Characteristic Modes Part I: Introduction Miloslav Capek - - PowerPoint PPT Presentation

Characteristic Modes

Part I: Introduction Miloslav ˇ Capek

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

Seminar May 2, 2018

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

Characteristic Modes

Conventionally, characteristic modes In are defined as XIn = λnRIn, in which Z = R + jX is the impedance matrix.

Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ka = 1/2, decomposed into characteristic modes in AToM in 47 s.

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

Characteristic Modes

Conventionally, characteristic modes In are defined as XIn = λnRIn, in which Z = R + jX is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? ◮ Why are they of our interest?

Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ka = 1/2, decomposed into characteristic modes in AToM in 47 s.

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

Characteristic Modes

Conventionally, characteristic modes In are defined as XIn = λnRIn, in which Z = R + jX is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? ◮ Why are they of our interest?

Dominant characteristic mode of helicopter model 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.

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

Characteristic Modes

Conventionally, characteristic modes In are defined as XIn = λnRIn, in which Z = R + jX is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? ◮ Why are they of our interest?

Dominant characteristic mode of helicopter model 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. . .

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

Outline

1 Necessary Background 2 Discretization and Method of Moments 3 Definition of Characteristic Modes 4 Properties of Characteristic Modes 5 Activities at the Department 6 Concluding Remarks This talk concerns: ◮ electric currents in vacuum (generalization is, however, straightforward), ◮ time-harmonic quantities, i.e., A (r, t) = Re {A (r) exp (jωt)}.

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

J 1 (r, t) J 2 (r, t)

Necessary Background

Electric Field Integral Equation1

Ω σ → ∞

(PEC)

Original problem.

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

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

Necessary Background

Electric Field Integral Equation1

Ω σ → ∞

(PEC)

k Ei (r) k Es (r)

Original problem.

ˆ n ×

• Es
• r′

+ Ei

• r′

= 0, r′ ∈ Ω

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

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

Necessary Background

Electric Field Integral Equation1

Ω σ → ∞

(PEC)

k Ei (r) k Es (r)

Original problem.

Ω ǫ0, µ0

Equivalent problem.

ˆ n ×

• Es
• r′

+ Ei

• r′

= 0, r′ ∈ Ω

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

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

Necessary Background

Electric Field Integral Equation1

Ω σ → ∞

(PEC)

k Ei (r) k Es (r)

Original problem.

Ω ǫ0, µ0 J (r′) k Es (r)

Equivalent problem.

ˆ n ×

• Es
• r′

+ Ei

• r′

= 0, r′ ∈ Ω −ˆ n × ˆ n × Ei

• r′

= Z (J) , J = J

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

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

Necessary Background

Electric Field Integral Equation – Problem Formalization

Key role of the impedance operator Z (J) ˆ n × ˆ n × Es

• r′

= Z (J) = −ˆ n × ˆ n × (jωA + ∇ϕ) . Substituting for Lorenz gauge-calibrated potentials2 A and ϕ gives Z (J) = jkZ0

• Ω

G

• r, r′

· J

• r′

dS

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

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

Necessary Background

Electric Field Integral Equation – Problem Formalization

Key role of the impedance operator Z (J) ˆ n × ˆ n × Es

• r′

= Z (J) = −ˆ n × ˆ 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, ◮ Impedance operator Z is linear, symmetric (reciprocal, thus non-Hermitian). ◮ Alternative formulation MFIE3, common extension towards CFIE3.

• 2J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 1998
• 3W. C. Gibson, The Method of Moments in Electromagnetics, 2nd ed. Chapman and Hall/CRC, 2014

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

Discretization and Method of Moments

Dicretization of the Problem

Only canonical bodies can typically be evaluated analytically. Problem −ˆ n × ˆ n × Ei (r′) = Z (J) has to be solved numerically! Ω ǫ0, µ0

Equivalent problem.

• 4J. A. De Loera, J. Rambau, and F. Santos, Triangulations – Structures for Algorithms and Applications. Berlin,

Germany: Springer, 2010

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

Discretization and Method of Moments

Dicretization of the Problem

Only canonical bodies can typically be evaluated analytically. Problem −ˆ n × ˆ n × Ei (r′) = Z (J) has to be solved numerically! ◮ Discretization4 Ω → ΩT is needed (nontrivial task!) Ω ǫ0, 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

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

Discretization and Method of Moments

Representation of the Operator (Recap.)

