On the Properties of Stored Electromagnetic Energy Miloslav Capek - - PowerPoint PPT Presentation

On the Properties of Stored Electromagnetic Energy

Miloslav Capek Lukas Jelinek

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

Progress In Electromagnetics Research Symposium Prague 2015

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 1 / 24

Outline

1 Motivation 2 Definition of Stored Energy 3 Selected Concepts of Quality Factor Q 4 Unification of Various Definitions 5 Comparison 6 Observations 7 Final Remarks

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 2 / 24

Motivation

Stored energy in EM field

Why we are interested?

◮ Stored energy poses interesting theoretical yet unsolved problem of classical electrodynamics.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 3 / 24

Motivation

Stored energy in EM field

Why we are interested?

◮ Stored energy poses interesting theoretical yet unsolved problem of classical electrodynamics.

• Potentially infinite total energy within a

time-harmonic steady state, for r → ∞ r c0

2π

ˆ

π

ˆ

• Efar (ϑ, ϕ)×H∗

far (ϑ, ϕ)

• ·n0 r2 sin ϑ dϑ dϕ → ∞.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 3 / 24

Motivation

Stored energy in EM field

Why we are interested?

◮ Stored energy Wsto is important1 for evaluation of antenna quality factor Q, i.e. Q = ω0 Wsto Prad . ◮ Stored energy can be used e.g. for convex

• ptimization2.
• Positive semi-definiteness is crucial.

◮ Knowledge of stored energy is essential for advanced technologies.

• Optics, nano-antennas, time-domain antennas. . .

1Standard definitions of terms for antennas 145 - 1993,

IEEE Antennas and Propagation Society

• 2M. Gustafsson and S. Nordebo, “Optimal antenna currents for Q, superdirectivity, and radiation

patterns using convex optimization”, IEEE Trans. Antennas Propag., vol. 61, no. 3, pp. 1109–1118, 2013. doi: 10.1109/TAP.2012.2227656

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 4 / 24

Motivation

Importance of stored energy

FBW

Q Wsto

Fractional bandwidth, quality factor Q and stored energy.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 5 / 24

Motivation

Importance of stored energy

FBW

Q Wsto

Fractional bandwidth, quality factor Q and stored energy.

We can/should study ◮ proportionality3 of various Q definitions to FBW, ◮ definition of Wsto, ◮ proportionality of Wsto to FBW (via Q).

• 3M. Capek, L. Jelinek, and P. Hazdra, “On the functional relation between quality factor and

fractional bandwidth”, IEEE Trans. Antennas Propag., vol. 63, no. 6, pp. 2787–2790, 2015. doi: 10.1109/TAP.2015.2414472

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 5 / 24

Motivation

Importance of stored energy

FBW

Q Wsto

Fractional bandwidth, quality factor Q and stored energy.

We can/should study ◮ proportionality3 of various Q definitions to FBW, ◮ definition of Wsto, ◮ proportionality of Wsto to FBW (via Q). Selected definitions of stored energy will be discussed.

• 3M. Capek, L. Jelinek, and P. Hazdra, “On the functional relation between quality factor and

fractional bandwidth”, IEEE Trans. Antennas Propag., vol. 63, no. 6, pp. 2787–2790, 2015. doi: 10.1109/TAP.2015.2414472

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 5 / 24

Definition of Stored Energy

What the stored energy is?

Proposed definition of stored energy4 Stored electromagnetic energy is that part of the total electromagnetic energy that is, in comparison with the radiated energy, bound to the sources of the field, being unable to escape towards infinity. ◮ In all cases, the statement above can symbolically be written as Wsto = F (Wtot, Wrad) . (1) ◮ In order to correctly define stored energy Wsto, radiation energy Wrad has to be completely understood. ◮ Note that any explicit mathematical definition of (1) automatically interprets radiated energy4.

