introduction 15 / 39

Page 38: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes – Generalization

Characteristic modes for arbitrarily shaped body are defined as GEP

X (Jn) = λnR (Jn).

Algebraic form17 is commonly used instead

XIn = λnRIn,

with Z = R + jX being the impedance matrix and In ∈ RN×1 being expansioncoefficients.

We know that GEP18 is capable to diagonalize both R and X operators19.

I Behavior solely described by the impedance operator/matrix.

I No feeding present (neither Ei, nor V)!

Capek, M. Characteristic Modes – Part I: Introduction 16 / 39

Page 39: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes – Generalization

Characteristic modes for arbitrarily shaped body are defined as GEP

X (Jn) = λnR (Jn).

Algebraic form17 is commonly used instead

XIn = λnRIn,

with Z = R + jX being the impedance matrix and In ∈ RN×1 being expansioncoefficients.

We know that GEP18 is capable to diagonalize both R and X operators19.

I Behavior solely described by the impedance operator/matrix.

I No feeding present (neither Ei, nor V)!

17Only six canonical bodies can, in principle, be solved analytically.

Capek, M. Characteristic Modes – Part I: Introduction 16 / 39

Page 40: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Characteristic Modes – Generalization

Characteristic modes for arbitrarily shaped body are defined as GEP

X (Jn) = λnR (Jn).

Algebraic form17 is commonly used instead

XIn = λnRIn,

with Z = R + jX being the impedance matrix and In ∈ RN×1 being expansioncoefficients.

We know that GEP18 is capable to diagonalize both R and X operators19.

I Behavior solely described by the impedance operator/matrix.

I No feeding present (neither Ei, nor V)!

17Only six canonical bodies can, in principle, be solved analytically.18G. H. Golub and C. F. Van Loan, Matrix Computations. Johns Hopkins University Press, 201219Generally, only two operators can simultaneously be diagonalized. Separable bodies are exceptional!

Capek, M. Characteristic Modes – Part I: Introduction 16 / 39

Page 41: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator byMontgomery et al.20

I proposal

1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jnwith orthogonal far-fields and radiating unitary power.

I analytical form

1971 Generalized by Harrington and Mautz22 for antenna problem usingimpedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R.

I algebraic form

23The literature will be closely reviewed later.20C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits. New York, United States:

McGraw-Hill, 1948

Capek, M. Characteristic Modes – Part I: Introduction 17 / 39

Page 42: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator byMontgomery et al.20

I proposal

1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jnwith orthogonal far-fields and radiating unitary power.

I analytical form

1971 Generalized by Harrington and Mautz22 for antenna problem usingimpedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R.

I algebraic form

23The literature will be closely reviewed later.20C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits. New York, United States:

McGraw-Hill, 194821R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 1968

Capek, M. Characteristic Modes – Part I: Introduction 17 / 39

Page 43: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator byMontgomery et al.20

I proposal

1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jnwith orthogonal far-fields and radiating unitary power.

I analytical form

1971 Generalized by Harrington and Mautz22 for antenna problem usingimpedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R.

I algebraic form

23The literature will be closely reviewed later.20C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits. New York, United States:

McGraw-Hill, 194821R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 196822R. 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

Capek, M. Characteristic Modes – Part I: Introduction 17 / 39

Page 44: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator byMontgomery et al.20

I proposal

1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jnwith orthogonal far-fields and radiating unitary power.

I analytical form

1971 Generalized by Harrington and Mautz22 for antenna problem usingimpedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R.

I algebraic form

23The literature will be closely reviewed later.20C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits. New York, United States:

McGraw-Hill, 194821R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 196822R. 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

Capek, M. Characteristic Modes – Part I: Introduction 17 / 39

Page 45: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Numbers λn

Rayleigh quotient24 is defined as

λn =

Ω

J∗n · X (Jn) dS

Ω

J∗n · R (Jn) dS

=2ω (Wm,n −We,n)

Prad,n≈ IHnXIn

IHnRIn.

I Notice Prad,n > 0⇒ R 0 is required.

I Eigenvalues λn represent the stationary points25.

24J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford University Press, 1988

Capek, M. Characteristic Modes – Part I: Introduction 18 / 39

Page 46: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Numbers λn

Rayleigh quotient24 is defined as

λn =

Ω

J∗n · X (Jn) dS

Ω

J∗n · R (Jn) dS

=2ω (Wm,n −We,n)

Prad,n≈ IHnXIn

IHnRIn.