Engineers like linear systems L (f) = h. ◮ Typically unsolvable for f in the present state (how to invert L?). f

L

g

Linear system with input f and

• utput g.

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

Discretization and Method of Moments

Representation of the Operator (Recap.)

Engineers like linear systems L (f) = h. ◮ Typically unsolvable for f in the present state (how to invert L?). f

L

g

Linear system with input f and

• utput g.

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

N

• n=1

InL (ψn) = h. ◮ One equation for N unknowns → still unsolvable.

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

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

Discretization and Method of Moments

Representation of the Operator (Recap.)

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

N

• n=1

Inχn, 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

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

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6 ψn (r) = ln 2A±

n

ρ± (r)

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x

RWG basis function ψn.

• 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

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

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6 ψn (r) = ln 2A±

n

ρ± (r) are applied to approximate J (r) as J (r) ≈

N

• n=1

Inψn (r) .

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x

RWG basis function ψn.

• 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

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

Discretization and Method of Moments

Algebraic Solution – Method of Moments

Piecewise basis functions6 ψn (r) = ln 2A±

n

ρ± (r) are applied to approximate J (r) as J (r) ≈

N

• n=1

Inψn (r) .

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

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

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

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.

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

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

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

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

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. ◮ We say8: “Matrix Z is the impedance operator Z represented in {ψn} basis.” ◮ Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

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

9(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

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

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. ◮ We say8: “Matrix Z is the impedance operator Z represented in {ψn} basis.” ◮ Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

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

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

9(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
• 11N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 2nd ed. Dover, 1993

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

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. ◮ We say8: “Matrix Z is the impedance operator Z represented in {ψn} basis.” ◮ Matrix Z can be calculated, e.g., in AToM9 (plenty of numerical techniques and

tricks should/can be used10).

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

• Symmetric, complex-valued.

◮ 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

9(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
• 11N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, 2nd ed. Dover, 1993

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

Discretization and Method of Moments

Two Hilbert Space Representations12

analytics numerics govering expression −ˆ n × ˆ n × Ei = Z (J) V = ZI type of solution exact approximate quantities

• perators, 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

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

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following.

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

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following. ◮ Continuous form13 using operator Z −1 2

• Ω

J∗ · Es dS = 1 2

• Ω

J∗ · Z (J) dS = Prad + 2jω (Wm − We) .

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

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

Discretization and Method of Moments

Example: Complex Power Balance

Quantity being important in the following. ◮ Continuous form13 using operator Z −1 2

• Ω

J∗ · Es dS = 1 2

• Ω

J∗ · Z (J) dS = Prad + 2jω (Wm − We) . ◮ 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
• 14R. F. Harrington, Field Computation by Moment Methods. Wiley – IEEE Press, 1993

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

Definition of Characteristic Modes

Diagonalization of the Impedance Operator/Matrix

Motivation Describe behavior of a scatterer Ω without feeding considered.

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

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 · · · ν2 · · · ν3 · · · . . . . . . . . . ... . . . · · · νN                      

Yet-unknown diagonalization of impedance matrix Z.

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

Definition of Characteristic Modes

Impedance Operator Represented in Spherical Harmonics

Spherical shell of radius a.

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

• Jsh

n (ϑ, ϕ, a)

• ∈ R
• Jsh

m, Z

• Jsh

n

• =
• Ω

Jsh

m · Z

• Jsh

n

• dS,

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

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

x y z r ϕ ϑ

Definition of Characteristic Modes

Impedance Operator Represented in Spherical Harmonics

Spherical shell of radius a.