• 4M. Capek and L. Jelinek, “Various interpretations of the stored and the radiated energy density”, ,

2015, submitted, arXiv: 1503.06752

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 6 / 24

Definition of Stored Energy

What conditions should be met?

Stored energy

?

≡ Physical quantity As a physical quantity5, stored energy has to poses i.a. ◮ uniqueness, ◮ positive semi-definiteness, ◮ gauge invariance, ◮ coordinate-independence, ◮ equality to total energy for PEC cavities.

• 5J. D. Jackson, Classical Electrodynamics, 3rd ed. John Wiley, 1998

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 7 / 24

Selected Concepts of Quality Factor Q

FBW

rev lost

W Q P ω =

in

2 Z Q R ω ω ∂ = ∂

2 2 2 sto tot rad

W W W = −

sto tot rad

W W W = −

sto lost

W Q P ω =

Fields Circuits

in in

2R Z ω ω ∂ ∂

{ }

X

Q Q = ℑ

Z

Q Q =

Source Concept Frequency domain

2 2

− E F

2

c − E S

Spectral decomposition Kajfez (1986) Yaghjian (2005) Gustafsson et al. (2014) Čapek & Jelínek (2014) Rhodes (1976) Grimes at al. (2000) Polevoi (1990) Direen (2010) Kaiser (2011) Chu (1948) Thal (2012) Gustafsson & Jonsson (2015) Mikki & Antar (2011) Rhodes (1972) Collin & Rothschild (1963) Geyi (2003) Vandenbosch (2010) Collin & Rothschild (1964) Gustafsson & Jonsson (2014) Rhodes (1977) Yaghjian (2004) Geyi (2003) Vandenbosch (2010) Uzsoky & Solymár (1955) Harrington (1965) , 2 , ω ω ∂ ∂ J ZJ J RJ , 2 , ω ω ∂ ∂ J XJ J RJ

Time domain

Čapek & Jelínek (2015) Vandenbosch (2013) Collin (1998)

FBW is parameter of primary importance.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 8 / 24

Selected Concepts of Quality Factor Q

FBW

rev lost

W Q P ω =

in

2 Z Q R ω ω ∂ = ∂

2 2 2 sto tot rad

W W W = −

sto tot rad

W W W = −

sto lost

W Q P ω =

Fields Circuits

in in

2R Z ω ω ∂ ∂

{ }

X

Q Q = ℑ

Z

Q Q =

Source Concept Frequency domain

2 2

− E F

2

c − E S

Spectral decomposition Kajfez (1986) Yaghjian (2005) Čapek & Jelínek (2014) Rhodes (1976) Grimes at al. (2000) Polevoi (1990) Direen (2010) Kaiser (2011) Chu (1948) Thal (2012) Gustafsson & Jonsson (2015) Mikki & Antar (2011) Rhodes (1972) Collin & Rothschild (1963) Geyi (2003) Vandenbosch (2010) Collin & Rothschild (1964) Gustafsson & Jonsson (2014) Rhodes (1977) Yaghjian (2004) Geyi (2003) Vandenbosch (2010) Uzsoky & Solymár (1955) Harrington (1965) Gustafsson et al. (2014) , 2 , ω ω ∂ ∂ J ZJ J RJ , 2 , ω ω ∂ ∂ J XJ J RJ

Time domain

Čapek & Jelínek (2015) Vandenbosch (2013) Collin (1998)

What Q (if any) is (inversely) proportional to FBW?