I Notice Prad,n > 0⇒ R 0 is required.

I Eigenvalues λn represent the stationary points25.

24J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford University Press, 198825J. Nocedal and S. Wright, Numerical Optimization. New York, United States: Springer, 2006

Capek, M. Characteristic Modes – Part I: Introduction 18 / 39

Page 47: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Physical Meaning of Characteristic Numbers λn

Characteristic modes can be classified as

Wm,n > We,n ⇒ λn > 0 mode is of inductive nature,

Wm,n < We,n ⇒ λn < 0 mode is of capacitive nature,

Wm,n = We,n ⇒ λn = 0 mode is in resonance26.

I To get current to the resonance, let us combine modes with λn < 0 and λn > 0.

Knowledge in group theory27 gives understanding of

I degeneracies,

I crossings,

I crossing avoidances.

26Resonance of a modal current impressed in vacuum is doubtful. It cannot be excited independently.

Capek, M. Characteristic Modes – Part I: Introduction 19 / 39

Page 48: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Physical Meaning of Characteristic Numbers λn

Characteristic modes can be classified as

Wm,n > We,n ⇒ λn > 0 mode is of inductive nature,

Wm,n < We,n ⇒ λn < 0 mode is of capacitive nature,

Wm,n = We,n ⇒ λn = 0 mode is in resonance26.

I To get current to the resonance, let us combine modes with λn < 0 and λn > 0.

Knowledge in group theory27 gives understanding of

I degeneracies,

I crossings,

I crossing avoidances.

26Resonance of a modal current impressed in vacuum is doubtful. It cannot be excited independently.27R. McWeeny, Symmetry, An Introduction to Group Theory and Its Applications. Dover, 2002, isbn: 0-486-42182-1

Capek, M. Characteristic Modes – Part I: Introduction 19 / 39

Page 49: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω 3 r′ of finite extent:

J2ω(Wm−We)

Prad

Jn λn

Mapping between current densities and their complex power ratios.

Characteristic modes can freely be combined as

J =∑

n

αnJn.

I Set of characteristic modes is infinite, but countable.

I Set of all currents has higher cardinality (uncountable).

Capek, M. Characteristic Modes – Part I: Introduction 20 / 39

Page 50: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω 3 r′ of finite extent:

J2ω(Wm−We)

Prad

Jn λn

Mapping between current densities and their complex power ratios.

Characteristic modes can freely be combined as

J =∑

n

αnJn.

I Set of characteristic modes is infinite, but countable.

I Set of all currents has higher cardinality (uncountable).

Capek, M. Characteristic Modes – Part I: Introduction 20 / 39

Page 51: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω 3 r′ of finite extent:

J2ω(Wm−We)

PradJn λn

Mapping between current densities and their complex power ratios.

Characteristic modes can freely be combined as

J =∑

n

αnJn.

I Set of characteristic modes is infinite, but countable.

I Set of all currents has higher cardinality (uncountable).

Capek, M. Characteristic Modes – Part I: Introduction 20 / 39

Page 52: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Numbers λn for Spherical Shell

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−15

−10

−5

0

5

10

15

20

Wm < We

TM

modes

ka

λn

Capek, M. Characteristic Modes – Part I: Introduction 21 / 39

Page 53: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Numbers λn for Spherical Shell

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−15

−10

−5

0

5

10

15

20

Wm < We

TM

modes

TE

modes

Wm > We

ka

λn

Capek, M. Characteristic Modes – Part I: Introduction 21 / 39

Page 54: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Eigenangles δn

Characteristic angles28 δn scale the dynamics of λn ∈ (−∞,∞)

δn = 180(

1− 1

πarctan (λn)

).

to δn ∈ (90, 270).

Similarly as for characteristic numbers:

λn > 0⇒ δn < 180 mode is of inductive nature,

λn < 0⇒ δn > 180 mode is of capacitive nature,

λn = 0⇒ δn = 180 mode is in resonance.

28E. Newman, “Small antenna location synthesis using characteristic modes”, IEEE Trans. Antennas Propag., vol. 27,no. 4, pp. 530–531, 1979. doi: 10.1109/TAP.1979.1142116

Capek, M. Characteristic Modes – Part I: Introduction 22 / 39

Page 55: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Eigenangles δn

Characteristic angles28 δn scale the dynamics of λn ∈ (−∞,∞)

δn = 180(

1− 1

πarctan (λn)

).

to δn ∈ (90, 270).