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

• Jsh

n (ϑ, ϕ, a)

• ∈ R
• Jsh

m, Z

• Jsh

n

• =
• Ω

Jsh

m · Z

• Jsh

n

• dS,

This representation gives diagonal matrix16, i.e.,

• Jsh

m, Z

• Jsh

n

• = 2
• P sh

rad,n + 2jω

• W sh

m,n − W sh e,n

• δmn,

δmn = 1 ⇔ m = n ∧ δmn = 0 ⇔ m = 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

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

x y z r ϕ ϑ

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

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

• Jsh

m, Z

• Jsh

n

• Jsh

m, R

• Jsh

n

=

• 1 + j2ω
• W sh

m,n − W sh e,n

• P sh

rad,n

• δmn,

where Z = R + jX

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

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

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

• Jsh

m, Z

• Jsh

n

• Jsh

m, R

• Jsh

n

=

• 1 + j2ω
• W sh

m,n − W sh e,n

• P sh

rad,n

• δmn,

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

• Jsh

m, R

• Jsh

n

• + jX
• Jsh

n

• =
• 1 + jλsh

n

Jsh

m, R

• Jsh

n

• δmn.

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

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

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

• Jsh

m, Z

• Jsh

n

• Jsh

m, R

• Jsh

n

=

• 1 + j2ω
• W sh

m,n − W sh e,n

• P sh

rad,n

• δmn,

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

• Jsh

m, R

• Jsh

n

• + jX
• Jsh

n

• =
• 1 + jλsh

n

Jsh

m, R

• Jsh

n

• δmn.

Linearity of the impedance operator allows to write

• Jsh

m, X

• Jsh

n

• = λsh

n

• Jsh

m, R

• Jsh

n

• δmn,

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

Definition of Characteristic Modes

Characteristic Modes of Spherical Shell

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

• Jsh

m, Z

• Jsh

n

• Jsh

m, R

• Jsh

n

=

• 1 + j2ω
• W sh

m,n − W sh e,n

• P sh

rad,n

• δmn,

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

• Jsh

m, R

• Jsh

n

• + jX
• Jsh

n

• =
• 1 + jλsh

n

Jsh

m, R

• Jsh

n

• δmn.

Linearity of the impedance operator allows to write

• Jsh

m, X

• Jsh

n

• = λsh

n

• Jsh

m, R

• Jsh

n

• δmn,

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

• Jsh

n

• = λsh

n R

• Jsh

n

• .

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

Definition of Characteristic Modes

Characteristic Modes – Generalization

Characteristic modes for arbitrarily shaped body are defined as GEP X (Jn) = λnR (Jn).

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

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 expansion coefficients.

17Only six canonical bodies can, in principle, be solved analytically. ˇ Capek, M. Characteristic Modes – Part I: Introduction 16 / 39

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 expansion coefficients. We know that GEP18 is capable to diagonalize both R and X operators19. ◮ Behavior solely described by the impedance operator/matrix. ◮ 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, 2012

19Generally, only two operators can simultaneously be diagonalized. Separable bodies are exceptional! ˇ Capek, M. Characteristic Modes – Part I: Introduction 16 / 39

Definition of Characteristic Modes

Historical Overview23

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

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

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

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator by Montgomery et al.20 1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jn with orthogonal far-fields and radiating unitary power.

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

• 21R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 1968

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

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator by Montgomery et al.20 1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jn with orthogonal far-fields and radiating unitary power. 1971 Generalized by Harrington and Mautz22 for antenna problem using impedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R.

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

• 21R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 1968

• 22R. 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

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

Definition of Characteristic Modes

Historical Overview23

1948 First mention of diagonalization of the scattering operator by Montgomery et al.20 ◮ proposal 1968 Rigorously introduced by Garbacz21 as field/current solutions En/Jn with orthogonal far-fields and radiating unitary power. ◮ analytical form 1971 Generalized by Harrington and Mautz22 for antenna problem using impedance matrix Z as ZIn = νnMIn, νn ≡ 1 + jλn, M ≡ R. ◮ 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

• 21R. J. Garbacz, “A generalized expansion for radiated and scattered fields”, PhD thesis, Department of Electrical

Engineering, Ohio State University, 1968

• 22R. 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

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

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 ≈ IH

n XIn

IH

n RIn

.

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

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

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 ≈ IH

n XIn

IH

n RIn

. ◮ Notice Prad,n > 0 ⇒ R ≻ 0 is required. ◮ Eigenvalues λn represent the stationary points25.