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 8 / 24

Selected Concepts of Quality Factor Q

FBW

rev lost

W Q P ω =

in

2 Z Q R ω ω ∂ = ∂

2 2 2 sto tot rad

W W W = −

sto tot rad

W W W = −

sto lost

W Q P ω =

Fields Circuits

in in

2R Z ω ω ∂ ∂

{ }

X

Q Q = ℑ

Z

Q Q =

Source Concept Frequency domain

2 2

− E F

2

c − E S

Spectral decomposition Geyi (2003) Vandenbosch (2010) Kajfez (1986) Yaghjian (2005) Uzsoky & Solymár (1955) Harrington (1965) Čapek & Jelínek (2014) Rhodes (1976) Grimes at al. (2000) Polevoi (1990) Direen (2010) Kaiser (2011) Chu (1948) Thal (2012) Gustafsson & Jonsson (2015) Mikki & Antar (2011) Rhodes (1972) Collin & Rothschild (1963) Geyi (2003) Vandenbosch (2010) Collin & Rothschild (1964) Gustafsson & Jonsson (2014) Rhodes (1977) Yaghjian (2004) Gustafsson et al. (2014) , 2 , ω ω ∂ ∂ J ZJ J RJ , 2 , ω ω ∂ ∂ J XJ J RJ

Time domain

Čapek & Jelínek (2015) Vandenbosch (2013) Collin (1998)

Recent concepts of Q that will be discussed. . .

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 8 / 24

Unification of Various Definitions

What concepts we selected?

Two different points of view can be distinguished. . .

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 9 / 24

Unification of Various Definitions

What concepts we selected?

Two different points of view can be distinguished. . . Extraction of radiated energy6 wsto (r) = wtot (r) − ǫ0 2 |F (r)|2 r2

• 6D. R. Rhodes, “A reactance theorem”,
• Proc. R. Soc. Lond. A., vol. 353, pp. 1–10, 1977. doi:

10.1098/rspa.1977.0018,

• A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas”,

IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. doi: 10.1109/TAP.2005.844443,

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

IEEE

• Trans. Antennas Propag., vol. 58, no. 4, pp. 1112–1127, 2010. doi: 10.1109/TAP.2010.2041166,
• M. Gustafsson and B. L. G. Jonsson, “Stored electromagnetic energy and antenna Q”, , Prog.
• Electromagn. Res., vol. 150, pp. 13–27, 2014

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 9 / 24

Unification of Various Definitions

What concepts we selected?

Two different points of view can be distinguished. . . Extraction of radiated energy wsto (r) = wtot (r) − ǫ0 2 |F (r)|2 r2 Differentiation of impedance7 Wsto ∝ ∂Z ∂ω

• 7M. Uzsoky and L. Solym´

ar, “Theory of super-directive linear arrays”, Acta Physica Academiae Scientiarum Hungaricae, vol. 6, no. 2, pp. 185–205, 1956. doi: 10.1007/BF03157322,

• R. F. Harrington, “Antenna excitation for maximum gain”,

IEEE Trans. Antennas Propag., vol. 13, no. 6,

• pp. 896–903, 1965. doi: 10.1109/TAP.1965.1138539,
• D. Kajfez and W. P. Wheless, “Invariant definitions of the unloaded Q factor”,

IEEE Antennas Propag. Magazine, vol. 34, no. 7, pp. 840–841, 1986,

• A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas”,

IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. doi: 10.1109/TAP.2005.844443,

• M. Capek, L. Jelinek, P. Hazdra, et al., “The measurable Q factor and observable energies of radiating

structures”, IEEE Trans. Antennas Propag., vol. 62, no. 1, pp. 311–318, 2014. doi: 10.1109/TAP.2013.2287519,

• M. Gustafsson, D. Tayli, and M. Cismasu. (2014), Q factors for antennas in dispersive media,

[Online]. Available: http://arxiv.org/abs/1408.6834

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 9 / 24

Unification of Various Definitions

What concepts we selected?

Two different points of view can be distinguished. . . Extraction of radiated energy wsto (r) = wtot (r) − ǫ0 2 |F (r)|2 r2 ◮ coordinate dependent ◮ can be negative for ka ≫ 1 Differentiation of impedance Wsto ∝ ∂Z ∂ω ◮ is not directly related to stored energy ◮ can be negative

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 9 / 24

Unification of Various Definitions

What concepts we selected?

