characteristic modes - part i: introduction...part i: introduction miloslav capek department of...
TRANSCRIPT
Characteristic ModesPart I: Introduction
Miloslav Capek
Department of Electromagnetic FieldCzech Technical University in Prague, Czech Republic
SeminarMay 2, 2018
Capek, M. Characteristic Modes – Part I: Introduction 1 / 39
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
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
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
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
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
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
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
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
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
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
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
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
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
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 g
Linear system with input f andoutput g.
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
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 g
Linear system with input f andoutput g.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Activities at the Department
Wikipedia Webpage
Visit wikipedia.org/wiki/Characteristic mode analysis
Capek, M. Characteristic Modes – Part I: Introduction 36 / 39
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
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
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