• 24J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford University Press, 1988
• 25J. Nocedal and S. Wright, Numerical Optimization. New York, United States: Springer, 2006

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

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

26Resonance of a modal current impressed in vacuum is doubtful. It cannot be excited independently. ˇ Capek, M. Characteristic Modes – Part I: Introduction 19 / 39

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. ◮ To get current to the resonance, let us combine modes with λn < 0 and λn > 0. Knowledge in group theory27 gives understanding of ◮ degeneracies, ◮ crossings, ◮ 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

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

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω ∋ r′ of finite extent: Jn λn

Mapping between current densities and their complex power ratios.

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

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω ∋ r′ of finite extent: Jn λn

Mapping between current densities and their complex power ratios.

Characteristic modes can freely be combined as J =

• n

αnJn.

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

Properties of Characteristic Modes

Cardinality of a Set of Characteristic Modes

For a radiator Ω ∋ r′ of finite extent: J

2ω(Wm−We) Prad

Jn λn

Mapping between current densities and their complex power ratios.

Characteristic modes can freely be combined as J =

• n

αnJn. ◮ Set of characteristic modes is infinite, but countable. ◮ Set of all currents has higher cardinality (uncountable).

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

Properties of Characteristic Modes

Characteristic Numbers λn for Spherical Shell

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −20 −15 −10 −5 5 10 15 20 Wm < We TM modes ka λn

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

Properties of Characteristic Modes

Characteristic Numbers λn for Spherical Shell

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −20 −15 −10 −5 5 10 15 20 Wm < We TM modes TE modes Wm > We ka λn

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

Properties of Characteristic Modes

Characteristic Eigenangles δn

Characteristic angles28 δn scale the dynamics of λn ∈ (−∞, ∞) δn = 180◦

• 1 − 1

π arctan (λn)

• .

to δn ∈ (90◦, 270◦).

• 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

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

Properties of Characteristic Modes

Characteristic Eigenangles δn

Characteristic angles28 δn scale the dynamics of λn ∈ (−∞, ∞) δn = 180◦

• 1 − 1

π arctan (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

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

Properties of Characteristic Modes

Characteristic Eigenangles δn for Spherical Shell

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 δn

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

Properties of Characteristic Modes

Characteristic Eigenangles δn for Spherical Shell

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 δn

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

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.

◮ degenerate eigenspace29, D (l) = 2l + 1, ◮ conformity of spherical surface with commonly used basis functions, ◮ internal resonances30 (λn → ∞), ◮ computationally demanding evaluation31.

• 29K. R. Schab and J. T. Bernhard, “A group theory rule for predicting eigenvalue crossings in characteristic mode

analyses”, 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 and

characteristic 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 basis

functions (unknowns) and triangles (3/2).

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

Properties of Characteristic Modes

Characteristic Modes Jn of Spherical Shell

Dominant capacitive characteristic mode (spherical harmonic TM10). Dominant inductive characteristic mode (spherical harmonic TE10).

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

Properties of Characteristic Modes

Properties of Characteristic Quantities

Orthogonality relations32 1 2IH

mZIn = (1 + jλn) δmn,

1 2 ǫ0 µ0

2π

• π
• F ∗

m · F n sin ϑ 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

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

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formula J =

• n

αnJn

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

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formula J =

• n

αnJn derived using linearity of the impedance operator and orthogonality of characteristic modes as J =

• n

Jn, Ei Jn, Z (Jn)Jn =

• n

V i

n

1 + jλn Jn

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

Properties of Characteristic Modes

Summation of Characteristic Modes

Summation formula J =

• n

αnJn derived using linearity of the impedance operator and orthogonality of characteristic modes as J =

• n

Jn, Ei Jn, Z (Jn)Jn =

• n

V i

n

1 + jλn Jn with V i

n being modal excitation coefficient33 and Mn = 1/ |1 + jλn| being modal

significance coefficient34. ◮ Connection between “external” and “modal” worlds.

• 33R. 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

• 34M. Cabedo-Fabres, E. Antonino-Daviu, A. Valero-Nogueira, et al., “The theory of characteristic modes revisited: A

contribution 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

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

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

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

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

|JnJn| Jn|Z|Jn| − ˆ n × ˆ n × Ei and compare with the defining formula | − ˆ n × ˆ n × Ei = Z|J.