Two different points of view can be distinguished. . . Extraction of radiated energy wsto (r) = wtot (r) − ǫ0 2 |F (r)|2 r2 ◮ coordinate dependent ◮ can be negative for ka ≫ 1 Differentiation of impedance Wsto ∝ ∂Z ∂ω ◮ is not directly related to stored energy ◮ can be negative Based on the recent work8, all related definitions can be unified.

• 8M. Capek, L. Jelinek, P. Hazdra, et al., “The measurable Q factor and observable energies of

radiating structures”, IEEE Trans. Antennas Propag., vol. 62, no. 1, pp. 311–318, 2014. doi: 10.1109/TAP.2013.2287519

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 9 / 24

Unification of Various Definitions

Source concept definition of Q

Pivotal concept derived by Vandenbosch9 Q(Van) = ω (Wm + We + Wrad) Prad (2) ◮ can be negative, is coordinate independent, is gauge variant ◮ yields good results in most cases!

Wm = 1 4πǫ0ω2 ˆ

V1

ˆ

V2

k2J (r1) · J∗ (r2) cos (kR) R dV2 dV1 We = 1 4πǫ0ω2 ˆ

V1

ˆ

V2

∇1 · J (r1) ∇2 · J∗ (r2) cos (kR) R dV2 dV1 Wrad = −k 4πǫ0ω2 ˆ

V1

ˆ

V2

• k2J (r1) · J∗ (r2) − ∇1 · J (r1) ∇2 · J∗ (r2)
• sin (kR) dV2 dV1
• 9G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”,

IEEE

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

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 10 / 24

Unification of Various Definitions

Measurable Q factors: QX and QZ

Input reactance10 QX = ω 2Rin ∂Xin ∂ω

• ∂I0/∂ω=0

(3)

• r input impedance11

QZ = ω 2Rin ∂Zin ∂ω

• ∂I0/∂ω=0

. (4) ◮ both concept represent measurable Q since they can be measured at input port

• 10R. F. Harrington, “Antenna excitation for maximum gain”,

IEEE Trans. Antennas Propag., vol. 13,

• no. 6, pp. 896–903, 1965. doi: 10.1109/TAP.1965.1138539, D. R. Rhodes, “Observable stored energies of

electromagnetic systems”,

• J. Franklin Inst., vol. 302, no. 3, pp. 225–237, 1976. doi:

10.1016/0016-0032(79)90126-1.

• 11D. R. Rhodes, “Observable stored energies of electromagnetic systems”,
• J. Franklin Inst., vol. 302,
• no. 3, pp. 225–237, 1976. doi: 10.1016/0016-0032(79)90126-1, A. D. Yaghjian and S. R. Best, “Impedance,

bandwidth and Q of antennas”, IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. doi: 10.1109/TAP.2005.844443.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 11 / 24

Unification of Various Definitions

Source concept definition of QX

Two recent attempts12,13 to express QX in source concept formalism Q(Cap)

X

= Q(Van) + ωW∂ω Prad (5) Q(Geyi)

X

= Q(Van) + ωWGeyi Prad (6)

W∂ω = 1 4πǫ0ω ˆ

V1

ˆ

V2

ℜ

• k2 ∂J (r1) · J∗ (r2)

∂ω − ∂∇1 · J (r1) ∇2 · J∗ (r2) ∂ω

• cos (kR)

R dV2 dV1 WGeyi = 1 4πǫ0ω ˆ

V1

ˆ

V2

ℑ

• k2 ∂J (r1) · J∗ (r2)

∂ω − ∂∇1 · J (r1) ∇2 · J∗ (r2) ∂ω

• sin (kR)

R dV2 dV1

• 12M. Capek, L. Jelinek, P. Hazdra, et al., “The measurable Q factor and observable energies of

radiating structures”, IEEE Trans. Antennas Propag., vol. 62, no. 1, pp. 311–318, 2014. doi: 10.1109/TAP.2013.2287519, W. Geyi, “On stored energies and radiation Q”, , IEEE Trans. Antennas Propag.,

• vol. 63, no. 2, pp. 636–645, 2015. doi: 10.1109/TAP.2014.2384028.