Similarly as for characteristic numbers:

λn > 0⇒ δn < 180 mode is of inductive nature,

λn < 0⇒ δn > 180 mode is of capacitive nature,

λn = 0⇒ δn = 180 mode is in resonance.

28E. Newman, “Small antenna location synthesis using characteristic modes”, IEEE Trans. Antennas Propag., vol. 27,no. 4, pp. 530–531, 1979. doi: 10.1109/TAP.1979.1142116

Capek, M. Characteristic Modes – Part I: Introduction 22 / 39

Page 56: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Eigenangles δn for Spherical Shell

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

100

120

140

160

180

200

220

240

260

Wm < We

TM

modes

ka

δ n

Capek, M. Characteristic Modes – Part I: Introduction 23 / 39

Page 57: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Eigenangles δn for Spherical Shell

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

100

120

140

160

180

200

220

240

260

Wm < We

TM

modes

TE

modes

Wm > We

ka

δ n

Capek, M. Characteristic Modes – Part I: Introduction 23 / 39

Page 58: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Spherical Shell Solved Numerically

While simplest canonical body, spherical shell has plenty of potential issues, e.g.,

Spherical shell made of 194 triangles.

I degenerate eigenspace29, D (l) = 2l + 1,

I conformity of spherical surface withcommonly used basis functions,

I internal resonances30 (λn →∞),

I computationally demanding evaluation31.

29K. R. Schab and J. T. Bernhard, “A group theory rule for predicting eigenvalue crossings in characteristic modeanalyses”, IEEE Antennas Wireless Propag. Lett., no. 16, pp. 944–947, 2017, K. R. Schab, J. M. Outwater Jr.,M. W. Young, et al., “Eigenvalue crossing avoidance in characteristic modes”, IEEE Trans. Antennas Propag., vol. 64, no. 7,pp. 2617–2627, 2016. doi: 10.1109/TAP.2016.2550098

30T. K. Sarkar, E. Mokole, and M. Salazar-Palma, “An expose on internal resonance, external resonance andcharacteristic modes”, IEEE Trans. Antennas Propag., vol. 64, no. 11, pp. 4695–4702, 2016. doi: 10.1109/TAP.2016.2598281

31Having no junctions, spherical shell (and other closed objects) has highest possible ratio between number of basisfunctions (unknowns) and triangles (3/2).

Capek, M. Characteristic Modes – Part I: Introduction 24 / 39

Page 59: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Characteristic Modes Jn of Spherical Shell

Dominant capacitive characteristic mode (sphericalharmonic TM10).

Dominant inductive characteristic mode (sphericalharmonic TE10).

Capek, M. Characteristic Modes – Part I: Introduction 25 / 39

Page 60: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Properties of Characteristic Quantities

Orthogonality relations32

1

2IHmZIn = (1 + jλn) δmn,

1

2

√ε0µ0

2π∫

0

π∫

0

F ∗m · F n sinϑ dϑ dϕ = δmn,

i.e., orthogonalization of modal complex power and modal far-fields.

32R. F. Harrington and J. R. Mautz, “Computation of characteristic modes for conducting bodies”, IEEE Trans.Antennas Propag., vol. 19, no. 5, pp. 629–639, 1971. doi: 10.1109/TAP.1971.1139990

Capek, M. Characteristic Modes – Part I: Introduction 26 / 39

Page 61: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formulaJ =

n

αnJn

derived using linearity of the impedance operator and orthogonality ofcharacteristic modes as

J =∑

n

〈Jn,Ei〉〈Jn,Z (Jn)〉Jn

=∑

n

V in

1 + jλnJn

with V in being modal excitation coefficient33 and Mn = 1/ |1 + jλn| being modal

significance coefficient34.

I Connection between “external” and “modal” worlds.

Capek, M. Characteristic Modes – Part I: Introduction 27 / 39

Page 62: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formulaJ =

n

αnJn

derived using linearity of the impedance operator and orthogonality ofcharacteristic modes as

J =∑

n

〈Jn,Ei〉〈Jn,Z (Jn)〉Jn =

n

V in

1 + jλnJn

with V in being modal excitation coefficient33 and Mn = 1/ |1 + jλn| being modal

significance coefficient34.