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

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

|JnJn| Jn|Z|Jn| − ˆ n × ˆ n × Ei and compare with the defining formula | − ˆ n × ˆ n × Ei = Z|J. We get Z−1 ≡

• n

|JnJn| Jn|Z|Jn. ◮ But, do not even try to calculate!

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

Properties of Characteristic Modes

Good theory needs time. . .

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 5 10 15 20 25

2 1 1 1 1 2 1 1 1 1 4 5 10 9 8 10 21 26 13

Year Number of publications

Number of publications in IEEE Trans. Antennas and Propagation and IEEE Antennas and Wireless Propagation Letters. WOS database queries: “characteristic modes”, “theory of characteristic modes”, and “characteristic mode analysis”, relevance checked manually.

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

Properties of Characteristic Modes

Notes on Characteristic Modes

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

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

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).
• 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 and

continuous radiating systems”, IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 571–575, 1982. doi: 10.1109/TAP.1982.1142866

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

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.
• 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 and

continuous 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. Society

Newsletter, vol. 28, no. 4, pp. 14–23, 1986. doi: 10.1109/MAP.1986.27868

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

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 = [f m, Z (f n)].

• 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 and

continuous 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. Society

Newsletter, 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 at

Urbana-Champaign, 2015

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

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 = [f m, Z (f n)].

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

continuous 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. Society

Newsletter, 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 at

Urbana-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

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

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 = [f m, Z (f n)].

• 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 and

continuous 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. Society

Newsletter, 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 at

Urbana-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

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

Activities at the Department

Topics Recently Solved at the Department

◮ Analytical properties of characteristic modes41, ◮ implementation of characteristic modes42, ◮ benchmarks of commercial and in-house solvers43, ◮ modal Q-factor for antennas44, ◮ radiation efficiency of characteristic modes45, ◮ 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”, IEEE

Antennas 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. Antennas

Propag., 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”, IEEE
• Trans. 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

ˇ Capek, M. Characteristic Modes – Part I: Introduction 31 / 39

Activities at the Department

Ongoing Research at the Department

◮ Improvement of characteristic modes decomposition47. ◮ tracking of modal data48, ◮ group theory (symmetries) for tracking and problem reducing49, ◮ characteristic modes for MLFMA50, ◮ interpolation using differentiated GEP, ◮ 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 European

Conference 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 of

electrically 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

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

Activities at the Department

AToM – Antenna Toolbox for MATLAB

AToM developed at the department between 2014 and 2018. ◮ Capable to calculate matrix Z (and many other matrices), ◮ capable to calculate the CMs, their tracking, post-processing.

Visit antennatoolbox.com

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

Activities at the Department

Special Interested Group

◮ Established by prof. Lau, Lund University, ◮ 77 groups worldwide, CTU/Elmag is an active member!

Visit characteristicmodes.org

SIG meeting at EuCAP in London, 2018.

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

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 |λn| TM modes TE modes exact AToM (1) FEKO AToM (8) KS WIPL-D IDA CEM One CMC Makarov

Visit elmag.org/CMbenchmark

• 52M. Capek, V. Losenicky, L. Jelinek, et al., “Validating the characteristic modes solvers”, IEEE Trans. Antennas

Propag., vol. 65, no. 8, pp. 4134–4145, 2017. doi: 10.1109/TAP.2017.2708094

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

Activities at the Department

Wikipedia Webpage

Visit wikipedia.org/wiki/Characteristic mode analysis

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

Activities at the Department

European School of Antennas

Course on Characteristic Modes: Theory and Applications Aimed at postgraduate research students and industrial engineers who want to acquire deep insight into the theory and applications

• f characteristic modes.

Visit esoa.webs.upv.es

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

Concluding Remarks

Summary

Characteristic modes decomposition XIn = λnRIn ◮ diagonalizes impedance matrix Z, ◮ constitutes entire domain basis {In}, ◮ generates orthogonal far-fields {Fn}, ◮ allows compact representation of the radiator ΩT . Ω σ → ∞

(PEC)

Ω ǫ0, µ0 J (r′) ΩT

J1 J2

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

Questions?

For a complete PDF presentation see

capek.elmag.org

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz May 3, 2018, v1.10

L

A

T E X, TikZ & PFG used

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