13Note that there is nowhere stated that Q(Cap) X

has any relationship to the stored energy.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 12 / 24

Unification of Various Definitions

Source concept definition of QX

Two recent attempts12,13 to express QX in source concept formalism Q(Cap)

X

= Q(Van) + ωW∂ω Prad

?

= QX (5) Q(Geyi)

X

= Q(Van) + ωWGeyi Prad

?

= QX (6)

W∂ω = 1 4πǫ0ω ˆ

V1

ˆ

V2

ℜ

• k2 ∂J (r1) · J∗ (r2)

∂ω − ∂∇1 · J (r1) ∇2 · J∗ (r2) ∂ω

• cos (kR)

R dV2 dV1 WGeyi = 1 4πǫ0ω ˆ

V1

ˆ

V2

ℑ

• k2 ∂J (r1) · J∗ (r2)

∂ω − ∂∇1 · J (r1) ∇2 · J∗ (r2) ∂ω

• sin (kR)

R dV2 dV1

• 12M. Capek, L. Jelinek, P. Hazdra, et al., “The measurable Q factor and observable energies of

radiating structures”, IEEE Trans. Antennas Propag., vol. 62, no. 1, pp. 311–318, 2014. doi: 10.1109/TAP.2013.2287519, W. Geyi, “On stored energies and radiation Q”, , IEEE Trans. Antennas Propag.,

• vol. 63, no. 2, pp. 636–645, 2015. doi: 10.1109/TAP.2014.2384028.

13Note that there is nowhere stated that Q(Cap) X

has any relationship to the stored energy.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 12 / 24

Unification of Various Definitions

Another concepts of QX

Another concepts14 are recalled to decide whether Q(Cap)

X

• r Q(Geyi)

X

is correct: Q(Yagh) = QX − ωW∂F Prad (7) Q(Gust) = Q(Van) + ωWF2 Prad (8)

W∂F = 2 Z0 ˛

S∞

ℑ

• F ∗ (r) · ∂F (r)

∂ω

• sin (ϑ) dϑ dϕ

WF2 = 1 4πǫ0ω2 ˆ

V1

ˆ

V2

ℑ

• k2J (r1) · J∗ (r2) − ∇1 · J (r1) ∇2 · J∗ (r2)
• G (r1, r2) dV2 dV1

G (r1, r2) = k2 r12 − r22 j1 (kR) R

• 14M. Gustafsson and B. L. G. Jonsson, “Stored electromagnetic energy and antenna Q”, , Prog.
• Electromagn. Res., vol. 150, pp. 13–27, 2014, A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and

Q of antennas”, IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. doi: 10.1109/TAP.2005.844443.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 13 / 24

Unification of Various Definitions

Another concepts of QX

Another concepts14 are recalled to decide whether Q(Cap)

X

• r Q(Geyi)

X

is correct: Q(Yagh) = QX − ωW∂F Prad (7) Q(Gust) = Q(Van) + ωWF2 Prad = Q(Yagh) (8)

W∂F = 2 Z0 ˛

S∞

ℑ

• F ∗ (r) · ∂F (r)

∂ω

• sin (ϑ) dϑ dϕ

WF2 = 1 4πǫ0ω2 ˆ

V1

ˆ

V2

ℑ

• k2J (r1) · J∗ (r2) − ∇1 · J (r1) ∇2 · J∗ (r2)
• G (r1, r2) dV2 dV1

G (r1, r2) = k2 r12 − r22 j1 (kR) R

• 14M. Gustafsson and B. L. G. Jonsson, “Stored electromagnetic energy and antenna Q”, , Prog.
• Electromagn. Res., vol. 150, pp. 13–27, 2014, A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and