I Connection between “external” and “modal” worlds.

Capek, M. Characteristic Modes – Part I: Introduction 27 / 39

Page 63: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formulaJ =

n

αnJn

derived using linearity of the impedance operator and orthogonality ofcharacteristic modes as

J =∑

n

〈Jn,Ei〉〈Jn,Z (Jn)〉Jn =

n

V in

1 + jλnJn

with V in being modal excitation coefficient33 and Mn = 1/ |1 + jλn| being modal

significance coefficient34.

I Connection between “external” and “modal” worlds.

33R. 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

34M. Cabedo-Fabres, E. Antonino-Daviu, A. Valero-Nogueira, et al., “The theory of characteristic modes revisited: Acontribution to the design of antennas for modern applications”, IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 52–68,2007. doi: 10.1109/MAP.2007.4395295

Capek, M. Characteristic Modes – Part I: Introduction 27 / 39

Page 64: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Bonus: On the Inversion of Z operator

Summation formula slightly rearranged (Dirac notation used, i.e., L|f〉 = |g〉)

|J〉 =∑

n

〈Jn|Ei〉〈Jn|Z|Jn〉

|Jn〉

Let’s do some magic. . .

|J〉 =∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

| − n× n×Ei〉

and compare with the defining formula | − n× n×Ei〉 = Z|J〉.We get

Z−1 ≡∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

.

I But, do not even try to calculate!

Capek, M. Characteristic Modes – Part I: Introduction 28 / 39

Page 65: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Bonus: On the Inversion of Z operator

Summation formula slightly rearranged (Dirac notation used, i.e., L|f〉 = |g〉)

|J〉 =∑

n

〈Jn|Ei〉〈Jn|Z|Jn〉

|Jn〉

Let’s do some magic. . .

|J〉 =∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

| − n× n×Ei〉

and compare with the defining formula | − n× n×Ei〉 = Z|J〉.

We get

Z−1 ≡∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

.

I But, do not even try to calculate!

Capek, M. Characteristic Modes – Part I: Introduction 28 / 39

Page 66: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Bonus: On the Inversion of Z operator

Summation formula slightly rearranged (Dirac notation used, i.e., L|f〉 = |g〉)

|J〉 =∑

n

〈Jn|Ei〉〈Jn|Z|Jn〉

|Jn〉

Let’s do some magic. . .

|J〉 =∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

| − n× n×Ei〉

and compare with the defining formula | − n× n×Ei〉 = Z|J〉.We get

Z−1 ≡∑

n

|Jn〉〈Jn|〈Jn|Z|Jn〉

.

I But, do not even try to calculate!

Capek, M. Characteristic Modes – Part I: Introduction 28 / 39

Page 67: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Good theory needs time. . .

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

5

10

15

20

25

21 1 1 1

21 1 1 1

45

109

8

10

21

26

13

Year

Number

ofpublications

Number of publications in IEEE Trans. Antennas and Propagation and IEEE Antennas and WirelessPropagation Letters. WOS database queries: “characteristic modes”, “theory of characteristic modes”, and

“characteristic mode analysis”, relevance checked manually.

Capek, M. Characteristic Modes – Part I: Introduction 29 / 39

Page 68: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

Capek, M. Characteristic Modes – Part I: Introduction 30 / 39

Page 69: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

37N. Inagaki and R. J. Garbacz, “Eigenfunctions of composite hermitian operators with application to discrete andcontinuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi:10.1109/TAP.1982.1142866

Capek, M. Characteristic Modes – Part I: Introduction 30 / 39

Page 70: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

37N. Inagaki and R. J. Garbacz, “Eigenfunctions of composite hermitian operators with application to discrete andcontinuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi:10.1109/TAP.1982.1142866

38C. Baum, “The singularity expansion method: Background and developments”, IEEE Antennas Propag. SocietyNewsletter, vol. 28, no. 4, pp. 14–23, 1986. doi: 10.1109/MAP.1986.27868

Capek, M. Characteristic Modes – Part I: Introduction 30 / 39

Page 71: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].

