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 → ∞ 2 π π r ˆ ˆ � � · n 0 r 2 sin ϑ d ϑ d ϕ → ∞ . E far ( ϑ, ϕ ) × H ∗ far ( ϑ, ϕ ) c 0 0 0 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 W sto is important 1 for evaluation of antenna quality factor Q, i.e. W sto Q = ω 0 . P rad ◮ Stored energy can be used e.g. for convex optimization 2 . • Positive semi-definiteness is crucial. ◮ Knowledge of stored energy is essential for advanced technologies. • Optics, nano-antennas, time-domain antennas. . . 1 Standard definitions of terms for antennas 145 - 1993 , IEEE Antennas and Propagation Society 2 M. 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 W sto 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 We can/should study FBW ◮ proportionality 3 of various Q definitions to FBW, ◮ definition of W sto , Q W sto ◮ proportionality of W sto to FBW (via Q ). Fractional bandwidth, quality factor Q and stored energy. 3 M. 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 We can/should study FBW ◮ proportionality 3 of various Q definitions to FBW, ◮ definition of W sto , Q W sto ◮ proportionality of W sto to FBW (via Q ). Fractional bandwidth, quality factor Q and stored energy. Selected definitions of stored energy will be discussed. 3 M. 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 energy 4 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 W sto = F ( W tot , W rad ) . (1) ◮ In order to correctly define stored energy W sto , radiation energy W rad has to be completely understood. ◮ Note that any explicit mathematical definition of (1) automatically interprets radiated energy 4 . 4 M. 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 quantity 5 , stored energy has to poses i.a. ◮ uniqueness, ◮ positive semi-definiteness, ◮ gauge invariance, ◮ coordinate-independence, ◮ equality to total energy for PEC cavities. 5 J. 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 Vandenbosch (2013) Grimes at al. (2000) Čapek & Jelínek (2015) ω FBW W = Q 0 rev Collin (1998) Polevoi (1990) P Direen (2010) Time lost domain ω W = Q 0 sto Kajfez (1986) = − W W W ω ∂ Z P Yaghjian (2005) sto tot rad Q = 0 lost ∂ ω Q = Q 2 R Z in Uzsoky & Solymár (1955) Harrington (1965) Chu (1948) Thal (2012) Fields Circuits { } ω ∂ J XJ , W 2 = W 2 − W 2 Q = ℑ Q Mikki & 0 X sto tot rad ∂ ω 2 J RJ , Antar (2011) Gustafsson & Jonsson (2015) Rhodes (1976) Source Collin & Rothschild (1964) S Concept Kaiser (2011) − E 2 ω ∂ Z Frequency 0 in c Geyi (2003) 0 2 R ∂ ω domain Vandenbosch (2010) in Rhodes (1977) Yaghjian (2004) Čapek & Jelínek (2014) Gustafsson et al. (2014) E 2 − F 2 Spectral Gustafsson & Jonsson (2014) ω ∂ J ZJ , decomposition 0 2 J RJ , ∂ ω Geyi (2003) Rhodes (1972) Vandenbosch (2010) Collin & Rothschild (1963) 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 Vandenbosch (2013) Grimes at al. (2000) Čapek & Jelínek (2015) ω FBW W = Q 0 rev Collin (1998) Polevoi (1990) P Direen (2010) Time lost domain ω W = Q 0 sto Kajfez (1986) = − W W W ω ∂ Z P Yaghjian (2005) = sto tot rad Q 0 lost ∂ ω Q = Q 2 R Z in Uzsoky & Solymár (1955) Harrington (1965) Chu (1948) Thal (2012) Fields Circuits { } = − ω ∂ J XJ , = ℑ W 2 W 2 W 2 Q Q Mikki & 0 X sto tot rad 2 J RJ , ∂ ω Antar (2011) Gustafsson & Jonsson (2015) Rhodes (1976) Source Collin & Rothschild (1964) S Concept Kaiser (2011) 2 − ω ∂ E Z Frequency c 0 in Geyi (2003) 0 2 R ∂ ω domain Vandenbosch (2010) in Rhodes (1977) Yaghjian (2004) Čapek & Jelínek (2014) Gustafsson et al. (2014) − E 2 F 2 Spectral Gustafsson & Jonsson (2014) ∂ ω J ZJ , decomposition 0 2 J RJ , ∂ ω Geyi (2003) Rhodes (1972) Vandenbosch (2010) Collin & Rothschild (1963) 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 Vandenbosch (2013) Grimes at al. (2000) Čapek & Jelínek (2015) ω FBW W = Q 0 rev Collin (1998) Polevoi (1990) P Direen (2010) Time lost domain ω W = Q 0 sto Kajfez (1986) = − W W W ω ∂ Z P Yaghjian (2005) sto tot rad Q = 0 lost ∂ ω Q = Q 2 R Z in Uzsoky & Solymár (1955) Harrington (1965) Chu (1948) Thal (2012) Fields Circuits { } ω ∂ J XJ , W 2 = W 2 − W 2 Q = ℑ Q Mikki & 0 X sto tot rad ∂ ω 2 J RJ , Antar (2011) Gustafsson & Jonsson (2015) Rhodes (1976) Source Collin & Rothschild (1964) S Concept Kaiser (2011) − E 2 ω ∂ Z Frequency 0 in c Geyi (2003) 0 2 R ∂ ω domain Vandenbosch (2010) in Rhodes (1977) Yaghjian (2004) Čapek & Jelínek (2014) Gustafsson et al. (2014) E 2 − F 2 Spectral Gustafsson & Jonsson (2014) ω ∂ J ZJ , decomposition 0 2 J RJ , ∂ ω Geyi (2003) Rhodes (1972) Vandenbosch (2010) Collin & Rothschild (1963) 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 energy 6 | F ( r ) | 2 w sto ( r ) = w tot ( r ) − ǫ 0 r 2 2 6 D. 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
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