Q of antennas”, IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. doi: 10.1109/TAP.2005.844443.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 13 / 24

Unification of Various Definitions

Independent test of QX

Combining previous relations, we arrive at QX = Q(Van) + ωW∂F Prad + ωWF2 Prad . (9) Observations: ◮ (9) forms second independent test of Q(Cap)

X

(5) and Q(Geyi)

X

(6) being equal to QX ◮ QX and Q(Van) are not coordinate dependent ◮ W∂F and WF2 are coordinate dependent

Based on (9), the following expression15 should be inspected (RHS is from e.q. (65), J′ ≡ ∂J (r) /∂ω):

W∂F + WF2 = 1 4 ˛

S∞

ℑ

• E
• J′

× H∗ (J) − E (J) × H∗ J′ · n0 dS (10)

• 15G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”,

IEEE

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

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 14 / 24

Comparison

Comparison of all methods

• 2
• 4
• 60

2 4 6 8 ka w ¶X / ¶w [ W ] ´104

L w

ka ´104 1.5 2 2.5 3 3.5 4 4.5

• 3
• 2
• 1

w L 0.903 L 1.09 L 0.498 L 0.249 L in

X w w ¶ ¶

Van

W W w

¶

+

e Van G yi

W W +

Van

2

F F

W W W

¶

+ +

Comparison of normalized frequency derivative of input impedance (left: dipole, right: yagi-uda).

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 15 / 24

Comparison

Comparison of all methods

in

X w w ¶ ¶ 2R

Van

Q Q

w ¶

+

e Van G yi

Q Q +

Van

2

F F

Q Q Q

¶

+ +

2 4 6 8 ka w ¶X / ¶w / (2R)

• 30
• 20
• 10

10 20

L w

port (8) (6) (9) ka 1.5 2 2.5 3 3.5 4 4.5 20 40 60

w L 0.903 L 1.09 L 0.498 L 0.249 L

port (8) (6) (9) Comparison of various Q factors (left: dipole, right: yagi-uda).

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 16 / 24

Observations

Results

Correct source definition of QX is QX = Q(Cap)

X

and thus, based on previous comparison W∂ω = W∂F + WF2. (11) ◮ W∂ω is not coordinate dependent, but W∂F and WF2 are ◮ is seems that Foster’s theorem cannot be used to derive stored energy in presence of radiation16 ◮ based on (11), the following expression17 should be inspected:

Based on (11), the following expression18 should be inspected (RHS is from e.q. (65), J′ ≡ ∂J (r) /∂ω):

W∂ω = 1 4 ˛

S∞

ℑ

• E
• J′

× H∗ (J) − E (J) × H∗ J′ · n0 dS (12)

• 18G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures”,

IEEE

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

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 17 / 24

Observations

Equality of analytic and algebraic relations

We can write19 QX ≈ ω 2ℜ {IHZI} ∂ℑ

• IHZI
• ∂ω

= ω

• IHXI

′ 2IHRI (13) and QX = Q(Van) + ωW∂ω Prad ≈ ωIHX′I 2IHRI + ω 2IHRI

• IH′ XI + IHXI′

. (14)

Q(Van) ≈ ωIHX′I 2IHRI , W∂ω ≈

• IH′

XI + IHXI′

• = 2ℜ
• IHXI′

, since

• IH′∗

X∗I∗ ∗ =

• IT′

XI∗ ∗ =

• IHXI′∗

. (15)

19Assumptions: Z = ZT, V = ZI, Zin = Rin + jXin, structure is fed by unitary current (I0 = 1 A). Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 18 / 24

Observations

Alternative expression for W∂ω

2ℜ

• IHXI′

≈ 1 4πǫ0ω ˆ

V1

ˆ

V2

ℜ

• k2 ∂J (r1) · J∗ (r2)

∂ω −∂∇1 · J (r1) ∇2 · J∗ (r2) ∂ω

• cos (kR)

R dV2 dV1 ◮ All analytic functionals can be dicretized to obtain algebraic expression. ◮ Final expressions are similar/same to those from MoM.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 19 / 24

Final Remarks

Final Remarks

All of the selected concept have problems with at least

• ne aspect of stored energy definition.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 20 / 24

Final Remarks

Final Remarks

All of the selected concept have problems with at least

• ne aspect of stored energy definition.