5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

37N. Inagaki and R. J. Garbacz, “Eigenfunctions of composite hermitian operators with application to discrete andcontinuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi:10.1109/TAP.1982.1142866

38C. Baum, “The singularity expansion method: Background and developments”, IEEE Antennas Propag. SocietyNewsletter, vol. 28, no. 4, pp. 14–23, 1986. doi: 10.1109/MAP.1986.27868

39A. J. King, “Characteristic mode theory for closely spaced dipole arrays”, PhD thesis, University of Illinois atUrbana-Champaign, 2015

Capek, M. Characteristic Modes – Part I: Introduction 30 / 39

Page 72: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

37N. Inagaki and R. J. Garbacz, “Eigenfunctions of composite hermitian operators with application to discrete andcontinuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi:10.1109/TAP.1982.1142866

38C. Baum, “The singularity expansion method: Background and developments”, IEEE Antennas Propag. SocietyNewsletter, vol. 28, no. 4, pp. 14–23, 1986. doi: 10.1109/MAP.1986.27868

39A. J. King, “Characteristic mode theory for closely spaced dipole arrays”, PhD thesis, University of Illinois atUrbana-Champaign, 2015

40L. 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. Characteristic Modes – Part I: Introduction 30 / 39

Page 73: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Properties of Characteristic Modes

Notes on Characteristic Modes

1. Characteristic modes derived for apertures35, dielectrics36, magnetics36, etc.

2. Generalization as Inagaki modes37 (orthogonal fields on arbitrary boundary).

3. Singular expansion method38, i.e., ZIn = 0.

4. Other representations39 applicable, i.e., Z = [〈fm,Z (fn)〉].5. Other bases40 exist, i.e., AIn = ξnBIn.

6. Nowadays implemented in FEKO, CST-MWS, WIPL-D, CEM One, HFSS.

35R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems”, IEEE Trans. Microw. Theory Techn.,vol. 33, no. 6, pp. 500–505, 1985. doi: 10.1109/TMTT.1985.1133105

36R. F. Harrington, J. R. Mautz, and Y. Chang, “Characteristic modes for dielectric and magnetic bodies”, IEEE Trans.Antennas Propag., vol. 20, no. 2, pp. 194–198, 1972. doi: 10.1109/TAP.1972.1140154

37N. Inagaki and R. J. Garbacz, “Eigenfunctions of composite hermitian operators with application to discrete andcontinuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi:10.1109/TAP.1982.1142866

38C. Baum, “The singularity expansion method: Background and developments”, IEEE Antennas Propag. SocietyNewsletter, vol. 28, no. 4, pp. 14–23, 1986. doi: 10.1109/MAP.1986.27868

39A. J. King, “Characteristic mode theory for closely spaced dipole arrays”, PhD thesis, University of Illinois atUrbana-Champaign, 2015

40L. 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. Characteristic Modes – Part I: Introduction 30 / 39

Page 74: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

Topics Recently Solved at the Department

I Analytical properties of characteristic modes41,

I implementation of characteristic modes42,

I benchmarks of commercial and in-house solvers43,

I modal Q-factor for antennas44,

I radiation efficiency of characteristic modes45,

I minimization of Q-factor using characteristic modes46.

41M. Capek, P. Hazdra, M. Masek, et al., “Analytical representation of characteristic modes decomposition”, IEEE Trans.Antennas Propag., vol. 65, no. 2, pp. 713–720, 2017. doi: 10.1109/TAP.2016.2632725

42M. Capek, P. Hamouz, P. Hazdra, et al., “Implementation of the theory of characteristic modes in Matlab”, IEEEAntennas Propag. Mag., vol. 55, no. 2, pp. 176–189, 2013. doi: 10.1109/MAP.2013.6529342

43M. Capek, V. Losenicky, L. Jelinek, et al., “Validating the characteristic modes solvers”, IEEE Trans. AntennasPropag., vol. 65, no. 8, pp. 4134–4145, 2017. doi: 10.1109/TAP.2017.2708094

44M. Capek, P. Hazdra, and J. Eichler, “A method for the evaluation of radiation Q based on modal approach”, IEEETrans. Antennas Propag., vol. 60, no. 10, pp. 4556–4567, 2012. doi: 10.1109/TAP.2012.2207329

45M. Capek, J. Eichler, and P. Hazdra, “Evaluating radiation efficiency from characteristic currents”, IET Microw.Antenna P., vol. 9, no. 1, pp. 10–15, 2015. doi: 10.1049/iet-map.2013.0473

46M. 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. Characteristic Modes – Part I: Introduction 31 / 39

Page 75: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

Ongoing Research at the Department

I Improvement of characteristic modes decomposition47.