There is no coherent definition of the stored energy based no selected approaches.

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 20 / 24

Final Remarks

Final Remarks

All of the selected concept have problems with at least

• ne aspect of stored energy definition.

There is no coherent definition of the stored energy based no selected approaches. As a consequence, new paradigm should probably be looked for. . .

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 20 / 24

Final Remarks

(Advanced) Contentious issues #1

◮ Locality20

• There is no local definition of outgoing directions.21 Thus, the

amount of radiated energy may not be interpreted strictly locally.

◮ Physical meaning of radiation inside transmission lines

• Being inside a perfectly matched transmission line the outgoing

power should be considered as radiated power.

20Questions raised during several discussions with Mats Gustafsson, Lars Jonsson and Lukas Jelinek. 21From correspondence with Gerald Kaiser. Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 21 / 24

Final Remarks

(Advanced) Contentious issues #2

◮ Energy stored by a scatterer20

• There is no feeding port since the structure is fed by an incident
• field. How incorporate such feeding scenario into stored energy

definition (and evaluation)?

◮ Definition of radiated energy in dissipative environment20

• Radiation cannot be defined as a part of total energy that reached

the far-field (σ = 0 ⇒ Efar × H∗

far = 0).

◮ Is there any radiation if the size of PEC cavity reaches infinity?

• This is combination of several tough problems mentioned above.

20Questions raised during several discussions with Mats Gustafsson, Lars Jonsson and Lukas Jelinek. Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 22 / 24

Final Remarks

FBW

rev lost

W Q P ω =

in

2 Z Q R ω ω ∂ = ∂

2 2 2 sto tot rad

W W W = −

sto tot rad

W W W = −

sto lost

W Q P ω =

Fields Circuits

in in

2R Z ω ω ∂ ∂

{ }

X

Q Q = ℑ

Z

Q Q =

Source Concept Time domain Frequency domain

2 2

− E F

2

c − E S

Spectral decomposition Kajfez (1986) Yaghjian (2005) Čapek & Jelínek (2014) Rhodes (1976) Grimes at al. (2000) Polevoi (1990) Direen (2010) Kaiser (2011) Chu (1948) Thal (2012) Gustafsson & Jonsson (2015) Mikki & Antar (2011) Čapek & Jelínek (2015) Vandenbosch (2013) Collin (1998) Rhodes (1972) Collin & Rothschild (1963) Geyi (2003) Vandenbosch (2010) Collin & Rothschild (1964) Gustafsson & Jonsson (2014) Rhodes (1977) Yaghjian (2004) Geyi (2003) Vandenbosch (2010) Uzsoky & Solymár (1955) Harrington (1965) Gustafsson et al. (2014) , 2 , ω ω ∂ ∂ J ZJ J RJ , 2 , ω ω ∂ ∂ J XJ J RJ

◮ Novel scheme22 satisfies all requirements for stored energy.

• It was presented at EuCAP2015, will be detailed at APS2015.
• 22M. Capek, L. Jelinek, G. A. E. Vandenbosch, et al., “A scheme for stored energy evaluation and a

comparison with contemporary techniques”, , 2015, submitted, arXiv:1309.6122

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 23 / 24

Thank you!

For complete PDF presentation see

capek.elmag.org

Miloslav Capek miloslav.capek@fel.cvut.cz July 24, 2015, v1.10

Capek & Jelinek, CTU in Prague On the Properties of Stored Electromagnetic Energy 24 / 24