I tracking of modal data48,

I group theory (symmetries) for tracking and problem reducing49,

I characteristic modes for MLFMA50,

I interpolation using differentiated GEP,

I characteristic modes for arrays51.

47D. Tayli, M. Capek, L. Akrou, et al., “Accurate and efficient evaluation of characteristic modes”,, 2017, submitted,arXiv:1709.09976. [Online]. Available: https://arxiv.org/abs/1709.09976

48M. Capek, P. Hazdra, P. Hamouz, et al., “A method for tracking characteristic numbers and vectors”, Prog.Electromagn. Res. B, vol. 33, pp. 115–134, 2011. doi: 10.2528/PIERB11060209

49M. Masek, M. Capek, and L. Jelinek, “Modal tracking based on group theory”, in Proceedings of the 12th EuropeanConference on Antennas and Propagation (EUCAP), 2018, pp. 1–5

50M. Masek, M. Capek, P. Hazdra, et al., “Characteristic modes of electrically small antennas in the presence ofelectrically large platforms”, in Progress In Electromagnetics Research Symposium, St. Petersburg, Russia: IEEE, 2017,pp. 3733–3738. doi: 10.1109/PIERS.2017.8262407

51T. Lonsky, P. Hazdra, and J. Kracek, “Characteristic modes of dipole arrays”, IEEE Antennas Wireless Propag. Lett.,vol. 4, pp. 1–4, 2018. doi: 10.1109/LAWP.2018.2828986

Capek, M. Characteristic Modes – Part I: Introduction 32 / 39

Page 76: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

AToM – Antenna Toolbox for MATLAB

AToM developed at the department between 2014 and 2018.

I Capable to calculate matrix Z (and many other matrices),

I capable to calculate the CMs, their tracking, post-processing.

Visit antennatoolbox.com

Capek, M. Characteristic Modes – Part I: Introduction 33 / 39

Page 77: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

Special Interested Group

I Established by prof. Lau, Lund University,

I 77 groups worldwide, CTU/Elmag is an active member!

Visit characteristicmodes.org

SIG meeting at EuCAP in London, 2018.

Capek, M. Characteristic Modes – Part I: Introduction 34 / 39

Page 78: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

Benchmarking of the CMs Decomposition52

63 48 35 24 15 8 3 3 8 15 24 35 48 63

5

10

15

TM/TE mode order

log10|λ

n|

TM modes TE modes

exact AToM (1)FEKO AToM (8)KS WIPL-DIDA CEM OneCMC Makarov

Visit elmag.org/CMbenchmark

52M. Capek, V. Losenicky, L. Jelinek, et al., “Validating the characteristic modes solvers”, IEEE Trans. AntennasPropag., vol. 65, no. 8, pp. 4134–4145, 2017. doi: 10.1109/TAP.2017.2708094

Capek, M. Characteristic Modes – Part I: Introduction 35 / 39

Page 79: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

Wikipedia Webpage

Visit wikipedia.org/wiki/Characteristic mode analysis

Capek, M. Characteristic Modes – Part I: Introduction 36 / 39

Page 80: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Activities at the Department

European School of Antennas

Course onCharacteristic Modes: Theory and Applications

Aimed at postgraduate research students and industrial engineerswho want to acquire deep insight into the theory and applicationsof characteristic modes.

Visit esoa.webs.upv.es

Capek, M. Characteristic Modes – Part I: Introduction 37 / 39

Page 81: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Concluding Remarks

Summary

Characteristic modes decomposition

XIn = λnRIn

I diagonalizes impedance matrix Z,

I constitutes entire domain basis In,I generates orthogonal far-fields Fn,I allows compact representation of the radiator ΩT .

Ω

σ →∞(PEC)

Ω

ε0, µ0

J (r′)

ΩT

J1

J2

Capek, M. Characteristic Modes – Part I: Introduction 38 / 39

Page 82: Characteristic Modes - Part I: Introduction · Characteristic Modes Conventionally, characteristic modes I nare de ned as XI n= nRI n; in which Z = R+ jX is the impedance matrix.

Questions?

For a complete PDF presentation see capek.elmag.org

Miloslav [email protected]

May 3, 2018, v1.10

LATEX, TikZ & PFG used

Capek, M. Characteristic Modes – Part I: Introduction 39 / 39